General relativity: Rindler space problems

[Funzies]

Homework Statement

I have derived the metric in for 2D Rindler space in a previous problem and it is explicitly given again here:
$$ds^2 = dx^2 - (dx^0)^2 = dw^2 - (1+gw/c^2)^2(dw^0)^2$$,
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.
In the question I have furthermore derived the solution for a particle starting at w=w~ and with velocity zero for w^0=0:
$$w(w^0) = \frac{c^2}{g}\left( (1+g\tilde w/c^2)\frac{1}{\cosh(gw^0/c^2)} -1 \right)$$

2. Question
Calculate the velocity $$v = c \frac{dw}{dw^0}$$. Find the maximum velocity of the particle and compare with the speed of light (calculate that too). What happens for $w^0 \rightarrow \infty$?

The Attempt at a Solution

I have solved all the questions and I am sure they are correct; only the interpretation is 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?

 

[clamtrox]

-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?

Yeah that would seem to make sense; in particular it depends on whether the directions of initial velocity and acceleration are same or opposite.

-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?

But the Rindler coordinates are not inertial, so there's no reason to expect things like velocities to follow usual rules of special relativity. For example, imagine a coordinate system which is rotating, in an ordinary Minkowski space. Now, test particles which are stationary in ordinary Minkowski coordinates appear to be moving at a velocity proportional to their distance from you.

-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.

Can you show the coordinate transformations? What's w⁰ in terms of x and t of Minkowski coordinates?

-What does it mean that the maximum speed of light is two times larger than the maximum particle velocity?

Probably nothing. Note that the maximum is evaluated at different coordinate values; maximum for speed of light is at w⁰=0 while maximum for massive particles is when w⁰ has some finite value.

 

[Funzies]

But the Rindler coordinates are not inertial, so there's no reason to expect things like velocities to follow usual rules of special relativity. For example, imagine a coordinate system which is rotating, in an ordinary Minkowski space. Now, test particles which are stationary in ordinary Minkowski coordinates appear to be moving at a velocity proportional to their distance from you.

What do you mean by this? Before I saw Rindler space I always believed that special relativity could not cope with accelerating frames, but apparently it can, if you define the frame at every proper time tau. I am a bit confused by all this.

Can you show the coordinate transformations? What's w⁰ in terms of x and t of Minkowski coordinates?

I have the following relations:
$$x = c^2/g(\cosh(gw^0/c^2)-1) + w\cosh(gw^0/c^2)$$
$$x^0 = c^2/g\sinh(gw^0/c^2) + w\sinh(gw^0/c^2)$$.

 

[clamtrox]

What do you mean by this? Before I saw Rindler space I always believed that special relativity could not cope with accelerating frames, but apparently it can, if you define the frame at every proper time tau. I am a bit confused by all this.

Well, there's nothing stopping you from making a coordinate transformation to an accelerating coordinate system, but the transformation is different from the usual Lorentz transformations. Or, as you say, you need to do an infinite number of them. This means that the accelerating frames do not respect same symmetries as inertial frames in SR. For example, speed of light need not be constant. I think the rotating frame serves as a good example. There all photons which are sufficiently far away move at an arbitrarily high speed. This should be pretty obvious once you think about it for a while.

I have the following relations:
$$x = c^2/g(\cosh(gw^0/c^2)-1) + w\cosh(gw^0/c^2)$$
$$x^0 = c^2/g\sinh(gw^0/c^2) + w\sinh(gw^0/c^2)$$.

I don't have a good answer here, but you can see that the coordinate transformation is pretty tricky when w⁰→∞. In particular, you see that the hypersurfaces w=const. approach the null surface x=x⁰ in this limit. In some sense, this corresponds to an infinitely accelerated reference frame. Perhaps you can think this result so, that all particles (regardless of their initial velocity) travel at the same speed once they've been accelerated so much.

 

[paranormal]

I would respond to this content by providing a thorough explanation and analysis of the results. Here are some possible points that could be addressed in response:

- The derived metric for 2D Rindler space is a valid solution in the context of general relativity, and it describes the geometry of spacetime in a system with constant acceleration.
- The calculated velocity for the particle is correct, and it shows that the speed of the particle is always less than the speed of light. This is consistent with the principles of special relativity, which state that the speed of light is the maximum speed that can be reached by any object in the universe.
- The maximum velocity of the particle and the maximum velocity of light are both dependent on the value of g, the constant acceleration. This is because the Rindler coordinates are defined in relation to the acceleration, and therefore the speed of any object in this system will also depend on the acceleration.
- The minus sign in the equation for the speed of the particle reflects the direction of motion in relation to the Rindler coordinates. As the particle moves in the positive direction of w, its velocity will also be positive. Similarly, as it moves in the negative direction of w, its velocity will be negative.
- The fact that the speed of light is not a fixed c in this system is a result of the chosen coordinates and the effects of acceleration. In Rindler space, the speed of light will vary depending on the direction and magnitude of the acceleration. However, this does not contradict the fundamental postulate of special relativity that the speed of light is always constant in any inertial frame of reference.
- As w^0 approaches infinity, both the speed of the particle and the speed of light approach zero. This is because as time passes in Rindler space, the effects of acceleration become less significant and the velocities of objects in this system approach their values in an inertial frame.
- The fact that the maximum speed of light is twice the maximum speed of the particle is simply a result of the chosen coordinates and the effects of acceleration. In other coordinate systems or in the absence of acceleration, the maximum speed of light and the maximum speed of particles would be equal.
- It is important to note that these calculations and interpretations are specific to the context of Rindler space and may not necessarily apply to other systems or scenarios. As a scientist, it is important to consider the limitations and assumptions of any model