E=mc²

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Image:E equals m plus c square at Taipei101.jpg Image:TaskForce One.jpg 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. In unit specific terms, E (Joules) = M (kilograms) · (299792458 m/s) 2.

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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 amount of (very large) 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 stored energy (such as chemical or thermal energy). That is why a body's 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 when the body is at rest.

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, correponding 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.

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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. To understand the significance of this relationship, compare the electromagnetic force with the gravitational force. In electromagnetism, energy is contained in the fields (electric and magnetic) associated with the force and not in the charges. In gravitation, the energy is contained in the matter itself. It is not a coincidence that mass bends spacetime, while the charges of the other three fundamental forces do not.

<math>\mathrm{Energy} = \mathrm{Mass}\,\times\,(\mathrm{speed\ of\ light})^2</math>

According to the equation, the maximum amount of energy obtainable from an object is the same as 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 enters and exits the reaction.

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

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Practical examples

A kilogram of mass completely converts into

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. In the equation, mass is energy, but for the sake of brevity, the word converted is used.

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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 obvects 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 <math>m_0</math>. 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 <math>E = mc^2 </math> is not in general correct for it, unless the mass is seen at rest.

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

<math>E = \frac{m_0 c^2}\sqrt{1-v^2/c^2},</math>

with <math> v </math> being the relative velocity. When <math> v=0 </math>, this become <math>E = {m_0 c^2}</math>, with E being the rest energy, <math>E_0</math>. This can be compared with the kinetic energy in newtonian mechanics:

<math>E = \frac{1}{2}m v^2</math>,

where <math>E_0 = 0</math>.

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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

<math>m_{\mathrm{rel}} \equiv \gamma m_0 \equiv \frac{m_0}{\sqrt{1-v^2/c^2}} . </math>

Using this form of the mass, we can again simply write <math>E=m_{\mathrm{rel}}c^2</math>, 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 <math>v=0</math> above, and get <math>m_{\mathrm{rel}}=m_0</math>.

Now, understanding the difference between rest mass and relativistic mass, we see that the equation <math>E = mc^2 </math> in the title must be rewritten: either <math>E = m_0 c^2 </math> for <math>v = 0</math>, or <math>E = m_{\mathrm{rel}} c^2</math> when <math>v \neq 0</math>.

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.

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Low-speed approximation

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

<math>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] . </math>

Higher-order terms in this expression (the ones farther to the right) get smaller and smaller since the velocity <math>v</math> is much smaller than <math>c</math>, so <math>v/c</math> is quite small. If the velocity is small enough, we can throw away all but the first two terms, and get

<math>E \approx m_0 c^2 + \frac{1}{2} m_0 v^2 . </math>

Thus, we see that Newton's form of the energy equation just ignores the part that Newton never knew about, the <math>m_0 c^2</math> part. This worked because Newton only saw objects move at speeds which were quite small compared to the speed of light, and never saw mass converted to energy, 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 mass could be converted to energy.

Interestingly, we could include the <math>m_0 c^2</math> 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 effect, as long as we don't allow things like nuclear reactions. Those "higher-order" terms that we left out show that Relativity is a high-order correction to Newtonian mechanics. The Newtonian version is actually wrong, but is close enough at low speeds.

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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, in this case, kinetic energy, and the mass being the ordinary concept of mass used in those times, the same one that we call today, rest energy or invariant mass, depending on the context.

It is the difference in the mass '<math> \Delta m\ </math>' before the ejection of energy and after it, that is equal to L/c², not the entire mass '<math> m\ </math>' of the object. At this moment it was only theoretical and not proven experimentally.

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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.

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Television biography

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

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See also

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References

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External links

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