En física , el espacio-tiempo es cualquier modelo matemático que fusiona las tres dimensiones del espacio y la dimensión única del tiempo en una única variedad de cuatro dimensiones . El tejido del espacio-tiempo es un modelo conceptual que combina las tres dimensiones del espacio con la cuarta dimensión del tiempo. Los diagramas de espacio-tiempo se pueden usar para visualizar efectos relativistas , como por qué diferentes observadores perciben de manera diferente dónde y cuándo ocurren los eventos.
Hasta el siglo XX, se asumía que la geometría tridimensional del universo (su expresión espacial en términos de coordenadas, distancias y direcciones) era independiente del tiempo unidimensional. El famoso físico Albert Einstein ayudó a desarrollar la idea del espacio-tiempo como parte de su teoría de la relatividad . Antes de su trabajo pionero, los científicos tenían dos teorías separadas para explicar los fenómenos físicos: las leyes de la física de Isaac Newton describían el movimiento de objetos masivos, mientras que los modelos electromagnéticos de James Clerk Maxwell explicaban las propiedades de la luz. Sin embargo, en 1905, Albert Einstein basó un trabajo sobre la relatividad especial en dos postulados:
- Las leyes de la física son invariantes (es decir, idénticas) en todos los sistemas inerciales (es decir, marcos de referencia no acelerados)
- La velocidad de la luz en el vacío es la misma para todos los observadores, independientemente del movimiento de la fuente de luz.
La consecuencia lógica de tomar juntos estos postulados es la unión inseparable de las cuatro dimensiones, hasta ahora asumidas como independientes, del espacio y el tiempo. Surgen muchas consecuencias contradictorias: además de ser independiente del movimiento de la fuente de luz, la velocidad de la luz es constante independientemente del marco de referencia en el que se mida; las distancias e incluso el orden temporal de pares de eventos cambian cuando se miden en diferentes marcos de referencia inerciales (esta es la relatividad de la simultaneidad ); y la aditividad lineal de las velocidades ya no es válida.
Einstein enmarcó su teoría en términos de cinemática (el estudio de los cuerpos en movimiento). Su teoría fue un avance sobre la teoría de los fenómenos electromagnéticos de Lorentz de 1904 y la teoría electrodinámica de Poincaré . Aunque estas teorías incluían ecuaciones idénticas a las que introdujo Einstein (es decir, la transformación de Lorentz ), eran esencialmente modelos ad hoc propuestos para explicar los resultados de varios experimentos, incluido el famoso experimento del interferómetro de Michelson-Morley , que eran extremadamente difíciles de encajar en paradigmas existentes.
En 1908, Hermann Minkowski —una vez uno de los profesores de matemáticas de un joven Einstein en Zúrich— presentó una interpretación geométrica de la relatividad especial que fusionaba el tiempo y las tres dimensiones espaciales del espacio en un único continuo de cuatro dimensiones ahora conocido como espacio de Minkowski . Una característica clave de esta interpretación es la definición formal del intervalo espaciotemporal. Aunque las mediciones de distancia y tiempo entre eventos difieren para las mediciones realizadas en diferentes marcos de referencia, el intervalo de espacio-tiempo es independiente del marco de referencia inercial en el que se registran. [1]
La interpretación geométrica de la relatividad de Minkowski resultó vital para el desarrollo de Einstein de su teoría general de la relatividad de 1915 , en la que mostró cómo la masa y la energía curvan el espacio-tiempo plano en una variedad pseudo-riemanniana .
Introducción
Definiciones
La mecánica clásica no relativista trata el tiempo como una cantidad universal de medida que es uniforme en todo el espacio y está separada del espacio. La mecánica clásica asume que el tiempo tiene una velocidad de paso constante, independiente del estado de movimiento del observador o de cualquier cosa externa. [2] Además, asume que el espacio es euclidiano ; asume que el espacio sigue la geometría del sentido común. [3]
En el contexto de la relatividad especial , el tiempo no puede separarse de las tres dimensiones del espacio, porque la velocidad observada a la que pasa el tiempo de un objeto depende de la velocidad del objeto en relación con el observador. La relatividad general también proporciona una explicación de cómo los campos gravitacionales pueden ralentizar el paso del tiempo de un objeto visto por un observador fuera del campo.
En el espacio ordinario, una posición se especifica mediante tres números, conocidos como dimensiones . En el sistema de coordenadas cartesianas , estos se denominan x, y y z. Una posición en el espacio-tiempo se llama evento y requiere que se especifiquen cuatro números: la ubicación tridimensional en el espacio, más la posición en el tiempo (Fig. 1). Un evento está representado por un conjunto de coordenadas x , y , z y t . Por tanto, el espacio-tiempo es de cuatro dimensiones . Los eventos matemáticos tienen una duración cero y representan un solo punto en el espacio-tiempo.
El camino de una partícula a través del espacio-tiempo puede considerarse una sucesión de eventos. La serie de eventos se puede unir para formar una línea que representa el progreso de una partícula a través del espacio-tiempo. Esa línea se llama línea del mundo de la partícula . [4] : 105
Matemáticamente, el espacio-tiempo es una variedad , es decir, aparece localmente "plano" cerca de cada punto de la misma manera que, a escalas suficientemente pequeñas, un globo parece plano. [5] Un factor de escala extremadamente grande,(convencionalmente llamada velocidad de la luz ) relaciona las distancias medidas en el espacio con las distancias medidas en el tiempo. La magnitud de este factor de escala (casi 300.000 kilómetros o 190.000 millas en el espacio equivalen a un segundo en el tiempo), junto con el hecho de que el espacio-tiempo es múltiple, implica que a velocidades ordinarias, no relativistas y ordinarias, a escala humana. distancias, hay poco que los humanos pudieran observar que sea notablemente diferente de lo que podrían observar si el mundo fuera euclidiano. Fue solo con el advenimiento de las mediciones científicas sensibles a mediados del siglo XIX, como el experimento de Fizeau y el experimento de Michelson-Morley , que comenzaron a notarse desconcertantes discrepancias entre la observación y las predicciones basadas en la suposición implícita del espacio euclidiano. [6]
En la relatividad especial, un observador, en la mayoría de los casos, significará un marco de referencia a partir del cual se mide un conjunto de objetos o eventos. Este uso difiere significativamente del significado común del término en inglés. Los marcos de referencia son construcciones inherentemente no locales y, de acuerdo con este uso del término, no tiene sentido hablar de un observador como si tuviera una ubicación. En la Fig. 1-1, imagine que el marco considerado está equipado con una densa red de relojes, sincronizados dentro de este marco de referencia, que se extiende indefinidamente a lo largo de las tres dimensiones del espacio. Cualquier ubicación específica dentro de la celosía no es importante. La celosía de los relojes se utiliza para determinar el tiempo y la posición de los eventos que tienen lugar dentro de todo el marco. El término observador se refiere al conjunto completo de relojes asociados con un marco de referencia inercial. [7] : 17-22 En este caso idealizado, cada punto en el espacio tiene un reloj asociado con él y, por lo tanto, los relojes registran cada evento instantáneamente, sin demora de tiempo entre un evento y su registro. Sin embargo, un observador real verá un retraso entre la emisión de una señal y su detección debido a la velocidad de la luz. Para sincronizar los relojes, en la reducción de datos que sigue a un experimento, la hora en que se recibe una señal se corregirá para reflejar su hora real si hubiera sido registrada por una red idealizada de relojes.
En muchos libros sobre relatividad especial, especialmente en los más antiguos, la palabra "observador" se usa en el sentido más común de la palabra. Por lo general, se desprende del contexto qué significado se ha adoptado.
Los físicos distinguen entre lo que uno mide u observa (después de que uno ha factorizado los retrasos en la propagación de la señal) y lo que uno ve visualmente sin tales correcciones. No comprender la diferencia entre lo que uno mide / observa y lo que ve es la fuente de muchos errores entre los estudiantes principiantes de la relatividad. [8]
Historia
A mediados del siglo XIX, se consideró que varios experimentos, como la observación del punto Arago y las mediciones diferenciales de la velocidad de la luz en el aire frente al agua, habían demostrado la naturaleza ondulatoria de la luz en contraposición a una teoría corpuscular . [9] Se asumió entonces que la propagación de ondas requería la existencia de un medio ondulante ; en el caso de las ondas de luz, se consideró que se trataba de un hipotético éter luminífero . [nota 1] Sin embargo, los diversos intentos de establecer las propiedades de este medio hipotético arrojaron resultados contradictorios. Por ejemplo, el experimento de Fizeau de 1851 demostró que la velocidad de la luz en el agua que fluye era menor que la suma de la velocidad de la luz en el aire más la velocidad del agua en una cantidad que dependía del índice de refracción del agua. Entre otras cuestiones, la dependencia del arrastre parcial del éter que implica este experimento en el índice de refracción (que depende de la longitud de onda) llevó a la desagradable conclusión de que el éter fluye simultáneamente a diferentes velocidades para diferentes colores de luz. [10] El famoso experimento de Michelson-Morley de 1887 (Fig. 1-2) no mostró ninguna influencia diferencial de los movimientos de la Tierra a través del hipotético éter sobre la velocidad de la luz, y la explicación más probable, el arrastre completo del éter, estaba en conflicto con el observación de aberraciones estelares . [6]
George Francis FitzGerald en 1889, y Hendrik Lorentz en 1892, propusieron de forma independiente que los cuerpos materiales que viajaban a través del éter fijo se veían afectados físicamente por su paso, contrayéndose en la dirección del movimiento en una cantidad que era exactamente la necesaria para explicar los resultados negativos de el experimento de Michelson-Morley. (No se producen cambios de longitud en direcciones transversales a la dirección del movimiento).
Para 1904, Lorentz había expandido su teoría de tal manera que había llegado a ecuaciones formalmente idénticas a las que Einstein derivaría más tarde (es decir, la transformada de Lorentz ), pero con una interpretación fundamentalmente diferente. Como teoría de la dinámica (el estudio de fuerzas y momentos de torsión y su efecto sobre el movimiento), su teoría asumía deformaciones físicas reales de los constituyentes físicos de la materia. [11] : 163-174 Las ecuaciones de Lorentz predijeron una cantidad que llamó hora local , con la que podría explicar la aberración de la luz , el experimento de Fizeau y otros fenómenos. Sin embargo, Lorentz consideraba que la hora local era solo una herramienta matemática auxiliar, un truco por así decirlo, para simplificar la transformación de un sistema a otro.
Otros físicos y matemáticos de principios de siglo estuvieron a punto de llegar a lo que actualmente se conoce como espacio-tiempo. El propio Einstein señaló que con tanta gente desentrañando piezas separadas del rompecabezas, "la teoría especial de la relatividad, si consideramos su desarrollo en retrospectiva, estaba lista para ser descubierta en 1905". [12]
Un ejemplo importante es Henri Poincaré , [13] [14] : 73–80,93–95 quien en 1898 argumentó que la simultaneidad de dos eventos es una cuestión de convención. [15] [nota 2] En 1900, reconoció que la "hora local" de Lorentz es en realidad lo que se indica mediante relojes en movimiento al aplicar una definición explícitamente operativa de sincronización de reloj asumiendo una velocidad constante de la luz. [nota 3] En 1900 y 1904, sugirió la indetectabilidad inherente del éter al enfatizar la validez de lo que llamó el principio de relatividad , y en 1905/1906 [16] perfeccionó matemáticamente la teoría de los electrones de Lorentz para llevarla a cabo de acuerdo con el postulado de la relatividad. Mientras discutía varias hipótesis sobre la gravitación invariante de Lorentz, introdujo el concepto innovador de un espacio-tiempo de 4 dimensiones definiendo varios cuatro vectores , a saber, cuatro posiciones , cuatro velocidades y cuatro fuerzas . [17] [18] Sin embargo, no siguió el formalismo de 4 dimensiones en artículos posteriores, afirmando que esta línea de investigación parecía "implicar un gran dolor por un beneficio limitado", y finalmente concluyó "que el lenguaje tridimensional parece el más adecuado a la descripción de nuestro mundo ". [18] Además, incluso en 1909, Poincaré siguió creyendo en la interpretación dinámica de la transformada de Lorentz. [11] : 163-174 Por estas y otras razones, la mayoría de los historiadores de la ciencia argumentan que Poincaré no inventó lo que ahora se llama relatividad especial. [14] [11]
En 1905, Einstein introdujo la relatividad especial (aunque sin utilizar las técnicas del formalismo del espacio-tiempo) en su comprensión moderna como teoría del espacio y el tiempo. [14] [11] Si bien sus resultados son matemáticamente equivalentes a los de Lorentz y Poincaré, Einstein demostró que las transformaciones de Lorentz no son el resultado de interacciones entre la materia y el éter, sino que más bien se refieren a la naturaleza del espacio y el tiempo en sí. Obtuvo todos sus resultados al reconocer que toda la teoría puede basarse en dos postulados: el principio de relatividad y el principio de constancia de la velocidad de la luz.
Einstein realizó su análisis en términos de cinemática (el estudio de cuerpos en movimiento sin referencia a fuerzas) en lugar de dinámica. Su trabajo al presentar el tema estuvo lleno de imágenes vívidas que involucran el intercambio de señales de luz entre relojes en movimiento, mediciones cuidadosas de la longitud de las varillas en movimiento y otros ejemplos similares. [19] [nota 4]
Además, Einstein en 1905 reemplazó los intentos anteriores de una relación masa-energía electromagnética al introducir la equivalencia general de masa y energía , que fue fundamental para su posterior formulación del principio de equivalencia en 1907, que declara la equivalencia de masa inercial y gravitacional. Al utilizar la equivalencia masa-energía, Einstein demostró, además, que la masa gravitacional de un cuerpo es proporcional a su contenido de energía, que fue uno de los primeros resultados en el desarrollo de la relatividad general . Si bien parece que al principio no pensó geométricamente sobre el espacio-tiempo, [21] : 219 en el desarrollo posterior de la relatividad general, Einstein incorporó completamente el formalismo del espacio-tiempo.
Cuando Einstein publicó en 1905, otro de sus competidores, su antiguo profesor de matemáticas Hermann Minkowski , también había llegado a la mayoría de los elementos básicos de la relatividad especial. Max Born relató una reunión que había hecho con Minkowski, buscando ser alumno / colaborador de Minkowski: [22]
Fui a Colonia, conocí a Minkowski y escuché su célebre conferencia 'Espacio y tiempo' pronunciada el 2 de septiembre de 1908. […] Me dijo más tarde que le sorprendió mucho cuando Einstein publicó su artículo en el que la equivalencia del se pronunciaron las diferentes horas locales de los observadores moviéndose entre sí; porque había llegado a las mismas conclusiones independientemente, pero no las publicó porque deseaba primero elaborar la estructura matemática en todo su esplendor. Nunca hizo una afirmación de prioridad y siempre le dio a Einstein toda su participación en el gran descubrimiento.
Minkowski se había preocupado por el estado de la electrodinámica después de los experimentos disruptivos de Michelson al menos desde el verano de 1905, cuando Minkowski y David Hilbert dirigieron un seminario avanzado al que asistieron notables físicos de la época para estudiar los artículos de Lorentz, Poincaré et al. Sin embargo, no está del todo claro cuándo Minkowski comenzó a formular la formulación geométrica de la relatividad especial que iba a llevar su nombre, o hasta qué punto fue influenciado por la interpretación cuatridimensional de Poincaré de la transformación de Lorentz. Tampoco está claro si alguna vez apreció plenamente la contribución crítica de Einstein a la comprensión de las transformaciones de Lorentz, pensando en el trabajo de Einstein como una extensión del trabajo de Lorentz. [23]
El 5 de noviembre de 1907 (poco más de un año antes de su muerte), Minkowski presentó su interpretación geométrica del espacio-tiempo en una conferencia en la sociedad matemática de Göttingen con el título El principio de relatividad ( Das Relativitätsprinzip ). [nota 5] El 21 de septiembre de 1908, Minkowski presentó su famosa charla, Espacio y tiempo ( Raum und Zeit ), [24] a la Sociedad Alemana de Científicos y Médicos. Las palabras iniciales de Espacio y tiempo incluyen la famosa declaración de Minkowski de que "de ahora en adelante, el espacio para sí mismo y el tiempo para sí mismo se reducirán por completo a una mera sombra, y sólo algún tipo de unión de los dos preservará la independencia". Espacio y tiempo incluyó la primera presentación pública de diagramas de espacio-tiempo (Fig. 1-4), e incluyó una demostración notable de que el concepto de intervalo invariante ( discutido a continuación ), junto con la observación empírica de que la velocidad de la luz es finita, permite derivación de la totalidad de la relatividad especial. [nota 6]
El concepto de espacio-tiempo y el grupo de Lorentz están estrechamente relacionados con ciertos tipos de geometrías esféricas , hiperbólicas o conformes y sus grupos de transformación ya desarrollados en el siglo XIX, en los que se utilizan intervalos invariantes análogos al intervalo espacio-temporal . [nota 7]
Einstein, por su parte, inicialmente desdeñó la interpretación geométrica de la relatividad especial de Minkowski, considerándola como überflüssige Gelehrsamkeit (aprendizaje superfluo). Sin embargo, para completar su búsqueda de la relatividad general que comenzó en 1907, la interpretación geométrica de la relatividad resultó ser vital, y en 1916, Einstein reconoció plenamente su deuda con Minkowski, cuya interpretación facilitó enormemente la transición a la relatividad general. [11] : 151-152 Dado que existen otros tipos de espaciotiempo, como el espaciotiempo curvo de la relatividad general, el espaciotiempo de la relatividad especial se conoce hoy como espaciotiempo de Minkowski.
El espacio-tiempo en la relatividad especial
Intervalo de espacio-tiempo
En tres dimensiones, la distancia entre dos puntos se puede definir usando el teorema de Pitágoras :
Aunque dos espectadores pueden medir la posición x , y , z de los dos puntos usando diferentes sistemas de coordenadas, la distancia entre los puntos será la misma para ambos (asumiendo que están midiendo usando las mismas unidades). La distancia es "invariable".
En relatividad especial, sin embargo, la distancia entre dos puntos ya no es la misma si la miden dos observadores diferentes cuando uno de los observadores se está moviendo, debido a la contracción de Lorentz . La situación se complica aún más si los dos puntos están separados tanto en el tiempo como en el espacio. Por ejemplo, si un observador ve que dos eventos ocurren en el mismo lugar, pero en diferentes momentos, una persona que se mueve con respecto al primer observador verá que los dos eventos ocurren en diferentes lugares, porque (desde su punto de vista) son estacionarios. , y la posición del evento se está alejando o acercándose. Por lo tanto, se debe utilizar una medida diferente para medir la "distancia" efectiva entre dos eventos.
En el espacio-tiempo de cuatro dimensiones, el análogo a la distancia es el intervalo . Aunque el tiempo se presenta como una cuarta dimensión, se trata de manera diferente a las dimensiones espaciales. Por tanto, el espacio de Minkowski difiere en aspectos importantes del espacio euclidiano de cuatro dimensiones . La razón fundamental para fusionar el espacio y el tiempo en el espacio-tiempo es que el espacio y el tiempo no son invariantes por separado, lo que quiere decir que, en las condiciones adecuadas, diferentes observadores no estarán de acuerdo sobre la duración del tiempo entre dos eventos (debido a la dilatación del tiempo ) o la distancia entre los dos eventos (debido a la contracción de la longitud ). Pero la relatividad especial proporciona un nuevo invariante, llamado intervalo espaciotemporal , que combina distancias en el espacio y en el tiempo. Todos los observadores que midan el tiempo y la distancia entre dos eventos terminarán calculando el mismo intervalo de espacio-tiempo. Suponga que un observador mide dos eventos separados en el tiempo por y una distancia espacial Entonces el intervalo de espacio-tiempo entre los dos eventos que están separados por una distancia en el espacio y por en el -coordinar es:
o para tres dimensiones de espacio,
- [28]
El constante la velocidad de la luz, convierte unidades de tiempo (como segundos) en unidades espaciales (como metros). Segundos por metros / segundo = metros.
Aunque por brevedad, con frecuencia se ven expresiones de intervalo expresadas sin deltas, incluso en la mayor parte de la siguiente discusión, debe entenderse que, en general, medio , etc. Siempre nos interesan las diferencias de valores de coordenadas espaciales o temporales que pertenecen a dos eventos, y dado que no hay un origen preferido, los valores de coordenadas individuales no tienen un significado esencial.
La ecuación anterior es similar al teorema de Pitágoras, excepto con un signo menos entre los y el condiciones. El intervalo de espacio-tiempo es la cantidad no sí mismo. La razón es que, a diferencia de las distancias en la geometría euclidiana, los intervalos en el espacio-tiempo de Minkowski pueden ser negativos. En lugar de tratar con raíces cuadradas de números negativos, los físicos suelen considerarcomo un símbolo distinto en sí mismo, en lugar del cuadrado de algo. [21] : 217
Debido al signo menos, el intervalo de espacio-tiempo entre dos eventos distintos puede ser cero. Sies positivo, el intervalo de espacio-tiempo es similar al tiempo , lo que significa que dos eventos están separados por más tiempo que espacio. Sies negativo, el intervalo de espacio-tiempo es similar a un espacio , lo que significa que dos eventos están separados por más espacio que tiempo. Los intervalos de espacio-tiempo son cero cuandoEn otras palabras, el intervalo de espacio-tiempo entre dos eventos en la línea del mundo de algo que se mueve a la velocidad de la luz es cero. Tal intervalo se denomina ligero o nulo . Un fotón que llega a nuestro ojo desde una estrella distante no habrá envejecido, a pesar de haber pasado (desde nuestra perspectiva) años en su paso.
Un diagrama de espacio-tiempo generalmente se dibuja con un solo espacio y una sola coordenada de tiempo. La figura 2-1 presenta un diagrama de espacio-tiempo que ilustra las líneas del mundo (es decir, trayectorias en el espacio-tiempo) de dos fotones, A y B, que se originan en el mismo evento y van en direcciones opuestas. Además, C ilustra la línea del mundo de un objeto a una velocidad menor que la de la luz. La coordenada de tiempo vertical se escala porpara que tenga las mismas unidades (metros) que la coordenada del espacio horizontal. Dado que los fotones viajan a la velocidad de la luz, sus líneas de mundo tienen una pendiente de ± 1. En otras palabras, cada metro que un fotón viaja hacia la izquierda o hacia la derecha requiere aproximadamente 3.3 nanosegundos de tiempo.
Hay dos convenciones de signos en uso en la literatura de la relatividad:
y
Estas convenciones de signos están asociadas con las firmas métricas (+ - - -) y (- + + +). Una pequeña variación es colocar la coordenada de tiempo en último lugar en lugar de primero. Ambas convenciones se utilizan ampliamente en el campo de estudio.
Marcos de referencia
Para obtener información sobre cómo las coordenadas del espacio-tiempo medidas por los observadores en diferentes marcos de referencia se comparan entre sí, es útil trabajar con una configuración simplificada con marcos en una configuración estándar. Con cuidado, esto permite simplificar las matemáticas sin pérdida de generalidad en las conclusiones a las que se llega. En la Fig. 2-2, dos marcos de referencia galileanos (es decir, marcos de 3 espacios convencionales) se muestran en movimiento relativo. La trama S pertenece a un primer observador O, y la trama S ′ (pronunciada "S prima") pertenece a un segundo observador O ′.
- Los ejes x , y , z del marco S están orientados paralelos a los respectivos ejes cebados del marco S '.
- La trama S ′ se mueve en la dirección x de la trama S con una velocidad constante v medida en la trama S.
- Los orígenes de las tramas S y S ′ son coincidentes cuando el tiempo t = 0 para la trama S y t ′ = 0 para la trama S ′. [4] : 107
La Fig. 2-3a vuelve a dibujar la Fig. 2-2 en una orientación diferente. La figura 2-3b ilustra un diagrama de espacio-tiempo desde el punto de vista del observador O. Dado que S y S ′ están en configuración estándar, sus orígenes coinciden en los momentos t = 0 en el marco S y t ′ = 0 en el marco S ′. El eje ct ′ pasa a través de los eventos en el marco S ′ que tienen x ′ = 0. Pero los puntos con x ′ = 0 se mueven en la dirección x del marco S con velocidad v , de modo que no coinciden con ct eje en cualquier momento que no sea cero. Por lo tanto, el eje ct ′ está inclinado con respecto al eje ct en un ángulo θ dado por
El eje x 'también está inclinado con respecto al eje x . Para determinar el ángulo de esta inclinación, recordamos que la pendiente de la línea universal de un pulso de luz es siempre ± 1. La figura 2-3c presenta un diagrama de espacio-tiempo desde el punto de vista del observador O ′. El evento P representa la emisión de un pulso de luz en x ′ = 0, ct ′ = - a . El pulso se refleja en un espejo situado a una distancia a de la fuente de luz (evento Q) y regresa a la fuente de luz en x ′ = 0, ct ′ = a (evento R).
Los mismos eventos P, Q, R se grafican en la figura 2-3b en el marco del observador O.Las trayectorias de luz tienen pendientes = 1 y −1, de modo que △ PQR forma un triángulo rectángulo con PQ y QR a 45 grados a las x y ct ejes. Dado que OP = OQ = OR, el ángulo entre x ′ y x también debe ser θ . [4] : 113–118
Mientras que el marco en reposo tiene ejes de espacio y tiempo que se encuentran en ángulos rectos, el marco en movimiento se dibuja con ejes que se encuentran en un ángulo agudo. Los marcos son realmente equivalentes. La asimetría se debe a distorsiones inevitables en la forma en que las coordenadas del espacio-tiempo se pueden mapear en un plano cartesiano , y no debe considerarse más extraña que la forma en que, en una proyección de Mercator de la Tierra, los tamaños relativos de las masas de tierra cerca de los polos (Groenlandia y Antártida) son muy exageradas en relación con las masas de tierra cercanas al ecuador.
Cono de luz
En la figura 2-4, el evento O está en el origen de un diagrama de espacio-tiempo y las dos líneas diagonales representan todos los eventos que tienen un intervalo de espacio-tiempo cero con respecto al evento de origen. Estas dos líneas forman lo que se llama el cono de luz del evento O, ya que al agregar una segunda dimensión espacial (figura 2-5) se hace la apariencia de dos conos circulares rectos que se encuentran con sus ápices en O. Un cono se extiende hacia el futuro (t> 0), el otro al pasado (t <0).
Un cono de luz (doble) divide el espacio-tiempo en regiones separadas con respecto a su vértice. El interior del cono de luz futuro consta de todos los eventos que están separados del vértice por más tiempo (distancia temporal) del necesario para cruzar su distancia espacial a la velocidad de la luz; estos eventos comprenden el futuro temporal del evento O. Asimismo, el pasado temporal comprende los eventos interiores del cono de luz pasado. Entonces, en intervalos de tiempo, Δ ct es mayor que Δ x , lo que hace que los intervalos de tiempo sean positivos. La región exterior al cono de luz consiste en eventos que están separados del evento O por más espacio del que se puede cruzar a la velocidad de la luz en el tiempo dado . Estos eventos comprenden la llamada región similar a un espacio del evento O, denotado "En otra parte" en la figura 2-4. Se dice que los eventos en el cono de luz son parecidos a la luz (o separados nulos ) de O.Debido a la invariancia del intervalo de espacio-tiempo, todos los observadores asignarán el mismo cono de luz a cualquier evento dado y, por lo tanto, estarán de acuerdo en esta división del espacio-tiempo. . [21] : 220
El cono de luz tiene un papel fundamental dentro del concepto de causalidad . Es posible que una señal de velocidad no más rápida que la de la luz viaje desde la posición y el tiempo de O a la posición y el tiempo de D (Fig. 2-4). Por lo tanto, es posible que el evento O tenga una influencia causal en el evento D. El cono de luz futuro contiene todos los eventos que podrían estar causalmente influenciados por O. De la misma manera, es posible que una señal de velocidad no más rápida que la luz viajar desde la posición y el tiempo de A, hasta la posición y el tiempo de O. El cono de luz pasado contiene todos los eventos que podrían tener una influencia causal en O. Por el contrario, suponiendo que las señales no pueden viajar más rápido que la velocidad de la luz, cualquier El evento, como por ejemplo, B o C, en la región similar a un espacio (en otros lugares), no puede afectar al evento O, ni pueden verse afectados por el evento O que emplea tal señalización. Bajo este supuesto, se excluye cualquier relación causal entre el evento O y cualquier evento en la región espacial de un cono de luz. [29]
Relatividad de la simultaneidad
Todos los observadores estarán de acuerdo en que para cualquier evento dado, un evento dentro del cono de luz futuro del evento dado ocurre después del evento dado. Del mismo modo, para cualquier evento dado, un evento dentro del cono de luz pasado del evento dado ocurre antes del evento dado. La relación antes-después observada para eventos separados en forma de tiempo permanece sin cambios sin importar cuál sea el marco de referencia del observador, es decir, sin importar cómo se esté moviendo el observador. La situación es bastante diferente para los eventos separados en forma de espacio. La figura 2-4 se extrajo del marco de referencia de un observador que se mueve en v = 0. Desde este marco de referencia, se observa que el evento C ocurre después del evento O, y se observa que el evento B ocurre antes del evento O. De una referencia diferente marco, el orden de estos eventos no relacionados causalmente se puede revertir. En particular, se observa que si dos eventos son simultáneos en un marco de referencia particular, están necesariamente separados por un intervalo similar a un espacio y, por lo tanto, no están relacionados causalmente. La observación de que la simultaneidad no es absoluta, sino que depende del marco de referencia del observador, se denomina relatividad de la simultaneidad . [30]
La figura 2-6 ilustra el uso de diagramas de espacio-tiempo en el análisis de la relatividad de la simultaneidad. Los eventos en el espacio-tiempo son invariantes, pero los marcos de coordenadas se transforman como se discutió anteriormente en la figura 2-3. Los tres eventos (A, B, C) son simultáneos desde el marco de referencia de un observador que se mueve en v = 0. Desde el marco de referencia de un observador que se mueve en v = 0.3 c , los eventos parecen ocurrir en el orden C, B , A. Desde el cuadro de referencia de un observador que se mueve a v = -0,5 c , los eventos parece ocurrir en el orden a, B, C . La línea blanca representa un plano de simultaneidad que se mueve del pasado del observador al futuro del observador, destacando los eventos que residen en él. El área gris es el cono de luz del observador, que permanece invariable.
Un intervalo de espacio-tiempo similar a un espacio da la misma distancia que mediría un observador si los eventos que se miden fueran simultáneos al observador. Por tanto, un intervalo de espacio-tiempo similar a un espacio proporciona una medida de la distancia adecuada , es decir, la distancia verdadera =Del mismo modo, un intervalo de espacio-tiempo similar al tiempo da la misma medida de tiempo que presentaría el tic-tac acumulativo de un reloj que se mueve a lo largo de una línea de mundo dada. Por tanto, un intervalo de espacio-tiempo similar al tiempo proporciona una medida del tiempo adecuado =[21] : 220–221
Hipérbola invariante
En el espacio euclidiano (que solo tiene dimensiones espaciales), el conjunto de puntos equidistantes (usando la métrica euclidiana) desde algún punto forma un círculo (en dos dimensiones) o una esfera (en tres dimensiones). En el espacio-tiempo de Minkowski (1 + 1) -dimensional (que tiene una dimensión temporal y una espacial), los puntos en algún intervalo de espacio-tiempo constante lejos del origen (usando la métrica de Minkowski) forman curvas dadas por las dos ecuaciones
con alguna constante real positiva. Estas ecuaciones describen dos familias de hipérbolas en un diagrama de espacio-tiempo x - ct , que se denominan hipérbolas invariantes .
En la figura 2-7a, cada hipérbola magenta conecta todos los eventos que tienen una separación espacial fija desde el origen, mientras que las hipérbolas verdes conectan eventos de igual separación temporal.
Las hipérbolas magenta, que cruzan el eje x , son curvas temporales, lo que quiere decir que estas hipérbolas representan trayectorias reales que pueden ser atravesadas por partículas (en constante aceleración) en el espacio-tiempo: entre dos eventos cualesquiera en una hipérbola es posible una relación de causalidad, porque la inversa de la pendiente, que representa la rapidez necesaria, para todas las secantes es menor que. Por otro lado, las hipérbolas verdes, que cruzan el eje ct , son curvas espaciales porque todos los intervalos a lo largo de estas hipérbolas son intervalos espaciales: no es posible causalidad entre dos puntos cualesquiera de una de estas hipérbolas, porque todas las secantes representan velocidades mayores que.
La figura 2-7b refleja la situación en el espacio-tiempo (1 + 2) -dimensional de Minkowski (una dimensión temporal y dos espaciales) con los hiperboloides correspondientes. Las hipérbolas invariantes desplazadas por intervalos espaciales desde el origen generan hiperboloides de una hoja, mientras que las hipérbolas invariantes desplazadas por intervalos temporales desde el origen generan hiperboloides de dos hojas.
El límite (1 + 2) -dimensional entre los hiperboloides espaciales y temporales, establecido por los eventos que forman un intervalo espaciotemporal cero con el origen, se compone degenerando los hiperboloides en el cono de luz. En las dimensiones (1 + 1), las hipérbolas degeneran en las dos líneas grises de 45 ° representadas en la figura 2-7a.
Dilatación del tiempo y contracción de la longitud
La figura 2-8 ilustra la hipérbola invariante para todos los eventos a los que se puede llegar desde el origen en un tiempo adecuado de 5 metros (aproximadamente 1,67 × 10 −8 s ). Diferentes líneas de mundo representan relojes que se mueven a diferentes velocidades. Un reloj que está estacionario con respecto al observador tiene una línea de mundo que es vertical, y el tiempo transcurrido medido por el observador es el mismo que el tiempo adecuado. Para un reloj que viaja a 0,3 c , el tiempo transcurrido medido por el observador es de 5,24 metros (1,75 × 10 −8 s ), mientras que para un reloj que viaja a 0,7 c , el tiempo transcurrido medido por el observador es de 7,00 metros (2,34 × 10 −8 s ). Esto ilustra el fenómeno conocido como dilatación del tiempo . Los relojes que viajan más rápido toman más tiempo (en el marco del observador) para marcar la misma cantidad de tiempo apropiado, y viajan más a lo largo del eje x dentro de ese tiempo apropiado de lo que lo hubieran hecho sin la dilatación del tiempo. [21] : 220-221 La medición de la dilatación del tiempo por dos observadores en diferentes marcos de referencia inerciales es mutua. Si el observador O mide que los relojes del observador O ′ corren más lento en su marco, el observador O ′ a su vez medirá los relojes del observador O como que corren más lento.
La contracción de la longitud , como la dilatación del tiempo, es una manifestación de la relatividad de la simultaneidad. La medición de la longitud requiere la medición del intervalo espaciotemporal entre dos eventos que son simultáneos en el marco de referencia de uno. Pero los eventos que son simultáneos en un marco de referencia, en general, no son simultáneos en otros marcos de referencia.
La figura 2-9 ilustra los movimientos de una varilla de 1 m que se desplaza a 0,5 c a lo largo del eje x . Los bordes de la banda azul representan las líneas del mundo de los dos extremos de la barra. La hipérbola invariante ilustra eventos separados del origen por un intervalo de espacio de 1 m. Los puntos finales O y B medidos cuando t ′ = 0 son eventos simultáneos en la trama S ′. Pero para un observador en el marco S, los eventos O y B no son simultáneos. Para medir la longitud, el observador en el cuadro S mide los puntos finales de la barra proyectada sobre el eje x a lo largo de sus líneas de mundo. La proyección de la hoja del mundo de la barra sobre el eje x produce la longitud OC acortada. [4] : 125
(no ilustrado) Dibujar una línea vertical que atraviese A de modo que cruce el eje x ′ demuestra que, incluso cuando OB se acorta desde el punto de vista del observador O, OA también se acorta desde el punto de vista del observador O ′. De la misma manera que cada observador mide que los relojes del otro corren lento, cada observador mide las reglas del otro como contraídas.
Con respecto a la contracción de la longitud mutua, 2-9 ilustra que los marcos cebados y no cebados se rotan mutuamente por un ángulo hiperbólico (análogo a los ángulos ordinarios en la geometría euclidiana). [nota 8] Debido a esta rotación, la proyección de una varilla de metro imprimada sobre el eje x no imprimado se acorta, mientras que la proyección de una varilla métrica no imprimada sobre el eje x ' imprimado también se acorta.
Dilatación mutua del tiempo y la paradoja de los gemelos
Dilatación mutua del tiempo
La dilatación mutua del tiempo y la contracción de la longitud tienden a sorprender a los principiantes como conceptos intrínsecamente contradictorios. Si un observador en el cuadro S mide un reloj, en reposo en el cuadro S ', como si fuera más lento que su', mientras que S 'se mueve a una velocidad v en S, entonces el principio de relatividad requiere que un observador en el cuadro S' también mida un reloj en el cuadro S, moviéndose a velocidad - v en S ', como si fuera más lento que el de ella. Cómo dos relojes pueden funcionar más lento que el otro, es una cuestión importante que "va al corazón de la comprensión de la relatividad especial". [21] : 198
Esta aparente contradicción se debe a que no se tienen en cuenta correctamente los diferentes ajustes de las medidas necesarias relacionadas. Estos escenarios permiten una explicación coherente de la única aparente contradicción. No se trata del tic-tac abstracto de dos relojes idénticos, sino de cómo medir en un cuadro la distancia temporal de dos tic-tac de un reloj en movimiento. Resulta que al observar mutuamente la duración entre los tics de los relojes, cada uno moviéndose en el marco respectivo, deben estar involucrados diferentes conjuntos de relojes. Para medir en el cuadro S la duración del tic de un reloj en movimiento W ′ (en reposo en S ′), se utilizan dos relojes sincronizados adicionales W 1 y W 2 en reposo en dos puntos arbitrariamente fijos en S con la distancia espacial d .
- Se pueden definir dos eventos mediante la condición "dos relojes están simultáneamente en un lugar", es decir, cuando W ′ pasa cada W 1 y W 2 . Para ambos eventos se registran las dos lecturas de los relojes colocados. La diferencia de las dos lecturas de W 1 y W 2 es la distancia temporal de los dos eventos en S, y su distancia espacial es d . La diferencia de las dos lecturas de W ′ es la distancia temporal de los dos eventos en S ′. En S ′ estos eventos solo están separados en el tiempo, ocurren en el mismo lugar en S ′. Debido a la invariancia del intervalo espaciotemporal abarcado por estos dos eventos, y la separación espacial distinta de cero d en S, la distancia temporal en S ′ debe ser menor que la de S: la distancia temporal más pequeña entre los dos eventos, resultante de la lecturas del reloj en movimiento W ′, pertenece al reloj de funcionamiento más lento W ′.
A la inversa, para juzgar en la trama S ′ la distancia temporal de dos eventos en un reloj en movimiento W (en reposo en S), se necesitan dos relojes en reposo en S ′.
- En esta comparación, el reloj W se mueve con una velocidad - v . Al registrar nuevamente las cuatro lecturas de los eventos, definidos por "dos relojes simultáneamente en un lugar", se obtienen las distancias temporales análogas de los dos eventos, ahora separados temporal y espacialmente en S ′, y solo temporalmente separados pero colocados en S. Para mantener invariante el intervalo de espacio-tiempo, la distancia temporal en S debe ser menor que en S ′, debido a la separación espacial de los eventos en S ′: ahora se observa que el reloj W corre más lento.
Las grabaciones necesarias para los dos juicios, con "un reloj en movimiento" y "dos relojes en reposo" en S o S ′ respectivamente, implican dos conjuntos diferentes, cada uno con tres relojes. Dado que hay diferentes conjuntos de relojes involucrados en las mediciones, no existe una necesidad inherente de que las mediciones sean recíprocamente "consistentes" de modo que, si un observador mide que el reloj en movimiento es lento, el otro observador mide que el reloj de uno es rápido. [21] : 198-199
La figura 2-10 ilustra la discusión previa de la dilatación mutua del tiempo con diagramas de Minkowski. La imagen superior refleja las medidas como se ve desde el cuadro S "en reposo" con ejes rectangulares sin imprimación, y el cuadro S ′ "moviéndose con v > 0", coordinado por ejes oblicuos con imprimación, inclinados hacia la derecha; la imagen inferior muestra la trama S ′ "en reposo" con coordenadas rectangulares preparadas, y la trama S "moviéndose con - v <0", con ejes oblicuos sin imprimación, inclinados hacia la izquierda.
Cada línea trazada paralela a un eje espacial ( x , x ′) representa una línea de simultaneidad. Todos los eventos en dicha línea tienen el mismo valor de tiempo ( ct , ct ′). Asimismo, cada línea trazada paralela a un eje temporal ( ct , ct ′ ) representa una línea de iguales valores de coordenadas espaciales ( x , x ′).
- Se puede designar en ambas imágenes el origen O (= O ′ ) como el evento, donde el "reloj en movimiento" respectivo se coloca con el "primer reloj en reposo" en ambas comparaciones. Obviamente, para este evento las lecturas en ambos relojes en ambas comparaciones son cero. Como consecuencia, las líneas de mundo de los relojes en movimiento están inclinadas hacia el eje ct ′ derecho (imágenes superiores, reloj W ′) y las inclinadas hacia el eje ct izquierdo (imágenes inferiores, reloj W). Las líneas de mundo de W 1 y W ′ 1 son los ejes de tiempo verticales correspondientes ( ct en las imágenes superiores y ct ′ en las imágenes inferiores).
- En la imagen superior, el lugar para W 2 se considera A x > 0, y por lo tanto la línea de mundo (no se muestra en las imágenes) de este reloj se cruza con la línea de mundo del reloj en movimiento (el eje ct ′) en el evento etiquetado A , donde "dos relojes están simultáneamente en un lugar". En la imagen inferior el lugar para W ' 2 se toma para ser C x ' <0, y así en esta medición el reloj en movimiento W pasa W ' 2 en el caso C .
- En la imagen superior, la ct -coordinada A t del evento A (la lectura de W 2 ) está etiquetada como B , dando así el tiempo transcurrido entre los dos eventos, medido con W 1 y W 2 , como OB . Para una comparación, la longitud del intervalo de tiempo OA , medido con W ′, debe transformarse a la escala del eje ct . Esto se hace mediante la hipérbola invariante (véase también la Fig. 2-8) a través de A , la conexión de todos los eventos con el mismo intervalo de espacio-tiempo desde el origen como A . Esto produce el evento C en el eje ct , y obviamente: OC < OB , el reloj "en movimiento" W ′ corre más lento.
Para mostrar la dilatación mutua del tiempo inmediatamente en la imagen superior, el evento D puede construirse como el evento en x ′ = 0 (la ubicación del reloj W ′ en S ′), que es simultáneo a C ( OC tiene el mismo intervalo de espacio-tiempo que OA ) en S ′. Esto muestra que el intervalo de tiempo OD es más largo que OA , lo que muestra que el reloj "en movimiento" corre más lento. [4] : 124
En la imagen inferior, el cuadro S se mueve con velocidad - v en el cuadro S ′ en reposo. La línea de mundo del reloj W es el eje ct (inclinado hacia la izquierda), la línea de mundo de W ′ 1 es el eje vertical de ct ′, y la línea de mundo de W ′ 2 es la vertical a través del evento C , con ct ′ -coordinada D . La hipérbola invariante a través del evento C escala el intervalo de tiempo OC a OA , que es más corto que OD ; además, B se construye (similar a D en las imágenes superiores) como simultáneo a A en S, en x = 0. El resultado OB > OC corresponde nuevamente al anterior.
La palabra "medir" es importante. En la física clásica un observador no puede afectar a un objeto observado, pero el estado del objeto de movimiento puede afectar del observador observaciones del objeto.
Paradoja de los gemelos
Muchas introducciones a la relatividad especial ilustran las diferencias entre la relatividad galileana y la relatividad especial planteando una serie de "paradojas". Estas paradojas son, de hecho, problemas mal planteados, que resultan de nuestra falta de familiaridad con velocidades comparables a la velocidad de la luz. El remedio es resolver muchos problemas de la relatividad especial y familiarizarse con sus llamadas predicciones contraintuitivas. El enfoque geométrico para estudiar el espacio-tiempo se considera uno de los mejores métodos para desarrollar una intuición moderna. [31]
La paradoja de los gemelos es un experimento mental que involucra a gemelos idénticos, uno de los cuales hace un viaje al espacio en un cohete de alta velocidad, regresa a casa y descubre que el gemelo que permaneció en la Tierra ha envejecido más. Este resultado parece desconcertante porque cada gemelo observa que el otro gemelo se mueve y, a primera vista, parecería que cada uno debería encontrar que el otro ha envejecido menos. La paradoja de los gemelos evita la justificación de la dilatación mutua del tiempo presentada anteriormente al evitar el requisito de un tercer reloj. [21] : 207 Sin embargo, la paradoja de los gemelos no es una paradoja verdadera porque se entiende fácilmente dentro del contexto de la relatividad especial.
La impresión de que existe una paradoja proviene de una mala comprensión de lo que dice la relatividad especial. La relatividad especial no declara que todos los marcos de referencia sean equivalentes, solo marcos inerciales. El marco del gemelo viajero no es inercial durante los períodos en los que está acelerando. Además, la diferencia entre los gemelos es detectable por observación: el gemelo que viaja necesita disparar sus cohetes para poder regresar a casa, mientras que el gemelo que se queda en casa no. [32] [nota 9]
Estas distinciones deberían resultar en una diferencia en las edades de los gemelos. El diagrama de espacio-tiempo de la figura 2-11 presenta el caso simple de un gemelo que sale directamente a lo largo del eje x y vuelve inmediatamente hacia atrás. Desde el punto de vista del gemelo que se queda en casa, la paradoja de los gemelos no tiene nada de extraño. El tiempo adecuado medido a lo largo de la línea mundial del gemelo que viaja de O a C, más el tiempo adecuado medido de C a B, es menor que el tiempo adecuado del gemelo que se queda en casa medido de O a A a B. Las trayectorias más complejas requieren la integración el tiempo adecuado entre los eventos respectivos a lo largo de la curva (es decir, la integral de trayectoria ) para calcular la cantidad total de tiempo adecuado experimentado por el gemelo que viaja. [32]
Las complicaciones surgen si la paradoja de los gemelos se analiza desde el punto de vista del gemelo viajero.
La nomenclatura de Weiss, que designa al gemelo que se queda en casa como Terence y al gemelo que viaja como Stella, se utiliza de ahora en adelante. [32]
Stella no está en un marco inercial. Dado este hecho, a veces se afirma incorrectamente que la resolución completa de la paradoja de los gemelos requiere la relatividad general: [32]
Un análisis de SR puro sería el siguiente: analizada en el marco de reposo de Stella, está inmóvil durante todo el viaje. Cuando dispara sus cohetes para dar la vuelta, experimenta una pseudo fuerza que se asemeja a una fuerza gravitacional. [32] Figs. 2-6 y 2-11 ilustran el concepto de líneas (planos) de simultaneidad: Las líneas paralelas al eje x del observador ( plano xy ) representan conjuntos de eventos que son simultáneos en el marco del observador. En la figura 2-11, las líneas azules conectan eventos en la línea del mundo de Terence que, desde el punto de vista de Stella , son simultáneos con eventos en su línea del mundo. (Terence, a su vez, observaría un conjunto de líneas horizontales de simultaneidad.) A lo largo de las etapas de ida y vuelta del viaje de Stella, ella mide los relojes de Terence como si fueran más lentos que los suyos. Pero durante el cambio (es decir, entre las líneas azules en negrita en la figura), se produce un cambio en el ángulo de sus líneas de simultaneidad, que corresponde a un salto rápido de los eventos en la línea del mundo de Terence que Stella considera simultáneos con su propio. Por lo tanto, al final de su viaje, Stella descubre que Terence ha envejecido más que ella. [32]
Aunque la relatividad general no es necesaria para analizar la paradoja de los gemelos, la aplicación del principio de equivalencia de la relatividad general proporciona una visión adicional del tema. Stella no está estacionaria en un marco inercial. Analizada en el marco de descanso de Stella, está inmóvil durante todo el viaje. Cuando está navegando, su marco de descanso es inercial, y el reloj de Terence parecerá correr lento. Pero cuando dispara sus cohetes para dar la vuelta, su marco de descanso es un marco acelerado y experimenta una fuerza que la empuja como si estuviera en un campo gravitacional. Terence parecerá estar muy arriba en ese campo y debido a la dilatación del tiempo gravitacional , su reloj parecerá correr rápido, tanto que el resultado neto será que Terence ha envejecido más que Stella cuando vuelvan a estar juntos. [32] Los argumentos teóricos que predicen la dilatación del tiempo gravitacional no son exclusivos de la relatividad general. Cualquier teoría de la gravedad predecirá la dilatación del tiempo gravitacional si respeta el principio de equivalencia, incluida la teoría de Newton. [21] : 16
Gravitación
Esta sección introductoria se ha centrado en el espacio-tiempo de la relatividad especial, ya que es la más fácil de describir. El espacio-tiempo de Minkowski es plano, no tiene en cuenta la gravedad, es uniforme en todas partes y no sirve más que como un fondo estático para los eventos que tienen lugar en él. La presencia de gravedad complica enormemente la descripción del espacio-tiempo. En la relatividad general, el espacio-tiempo ya no es un fondo estático, sino que interactúa activamente con los sistemas físicos que contiene. El espacio-tiempo se curva en presencia de materia, puede propagar ondas, desviar la luz y exhibe una serie de otros fenómenos. [21] : 221 Algunos de estos fenómenos se describen en las secciones posteriores de este artículo.
Matemáticas básicas del espacio-tiempo
Transformaciones galileanas
Un objetivo básico es poder comparar las mediciones realizadas por los observadores en movimiento relativo. Si hay un observador O en el cuadro S que ha medido las coordenadas de tiempo y espacio de un evento, asignando a este evento tres coordenadas cartesianas y el tiempo medido en su red de relojes sincronizados ( x , y , z , t ) (ver Fig. .1-1 ). Un segundo observador O ′ en un marco diferente S ′ mide el mismo evento en su sistema de coordenadas y su red de relojes sincronizados ( x ′ , y ′ , z ′ , t ′ ) . Con los marcos inerciales, ninguno de los observadores está acelerado, y un conjunto simple de ecuaciones nos permite relacionar las coordenadas ( x , y , z , t ) con ( x ′ , y ′ , z ′ , t ′ ) . Dado que los dos sistemas de coordenadas están en configuración estándar, lo que significa que están alineados con coordenadas paralelas ( x , y , z ) y que t = 0 cuando t ′ = 0 , la transformación de coordenadas es la siguiente: [33] [34]
Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light.[35]:36–37 Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4 c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4 c in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.
More generally, assuming that frame S′ is moving at velocity v with respect to frame S, then within frame S′, observer O′ measures an object moving with velocity u′. Velocity u with respect to frame S, since x = ut, x′ = x − vt, and t = t′, can be written as x′ = ut − vt = (u − v)t = (u − v)t′. This leads to u′ = x′/t′ and ultimately
- or
which is the common-sense Galilean law for the addition of velocities.
Relativistic composition of velocities
The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,
Fig. 3-2a illustrates a red train that is moving forward at a speed given by v/c = β = s/a. From the primed frame of the train, a passenger shoots a bullet with a speed given by u′/c = β′ = n/m, where the distance is measured along a line parallel to the red x′ axis rather than parallel to the black x axis. What is the composite velocity u of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. 3-2b:
- From the platform, the composite speed of the bullet is given by u = c(s + r)/(a + b).
- The two yellow triangles are similar because they are right triangles that share a common angle α. In the large yellow triangle, the ratio s/a = v/c = β.
- The ratios of corresponding sides of the two yellow triangles are constant, so that r/a = b/s = n/m = β′. So b = u′s/c and r = u′a/c.
- Substitute the expressions for b and r into the expression for u in step 1 to yield Einstein's formula for the addition of velocities:[35]:42–48
The relativistic formula for addition of velocities presented above exhibits several important features:
- If u′ and v are both very small compared with the speed of light, then the product vu′/c2 becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: u = u′ + v. The Galilean formula is a special case of the relativistic formula applicable to low velocities.
- If u′ is set equal to c, then the formula yields u = c regardless of the starting value of v. The velocity of light is the same for all observers regardless their motions relative to the emitting source.[35]:49
Time dilation and length contraction revisited
It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig. 3-3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.
To reduce the complexity of the equations slightly, there are a variety of different shorthand notations for ct:
- and are common.
- One also sees very frequently the use of the convention
In Fig. 3-3a, segments OA and OK represent equal spacetime intervals. Time dilation is represented by the ratio OB/OK. The invariant hyperbola has the equation w = √x2 + k2 where k = OK, and the red line representing the world line of a particle in motion has the equation w = x/β = xc/v. A bit of algebraic manipulation yields
The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma :[36]
If v is greater than or equal to c, the expression for becomes physically meaningless, implying that c is the maximum possible speed in nature. For any v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.
In Fig. 3-3b, segments OA and OK represent equal spacetime intervals. Length contraction is represented by the ratio OB/OK. The invariant hyperbola has the equation x = √w2 + k2, where k = OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/β = c/v. Event A has coordinates (x, w) = (γk, γβk). Since the tangent line through A and B has the equation w = (x − OB)/β, we have γβk = (γk − OB)/β and
Lorentz transformations
The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.
Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity.
The Lorentz factor appears in the Lorentz transformations:
The inverse Lorentz transformations are:
When v ≪ c and x is small enough, the v2/c2 and vx/c2 terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.
etc., most often really mean etc. Although for brevity the Lorentz transformation equations are written without deltas, x means Δx, etc. We are, in general, always concerned with the space and time differences between events.
Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the S frame can only be moving forwards or reverse with respect to S′. So inverting the equations simply entails switching the primed and unprimed variables and replacing v with −v.[37]:71–79
Example: Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time t = t′ = 0, Stella's spaceship accelerates instantaneously to a speed of 0.5 c. The distance from Earth to Mars is 300 light-seconds (about 90.0×106 km). Terence observes Stella crossing the finish-line clock at t = 600.00 s. But Stella observes the time on her ship chronometer to be as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about 77.9×106 km). 1).
Deriving the Lorentz transformations
There have been many dozens of derivations of the Lorentz transformations since Einstein's original work in 1905, each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different expressions of the underlying principle of locality, which states that the influence that one particle exerts on another can not be transmitted instantaneously.[38]
The derivation given here and illustrated in Fig. 3-5 is based on one presented by Bais[35]:64–66 and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections. Event P has coordinates (w, x) in the black "rest system" and coordinates (w′, x′) in the red frame that is moving with velocity parameter β = v/c. To determine w′ and x′ in terms of w and x (or the other way around) it is easier at first to derive the inverse Lorentz transformation.
- There can be no such thing as length expansion/contraction in the transverse directions. y' must equal y and z′ must equal z, otherwise whether a fast moving 1 m ball could fit through a 1 m circular hole would depend on the observer. The first postulate of relativity states that all inertial frames are equivalent, and transverse expansion/contraction would violate this law.[37]:27–28
- From the drawing, w = a + b and x = r + s
- From previous results using similar triangles, we know that s/a = b/r = v/c = β.
- Because of time dilation, a = γw′
- Substituting equation (4) into s/a = β yields s = γw′β.
- Length contraction and similar triangles give us r = γx′ and b = βr = βγx′
- Substituting the expressions for s, a, r and b into the equations in Step 2 immediately yield
The above equations are alternate expressions for the t and x equations of the inverse Lorentz transformation, as can be seen by substituting ct for w, ct′ for w′, and v/c for β. From the inverse transformation, the equations of the forwards transformation can be derived by solving for t′ and x′.
Linearity of the Lorentz transformations
The Lorentz transformations have a mathematical property called linearity, since x′ and t′ are obtained as linear combinations of x and t, with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that was tacitly assumed in the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere.[35]:67 All inertial observers will agree on what constitutes accelerating and non-accelerating motion.[37]:72–73 Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well.[21]:190
A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.
Example: Terence observes Stella speeding away from him at 0.500 c, and he can use the Lorentz transformations with β = 0.500 to relate Stella's measurements to his own. Stella, in her frame, observes Ursula traveling away from her at 0.250 c, and she can use the Lorentz transformations with β = 0.250 to relate Ursula's measurements with her own. Because of the linearity of the transformations and the relativistic composition of velocities, Terence can use the Lorentz transformations with β = 0.666 to relate Ursula's measurements with his own.
Doppler effect
The Doppler effect is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at right angles to the said line (transverse Doppler effect). We are ignoring scenarios where they move along intermediate angles.
Longitudinal Doppler effect
The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of vs for a velocity parameter of βs, the wavelength is increased, and the observed frequency f is given by
On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of vr for a velocity parameter of βr, the wavelength is not changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency f is given by
Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig. 3-6 illustrates a relativistic spacetime diagram showing a source separating from the receiver with a velocity parameter β, so that the separation between source and receiver at time w is βw. Because of time dilation, . Since the slope of the green light ray is −1, . Hence, the relativistic Doppler effect is given by[35]:58–59
Transverse Doppler effect
Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:[39]:541–543
- Fig. 3-7a. What is the frequency measurement when the receiver is geometrically at its closest approach to the source? This scenario is most easily analyzed from the frame S' of the source.[note 10]
- Fig. 3-7b. What is the frequency measurement when the receiver sees the source as being closest to it? This scenario is most easily analyzed from the frame S of the receiver.
Two other scenarios are commonly examined in discussions of transverse Doppler shift:
- Fig. 3-7c. If the receiver is moving in a circle around the source, what frequency does the receiver measure?
- Fig. 3-7d. If the source is moving in a circle around the receiver, what frequency does the receiver measure?
In scenario (a), the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt = 0 where r is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency f′, but also observes the receiver as having a time-dilated clock. In frame S, the receiver is therefore illuminated by blueshifted light of frequency
In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is redshifted with frequency
Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of , and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.)[39]:541–543 Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d).[note 11]
Energy and momentum
Extending momentum to four dimensions
In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. Linear momentum, the product of a particle's mass and velocity, is a vector quantity, possessing the same direction as the velocity: p = mv. It is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.
In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector . In exploring the properties of the spacetime momentum, we start, in Fig. 3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. p = 0, but the time component equals mc.
We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that and , since the red axes are rescaled by gamma. Fig. 3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches c.[35]:84–87
We will use this information shortly to obtain an expression for the four-momentum.
Momentum of light
Light particles, or photons, travel at the speed of c, the constant that is conventionally known as the speed of light. This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a light-like world line and, in appropriate units, have equal space and time components for every observer.
A consequence of Maxwell's theory of electromagnetism is that light carries energy and momentum, and that their ratio is a constant: . Rearranging, , and since for photons, the space and time components are equal, E/c must therefore be equated with the time component of the spacetime momentum vector.
Photons travel at the speed of light, yet have finite momentum and energy. For this to be so, the mass term in γmc must be zero, meaning that photons are massless particles. Infinity times zero is an ill-defined quantity, but E/c is well-defined.
By this analysis, if the energy of a photon equals E in the rest frame, it equals in a moving frame. This result can be derived by inspection of Fig. 3-9 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.[35]:88
Mass-energy relationship
Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several famous conclusions.
- In the low speed limit as β = v/c approaches zero, γ approaches 1, so the spatial component of the relativistic momentum approaches mv, the classical term for momentum. Following this perspective, γm can be interpreted as a relativistic generalization of m. Einstein proposed that the relativistic mass of an object increases with velocity according to the formula .
- Likewise, comparing the time component of the relativistic momentum with that of the photon, , so that Einstein arrived at the relationship . Simplified to the case of zero velocity, this is Einstein's famous equation relating energy and mass.
Another way of looking at the relationship between mass and energy is to consider a series expansion of γmc2 at low velocity:
The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy.[35]:90–92[37]:129–130,180
The concept of relativistic mass that Einstein introduced in 1905, mrel, although amply validated every day in particle accelerators around the globe (or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes,[40] old-fashioned color television sets, etc.), has nevertheless not proven to be a fruitful concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity.
For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy.[41] "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or invariant mass, and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula,
This formula applies to all particles, massless as well as massive. For massless photons, it yields the same relationship as established earlier, .[35]:90–92
Four-momentum
Because of the close relationship between mass and energy, the four-momentum (also called 4-momentum) is also called the energy–momentum 4-vector. Using an uppercase P to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as
- or alternatively,
- using the convention that [37]:129–130,180
Conservation laws
In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, Emmy Noether discovered that underlying each conservation law is a fundamental symmetry of nature.[42] The fact that physical processes don't care where in space they take place (space translation symmetry) yields conservation of momentum, the fact that such processes don't care when they take place (time translation symmetry) yields conservation of energy, and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective.
Total momentum
To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.
In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity:
- (1) The two bodies rebound from each other in a completely elastic collision.
- (2) The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision.
For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat.
In case (2), two masses with momentums and collide to produce a single particle of conserved mass traveling at the center of mass velocity of the original system, . The total momentum is conserved.
Fig. 3-10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components and add up to total E/c of the resultant vector, meaning that energy is conserved. Likewise, the space components and add up to form p of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: .[35]:94–97
Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable elementary particle spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.[37]:134–138
Choice of reference frames
The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the "center-of-momentum frame" (also called the zero-momentum frame, or COM frame). This is the frame in which the space component of the system's total momentum is zero. Fig. 3-11 illustrates the breakup of a high speed particle into two daughter particles. In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same.
Energy and momentum conservation
In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since , the momentum . If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame.[37]:241–245
Conservation of momentum in the COM frame amounts to the requirement that p = 0 both before and after collision. In the Newtonian analysis, conservation of mass dictates that . In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero.[37]:241–245
Newtonian momenta, calculated as , fail to behave properly under Lorentzian transformation. The linear transformation of velocities is replaced by the highly nonlinear so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. This second option was what he chose.[35]:104
The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.[37]:127
Example: Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where 1 MeV = 106 electron volts. A charged pion is a particle of mass 139.57 MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66 MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion mass and the muon mass is 33.91 MeV.
π−
→ μ− + νμ
Fig. 3-12a illustrates the energy–momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.
Algebraic analyses of the energetics of this decay reaction are available online,[43] so Fig. 3-12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is 33.91 MeV − 29.79 MeV = 4.12 MeV. Most of the energy is carried off by the near-zero-mass neutrino.
Mas allá de lo básico
The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding Introduction to curved spacetime.
Rapidity
Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas.
This nonlinearity is an artifact of our choice of parameters.[7]:47–59 We have previously noted that in an x–ct spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.
The natural functions for expressing these relationships are the hyperbolic analogs of the trigonometric functions. Fig. 4-1a shows a unit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that a is interpreted, not as the angle between the ray and the x-axis, but as twice the area of the sector swept out by the ray from the x-axis. (Numerically, the angle and 2 × area measures for the unit circle are identical.) Fig. 4-1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area.[44] Fig. 4-2 presents plots of the sinh, cosh, and tanh functions.
For the unit circle, the slope of the ray is given by
In the Cartesian plane, rotation of point (x, y) into point (x', y') by angle θ is given by
In a spacetime diagram, the velocity parameter is the analog of slope. The rapidity, φ, is defined by[37]:96–99
where
The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;[7]:47–59
or in other words,
The Lorentz transformations take a simple form when expressed in terms of rapidity. The γ factor can be written as
Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts.
Substituting γ and γβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the x-direction may be written as
and the inverse Lorentz boost in the x-direction may be written as
In other words, Lorentz boosts represent hyperbolic rotations in Minkowski spacetime.[37]:96–99
The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.[7][45][note 12]
4‑vectors
Four‑vectors have been mentioned above in context of the energy–momentum 4‑vector, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, 4‑vectors, and more generally tensors, greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation (really no more than an observation) using the field strength tensor formulation. On the other hand, general relativity, from the outset, relies heavily on 4‑vectors, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such 4‑vectors even within a curved spacetime, and not just within a flat one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime.
Definition of 4-vectors
A 4-tuple, is a "4-vector" if its component A i transform between frames according to the Lorentz transformation.
If using coordinates, A is a 4–vector if it transforms (in the x-direction) according to
which comes from simply replacing ct with A0 and x with A1 in the earlier presentation of the Lorentz transformation.
As usual, when we write x, t, etc. we generally mean Δx, Δt etc.
The last three components of a 4–vector must be a standard vector in three-dimensional space. Therefore, a 4–vector must transform like under Lorentz transformations as well as rotations.[31]:36–59
Properties of 4-vectors
- Closure under linear combination: If A and B are 4-vectors, then is also a 4-vector.
- Inner-product invariance: If A and B are 4-vectors, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a 3-vector. In the following, and are 3-vectors:
- In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in 3-space.
- Two vectors are said to be orthogonal if Unlike the case with 3-vectors, orthogonal 4-vectors are not necessarily at right angles with each other. The rule is that two 4-vectors are orthogonal if they are offset by equal and opposite angles from the 45° line which is the world line of a light ray. This implies that a lightlike 4-vector is orthogonal with itself.
- Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a 4-vector with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval and the invariant length of the relativistic momentum vector [37]:178–181[31]:36–59
Examples of 4-vectors
- Displacement 4-vector: Otherwise known as the spacetime separation, this is (Δt, Δx, Δy, Δz), or for infinitesimal separations, (dt, dx, dy, dz).
- Velocity 4-vector: This results when the displacement 4-vector is divided by , where is the proper time between the two events that yield dt, dx, dy, and dz.
- The 4-velocity is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.
- An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the momentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles.
- Since photons move on null lines, for a photon, and a 4-velocity cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.
- Energy–momentum 4-vector:
- As indicated before, there are varying treatments for the energy-momentum 4-vector so that one may also see it expressed as or The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy-momentum 4-vector is a conserved quantity.
- Acceleration 4-vector: This results from taking the derivative of the velocity 4-vector with respect to
- Force 4-vector: This is the derivative of the momentum 4-vector with respect to
As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum, 3-force etc.[37]:178–181[31]:36–59
4-vectors and physical law
The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving 4-vectors rather than give up on conservation of momentum.
Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving 4-vectors require the use of tensors with appropriate rank, which themselves can be thought of as being built up from 4-vectors.[37]:186
Acceleration
It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required.[46]
Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.[46]
In this section, we analyze several scenarios involving accelerated reference frames.
Dewan–Beran–Bell spaceship paradox
The Dewan–Beran–Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.
In Fig. 4-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration.[note 13] Will the string break?
When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.[37]:106,120–122
- To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance L' = γL in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break.
- Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.[37]:106,120–122
The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.[37]:106,120–122
A spacetime diagram (Fig. 4-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude acceleration for proper time (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length of the spacelike line segment turns out to be greater than the length of the spacelike line segment .
The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 4-5, the acceleration is finished, the ships will remain at a constant offset in some frame If and are the ships' positions in the positions in frame are:[47]
The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame . As shown in Fig. 4-5, Bell's example asserts the moving lengths and measured in frame to be fixed, thereby forcing the rest frame length in frame to increase.
Accelerated observer with horizon
Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons. In the text accompanying Fig. 2-7, the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just approaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.
Fig. 4-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter approaches a limit of one as increases. Likewise, approaches infinity.
The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:
- We remember that
- Since we conclude that
- From the relativistic force law,
- Substituting from step 2 and the expression for from step 3 yields which is a constant expression.[35]:110–113
Fig. 4-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01 c per hour. Every twenty hours, Terence radios updates to Stella about the situation at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she never receives any communications from Terence after 100 hours on his clock (dashed green lines).[35]:110–113
After 100 hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to receive Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an apparent event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.[35]:110–113
Introducción al espacio-tiempo curvo
Basic propositions
Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean reference frame that extends throughout all space and all time. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space.[note 14] In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.[7]:175–190
In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a geodesic. No evidence of gravitation can be discovered following alongside the motions of a single particle.[7]:175–190
In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of two bodies or two separated particles. In Fig. 5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the appearance of a gravitational force acting at a long range from Earth.[7]:175–190
Two central propositions underlie general relativity.
- The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."[48]:113 This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.[49]:137–149
- The equivalence principle states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration.
- In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of g. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be Lilliputians), there are no experiments that A and B can perform which will enable them to tell which setting they are in. [49]:141–149
- An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, F = GMmg /r2 =mgg and in Newton's second law, F = m ia, there is no a priori reason why the gravitational massmg should be equal to the inertial massm i. The equivalence principle states that these two masses are identical. [49]:141–149
To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write about general relativity, but it is not possible to demonstrate any non-trivial derivations.
Curvature of time
In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a global inertial frame. In small enough regions of spacetime, local inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.[31]:118–126
Shortly after the publication of the general theory in 1916, a number of scientists pointed out that general relativity predicts the existence of gravitational redshift. Einstein himself suggested the following thought experiment: (i) Assume that a tower of height h (Fig. 5-3) has been constructed. (ii) Drop a particle of rest mass m from the top of the tower. It falls freely with acceleration g, reaching the ground with velocity v = (2gh)1/2, so that its total energy E, as measured by an observer on the ground, is (iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon E' back into a particle of rest mass m'.[31]:118–126
It must be that m = m', since otherwise one would be able to construct a perpetual motion device. We therefore predict that E' = m, so that
A photon climbing in Earth's gravitational field loses energy and is redshifted. Early attempts to measure this redshift through astronomical observations were somewhat inconclusive, but definitive laboratory observations were performed by Pound & Rebka (1959) and later by Pound & Snider (1964).[50]
Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side by side with the ground clock.[21]:16–18 For a 1 km tower, the discrepancy would amount to about 9.4 nanoseconds per day, easily measurable with modern instrumentation.
Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime. The Pound–Rebka experiment says nothing about curvature of the space component of spacetime. But the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence.[21]:16 This includes Newtonian gravitation. A standard demonstration in general relativity is to show how, in the "Newtonian limit" (i.e. the particles are moving slowly, the gravitational field is weak, and the field is static), curvature of time alone is sufficient to derive Newton's law of gravity.[51]:101–106
Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space. Given G as the gravitational constant, M as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance r from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable:[21]:229–232
Curvature of space
The coefficient in front of describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to and , and because of the in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved.
But general relativity is a theory of curved space and curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, shouldn't their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?
The answer is that they are seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the solar system, the term dwarfs the spatial terms.[21]:234–238
Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, Urbain Le Verrier, in an analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets.[52] The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry.
As the famous astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing wobbles in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.[53]
In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.
The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of ±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.[21]:234–238
In Newton's gravitation, the coefficient in front of predicts bending of light around a star. In general relativity, the coefficient in front of predicts a doubling of the total bending.[21]:234–238
The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.[54]
Sources of spacetime curvature
In Newton's theory of gravitation, the only source of gravitational force is mass.
In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations, the sources of gravity are presented on the right-hand side in the stress–energy tensor.
Fig. 5-5 classifies the various sources of gravity in the stress–energy tensor:
- (red): The total mass–energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions.
- and (orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum.
- are the rates of flow of the i-component of momentum per unit area in the j-direction. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the i = j terms (green) represent isotropic pressure, and the i ≠ j terms (blue) represent shear stresses.[55]
One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity.[note 15] Energy has mass. Even in Newtonian gravity, the gravitational field is associated with an energy, called the gravitational potential energy. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field. This makes the equations nonlinear and hard to solve in anything other than weak field cases.[21]:240 Numerical relativity is a branch of general relativity using numerical methods to solve and analyze problems, often employing supercomputers to study black holes, gravitational waves, neutron stars and other phenomena in the strong field regime.
Energy-momentum
In special relativity, mass-energy is closely connected to momentum. Just as space and time are different aspects of a more comprehensive entity called spacetime, mass–energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum. In consequence, if mass–energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism.[56]
It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, chapter 13–6 of his Lectures on Physics, available online.[57]) Analogous logic can be used to demonstrate the origin of gravitomagnetism. In Fig. 5-7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −v and +v relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume so that velocities are simply additive. Fig. 5-7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +v, and the bottom stream has a velocity of +2v. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream. But it is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream.
The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle that is moving in the same direction as the bottom stream. This velocity-dependent gravitational effect is gravitomagnetism.[21]:245–253
Matter in motion through a gravitomagnetic field is hence subject to so-called frame-dragging effects analogous to electromagnetic induction. It has been proposed that such gravitomagnetic forces underlie the generation of the relativistic jets (Fig. 5-8) ejected by some rotating supermassive black holes.[58][59]
Pressure and stress
Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal pressure and stress. Taken together, mass-energy, momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve.
General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass–energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a neutron star. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass. Above a certain mass determined by the Tolman–Oppenheimer–Volkoff limit, the process becomes runaway and the neutron star collapses to a black hole.[21]:243,280
The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae.[60]
These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory. In regards to pressure, the early universe was radiation dominated,[61] and it is highly unlikely that any of the relevant cosmological data (e.g. nucleosynthesis abundances, etc.) could be reproduced if pressure did not contribute to gravity, or if it did not have the same strength as a source of gravity as mass–energy. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms did not contribute as a source of gravity.
Experimental test of the sources of spacetime curvature
Definitions: Active, passive, and inertial mass
Bondi distinguishes between different possible types of mass: (1) active mass () is the mass which acts as the source of a gravitational field; (2)passive mass () is the mass which reacts to a gravitational field; (3) inertial mass () is the mass which reacts to acceleration.[62]
- is the same as gravitational mass () in the discussion of the equivalence principle.
In Newtonian theory,
- The third law of action and reaction dictates that and must be the same.
- On the other hand, whether and are equal is an empirical result.
In general relativity,
- The equality of and is dictated by the equivalence principle.
- There is no "action and reaction" principle dictating any necessary relationship between and .[62]
Pressure as a gravitational source
The classic experiment to measure the strength of a gravitational source (i.e. its active mass) was first conducted in 1797 by Henry Cavendish (Fig. 5-9a). Two small but dense balls are suspended on a fine wire, making a torsion balance. Bringing two large test masses close to the balls introduces a detectable torque. Given the dimensions of the apparatus and the measurable spring constant of the torsion wire, the gravitational constant G can be determined.
To study pressure effects by compressing the test masses is hopeless, because attainable laboratory pressures are insignificant in comparison with the mass-energy of a metal ball.
However, the repulsive electromagnetic pressures resulting from protons being tightly squeezed inside atomic nuclei are typically on the order of 1028 atm ≈ 1033 Pa ≈ 1033 kg·s−2m−1. This amounts to about 1% of the nuclear mass density of approximately 1018kg/m3 (after factoring in c2 ≈ 9×1016m2s−2).[63]
If pressure does not act as a gravitational source, then the ratio should be lower for nuclei with higher atomic number Z, in which the electrostatic pressures are higher. L. B. Kreuzer (1968) did a Cavendish experiment using a Teflon mass suspended in a mixture of the liquids trichloroethylene and dibromoethane having the same buoyant density as the Teflon (Fig. 5-9b). Fluorine has atomic number Z = 9, while bromine has Z = 35. Kreuzer found that repositioning the Teflon mass caused no differential deflection of the torsion bar, hence establishing active mass and passive mass to be equivalent to a precision of 5×10−5.[64]
Although Kreuzer originally considered this experiment merely to be a test of the ratio of active mass to passive mass, Clifford Will (1976) reinterpreted the experiment as a fundamental test of the coupling of sources to gravitational fields.[65]
In 1986, Bartlett and Van Buren noted that lunar laser ranging had detected a 2 km offset between the moon's center of figure and its center of mass. This indicates an asymmetry in the distribution of Fe (abundant in the Moon's core) and Al (abundant in its crust and mantle). If pressure did not contribute equally to spacetime curvature as does mass–energy, the moon would not be in the orbit predicted by classical mechanics. They used their measurements to tighten the limits on any discrepancies between active and passive mass to about 10−12.[66]
Gravitomagnetism
The existence of gravitomagnetism was proven by Gravity Probe B (GP-B), a satellite-based mission which launched on 20 April 2004.[67] The spaceflight phase lasted until . The mission aim was to measure spacetime curvature near Earth, with particular emphasis on gravitomagnetism.
Initial results confirmed the relatively large geodetic effect (which is due to simple spacetime curvature, and is also known as de Sitter precession) to an accuracy of about 1%. The much smaller frame-dragging effect (which is due to gravitomagnetism, and is also known as Lense–Thirring precession) was difficult to measure because of unexpected charge effects causing variable drift in the gyroscopes. Nevertheless, by , the frame-dragging effect had been confirmed to within 15% of the expected result,[68] while the geodetic effect was confirmed to better than 0.5%.[69][70]
Subsequent measurements of frame dragging by laser-ranging observations of the LARES, LAGEOS-1 and LAGEOS-2 satellites has improved on the GP-B measurement, with results (as of 2016) demonstrating the effect to within 5% of its theoretical value,[71] although there has been some disagreement on the accuracy of this result.[72]
Another effort, the Gyroscopes in General Relativity (GINGER) experiment, seeks to use three 6 m ring lasers mounted at right angles to each other 1400 m below the Earth's surface to measure this effect.[73][74]
Temas técnicos
Is spacetime really curved?
In Poincaré's conventionalist views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity. A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it more convenient to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime really is.[75]
Such being said,
- 1. Is it possible to represent general relativity in terms of flat spacetime?
- 2. Are there any situations where a flat spacetime interpretation of general relativity may be more convenient than the usual curved spacetime interpretation?
In response to the first question, a number of authors including Deser, Grishchuk, Rosen, Weinberg, etc. have provided various formulations of gravitation as a field in a flat manifold. Those theories are variously called "bimetric gravity", the "field-theoretical approach to general relativity", and so forth.[76][77][78][79] Kip Thorne has provided a popular review of these theories.[80]:397–403
The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate. The flat spacetime paradigm is fully equivalent to the curved spacetime paradigm in that they both represent the same physical phenomena. However, their mathematical formulations are entirely different. Working physicists routinely switch between using curved and flat spacetime techniques depending on the requirements of the problem. The flat spacetime paradigm turns out to be especially convenient when performing approximate calculations in weak fields. Hence, flat spacetime techniques will be used when solving gravitational wave problems, while curved spacetime techniques will be used in the analysis of black holes.[80]:397–403
Asymptotic symmetries
The spacetime symmetry group for Special Relativity is the Poincaré group, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group.
In 1962 Hermann Bondi, M. G. van der Burg, A. W. Metzner[81] and Rainer K. Sachs[82] addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no a priori assumptions about the nature of the asymptotic symmetry group — not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies the conclusion that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances.[83]:35
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" [84] ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R 3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.
Curved manifolds
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold . This means the smooth Lorentz metric has signature . The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light is equal to 1.[85]
A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event . Another reference frame may be identified by a second coordinate chart about . Two observers (one in each reference frame) may describe the same event but obtain different descriptions.[85]
Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing (representing an observer) and another containing (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.[85]
For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.
Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.[85]
Privileged character of 3+1 spacetime
There are two kinds of dimensions: spatial (bidirectional) and temporal (unidirectional).[86] Let the number of spatial dimensions be N and the number of temporal dimensions be T. That N = 3 and T = 1, setting aside the compactified dimensions invoked by string theory and undetectable to date, can be explained by appealing to the physical consequences of letting N differ from 3 and T differ from 1. The argument is often of an anthropic character and possibly the first of its kind, albeit before the complete concept came into vogue.
The implicit notion that the dimensionality of the universe is special is first attributed to Gottfried Wilhelm Leibniz, who in the Discourse on Metaphysics suggested that the world is "the one which is at the same time the simplest in hypothesis and the richest in phenomena".[87] Immanuel Kant argued that 3-dimensional space was a consequence of the inverse square law of universal gravitation. While Kant's argument is historically important, John D. Barrow says that it "gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa" (Barrow 2002: 204).[note 16]
In 1920, Paul Ehrenfest showed that if there is only one time dimension and greater than three spatial dimensions, the orbit of a planet about its Sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy.[88] Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a wave impulse will travel at different speeds. If there are spatial dimensions, where k is a positive whole number, then wave impulses become distorted. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only with three dimensions of space and one of time.[89] Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.[90]
Max Tegmark expands on the preceding argument in the following anthropic manner.[91] If T differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) N = 1 and T = 3 has the peculiar property that the speed of light in a vacuum is a lower bound on the velocity of matter; all matter consists of tachyons.[91]
Lastly, if N < 3, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, when N < 3, nerves cannot cross without intersecting.[91]
Hence anthropic and other arguments rule out all cases except N = 3 and T = 1, which happens to describe the world around us.Ver también
- Basic introduction to the mathematics of curved spacetime
- Complex spacetime
- Einstein's thought experiments
- Global spacetime structure
- Metric space
- Philosophy of space and time
- Present
Notas
- ^ luminiferous from the Latin lumen, light, + ferens, carrying; aether from the Greek αἰθήρ (aithēr), pure air, clear sky
- ^ By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break from Newton, who conceived of an absolute, true time that was independent of the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher Henri Bergson, who argued that time, simultaneity, and duration were matters of intuitive understanding.[15]
- ^ The operational procedure adopted by Poincaré was essentially identical to what is known as Einstein synchronization, even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive.[15]
- ^ A hallmark of Einstein's career, in fact, was his use of visualized thought experiments (Gedanken–Experimente) as a fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his great Solvay Debates with Bohr on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the Heisenberg uncertainty principle might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement. [20]:26–27;122–127;145–146;345–349;448–460
- ^ In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the Annals of Physics (Annalen der Physik) was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski's judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor.[22]
- ^ (In the following, the group G∞ is the Galilean group and the group Gc the Lorentz group.) "With respect to this it is clear that the group Gc in the limit for c = ∞, i.e. as group G∞, exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since Gc is mathematically more intelligible than G∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group G∞, but rather for a group Gc, where c is definitely finite, and only exceedingly large using the ordinary measuring units."[24]
- ^ For instance, the Lorentz group is a subgroup of the conformal group in four dimensions.[25]:41–42 The Lorentz group is isomorphic to the Laguerre group transforming planes into planes,[25]:39–42 it is isomorphic to the Möbius group of the plane,[26]:22 and is isomorphic to the group of isometries in hyperbolic space which is often expressed in terms of the hyperboloid model.[27]:3.2.3
- ^ In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric.
- ^ Even with no (de)acceleration i.e. using one inertial frame O for constant, high-velocity outward journey and another inertial frame I for constant, high-velocity inward journey – the sum of the elapsed time in those frames (O and I) is shorter than the elapsed time in the stationary inertial frame S. Thus acceleration and deceleration is not the cause of shorter elapsed time during the outward and inward journey. Instead the use of two different constant, high-velocity inertial frames for outward and inward journey is really the cause of shorter elapsed time total. Granted, if the same twin has to travel outward and inward leg of the journey and safely switch from outward to inward leg of the journey, the acceleration and deceleration is required. If the travelling twin could ride the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly. The asymmetry is because of the comparison of sum of elapsed times in two different inertial frames (O and I) to the elapsed time in a single inertial frame S.
- ^ The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis. In this linked image, we present alternative views of the transverse Doppler shift scenario where source and receiver are at their closest approach to each other. (a) If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source. (b) It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of gamma.
- ^ Not all experiments characterize the effect in terms of a redshift. For example, the Kündig experiment was set up to measure transverse blueshift using a Mössbauer source setup at the center of a centrifuge rotor and an absorber at the rim.
- ^ Rapidity arises naturally as a coordinates on the pure boost generators inside the Lie algebra algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo 2π) on the pure rotation generators in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.
- ^ In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured.
- ^ Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it."
- ^ More precisely, the gravitational field couples to itself. In Newtonian gravity, the potential due to two point masses is simply the sum of the potentials of the two masses, but this does not apply to GR. This can be thought of as the result of the equivalence principle: If gravitation did not couple to itself, two particles bound by their mutual gravitational attraction would not have the same inertial mass (due to negative binding energy) as their gravitational mass.[51]:112–113
- ^ This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the proportional relationship of flux density and the strength of field. If N = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius r has area of 4πr 2. More generally, in a space of N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to rN−1.
Detalles adicionales
- ^ Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles 2 and 3 but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second reporter, aware of the large mass 1, smiles at the first reporter's naiveté. This second reporter knows that in reality, the apparent forces between particles 2 and 3 really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesics in spacetime.
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...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages.
Cite journal requires|journal=
(help) - ^ http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/
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- ^ Ehrenfest, Paul (1920). "How do the fundamental laws of physics make manifest that Space has 3 dimensions?". Annalen der Physik. 61 (5): 440–446. Bibcode:1920AnP...366..440E. doi:10.1002/andp.19203660503.. Also see Ehrenfest, P. (1917) "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?" Proceedings of the Amsterdam Academy20: 200.
- ^ Weyl, H. (1922). Space, time, and matter. Dover reprint: 284.
- ^ Tangherlini, F. R. (1963). "Atoms in Higher Dimensions". Nuovo Cimento. 14 (27): 636. doi:10.1007/BF02784569. S2CID 119683293.
- ^ a b c Tegmark, Max (April 1997). "On the dimensionality of spacetime" (PDF). Classical and Quantum Gravity. 14 (4): L69–L75. arXiv:gr-qc/9702052. Bibcode:1997CQGra..14L..69T. doi:10.1088/0264-9381/14/4/002. S2CID 15694111. Retrieved 16 December 2006.
Otras lecturas
- Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
- George F. Ellis and Ruth M. Williams (1992) Flat and curved space–times. Oxford Univ. Press. ISBN 0-19-851164-7
- Lorentz, H. A., Einstein, Albert, Minkowski, Hermann, and Weyl, Hermann (1952) The Principle of Relativity: A Collection of Original Memoirs. Dover.
- Lucas, John Randolph (1973) A Treatise on Time and Space. London: Methuen.
- Penrose, Roger (2004). The Road to Reality. Oxford: Oxford University Press. ISBN 0-679-45443-8. Chpts. 17–18.
- Taylor, E. F.; Wheeler, John A. (1992). Spacetime Physics, Second Edition. Internet Archive: W. H. Freeman. ISBN 0-7167-2327-1.
enlaces externos
- Albert Einstein on space–time 13th edition Encyclopædia Britannica Historical: Albert Einstein's 1926 article
- Encyclopedia of Space–time and gravitation Scholarpedia Expert articles
- Stanford Encyclopedia of Philosophy: "Space and Time: Inertial Frames" by Robert DiSalle.