Can someone prove E = m c^2 ?

Question

I want a detailed mathematical proof so please send a Web reference because equation editing must be hard on this system.

Answer 1

In relativity, the transformation of coordinates between coordinate systems at relative speed v has the form

t' = g(t - Bx/c)
x' = g(x + Bct)

Here, B is the ratio B = v/c (between 0 and 1), and g = 1/sqrt(1 - B^2) (between 1 and infinity).

For a Lorentz covariant theory, all physical vectors must obey this same transformation law. In particular, the "momentum four-vector" (E/c, p)

must obey the relations

E' = g(E - Bpc)
p' = g(p + BE/c)

Any moving particle is at rest in its own coordinate system: p = 0. We will assume that it has a "rest energy" E = E0. For a stationary observer who sees the particle move at speed v = Bc, the transformation formula shows

E' = g(E - Bpc) = g E0
p' = g(p + BE/c) = g B E0/c

The observed energy E' is greater than the rest energy E0, because for the observer the particle has kinetic energy. Classically, the total energy would be

E' = E0 + Ek = E0 + 1/2 mv^2

However, relativity tells us that

E' = g E0 = E0/sqrt(1 - B^2)

At least for low velocities (B << 1), the two theories should agree. Let us therefore expand E' = g E0 in powers of B:

E' = g E0 = E0 . (1 + B^2/2 + B^4/4 + ...)

The terms with B^4 and higher powers can be omitted for anything with low velocities. We are left with

E = E0 . (1 + B^2/2) = E0 + 1/2 E0 B^2

The second term must be the kinetic energy:

1/2 E0 B^2 = 1/2 m v^2
1/2 E0 B^2 = 1/2 m (cB)^2
1/2 E0 B^2 = 1/2 m c^2 B^2

Divide by 1/2 B^2 to find

E0 = m c^2

In order for the relativistic idea of energy to agree with the classical definition, this must be the rest energy of a particle.

Answer 2

Is this what you are looking for??? The Formula:
http://www.karlscalculus.org/einstein.html

The Proof:
Yes, It's called an "Atom Bomb". Just in case you have been living under a rock for all the years you've been alive check out these websites to get up to date LOL:

E=mc² was derived from Einsteins Theory of Special Relativity which was the foundation for the development of the Atomic Bomb.

http://en.wikipedia.org/wiki/E%3Dmc%C2%B2

http://www.firstscience.com/SITE/articles/kaku.asp

So, the Atomic Bomb pretty much proves E=mc², if you have doubts please stand near the next test detonation and get back to us with your findings LOL *-)

Here are some pictures for your viewing pleasure: http://zvis.com/nuclear/nukimgs.shtml

The YESO pics (near bottom of page) are pretty awesome, even though they are "technically" Tritium Gadgets.

Video of the Largest Atomic Bomb ever made and detonated: http://www.militaryfix.com/videos/worlds-largest-nuclear-explosion/

Answer 3

1. Take an electron.
2. Shoot it at a positron, its anti-particle.
3. Measure energy.
4. Check in books for the mass of the 2 particles.
5. Check if e=mc^2

Answer 4

If I accelerate the mass of my fist the energy produced will hurt you. So in space the relative answer is me jumping do to gravity's effect on acceleration and producing enough energy to do the same under loss of revolution and still hurting you.

Answer 5

Does this help? Its kind of an online reader thing where you can read this book without having to buy it.

Answer 6

Ask Einstein.

Answer 7

Ask your local Combine official and take what ever drug he perscribes to you.

Answer 8

http://en.wikipedia.org/wiki/E%3DMC2

Answer 9

E = mc2 is one of the most famous equations in physics, even to non-physicists. It states a relationship between energy (E), in whatever form, and mass (m). In this formula, c², the square of the speed of light in a 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.

The equation was first published in a slightly different formulation by Albert Einstein in 1905 in one of his famous articles. He derived it as a consequence of the special theory of relativity which he had proposed the same year.

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. 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.


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%.

Energy=mass *(speed of light)^2 in vacua

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 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 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 (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 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 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 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 constant rest energy of the object, and will not change with the object's motion).

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