Mass–energy equivalence

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molecules), since the system's total energy and invariant mass are the same in any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed. In a similar way, the theory of special relativity posits that the thermal energy in all objects, including solids, contributes to their total masses, even though this energy is present as the kinetic and potential energies of the atoms in the object, and it (in a similar way to the gas) is not seen in the rest masses of the atoms that make up the object. Similarly, even photons, if trapped in an isolated container, would contribute their energy to the mass of the container. Such extra mass, in theory, could be weighed in the same way as any other type of rest mass, even though individually photons have no rest mass. The property that trapped energy in any form adds weighable mass to systems that have no net momentum is one of the consequences of relativity. It has no counterpart in classical Newtonian physics, where energy never exhibits weighable mass.

2643:), 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 2659:) 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 5732: 653: 3614: 933:; in general, the mass of an object is not the sum of the masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as observed from the center of momentum frame, and potential energy. The masses add up only if the constituents are at rest (as observed from the center of momentum frame) and do not attract or repel, so that they do not have any extra kinetic or potential energy. Massless particles are particles with no rest mass, and therefore have no intrinsic energy; their energy is due only to their momentum. 2987: 5654:, 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." There are other views on the equation's importance to nuclear reactions. In late 1938, the Austrian-Swedish and British physicists 5203:. 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 5558:"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 835: 1076:. This frequency and thus the relativistic energy are frame-dependent. If an observer runs away from a photon in the direction the photon travels from a source, and it catches up with the observer, the observer sees it as having less energy than it had at the source. The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon would be seen to have. As an observer approaches the speed of light with regard to the source, the 292: 11622: 10762: 666: 5662:—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 4009:

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.

3698: 38: 5504: 1225:, for instance, about 0.1% of the mass of the original atom is lost. In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into heat and light, but none of the theoretically known methods are practical. One way to harness all the energy associated with mass is to annihilate matter with 5495:. 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. 10306: 1280:, and Yu. S. Tyupkin. This process, can in principle destroy matter and convert all the energy of matter into neutrinos and usable energy, but it is normally extraordinarily slow. It was later shown that the process occurs rapidly at extremely high temperatures that would only have been reached shortly after the 5577:, 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 1148:

is a special case of the center of momentum frame where the center of mass is put at the origin. A simple example of an object with moving parts but zero total momentum is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas

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Conservation of energy is a universal principle in physics and holds for any interaction, along with the conservation of momentum. The classical conservation of mass, in contrast, is violated in certain relativistic settings. This concept has been experimentally proven in a number of ways, including

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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,

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that make it up. This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons. This effect can be understood by looking at the potential energy of the individual components. The individual particles have a force attracting them together,

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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

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In some reactions, matter particles can be destroyed and their associated energy released to the environment as other forms of energy, such as light and heat. One example of such a conversion takes place in elementary particle interactions, where the rest energy is transformed into kinetic energy.

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is postulated: the gravitational and the inertial mass of every object are the same. Thus, the mass–energy equivalence, combined with the weak equivalence principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object. This observation is

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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. He has stated that the laws of conservation of energy and conservation of mass are "one and the same". Einstein

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In his memoirs Frisch recalled..."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

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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

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generated by an object, as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by other bodies. The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is applied to it. The mass–energy equivalence in

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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

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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. Once discovered, Einstein's formula was

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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

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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

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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

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Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. The relativistic mass of a moving object is larger than the relativistic mass of an object at rest, because a moving object has kinetic energy. If the object moves

4008:

In older physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass. 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

1303:. This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-monopoles, whose production is expected to be inefficient. Another method of completely annihilating matter uses the gravitational field of black holes. The British 1203:

of the light detected was higher than the light emitted. This result confirms that the energy of photons increases when they fall in the gravitational field of the Earth. The energy, and therefore the gravitational mass, of photons is proportional to their frequency as stated by the Planck's

950:). If the object moves quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. Massless particles also have relativistic mass derived from their kinetic energy, equal to their relativistic energy divided by 3688:

had been combined with the principle of the conservation of heat . 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."

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observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational attraction of light by the Sun. The observation confirmed that the energy carried by light indeed is equivalent to a gravitational mass. Another seminal experiment, the

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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

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depending on the convention. A particle ether was usually considered unacceptably speculative science at the time, and since these authors did not formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames.

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As most of the mass which comprises ordinary objects resides in protons and neutrons, converting all the energy of ordinary matter into more useful forms requires that the protons and neutrons be converted to lighter particles, or particles with no mass at all. In the

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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

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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.

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Trotzdem die einfachen formalen Betrachtungen, die zum Nachweis dieser Behauptung durchgeführt werden müssen, in der Hauptsache bereits in einer Arbeit von H. Poincaré enthalten sind, werde ich mich doch der Übersichtlichkeit halber nicht auf jene Arbeit

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are already mostly contained in a work by H. Poincaré, for the sake of clarity I will not rely on that work." 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

929:, as it is independent of the motion of the observer, it is the smallest possible value of the relativistic mass of the object. Because of the attraction between components of a system, which results in potential energy, the rest mass is almost never 2911:

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.

1125:

For an isolated system of particles moving in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers, even those in relative motion. It is defined as the total energy (divided by

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is largely conventional in prerelativistic physics. 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

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Einstein was unequivocally against the traditional idea of conservation of mass. He had concluded that mass and energy were essentially one and the same; 'inert mass is simply latent energy.' He made his position known publicly time and

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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. Max Planck rewrote Einstein's mass–energy relationship as

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mass increases whenever it is put into compression or tension. Its mass increase arises from the increased potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the

1396: 3014:, 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?" Swedish scientist and theologian 774:

is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same equivalent energy as the difference. In analyzing these extreme events, Einstein's formula can be used with

3213: 5903:

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

5189:. 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 5931:

of 25.8, 5.134 moles of metal, and 132 J⋅K 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.

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and forcing them apart increases the potential energy of the particles in the same way that lifting an object up on earth does. This energy is equal to the work required to split the particles apart. The mass of the

5484: 1084:. The energy of the photon is reduced and as the wavelength becomes arbitrarily large, the photon's energy approaches zero, because of the massless nature of photons, which does not permit any intrinsic energy. 2621:

and is related to the binding energy through Einstein's formula. The principle is used in modeling nuclear fission reactions, and it implies that a great amount of energy can be released by the nuclear fission

5631:

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

2499: 4623: 1108:

by attractive forces, and the energy gained in excess of the work done is removed from the system, then mass is lost with this removed energy. The mass of an atomic nucleus is less than the total mass of the

2612:

is the minimum energy that is required to disassemble the nucleus of an atom into its component parts. The mass of an atom is less than the sum of the masses of its constituents due to the attraction of the

2235:

term is ignored in classical physics. While the higher-order terms become important at higher speeds, the Newtonian equation is a highly accurate low-speed approximation; adding in the third term yields:

2202: 3600: 3438: 2675:(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: 2637:

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

1751:(total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation is called the 2568: 173: mi/s), the formula implies that a small amount of "rest mass", measured when the system is at rest, corresponds to an enormous amount of energy, which is independent of the composition of the 4244: 4122:

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

1610: 1401: 5318: 3987:

Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906. In Einstein's first 1905 paper on

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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

5548:

Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify them. The theoretical explanation for radioactive decay is given by the

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indicates how much mass is lost when energy is removed. In the same way, when any energy is added to an isolated system, the increase in the mass is equal to the added energy divided by

794:. If an isolated box of ideal mirrors could contain light, the individually massless photons would contribute to the total mass of the box by the amount equal to their energy divided by 3723: 3680:

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

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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

7414: 6808: 1804: 790:, all the energy that moves with an object (i.e., the energy as measured in the object's rest frame) contributes to the total mass of the body, which measures how much it resists 5854: 5595:, 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 2095: 1102:, planet, or star, the relativistic energy is given by the sum of the relativistic energies of each of the parts, because energies are additive in these systems. If a system is 3207:, 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: 2543: 1881: 895:, in both Newtonian mechanics and relativity, is 'frame dependent', so that the amount of relativistic energy that an object is measured to have depends on the observer. The 1361: 1174:

The prediction that all forms of energy interact gravitationally has been subject to experimental tests. One of the first observations testing this prediction, called the

1030:. Mass conservation breaks down when the energy associated with the mass of a particle is converted into other forms of energy, such as kinetic energy, thermal energy, or 2242: 1605: 1300: 9512: 1745: 115: 5874: 1837: 11325: 7862: 1388: 4847:

the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body. Subsequently, in October 1907, this was rewritten as

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itself is more massive due to its rotation, than it would be with no rotation. The rotational energy of the Earth is greater than 10 Joules, which is over 10 kg.

2426:

was the fastest ever, with a speed of 153,454 miles per hour (68,600 m/s). The difference between the approximations for the Parker Solar Probe in 2018 is

7738: 7187: 4414: 3020: 7699: 4023: 10677: 1590:{\displaystyle {\begin{aligned}E_{\rm {rel}}^{2}-|\mathbf {p} |^{2}c^{2}&=m_{0}^{2}c^{4}\\E_{\rm {rel}}^{2}-(pc)^{2}&=(m_{0}c^{2})^{2}\end{aligned}}} 3982: 3897:

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

10702: 1157:

Physics has two concepts of mass, the gravitational mass and the inertial mass. The gravitational mass is the quantity that determines the strength of the

921:

of an object is defined as the mass an object has in its rest frame, when it is not moving with respect to the observer. Physicists typically use the term

697: 6556: 8453: 7764: 925:, though experiments have shown an object's gravitational mass depends on its total energy and not just its rest mass. The rest mass is the same for all 4017:

Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:

3359:{\displaystyle m_{L}={\frac {m_{0}}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{3}}},\quad m_{T}={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} 10288: 3066: 2223:

term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured and so the

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is the rest mass. The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side:

4732:

There were several further developments in the first decade of the twentieth century. In May 1907, Einstein explained that the expression for energy

8636:

There followed also the principle of the equivalence of mass and energy, with the laws of conservation of mass and energy becoming one and the same.

9259: 7350: 7071: 10062: 5618:

method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem.

10313: 9959: 7585: 6830: 6769: 6510: 6162: 4691: 2672: 219:, a corresponding amount of energy will be released. The energy can be released to the environment (outside of the system being considered) as 9140: 8776: 6846:"IX. A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919" 76:, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist 9891: 6213: 1237:

estimated in 2011 that over a billion times more energy is required to make and store antimatter than could be released in its annihilation.

10630: 7129: 3073:

linking De Pretto to Einstein, concluding that Einstein was probably aware of De Pretto's work. Preston and De Pretto, following physicist

979:

and the relativistic mass and energy would be equal in value and dimension. As it is just another name for the energy, the use of the term

11372: 7406: 905:. Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly 8027:

The principia: or, The first principles of natural things, being new attempts toward a philosophical explanation of the elementary world

2429: 1221:

allows a tiny fraction of the energy associated with the mass to be converted into usable energy such as radiation; in the decay of the

11607: 10707: 10233: 8210: 4757:

is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula

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See Taylor and Wheeler for a discussion of mass remaining constant after detonation of nuclear bombs, until heat is allowed to escape.

5401: 4694:, in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons. 11396: 11142: 10263: 10123: 8496: 5578: 1093: 9592: 4180:." A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a 10747: 2128: 690: 6946: 1310:

theorized it is possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of

257:(1854–1912). Einstein was the first to propose the equivalence of mass and energy as a general principle and a consequence of the 10636: 5591:

became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured on page 2 of the

3538: 3521:

By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.

3376: 8624: 3028:

During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various

2978:

initially written in many different notations, and its interpretation and justification was further developed in several steps.

291: 11178: 11132: 10491: 5565:

This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single

4381:{\displaystyle \left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)} 8554: 7957: 7443: 6850:

Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character

3642:"Does the Inertia of an object Depend Upon Its Energy Content?"; rather, the paper states that if a body gives off the energy 1314:, however, larger black holes radiate less than smaller ones, so that usable power can only be produced by small black holes. 1217:

lose a small fraction of their original mass, though the mass lost is not due to the destruction of any smaller constituents.

11316: 9984: 9923: 9815: 9788: 9753: 9628: 8878: 8659: 8462: 8391: 8078: 7896: 7848: 7808: 7773: 7674: 7635: 7561: 7523: 7475: 7382: 7336: 6794: 6745: 6707: 6658: 6627: 6588: 6542: 6486: 6459: 6265: 6199: 6138: 6082: 2950:

A spinning ball has greater mass than when it is not spinning. Its increase of mass is exactly the equivalent of the mass of

9486: 9415: 9033: 6664: 5169:. Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy 3907: 3091:

up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of

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noted that somehow "the popular notion took hold long ago that Einstein's theory of relativity, in particular his equation

3714: 8422: 7504:"The Basics of Nuclear Chemistry and Radiochemistry: An Introduction to Nuclear Transformations and Radioactive Emissions" 6179:

Günther, Helmut; Müller, Volker (2019), "Einstein 's Energy–Mass Equivalence", in Günther, Helmut; Müller, Volker (eds.),

1199:, was performed in 1960. In this test a beam of light was emitted from the top of a tower and detected at the bottom. The 4667:

in 1907, who argued that it is only valid to first approximation. Another criticism was formulated by American physicist

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this gram of mass from the resulting heating, thus, in this case, the mass "loss" would represent merely its relocation.

987:

to refer to rest mass, or invariant mass, as opposed to relativistic mass. A consequence of this terminology is that the

683: 9551: 11053: 10697: 10520: 10362: 10203: 9584: 5693: 5269: 3457: 2100:

For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because

1242: 311: 7986: 11424: 11058: 10381: 10258: 3685: 1233:, however, and the known mechanisms of production require more usable energy than would be released in annihilation. 7722: 10496: 9937: 5700:, there remain only four known copies of this equation as written by Einstein. One of these is a letter written in 3973:

Without this second term, there would be an additional contribution in the energy when the particle is not moving.

3859:{\displaystyle E_{k}=m_{0}c^{2}(\gamma -1)=m_{0}c^{2}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right),} 3157:

speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative

331: 11378: 5731: 4690:, have argued that Ives' criticism was wrong, and that Einstein's derivation was correct. American physics writer 652: 11091: 10541: 5343: 3184: 3004:

Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist

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was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy

11194: 10351: 10223: 5697: 2372: 1162:

special relativity refers to the inertial mass. However, already in the context of Newtonian gravity, the weak

9807:

Atomic Bomb: The Story of the Manhattan Project: How nuclear physics became a global geopolitical game-changer

5610: 11625: 11367: 11272: 11202: 10116: 8745:"Über eine Methode zur Bestimmung des Verhältnisses der transversalen und longitudinalen Masse des Elektrons" 7691: 4883:, who assumed its validity and correctness. In December 1907, Einstein expressed the equivalence in the form 3008:

in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the

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had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of

2686: 1179: 508: 11335: 3175:

There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists

11189: 10329: 10218: 10213: 6286: 5781: 4738:

of a moving mass point assumes the simplest form when its expression for the state of rest is chosen to be

1752: 1390:) and the total momentum of the system. The extension of Einstein's equation to these systems is given by: 1323: 1273: 1168: 1043: 269: 261:. The principle first appeared in "Does the inertia of a body depend upon its energy-content?", one of his 8749:

On a method for the determination of the ratio of the transverse and the longitudinal mass of the electron

8010:

Principia rerum naturalium sive Novorum tentaminum phaenomena mundi elementaris philosophice explicandi...

1758: 11679: 11644: 11230: 10964: 10486: 10445: 10389: 10238: 9823:

talking moonshine. But the subject was scientifically interesting because it gave insight into the atoms.

5813: 5536:

existence of a huge amount of latent energy, stored within matter, was proposed by New Zealand physicist

1213:

Such conversions between types of energy happen in nuclear weapons, in which the protons and neutrons in

1081: 670: 413: 398: 336: 193: 9060:

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

7063: 2931:) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of 1921: 11517: 10959: 10912: 10413: 10228: 5776: 3078: 2939:. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = 2915:

Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged:

2512: 1196: 947: 829: 393: 383: 127: 9554:[On the integral form of the conservation laws and the theory of the spatially closed world], 1842: 31: 11664: 11547: 11173: 10854: 10321: 10268: 10248: 3070: 575: 9364: 9240: 9174: 8806: 8729: 7800:

Albert Einstein's special theory of relativity: emergence (1905) and early interpretation, 1905–1911

6405: 6322: 2954:, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, 1103: 11659: 11654: 11649: 11572: 11122: 10882: 10793: 10766: 10717: 10712: 10692: 10682: 10642: 10337: 10193: 10109: 10094: 5751: 5107: 3898: 2358:{\displaystyle E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}\left(1+{\frac {3v^{2}}{4c^{2}}}\right)} 1702:{\displaystyle {\begin{aligned}E_{\rm {rel}}={\sqrt {(m_{0}c^{2})^{2}+(pc)^{2}}}\,\!\end{aligned}}} 1331: 1292: 1133: 1022:. While modern physics has discarded the expression 'conservation of mass', in older terminology a 268:, published on 21 November 1905. The formula and its relationship to momentum, as described by the 17: 9883: 5801:

They can also have a positive kinetic energy and a negative potential energy that exactly cancels.

800:. For an observer in the rest frame, removing energy is the same as removing mass and the formula 11301: 11223: 10843: 10834: 10722: 9325: 9055: 4721: 4005:, and it has been noted that in his later years he did not like the idea of "relativistic mass". 2585: 2573: 527: 301: 7630:. Wheeler, John Archibald, 1911–2008. (2nd ed.). New York: W.H. Freeman. pp. 248–249. 3451:

associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass

11527: 11390: 11296: 11254: 10928: 10892: 10654: 10562: 10534: 10437: 10296: 8958: 6895:"'An Expedition to Heal the Wounds of War' The 1919 Eclipse and Eddington as Quaker Adventurer" 5756: 5684:

allowed them to realize on the spot that the basic fission process was energetically possible.

4928:. It appears far more natural to consider every inertial mass as a store of energy." American 4123: 3636: 3196: 2609: 2579: 2559: 2502: 1018:

the conversion of mass into kinetic energy in nuclear reactions and other interactions between

1008: 996: 351: 346: 262: 7797:

Miller, Arthur I. (1981). "Some Others Who Discussed an Association Between Energy and Mass".

6650: 6619: 6451: 3525: 1714: 84: 11532: 11137: 11076: 10727: 10687: 10527: 10476: 10471: 10397: 10208: 10188: 10178: 9804:

Reed, Bruce Cameron (2015-06-01). "The neutrino, artificial radioactivity and new elements".

9349: 9225: 9159: 8791: 8714: 8649: 7666: 6390: 6307: 5859: 5717: 5395:

The momentum of the object in the moving frame after the emission is reduced to this amount:

3613: 3158: 3074: 3055: 2879: 1809: 1304: 1163: 82: 10057: 9193: 9104: 8748: 8194: 6420: 4184:. Einstein used a body emitting two light pulses in opposite directions, having energies of 770:

for a mass of one kilogram. Due to this principle, the mass of the atoms that come out of a

11437: 11363: 11207: 10809: 10606: 10555: 10548: 10501: 10173: 10146: 9552:"Über die Integralform der Erhaltungssätze und die Theorie der räumlich-geschlossenen Welt" 9524: 9391: 9337: 9271: 9205: 9116: 9067: 9009: 8915: 8833: 8760: 8692: 8592: 8530: 8297: 7933: 7291: 7221: 7163: 7105: 7029: 6980: 6906: 6857: 6360: 4676: 4181: 3204: 3170: 2822: 2614: 2506: 1366: 1175: 1012: 988: 787: 472: 306: 258: 9105:

The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy

8580: 8486: 8383: 5940:

See the sentence on the last page 641 of the original German edition, above the equation

5503: 3605:

to the cavity's mass. He argued that this implies mass dependence on temperature as well.

8: 11512: 11507: 11497: 11329: 11278: 11096: 11081: 10907: 10838: 10786: 9579: 9570: 8266: 6240: 4505:{\displaystyle K_{0}-K_{1}=E\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)} 3200: 2904: 2208: 1158: 1019: 910: 737: 565: 522: 123: 11502: 9778: 9528: 9395: 9341: 9298: 9275: 9209: 9120: 9071: 9013: 8919: 8837: 8764: 8696: 8596: 8534: 8332: 8301: 7937: 7295: 7225: 7167: 7109: 7033: 6984: 6910: 6861: 6364: 5131:

in the title of his article intended as an explanation for a general reader by analogy.

4111:{\displaystyle E_{k}=mc^{2}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)} 11669: 11577: 11477: 11401: 11259: 11240: 11234: 11185: 11127: 11036: 10954: 10872: 10849: 10817: 10569: 10453: 10429: 10273: 10156: 10151: 9953: 9865: 9844:"The Transformation of Lithium by Protons and by Ions of the Heavy Isotope of Hydrogen" 9724: 9457: 9132: 8939: 8680: 8100: 8084: 7579: 7529: 7307: 7264: 6938: 6894: 6824: 6763: 6504: 6205: 6156: 6088: 5766: 5705: 5659: 5615: 5140: 3710: 3444: 3015: 2423: 1250: 888: 378: 283: 243: 10051: 3983:

Mass in special relativity § Early developments: transverse and longitudinal mass

1026:

can also be defined to be equivalent to the energy of a moving system, allowing for a

192:, even at extreme speeds approaching the speed of light. Its value is the same in all 11587: 11419: 11411: 11012: 10969: 10660: 9990: 9980: 9929: 9919: 9839: 9811: 9784: 9759: 9749: 9716: 9685: 9634: 9624: 9407: 9136: 9101:"Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie" 9025: 8978: 8943: 8931: 8884: 8874: 8849: 8655: 8616: 8608: 8546: 8468: 8458: 8430: 8397: 8387: 8352: 8313: 8242: 8238:

Albert Einstein e Olinto De Pretto: la vera storia della formula più famosa del mondo

8202: 8169: 8165:

Albert Einstein e Olinto De Pretto: la vera storia della formula più famosa del mondo

8137: 8104: 8092: 8074: 8035: 8031: 7949: 7902: 7892: 7854: 7844: 7814: 7804: 7779: 7769: 7730: 7670: 7641: 7631: 7606: 7567: 7557: 7533: 7519: 7502:

Rösch, Frank (2019), Lewis, Jason S.; Windhorst, Albert D.; Zeglis, Brian M. (eds.),

7481: 7471: 7388: 7378: 7342: 7332: 7268: 7260: 7179: 7175: 7121: 7045: 6998: 6930: 6922: 6875: 6800: 6790: 6751: 6741: 6713: 6703: 6654: 6623: 6594: 6584: 6548: 6538: 6492: 6482: 6455: 6378: 6348: 6261: 6227: 6209: 6195: 6144: 6134: 6098: 6097:. The quotation is taken from Serber's 1992 version, and is not in the original 1943 6078: 5771: 5761: 5606: 5537: 5050:

being the total energy (rest energy plus kinetic energy) of a moving material point,

4936: 3108: 2986: 2951: 1311: 1288: 1269: 1023: 930: 630: 595: 560: 447: 418: 408: 403: 212: 197: 185: 46: 11457: 9869: 9321: 7803:. Reading, Mass.: Addison-Wesley Pub. Co., Advanced Book Program. pp. 339–340. 6942: 6092: 5014:

is the relativistic mass (the rest mass and the extra mass gained when moving), and

3448: 590: 254: 11492: 11482: 11429: 11406: 10984: 10618: 10600: 10278: 9855: 9675: 9532: 9483:

The principle of relativity: three lectures given in Teyler's foundation in Haarlem

9449: 9399: 9279: 9213: 9124: 9075: 9017: 8970: 8923: 8841: 8768: 8700: 8600: 8538: 8423:"E = mc2 l'équation de Poincaré, Einstein et Planck: Henri Poincare et la physique" 8344: 8305: 8066: 7941: 7511: 7311: 7299: 7256: 7229: 7171: 7113: 7041: 7037: 6988: 6914: 6865: 6368: 6187: 6070: 5001:

being the relativistic energy (the energy of an object when the object is moving),

4932: 4929: 4705: 4687: 3681: 3529: 3443:

Another way of deriving a type of electromagnetic mass was based on the concept of

3180: 3062: 2722: 1230: 1191: 771: 760:. These energies tend to be much smaller than the mass of the object multiplied by 753: 600: 570: 236: 216: 11552: 9299:"Elementarquantum der Energie, Modell der negativen und der positiven Elekrizitat" 8518: 8285: 7921: 7151: 11567: 11542: 11467: 11462: 11345: 11306: 11268: 11212: 11086: 11022: 10648: 10580: 10345: 10253: 10198: 10168: 10132: 9805: 9478:

Das Relativitätsprinzip: drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem

9476: 8448: 8427:

E = mc2 l'équation de Poincaré, Einstein et Planck: Henri Poincare et la physique

8008: 7759: 7660: 7209: 7017: 6644: 6613: 6445: 6255: 5701: 5597: 5541: 5523: 3702: 3618: 3029: 2974: 2920: 2501:, which accounts for an energy correction of four parts per hundred million. The 1307: 1218: 1065: 757: 745: 585: 580: 555: 532: 442: 232: 159:). Because the speed of light is a large number in everyday units (approximately 119: 77: 11350: 10588: 9508: 9379: 9255: 9189: 9100: 9079: 8997: 8744: 8604: 7515: 7435: 6789:. Jewett, John W., Peroomian, Vahé. (Ninth ed.). Boston, MA. p. 1386. 6740:. Jewett, John W., Peroomian, Vahé. (Ninth ed.). Boston, MA. p. 1219. 6481:. Jewett, John W., Peroomian, Vahé. (Ninth ed.). Boston, MA. p. 1386. 6191: 5489:

So the change in the object's mass is equal to the total energy lost divided by

992: 11592: 11264: 11248: 11244: 11147: 11117: 10974: 10877: 10405: 10183: 10066: 6291: 5877: 5746: 5737: 5671: 5637: 5602: 5574: 5518: 5148: 5144: 4880: 3154: 2928: 2843: 2648: 2627: 2623: 1892: 1748: 1277: 1141: 1099: 1031: 926: 918: 892: 872: 749: 741: 727: 718:, have a corresponding intrinsic energy, even when they are stationary. In the 657: 625: 486: 341: 228: 220: 181: 150: 10082: 8903: 8096: 7556:. Jewett, John W., Peroomian, Vahé. (9th ed.). Boston, MA. p. 1419. 7503: 7150:

Belavin, A.A.; Polyakov, A.M.; Schwartz, A.S.; Tyupkin, Yu.S. (October 1975).

7093: 6993: 6968: 975:. The speed of light is one in a system where length and time are measured in 726:, the mass and energy are equal or they differ only by a constant factor, the 11674: 11638: 11582: 11562: 11557: 11472: 11340: 11168: 11112: 10944: 10897: 10624: 10594: 10481: 9835: 9763: 9720: 9689: 9536: 9411: 9283: 9217: 9128: 9029: 8982: 8935: 8888: 8853: 8772: 8705: 8676: 8612: 8550: 8491: 8434: 8401: 8356: 8317: 8206: 7953: 7858: 7734: 7571: 7346: 7233: 7183: 7125: 7117: 7094:"Computation of the quantum effects due to a four-dimensional pseudoparticle" 7049: 7002: 6926: 6879: 6804: 6755: 6717: 6496: 6382: 6373: 6344: 6285: 6148: 5641: 5549: 5532: 5204: 4539: 3176: 3065:, presented a mass–energy relation. Italian mathematician and math historian 2804: 2631: 1246: 1214: 1183: 976: 491: 373: 205: 10071: 9994: 9933: 8472: 8246: 8173: 8088: 8060: 7906: 7783: 7645: 7485: 7392: 6598: 6552: 5993:. See also the sentence above the last equation in the English translation, 5605:

emblazoned with the equation. Einstein himself had only a minor role in the

3000:, published in 1717, Newton speculated on the equivalence of mass and light. 11602: 11522: 11487: 11017: 10979: 10732: 10612: 9860: 9843: 8974: 8620: 8348: 8141: 7978: 7818: 6934: 6870: 6845: 5655: 5592: 5118: 5071: 4683: 4668: 4202:

after the emission as seen in its rest frame. As seen from a moving frame,

3188: 3033: 3005: 2991: 2900: 1912: 1296: 1254: 1119: 946:

and both are nearly equal to the classical inertial mass (as it appears in

834: 791: 744:, and it may or may not have other amounts of internal stored energy, like 620: 9743: 9638: 9322:"Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen" 8868: 8377: 8039: 7836: 7610: 7326: 6784: 6476: 5601:

magazine prominently featured a picture of Einstein next to an image of a

5155:. If the body is examined in a frame moving with nonrelativistic velocity 11386: 11355: 10902: 10737: 10421: 10243: 10090: 9974: 9913: 8286:"Why is the energy of motion proportional to the square of the velocity?" 8236: 8163: 8070: 7886: 7625: 7465: 7407:"Parker Solar Probe Becomes Fastest-Ever Spacecraft – Parker Solar Probe" 7372: 6578: 6532: 6064: 5554: 5323:

The left-moving light carries a little less momentum, by the same amount

5111: 4680: 3192: 2828: 2594:

crew members are spelling out Einstein's mass–energy equivalence formula

2563: 605: 481: 9461: 9437: 8131: 7798: 7064:"Making antimatter | Angels & Demons – The science behind the story" 6074: 3036:

pointed out a relation between mass and energy for ether in the form of

1328:

Unlike a system's energy in an inertial frame, the relativistic energy (

11597: 11163: 11007: 11002: 10742: 9614: 8927: 8059:

Kragh, Helge (1999). "Fin-de-Siècle Physics: A World Picture in Flux".

8025: 7662:

Megawatts and Megatons: The Future of Nuclear Power and Nuclear Weapons

7600: 7331:. Smith, A. Gavin. Chichester, UK: John Wiley & Sons. p. 259. 5713: 5200: 4672: 4664: 2422:, a number very small for everyday objects. In 2018 NASA announced the 1226: 1187: 719: 635: 615: 457: 452: 73: 50: 9728: 9704: 8821: 8542: 8309: 7945: 6537:(3rd ed.). Upper Saddle River, N.J.: Prentice Hall. p. 512. 3199:

in 1904—to understand how the mass of a charged object depends on the

2122:

is small. For low speeds, all but the first two terms can be ignored:

10994: 9453: 8845: 8007:

Swedenborg, Emanuel (1734). "De Simplici Mundi vel Puncto naturali".

7303: 6129:

Serway, Raymond A.; Jewett, John W.; Peroomian, Vahé (5 March 2013).

5570: 5559: 3059: 2955: 2908: 2644: 1265: 1253:

showed that there is a process that converts protons and neutrons to

1200: 1073: 943: 868: 433: 131: 9680: 9653: 9403: 9021: 4142:

lost by a body in the form of radiation. Consequently, the equation

231:. The principle is fundamental to many fields of physics, including 11537: 10887: 6918: 6345:"Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" 6180: 5196: 3876: 2932: 1281: 1258: 1077: 848: 723: 610: 251: 189: 10101: 9190:"Über die vom Relativitätsprinzip geforderte Trägheit der Energie" 8062:

Quantum generations: a history of physics in the twentieth century

6844:

Dyson, F.W.; Eddington, A.S. & Davidson, C.R. (January 1920).

6843: 5159:, it is no longer at rest and in the moving frame it has momentum 4650:, and that the mass of a body is a measure of its energy content. 3697: 2885:

Any time energy is released, the process can be evaluated from an

2494:{\displaystyle {\tfrac {3v^{2}}{4c^{2}}}\approx 3.9\times 10^{-8}} 8581:"Considerations Concerning the Fundaments of Theoretical Physics" 6181:

Special Theory of Relativity: Einstein’s World in New Axiomatics

5709: 5640:, at that time). The physicist and Manhattan Project participant 5566: 4618:{\displaystyle K_{0}-K_{1}={\frac {E}{c^{2}}}{\frac {v^{2}}{2}}.} 3010: 2996: 2936: 2896: 2839: 2815: 1222: 1114: 906: 722:

of an object, where by definition it is motionless and so has no

247: 57: 9810:. Morgan & Claypool Publishers. Second page of section 2.2. 7212:(1984). "A Saddle Point Solution in the Weinberg Salam Theory". 5479:{\displaystyle P'=Mv-2\Delta P=\left(M-{E \over c^{2}}\right)v.} 4628:

Einstein concluded that the emission reduces the body's mass by

2808: 1140:

is defined so that the system has zero total momentum; the term

11217: 10010:, and… the mass was just equivalent to 200 MeV; it all fitted!" 9884:"TIME Magazine – U.S. Edition – July 1, 1946 Vol. XLVIII No. 1" 4139: 3684:

principle—just as, about 60 years before, the principle of the

2875: 2798: 2712: 1110: 1094:

Mass in special relativity § The mass of composite systems

1047: 880: 876: 858: 201: 174: 69: 9979:. Berkeley: University of California Press. pp. 236–237. 9194:

On the Inertial of Energy Required by the Relativity Principle

7843:, Damour, Thibault., Basel: Birkhäuser Verlag, pp. 1–22, 7149: 4697: 2572:

Task Force One, the world's first nuclear-powered task force.

10778: 9834: 9326:

On the Relativity Principle and the Conclusions Drawn From it

7692:"The Yields of the Hiroshima and Nagasaki Nuclear Explosions" 7665:(illustrated ed.). University of Chicago Press. p. 7247:

Rubakov, V. A. (1988). "Monopole Catalysis of Proton Decay".

4720:, because mass conservation appears as a special case of the 2861: 2197:{\displaystyle E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}.} 1262: 767: 224: 9703:

Badash, Lawrence; Hodes, Elizabeth; Tiddens, Adolph (1986).

9380:"Einstein's comprehensive 1907 essay on relativity, part II" 6646:

Introduction to Electromagnetic Theory: A Modern Perspective

5330:. So the total right-momentum in both light pulses is twice 899:

of an object is given by the relativistic energy divided by

891:, depending on the motion of the observer. This implies the 211:

The equivalence principle implies that when mass is lost in

11198: 9748:(2nd ed.). New York: Juniper Grove. pp. 336–338. 9621:

Bibliography of the Writings of Albert Einstein to May 1951

8873:. Princeton, N.J.: Princeton University Press. p. 51. 7979:"Selected Queries from Isaac Newton's Opticks | Inters.org" 7510:, Cham: Springer International Publishing, pp. 27–61, 4922:

is equivalent, as regards inertia, to a quantity of energy

3595:{\displaystyle m_{0}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}} 3433:{\displaystyle m_{0}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}} 1234: 711: 65: 42: 9438:"The Principle of Relativity, and Non-Newtonian Mechanics" 6349:

Does the Inertia of a Body Depend Upon its Energy-Content?

5256:. So the right-moving light is carrying an extra momentum 4675:

in 1961, asserting that Einstein's derivation is based on

2369:

The difference between the two approximations is given by

37: 10065:– Conversations About Science with Theoretical Physicist 10023:"Handwritten example of Einstein equation fetches $ 1.2M" 8231:

Bartocci, U; Bonicelli, Bianca Maria (1999). "Pretto, O.

7762:(1989). "The relativity theory of Poincaré and Lorentz". 6702:(2nd, rev. ed.). Weinheim : Wiley-VCH. p. 101. 6296:

Archives Néerlandaises des Sciences Exactes et Naturelles

5883: 5633: 9619:(3d ed.). La Salle, Ill.: Open Court. M.C. Shields 9442:

Proceedings of the American Academy of Arts and Sciences

8998:"Einstein's first derivation of mass–energy equivalence" 8034:. London; Boston: W. Newbery; O. Clapp. pp. 55–57. 7554: