E=mc²

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

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

One of the most famous equations in the world, even to non-physicists, the theoretical physics equation E = mc² states a relationship between energy (E), in whatever form, and mass (m). In this formula, c², the square of the speed of light in vacuum, is the conversion factor required to formally convert from units of mass to units of energy, i.e. the energy per unit mass. In unit specific terms, E (joules) = M (kilograms) multiplied by (299792458 m/s) ².

Meanings of the formula

This formula, which was discovered by Albert Einstein, 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. Photons are generally considered to be "massless," (i.e., they have no rest mass or invariant mass) even though they have varying amounts of energy.

This formula also gives the quantitative relation of the quantity of mass lost from a resting body or an initially resting system, 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 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.

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

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.

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 can obtain an estimate of the binding energy available within an atomic nucleus. This can be 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).

Proof

In this proof $W$ represents Work (or Kinetic Energy), $m$ is relativistic mass (meaning it is a function of v, which is why I used two measures of v as my limits for the integrals), $m_{0}$ is rest mass, $c$ is the speed of light in a vacuum (Einstein's upper limit for velocity in his Special Theory of Relativity), $v$ is the velocity of the object, $E_{0}$ is rest energy, and $E$ is total energy.

We begin with the formula that yields the work that a force F produces while accelerating any mass on a infinitesimal distance ds.

$W=\int _{0}^{v}F\ ds$

We also know

$F={\frac {d}{dt}}mv$

Therefore,

$W=\int _{0}^{v}{\frac {d}{dt}}mv\ ds$

$W=\int _{0}^{v}{\frac {ds}{dt}}\ d(mv)$

$W=\int _{0}^{v}v\ d(mv)$

We know from the product rule that

$\ d(mv)=v(\ dm)+m(\ dv)$

$d(mv)=(v+m{\frac {dv}{dm}})dm$

Now we must find ${\frac {dv}{dm}}$

According to the Lorentz transformation for mass ($m$ is relativistic mass, $m_{0}$ is rest mass),

$m={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

where $c$ is the speed of light in a vacuum (Einstein's upper limit on velocity in his Special Theory of Relativity)

${\frac {dm}{dv}}={\frac {m_{0}v}{c^{2}({\sqrt {1-{\frac {v^{2}}{c^{2}}}}})^{3}}}$

${\frac {dm}{dv}}=({\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}})({\frac {\frac {v}{c^{2}}}{1-{\frac {v^{2}}{c^{2}}}}})$

${\frac {dm}{dv}}=m{\frac {\frac {v}{c^{2}}}{{\frac {c^{2}}{c^{2}}}-{\frac {v^{2}}{c^{2}}}}}$

${\frac {dm}{dv}}=m{\frac {v}{c^{2}-v^{2}}}$

To find ${\frac {dv}{dm}}$ we will simply find the reciprocal of ${\frac {dm}{dv}}$

${\frac {dv}{dm}}={\frac {c^{2}-v^{2}}{mv}}$

$d(mv)=(v+m({\frac {c^{2}-v^{2}}{mv}})dm$

$d(mv)=(v+{\frac {c^{2}-v^{2}}{v}})dm$

$W=\int _{0}^{v}v\ d(mv)$

$W=\int _{0}^{v}v(v+{\frac {c^{2}-v^{2}}{v}})\ dm$

$W=\int _{0}^{v}v^{2}+c^{2}-v^{2}\ dm$

$W=\int _{0}^{v}c^{2}\ dm$

$\Delta W=\Delta E=\Delta mc^{2}\,\!$

Therefore, the work done by the force acting on ds is the previous expression. ΔE is the change in energy due to kinetic energy, and Δm is the change in the object's relativistic mass associated with that energy.

This is very close to the form in which Einstein first introduced this equation in 1905, although his original derivation was concerned with rest mass change due to an increase in internal energy in a body, and not relativistic mass change due to a body's kinetic energy, which is the route used above, and which Einstein explored in 1907. Note that modern physicists often prefer not to use the relativistic mass term $m$ (see mass in special relativity) even though it allows Einstein's $E=mc^{2}$ equation to remain correct, even when $E$ is the total energy. In general, however, Einstein did not use $m$ as above and below, and his equation when used with ordinary mass $m_{0}$ is not correct, except in the special case of objects at rest, where $E=E_{0}$, the rest energy.

However, continuing to use the notation of $m$ as relativistic mass:

$W=c^{2}\left(m-m_{0}\right)$

$\,\!W=mc^{2}-m_{0}c^{2}$

At rest, $\,\!v=0$

$W=-m_{0}c^{2}\,\!$

but at rest, $W=0\,\!$

Therefore, $W=-m_{0}c^{2}+E_{0}\,\!$

$\,\!0=-m_{0}c^{2}+E_{0}$

$\,\!E_{0}=m_{0}c^{2}$

$E=W+E_{0}\,\!$

$\,\!E=mc^{2}-m_{0}c^{2}+m_{0}c^{2}$

$E=mc^{2}\,\!$

Practical examples

A kilogram of mass completely converts into

• 89,875,517,873,681,764 joules (exactly) or
• 24,965,421,632 kilowatt-hours or
• 21.48076431 megatons of TNT
• approximately 0.0851900643 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; for most cases, byproducts are produced instead of energy, and therefore very little mass is actually converted. For example, in nuclear fission ca. 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 ca. 0.3% of the mass of fused atoms is converted to energy.

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).

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 to objects which are not in motion, or systems of objects in which objects are moving, but in different directions such as to cancel momentums (such as a container of gas). The equation is a special case of a more general equation in which both energy and momentum are taken into account, so it is only correct for situations in which momentum cancels or is zero.

This equation applies to an object that is not moving as seen from a reference point. But this same object can be moving from the standpoint of other frames of reference. In such cases, the equation becomes more complicated, as the energy changes, and momentum terms must be added to keep the mass constant. (Alternative formulations of relativity, see below, allow the mass to vary with energy and ignore momentum, but this is a second definition of mass, called relativistic mass because it causes 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 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 constant rest energy of the object, and will not change with the object's 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 math might seem simpler if we define a different type of mass. The relativistic mass is defined by

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

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

For speeds much smaller than the speed of light, we can rewrite the expression above as a Taylor series:

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

Higher-order terms in this expression (the ones farther to the right) get smaller and smaller since 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

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

Thus, we see that Newton's form of the energy equation just ignores the part that Newton never knew about, the $m_{0}c^{2}$ part. This worked because Newton only saw objects move at speeds which were quite small compared to the speed of light, and never saw an object lose enough energy to measurably change its rest mass, as in a nuclear process. 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 high voltage 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 celestrial 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 (such as visible light or X-rays), 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. According to Umberto Bartocci (University of Perugia historian of mathematics), the equation was first published two years earlier by Olinto De Pretto, an industrialist from Vicenza, Italy, though this is not generally regarded as true or important by mainstream historians. Even if De Pretto introduced the formula, it was Einstein who connected it with the theory of relativity.

Television biography

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

See also

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

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