# Newbie question: Why is the speed-of-light squared equal to energy/mass?

1. Dec 18, 2007

### trewsx7

Hello all! I am a newbie here, and even though I have a general interest in cosmology and theoretical physics, I am by no means an expert nor do I even possess a firm grasp on the fundamentals of physics. I am a political science/international affairs major and thus I am largely ignorant of the higher mathematics needed to understand higher physics.

My question is this: in layman's terms, why is it that Einstein's famous equation showing that matter is just concentrated energy (E=mc2) and matter-energy can be converted into eachother, why is it that it so happens to be the speed-of-light squared (times mass) that is equal to energy??? What is so special about the speed of light, which is just how fast light photons move?

Why isn't it 122,000 miles per second squared x mass = energy or 1,000,0000 miles per second squared x mass = energy???

What, exactly, is so darn special about the speedlimit of photon particles that somehow fits into an perfect equation relating to energy/mass equivalence? Is this just a miraculous, unbeliavable coincidence that the number just so happens to be the speed of light that is squared?

2. Dec 18, 2007

### Janus

Staff Emeritus
To start off, c isn't just the speed limit for light, it is the speed limit for everything.

What is so special about c is that it is a speed that everyone measures as the same relative to themselves. For instance if you turn on a flashlight you will measure that the photons leaving the light travel at c with respect to you. Someone running past you in the same direction as you shine the light will measure the same photons as traveling at c with respect to himself, as will someone running in the opposite direction.

As a result of this all three of theses people will measure time and space differently.(each will measure the others as being lenght contracted in the direction of travel and as aging slower.)

How does this relate to energy? well energy is measured in units of Mass times Distance squared divided by Time squared.

If you use the time and space distortions mentioned above to determine the energy of an object moving realtive to you, you get a equation, that when a velocity of zero is plugged in leaves a value of E = mc^2.

So there is a logical progression from the speed limit of c and mass-energy conversion.

Of course, I glossed over a lot of details above, but this is the general gist of it.

3. Dec 18, 2007

### JeffKoch

I think you're thinking of it backwards, what's true is that the rest energy of a massive object is mc^2. This is how you define rest energy, so saying it's a miraculous coincidence that E/m = c^2 is putting the chicken before the egg, so to speak. A subtle but important distinction. As to "why" c appears in the equation rather than some other number, I'm not sure it's really relevant since the equation works and can predict the outcomes of experiments, but I believe it's the only fundamental frame-invariant speed we know of, and therefore the only speed we can use in the equation (it has to be a speed for the units to work out) that produces a frame-invariant amount of rest energy. So it's logical that the equation should contain c. The total energy is frame-dependent, however, since that is gamma*m*c^2, and gamma contains a velocity relative to an observer frame.

4. Dec 18, 2007

### Mephisto

it kinda comes in from special relativity. Einstein postulated that the highest speed a particle with some mass can travel is the speed of light. You cant travel faster than that. Keep in mind that he didn't find this out from anywhere. He simply supposed that its true, and then went on to derive lots of things from that, and it turns out that all of the stuff is correct if you assume that speed of light is indeed the maximum. The most amazing thing about it is that it is maximum in ALL reference frames. This gets a little more tricky to explain in Layman's terms... think of it this way:
If i ride down the street at 10 km/h and a car comes from ahead at 20km/h, YOU will actually see it approaching at 30km/h right? That's correct, but it turns out its correct only for really small speeds. But now if you are in spaceship traveling 90% of speed of light, and there is another spaceship traveling towards you at 50% speed of light, you will NOT see it travel towards you at 140% speed of light, because nothing can travel that fast. (if you do the math the result ends up being that you see him approach at like 99.9% speed of light). Nothing can ever travel faster than speed of light, in ANY reference frame... no matter how fast you are going, what direction, etc, etc. Einstein postulated this simple (not really intuitive) fact, and all this math popped out that noone was able to disprove yet.
Therefore all of his equations contain the speed of light, c, as a limit of the highest speed you can travel at. Then you can do some math, plugging from one equation to another, and at the end everything cancels, but you are left with E = mc^2

hope that clears it up a little

5. Dec 19, 2007

### rbj

from what i understood from many previous discussions, c is the speed of propagation of any "instantaneous" interaction. if the interaction is E&M, then c is the speed of E&M or, more commonly, the "speed of light". if the interaction is gravity, then c is the speed of gravity. i s'pose c is the speed of the strong and weak nuclear interactions also, but the distances involved there are so small, I don't think anyone could measure it.

i don't think that any physical interaction propagates at a speed any faster, lest information can move faster than c.

anyway, to address the OP's question, from how it was taught to me in the SR section of my intro to modern physics class, it's this: when relativistic considerations are made regarding the Kinetic Energy of a body, the equation for KE comes out to be:

$$KE = m c^2 - m_0 c^2$$

where

$$m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$$

and is sometimes called the "relativistic mass".

anyway the top equation was interpreted as

$$m_0 c^2 + KE = m c^2$$

or

$$E_0 + KE = E$$

rest energy plus kinetic energy equals the total energy of the body.

where

$$E_0 = m_0 c^2$$
and
$$E = m c^2$$

6. Dec 19, 2007

### vanesch

Staff Emeritus
First of all, Einstein's formula doesn't say that "mass is concentrated energy" and all that. In fact, the relationship E = m c^2 is an abreviation of a more fundamental relationship, which is the dispersion relation of particles:

E^2 = m0^2 c^4 + p^2 c^2

which gives us a relationship between a particle's energy (E) and a particle's momentum (p). When you look at a particle at rest, p = 0, and you find back the famous E = m c^2. Why is the dispersion relation the way it is ? The dispersion relation fixes the kinds of "world lines" one can have that respect the symmetries of space and time, and there is a free parameter in it, namely m0. That is: for every world line, you can pic a number, m0, and the dispersion relation gives you then an acceptable world line that respects the symmetries of spacetime.

As such, the dispersion relation is a fundamental property of the symmetries of space and time, and "energy" is nothing else but something which has to do with translations in time, while "momentum" is something which has to do with translations in space. Our spacetime is such, that if you "shift" everything 1 second forward, that everything is going to be the same. In the same way, if you shift everything one cm to the left, again, everything will stay the way it is. It is a *symmetry* (or an assumed symmetry) of space and time. This was actually already known since Galileo, but the implications of this symmetry weren't established until the end of the 19th century.

In fact, there is still a "freedom" of choice when just stating the above, and most people before Einstein fixed that freedom by saying that space and time were distinct and independent entities. If you do that, then you find the following dispersion relation:

E = p^2/(2m).

Again, m is a free parameter.

But in the 19th century, people established the laws of electromagnetism, and darn, they didn't seem to obey the symmetries of space and time the way people thought about them until then, and Einstein found the culprit: the assumption of the independence of time and space. The reason was that the Maxwell equations of electromagnetism seemed to define an "absolute velocity", something which is at odds with the symmetries of space and time at first sight - but Einstein could restore this by assuming that space and time were aspects of a single entity, spacetime. As such, he could KEEP the symmetries of spacetime (the translation symmetries), and KEEP the Maxwell equations with its fixed speed in it.

But this modified the dispersion relation (and many other things), from E = p^2/(2m) into E^2 = m0^2 c^4 + p^2 c^2.

The whole working out of this is what's called special relativity.

So Einstein's relation is just a consequence of the symmetries of space and time the way we understand it in the frame of relativity. If you want to know why it is this way, ask yourself why it was E = p^2/(2m) in classical physics...