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Einstein's derivation of E=mc^{2} is invalid |
Qing-Ping Ma |
Abstract It is generally accepted that Einstein correctly derived the mass-energy relation formula E=mc^{2} in 1905. However, his equations for total energy changes omitted the important component of changes in kinetic energy. The omission makes Einstein’s derivation invalid. (This manuscript has been submitted for publication) PACS-code: 03.30.+p: Special relativity Key words: mass, energy, Einstein, special relativity.
It is generally accepted that Einstein correctly derived the relationship between mass and energy [1]. A close examination of Einstein’s derivation suggests that Einstein has made an important omission in his equations for changes in total energy. This omission invalidates Einstein’s derivation. The aim of the present work is to demonstrate how the mistake was made and its implications. Einstein made the following assumptions in his derivation: “Let a system of plane waves of light, referred to the system of co-ordinates (x, y, z), possess the energy l; let the direction of the ray (the wave-normal) make an angle f with the axis of x of the system. If we introduce a new system of co-ordinates (x, h, z) moving in uniform parallel translation with respect to the system (x, y, z), and having its origin of co-ordinates in motion along the axis of x with the velocity v, then this quantity of light – measured in the system (x, h, z) – possesses the energy (1) where c denotes the velocity of light. We shall make use of this result in what follows. Let there be a stationary body in the system (x, y, z), and let its energy – referred to the system (x, y, z) – be E_{0}. Let the energy of the body relative to the system (x, h, z), moving as above with the velocity v, be H_{0}. Let this body send out, in a direction making an angle f with the axis of x, plane waves of light, of energy L/2 measured relative to (x, y, z), and simultaneously an equal quantity of light in the opposite direction. Meanwhile the body remains at rest with respect to the system (x, y, z). The principle of energy must apply to this process, and in fact (by the principle of relativity) with respect to both systems of co-ordinates. If we call the energy of the body after the emission of light E_{1} or H_{1} respectively, measured relatively to the system (x, y, z) or (x, h, z) respectively, then by employing the relation given above we obtain (2) (3)
By subtraction we obtain from these equations (4)”
(English translation was quoted from The Principle of Relativity, Dover Publications, New York, 1952) Since the two light waves have equal energy and equal momentum in the system (x, y, z), Einstein’s equation (2) is correct. However, the two light waves do not have equal energy or equal momentum in the system (x, h, z), which will cause a change in kinetic energy. Einstein appeared aware this because later he would use the difference in kinetic energy to derive the renowned mass-energy formula. Therefore, the equation (3) is incorrect because the change in total energy should include both the energy carried by the light waves and the change in kinetic energy. The correct equation should be (5)
where K_{0} and K_{1} are the kinetic energy of the body before and after the light emission in the system (x, h, z) respectively. The correct form of equation (4) should be (6)
It is clear that Einstein would not be able to derive the mass-energy relation E=mc^{2} from the correct equations (5) and (6). Einstein’s mistake has not been spotted so far, largely because people mistakenly believe that the energy carried by the emitted light includes the change in kinetic energy. Since in the system (x, y, z) the energy carried by the light waves comes from non-kinetic energy, in the system (x, h, z) it should also come from non-kinetic energy. Only when a moving body is emitting light by consuming its kinetic energy, can we assume that the energy carried by the emitted light includes the change in kinetic energy. Without interaction with a field or an object, however, a moving body cannot transform its kinetic energy into light waves. Therefore, the change in kinetic energy is a consequence of light emission rather than its cause or energy source, and the change in total energy should include both the energy carried by the light waves and the change in kinetic energy of the body. In a broader sense, our finding indicates that it is impossible to derive mass-energy relation using two reference systems with a relative velocity v between them. The correct equations (5) and (6) may have an important impact on the validity of the relativistic formulation of Doppler’s Principle. Einstein stated further in his derivation that
(7) When the change in kinetic energy in equation (6) was omitted (which led to the incorrect equation (4)), combining equations (4) and (7) will give (8) Einstein derived the mass-energy relation from this incorrect equation. When the correct equation (6) is used instead of the incorrect equation (4), we will have (9) which cannot be true except for L=0 or v=0. This leaves us with two possibilities: either the relativistic formulation of Doppler’s Principle is wrong or Einstein’s equations in (7) are wrong. If the equations in (7) are correct as generally believed, the relativistic formulation of Doppler’s Principle has to be rejected. However, as Ives pointed out, the equations in (7) should be proved instead of being assumed as Einstein did [2]. If equations in (7) are incorrect, the relativistic formulation of Doppler’s Principle may still be correct. In conclusion, Einstein did not correctly derive the mass-energy formula in his 1905 paper, and any derivation using two reference systems with a relative velocity v between them is likely to be incorrect. References [1] A. Einstein, Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig? Annalen der Physik 17 (1905) 891. [2] H. E. Ives, Derivation of the mass-energy relation. J. Optical Soc. Am. 42 (1952) 540. |
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