En física , la equivalencia masa-energía es la relación entre masa y energía en el marco de reposo de un sistema , donde los dos valores difieren solo por una constante y las unidades de medida. [1] [2] El principio es descrito por la famosa fórmula del físico Albert Einstein : [3]
La fórmula define la energía E de una partícula en su marco de reposo como el producto de la masa ( m ) por la velocidad de la luz al cuadrado ( c 2 ). Debido a que la velocidad de la luz es un gran número en unidades diarias (aproximadamente3 × 10 8 metros por segundo), la fórmula implica que una pequeña cantidad de masa en reposo corresponde a una enorme cantidad de energía, que es independiente de la composición de la materia . La masa en reposo, también llamada masa invariante , es la masa que se mide cuando el sistema está en reposo. Es una propiedad física fundamental que es independiente del momento , incluso a velocidades extremas que se acercan a la velocidad de la luz (es decir, su valor es el mismo en todos los marcos de referencia inerciales ). Las partículas sin masa , como los fotones, tienen masa invariante cero, pero las partículas libres sin masa tienen tanto impulso como energía. El principio de equivalencia implica que cuando se pierde energía en reacciones químicas , reacciones nucleares y otras transformaciones de energía , el sistema también perderá una cantidad correspondiente de masa. La energía y la masa se pueden liberar al medio ambiente como energía radiante , como luz , o como energía térmica . El principio es fundamental para muchos campos de la física, incluida la física nuclear y de partículas .
La equivalencia masa-energía surgió de la relatividad especial como una paradoja descrita por el erudito francés Henri Poincaré . [4] Einstein fue el primero en proponer la equivalencia de masa y energía como un principio general y una consecuencia de las simetrías del espacio y el tiempo . El principio apareció por primera vez en "¿Depende la inercia de un cuerpo de su contenido de energía?", Uno de sus artículos Annus Mirabilis (Año milagroso) , publicado el 21 de noviembre de 1905. [5] La fórmula y su relación con el impulso, como descritos por la relación energía-momento , fueron desarrollados más tarde por otros físicos.
Descripción
La equivalencia masa-energía establece que todos los objetos que tienen masa , u objetos masivos , tienen una energía intrínseca correspondiente, incluso cuando están estacionarios. En el marco de reposo de un objeto, donde por definición está inmóvil y por lo tanto no tiene momento , la masa y la energía son equivalentes y difieren solo por una constante, la velocidad de la luz al cuadrado ( c 2 ). [1] [2] En la mecánica newtoniana , un cuerpo inmóvil no tiene energía cinética , y puede tener o no otras cantidades de energía almacenada interna, como energía química o energía térmica , además de cualquier energía potencial que pueda tener de su posición en un campo de fuerza . Estas energías tienden a ser mucho más pequeñas que la masa del objeto multiplicada por c 2 , que es del orden de 10 17 julios para una masa de un kilogramo. Debido a este principio, la masa de los átomos que salen de una reacción nuclear es menor que la masa de los átomos que entran, y la diferencia de masa se muestra como calor y luz con la misma energía equivalente que la diferencia. Al analizar estas explosiones, la fórmula de Einstein se puede utilizar con E como la energía liberada y eliminada, ym como el cambio de masa.
En relatividad , toda la energía que se mueve con un objeto (es decir, la energía medida en el marco de reposo del objeto) contribuye a la masa total del cuerpo, que mide cuánto resiste la aceleración . Si una caja aislada de espejos ideales pudiera contener luz, los fotones sin masa individualmente contribuirían a la masa total de la caja, por la cantidad igual a su energía dividida por c 2 . [6] Para un observador en el marco de reposo, eliminar energía es lo mismo que eliminar masa y la fórmula m = E / c 2 indica cuánta masa se pierde cuando se elimina energía. [7] De la misma manera, cuando se agrega cualquier energía a un sistema aislado, el aumento en la masa es igual a la energía agregada dividida por c 2 . [8]
Masa en relatividad especial
Un objeto se mueve con diferentes velocidades en diferentes marcos de referencia , dependiendo del movimiento del observador. Esto implica que la energía cinética, tanto en la mecánica newtoniana como en la relatividad, es "dependiente del marco", de modo que la cantidad de energía relativista que se mide que tiene un objeto depende del observador. La masa relativista de un objeto está dada por la energía relativista dividida por c 2 . [9] Debido a que la masa relativista es exactamente proporcional a la energía relativista, la masa relativista y la energía relativista son casi sinónimos ; la única diferencia entre ellos son las unidades . La masa en reposo o masa invariante de un objeto se define como la masa que tiene un objeto en su marco de reposo, cuando no se mueve con respecto al observador. Los físicos suelen utilizar el término masa , aunque los experimentos han demostrado que la masa gravitacional de un objeto depende de su energía total y no solo de su masa en reposo. [ cita requerida ] La masa en reposo es la misma para todos los marcos inerciales , ya que es independiente del movimiento del observador, es el valor más pequeño posible de la masa relativista del objeto. Debido a la atracción entre los componentes de un sistema, que da como resultado energía potencial, la masa en reposo casi nunca es aditiva ; en general, la masa de un objeto no es la suma de las masas de sus partes. [8] La masa en reposo de un objeto es la energía total de todas las partes, incluida la energía cinética, observada desde el centro del marco del momento, y la energía potencial. Las masas se suman solo si los constituyentes están en reposo (como se observa desde el centro del marco del momento) y no se atraen ni se repelen, de modo que no tienen energía cinética o potencial adicional. [nota 1] Las partículas sin masa son partículas sin masa en reposo y, por lo tanto, no tienen energía intrínseca; su energía se debe únicamente a su impulso.
Masa relativista
La masa relativista depende del movimiento del objeto, por lo que diferentes observadores en movimiento relativo ven diferentes valores para ella. La masa relativista de un objeto en movimiento es mayor que la masa relativista de un objeto en reposo, porque un objeto en movimiento tiene energía cinética. Si el objeto se mueve lentamente, la masa relativista es casi igual a la masa en reposo y ambas son casi iguales a la masa inercial clásica (como aparece en las leyes del movimiento de Newton ). Si el objeto se mueve rápidamente, la masa relativista es mayor que la masa en reposo en una cantidad igual a la masa asociada con la energía cinética del objeto. Partículas sin masa también tienen masa relativista deriva de su energía cinética, igual a su energía relativista dividida por c 2 , o m rel = E / c 2 . [10] [11] La velocidad de la luz es una en un sistema donde la longitud y el tiempo se miden en unidades naturales y la masa y la energía relativistas serían iguales en valor y dimensión. Como es solo otro nombre para la energía, el uso del término masa relativista es redundante y los físicos generalmente reservan masa para referirse a masa en reposo, o masa invariante, en oposición a masa relativista. [12] [13] Una consecuencia de esta terminología es que la masa no se conserva en la relatividad especial, mientras que la conservación del momento y la conservación de la energía son leyes fundamentales. [12]
Conservación de masa y energía
La conservación de la energía es un principio universal en física y se aplica a cualquier interacción, junto con la conservación del impulso. [12] La conservación clásica de la masa, por el contrario, se viola en ciertos contextos relativistas. [13] [12] Este concepto ha sido probado experimentalmente de varias formas, incluida la conversión de masa en energía cinética en reacciones nucleares y otras interacciones entre partículas elementales . [13] Si bien la física moderna ha descartado la expresión "conservación de la masa", en terminología más antigua una masa relativista también se puede definir como equivalente a la energía de un sistema en movimiento, lo que permite la conservación de la masa relativista . [12] La conservación de masa se rompe cuando la energía asociada con la masa de una partícula se convierte en otras formas de energía, como energía cinética, energía térmica o energía radiante . Del mismo modo, la energía cinética o radiante se puede utilizar para crear partículas que tengan masa, siempre conservando la energía total y el momento. [12]
Partículas sin masa
Las partículas sin masa tienen masa en reposo cero. La relación de Planck-Einstein para la energía de los fotones está dada por la ecuación E = hf , donde h es la constante de Planck y f es el fotón de frecuencia . Esta frecuencia y, por tanto, la energía relativista dependen del marco. Si un observador se aleja de un fotón en la dirección en la que el fotón viaja desde una fuente y alcanza al observador, el observador ve que tiene menos energía que la que tenía en la fuente. Cuanto más rápido esté viajando el observador con respecto a la fuente cuando el fotón se ponga al día, menos energía se verá que tiene el fotón. A medida que un observador se acerca a la velocidad de la luz con respecto a la fuente, aumenta el corrimiento al rojo del fotón, de acuerdo con el efecto Doppler relativista . La energía del fotón se reduce y, a medida que la longitud de onda se vuelve arbitrariamente grande, la energía del fotón se acerca a cero, debido a la naturaleza sin masa de los fotones, que no permite ninguna energía intrínseca.
Sistemas compuestos
Para los sistemas cerrados formados por muchas partes, como un núcleo atómico , un planeta o una estrella, la energía relativista viene dada por la suma de las energías relativistas de cada una de las partes, porque las energías son aditivas en estos sistemas. Si un sistema está limitado por fuerzas de atracción y la energía ganada en exceso del trabajo realizado se elimina del sistema, entonces se pierde masa con esta energía eliminada. La masa de un núcleo atómico es menor que la masa total de los protones y neutrones que lo componen. [14] Esta disminución de masa también es equivalente a la energía necesaria para dividir el núcleo en protones y neutrones individuales. La masa del Sistema Solar es ligeramente menor que la suma de sus masas individuales.
Para un sistema aislado de partículas que se mueven en diferentes direcciones, la masa invariante del sistema es análoga a la masa en reposo y es la misma para todos los observadores, incluso aquellos en movimiento relativo. Se define como la energía total (dividida por c 2 ) en el centro del marco del momento . El centro del marco de la cantidad de movimiento se define de modo que el sistema tenga una cantidad de movimiento total cero; El término marco del centro de masa también se usa a veces, donde el marco del centro de masa es un caso especial del marco del centro de momento en el que el centro de masa se coloca en el origen. Un ejemplo simple de un objeto con partes móviles pero momento total cero es un contenedor de gas. En este caso, la masa del contenedor viene dada por su energía total (incluida la energía cinética de las moléculas de gas), ya que la energía total del sistema y la masa invariante son las mismas en cualquier sistema de referencia donde el momento es cero, y tal El marco de referencia es también el único marco en el que se puede pesar el objeto. De manera similar, la teoría de la relatividad especial postula que la energía térmica en todos los objetos, incluidos los sólidos, contribuye a sus masas totales, aunque esta energía está presente como las energías cinética y potencial de los átomos en el objeto, y ( de forma similar al gas) no se ve en las masas en reposo de los átomos que componen el objeto. [8] De manera similar, incluso los fotones, si quedan atrapados en un contenedor aislado, contribuirían con su energía a la masa del contenedor. Esta masa adicional, en teoría, podría pesarse de la misma manera que cualquier otro tipo de masa en reposo, aunque los fotones individualmente no tengan masa en reposo. La propiedad de que la energía atrapada en cualquier forma agrega masa pesada a los sistemas que no tienen momento neto es una de las consecuencias de la relatividad. No tiene contraparte en la física newtoniana clásica, donde la energía nunca exhibe una masa pesada. [8]
Relación con la gravedad
La física tiene dos conceptos de masa, la masa gravitacional y la masa inercial. La masa gravitacional es la cantidad que determina la fuerza del campo gravitacional generado por un objeto, así como la fuerza gravitacional que actúa sobre el objeto cuando está inmerso en un campo gravitacional producido por otros cuerpos. La masa inercial, por otro lado, cuantifica cuánto acelera un objeto si se le aplica una fuerza determinada. La equivalencia masa-energía en relatividad especial se refiere a la masa inercial. Sin embargo, ya en el contexto de la gravedad de Newton, se postula el Principio de Equivalencia Débil : la masa gravitacional y la inercial de todos los objetos son iguales. Por lo tanto, la equivalencia masa-energía, combinada con el principio de equivalencia débil, da como resultado la predicción de que todas las formas de energía contribuyen al campo gravitacional generado por un objeto. Esta observación es uno de los pilares de la teoría general de la relatividad .
La predicción de que todas las formas de energía interactúan gravitacionalmente se ha sometido a pruebas experimentales. Una de las primeras observaciones que probaron esta predicción, llamada experimento de Eddington , se realizó durante el eclipse solar del 29 de mayo de 1919 . [15] [16] Durante el eclipse solar , el astrónomo y físico inglés Arthur Eddington observó que la luz de las estrellas que pasaban cerca del Sol estaba doblada. El efecto se debe a la atracción gravitacional de la luz por parte del Sol. La observación confirmó que la energía transportada por la luz es equivalente a una masa gravitacional. Otro experimento fundamental, el experimento de Pound-Rebka , se realizó en 1960. [17] En esta prueba, se emitió un haz de luz desde la parte superior de una torre y se detectó en la parte inferior. La frecuencia de la luz detectada fue más alta que la luz emitida. Este resultado confirma que la energía de los fotones aumenta cuando caen en el campo gravitacional de la Tierra. La energía, y por lo tanto la masa gravitacional, de los fotones es proporcional a su frecuencia según lo establecido por la relación de Planck.
Eficiencia
En algunas reacciones, las partículas de materia pueden destruirse y su energía asociada puede liberarse al medio ambiente como otras formas de energía, como la luz y el calor. [1] Un ejemplo de tal conversión tiene lugar en las interacciones de partículas elementales, donde la energía en reposo se transforma en energía cinética. [1] Tales conversiones entre tipos de energía ocurren en las armas nucleares, en las que los protones y neutrones en los núcleos atómicos pierden una pequeña fracción de su masa original, aunque la masa perdida no se debe a la destrucción de componentes más pequeños. La fisión nuclear permite que una pequeña fracción de la energía asociada con la masa se convierta en energía utilizable, como la radiación; en la desintegración del uranio , por ejemplo, se pierde aproximadamente el 0,1% de la masa del átomo original. [18] En teoría, debería ser posible destruir la materia y convertir toda la energía en reposo asociada con la materia en calor y luz, pero ninguno de los métodos teóricamente conocidos es práctico. Una forma de aprovechar toda la energía asociada con la masa es aniquilar la materia con antimateria . La antimateria es rara en nuestro universo , sin embargo, y los mecanismos de producción conocidos requieren más energía utilizable de la que se liberaría en la aniquilación. El CERN estimó en 2011 que se requiere más de mil millones de veces más energía para producir y almacenar antimateria de la que podría liberarse en su aniquilación. [19]
Como la mayor parte de la masa que comprende los objetos ordinarios reside en protones y neutrones, convertir toda la energía de la materia ordinaria en formas más útiles requiere que los protones y neutrones se conviertan en partículas más ligeras, o partículas sin masa en absoluto. En el modelo estándar de física de partículas , la cantidad de protones más neutrones se conserva casi exactamente. A pesar de esto, Gerard 't Hooft demostró que existe un proceso que convierte protones y neutrones en antielectrones y neutrinos . [20] Esta es la debilidad del SU (2) instantón propuesto por los físicos Alexander Belavin , Aleksandr Poliakov , Albert Schwarz , y Yu. S. Tyupkin. [21] Este proceso, en principio, puede destruir la materia y convertir toda la energía de la materia en neutrinos y energía utilizable, pero normalmente es extraordinariamente lento. Más tarde se demostró que el proceso ocurre rápidamente a temperaturas extremadamente altas que solo se habrían alcanzado poco después del Big Bang . [22]
Muchas extensiones del modelo estándar contienen monopolos magnéticos y, en algunos modelos de gran unificación , estos monopolos catalizan la desintegración de protones , un proceso conocido como efecto Callan-Rubakov . [23] Este proceso sería una conversión de masa-energía eficiente a temperaturas ordinarias, pero requiere la fabricación de monopolos y antimonopolos, cuya producción se espera que sea ineficiente. Otro método para aniquilar completamente la materia utiliza el campo gravitacional de los agujeros negros. El físico teórico británico Stephen Hawking teorizó [24] que es posible arrojar materia a un agujero negro y utilizar el calor emitido para generar energía. Sin embargo, según la teoría de la radiación de Hawking , los agujeros negros más grandes irradian menos que los más pequeños, por lo que la energía utilizable solo puede ser producida por agujeros negros pequeños.
Ampliación para sistemas en movimiento
A diferencia de la energía de un sistema en un marco inercial, la energía relativista () de un sistema depende tanto de la masa en reposo () y el impulso total del sistema. La extensión de la ecuación de Einstein a estos sistemas viene dada por: [25] [26] [nota 2]
o
donde el término representa el cuadrado de la norma euclidiana (longitud total del vector) de los diversos vectores de momento en el sistema, que se reduce al cuadrado de la magnitud de momento simple, si solo se considera una sola partícula. Esta ecuación se llama relación energía-momento y se reduce acuando el término de la cantidad de movimiento es cero. Para fotones donde, la ecuación se reduce a .
Expansión a baja velocidad
Usando el factor de Lorentz , γ , la energía-momento se puede reescribir como E = γmc 2 y expandirse como una serie de potencias :
Para velocidades mucho más pequeñas que la velocidad de la luz, los términos de orden superior en esta expresión se vuelven cada vez más pequeños porque v/Ces pequeño. Para velocidades bajas, se pueden ignorar todos los términos excepto los dos primeros:
En la mecánica clásica , se ignoran tanto el término m 0 c 2 como las correcciones de alta velocidad. El valor inicial de la energía es arbitrario, ya que solo se puede medir el cambio de energía, por lo que el término m 0 c 2 se ignora en la física clásica. Mientras que los términos de orden superior se vuelven importantes a velocidades más altas, la ecuación newtoniana es una aproximación de baja velocidad muy precisa; agregando en el tercer término los rendimientos:
- .
La diferencia entre las dos aproximaciones viene dada por , a number very small for everyday objects. In 2018 NASA announced the Parker Solar Probe was the fastest ever, with a speed of 153,454 miles per hour (68,600 m/s).[27] The difference between the approximations for the Parker Solar Probe in 2018 is , which accounts for an energy correction of four parts per hundred million. The gravitational constant, in contrast, has a standard relative uncertainty of about .[28]
Aplicaciones
Application to nuclear physics
The nuclear binding energy is the minimum energy that is required to disassemble the nucleus of an atom into its component parts.[29] The mass of an atom is less than the sum of the masses of its constituents due to the attraction of the strong nuclear force.[30] The difference between the two masses is called the mass defect and is related to the binding energy through Einstein's formula.[30][31][32] The principle is used in modeling nuclear fission reactions and it implies a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power.
A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by c2), which was given off as heat when the molecule formed (this heat had mass). Similarly, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion; in this case the mass difference is the energy and heat that is released when the dynamite explodes. Such a change in mass may only happen when the system is open, and the energy and mass are allowed to escape. Thus, if a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation.[note 3] Thus, a 21.5 kiloton (9×1013 joule) nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If a transparent window passing only electromagnetic radiation were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass that absorbed the X-rays (and other "heat") would gain this gram of mass from the resulting heating, thus, in this case, the mass "loss" would represent merely its relocation.
Practical examples
Einstein used the centimeter gram second system of units (cgs), but the formula is independent of the system of units. In natural units, the numerical value of the speed of light is set to equal 1, and the formula expresses an equality of numerical values: E = m. In the SI system (expressing the ratio E/m in joules per kilogram using the value of c in meters per second):[34]
- E/m =c2 = (299792458 m/s)2 =89875517873681764 J/kg (≈ 9.0 × 10 16 joules per kilogram).
So the energy equivalent of one kilogram of mass is
- 89.9 petajoules
- 25.0 billion kilowatt-hours (≈ 25,000 GW·h)
- 21.5 trillion kilocalories (≈ 21 Pcal)[note 4]
- 85.2 trillion BTUs[note 4]
- 0.0852 quads
or the energy released by combustion of the following:
- 21 500 kilotons of TNT-equivalent energy (≈ 21 Mt)[note 4]
- 2630000000 litres or 695000000 US gallons of automotive gasoline
Any time energy is released, the process can be evaluated from an E = mc2 perspective. For instance, the "Gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT.[35] About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling. The electromagnetic radiation and kinetic energy (thermal and blast energy) released in this explosion carried the missing gram of mass.
Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged:
- A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring.
- Raising the temperature of an object (increasing its heat energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum and iridium. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = 1×10−12 g).[note 5]
- A spinning ball weighs more than a ball that is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, the Earth itself is more massive due to its rotation, than it would be with no rotation. The rotational energy of the earth is greater than 1024 Joules, which is over 107 kg.[36]
Historia
While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass, though nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields.[37][38][39] Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.[40][41]
Developments prior to Einstein
Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?"[42] Swedish scientist and theologian Emanuel Swedenborg, in his Principia of 1734 theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it.[43][44]
During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories.[45] In 1873 the Russian physicist and mathematician Nikolay Umov pointed out a relation between mass and energy for ether in the form of Е = kmc2, where 0.5 ≤ k ≤ 1.[46] The writings of the English engineer Samuel Tolver Preston,[47] and a 1903 paper by the Italian industrialist and geologist Olinto De Pretto,[48][49] presented a mass–energy relation. Italian mathematician and math historian Umberto Bartocci observed that there were only three degrees of separation linking De Pretto to Einstein, concluding that Einstein was probably aware of De Pretto's work.[50] Preston and De Pretto, following physicist Georges-Louis Le Sage, imagined that the universe was filled with an ether of tiny particles that always move at speed c. Each of these particles has a kinetic energy of mc2 up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of 1/2, since German polymath Gottfried Leibniz introduced kinetic energy without it, and the 1/2 is largely conventional in prerelativistic physics.[51] By assuming that every particle has a mass that is the sum of the masses of the ether particles, the authors concluded that all matter contains an amount of kinetic energy either given by E = mc2 or 2E = mc2 depending on the convention. A particle ether was usually considered unacceptably speculative science at the time,[52] and since these authors did not formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames.
In 1905, and independent of Einstein, French polymath Gustave Le Bon speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics.[53][54]
Electromagnetic mass
There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists J. J. Thomson in 1881 and Oliver Heaviside in 1889, and George Frederick Charles Searle in 1897, German physicists Wilhelm Wien in 1900 and Max Abraham in 1902, and the Dutch physicist Hendrik Antoon Lorentz in 1904—to understand how the mass of a charged object depends on the electrostatic field.[55] This concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well. Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass:
- ,
where
Another way of deriving a type of electromagnetic mass was based on the concept of radiation pressure. In 1900, French polymath Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass[4]
By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.[39]
Austrian physicist Friedrich Hasenöhrl showed in 1904 that electromagnetic cavity radiation contributes the "apparent mass"
to the cavity's mass. He argued that this implies mass dependence on temperature as well.[56]
Einstein: mass–energy equivalence
Einstein did not write the exact formula E = mc2 in his 1905 Annus Mirabilis paper "Does the Inertia of an object Depend Upon Its Energy Content?";[5] rather, the paper states that if a body gives off the energy L in the form of radiation, its mass diminishes by L/c2.[note 6] This formulation relates only a change Δm in mass to a change L in energy without requiring the absolute relationship. The relationship convinced him that mass and energy can be seen as two names for the same underlying, conserved physical quantity.[57] He has stated that the laws of conservation of energy and conservation of mass are "one and the same".[58] Einstein elaborated in a 1946 essay that "the principle of the conservation of mass… proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy conservation principle—just as, about 60 years before, the principle of the conservation of mechanical energy had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone."[59]
Mass–velocity relationship
In developing special relativity, Einstein found that the kinetic energy of a moving body is
with v the velocity, m0 the rest mass, and γ the Lorentz factor.
He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the correspondence principle:
Without this second term, there would be an additional contribution in the energy when the particle is not moving.
Einsteins's view on mass
Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906.[60][61] In Einstein's first 1905 paper on E = mc2, he treated m as what would now be called the rest mass,[5] and it has been noted that in his later years he did not like the idea of "relativistic mass".[62]
In older physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass.[12] Historically, there has been considerable debate over the use of the concept of "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. One view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. Another view, attributed to Norwegian physicist Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.[63][64]
Einstein's 1905 derivation
Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:
- .
Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", one of his Annus Mirabilis papers. Here, Einstein used V to represent the speed of light in a vacuum and L to represent the energy lost by a body in the form of radiation.[5] Consequently, the equation E = mc2 was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy L in the form of radiation, its mass diminishes by L/V2." A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a series expansion.[note 7] Einstein used a body emitting two light pulses in opposite directions, having energies of E0 before and E1 after the emission as seen in its rest frame. As seen from a moving frame, this becomes H0 and H1. Einstein obtained, in modern notation:
- .
He then argued that H − E can only differ from the kinetic energy K by an additive constant, which gives
- .
Neglecting effects higher than third order in v/c after a Taylor series expansion of the right side of this yields:
Einstein concluded that the emission reduces the body's mass by E/c2, and that the mass of a body is a measure of its energy content.
The correctness of Einstein's 1905 derivation of E = mc2 was criticized by German theoretical physicist Max Planck in 1907, who argued that it is only valid to first approximation. Another criticism was formulated by American physicist Herbert Ives in 1952 and the Israeli physicst Max Jammer in 1961, asserting that Einstein's derivation is based on begging the question.[40][65] Other scholars, such as American and Chilean philosophers John Stachel and Roberto Torretti, have argued that Ives' criticism was wrong, and that Einstein's derivation was correct.[66] American physics writer Hans Ohanian, in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons.[67]
Relativistic center-of-mass theorem of 1906
Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote: "Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré2, for the sake of clarity I will not rely on that work."[68] In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the perpetual motion problem because, on the basis of the mass–energy equivalence, he could show that the transport of inertia that accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's E = mc2, because mass conservation appears as a special case of the energy conservation law.
Further developments
There were several further developments in the first decade of the twentieth century. In May 1907, Einstein explained that the expression for energy ε of a moving mass point assumes the simplest form when its expression for the state of rest is chosen to be ε0 = μV2 (where μ is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula μ = E0/V2, with E0 being the energy of a system of mass points, to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased.[69] Max Planck rewrote Einstein's mass–energy relationship as M = E0 + pV0/c2 in June 1907, where p is the pressure and V0 the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body.[70] Subsequently, in October 1907, this was rewritten as M0 = E0/c2 and given a quantum interpretation by German physicist Johannes Stark, who assumed its validity and correctness.[71] In December 1907, Einstein expressed the equivalence in the form M = μ + E0/c2 and concluded: "A mass μ is equivalent, as regards inertia, to a quantity of energy μc2. […] It appears far more natural to consider every inertial mass as a store of energy."[72][73] American physical chemists Gilbert N. Lewis and Richard C. Tolman used two variations of the formula in 1909: m = E/c2 and m0 = E0/c2, with E being the relativistic energy (the energy of an object when the object is moving), E0 is the rest energy (the energy when not moving), m is the relativistic mass (the rest mass and the extra mass gained when moving), and m0 is the rest mass.[74] The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side: ε = Mc2 and ε0 = mc2, with ε being the total energy (rest energy plus kinetic energy) of a moving material point, ε0 its rest energy, M the relativistic mass, and m the invariant mass.[75]
In 1911, German physicist Max von Laue gave a more comprehensive proof of M0 = E0/c2 from the stress–energy tensor,[76] which was later generalized by German mathematician Felix Klein in 1918.[77]
Einstein returned to the topic once again after World War II and this time he wrote E = mc2 in the title of his article[78] intended as an explanation for a general reader by analogy.[79]
Alternative version
An alternative version of Einstein's thought experiment was proposed by American theoretical physicist Fritz Rohrlich in 1990, who based his reasoning on the Doppler effect.[80] Like Einstein, he considered a body at rest with mass M. If the body is examined in a frame moving with nonrelativistic velocity v, it is no longer at rest and in the moving frame it has momentum P = Mv. Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy E/2. In its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum. However, if the same process is considered in a frame that moves with velocity v to the left, the pulse moving to the left is redshifted, while the pulse moving to the right is blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right. The object has not changed its velocity before or after the emission. Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox. The velocity is small, so the right-moving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor 1 − v/c. The momentum of the light is its energy divided by c, and it is increased by a factor of v/c. So the right-moving light is carrying an extra momentum ΔP given by:
The left-moving light carries a little less momentum, by the same amount ΔP. So the total right-momentum in both light pulses is twice ΔP. This is the right-momentum that the object lost.
The momentum of the object in the moving frame after the emission is reduced to this amount:
So the change in the object's mass is equal to the total energy lost divided by c2. Since any emission of energy can be carried out by a two-step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass.
Radioactivity and nuclear energy
It was quickly noted after the discovery of radioactivity in 1897, that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change, raising the question of where the energy comes from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by New Zealand physicist Ernest Rutherford and British radiochemist Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904: "If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter."[81][82]
Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify it. The theoretical explanation for radioactive decay is given by the nuclear forces responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which was known by then to release enough energy to possibly be "weighed," when missing from the system. However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."[83]
This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single nuclides and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of lithium-7 plus protons giving rise to two alpha particles, allowed Einstein's equation to be tested to an error of ±0.5%. However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles. After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945, the equation E = mc2 became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured on page 2 of the Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of Time magazine prominently featured a picture of Einstein next to an image of a mushroom cloud emblazoned with the equation.[84] Einstein himself had only a minor role in the Manhattan Project: he had cosigned a letter to the U.S. president in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem.[85]
While E = mc2 is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200 MeV (which was directly possible, using a quantitative Geiger counter, at that time). The physicist and Manhattan Project participant Robert Serber noted that somehow "the popular notion took hold long ago that Einstein's theory of relativity, in particular his famous equation E = mc2, plays some essential role in the theory of fission. Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly."[note 8] There are other views on the equation's importance to nuclear reactions. In late 1938, the Austrian-Swedish and British physicists Lise Meitner and Otto Robert Frisch—while on a winter walk during which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission—directly used Einstein's equation to help them understand the quantitative energetics of the reaction that overcame the "surface tension-like" forces that hold the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic fission. To do this, they used packing fraction, or nuclear binding energy values for elements. These, together with use of E = mc2 allowed them to realize on the spot that the basic fission process was energetically possible.[note 9]
Einstein's equation written
According to the Einstein Papers Project at the California Institute of Technology and Hebrew University of Jerusalem, only four known copies of the equation as written by Einstein are known. One of these is a letter written in German to Ludwik Silberstein, which was in Silberstein's archives, and sold at auction for $1.2 million, RR Auction of Boston, Massachusetts said on May 21, 2021.[87]
Ver también
- Energy density
- Index of energy articles
- Index of wave articles
- Lorentz transform
- Length contraction
- Outline of energy
- Relativity of simultaneity
Notas
- ^ They can also have a positive kinetic energy and a negative potential energy that exactly cancels.
- ^ Some authors state the expression equivalently as where is the Lorentz factor.
- ^ See Taylor and Wheeler[33] for a discussion of mass remaining constant after detonation of nuclear bombs, until heat is allowed to escape.
- ^ a b c Conversions used: 1956 International (Steam) Table (IT) values where one calorie ≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers' conversion value of one gram TNT ≡ 1000 calories used.
- ^ Assuming a 90/10 alloy of Pt/Ir by weight, a Cp of 25.9 for Pt and 25.1 for Ir, a Pt-dominated average Cp of 25.8, 5.134 moles of metal, and 132 J⋅K−1 for the prototype. A variation of ±1.5 picograms is much smaller than the uncertainty in the mass of the international prototype, which is ±2 micrograms.
- ^ Here, "radiation" means electromagnetic radiation, or light, and mass means the ordinary Newtonian mass of a slow-moving object.
- ^ See the sentence on the last page 641 of the original German edition, above the equation K0 − K1 = L/V2v2/2. See also the sentence above the last equation in the English translation, K0 − K1 = 1/2( L/c2)v2, and the comment on the symbols used in About this edition that follows the translation.
- ^ Serber, Robert (2020-04-07). The Los Alamos Primer. University of California Press. p. 7. doi:10.2307/j.ctvw1d5pf. ISBN 978-0-520-37433-1.. The quotation is taken from Serber's 1992 version, and is not in the original 1943 Los Alamos Primer of the same name.
- ^
We walked up and down in the snow, I on skis and she on foot… and gradually the idea took shape… explained by Bohr's idea that the nucleus is like a liquid drop; such a drop might elongate and divide itself… We knew there were strong forces that would resist, ..just as surface tension. But nuclei differed from ordinary drops. At this point we both sat down on a tree trunk and started to calculate on scraps of paper… the Uranium nucleus might indeed be a very wobbly, unstable drop, ready to divide itself… But… when the two drops separated they would be driven apart by electrical repulsion, about 200 MeV in all. Fortunately Lise Meitner remembered how to compute the masses of nuclei… and worked out that the two nuclei formed… would be lighter by about one-fifth the mass of a proton. Now whenever mass disappears energy is created, according to Einstein's formula E = mc2, and… the mass was just equivalent to 200 MeV; it all fitted!
— Lise Meitner[86]
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We might in these processes obtain very much more energy than the proton supplied, but on the average we could not expect to obtain energy in this way. It was a very poor and inefficient way of producing energy, and anyone who looked for a source of power in the transformation of the atoms was talking moonshine. But the subject was scientifically interesting because it gave insight into the atoms.
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enlaces externos
- Einstein on the Inertia of Energy – MathPages
- Mass and Energy – Conversations About Science with Theoretical Physicist Matt Strassler
- The Equivalence of Mass and Energy – Entry in the Stanford Encyclopedia of Philosophy
- Merrifield, Michael; Copeland, Ed; Bowley, Roger. "E=mc2 – Mass–Energy Equivalence". Sixty Symbols. Brady Haran for the University of Nottingham.