Relativistic Velocity Addition, Curved Spacetime, and the Limits of Inertial Frames

[jaketodd]

Moderator's note: This was originally posted in another thread but has been spun off.

"It is shown how, within the framework of general relativity ... motion faster than the speed of light as seen by observers outside the disturbed region is possible."

The warp drive: hyper-fast travel within general relativity

 

[PeterDonis]

"It is shown how, within the framework of general relativity ... motion faster than the speed of light as seen by observers outside the disturbed region is possible."

The warp drive: hyper-fast travel within general relativity

Irrelevant since this is a case of curved spacetime, not flat spacetime, and in curved spacetime the relativistic velocity addition law only holds within a local inertial frame. The effect you describe cannot be seen within a local inertial frame.

 

[jaketodd]

Irrelevant since this is a case of curved spacetime, not flat spacetime, and in curved spacetime the relativistic velocity addition law only holds within a local inertial frame. The effect you describe cannot be seen within a local inertial frame.

So, in non-locally inertial reference frames, you can see something go faster than light? Is curved spacetime, which one example is what Alcubierre describes, the only way to see someone go faster than light, or can you ever see someone go faster than light without spacetime curvature/gravity?

Thanks,

Jake

 

[jbriggs444]

So, in non-locally inertial reference frames, you can see something go faster than light?

Even in an non-inertial reference frame, material objects cannot go faster than light. They can exceed a coordinate velocity of $c$. But that is not the same thing. A material object cannot catch up to a light ray. That light ray can exceed a coordinate velocity of $c$ in the coordinate system that you happen to be using.

 

[PeterDonis]

in non-locally inertial reference frames

What does this even mean?

 

[Nugatory]

So, in non-locally inertial reference frames, you can see something go faster than light?

Yes. For example, using the very comfortable and natural but non-inertial (because the earth is rotating) frame in which I and my desk are at rest, everything outside the solar system is moving faster than light - just 24 hours to go around a circle many light years in circumference.

 

[Ibix]

So, in non-locally inertial reference frames, you can see something go faster than light? Is curved spacetime, which one example is what Alcubierre describes, the only way to see someone go faster than light, or can you ever see someone go faster than light without spacetime curvature/gravity?

Nothing ever goes faster than light.

In a non-inertial reference frame you can have things exceed $c$, but they do not go faster than light. The classic example is to turn around 360°. In the rotating reference frame where you were at rest, Alpha Centauri, 4 light years away, moved $2\pi\times 4\mathrm{ly}$ in the second or so it took you to turn around - far faster than $c$, but nothing overtook light.

You can't travel faster than light in curved spacetime either. However, you can sometimes find shortcuts that let you get somewhere before light that has to take the long route. A naturally occurring example would be hovering above a black hole at the altitude where light can orbit. For a reasonably large black hole, I can fire a light pulse in one direction, travel 100m in the opposite direction, and wait for the light to finish orbiting the hole and reach me. I didn't travel faster than light, but I used a shortcut to reach my destination before light that takes the long route.

The warp drive is basically a "build your own shortcut" kit. Nothing travels faster than light; you just arrange a shortcut to your destination. You will get there before light that takes the long route, but you did not exceed the speed of light at any time.

 

[Halc]

So, in non-locally inertial reference frames, you can see something go faster than light?

Some examples:

Relative to Paris (a rotating accelerating reference frame), Neptune is moving faster than c. This frame is not inertial, not even locally, as evidenced by an accelerometer placed at Paris.

Relative to the cosmic frame (proper distances), GN-z11 (no longer the 'most distant visible object') is receding from Earth at about 2.3c

Neither of these things moves faster than light, else something ahead of its path would not be able to see it coming, and that is not the case for any illuminated object.

 

[jaketodd]

Inside the warp bubble/for the spaceship described by Alcubierre, the spacetime is flat.

So, therefore, wouldn't that be compatible to an observer, in a local inertial frame/no curvature difference between reference frames? If so, and if Alcubierre is right, wouldn't that throw a wrench in the gears of the relativistic velocity addition law holding within locally inertial frames?

What does this even mean?

A reference frame that is in curved spacetime.

Yes. For example, using the very comfortable and natural but non-inertial (because the earth is rotating) frame in which I and my desk are at rest, everything outside the solar system is moving faster than light - just 24 hours to go around a circle many light years in circumference.

Beautiful!

The warp drive is basically a "build your own shortcut" kit. Nothing travels faster than light; you just arrange a shortcut to your destination. You will get there before light that takes the long route, but you did not exceed the speed of light at any time.

I'm confused. Wouldn't the spaceship get to another solar system before a photon, on the same trajectory, would? Wouldn't the spaceship outrun a photon on the same trajectory? The photon and the ship would be in inertial frames to each other because both are in flat spacetime. Which brings me back to violating the relativistic velocity addition law, because both the ship, and the photon it outruns, are in flat spacetime. But then photons aren't a reference frame, right? Well then how about an object moving at near the speed of light, watching the spaceship, on the same trajectory, keep accelerating beyond the speed of light, and get to the target solar system very quickly?

Thanks All!

 

[Ibix]

Wouldn't the spaceship get to another solar system before a photon, on the same trajectory, would?

Advice: don't say photon unless you mean photon, or you are letting yourself in for a world of trouble when you actually try to understand what a photon is. Say "light pulse".

You mean, would the ship, going through the shortcut built for it, arrive before a light pulse that started right next to it but didn't take the shortcut? Sure. That's pretty much what I said.

The photon and the ship would be in inertial frames to each other because both are in flat spacetime. Which brings me back to violating the relativistic velocity addition law, because both the ship, and the photon it outruns, are in flat spacetime. But then photons aren't a reference frame, right? Well then how about an object moving at near the speed of light, watching the spaceship, on the same trajectory, keep accelerating beyond the speed of light, and get to the target solar system very quickly?

This makes no sense.

Inside the warp bubble/for the spaceship described by Alcubierre, the spacetime is flat.

The particular spatial slice illustrated in that diagram is approximately flat, yes. I'm not sure about spacetime.

So, therefore, wouldn't that be compatible to an observer, in a local inertial frame/no curvature difference between reference frames? If so, and if Alcubierre is right, wouldn't that throw a wrench in the gears of the relativistic velocity addition law holding within locally inertial frames?

I don't think there's a clear chain of reasoning here.

You can certainly use relativistic velocity addition over a small region in any arbitratily curved spacetime. You can't use it beyond that region because of the path dependence of parallel transport. The Alcubierre spacetime is not special in this regard; at any event, inside or outside the bubble or in the bubble wall, you can use relativistic velocity addition. You just can't use it to compare velocities of things not at the same event.

 

[PeterDonis]

Inside the warp bubble/for the spaceship described by Alcubierre, the spacetime is flat.

Yes. So if you use a local inertial frame that is restricted to the region inside the warp bubble, you can use the relativistic velocity addition law within that frame and it works fine. But only within that frame. There is no "link" between that frame and an inertial frame covering the flat spacetime region outside the warp bubble. They are two different frames and you can't mix anything between them.

wouldn't that throw a wrench in the gears of the relativistic velocity addition law holding within locally inertial frames?

No. See above.

Wouldn't the spaceship get to another solar system before a photon, on the same trajectory, would?

No. A light ray emitted from the spaceship inside the warp bubble will arrive at the destination before the ship itself does. Nothing can outrun a light ray. That's true in any spacetime.

The photon and the ship would be in inertial frames to each other because both are in flat spacetime.

If the photon is outside the bubble, and the ship is inside the bubble, they are not in the same region of flat spacetime. They are in different, disconnected regions of flat spacetime. The overall spacetime is curved, and in curved spacetimes your intuitive arguments are not valid.

If the photon is emitted from the ship inside the bubble, then see my remarks above.

 

[jaketodd]

Yes. So if you use a local inertial frame that is restricted to the region inside the warp bubble, you can use the relativistic velocity addition law within that frame and it works fine. But only within that frame. There is no "link" between that frame and an inertial frame covering the flat spacetime region outside the warp bubble. They are two different frames and you can't mix anything between them.

So the curved spacetime surrounding the spaceship prevents "linking" the reference frame of the spaceship to an observer outside the Alcubierre drive region via the relativistic velocity addition law? If so, reference frames are linked using that law if you can draw a line between them that does not encounter spacetime curvature?

No. A light ray emitted from the spaceship inside the warp bubble will arrive at the destination before the ship itself does. Nothing can outrun a light ray. That's true in any spacetime.

And that leads to why there is time dilation right? Thanks!

You can certainly use relativistic velocity addition over a small region in any arbitratily curved spacetime. You can't use it beyond that region because of the path dependence of parallel transport. The Alcubierre spacetime is not special in this regard; at any event, inside or outside the bubble or in the bubble wall, you can use relativistic velocity addition. You just can't use it to compare velocities of things not at the same event.

I'm having trouble understanding path dependent parallel transport. It seems to me that as you slide a topology from one region to another, as long as the beginning location and ending location are of the same spacetime curvature environment, then the topology would be identical from before and after that transition, as long as the resolution of the topology is preserved and not distorted by the differing spacetime curvatures in-between the start and end. Thanks

 

[PeterDonis]

So the curved spacetime surrounding the spaceship prevents "linking" the reference frame of the spaceship to an observer outside the Alcubierre drive region via the relativistic velocity addition law?

Yes.

If so, reference frames are linked using that law if you can draw a line between them that does not encounter spacetime curvature?

Yes. Which is an extremely rare occurrence. Note that the Alcubierre solution as it is presented in the literature cannot exist in our actual universe, because the spacetime of our actual universe is not flat, so even if spacetime were flat inside a warp bubble in our actual universe (supposing one could be created, which is extremely improbable), spacetime would not be flat outside the bubble.

And that leads to why there is time dilation right?

I don't know what you mean. A spaceship traveling inside a warp bubble does not experience time dilation.

I'm having trouble understanding path dependent parallel transport.

That belongs in a separate thread since it is a different topic from the one we are discussing here. However...

topology

Path dependence of parallel transport in curved spacetime has nothing to do with topology. So I think you need to take some time to review a textbook on the topic.

 

[jaketodd]

If so, reference frames are linked using that law if you can draw a line between them that does not encounter spacetime curvature?

Yes. Which is an extremely rare occurrence. Note that the Alcubierre solution as it is presented in the literature cannot exist in our actual universe, because the spacetime of our actual universe is not flat, so even if spacetime were flat inside a warp bubble in our actual universe (supposing one could be created, which is extremely improbable), spacetime would not be flat outside the bubble.

So then technically, since spacetime is curved everywhere in the universe, then it's always impossible, in-practice, to use the relativistic velocity addition law for any two points, anywhere?

I don't know what you mean. A spaceship traveling inside a warp bubble does not experience time dilation.

I meant the invariance of the speed of light, even for light emanating from something moving faster than light, which is the spaceship inside the warp bubble.

Path dependence of parallel transport in curved spacetime has nothing to do with topology. So I think you need to take some time to review a textbook on the topic.

I apologize, thanks.

Thank you for your patience, Peter!

Jake

 

[PeterDonis]

So then technically, since spacetime is curved everywhere in the universe, then it's always impossible, in-practice, to use the relativistic velocity addition law for any two points, anywhere?

Unless you are within the confines of a local inertial frame, as has already been said. In cases where you have a region of spacetime that is flat, you can always construct a local inertial frame that covers the entire region, even if it is large. But if spacetime is not flat in a region, then local inertial frames are restricted to regions that are small enough that the effects of spacetime curvature can be ignored.

I meant the invariance of the speed of light, even for light emanating from something moving faster than light, which is the spaceship inside the warp bubble.

The spaceship is not moving "faster than light". It has a timelike worldline and is within the local light cones in its region of spacetime. It cannot catch up to a light ray that it emits.

The invariance of the light cone structure of spacetime is the correct meaning of "invariance of the speed of light" in a curved spacetime.

 

[jaketodd]

If so, reference frames are linked using that law if you can draw a line between them that does not encounter spacetime curvature?

Yes. Which is an extremely rare occurrence. Note that the Alcubierre solution as it is presented in the literature cannot exist in our actual universe, because the spacetime of our actual universe is not flat, so even if spacetime were flat inside a warp bubble in our actual universe (supposing one could be created, which is extremely improbable), spacetime would not be flat outside the bubble.

So then technically, since spacetime is curved everywhere in the universe, then it's always impossible, in-practice, to use the relativistic velocity addition law for any two points, anywhere?

Unless you are within the confines of a local inertial frame, as has already been said. In cases where you have a region of spacetime that is flat, you can always construct a local inertial frame that covers the entire region, even if it is large. But if spacetime is not flat in a region, then local inertial frames are restricted to regions that are small enough that the effects of spacetime curvature can be ignored.

But you said spacetime is curved everywhere in the universe. Wouldn't that make it impossible to define any region of two or more points as inertial? A line between any two points would always traverse spacetime curvature. I guess I'm splitting hairs. Sounds like the relativistic velocity addition law is never accurate, but accurate-enough to give good approximations. Interesting though that inertial reference frames can't technically exist in this universe with a size of more than one point. Thanks

 

[Vanadium 50]

Now this thread is well and truly hijacked.

Complaining that you can never find a region of spcaetime that is truly flat is like complaining you can't find a massless pulley, a frictionless plane, a strestchless rope, a perfect crystal, etc. One does not look like a genius for pointing out that something everybody knows is an idealization is in fact, an idealization. One looks like something else.

We use idealizations to aid understanding. Filling the air with "yeahbuts" does not aid in understanding.

 

[jaketodd]

Now this thread is well and truly hijacked.

Complaining that you can never find a region of spcaetime that is truly flat is like complaining you can't find a massless pulley, a frictionless plane, a strestchless rope, a perfect crystal, etc. One does not look like a genius for pointing out that something everybody knows is an idealization is in fact, an idealization. One looks like something else.

We use idealizations to aid understanding. Filling the air with "yeahbuts" does not aid in understanding.

Like I said, I realize I'm splitting hairs. But it's still kind of interesting.

I did not know inertial reference frames larger than one point were an idealization until today, for example. Sorry to those that already knew that.

 

[PeterDonis]

you said spacetime is curved everywhere in the universe.

Yes.

Wouldn't that make it impossible to define any region of two or more points as inertial?

If you assume that our ability to measure spacetime curvature has infinite accuracy, yes. But of course that is not the case. We can only measure spacetime curvature with a finite accuracy. That means that, if we pick any point in spacetime, there will be some finite region surrounding that point in which we cannot detect any spacetime curvature. But how large that region will be will depend on how strong the spacetime curvature is at that point: the stronger the curvature, the smaller the region.

Sounds like the relativistic velocity addition law is never accurate, but accurate-enough to give good approximations.

You're looking at it wrong. It's not a question of accuracy, but of region application: in a curved spacetime you can't apply the relativistic velocity addition law globally, but only within the confines of a local inertial frame. How large such a frame can be will depend on how strong the spacetime curvature is, as above.

 

[Vanadium 50]

You're splitting hairs on someone else's thread. But hey, why should he get a good answer to his question whne Jake wants to split hairs?

Do you understand why people might see this as antisocial?

If I counted correctly, 8 of your last 10 threads have ended up locked. Not an enviable record. You will likely get more out of the forum if you adjusted your approach. Or you can quibble about perfect spheres and massless springs.

 

[PeterDonis]

You're splitting hairs on someone else's thread.

I have fixed that by spinning off this whole subthread into its own separate thread. (The original thread's last post was more than a year ago anyway, so this should have been a new thread to begin with.)

If I counted correctly, 8 of your last 10 threads have ended up locked.

It is now 9 of 11 since I have locked this thread.