# Rindler space in semi-general relativity

1. Oct 7, 2012

### Funzies

Hey guys, I am considering a Rindler space in which the metric is given by:
$$ds^2 = dx^2 - (dx^0)^2 = dw^2 - (1+gw/c^2)^2(dw^0)^2$$,
where $(x^0, x)$ are Minkowski coordinates in an intertial system I and $(w^0,w)$ the Rindler coordinates of a system of reference R with constant acceleration g relative to I.
From this I have derived the solution for a particle starting at w=w~ and with velocity zero for w^0=0:
$$w(w^0) = \frac{c^2}{g}\left( (1+g\tilde w/c^2)\frac{1}{\cosh(gw^0/c^2)} -1 \right)$$
The velocity in Rindler space is defined as $$v = c \frac{dw}{dw^0}$$. From this I have calculated several velocities, from which I am certain that they are correct, but the interpretation is rather lacking:
$$v_{\mathrm{particle}} = -c(1+g\tilde w/c^2)\frac{\sinh(gw^0/c^2)}{\cosh^2(gw^0/c^2)}$$
$$v_{\mathrm{particle, max}} = \pm c/2(1+g\tilde w/c^2)$$
$$v_{\mathrm{light}} = \pm c(1+g\tilde w/c^2)\frac{1}{\cosh(gw^0/c^2)}$$
$$v_{\mathrm{light, max}} = \pm c(1+g\tilde w/c^2)$$

Now for the interpretation:
-First of all: I presume the minus sign in the equation for the speed of the particle just reflects the fact that for w^0<0 it travels in one direction and for $w^0>0$ in another direction?
-I find it weird that the speed of light is not a fixed c. I know that the for light $ds^2 \equiv 0$, but I still don't find this answer rather comforting. Can anyone elaborate on this?
-In the limit $w^0 \rightarrow \infty$ the speed of the particle and of the light go to zero. I completely do not understand what is happening here.
-What does it mean that the maximum speed of light is two times larger than the maximum particle velocity?

2. Oct 7, 2012

### pervect

Staff Emeritus
It might be interesting for you to compare your equations with those of the relativistic rocket

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]

A few oddities (not necessarily problems) strike me about your treatment. You start out with assuming that c=1 to get the minkowski metric, but then put it back in (partially) in the Rindler metric. It'd be better (and possibly less misleading) to assume c=1 there and have only the one constant g (which would be in geometric units).

Also, you use $x^0$ for time but x rather than $x^1$ for distance. This isn't wrong - just funny lookig.

Anyway, you should be able to see right from your metric that $dw / dw^0$ can't possibly be constant just by the defintion that ds must be zero for a light beam.

"c" will always be constant if you use local clocks and local rulers, i.e. proper clocks and proper rulers. dw/dw0 represents a measurement based on coordinate clocks so it doesn't have to be (and in fact can't be) constant.

Knowing, however, that if you used proper clocks and proper rulers, that you would get "c", you can explain what coordinate clocks and rulers are telling you in terms of the more fundamental proper clocks.

Specifically, you can see that the relation between coordinate rulers and proper rulers isn't an issue in this case, it's the relation between the coordinate clocks and the proper clocks that causes dw/dw0 to vary.

The relation between coordinate clocks and proper clocks is commonly known as "time dilation" - just consider the case where when coordinate time at some location advances by 1 second, proper time (as measured by a clock on a worldline of constant spatial coordinates) advances by something other than one second.

I personally avoid calling quantities like dw/dw0 "speeds", I find it invites too much confusion.

Last edited by a moderator: May 6, 2017