# Acceleration in Special Relativity

1. Oct 21, 2005

### masudr

It greatly annoys me every time I read somewhere that Special Relativity deals with objects moving at constant speed and the generalisation of this is General Relativity which considers all kinds of motion. What's even more annoying is that you can find such nonsense written in all kinds of scientific literature, from lecture notes, to popular science to websites etc.

This is, of course, ridiculous. Special relativity easily deals with 4-forces and accelerations (well, the second derivative of the position four-vector with respect to proper time), and so on.

The correct distinction is that SR deals with non-gravitational physics in inertial reference frames.

The generalisation of this to non-inertial reference frames is straightforward really (and this isn't GR): replacing $\partial_x$ with covariant derivatives and so on. However in doing so we see some terms appear in our equations of motion (which are related to the curvature tensor).

What GR does is it associates these extra terms (and so the curvature tensor and it's various contractions) with the stress-energy tensor, and so finally solves the problem of gravitation in the relativistic limit, which is valid in all frames, inertial or not.

Apologies for the rant, but it annoyed me enough and this has helped to vent my anger.

2. Oct 22, 2005

### Mortimer

Agreed that SRT can handle instantaneous accelerations and forces with it's 4-vectors but I believe that certain aspects of such frames can not be explained properly by SRT's Lorentz transforms. These come forward in e.g. the clock behaviour during periods of acceleration that is also the basis for endless disussions on the twin paradox (for an example see John Baez' page at http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html#gap).

Last edited: Oct 22, 2005
3. Oct 22, 2005

### masudr

The twin "paradox" is easy to resolve in SR - the proper time integral is path dependent (i.e. the path in spacetime) and if you integrate the required expression along two paths you will see that one ages more than the other, as stated in Carroll's lecture notes on GR.

4. Oct 22, 2005

### jcsd

Essientially what you are doing in SR when you look at non-inertial frames is examining a certain class of non-constant coordinate bases (which mean the Christoffel symbols do not vanish). You are still doing this on a manifold with no curvature, it is the coordinate system that is 'curved'.

5. Oct 22, 2005

### pmb_phy

Whether this is GR of not will depend on how "GR" is defined. Einstein defined SR as relativity in inertial frames. I always take Einstein's definitions. What you've done is to take a principle of GR and call it SR. I.e. when you replaced partial derivatives with the covariant derivative (i.e. comma goes to colon rule) do you know what this is called? Its called the strong form of the equivalence principle. This is one of the basic postulate of GR. Another basic postulate of GR has to do with the law which says that the laws of nature must be able to be expressed in tensor form. There are plenty of reasons I go by Einstein's definition but I think I've debated that here before and I'm not healthy enough to want to repeat that debate right now.

So basically what you've done is to define GR away to meet your preference. But you're correct in saying that things like particle acceleration as observed in an inertial frame belongs to SR.

Pete