Mass–energy equivalence Totally Explained

Everything about Mass Energy Equivalence totally explained

ť For other uses, see E=MC2 (disambiguation).

In physics, mass–energy equivalence is the concept that any mass has an associated energy and vice versa. In special relativity this relationship is expressed using the mass–energy equivalence formula ť :

E = mc^2 ,

where ť * E = energy,

   * m = mass, ť * c = the speed of light in a vacuum (celeritas),

   * and the superscript 2 indicates the squaring of c.
   Two definitions of mass in special relativity may be validly used with this formula. If the mass in the formula is the rest mass, the energy in the formula is called the rest energy. If the mass is the relativistic mass, then the energy is the total energy.
   The formula was derived by Albert Einstein, who arrived at it in 1905 in the paper "Does the inertia of a body depend upon its energy-content?", one of his Annus Mirabilis ("Wonderful Year") Papers. While Einstein wasn't the first to propose a mass–energy relationship, and various similar formulas appeared before Einstein's theory, Einstein was the first to propose that the equivalence of mass and energy is a general principle, which is a consequence of the symmetries of space and time.
   In the formula, c² is the conversion factor required to convert from to . The formula doesn't depend on a specific system of units. In the International System of Units, the unit for energy is the joule, for mass the kilogram, and for speed meters per second. Note that 1 joule equals 1 kg·m²/s². In unit-specific terms, E (in joules) = m (in kilograms) multiplied by (299,792,458 m/s)².

Conservation of mass and energy

The concept of mass–energy equivalence unites the concepts of conservation of mass and conservation of energy, allowing rest mass to be converted to forms of active energy (such as kinetic energy, heat, or light) while still retaining mass. Conversely, active energy in the form of kinetic energy or radiation can be converted to particles which have rest mass. The total amount of mass/energy in a closed system (as seen by a single observer) remains constant because energy can't be created or destroyed and, in all of its forms, trapped energy exhibits mass. In relativity, mass and energy are two forms of the same thing, and neither one appears without the other.

Fast-moving object

If a force is applied to an object in the direction of motion, the object gains energy because the force is doing work, but an object can't be accelerated to the speed of light, regardless of how much energy it absorbs. Its kinetic energy continues to increase without bounds, whereas its speed approaches the (finite) speed of light. This means that in relativity the kinetic energy isn't given by 1/2 mv².
   The relativistic mass is the ratio of the momentum of an object to its speed, and it's a quantity that depends on the motion of the observer. If the observer is moving at nearly the same velocity as the object, the relativistic mass is nearly equal to the rest mass, which is also the usual Newtonian mass. If the observer is moving quickly relative to the object, the relativistic mass is bigger than the rest mass.
   The relativistic mass is always equal to the total energy divided by c². The difference between the relativistic mass and the rest mass is the relativistic kinetic energy (divided by c²). Because the relativistic mass is exactly proportional to the energy, relativistic mass and relativistic energy are terms which can be used interchangeably. For this reason, when people talk about the mass of a particle, they're usually talking about its rest mass, which is the same in all inertial frames.
   For a system made up of many parts, linked in (nucleus, atom, common object, planet, star . . .), the relativistic mass is the sum of the relativistic masses of the parts, because the energy adds up.

Meanings of the mass–energy equivalence formula

Mass–energy equivalence says that when a body has a mass, it has a certain energy, even when it isn't moving. In Newtonian mechanics, a massive body at rest has no kinetic energy, and it may or may not have other (relatively small) amounts of internal stored energy such as chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. In Newtonian mechanics, none of these energies contributes to the mass.
   In relativity, all the energy which moves along with the body adds up to the rest energy of the body, which is proportional to the rest mass of the body. Even a single photon traveling in empty space has a relativistic mass, which is its energy divided by c². If a box of mirrors contains light, the mass of the box is increased by the energy of the light, since the total energy of the box is its mass.
   Although a photon is never "at rest", it still has a rest mass, which is zero. If an observer chases a photon faster and faster, the observed energy of the photon approaches zero as the observer approaches the speed of light. This is why photons are massless. They have zero rest mass even though they've varying amounts of energy and relativistic mass. But, systems of two or more photons moving in different directions (as for example from an electron–positron annihilation) may have zero momentum over all. Their energy E then adds up to an invariant mass m = E/c², when they're considered as a system.
   This formula also gives the amount of mass lost from a body when energy is removed. In a chemical or nuclear reaction, when heat and light are removed, the mass is decreased. So the E in the formula is the energy released or removed, corresponding to a mass m which is lost. In those cases, the energy released and removed is equal in quantity to the mass lost, times c². Similarly, when energy of any kind is added to a resting body, the increase in the mass is equal to the energy added, divided by c².

Consequences for nuclear physics

Max Planck pointed out that the mass–energy equivalence formula implied that bound systems would have a mass less than the sum of their constituents, once the binding energy had been allowed to escape. However, Planck was thinking about chemical reactions, where the binding energy is too small to measure. Einstein suggested that radioactive materials such as radium would provide a test of the theory, but even though a large amount of energy is released per atom, only a small fraction of the atoms decay.
Once the nucleus was discovered, experimenters realized that the very high binding energies of the atomic nuclei should allow calculation of their binding energies from mass differences. But it wasn't until the discovery of the neutron in 1932, and the measurement of its mass, that this calculation could actually be performed (see nuclear binding energy for example calculation). A little while later, the first transmutation reactions (such as

scriptstyle  m_0 v^2 .

The total energy is a sum of the rest energy and the Newtonian kinetic energy.
The classical energy equation ignores both the

m_0 c^2

part, and the high-speed corrections. This is appropriate, because all the high order corrections are small. Since only changes in energy affect the behavior of objects, whether we include the

m_0 c^2

part makes no difference, since it's constant. For the same reason, it's possible to subtract the rest energy from the total energy in relativity. In order to see if the rest energy has any physical meaning, it's essential to consider emission and absorption of energy in different frames.
The higher-order terms are extra correction to Newtonian mechanics which become important at higher speeds. The Newtonian equation is only a low speed approximation, but an extraordinarily good one. All of the calculations used in putting astronauts on the moon, for example, could have been done using Newton's equations without any of the higher order corrections.

History

While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he wasn't the first to have related energy with mass. But nearly all previous authors thought that the energy which contributes to mass comes only from electromagnetic fields.

Newton: Matter and light

In 1717 Isaac Newton speculated that light particles and matter particles were inter-convertible in "Query 30" of the Opticks, where he states:
Since Newton didn't understand light as the motion of a field, he wasn't speculating about the conversion of motion into matter. Since he didn't know about energy, he couldn't have understood that converting light to matter is turning work into mass.

Electromagnetic rest mass

There were many attempts in the 19th and the beginning of the 20th century - like those of J. J. Thomson (1881), Oliver Heaviside (1888), George Frederick Charles Searle (1896), - to understand how the mass of a charged object varied with the velocity. Because the electromagnetic field carries part of the momentum of a moving charge, it was suspected that the mass of an electron would vary with velocity near the speed of light.
   Following Searle (1896), Wilhelm Wien (1900), Max Abraham (1902),
   and Hendrik Lorentz (1904)
   concluded that the velocity dependant electromagnetic mass of a body at rest is

m=(4/3)E/c^2

. According to them, this relation applies to the complete mass of bodies, because any form of inertial mass was considered to be of electromagnetic origin. Wien went on by stating, that if it's assumed that gravitation is an electromagnetic effect too, than there has to be a strict proportionality between (electromagnetic) inertial mass and (electromagnetic) gravitational mass. To explain the stability of the matter-electron configuration, Poincaré in 1906 introduced some sort of pressure of non-electrical nature, which contributes the amount

-(1/3)E/c^2

to the mass of the bodies, and therefore the 4/3-factor vanishes.

Inertia of energy and radiation

Maxwell, Bartoli, Lorentz James Clerk Maxwell (1874)
and Adolfo Bartoli (1876) found out that the existence of tensions in the ether like the radiation pressure follows from the electromagnetic theory. However, Lorentz (1895)
recognized that this led to a conflict between the action/reaction principle and Lorentz's ether theory.

Poincaré In 1900 Henri Poincaré studied this conflict and tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included. He noticed that the action/reaction principle doesn't hold for matter alone, but that the electromagnetic field has its own momentum. The electromagnetic field energy behaves like a fictitious fluid ("fluide fictif") with a mass density of $E/c^2$ (in other words $m=E/c^2$ ). If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible - it's neither created or destroyed - then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one shouldn't be too surprised by these assumptions, since they're only mathematical fictions.
   But Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it'll suffer a recoil from the inertia of the fictitious fluid. In the framework of Lorentz ether theory Poincaré performed a Lorentz boost to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow a perpetuum mobile, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle wouldn't hold.
   Poincaré's paradox was resolved by Einstein's insight that a body losing energy as radiation or heat was losing a mass of the amount $m=E/c^2$ . The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame.
   Einstein noted in 1906 that Poincaré's solution to the center of mass problem and his own were mathematically equivalent (see below).
   Poincaré came back to this topic in "Science and Hypothesis" (1902) and "The Value of Science" (1905). This time he rejected the possibility that energy carries mass: "... [therecoil] is contrary to the principle of Newton since our projectile here has no mass, it isn't matter, it's energy". He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass $gamma m$ , Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Madame Curie.

Abraham and Hasenöhrl Following Poincaré, Max Abraham in 1902-1904
   introduced the term "electromagnetic momentum" to maintain the action/reaction principle. Poincaré's result, who according to Abraham gave no proof of his result, was verified by him, whereby the field density of momentum per cm³ is $E/c^2$ and $E/c$ per cm².
   In 1904, Friedrich Hasenöhrl specifically associated inertia with radiation in a paper, which was according to his own words very similar to some papers of Abraham. Hasenöhrl suggested that part of the mass of a body (which he called apparent mass) can be thought of as radiation bouncing around a cavity. The apparent mass of radiation depends on the temperature (because every heated body emits radiation) and is proportional to its energy, and he first concluded that $m=(8/3)E/c^2$ . However, in 1905 Hasenöhrl published a summary of a letter, which was written by Abraham to him. Abraham concluded that Hasenöhrl's formula of the apparent mass of radiation isn't correct, and based on his definition of electromagnetic momentum and longitudinal electromagnetic mass Abraham changed it to $m=(4/3)E/c^2$ , the same value for the electromagnetic mass for a body at rest. Hasenöhrl re-calculated his own derivation and verified Abraham's result. He also noticed the similarity between the apparent mass and the electromagnetic mass. However, Hasenöhrl stated that this energy-apparent-mass relation only holds as long a body radiates, for example if the temperature of a body is greater than 0 K.
   However, it was suggested that Hasenöhrl had made an error in that he didn't include the pressure of the radiation on the cavity shell. If he'd included the shell pressure and inertia as it would be included in the theory of relativity, the factor would have been equal to 1 or $m=E/c^2$ . This calculation assumes that the shell properties are consistent with relativity, otherwise the mechanical properties of the shell including the mass and tension wouldn't have the same transformation laws as those for the radiation. Nobel Prize-winner and Hitler advisor Philipp Lenard claimed that the mass–energy equivalence formula needed to be credited to Hasenöhrl to make it an aryan creation.

Einstein: Mass–energy equivalence

Albert Einstein didn't formulate exactly this formula in his 1905 paper "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" ("Does the Inertia of a Body Depend Upon Its Energy Content?", published in Annalen der Physik on September 27), one of the articles now known as his Annus Mirabilis Papers.
However, Einstein didn't have to introduce fictitious masses and could also avoid the perpetuum mobile problem, because based on the mass–energy equivalence he could show that emission and absorption of em-radiation and therefore the transport of inertia solves the problem. Also Poincaré's rejection of the reaction principle due to the violation of the mass conservation law (as discussed in the preceding section) can be avoided through Einstein's

E=mc^2

, because mass conservation appears as a special case of the energy conservation law.

Others

During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various discredited ether theories. In particular, the writings of S. Tolver Preston, and a 1903 paper by Olinto De Pretto,
Preston and De Pretto, following Le Sage, imagined that the universe was filled with an ether of tiny particles which are always moving at speed c. Each of these particles have a kinetic energy of mc² up to a small numerical factor. The nonrelativistic kinetic energy formula didn't always include the traditional factor of 1/2, since Leibniz introduced kinetic energy without it, and the 1/2 is largely conventional in prerelativistic physics. By assuming that every particle has a mass which is the sum of the masses of the ether particles, the authors would conclude 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, and since these authors didn't formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames.
Independently, Gustave Le Bon in 1905 speculated that atoms could release large amounts of latent energy, reasoning from an all encompassing qualitative philosophy of physics.

Nuclear energy and popular culture

Radioactivity was discovered in 1896, and the source of the energy was initially a mystery. By 1903, Ernest Rutherford and Frederick Soddy had proved that radioactive elements was due to the fact that they decayed into other elements, releasing a great deal of energy in the process. Einstein mentions in his 1905 paper that mass-energy equivalence might perhaps be tested with radioactive decay, which releases enough energy (the quantitative amount known roughly even by 1905) to possibly be "weighed," when missing. But the idea that great amounts of usable energy could be liberated from matter, however, proved initially difficult to substantiate in a practical fashion. Because it had been used as the basis of much speculation, Rutherford himself was once reported in the 1930s to have said that: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."
This changed dramatically after the demonstration of energy 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 as early as 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 close-enough linked 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. Einstein himself had only a minor role in the Manhattan Project: he'd cosigned a letter to the US 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 based on the rate of molecular diffusion through pores, a now obsolete process that was then competitive and contributed a fraction of the enriched uranium used in the project.
While E=mc² is useful for understanding the amount of energy released in a fission reaction, it wasn't strictly necessary to develop the weapon. As the physicist and Manhattan Project participant Robert Serber put it: "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. Albert Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity isn't 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." However the association between E=mc² and nuclear energy has since stuck, and because of this association, and its simple expression of the ideas of Albert Einstein himself, it has become "the world's most famous equation".

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