E=mc²

A display of the famous equation on Taipei 101 during the event of the World Year of Physics 2005.

In physics, E = mc² is an important and well-known equation, which states an equivalence between energy (E) and mass (m), in direct proportion to the square of the speed of light in a vacuum (c²). See mass in special relativity for a discussion of the various definitions of mass which may be validly used with this equation.

The equation was first derived (in a slightly different formulation) in 1905 by Albert Einstein, in what are known as his Annus Mirabilis ("Wonderful Year") Papers. In these, he showed that a unified four-dimensional model of space and time ("spacetime") could accurately describe observable phenomena in a way that was consistent with Galileo's Principle of Relativity, but also accounted for the constant speed of light. His special theory of relativity ultimately showed that the traditional (Euclidean-Galilean) assumption of absolute time and distance was incorrect, and, as a consequence, that mass and energy are different only in form.

Thus c² is the conversion factor required to sometimes convert from units of mass to units of energy, i.e. the energy per unit mass. In unit-specific terms, E (joules or kg·m²/s²) = m (kilograms) multiplied by (299,792,458 m/s)².

Meanings of the formula

Main article: mass-energy equivalence

This formula proposes that when a body has a mass (measured at rest), it has a certain (very large) amount of energy associated with this mass. This is opposed to the Newtonian mechanics, in which a massive body at rest has no kinetic energy, and 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. That is why a body's rest mass, in Einstein's theory, is often called the rest energy of the body. The E of the formula can be seen as the total energy of the body, which is proportional to the mass of the body.

Conversely, a single photon travelling in empty space cannot be considered to have an effective mass, m, according to the above equation. The reason is that such a photon cannot be measured in any way to be at "rest" and the formula above applies only to single particles when they are at rest, and also systems at rest (i.e., systems when seen from their center of mass frame). Individual photons are generally considered to be "massless," (that is, they have no rest mass or invariant mass) even though they have varying amounts of energy and relativistic mass. Systems of two or more photons moving in different directions (as for example from an electron-positron annihilation) will have an invariant mass, and the above equation will then apply to them, as a system, if the invariant mass is used.

This formula also gives the quantitative relation of the quantity of mass lost from a resting body or a resting system (a system with no net momentum, where invariant mass and relativistic mass are equal), when energy is removed from it, such as in a chemical or a nuclear reaction where heat and light are removed. Then this E could be seen as the energy released or removed, corresponding with a certain amount of relativistic or invariant mass m which is lost, and which corresponds with the removed heat or light. In those cases, the energy released and removed is equal in quantity to the mass lost, times the speed of light squared. Similarly, when energy of any kind is added to a resting body, the increase in the resting mass of the body will be the energy added, divided by the speed of light squared.

History and consequences

Albert Einstein derived the formula based on his 1905 inquiry into the behavior of objects moving at nearly the speed of light. The famous conclusion he drew from this inquiry is that the mass of a body is actually a measure of its energy content. Conversely, the equation suggests (see below) that all of the energies present in closed systems affect the system's resting mass.

$\mathrm {Energy} =\mathrm {Mass} \,\times \,(\mathrm {speed\ of\ light\ in\ vacuum} )^{2}$

According to the equation, the maximum amount of energy "obtainable" from an object to do active work, is the mass of the object multiplied by the square of the speed of light.

It was actually Max Planck who first pointed out that Einstein's equation 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 in terms of chemical reactions, which have binding energies too small for the measurement to be practical. Early experimenters also realized that the very high binding energies of the atomic nuclei should allow calculation of their binding energies from mass differences, however it was not till the discovery of the neutron and its mass in 1932 that this calculation could actually be performed. Very shortly thereafter, the first transmutation reactions (such as ${}^{7}\mathrm {Li} +\mathrm {p} \rightarrow 2\,{}^{4}\mathrm {He}$) were able to verify the correctness of Einstein's equation to an accuracy of 1%.

This equation was used in the development of the atomic bomb. By measuring the mass of different atomic nuclei and subtracting from that number the total mass of the protons and neutrons as they would weigh separately, one could obtain an estimate of the binding energy available within an atomic nucleus. This could be and was used in estimating the energy released in the nuclear reaction, by comparing the binding energy of the nuclei that enter and exit the reaction.

It is a little known piece of trivia that Einstein originally wrote the equation in the form Δm = L/c² (with an "L", instead of an "E", representing energy, the E being utilised elsewhere in the demonstration to represent energy too).

Practical examples

A kilogram of mass could (theoretically) convert completely into approximately : -

• 90 000 000 000 000 000 joules or 90 PJ or
• 25 000 000 000 kilowatt-hours or
• The energy in 21 megatons of TNT
• approximately 0.0850 Quads (quadrillion British thermal units)

It is important to note that practical conversions of "mass" to energy are seldom 100 percent efficient. One theoretically perfect conversion would result from a collision of matter and antimatter (e.g. in positronium experiments); for most cases, byproducts are produced instead of energy, and therefore very little mass is actually converted. For example, in nuclear fission roughly 0.1% of the mass of fissioned atoms is converted to energy. In turn, the mass of fissioned atoms is only part of the mass of the fissionable material: e.g. in a nuclear fission weapon, the efficiency is 40% at most. In nuclear fusion roughly 0.3% of the mass of fused atoms is converted to energy. In actual thermonuclear weapons (see nuclear weapon yield) some of the total bomb mass is casing and non-reacting components, so the efficiency in converting passive energy to active energy, at 7 kilotons/kg, does not exceed 0.03% of the bomb mass.

USS Enterprise, Long Beach and Bainbridge in formation in the Mediterranean, 18 June 1964. Enterprise crewmembers spelled out Einstein's famous equation on the flight deck to commemorate the first all-nuclear battle formation.

In the equation, mass is energy, but for the sake of brevity, the word "converted" is used; in practice, one kind of energy is converted to another, but it continues to contribute mass to systems so long as it is trapped in them (active energy is associated with mass also, as seen by single observers). Thus, the total mass of any system is conserved and remains unchanged (for any single observer) unless energy (such as heat, light, or other radiation) is allowed to escape the system. In any cases, the use of the phrase "converted" is intended to signify energy which has gone from passive potential energy, into heat or kinetic energy which can be used to do work (as in a nuclear reactor or even in a heat-producing chemical reaction). To use the quantitative examples above, if a kilogram of rest mass was converted to 90 PJ of light, heat, or other forms of kinetic energy, it would still continue to weigh one kilogram, so long as it was trapped in any system which allowed it to be weighed.

Background

E = mc² where m stands for rest mass (invariant mass), applies to all objects or systems with mass but no net momentum. Thus, it applies most simply to particles which are not in motion. However, in a more general case, it also applies to particle systems (such as ordinary objects) in which particles are moving but in different directions so as to cancel momentums. In the later case, both the mass and energy of the object include contributions from heat and particle motion, but the equation continues to hold.

The equation is a special case of a more general equation in which both energy and net momentum are taken into account. This equation always applies to a particle that is not moving as seen from a reference point, but this same particle can be moving from the standpoint of other frames of reference (where it has a net momentum). In such cases, the equation (if the mass used is invariant mass must becomes more complicated as the energy changes, since momentum-containing terms must be added so that the invariant mass remains constant from any reference frame (as it must, given the definition of invariant mass).

Alternative formulations of relativity, see below, allow the mass to vary with energy and simply ignore momentum, but this involves use of a second definition of mass, called relativistic mass because it causes mass (which is now relativistic mass, not invariant mass) to differ in different reference frames.

A key point to understand is that there may be two different meanings used here for the word "mass". In one sense, mass refers to the usual mass that someone would measure if sitting still next to the mass, for example. This is the concept of rest mass, which is often denoted $m_{0}$. It is also called invariant mass. In relativity, this type of mass does not change with the observer, but it is computed using both energy and momentum, and (unless momentum happens to be zero) the equation $E=mc^{2}$ is not in general correct for it, if the total energy is wanted. (In other words, if this equation is used with constant invariant mass or rest mass of the object, the $E$ given by the equation will always be the rest energy of the object, and will change with the object's internal energy, such as heating, but will not change with the object's overall motion).

In developing special relativity, Einstein found that the total energy of a moving body is

$E={\frac {m_{0}c^{2}}{\sqrt {1-(v^{2}/c^{2})}}},$

with $v$ being the relative velocity. This can be shown to be equivalent to

$E={\sqrt {m_{0}^{2}c^{4}+p^{2}c^{2}}}$

with p being the relativistic momentum (ie. $p=\gamma p_{0}=m_{\mathrm {rel} }*v$).

When $v=0$, then $p=0$, and both formulas above reduce to $E={m_{0}c^{2}}$, with E now representing the rest energy, $E_{0}$. This can be compared with the kinetic energy in newtonian mechanics:

$E={\frac {1}{2}}mv^{2}$,

where $E_{0}=0$ (in Newtonian mechanics only kinetic energy is treated, and thus "rest energy" is zero).

Relativistic mass

After Einstein first made his proposal, some suggested that the mathematics might seem simpler if we define a different type of mass. The relativistic mass is defined by

$m_{\mathrm {rel} }\;=\;\gamma m_{0}\;=\;{\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.$

Using this form of the mass, we can again simply write $E=m_{\mathrm {rel} }c^{2}$, even for moving objects. Now, unless the velocities involved are comparable to the speed of light, this relativistic mass is almost exactly the same as the rest mass. That is, we set $v=0$ above, and get $m_{\mathrm {rel} }=m_{0}$.

Now, understanding the difference between rest mass and relativistic mass, we see that the equation $E=mc^{2}$ in the title must be rewritten: either $E=m_{0}c^{2}$ for $v=0$, or $E=m_{\mathrm {rel} }c^{2}$ when $v\neq 0$.

Einstein's original papers (see, e.g. [1]) treated m as what would now be called the rest mass or invariant mass and he did not like the idea of "relativistic mass" (see usenet physics FAQ [2]). When a modern physicist refers to "mass," he or she is almost certainly speaking about rest mass, also. This can be a confusing point, though, because students are sometimes still taught the concept of "relativistic mass" in order to be able to keep Einstein's simple equation correct, even for moving bodies.

Low-speed approximation

We can rewrite the expression above as a Taylor series:

$E=m_{0}c^{2}\left[1+{\frac {1}{2}}\left({\frac {v}{c}}\right)^{2}+{\frac {3}{8}}\left({\frac {v}{c}}\right)^{4}+{\frac {5}{16}}\left({\frac {v}{c}}\right)^{6}+\ldots \right].$

For speeds much smaller than the speed of light, higher-order terms in this expression (the ones farther to the right) get smaller and smaller. The reason for this is that the velocity $v$ is much smaller than $c$, so $v/c$ is quite small. If the velocity is small enough, we can throw away all but the first two terms, and get

$E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}.$

This expresses energy as the sum of Einstein's term for a resting object and the usual kinetic energy which Newton knew about. Thus, we see that Newton's form of the energy equation just ignores the parts that he never knew about: the $m_{0}c^{2}$ part, and the high-speed parts. This worked because Newton never saw an object lose enough energy to measurably change its rest mass--as in a nuclear process--and only saw objects move at speeds which were quite small compared to the speed of light. Einstein needed to add the extra terms to make sure his formula was right, even at high speeds. In doing so, he discovered that rest mass could be "converted" to energy (or more correctly, converted to active energy which retained mass, but which could be drained away as heat or radiation, so that it subtracted from rest mass when gone).

Interestingly, we could include the $m_{0}c^{2}$ part in Newtonian mechanics because it is constant, and only changes in energy have any influence on what objects actually do. This would be a waste of time, though, precisely because this extra term would not have any noticeable effect, except at the very high energies characteristic of nuclear reactions or particle accelerators. The "higher-order" terms that we left out show that special relativity is a high-order correction to Newtonian mechanics. The Newtonian version is actually wrong, but is close enough to use at "low" speeds, meaning low compared with the speed of light. For example, all of the celestial mechanics involved in putting astronauts on the moon could have been done using only Newton's equations.

Einstein and his 1905 paper

Albert Einstein did not formulate exactly this equation 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.

That paper says: If a body gives off the energy L in the form of radiation, its mass diminishes by L/c², "radiation" being electromagnetic radiation in Einstein's example (the paper specifies "light"), and the mass being the ordinary concept of mass used in those times, the same one that today we call rest energy or invariant mass, depending on the context. Einstein's very first formulation of this equation asserts that the invariant mass of a body does not change until the system is opened and light or heat is removed.

In Einstein's first formulation, it is the difference in the mass '$\Delta m\$' before the ejection of energy and after it, that is equal to L/c², not the entire mass '$m\$' of the object. At that moment in 1905, even this was only theoretical and not proven experimentally. Not until the discovery of the first type of antimatter (the positron in 1932) was it found that entire pairs of resting particles could be converted to radiation moving away at the speed of light.

Contributions of others

Einstein was not the only one to have related energy with mass, but he was the first to have presented that as a part of a bigger theory, and even more, to have deduced the formula from the premises of this theory. During the nineteenth century in particular there were many speculative attempts to show that mass and energy were equivalent, often within the premises of the electromagnetic worldview, though they were not regarded as theoretically successful.

Sir Isaac Newton published Opticks in 1704, in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, ordinary matter of grosser corpuscles, and speculated that a kind of alchemical transmutation existed between them. "Are not gross bodies and light convertible into one another; and may not bodies receive much of their activity from the particles of light which enter into their composition? The changing of bodies into light, and light into bodies, is very conformable to the course of Nature, which seems delighted with transmutations."

In 1904, Friedrich Hasenöhrl specifically associated mass via inertia with the energy concept through an equation. Hasenöhrl first concluded that m = (8/3)E/c^2.

In a later paper, Hasenöhrl re-calculated this result and arrived at m = (4/3)E/c^2. Hasenöhrl indicated that if the internal energy of a system consists of radiation, then, in general, the inertial mass of the system would depend upon that energy. This would be in accordance with his calculation. Thus, this new Hasenöhrl calculation establishes that due to the radiant energy E contained in his system, to that inertial mass must be added an apparent mass m. Indeed, in 1914 Cunningham showed that Hasenöhrl had made a slight error in that he did not include the shell. If he had included the shell in his calculations, the factor would have been 1 or m = E/c^2.

According to Umberto Bartocci (University of Perugia historian of mathematics), the correct equation E=mc² was first published on June 16, 1903 by Olinto De Pretto, an industrialist from Vicenza, Italy, though this is not generally regarded as important by mainstream historians. Even though De Pretto was first to introduce the formula and to understand it, it was Einstein who properly derived it.

In a paper of 1900 the French mathematician Henri Poincaré discussed the recoil of a physical object when it emits a burst of radiation in one direction, as predicted by Maxwell-Lorentz electrodynamics. He remarked that the stream of radiation appeared to act like a "fictitious fluid" with a mass per unit volume of e/c², where e is the energy density; in other words, the equivalent mass of the radiation is $m=E/c^{2}$. Poincaré considered the recoil of the emitter to be an unresolved feature of Maxwell-Lorentz theory, which he discussed again in "Science and Hypothesis" (1902) and "The Value of Science" (1904). In the latter he said the recoil "is contrary to the principle of Newton since our projectile here has no mass, it is not matter, it is energy", and 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. It was Einstein's insight that a body losing energy as radiation or heat was losing mass of amount $m=E/c^{2}$, and the corresponding mass-energy conservation law, which resolved these problems.

Television biography

E=mc² was used as the title of a 2005 television biography about Einstein concentrating on the year 1905.

Derivation

Newton's second law as it appears in nonrelativistic classical mechanics reads

$\mathbf {F} ={\frac {d(m\mathbf {v} )}{dt}},$

where mv is the nonrelativistic momentum of a body, F is the force acting upon it, and t is the coordinate of absolute time. In this form, the law is incompatible with the principles of relativity; the law does not change covariantly under Lorentz transformations. This situation is naturally remedied by modifying the law to read

$\mathbf {F} ={\frac {d\mathbf {p} }{d\tau }},$

where now p=mγc is the relativistic momentum of the body, F is the force acting on a body as measured in its rest frame, and τ is the proper time of the body, the time measured by a clock in its rest frame. This equation agrees with the Newtonian form in the low velocity limit as required by the correspondence principle. Moreover it is covariant under Lorentz transformations; if this law holds in one reference frame, then it holds in all reference frames.

The relativistic momentum p=mγc is the spatial part of p, the energy-momentum Minkowski vector and therefore F must also be the spatial part of a Minkowski vector, F. The full covariant relativistic version of Newton's second law must include the full four vectors:

$F={\frac {dp}{d\tau }}.$

Here we have the momentum-energy Minkowski vector

$p=(m\gamma c,\mathbf {p} )^{T}$

which satisfies

$p^{2}=m^{2}c^{2}$

from which we may infer

$F\cdot p=0.$

In the particle's rest frame, the momentum is (mc,0) and so for the force four-vector to be orthogonal, its time component must be zero in the rest frame as well, so F = (0,F). Applying a Lorentz transformation to an arbitrary frame, we find

$F=\left({\frac {\gamma }{c}}(\mathbf {F} \cdot \mathbf {v} ),\mathbf {F} +{\frac {\gamma ^{2}}{\gamma +1}}(\mathbf {F} \cdot \mathbf {v} )\right)^{T}.$

Thus the time component of the relativistic version of Newton's second law is

${\frac {\gamma }{c}}(\mathbf {F} \cdot \mathbf {v} )={\frac {d(m\gamma c)}{d\tau }}.$

Recalling the definition of work done by the applied force as

$W=\int \mathbf {F} \cdot \,d\mathbf {r} =\int \mathbf {F} \cdot \mathbf {v} \,dt,$

and since the change in energy is given by the work done, we have

${\frac {dE}{d\tau }}=\gamma \mathbf {F} \cdot \mathbf {v} ,$

and so finally we see that, up to an additive constant,

$E=m\gamma c^{2}.$

The energy is only defined up to an additive constant, so it is conceivable that we could define the total energy of a free particle to be given simply by the kinetic energy T = mc²(γ – 1) which differs from E by a constant, which is afterall the case in nonrelativistic mechanics. To see that the rest energy must be included, the law of conservation of momentum (which will serve as the relativistic replacement for Newton's third law) must be invoked, which dictates that the quantity mγc² = mc² + T be conserved and allows that rest energy can be converted into kinetic energy and vice versa, a phenomenon that is observed in many experiments.

See also

• Albert Einstein
• Celeritas for the origins of using the c notation in E=mc².
• Energy-momentum relation
• Inertia
• Mass-energy equivalence
• Mass, momentum, and energy
• Relativistic mass
• Spacetime

References

• Bodanis, David (2001). E=mc²: A Biography of the World's Most Famous Equation. Berkley Trade. ISBN 0425181642.
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0716743450.{{cite book}}: CS1 maint: multiple names: authors list (link)

External links

• [3] Physics FAQ discussion of the two kinds of masses in relativity, and which Einstein and today's physicists prefer.
• Happy 100th Birthday E=mc² BBC
• Edward Muller's Homepage > Antimatter Calculator
• Energy of a Nuclear Explosion
• Albert Einstein’s 27 September 1905 paper
• Einstein's 1912 manuscript page displaying E=mc²
• NOVA Einstein's Big Idea (PBS Television)
• Simple derivation of E=mc²
• E=mc² at IMDb
• Translation to english from the original article "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?"