# Thread: Does the inside of a neutron star rotate slower than the outer parts, due to t-dilat?

1. Originally Posted by kzb
Just to point out, relativity does apparently result in real forces being generated under certain circumstances.

In Bell's paradox, two accelerating rockets are linked by a rope. Lorentz contraction does indeed cause the rope to break.

I believe, in translating from Rindler to Schwarzschild, that is the equivalent of the tidal force - the two ships need to accelerate at different rates to stay a constant distance apart in their instantaneous shared reference frame; if they accelerate at the same rate, they generate tension in the rope.

Grant Hutchison

2. Order of Kilopi
Join Date
Oct 2005
Posts
25,461
GR is always tricky, and experts use language that often sounds incomprehensible to us. So I don't know if experts would say the issue in the Bell spaceship paradox is a tidal force, indeed I wouldn't be surprised if they would argue over the point! I've heard it said that tidal forces are what distinguish real gravity from the fictitious gravity that comes from the choice of reference frame, where you can play with the latter but the former is the same in all reference frames. But it is certainly true that if we choose a reference frame that is uniformly accelerating when regarded from an inertial frame like the launchpad frame, we will encounter a spatially uniform fictitious gravity in the frame of the rope, meaning that there is no single inertial frame in which to insert the whole rope. That's the source of the stress that breaks the rope-- a coordinate system that is regarded as having a spatially uniform acceleration when regarded from a global inertial frame is not itself a global inertial frame, so cannot allow an extended object to remain stress free if that object is to be regarded as rigid in that accelerating coordinate system. But I would call the forces needed to keep it rigid in that frame as coordinate forces rather than tidal forces, so they'd be similar to centrifugal forces. In other words, tidal forces apply specifically to real gravity, whereas coordinate forces are much less constrained and stem just from one's choice of coordinates and the associated quantitative language those coordinates invoke. So I think the best way to explain why the rope breaks is to say that in a uniformly accelerating reference frame, by which I mean a frame in which we obtain distances and times by using rulers and clocks that are pinned everywhere to accelerometers that read the same fixed value A, the distances all increase along the direction that A points. That's just a rule about accelerating coordinates that correspond to what we mean by measurements of lengths and times, and that rule says that two rockets that both register A must have their distance between them increase with time when using rulers and clocks that also all register A.

Trying to translate that last statement, which is simply found empirically to be true, into the language of "why this is happening", provides yet another classic example of how one's perspective can alter what one regards as the reality of the situation. In the global inertial frame of the launchpad, we say the rope breaks because it needs to "length contract" if we contrast "initial" and "final" global inertial reference frames. So our game there is to mentally insert a rope into those two global inertial frames, but the rope doesn't really fit in those two reference frames without being able to support the necessary internal stresses to fit into both those frames, so it expresses its objection by breaking. Those frames are simply the template we hold to the reality, if we are on the launchpad, and a stressless rope simply can't fit them. But in a different global inertial frame, the "final frame" of the rockets, we fit a different template-- now fitting the rope into the "initial" global inertial frame means it starts out length contracted, and so its length contraction is going away, which would seem to make the rope go slack rather than break, but something else happens in that frame-- the leading rocket takes off first, enough so that the distance between rockets increases even more than the un-contraction of the rope, so it breaks! Those two observers could argue that reality until they are blue in the face, one saying the rope needed to length contract and the other saying the leading rocket launched first, until an observer clinging to the middle of the rope comes along and says that both rockets, and the rope, all have accelerometers that read A, but that requires that the distance between the rockets increases, and the reason given for that distance increase is that time is passing more rapidly for the leading rocket, as reckoned from the center of the rope.

So what is the "real reason" that the rope breaks-- is it that there exists a tidal force? Well, there is not a tidal force here in any formal or coordinate independent sense, because there is not real gravity here, but you can probably take a meaning for tidal forces that is relaxed enough to allow you to frame it that way. Is it that the rope needs to length contract so cannot span the constant distance between the rockets? Is if that the distance between the rockets actually increases, and if so, does it increase the amount the observer on the rope perceives it to increase, or the greater amount that the observer in the "final" inertial frame of the rockets perceives that distance to increase? The answer to all those questions is, it depends on the perspective you choose, the coordinate language you invoke. The language is not some universal language that is handed to you that you must figure it, it is simply whatever you have chosen it to be.

What is so fascinating about all this is that for anyone who insists that "things happen for a reason" (and perhaps they might go further to say that the analysis of that reason is what distinguishes reality from fantasy), they cannot also insist that the "actual reasons" why things happen are part of some "observer-independent reality"-- insisting both those things is shown to be inconsistent by the above example. So you have to either choose one or the other, or more resonably, drop them both, because neither without the other makes much sense-- at least not the sense we'd like.
Last edited by Ken G; 2017-Aug-13 at 12:29 PM.

3. kzb
Established Member
Join Date
Apr 2005
Posts
2,162
But in the Bell's paradox example, the rockets launch simultaneously? The lead rocket does not start accelerating before the trailing rocket. They are joined by a taut rope and both start accelerating together, at the same rate.

So, naively, I think both rockets are in the same reference frame, and the crews of each should measure things exactly the same as each other. But apparently this isn't the case.

I think what makes me uncomfortable is that work is being done (breaking the rope), with no apparent source of energy to do that work.

4. But how do the rockets start at the same time? What is the trigger; an agreed time? from behind the rockets, it would seem like the front one started after the rear one, but infront of the rockets, it may appear like the front one started first.

5. Originally Posted by kzb
But in the Bell's paradox example, the rockets launch simultaneously?
Since they have zero relative velocity in their instantanous reference frames at the moment of launch, I think that's true. But the relativity of simultaneity means that, after the launch, the lead rocket is simultaneous with an earlier moment in the trailing rocket's trajectory, and the trailing rocket is simultaneous with a later moment in the lead rocket's trajectory (these moments "later" or "earlier" as judged by an inertial observer in their original rest frame). So the lead rocket is simultaneous with a trailing rocket that has been accelerating for less time than it has, while the trailing rocket is simultaneous with a lead rocket that has been accelerating for longer than it has. So they have a relative velocity that moves them apart, breaking the rope.
In the succession of instantaneous rest frames of the lead rocket, the trailing rocket needs to accelerate harder to keep up, or the lead rocket needs to reduce its acceleration to avoid moving farther ahead. If they adjust their acceleration in this way (thereby avoiding breaking the rope), then in the original rest frame the trailing rocket will gain steadily on the lead rocket, with just the right speed to match the length contraction of the rope.

Grant Hutchison

6. Originally Posted by Mudskipper
But how do the rockets start at the same time? What is the trigger; an agreed time? from behind the rockets, it would seem like the front one started after the rear one, but infront of the rockets, it may appear like the front one started first.
There's no problem with starting them at the same time, since they start in the same inertial reference frame. Einstein told us how to synchronize clocks in a single inertial frame. The disagreement about simultaneity only begins once they're in motion.

Grant Hutchison

7. kzb
Established Member
Join Date
Apr 2005
Posts
2,162
Originally Posted by grant hutchison
There's no problem with starting them at the same time, since they start in the same inertial reference frame. Einstein told us how to synchronize clocks in a single inertial frame. The disagreement about simultaneity only begins once they're in motion.

Grant Hutchison
Yeah surely there is no issue with this. A signal light equidistant from the control room of each rocket, for example.

8. Order of Kilopi
Join Date
Oct 2005
Posts
25,461
Originally Posted by kzb
But in the Bell's paradox example, the rockets launch simultaneously? The lead rocket does not start accelerating before the trailing rocket.
Simultaneity is always judged in the reference frame of an observer, not the reference frame of an object. Observers have reference frames, objects don't. Indeed, we should probably distinguish between a "reference frame" (which is entirely local to the observer, as it is defined by the observations that observer makes) and a "coordinate system" (which is spatially extended away from the location of the observer, and has the purpose of quantifying conceptual distances and times that are never actually measured by the observer in question). Note that in this language, the reference frame is both local and uniquely determined for each observer (so the two crews are not in the "same reference frame" as they cannot make the same observations), yet there are many coordinate systems that can be used by the same observer (the coordinates become the same locally to the observer, as a good coordinate system should correspond to the local distance and time measurements the observer can actually make, but can be extended nonlocally in arbitrary ways to generate a quantitative language involving distances and times that can be used to say "what is going on" in a very nonunique way).
So, naively, I think both rockets are in the same reference frame, and the crews of each should measure things exactly the same as each other.
Yes, but my point is that there are many observers here, not just observers "in the crew." The different observers have different accounts of what is going on, and the special set of observers I mentioned (those in the global inertial frame of the final velocity of both rockets) will say that the lead rocket took off first, breaking the rope.
But apparently this isn't the case.
No, you're right about the observers in the crew, they think the rockets took off simultaneously. But another way to frame what Grant said is to point out the the crews are in accelerating reference frames, which are not global inertial frames. As such, they experience fictitious forces that make the lead crew think time is going by more slowly for the trailing crew, so the trailing rocket lags, and the trailing crew thinks time is going by faster for the lead crew, so the lead rocket advances-- breaking the rope either way. My point is, notice how different that account of the situation is from the account in the global inertial frames of either the launchpad (where the rope length contracts and breaks), or the final rocket frame (where the lead rocket took off first).

It sounds like you are trying to give some kind of preferred status to the crews on the rockets, or more correctly, to the rope itself. There is probably some value in doing that-- in saying that if we want to know what is happening to the rope, we should ask the rope (or more correctly, an observer on the rope). But note that relativity makes no such claim-- it says that all observers have a valid account, as the laws of physics must work the same for all. If we are to give preference to the rope frame, we must toss our predilection for choosing global inertial frames in which to describe reality, and fragment reality into a set of different reference frames for every different object. I'm not against that, indeed it is very much in the spirit of the "proper time" concept, but given that many objects (and the hypothetical observers that accompany them) are accelerating, your approach requires much more complicated accounting than what is preferred by the global inertial frames popular in special relativity. In particular, there are no longer any global frames of reference because an accelerating frame is a Rindler frame and has a Rindler horizon beyond which the coordinates cannot be applied. Nevertheless, if we do take the Rindler frame of the rope, the "answer" is that the rope breaks because time is going by faster for the lead rocket, so its identical engine causes it to go farther than the trailing rocket does. There are those who might argue that the global inertial frames we could instead use will provide a simpler explanation for the "why" of the breaking rope, and that's why language like "length contraction" is so popular in the first place!
I think what makes me uncomfortable is that work is being done (breaking the rope), with no apparent source of energy to do that work.
In the rope frame, the source of energy is coming from the extra work done by the lead rocket, which burns more fuel in the more time allotted to it. Having a rope attached to it will slightly slow the lead rocket-- an answer that all observers in all frames will agree with, but for very different reasons as I said.
Last edited by Ken G; Yesterday at 03:30 PM.

9. kzb
Established Member
Join Date
Apr 2005
Posts
2,162
Originally Posted by grant hutchison
Since they have zero relative velocity in their instantanous reference frames at the moment of launch, I think that's true. But the relativity of simultaneity means that, after the launch, the lead rocket is simultaneous with an earlier moment in the trailing rocket's trajectory, and the trailing rocket is simultaneous with a later moment in the lead rocket's trajectory (these moments "later" or "earlier" as judged by an inertial observer in their original rest frame). So the lead rocket is simultaneous with a trailing rocket that has been accelerating for less time than it has, while the trailing rocket is simultaneous with a lead rocket that has been accelerating for longer than it has. So they have a relative velocity that moves them apart, breaking the rope.
In the succession of instantaneous rest frames of the lead rocket, the trailing rocket needs to accelerate harder to keep up, or the lead rocket needs to reduce its acceleration to avoid moving farther ahead. If they adjust their acceleration in this way (thereby avoiding breaking the rope), then in the original rest frame the trailing rocket will gain steadily on the lead rocket, with just the right speed to match the length contraction of the rope.

Grant Hutchison
But as a stationary observer, I could be watching perpendicular to the direction of travel. I could watch the joined rockets go past and at one point each rocket would be equidistant from me.

The whole setup, rocket-rope-rocket, would be Lorentz-contracted from my point of view, I've no problem with that.

If they both set off together, I don't see how the trailing rocket is simultaneous with a rocket that has been accelerating for longer. They've both been accelerating for the same time. I, as stationary observer, would agree with both rocket pilots on the time they set off. All three of us would agree.

10. kzb
Established Member
Join Date
Apr 2005
Posts
2,162
It's all very difficult ! Ken G please note I posted #39 before your post #38 appeared.

Anyhow we seem to have established that relativity causes real stresses in physical objects.

11. Order of Kilopi
Join Date
Oct 2005
Posts
25,461
Originally Posted by kzb
But as a stationary observer, I could be watching perpendicular to the direction of travel. I could watch the joined rockets go past and at one point each rocket would be equidistant from me.

The whole setup, rocket-rope-rocket, would be Lorentz-contracted from my point of view, I've no problem with that.
No, that's wrong, the distance between the two rockets is not length contracted. That would require a strong rigid connection between them, instead of two identical rockets doing identical things (in the frame you describe), with essentially no connectors between them. So what actually happens is the length L between the rockets stays always constant in the frame you describe. (The rockets themselves are rigid, and will length contract, but the puzzle is best stated with point-sized rockets and an extended distance L between them, so length contraction in the rockets plays no role.) Indeed, the reason many people imagine the rope doesn't break is that they think the distance between the two rockets should length contract along with the rope, but that's not true.
If they both set off together, I don't see how the trailing rocket is simultaneous with a rocket that has been accelerating for longer. They've both been accelerating for the same time. I, as stationary observer, would agree with both rocket pilots on the time they set off. All three of us would agree.
All observers in the global inertial frame of the launch pad will agree that the rockets take off simultaneously. Also, the crews on the rockets will think that too. But there are other observers who will not agree. Statements about "what actually happened" must either restrict entirely to invariants (and the time the rockets took off is not one of those), or we must allow that language about "what really happened" is observer dependent. There's simply no other choice, this is the often-missed core lesson of relativity.

12. Order of Kilopi
Join Date
Oct 2005
Posts
25,461
Originally Posted by kzb
It's all very difficult ! Ken G please note I posted #39 before your post #38 appeared.
No apologies needed, we can all agree that relativity is bizarre and counterintuitive! It challenges much of what we would otherwise think should be true about reality, that's what is so fascinating about it.

Anyhow we seem to have established that relativity causes real stresses in physical objects.
Correct, that's the bottom line. Relativity is a real aspect of nature, and is associated with real effects. A similar issue comes up when we talk about objects being destroyed by "spaghettification" near the event horizon of a stellar-mass black hole-- it's not just that some coordinates say the object gets elongated, it would really rip you limb from limb! (And in some ways, that's even more bizarre, because accelerometers on your body would always measure zero the entire time-- so we have a real effect without proper acceleration, that real effect is gravity.)
Last edited by Ken G; Yesterday at 04:03 PM.

13. Order of Kilopi
Join Date
Oct 2005
Posts
25,461
On the matter of length contraction of the whole two-rocket system, it is informative to contrast this puzzle with the situation where the rockets never take off, but an observer in a third rocket accelerates past. That situation will look superficially similar, as such an observer will see the two rockets appear to accelerate. But of course the rope does not break in that situation, and it is "because" in that situation the length L between the two rockets does length contract along with the rope. So coordinate acceleration (acceleration that appears in our mathematical bean counting of what speed everything is moving at) of the rockets cannot break a rope, instead it is proper acceleration (acceleration measured by accelerometers on the rockets, what we could call real acceleration) that is associated with the stresses that can break a rope.

The point being, again we see the crucial importance of distinguishing the outcomes of local measurements, which we might regard as "real", from the quantitative language we use to say what is going on, which we should call a coordinate system. Confusing the ramifications of an arbitrarily chosen coordinate system with the outcomes of real measurements causes a lot of misconceptions in relativity, leading people to say things like "space is actually expanding" in the Big Bang model, and so forth. But in fact, anything that is not a measurement, when you are doing physics, should be regarded as a picture. There have been entire threads where people continuously mistake their pictures for what could be regarded as some kind of invariant "actual reality," it's an ongoing problem that is responsible for a significant fraction of the mistakes people make when thinking about relativity.
Last edited by Ken G; Yesterday at 03:55 PM.

14. Originally Posted by kzb
If they both set off together, I don't see how the trailing rocket is simultaneous with a rocket that has been accelerating for longer. They've both been accelerating for the same time. I, as stationary observer, would agree with both rocket pilots on the time they set off. All three of us would agree.
As soon as the rockets change velocity, their standard of simultaneity changes. If they set up an array of clocks reflecting their own standard of simultaneity, it would not agree with those similarly configured by an observer at rest in the launch frame. In particular the rocket crew's clocks would show an earlier time in the direction of their flight (and a later time behind them) than the ground crew's.
So any given event in the rear rocket's trajectory it is simultaneous (as measured by its crew) with an event that occurs after more elapsed time along the front rocket's trajectory (both by the judgement of the ground crew and the crew of the leading rocket). And any event in the leading rocket is simultaneous (as measured by its crew) with an event that occurred earlier in the trajectory of the rear rocket (both by the judgement of crew of the trailing rocket and by the ground crew).
I've been framing this as creating a difference in the velocities of the two rockets, as measured by the other rocket. Ken has framed it as less proper time passing for the rear rocket than the leading rocket. Either argument works, I think.

Grant Hutchison

15. Here's a space-time diagram that might make the simultaneity issues clearer (click to enlarge):
Bell.jpg

Time on the vertical axis, space on the horizontal. The two hyperbolae represent the worldlines of two spacecraft that start to accelerate in the x direction simultaneously, starting from rest in these coordinates. They continue at constant proper acceleration, but their velocities approach the speed of light asymptotically - reflected in the way the hyperbolae approach a 45-degree slope asymptotically in the diagram. I've marked the two worldlines "Leader" and "Trailer".

So:
When Leader has reached Event A on its trajectory, an observer in the original rest frame measures Event A to be simultaneous with Event B on Trailer's worldline. This line of simultaneity is marked by the dashed black line, parallel to the x axis, connecting A and B along a constant rest-frame t coordinate. Equal proper time has elapsed along both worldlines, so the original rest frame finds that Leader and Trailer are moving at the same (instantaneous) velocity.
But when Leader is at Event A, its (instantaneous) velocity gives it a different standard of simultaneity, represented by the sloping red dashed line. In Leader's (instantaneous) reference frame, Event A is simultaneous with Event C on Trailer's worldline. Less proper time has elapsed for Trailer at Event C than has elapsed for Leader at Event A. So as far as Leader is concerned, less acceleration time has passed for Trailer than for Leader, so Trailer has a lower velocity than Leader, and Trailer is therefore lagging behind.
But if we move to Event C on Trailer's worldline, that is associated with another standard of simultaneity, marked by the blue dashed line. Trailer measures Event C on its worldline to be simultaneous with Event D on Leader's worldline. More proper time has elapsed for Leader at Event D than has elapsed for Trailer at Event C, so in this (instantaneous) reference frame, Leader has a higher velocity than Trailer, and is pulling ahead.

We see that Trailer will always find Leader pulling ahead, and Leader will always find Trailer falling behind, although they will always disagree with each other (and with the original rest frame) about which events on the two worldlines are simultaneous.

Grant Hutchison
Last edited by grant hutchison; Yesterday at 11:48 PM. Reason: Removed extra text

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•