Exploring the Equation: Why is e=mc^2 and Why c is Squared?

[jhe1984]

In e=mc^2, why is c squared?

The (mc) isn't squared, right?

Please don't be insulted by the remedialness of this question: I simply don't know.

 

[masudr]

Try and get the units of the right hand side the same as the units of joules. Remember, a joule has the units of

$$[M L^2 T^{-2}].$$

 

[dextercioby]

Along the same line with masud, what's the expression for nonrelativistic KE ?

Daniel.

 

[James R]

To explain the above responses a little: this is called dimensional analysis.

If you think energy E depends on mass m and the speed of light c, then how must those things be combined? The most general way is like this:

$$E = km^\alpha c^\beta$$

where $\alpha$ and $\beta$ are constants to be determined, and k is a dimensionless constant (a number without units).

Now, energy, in SI units, is in Joules, and 1 Joule is 1 kilogram (metre/second)^2. The dimensions of energy are therefore mass.(length/time)^2, often written: $[M][L]^2[T]^{-2}$.

Similarly, the dimensions of m are: mass, or $[M]$
The dimensions of c are: (length/time), or $[L][T]^{-1}$

Putting these dimensions into the general equation, we get:

$$[M][L]^2[T]^{-2} = k([M]^\alpha)([L]^\beta [T]^{-\beta})$$

We want to solve for alpha and beta. Matching the dimensions on the left and right hand sides gives:

$$[M]: \alpha = 1$$
$$[L]: \beta = 2$$
$$[T]: -\beta = -2$$

Therefore, our expression must be:

$$E = kmc^2$$

This doesn't tell us what k is. k could be 1 or 17 or $pi$, or some other number. To find k, we need to derive the equation from physical arguments. If we do that, we find that k=1.

However, this argument is enough to show you that the speed of light must be squared.

 

[bernhard.rothenstein]

mc has the dimensions of momentum and has no physical meaning because a
tardyon never moves with speed c.
mcc has the dimensions of energy. the invariance of c makes that it transforms as mass does. avoiding the concept of mass. the concept of relativistic energy mcc and rest energy avoids the endless discussion concerning the use of the concept of relativistic mass.

 

[Mk]

The tardyon Mr. Rothenstein is referring to is a theoretical particle with imaginary (i) rest mass, and therefore travels faster than light.

 

[bernhard.rothenstein]

A tardyon or bradyon is a particle that travels slower than light. This includes all known particles (except luxons). The term "tardyon" is constructed to contrast with "tachyon", which refers to hypothetical particles that travel faster than light.

Tardyons have positive masses, whereas luxons have zero mass.
(quoted from Wikipedia)

 

[bernhard.rothenstein]

A tardyon or bradyon is a particle that travels slower than light. This includes all known particles (except luxons). The term "tardyon" is constructed to contrast with "tachyon", which refers to hypothetical particles that travel faster than light.

Tardyons have positive masses, whereas luxons have zero mass.

 

[Mk]

I apologize, I got them mixed up.

Tardyon speed < c
Luxon speed = c
Tachyon speed > c

 

[bernhard.rothenstein]

there are authors who consider that p=mc works in the case of a luxon (photon) p and m representing its momentum and mass repectively.