# Non inertial frames

## Recommended Posts

If you could get into a really fast centrifuge with a device that measures the speed of light, would you see different values for c?

If so, would the measurements be both higher and lower than c or just one or the other?

##### Share on other sites

Sorry if I've misunderstood, but I thought it was impossible to go higher than c?, isn't c always the same no matter what?

By c you mean the speed of light right?

##### Share on other sites

I was thinking about a high rate of acceleration more than high velocity. When the speed of light is discussed in Relativity, they always specify "in a vacuum when measured from an inertial frame of reference". I was wondering if there are any predictions of what velocity would be measured for the speed of light from a non-inertial frame.

##### Share on other sites

4 minutes ago, moth said:

I was thinking about a high rate of acceleration more than high velocity. When the speed of light is discussed in Relativity, they always specify "in a vacuum when measured from an inertial frame of reference". I was wondering if there are any predictions of what velocity would be measured for the speed of light from a non-inertial frame.

Even in a non-inertial frame or in the presence of gravity (which the continuous acceleration of a centrifuge is equivalent to) the local speed of light is still c. You might determine the speed of light outside the centrifuge to be different(*), but you will always measure it locally as c.

(*) Off the top of my head, I am not sure if that is the case or not.

##### Share on other sites

I see hints that the velocity of light is different than c in non-inertial frames, but not how it would be different.

Quote

Accelerating reference frames are a different matter. In GR the physical equations take the same form in any co-ordinate system.  In SR they don't, but it's still possible to use co-ordinate systems corresponding to accelerating or rotating frames of reference, just as it is possible to solve ordinary mechanics problems in curvilinear co-ordinate systems.  This is done by introducing a metric tensor.  The formalism is very similar to that of many general relativity problems, but it is still special relativity as long as the space-time is constrained to be flat and minkowskian.  Note that the speed of light is rarely a constant in non-inertial frames, and this has been known to cause confusion.

##### Share on other sites

52 minutes ago, moth said:

I see hints that the velocity of light is different than c in non-inertial frames, but not how it would be different.

When you start working in multidimensional frames (matrix, tensor etc) many quantities which are a plain numerical constant in simpler constructs become themselves matrix or tensor objects.
This is true for instance in GR.

##### Share on other sites

9 minutes ago, studiot said:

When you start working in multidimensional frames (matrix, tensor etc) many quantities which are a plain numerical constant in simpler constructs become themselves matrix or tensor objects.
This is true for instance in GR.

Does this mean c might be the same value but velocity vector has changed?

Edit: Velocity vector sounds redundant. i mean the magnitude of c is the same,  but the trajectory is different?

Edited by moth
##### Share on other sites

c itself is still constant under rapidity to all observers.

You can see that via the inverse hyperbolic function used to describe rapidity (acceleration causes a rotation of the Minkowskii metric).

$w=arctanh(v/c)$

Edited by Mordred
##### Share on other sites

1 hour ago, Mordred said:

c itself is still constant under rapidity to all observers.

You can see that via the inverse hyperbolic function used to describe rapidity (acceleration causes a rotation of the Minkowskii metric).

w=arctanh(v/c)

I was just looking at the wiki for rapidity and realized this is about where I got stuck on vector calculus.

Just a quick question: if y=arctanh x,  does that mean x=tanh y ?

edit2:Another quick innumeracy related question, hyperbolic trig functions map points on a hyperbolic curve to points on the x,y axis the same as trig functions map

points on the unit circle to the x,y axis?

Thanks

Edited by moth
##### Share on other sites

Yes to the first but on the second (which affects the first) instead of the y axis replace the y axis with ct.

Your Lorentz transforms being two frames of importance is the x and x prime and ct and ct primed.

Here is a primer

This one is handy as it includes the Minkowskii diagram showing the hyperbolic rotation.

##### Share on other sites

7 hours ago, Strange said:

Even in a non-inertial frame or in the presence of gravity (which the continuous acceleration of a centrifuge is equivalent to) the local speed of light is still c. You might determine the speed of light outside the centrifuge to be different(*), but you will always measure it locally as c.

(*) Off the top of my head, I am not sure if that is the case or not.

Local would have to be a lot smaller than that, eg. "measured at the center of the centrifuge" might work.

If you're moving fast enough for it to not be negligible, and you send light in a path around the edge of the centrifuge, then an observer moving around the centrifuge will measure that light will take more time to make a round trip (from observer back to observer) in the direction the observer is going, and less than usual in the opposite direction. Both path lengths are measured to be the same in either direction. You can confirm the timing in the "outside" inertial frame: by the time the light has made a full circle in the outside frame, the observer has moved on from its original position, and light must make more than a full circle when sent in the same direction the observer is moving, and less in the opposite direction.

However, this is not really a valid measure of the speed of light. You could call it the "coordinate speed" of light, and it's been argued on these forums that it's a meaningless measure. I'd say it'd be like making individual measurements in different momentary inertial frames of the revolving observer; you can get similar invalid measures of the speed of light if you switch between inertial frames in SR without properly accounting for the switch.

##### Share on other sites

25 minutes ago, md65536 said:

Local would have to be a lot smaller than that, eg. "measured at the center of the centrifuge" might work.

If you're moving fast enough for it to not be negligible, and you send light in a path around the edge of the centrifuge, then an observer moving around the centrifuge will measure that light will take more time to make a round trip (from observer back to observer) in the direction the observer is going, and less than usual in the opposite direction. Both path lengths are measured to be the same in either direction. You can confirm the timing in the "outside" inertial frame: by the time the light has made a full circle in the outside frame, the observer has moved on from its original position, and light must make more than a full circle when sent in the same direction the observer is moving, and less in the opposite direction.

However, this is not really a valid measure of the speed of light. You could call it the "coordinate speed" of light, and it's been argued on these forums that it's a meaningless measure. I'd say it'd be like making individual measurements in different momentary inertial frames of the revolving observer; you can get similar invalid measures of the speed of light if you switch between inertial frames in SR without properly accounting for the switch.

I think i remember that thread. I can't remember if it got into shooting the light across the diameter. At high enough angular velocity would the beam hit the detector directly at the other side, or would it appear to curve (from the rotating frames perspective)?

##### Share on other sites

6 hours ago, md65536 said:

If you're moving fast enough for it to not be negligible, and you send light in a path around the edge of the centrifuge, then an observer moving around the centrifuge will measure that light will take more time to make a round trip (from observer back to observer) in the direction the observer is going, and less than usual in the opposite direction. Both path lengths are measured to be the same in either direction. You can confirm the timing in the "outside" inertial frame: by the time the light has made a full circle in the outside frame, the observer has moved on from its original position, and light must make more than a full circle when sent in the same direction the observer is moving, and less in the opposite direction.

That's the Sagnac effect. As seen in a Mach-Zehnder interferometer or laser gyroscope, among other places.

##### Share on other sites

11 hours ago, moth said:

I think i remember that thread. I can't remember if it got into shooting the light across the diameter. At high enough angular velocity would the beam hit the detector directly at the other side, or would it appear to curve (from the rotating frames perspective)?

You could set it up so that in the "lab" frame, the light is emitted from one point along the circumference, passes through the center, and hits a detector on the opposite side, all in a straight line. This is true no matter what frame the light source is in (it would just be pointed differently in different frames, due to "aberration of light").

If the detector was rotating around the circumference, it wouldn't be hit directly opposite the emitter, in the rotating frame. The observer moving around the circumference would see the same beam of light aimed "backward" of center, curve "forward" (in the direction the observer is moving) so that it hits the center, and continue curving so that it hits the circumference a bit "closer" along the circumference rather than directly across from it. All observers would agree on this: if the light is emitted as the observer passes by, the rotating frame turns during the time it takes the light to travel across the diameter, and "directly opposite the observer in the rotating frame" has moved on by the time the light reaches the other side.

There's probably a better way to describe it. You can figure out a lot by describing things in terms of events (happening at one time and place, so all observers agree that it happens (like if a particular beam passes through the center, it does so in all frames)), adding extra measuring tools (like a marked disk that turns with the centrifuge), replacing the accelerating observer with an inertial one that shares a momentary inertial frame at an event, etc.

## Create an account

Register a new account