# Hm... well... 12 replies

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Chris

I pretend I'm cooler than AzH

50 XP

20th February 2006

5,663 Posts

#1 13 years ago

In physics E = mc2 is an important and well-known equation, which states an equivalence between energy (E) and relativistic mass (m), in direct proportion to the square of the speed of light in a vacuum (c2). The equation was first published (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 formally 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.

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 anihilation) 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.

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. 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 experimentors 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 7Li + p → 2 4He) 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).

A kilogram of mass could (theoretically) convert completely into

• 89,875,517,873,681,764 joules (exactly) or
• 24,965,421,632 kilowatt-hours or
• 21.48076431 megatons of TNT

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.

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 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. In such cases, the equation must become more complicated as the energy changes, since momentum terms must be added so that the mass remains constant from any reference frame. Alternative formulations of relativity, see below, allow the mass to vary with energy and ignore momentum, but this involves use of 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 m0. 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 = mc2 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 with v being the relative velocity. This can be shown to be equivalent to with p being the relativistic momentum (ie. p = γp0 = mrel * v). When v = 0, then p = 0, and both formulas above reduce to E = m0c2, with E now representing the rest energy, E0. This can be compared with the kinetic energy in newtonian mechanics: , where E0 = 0 (in Newtonian mechanics only kinetic energy is treated, and thus "rest energy" is zero).

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 Using this form of the mass, we can again simply write E = mrelc2, 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 mrel = m0. Now, understanding the difference between rest mass and relativistic mass, we see that the equation E = mc2 in the title must be rewritten: either E = m0c2 for v = 0, or E = mrelc2 when . 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.

We can rewrite the expression above as a Taylor series: 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 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 m0c2 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 m0c2 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.

Newton's second law as it appears in nonrelativistic classical mechanics reads 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 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: Here we have the momentum-energy Minkowski vector which satisfies p2 = m2c2 from which we may infer 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 Thus the time component of the relativistic version of Newton's second law is Recalling the definition of work done by the applied force as and since the change in energy is given by the work done, we have and so finally we see that, up to an additive constant, E = mγc2. 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.

... :hump:?

Sh0wdowN

Skeptic Extraordinaire.

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31st December 2003

7,942 Posts

#2 13 years ago

So, nice weather?

DavetheFo

RogueDevil / Rogue Angel

50 XP

29th May 2003

8,689 Posts

#3 13 years ago

Actually, I proved that wrong in chemistry today.

E=mc7

116,900 XP

10th April 2006

11,075 Posts

#4 13 years ago

I agree with the Doctor on this one. He has more experience in this sorta stuff...

Cap'n Rommel

The Good

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7th August 2004

8,766 Posts

#5 13 years ago

looks like a post I posted as an answer for something a while ago

Edit: No wait, that was about anit gravity, I think

Guest

I didn't make it!

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#6 13 years ago

penis

Cap'n Rommel

The Good

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7th August 2004

8,766 Posts

#7 13 years ago

correct

116,900 XP

10th April 2006

11,075 Posts

#8 13 years ago
Cap'n Rommelerect

that's better :)

Chris

I pretend I'm cooler than AzH

50 XP

20th February 2006

5,663 Posts

#9 13 years ago

reflect.

Guest

I didn't make it!

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#10 13 years ago
Chris Downreflect.

lies

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