Speed of light for a Rindler observer

[PeterDonis]

if now I use different coordinates defined as $\tau' = 2\tau$ you coordinate velocity changes by a factor 2, but my velocity stays the same, because there's also a factor 2 in the relation between "time" and $\tau'$ that cancels.

You have it backwards. If you want to remove the factor of 2 between your "time" and $\tau$, you need to define $\tau' = 0.5 \tau$. In other words, you want the $\tau$ intervals along your chosen worldline, the one with $\rho = 0.5$, to be marked off the same as the proper time intervals along that worldline; that means marking them off with the same scaling as $\rho_0$.

It is true that this rescaling of $\tau$ will change the coordinate speed by a factor of $2$, without changing your "speed", since you have defined your "speed" using the proper time scaling along your chosen worldline. Now try rescaling $\rho$.

 

[Gaussian97]

No, this is wrong. If you just rescale $\tau$ but don't change $\rho$, then your observer is still at $\rho_0 = 0.5$ and his proper time is still $\tau' / 2$. So, in these new coordinates, $\Delta \rho$ stays the same and $\Delta \tau$ gets multiplied by $2$, so coordinate speed is half what it was in the old coordinates. And the "speed" for your observer by your definition is still twice the coordinate speed, so it also is half what it was in the old coordinates.

In other words, in these new coordinates, the coordinate speed will be $\approx 0.6664$ and the "speed" for your observer by your definition will be $\approx 1.3328$.

Let me do it in detail, I change from $(\tau, \rho)$ to $(\tau', \rho)$, therefore my new metric is $ds^2=\frac{1}{4}\rho^2d\tau'^2 - d\rho^2$. And I'm still at $\rho=0.5$ so my proper time is related to $\tau'$ as follows:
$$ds^2=\frac{1}{4}\rho_0^2d\tau'^2 \Longrightarrow \Delta T = \frac{\rho_0}{2}\Delta \tau'$$
Now before the two event were separated by $\Delta \tau = 0.55 \Longrightarrow \Delta \tau'=1.1$. Therefore the time between the two events is still $\Delta T = \frac{0.5 \cdot 1.1}{2}=0.27$ as before, I haven't change the way I define distance, so velocity is still $2.7$.
However, as you say, coordinate velocity has changed from $1.33$ to $0.66$, just because we have changed how we describe mathematically the real world.

Now if I rescale $\rho'=2\rho$ now the metric becomes $ds^2=\frac{{\rho'}^{2}}{4}d\tau^2-\frac{1}{4}d\rho'^2$. Now of course I'm at $\rho'_0=2\rho_0=1$ and the proper time is
$$ds^2=\frac{{\rho'_0}^2}{4}d\tau^2 \Longrightarrow \Delta T = \frac{\rho'_0}{2}\Delta \tau = \frac{\Delta \tau}{2}$$. So again, the time between the two events is still $0.27$. But now we have also modify how we measure distance, because at constant times
$-ds^2=\frac{1}{4}d\rho'^2 \Longrightarrow \Delta L = \frac{\Delta \rho'}{2}$. So the two event are separated by $\Delta \rho' = 1.46$, which means a distance of $\Delta L = 0.73$, the same as before, if I compute the velocity I get, again $2.7$. And you're right now the coordinate velocity is in fact $2.7$ so both are equal, but I don't understand why you say that coordinate velocity stays the same while mine changes...

I challenge you to prove this.

Well, then probably I'm wrong, I haven't proved rigorously, but my intuition was something like this:
The proper time between two points of my worldline is just the relativistic distance between them, this is a scalar quantity. I also define the physical distance as the relativistic distance between two points at the same time (because we have a scalar notion of time, there no problem there, and because the relativistic distance is invariant, we should have an invariant "physical" distance). Therefore the velocity as I compute it would be invariant. As I said, this was my reasoning and probably it's wrong. But surely, for the easy cases that I've done before, I still don't understand why the coordinate velocity is better.

Everybody all along has been telling you that the things you are calculating are coordinate speeds, not physical speeds. Apparently you haven't been paying attention.

Well, I'm sorry, English is not my main language, and I really thought that everyone before message #17 was ok with my calculations.

 

[Dale]

until #17 nobody seem to have any problem with

I still don’t have a problem with it, but I would not call it a physical speed.

I would be fine calling it a coordinate speed even though it is not the standard Rindler coordinate speed because you can simply rescale the time coordinate to get coordinates where it is the standard coordinate speed.

 

[PeterDonis]

Let me do it in detail

I agree with your metrics and with the coordinates you obtain for both cases.

now we have also modify how we measure distance

This, however, is a new element. Before you had $\Delta L = \Delta \rho$. Now you have $\Delta L = \Delta \rho' / 2$.

So it appears that what you are actually trying to define is something like: $ds$ along a spacelike curve of constant time / $ds$ along a timelike curve of constant spatial location. And you are picking the curves and intervals as follows:

The spacelike curve is a curve of constant time between the event of emission of the light ray, and the spatial location where the light ray is received (but the event you end up at will not be the same as the reception event).

The timelike curve is the segment of your chosen observer's worldline between the event that is simultaneous, in your chosen coordinates, with the emission of the light ray, and the event that is simultaneous, in your chosen coordinates, with the reception of the light ray.

If you define things as above, then you are correct that the ratio of these two $ds$ values is an invariant, since both of them are arc lengths along well-defined segments of well-defined curves in spacetime, and arc lengths along specific curves in spacetime are invariants.

However, your interpretation of this ratio as a "physical speed" is still not correct, because it does not correspond to any direct observation that any observer would make. It is still a calculation that your observer does about events none of which take place on his worldline, meaning that he cannot observe any of them directly. Strictly speaking, I suppose, since your ratio, if defined as above, is a ratio of invariants, there would be some way of constructing a measurement that gave that ratio directly; but such a measurement would be obviously gerrymandered and would not meet any reasonable definition of a direct measurement of "physical speed".

Note also that the conditions under which it is even possible to find curves that meet your specifications is extremely limited. Notice the limitations in how your curves are defined: the observer whose worldline we are using has to be at rest in your chosen coordinates; and both of the spatial locations for the light ray (emission and reception) also have to be at rest in your chosen coordinates. Also, the "physical distance" between the two spatial locations, of emission and reception, has to be constant (since you're only calculating it at one time, the time of emission, so you are assuming that it doesn't change). That means there must be three timelike worldlines that can all be viewed as at rest relative to each other in suitable coordinates, and physical distances along spacelike curves of constant time in those coordinates must be constant in time. That is an extremely restrictive condition (the technical way of stating it is that all three curves must be integral curves of the same timelike Killing vector field). In most spacetimes, no such set of curves even exists.

 

[cianfa72]

So it appears that what you are actually trying to define is something like: $ds$ along a spacelike curve of constant time / $ds$ along a timelike curve of constant spatial location. And you are picking the curves and intervals as follows:

The spacelike curve is a curve of constant time between the event of emission of the light ray, and the spatial location where the light ray is received (but the event you end up at will not be the same as the reception event).

The timelike curve is the segment of your chosen observer's worldline between the event that is simultaneous, in your chosen coordinates, with the emission of the light ray, and the event that is simultaneous, in your chosen coordinates, with the reception of the light ray.

Just to point out that Peter is talking about the 'coordinate time' and 'coordinate spatial location' of events in the chosen coordinate system (Rindler coordinate system for flat spacetime).

Thus for instance in the sententence "The spacelike curve is a curve of constant time between the event of emission of the light ray, and the spatial location where the light ray is received" constant time is really constant coordinate time.

 

[PeterDonis]

in the sententence "The spacelike curve is a curve of constant time between the event of emission of the light ray, and the spatial location where the light ray is received" constant time is really constant coordinate time.

Yes, this is correct.

 

[pervect]

Well, I don't observe them, of course, but I know that they are occurring at those coordinates. Doesn't that allow me to say that the lightray has traveled 0.7 meters in 0.3 seconds? (Well, of course, I'm using $c=1$ so my units wouldn't be meters and seconds, but that's not the important here)

That is true, but only in the particular coordinate system you chose.

The assignment of coordinates (typically, though not always, the relativity of simultaneity) affects any sort of calculation that you make.

But, basically by definition (at least mine, I'm not sure where , or even if, you'd find a formal definition) "physical" thigns are indepenent of how you assign coordinates.

It's possible you have some diferent definition of the distinction we are making between "physical" quantities and "non-physical" or "coordinate" quantities? If so, you'd need to explain.

I assume you are familiar with the relativity of simultaneity - aka "Einstein's train?" If not, we might have to take a step back, as that (as usual) is the non-intuitive part of the problem. Typically, things like changing the scale factor also affect coordinate speeds, but we are used to that, and don't regard it as significant. But the simultaneity convention also affects coordinate speeds, and that is typically not intuitive.

 

[cianfa72]

Notice the limitations in how your curves are defined: the observer whose worldline we are using has to be at rest in your chosen coordinates; and both of the spatial locations for the light ray (emission and reception) also have to be at rest in your chosen coordinates. Also, the "physical distance" between the two spatial locations, of emission and reception, has to be constant (since you're only calculating it at one time, the time of emission, so you are assuming that it doesn't change). That means there must be three timelike worldlines that can all be viewed as at rest relative to each other in suitable coordinates, and physical distances along spacelike curves of constant time in those coordinates must be constant in time.

Not sure to grasp which are the two other timelike worldlines (the first is ok because it is just the observer worldline ) at the spatial location for the light ray (emission and reception) required to be at rest in the chosen coordinate system

 

[Gaussian97]

Ok, I think I see what's the problem, so you are saying that, in general, I can always define a proper time between two points of my worldline, but that this concept of "physical time" is not generalizable to any arbitrary points in space-time, right? And therefore is also not possible to define a consistent concept of distance between any two points, which make it impossible to define a consistent concept of "physical velocity".

But, basically by definition (at least mine, I'm not sure where , or even if, you'd find a formal definition) "physical" thigns are indepenent of how you assign coordinates.

It's possible you have some diferent definition of the distinction we are making between "physical" quantities and "non-physical" or "coordinate" quantities? If so, you'd need to explain.

I assume you are familiar with the relativity of simultaneity - aka "Einstein's train?" If not, we might have to take a step back, as that (as usual) is the non-intuitive part of the problem. Typically, things like changing the scale factor also affect coordinate speeds, but we are used to that, and don't regard it as significant. But the simultaneity convention also affects coordinate speeds, and that is typically not intuitive.

Yes, I actually have no formal definition of what I'm calling "physical" but for sure, for me is something that doesn't depend on the coordinate choice we use to describe the world.
About relativity of simultaneity, I would say that I understand it, at least in the case of inertial observers in Minkowski space, although I'm not sure about nothing now. But, as far as I understand, this simultaneity is something that appears when comparing the "physical" observables between two observers, right? Here I'm talking about a single observer, and I'm not sure how we would apply the relativity of simultaneity here.

 

[Dale]

I can always define a proper time between two points of my worldline, but that this concept of "physical time" is not generalizable to any arbitrary points in space-time, right?

Yes. You can define the invariant spacetime interval along any path between two endpoints. The issue is that it depends on both the path and the endpoints.

In this case the path of the light was fixed, but the endpoints were determined by your coordinates which makes the result coordinate dependent.

 

[PeterDonis]

Not sure to grasp which are the two other timelike worldlines

The worldlines of the things that emit and receive the light ray. @Gaussian97 is making implicit assumptions about what those worldlines are in his calculation, when he defines the coordinates $\rho$ of emission and reception and defines the distance between them as unchanging according to his chosen observer.

 

[PeterDonis]

this concept of "physical time" is not generalizable to any arbitrary points in space-time, right?

Yes. Proper time along your worldline only defines a "physical time" along your worldline. Extending that notion of "time" off of your worldline requires adopting a simultaneity convention, and there are always an infinite number of possible simultaneity conventions you can adopt.

is also not possible to define a consistent concept of distance between any two points

Yes, because this also requires adopting a simultaneity convention, since "distance" has to be between two points at the same time.

 

[PeterDonis]

this simultaneity is something that appears when comparing the "physical" observables between two observers, right?

No. Simultaneity is a convention that you must adopt in order to assign "time" values to events that aren't on your worldline, by picking, for each event on your worldline, the set of events off your worldline that happen at the same time as that event.

ere I'm talking about a single observer

But you're talking about events that aren't on that observer's worldline, so you have to adopt a simultaneity convention in order to assign a "time" value to those events.

 

[PeterDonis]

I would say that I understand it, at least in the case of inertial observers in Minkowski space

Discussions of simultaneity in SR often obscure the fact that it is a convention, by implicitly treating the simultaneity convention of global inertial frames in flat spacetime as though it were the only one that could be chosen. It isn't. Rindler coordinates make that clear: their simultaneity convention is different from that of inertial frames in flat spacetime. Note, for example, that the two "end" events in your scenario--the reception of the light ray, and the event on your chosen observer's worldline whose Rindler coordinate time $\tau$ is the same as that reception event--are not simultaneous in the global inertial frame in which your Minkowski coordinates $t$, $x$ are given.

 

[cianfa72]

Rindler coordinates make that clear: their simultaneity convention is different from that of inertial frames in flat spacetime. Note, for example, that the two "end" events in your scenario--the reception of the light ray, and the event on your chosen observer's worldline whose Rindler coordinate time $\tau$ is the same as that reception event--are not simultaneous in the global inertial frame in which your Minkowski coordinates $t$, $x$ are given.

To me this is a crucial point: a chosen coordinate system (Rindler coordinates in this case) defines the notion of simultaneity for events (points) separated in spacetime.

 

[PeterDonis]

a chosen coordinate system (Rindler coordinates in this case) defines the notion of simultaneity for events (points) separated in spacetime.

Yes.

 

[cianfa72]

The worldlines of the things that emit and receive the light ray. @Gaussian97 is making implicit assumptions about what those worldlines are in his calculation, when he defines the coordinates $\rho$ of emission and reception and defines the distance between them as unchanging according to his chosen observer.

Actually I assumed that the observer's worldline was the same as the worldline of the 'thing' emitting the light ray

 

[PeterDonis]

I assumed that the observer's worldline was the same as the worldline of the 'thing' emitting the light ray

It isn't. The observer is at $\rho = 0.5$. The emitter is at $\rho = 1$.

 

[cianfa72]

Also, the "physical distance" between the two spatial locations, of emission and reception, has to be constant (since you're only calculating it at one time, the time of emission, so you are assuming that it doesn't change). That means there must be three timelike worldlines that can all be viewed as at rest relative to each other in suitable coordinates, and physical distances along spacelike curves of constant time in those coordinates must be constant in time.

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.

 

[Dale]

the spacelike surfaces of constant coordinate time τ -- in the chosen coordinate system. ... it is actually the 'physical distance' we are talking about.

Sure, but that quantity clearly depends on the coordinates. Calling it a “physical distance” is suspect.

 

[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.