Speed of Light in Vacuum in Non-Inertial Frame

[cianfa72]

TL;DR

about the significance of speed to attach to the 'speed of light in vacuum' in the context of SR non inertial frame of reference

Hi,

I read various threads in PF about the concept of invariant speed and the speed of light in vacuum that in our universe happens to be the same as the 'invariant speed'.

My doubt is about the speed of the light in vacuum as measured from a non-inertial frame (basically in the context of SR a physical frame such that an accelerometer attached to it reads a non-zero acceleration).

First point: how can we define it in a non-inertial frame of reference ? I guess the only way to define it is via 'coordinate labels ' in order to get a 'coordinate speed' for it

Second point: does it exist an 'invariant way' to define it ?

Thanks.

 

[Orodruin]

First point: how can we define it in the non-inertial frame of reference ? I guess the only way to define it is via 'coordinate labels ' in order to get a 'coordinate speed' for it

Relative speed is generally defined at a single event and as a relation between the 4-velocities of two observers, i.e., the tangents of their world-lines. This does not depend on coordinates whatsoever. The relative speed between observers with 4-velocities $U$ and $V$, respectively, is given by
$$
\gamma = \frac{1}{\sqrt{1-v^2}} = V\cdot U = g(V,U),
$$
i.e., the gamma factor is the inner product between the 4-velocities. This is coordinate independent.

Second point: does it exist an 'invariant way' to define it ?

See the previous point.

 

[cianfa72]

$\gamma = \frac{1}{\sqrt{1-v^2}} = V\cdot U = g(V,U),$

what is the meaning of $v$ in this formula ?

 

[jbriggs444]

a non-inertial frame (basically in the context of SR a physical frame such that an accelerometer attached to it reads a non-zero acceleration).

[Here I will use "frame" as a synonym for "cartesian coordinate system"]
In special relativity, it is fairly straightforward to define an inertial frame. There are not a lot of choices. Once you've settled on a position, a start time and a state of motion for the origin, you are done. Rotation is locked in at zero and Einstein synchronization nails down the time coordinates for all of the space-time. I count seven degrees of freedom -- four for the coordinates of the origin and three for the state of motion of the frame. [We might quibble about counting handedness as another free choice]

For a non-inertial frame, life is not so simple. You have infinitely many degrees of freedom. You'll have to be more specific than "a physical frame".

 

[Orodruin]

what is the meaning of $v$ in this formula ?

The relative speed between the two observers.

For a light-like world line, it is characterised by having a tangent 4-vector $N$ for which $N\cdot N = N^2 = g(N,N) = 0$, which is also completely coordinate independent.

 

[cianfa72]

The relative speed between the two observers.

For a light-like world line, it is characterized by having a tangent 4-vector $N$ for which $N\cdot N = N^2 = g(N,N) = 0$, which is also completely coordinate independent.

Sorry, not sure to fully understand: relative to the observer (following a non straight path in flat spacetime and having 4-velocity U) which is the measured speed of a light ray ?

 

[cianfa72]

For a non-inertial frame, life is not so simple. You have infinitely many degrees of freedom. You'll have to be more specific than "a physical frame".

Take for instance the case of constant proper acceleration (Rindler reference frame): which would be the speed of a light ray in vacuum as measured by a Rindler observer ?

 

[Orodruin]

Take for instance the case of constant proper acceleration (Rindler reference frame): which would be the speed of a light ray in vacuum as measured by a Rindler observer ?

c

 

[jbriggs444]

Take for instance the case of constant proper acceleration (Rindler reference frame): which would be the speed of a light ray in vacuum as measured by a Rindler observer ?

Locally, the speed is still c. For an observer far from the horizon and a light ray near, the coordinate speed is near zero, of course.

 

[pervect]

Summary:: about the significance of speed to attach to the 'speed of light in vacuum' in the context of SR non inertial frame of reference

Hi,

I read various threads in PF about the concept of invariant speed and the speed of light in vacuum that in our universe happens to be the same as the 'invariant speed'.

My doubt is about the speed of the light in vacuum as measured from a non-inertial frame (basically in the context of SR a physical frame such that an accelerometer attached to it reads a non-zero acceleration).

First point: how can we define it in a non-inertial frame of reference ? I guess the only way to define it is via 'coordinate labels ' in order to get a 'coordinate speed' for it

Second point: does it exist an 'invariant way' to define it ?

Thanks.

In the context of SR, the answer can be simple. You have some accelerating and rotating observer O_a At any particular point P on the observer's worldline, you can create a non-accelerated observer moving at the same velocity by having them not accelerate and not rotate. This observer is the co-moving inertial observer O_i.

Then O_i has an inertial coordinate system, always measures the speed of light at the point P as "c", as we'll assume you know how to measure the speed of light in an inertial coordinate system.

The measurement of the speed of light at some other point than point P in the accelerated frame depends on exactly how you are defining what you mean by an accelerated frame. Presumably, you have some set of coordinates that you think of as representing "position" in the accelerated frame, and some other coordinate that represents "time" in the accelerated frame. If that's what you mean by an accelerated frame, the issue of the speed of light in your accelerated frame at some point that is not P depends on how you define the notion of "at the same time" in said accelerated frame. Presumably you do this by whatever time coordinate you use to represent time in your accelrated frame, then "at the same time" means the set of points with the same time coordinate. This in general doesn't have any physical significance, because it changes depending on how you define your "accelerated frame". Note that even for an inertial frame, different inertial frames have different notions of "at the same time". This is likely to be a stumbling block in your understanding, the issue is known as "the relativity of simultaneity".

If you're not famliiar with the reltivity of simultaneity, you may need to look into it. This is easier said than done - the idea itself isn't hard once you accept it, but it is notoriously hard to get people to accept it.

As an aside, if you are "measuring" the speed of light, you need some way to define the meter and the second, or more generally the unit of distance, and the unit of time. In the days past, this used to be a genuine measurement, because the meter stick was a physical artifact, and at that time the measurement of the speed of light was an actual measurement.

Nowadays things are a bit different, I can go into the details if you ask, but it may not be relevant to your question.

If you want to move to the context of GR, you probably want the idea of a manifold, representing the curved space-time, and the tangent space to the manifold. These concepts will also be helpful in a more rigorous treatment of the SR case as well. I'm not sure what background you have, or how deep you want to get into it.

 

[cianfa72]

The measurement of the speed of light at some other point than point P in the accelerated frame depends on exactly how you are defining what you mean by an accelerated frame. Presumably, you have some set of coordinates that you think of as representing "position" in the accelerated frame, and some other coordinate that represents "time" in the accelerated frame. If that's what you mean by an accelerated frame, the issue of the speed of light in your accelerated frame at some point that is not P depends on how you define the notion of "at the same time" in said accelerated frame.

Not sure to understand your argument: take Rindler coordinates in flat spacetime (SR). The measurement of speed of light at P for the Rindler accelerating observer O_a is the same as measured by co-moving inertial observer O_i at P itself. What about the measurement at some other point than P ?

 

[Nugatory]

The measurement of speed of light at P for the Rindler accelerating observer O_a is the same as measured by co-moving inertial observer O_i at P itself. What about the measurement at some other point than P ?

Look at the part of @pervect’s post #10 following the word “presumably” - (if that’s not what you meant, you’ll want to clarify what you did mean)...

To measure the speed of light at any point off our worldline, we must have some rule for assigning times to events happening off our worldline (we need this because any speed of light measurement is going to involve statements about the times of emission and detections, or the difference between them). Thus, our measurement attempt isn’t telling us anything about the speed of light, or even what an inertial observer at the point of measurement would consider the speed of light; the value we come up with was determined by how we assigned times to the remote detection and emission events.

We do not have this problem when measuring the speed of light at our own location because we can measure time along our worldline, and that is a frame-independent invariant.

 

[cianfa72]

We do not have this problem when measuring the speed of light at our own location because we can measure time along our worldline, and that is a frame-independent invariant.

In this case, namely using our wristwatch time to measure the time elapsed along our wordline, the measurement of speed of light has to involve a round-trip path between emission and detection of it (at out own location) I believe..

 

[jbriggs444]

In this case, namely using our wristwatch time to measure the time elapsed along our wordline, the measurement of speed of light has to involve a round-trip path between emission and detection of it (at out own location) I believe..

It involves assigning a coordinate for an event remote from the world line of your wristwatch. Round trip time constrains the assignment of time coordinates to the remote event -- if you can send a signal from your world line to the event and get a response back, the time coordinate of the event has to lie between the wristwatch time when the signal was sent and the wristwatch time when the response was received.

Einstein synchronization means using an inertial watch and assigning the mean of the sent time and the response time.

Other conventions are possible.

 

[Nugatory]

In this case, namely using our wristwatch time to measure the time elapsed along our wordline, the measurement of speed of light has to involve a round-trip path between emission and detection of it (at out own location) I believe..

It is a postulate that the speed of light is the same in both directions. That postulate allows us to calcuate the speed of light from the round-trip time based on the proper time between emission and reception events on the same worldine - bounce the light signal off a mirror and wait for the refection to get back to you.

The reflection event is off that worldline so assigning a time to it is an arbitrary choice (although if we interpret "happening at the same time" to mean "has the same time coordinate" that limits us to values between the coordinate times of the emission and detection events). If we are using inertial coordinates in which the clock doing the measurement is at rest then it is natural to assign a time coordinate to the reflection event that is halfway between the time coordinates of the emission and detection events. However, that's not necessary for the round-trip speed of light measurement.

 

[PeterDonis]

What about the measurement at some other point than P ?

There is no invariant way to define the "speed" of anything relative to an observer unless the thing and the observer are co-located. If they are co-located, the invariant definition of relative speed is the one given by @Orodruin (the inner product of two tangent 4-vectors at the same event). If they are not co-located, any definition of "relative speed" requires a choice of coordinates and is thus not invariant. Note that this is just as true for inertial coordinates as for non-inertial coordinates; inertial coordinates, however "natural" they seem to be in flat spacetime, are a choice of coordinates like any other.

Also note that the proper definition of "invariant speed" is in terms of @Orodruin's definition of relative speed for two objects that are co-located. Basically the "invariant speed" is the speed, using @Orodruin's definition, that any object whose tangent 4-vector has zero norm will have relative to any object whose tangent 4-vector has nonzero norm.