# Derivation of the Equation for Relativistic Momentum

1. Feb 8, 2007

### NanakiXIII

I asked a quite similar question about relativistic mass and the reason for this question is identical: I can't seem to dig up any derivation for the equation for relativistic momentum:

$$p=\gamma mv$$

If anyone could point me in the right direction, I'd much appreciate it.

2. Feb 8, 2007

### nakurusil

Because this is a definition. And so is the relativistic total energy:

$$E=\gamma mc^2$$

3. Feb 8, 2007

### NanakiXIII

I see. Why is it defined as such, exactly? There must be a reason for adding the so frequently seen gamma, no?

Another question on the side: I've seen two different versions of the equation for the mass-energy equivalence, one with and one without the gamma factor. What's the difference?

4. Feb 8, 2007

### nakurusil

I explained that to you in the other thread (relativisic mass) a few minutes ago.

Same reason as momentum, I also explained that to you in the other thread.

5. Feb 8, 2007

### NanakiXIII

You mentioned the following:

You mean the reason for the gamma is because of the use of relativistic mass? And yet in the other thread you tell me to base relativistic mass on relativistic momentum. That's circular. Are you saying that relativistic mass is a definition and thus relativistic momentum is as well? That would still leave the question of "why the gamma?".

6. Feb 8, 2007

### nakurusil

"Relativistic mass" is just a misnomer, ok? Try to learn to forget about it, it has no meaning. Tolman's derivation from the other thread has no "relativistic momentum" in it, ok?

Can't you read ? Relativistic momentum is a definition.
"Relativistic mass" is an unfortunate misnomer that corresponds to the result of multiplying $$\gamma$$ by proper mass. It has no physical meaning.

7. Feb 8, 2007

### robphy

Given the 4-momentum vector $$\tilde P$$,
an inertial observer (with unit 4-velocity $$\tilde t$$
can write that vector as the sum of two vectors, a "temporal" one parallel to $$\tilde t$$ and a "spatial" one perpendicular to $$\tilde t$$.
$$\gamma$$ is $$\cosh{\theta}$$, the analogue of cosine(angle) , and $$\gamma\beta$$ is $$\sinh{\theta}$$, the analogue of sine(angle), where the [Minkowski-]angle $$\theta$$ (called the rapidity) is between $$\tilde P$$ and $$\tilde t$$.

8. Feb 8, 2007