I Speed of light for a Rindler observer

[PAllen]

Here with "physical distance" between the two spatial locations I understand the following:

Consider the timelike worldlines of the 'things' emitting and receiving the light ray, respectively. They cross in two points (events) the spacelike surfaces of constant coordinate time $\tau$ -- in the chosen coordinate system. Then take the spacelike curve on each of these spacelike surfaces joining the points of intersection above.

The spacetime 'length' along these spacelike curves is required to be constant in coordinate time (in other words on each of the above spacelike surfaces) and it is actually the 'physical distance' we are talking about.

Consider two comoving inertial world lines in special relativity. Consider just standard inertial coordinates. This 'physical distance' so defined, between these two bodies, can take any value in (0,L] , where L is the proper distance between them. Note, that in all such coordinates this so called physical distance is constant in coordinate time.

 

[cianfa72]

Consider two comoving inertial world lines in special relativity.

Do you mean two straight worldlines having same 'velocity' in standard inertial coordinates (assuming of course flat spacetime) ?

This 'physical distance' so defined, between these two bodies, can take any value in (0,L] , where L is the proper distance between them

As far as I can understand, here the point is that given a spacelike surface and taken two points (events) on it the set of spacelike curves on it joining them have a infimum 'lenght' of 0. Se for instance here

 

[PAllen]

Do you mean two straight worldlines having same 'velocity' in standard inertial coordinates (assuming of course flat spacetime) ?

Yes

As far as I can understand, here the point is that given a spacelike surface and taken two points (events) on it the set of spacelike curves on it joining them have a infimum 'lenght' of 0. Se for instance here

Well, (0,L] means the interval from >0 but not including zero to L, inclusive. In any given inertial frame, the distance is constant, but different inertial frames can have any distance value in this interval.

 

[cianfa72]

Well, (0,L] means the interval from >0 but not including zero to L, inclusive. In any given inertial frame, the distance is constant, but different inertial frames can have any distance value in this interval.

ok, thus I believe the scenario is the following (in blue the two worldlines):

In all inertial frames (coordinates) the 'distance' between worldlines -- as defined in the above posts-- is actually in the range (0,L]. In the inertial frame in which the two worldlines are (both) at rest that 'distance' is maximum (= proper length = L).

 

[PAllen]

Yes.

 

[haushofer]

I shared them before somewhere else, but recently I've written up some notes on Rindler observers and why they perceive a varying speed of light. Maybe they help. :) They still contain some slight errors, as a user already noted. Having a milk-craving creature recently at home has obstructed my scientific duties concerning decent-notes writing :P

 

[lerus]

Sorry for joining late to this discussion, just trying to verify that I understood correctly:
Rindler metric:
${ds}^2 = -(\alpha x)^2 {dt}^2 + {dx}^2$
for light cone we have: ${ds} = 0$
It means that: $\frac {dx} {dt} = \alpha x$
That is a coordinate speed, it tells us how fast coordinate x changes with coordinate t, it depends on x and could be more or less then 1.
The physical speed is the rate of change of physical distance, $l$ , in the physical time $\tau$

${dl} = {\sqrt(g_{11})}{dx} = {dx}$

${d\tau} = {\sqrt(g_{00})}{dt} = \alpha x {dt}$

$\frac {dl} {d\tau} = \frac {\sqrt(g_{11}){dx}} {\sqrt(g_{00}){dt}} = 1$
It means that speed of light = 1.

Is it correct?

 

[pervect]

Sorry for joining late to this discussion, just trying to verify that I understood correctly:
Rindler metric:
${ds}^2 = -(\alpha x)^2 {dt}^2 + {dx}^2$
for light cone we have: ${ds} = 0$
It means that: $\frac {dx} {dt} = \alpha x$
That is a coordinate speed, it tells us how fast coordinate x changes with coordinate t, it depends on x and could be more or less then 1.
The physical speed is the rate of change of physical distance, $l$ , in the physical time $\tau$

${dl} = {\sqrt(g_{11})}{dx} = {dx}$

${d\tau} = {\sqrt(g_{00})}{dt} = \alpha x {dt}$

$\frac {dl} {d\tau} = \frac {\sqrt(g_{11}){dx}} {\sqrt(g_{00}){dt}} = 1$
It means that speed of light = 1.

Is it correct?

Yes. You've got it.