A variation on the twin paradox

[RandallB]

Maybe I'm just being too nitpicky, but that's incorrect. During each flyby, each satellite will observe the other to be running slowly. Of course, they will both agree on the time between flybys.

You’re not being to nitpicky. You just haven’t looked at the Twins issue closely when you send them BOTH on the trip in opposite directions.

Don’t they see each other as aging more slowing at the start? Even on the return just making direct observations at a distance don’t they see the same thing? With the exception of when they do fly by Earth on the return comparing each other total age shows they are both THE SAME, only those on Earth have aged much more.
Since they are already flying-by let them retrace the other path and won’t you get the same result again on the next return? – and the next?
Just like counter orbiting GPS satellites.
Just like circumventing a “compact space”,
be it a living room, the Fermi Lab Ring, or galaxy etc. etc.

I see no reason for any of these to behave differently then the twins do.
Other than the acceleration to bring them back being applied differently,
what do they get the twins don’t?
None of them remain in a single reference frame.

 

[JesseM]

Since they are already flying-by let them retrace the other path and won’t you get the same result again on the next return? – and the next?
Just like counter orbiting GPS satellites.
Just like circumventing a “compact space”,
be it a living room, the Fermi Lab Ring, or galaxy etc. etc.

I see no reason for any of these to behave differently then the twins do.
Other than the acceleration to bring them back being applied differently,
what do they get the twins don’t?
None of them remain in a single reference frame.

There is no "acceleration to bring them back" in a compact universe, that's the whole point. They both travel inertially in opposite directions, each remaining in the same inertial rest frame, but they can still meet up again due to the weird topology of the universe. Again, just think of the game Asteroids, where you can fly away from the center of the screen to the right, then when you hit the right edge you reappear on the left edge of the screen still traveling to the right, so if you keep going you'll end up back at the center of the screen without ever having turned around or changed velocity.

 

[RandallB]

There is no "acceleration to bring them back" in a compact universe, that's the whole point.

Now you’re back to hypothetical universes that don’t have to deal with the light horizon of the Big Bang – As I said before I’m dealing with real universes here. Traveling inertially won’t get past that line and that’s not even ¼ the way around!

 

[JesseM]

Now you’re back to hypothetical universes that don’t have to deal with the light horizon of the Big Bang – As I said before I’m dealing with real universes here. Traveling inertially won’t get past that line and that’s not even ¼ the way around!

As I said, most physicists would still consider a paradox interesting even if it could only happen in a universe with different initial conditions. And you never addressed my point that the size of a compact universe can in fact be smaller than the horizon created by the big bang, and that in fact physicists are looking for evidence of this possibility in the cosmic microwave background radiation. For now we can't rule out the possibility that the universe could be small enough to circumnavigate, even with that horizon.

In any case, your argument is inconsistent. First you say, "the cosmological twin paradox is just like any other version of the twin paradox" and then I say "no it isn't, the special feature of the cosmological twin paradox is that the twins can depart and then later reunite without either accelerating" and your response is "yeah, but you could never circumnavigate the universe anyway!" Circumnavigating the universe without accelerating is the essential feature of the cosmological twin paradox, so you're free to dismiss the cosmological twin paradox as irrelevant if you think it'll turn out to be impossible to do this in our universe, but your comparison with the GPS satellites or other situations involving acceleration is still off-base.

 

[RandallB]

Garth
I see my problem
The term “Compact Space” must mean:
A large enough periodic orbit moving fast enough in an inertial straight line to circumvent the complete universe.

Then it’s the term “Compact Space” I’ve been having a problem with – my mistake.
Just substitute “any periodic orbit circumventing any part of the real universe large or small” where I may have used the term.

Until someone shows they can even be such a thing, no need for me to deal with “Compact Space”. Sorry if I intruded on just a hypothetical.
Thought you were dealing with a real thought experiment.

 

[dicerandom]

Until someone shows they can even be such a thing, no need for me to deal with “Compact Space”. Sorry if I intruded on just a hypothetical.
Though you were dealing with a real thought experiment.

There certainly can be, the question is just whether or not the universe we live in is closed. Although just because it's closed doesn't mean that you can circumnavigate it, the space could have an accelerating expansion such that you'd never be able to circumnavigate it. In the Robertson-Walker metric I posted earlier the parameter \kappa dictates whether the space is positively curved and closed, flat and open, or negatively curved and open.

Which of these three possibilities our universe truly is depends on the relative densities of matter, radiation, and a possible value for a cosmological constant. As it turns out we are somewhere extremely near the critical value which is the "tripple point", meaning that our universe is basically flat on large scales, but it could be very slightly positively or negatively curved and we simply don't have the accuracy available to measure which it is.

 

[JesseM]

Which of these three possibilities our universe truly is depends on the relative densities of matter, radiation, and a possible value for a cosmological constant. As it turns out we are somewhere extremely near the critical value which is the "tripple point", meaning that our universe is basically flat on large scales, but it could be very slightly positively or negatively curved and we simply don't have the accuracy available to measure which it is.

But as I said before, the question of the topology of the universe is actually independent of the curvature issue. It's true that if you assume the simplest possible topology, a positively-curved universe would be finite while a flat or negatively-curved universe would be infinite; but by choosing other topologies you can have a finite universe that is flat or negatively curved (not sure whether a positively-curved and infinite universe is possible, though).

 

[Garth]

MTW included no such caveat about there needing to be matter in the universe, though. Certainly a flat and empty infinite universe is a valid GR solution, with an arbitrary flat hypersurface qualifying as a "hypersurface of homogeneity", so why can't you do the same trick they mentioned of identifying faces on a cube in such a universe?

You can; however, I would question the physical reality of such a hypothetical extrapolation of testable physics. On the other hand, if 'circles in the sky' are observed I would have to revise this opinion. I am ready to acknowledge that not only is the universe more weird than I imagined, but more weird than I can imagine!

You are misunderstanding the principle of relativity here, I think. After all, the laws of physics would still work the same in every local inertial frame, and GR doesn't say anything about the equations of physics (written without using tensors) working the same in every non-local coordinate system, so that's all you need to satisfy the principle of relativity.

You have a local laboratory belonging to one observer A. A second observer B momentarily passes through at high speed and both synchronise clocks. A very long time later B passes through A's laboratory again after inertial circumnavigation of the universe, and clocks are compared.
The fact that the global topology imparts a preferred frame which says that it is A's clock that will register the greatest time elapse means that A and not B can, at the initial local encounter, think of their time as being 'absolute' in some sense. This I understand to be in contradiction to the Principle of Relativity.

But anyway, if you are sure to include the correct initial clock settings for each copy as seen in a given observer's coordinate system, then the laws of physics do work the same in every global coordinate system in the flat spacetime version of the cosmological twin paradox--when one twin leaves Earth and returns to Earth to find that his twin has aged more, he can say that he actually traveled from one copy of Earth to another, and although each copy was aging more slowly than himself (as required if the law of time dilation is to work the same way in his frame), the copy he was traveling towards started out older than the copy he left.

Are these 'copies' of the Earth the actual one Earth experienced after circumnavigations of the universe, or are we saying that the world we know is itself replicated many/infinite number of times? I cannot swallow the second interpretation. That interpretation, IMHO, seems too high a price to pay, stretching physical reality too far, in order to resolve the paradox.

I would argue, contrary to MTW, that mass in the universe is essential to resolve this paradox and that A's 'absolute' frame of reference is that defined by the Centre of Mass/momentum of the matter in the universe at large.

Garth

 

[RandallB]

I am ready to acknowledge that not only is the universe more weird than I imagined, but more weird than I can imagine!

Don’t give up on your imagination based on a hypothetical paradox.
If a paradox only exists in a hypothetical universe then the paradox isn’t real.
Just because a parameter can define a “hyperspace” curved universe, doesn’t make the idea of a curved universe real. No more than being shown a simple SR-GPS problem masquerading as a house of mirrors.
Until there is REAL evidence of ‘compact space’ there is no reason to let this “paradox” control your imagination no matter how good a hypothetical argument may sound. The idea that reality might act like a computer screen has only the idea to support it, nothing real.

I would argue, contrary to MTW, that mass in the universe is essential to resolve this paradox and that A's 'absolute' frame of reference is that defined by the Center of Mass/momentum of the matter in the universe at large.

Very good point, and the way to bring your idea into reality is to use the CBR. And that 'absolute' frame will look like a “preferred” frame. But, contrary to the Lorentz R fans out there, it cannot be preferred because if you move a significant distance away to define another 'absolute' frame using the same CBR, it will not be the same as the first CBR defined 'absolute' frame. Thus still no “preferred” frame, just as relativity demands.
Using your imagination on a real paradox like this is much more valuable than giving up on it over some hypothetical paradox.

I disagree with Eddington on what we can do with imagination.

 

[Garth]

Rather than take the hypothetical case of non-trivial topologies I prefer to keep it simple and consider this paradox in the case of the topologically simple compact space of a closed universe, i.e. the spherical or cylindrical universes of Friedmann or Einstein.

I am willing to ignore the practicality of circumnavigating such a universe for the sake of the 'gedanken'.

Garth

 

[dicerandom]

There were some comments earlier about the non-physicalness of the cylindrical spacetime, I'd just like to point out that it is a case of the Friedman-Robertson-Walker metric, it just hapens to be a 1+1 temporal-spatial case. The problem breaks down to a two dimensional one anyway if you only consider inertial observers.

 

[JesseM]

You can; however, I would question the physical reality of such a hypothetical extrapolation of testable physics.

Sure, I don't have a problem with that, as long as you acknowledge that the mathematical theory of GR allows such things. What you're saying is that not all spacetimes that are valid according to GR may actually be possible, and I agree we have no way of knowing for sure.

You are misunderstanding the principle of relativity here, I think. After all, the laws of physics would still work the same in every local inertial frame, and GR doesn't say anything about the equations of physics (written without using tensors) working the same in every non-local coordinate system, so that's all you need to satisfy the principle of relativity.

You have a local laboratory belonging to one observer A. A second observer B momentarily passes through at high speed and both synchronise clocks. A very long time later B passes through A's laboratory again after inertial circumnavigation of the universe, and clocks are compared.

No, I don't think that counts as a valid test of the "principle of relativity" in GR. After all, even without picking out weird topologies you can do something similar in the neighborhood of the earth--have two observers, one orbiting the Earth and the other shot up from the surface at slightly less than escape velocity, in such a way that the shot-up observer passes right next to the orbiting observer on his way up, moves away from the Earth for a while and finally begins to fall back down, with everything timed so that on his way down he will again pass right next to the orbiting observer who has just completed one orbit. In this case both observers are moving on geodesics, so in any local neighborhood of a point on their path it should look like they are moving inertially with all the normal rules of SR applying in this local neighborhood; and yet, if they synchronized their clocks at the first moment they passed, I don't think their clocks would still be synchronized at the second moment they pass. Surely this does not mean that this simple situation violates the principle of relativity, or implies a "preferred frame" in terms of the fundamental laws of physics? I'm pretty sure you can't compare two separate local regions like this as if they are two crossings in SR flat spacetime, the principle of relativity as applied to GR just means that if you look at a single local region of spacetime, an observer following a geodesic through that region will locally observe the laws of physics to work just like an inertial observer would see them work in SR.

The fact that the global topology imparts a preferred frame which says that it is A's clock that will register the greatest time elapse means that A and not B can, at the initial local encounter, think of their time as being 'absolute' in some sense. This I understand to be in contradiction to the Principle of Relativity.

I'm pretty sure you're wrong. Again, in GR the principle of relativity as I understand it only says that if you look at a single local region of spacetime, within that region the laws of physics must work just like in SR, including the symmetry between different locally inertial observers' view of events within that local region. But if you look at things non-locally, then even without invoking weird topologies you can still have situations where two different geodesic paths cross at two different points, and the geometry of spacetime tells you which of two observers traveling along these paths will have elapsed more time on their clock. If the cosmological twin paradox was a violation of the principle of relativity, then any such situation would have to be one too, even the simple one I outlined above with one observer orbiting the Earth and the other shot upwards from the surface and then falling back down.

Are these 'copies' of the Earth the actual one Earth experienced after circumnavigations of the universe, or are we saying that the world we know is itself replicated many/infinite number of times?

The actual one earth. I was just describing how things would look in each observer's coordinate system, assuming they construct their coordinate systems in the same way as in SR, but allow the spatial axes to keep wrapping around the closed space over and over, so that each event would have multiple coordinates. For example, the departure of the rocket from the Earth in the Earth's coordinate system might have coordinates x=0 l.y., t=0 y, but also x=5 l.y., t=0 y, x=10 l.y., t=0 y, x=15 l.y., t=0 y, and so on.

Also, with a sufficiently powerful telescope you could see multiple images of the same object at different distances from you, with each image being caused by light that has circumnavigated the universe a different number of times before reaching your telescope, so this is another sense in which there'd be "copies" of the earth. Visually, if a rocket circumnavigated the universe and returned to earth, it would look like the rocket that departed "my" Earth landed on the distant image of the Earth on my right, while the rocket that landed on my Earth would appear to be the one that had departed the distant image of the Earth on my left.

I would argue, contrary to MTW, that mass in the universe is essential to resolve this paradox and that A's 'absolute' frame of reference is that defined by the Centre of Mass/momentum of the matter in the universe at large.

But if you agree that an empty compact universe is a valid solution in GR--whatever the definition of "valid solution" is used, perhaps a spacetime manifold where the Einstein field equations are obeyed at every point--then how can you argue that there's a genuine paradox without saying that the paradox is inherent to GR itself, or denying that GR really does respect the principle of relativity?

 

[Garth]

No, I don't think that counts as a valid test of the "principle of relativity" in GR. After all, even without picking out weird topologies you can do something similar in the neighborhood of the earth--have two observers, one orbiting the Earth and the other shot up from the surface at slightly less than escape velocity, in such a way that the shot-up observer passes right next to the orbiting observer on his way up, moves away from the Earth for a while and finally begins to fall back down, with everything timed so that on his way down he will again pass right next to the orbiting observer who has just completed one orbit. In this case both observers are moving on geodesics, so in any local neighborhood of a point on their path it should look like they are moving inertially with all the normal rules of SR applying in this local neighborhood; and yet, if they synchronized their clocks at the first moment they passed, I don't think their clocks would still be synchronized at the second moment they pass. Surely this does not mean that this simple situation violates the principle of relativity, or implies a "preferred frame" in terms of the fundamental laws of physics?

I'm pretty sure you can't compare two separate local regions like this as if they are two crossings in SR flat spacetime, the principle of relativity as applied to GR just means that if you look at a single local region of spacetime, an observer following a geodesic through that region will locally observe the laws of physics to work just like an inertial observer would see them work in SR. I'm pretty sure you're wrong. Again, in GR the principle of relativity as I understand it only says that if you look at a single local region of spacetime, within that region the laws of physics must work just like in SR, including the symmetry between different locally inertial observers' view of events within that local region.

This experiment relies on the Earth's gravitational field. Extend this gedanken by drilling holes through the centre of the Earth. Now include in these inertial observers the one at the centre of the Earth, in free fall and yet stationary wrt the Earth. Let the orbiting observers now fly through this COM laboratory. The observer whose clock records the longest interval between all such encounters will be the stationary one, and this may be defined therefore as an 'absolute' frame, different from all the rest as having the greatest proper time interval. It is a frame of reference that is defined by the presence of the Earth's mass. At the COM laboratory the field is locally flat, in a 'small enough' region around the COM, and a particular frame of reference is different from all the rest. The Prinicple of Special Relativity does not hold!

But if you agree that an empty compact universe is a valid solution in GR--whatever the definition of "valid solution" is used, perhaps a spacetime manifold where the Einstein field equations are obeyed at every point--then how can you argue that there's a genuine paradox without saying that the paradox is inherent to GR itself, or denying that GR really does respect the principle of relativity?

I believe the paradox is inherent to GR itself, that is why in my work http://en.wikipedia.org/wiki/Self_creation_cosmology I include Mach's Principle and violate the Principle of Relativity - the COM Machian frame is the preferred frame.

Garth

 

[JesseM]

This experiment relies on the Earth's gravitational field. Extend this gedanken by drilling holes through the centre of the Earth. Now include in these inertial observers the one at the centre of the Earth, in free fall and yet stationary wrt the Earth. Let the orbiting observers now fly through this COM laboratory. The observer whose clock records the longest interval between all such encounters will be the stationary one, and this may be defined therefore as an 'absolute' frame, different from all the rest as having the greatest proper time interval. It is a frame of reference that is defined by the presence of the Earth's mass. At the COM laboratory the field is locally flat, in a 'small enough' region around the COM, and a particular frame of reference is different from all the rest. The Prinicple of Special Relativity does not hold!

OK, then you are defining the "principle of special relativity" in such a way that it is violated in GR. However, most physicists would not define it this way, I think. I don't think it even makes sense to talk about the "principle of special relativity" in GR except in a local sense, since this principle only says the laws of physics should look the same in coordinate systems constructed in a certain way (a system of measuring-rods and clocks at rest with respect to each other, with the clocks synchronized using the Einstein synchronization convention) which is not really possible in curved spacetime where the measuring-rods cannot remain rigid. It is still true that if you look at only a single local region of spacetime, and don't compare multiple regions as you are doing, that within that region the laws of physics work just like they do in SR (in the limit as the size of the region goes to zero).

 

[pervect]

I'll have to take back my previous remark (which I've deleted to minimize confusion, since I think now I was wrong).

Lets' try to do the calculation of which time is longer.

(I tried to be careful, but I could use a double check. Having an annoying cold isn't helping my accuracy).

First we need a maetric for an object inside of the Earth. We can use the Newtonian approximation, where

goo = 1-2U
g11 = 1

-U is the potential enregy, which in the inside of the planet is
<br /> -U = -\frac{3}{2}\frac{M}{r_0}+\frac{1}{2}\frac{M}{r_0} \left( \frac{r}{r_0} \right)^2<br />

Geometric units are used so G=c=1

M is the mass of the planet
r_0 is the radius of the planet
0<r<r_0 is the position coordinate

This gives a force, dU/dr, of
<br /> \frac{M}{{r_0}^2}\frac{r}{r_0}<br />

which is a hooke's law force with the correct value at the surface r=r_0. In addition U has the proper value at the surface as well.

So we have

g_{00} = 1 -\frac{3M}{r_0} + \frac{M}{r_0}\left(\frac{r}{r_0}\right) ^2<br />

Now we need to integrate d\tau = \sqrt{g_{00}(t) - v^2(t)}

We can approximate \sqrt{1+a+b+c} = 1 + 1/2(a+b+c) to make our job of integration easier.

We know that an object experience a hooke's law force will move in sinusoidal motion, thus r(t) = r_0 sin(wt).

Thus

<br /> \tau = \int (1 - \frac{3}{2}\frac{M}{r_0}) dt + \frac{1}{2}\int \frac{M}{r_0}\left( \frac{r(t)}{r_0}\right)^2 dt - \frac{1}{2} \int v^2(t) dt<br />

This adds a positive term that's equal to

<br /> \frac{1}{2}\frac{M}{r_0} T \int_0^\pi sin^2 <br />

and a negative term equal to

<br /> -\frac{1}{2}v_{max}^2 T \int_0^\pi cos^2<br />

where T = \int dt

Now \frac{v_{max}^2}{2} = \frac{M}{2r_0} therefore v_{max}^2 = M/r_0

This means that the postive and negative terms are equal, and the two times are the same (?!).

I think this makes sense from the virial theorem. T=V for a hooke's law force, T=-V/2 for an inverse square law force (I looked this up in Goldstein). Here T = time avg of kinetic energy, V = time avg of potential energy.

Under the square root, we have 2U twice the potential energy and v^2 which is twice the kinetic energy. Since T=V for a hooke's law force, the terms cancel.

Outside the planet, with an inverse square law the terms shouldn't cancel, and I believe the object thrown upwards will have the longer time. But I haven't double checked this.

 

[Garth]

pervect That is an interesting calculation, however I don't think it has correctly dealt with the situation:

This means that the postive and negative terms are equal, and the two times are the same (?!)

Is not this result simply the consequence of:

First we need a metric for an object inside of the Earth. We can use the Newtonian approximation, where

goo = 1-2U
g11 = 1

i.e. by starting with the Newtonian approximation you end up with a Newtonian result?

I am quite sure that with a (-+++) metric any extra motion of a particle on a geodesic between two events will result in its proper time duration being shortened.

OK, then you are defining the "principle of special relativity" in such a way that it is violated in GR. However, most physicists would not define it this way, I think. I don't think it even makes sense to talk about the "principle of special relativity" in GR except in a local sense, since this principle only says the laws of physics should look the same in coordinate systems constructed in a certain way (a system of measuring-rods and clocks at rest with respect to each other, with the clocks synchronized using the Einstein synchronization convention) which is not really possible in curved spacetime where the measuring-rods cannot remain rigid. It is still true that if you look at only a single local region of spacetime, and don't compare multiple regions as you are doing, that within that region the laws of physics work just like they do in SR (in the limit as the size of the region goes to zero).

I do think my understanding of the Principle of Relativity as being "no preferred frames of reference" ( in a local laboratory) is enshrined in the foundations of GR, particularly in the conservation laws and the use of the principle of Least Action in 4D space-time.

The point about using Lagrangian methods is that they work for generalised coordinates, in particular in a (-+++) metric, they work for all Lorentzian frames of reference, and they thereby guarantee the conservation of energy-momentum and the conservation, wrt covariant differentiation, of the stress-energy tensor.

The Bianchi identities guarantee the conservation, wrt covariant diffrentiation, of the Einstein tensor in the GR field equation and the constant G guarantees the consistency of that field equation's conservation properties.

NB: the Brans-Dicke theory allows G to vary and has to introduce extra scalar field terms to maintain the conservation qualities of its field equation.

If, however, there are preferred frames of reference, defined by the presence of local or cosmological mass, or the global topology of a compact space, then this restriction of the conservation of energy-momentum need not necessarily be enforced. This might then allow mass creation for example as it appears in http://en.wikipedia.org/wiki/Self_creation_cosmology .

Garth

 

[pervect]

pervect That is an interesting calculation, however I don't think it has correctly dealt with the situation: Is not this result simply the consequence of:i.e. by starting with the Newtonian approximation you end up with a Newtonian result?

I don't agree - the effect of g11 should be second order, as it only modifies the velocity term slightly. The approximations should be perfectly resonable, and adequate to demonstrate relativistic effects (the apprxoimations include the two important terms of gravitational time dilation and relativistic time dilation).

I am quite sure that with a (-+++) metric any extra motion of a particle on a geodesic between two events will result in its proper time duration being shortened.

BOTH paths are following geodesics. They are local maximums of the elapsed time, but you can't tell which is the longest unles you calculate them and compare them.

Consider the following example on the 2D surface of a curved 3D space

           x
          xxx
         xxxxx
       xxxxxxxx
     xxxxxxxxxxx
    xxxxxxxxxxxxx
A  xxxxxxxxxxxxxxx  B

Points A and B are separated by a tall mountain. There is a "straight line" path from A to B over the mountain. It is satisfies the geodesic equation. However, the existence of this geodesic path does not mean that there is not a shorter path around the mountain!

This shorter path must also be a geodesic, of course, if it is truly the shortest.

The shortest distance between two points is always a straight line, but a given straight line connecting two points is not always the shortest distance - there may be another line connecting them that is shorter in curved spaces.

Another example. On a sphere, a great circle is a geodesic path. You can get from point A on the great circle to point B on the great circle by heading in either direction. One direction will generally be shorter than the other, however. Both paths are geodesics - but one path is shorter than the other.

 

[Garth]

pervect I see what you are saying, both observers, the one remaining at the Centre of the Earth and the other one on a rectilinear orbit are in free fall and think it is the other that is moving. However, the metric of the Schwarzschild solution used is anchored to the COM of the system, and I am sure that the COM inertial observer will have the longest proper time duration.

Therefore, I would like to redo the calculation in the Post Newtonian approximation. Remember in the calculation of the deflection of light the spatial curvature term, g₁₁, makes an equal contribution as the time dilation term, g₀₀, to the total result.

Garth

 

[JesseM]

I do think my understanding of the Principle of Relativity as being "no preferred frames of reference" ( in a local laboratory) is enshrined in the foundations of GR, particularly in the conservation laws and the use of the principle of Least Action in 4D space-time.

If I'm understanding your argument correctly, I think your interpretation of what "in a local laboratory" means is incorrect. You can't have two small laboratories pass each other, then travel on different paths for significant time periods, then reunite later, and compare what has happened to each laboratory between the two meetings, saying that any asymmetry in how much time has elapsed on each lab's clock between these meetings indicates a failure of the principle of relativity. The principle of relativity only applies to measurements in a single small region of spacetime, not a comparison of two distinct small regions of spacetime separated by a large spacetime interval.

 

[Garth]

If I'm understanding your argument correctly, I think your interpretation of what "in a local laboratory" means is incorrect. You can't have two small laboratories pass each other, then travel on different paths for significant time periods, then reunite later, and compare what has happened to each laboratory between the two meetings, saying that any asymmetry in how much time has elapsed on each lab's clock between these meetings indicates a failure of the principle of relativity. The principle of relativity only applies to measurements in a single small region of spacetime, not a comparison of two distinct small regions of spacetime separated by a large spacetime interval.

I'm not using two laboratories.
Barrow and Levin say in The twin paradox in compact spaces

The resolution hinges on the existence of a preferred frame introduced by the topology

When the two observers pass close by the first time in the single local laboratory already one of them is maked off as being in the 'preferred frame'. It is true that you have to wait until the second encounter to do the experiment and discover which observer it is, or you could simply look out and see what the matter in the rest of the scenario is doing and discover which observer is at 'rest'.

Garth

 

[JesseM]

I'm not using two laboratories.

Well, you're at least using one laboratory at two significantly different points in time, as opposed to one arbitrarily small laboratory looked at during a single arbitrarily small time interval.

Barrow and Levin say in The twin paradox in compact spaces

The resolution hinges on the existence of a preferred frame introduced by the topology

Yes, but they aren't talking about a local preferred frame, they're just saying that if you create global coordinate systems for each twin using the same procedure that's used in SR, then one of these coordinate systems is in some sense "preferred" (for example, they'll be only one such coordinate system where each apparent copy of the same clock is synchronized with every other copy). Also, my understanding is that because of diffeomorphism invariance, if you express the laws of physics in different global coordinate systems using tensor equations, the laws will be the same in each coordinate system--it's only when you try to express how things look without using tensor equations that they may look different in different coordinate systems. And this would probably be true in any curved spacetime, it wouldn't be specific to the compact universe scenario.

When the two observers pass close by the first time in the single local laboratory already one of them is maked off as being in the 'preferred frame'.

Not by any experiment that can be observed within that laboratory. If you look only at the results of experiments in the lab, you see complete symmetry.

It is true that you have to wait until the second encounter to do the experiment and discover which observer it is, or you could simply look out and see what the matter in the rest of the scenario is doing and discover which observer is at 'rest'.

Yes, and because you have to do these things, I don't think you are testing the "principle of relativity" as physicists define it. As I understand it, the principle of relativity in GR only says that within a single small region of spacetime, ignoring everything outside that region, the laws of physics look just like they do in SR, including the symmetry between different inertial observers.

 

[pervect]

Therefore, I would like to redo the calculation in the Post Newtonian approximation. Remember in the calculation of the deflection of light the spatial curvature term, g₁₁, makes an equal contribution as the time dilation term, g₀₀, to the total result.

Garth

I believe that g11 is 1+2U in the PPN approximation? Any g11 > 1 will tip the balance towards favoring staying still, but the effect is extremely small - because the velocities involved are extremely small (nowhere near lightspeed, where the contribution does become equal).

Changing the density profile of the planet will probably have a bigger effect than including g11.

The most important point, IMO, is that if one changes the problem so that the clock is allowed to reach an altitude above the surface of the planet, rather than being confined to the interior, I would predict that the lowest time would definitely be seen by the "moving" clock.

 

[Garth]

I believe that g11 is 1+2U in the PPN approximation? Any g11 > 1 will tip the balance towards favoring staying still, but the effect is extremely small - because the velocities involved are extremely small (nowhere near lightspeed, where the contribution does become equal).

Changing the density profile of the planet will probably have a bigger effect than including g11.

The most important point, IMO, is that if one changes the problem so that the clock is allowed to reach an altitude above the surface of the planet, rather than being confined to the interior, I would predict that the lowest time would definitely be seen by the "moving" clock.

I concur.

Garth

 

[Garth]

Well, you're at least using one laboratory at two significantly different points in time, as opposed to one arbitrarily small laboratory looked at during a single arbitrarily small time interval. Yes, but they aren't talking about a local preferred frame, they're just saying that if you create global coordinate systems for each twin using the same procedure that's used in SR, then one of these coordinate systems is in some sense "preferred" (for example, they'll be only one such coordinate system where each apparent copy of the same clock is synchronized with every other copy). Also, my understanding is that because of diffeomorphism invariance, if you express the laws of physics in different global coordinate systems using tensor equations, the laws will be the same in each coordinate system--it's only when you try to express how things look without using tensor equations that they may look different in different coordinate systems. And this would probably be true in any curved spacetime, it wouldn't be specific to the compact universe scenario. Not by any experiment that can be observed within that laboratory. If you look only at the results of experiments in the lab, you see complete symmetry. Yes, and because you have to do these things, I don't think you are testing the "principle of relativity" as physicists define it. As I understand it, the principle of relativity in GR only says that within a single small region of spacetime, ignoring everything outside that region, the laws of physics look just like they do in SR, including the symmetry between different inertial observers.

It depends on what you are prepared to call a "law of physics". I understand about the Principle of Equivalence! IMHO the 'compact space twin paradox' is interesting because clock rate is fundamental to physical experiment and it seems that at the second encounter, also determined by observation of the rest of the universe, one clock measured duration will definitely be greater than the other.

Garth

 

[ralfsmith]

Oh its very interesting. Could you provide me more information ?

johntvery@operamail.com

johntvery@hotmail.com

 

[yogi]

IMHO the 'compact space twin paradox' is interesting because clock rate is fundamental to physical experiment and it seems that at the second encounter, also determined by observation of the rest of the universe, one clock measured duration will definitely be greater than the other.

Garth

But can't the same reasoning be applied to any pair of clocks that pass with relative velocity v - there is no guarantee that the proper rate of the two clocks is the same - and in general it will not be. For example if at some point in the distant past two clocks had been synchronized and put in relative uniform motion toward one another by a short duration acceleration, which could be applied to one or the other (or both but with a different magnitude), then upon passing each could mark the time on the other clock and after they had separated some distance L either could be decelerated to bring their relative velocity to zero - at this point both clocks are in the same frame and they can be stopped and compared - depending upon which clock accumulated the most time between the point at which they passed, the difference between the proper rates would be revealed.

 

[dicerandom]

yogi: I believe what you have described is essentially a more complicated version of the flat space twin paradox. Which clock reads more is determined by the nature of the accelerations and can be determined using k-calculus on spacetime diagrams.

 

[Garth]

Oh its very interesting. Could you provide me more information ?

johntvery@operamail.com
johntvery@hotmail.com

Hi ralfsmith! Have you read all the material already posted and linked? on
http://www.findarticles.com/p/articles/mi_qa3742/is_200108/ai_n8954244
cosmological twin paradox,
The twin paradox in compact spaces
Twin paradox and space topology?

Read these and if you have any further questions come back and ask again!

Garth

 

[yogi]

dicerandom - correct - what i was attempting to illustrate was a common misconception at the outset that leads to an apparent paradox - the reasoning goes like this - two clocks pass each other and each observes the other clock to be running slow - but it is impossible that each can be actually running slower than the other - yet when this initial assumption is followed by a round trip analysed which includes a turn-around accceleration, there is a real time difference when the clocks are ultimately brought to rest and compared in the same frame. I maintain you cannot synchronize clocks in relative motion - you can read a passing clock - but you cannot be assured that the proper rate of two clocks in relative motion will be equal. In the cosmological twin paradox, there is a similar ambiguity at the outset (first passing) and the clocks will continue to run at different proper rates until they re-encounter, ergo there is no difference in the cosmological case than the local round trip case - each involved an initial difference in velocity due to some past acceleration; the root of both the local and global paradox lies in the failure to properly consider this initial condition.

 

[Garth]

Hi yogi! I also agree with dicerandom, the point about the cosmological twin paradox is that the two observers are always in inertial frames of reference, there is no acceleration, they are both on geodesics all the time.

I don't understand what you mean by your 'initial condition'. You don't synchronise the clocks, just read them twice, which you can do so while in relative motion by passing a light signal, as long as they pass arbitrarily close to each other.

Consider two clocks, one freely floating, 'sitting', in a cave at the COM of the Earth and the other traveling at high speed through that cave along a tunnel in a vacuum tube on a rectilinear orbit, they compare times at the encounter.

They both think the other is moving and they are stationary, so each thinks the other clock is running 'slow'.

Then they encounter each other again and compare times. Each thinks they have recorded the greatest time elapse between encounters and yet only one can actually do so, so which one is it?

The truly 'stationary' clock must be defined by the mass of the Earth.


Garth

 

[yogi]

If the truly stationary clock is defined by the mass of the Earth - then you are implying the Earth's mass creates a preferred frame - this as you already know smacks of LR ...the notion that MMX and other experiments are better explained by considering the Earth mass as creating a preferred frame as opposed to SR which treats all inertial frames as equivalent. LR is a competing theory of SR usually associated with relativity dissidents ...that is not a problem from my perspective, but it is a yet to be verified theory
But let's take the case you have proposed - what were the initial conditions? - was the oscillating clock originally at rest at the center of the Earth and synchronized with the clock which remained at the center - then raised to the surface potential and dropped to be forever in oscillatory motion? And is this any different than a GPS satellite clock that is in a free float inertial frame. In the latter case its clear that, if we ignor the height factor, the Earth centered clock will run faster than an uncompensated GPS clock - in other words the proper rate of the satellite clock is slower than the proper rate of the Earth centered clock - if we take the Earth centered clock to a tower at the North pole, each time the GPS clock passes by, it will believe the Earth clock to be running slow by making overservational experiments as it passes by - and likewise the clock at the top of the tower will believe the satellite clock to be running slow - but the conflict as I have maintained in other threads lies in the fact that these measurements determine what is apparent - neither is actually measuring the proper rate of the other clock - they are only measuring the apparent rate of the other clock in relation to their own.

 

[yogi]

Garth - to further embellish on what I think I am trying to say, consider a non rotating Earth centered clock A and a GPS satellite clock B that is compensated only for its height. If A sends out radar pulses every 10 microseconds according to A's measurement of time 10 and B sends out radar pulses every 10 microsceonds according to B's measurment of time, and both A and B measure the times of arrival of the return pulses - then since the distance between A and B is always constant for a circular orbit, A will be able to determine that the proper rate of B clock is always less than the proper rate of A clock. In other words, this geometry provides a convenient method of determining the difference in the proper rate of both clocks. Real time dilation involves a difference between proper rates - there is no paradox here. I do not see why the same reasoning cannot be applied to the cosmological case i.e., assuming the universe to be a Hubble sphere - put a clock at some point C and establish two other clocks J and K at R = c/H that pass each other as they circumscribe the universe following a geodesic - although we would have to wait a long time for the signals to be returned - in our imagination we could surmise that the C clock would reveal the difference between the proper rates of J and K

 

[Garth]

they are only measuring the apparent rate of the other clock in relation to their own.

That is all that they can ever measure. However they can each make the two measurements of time at consequtine encounters. And then radio each other the results. One result will definitely be a longer duration than the other, but which one and how is that to be determined.

I do not see how the inital conditions resolve the paradox in GR, for the gedanken experiment both clocks are simply dropped, one at the COM and the other down the vacuum tube from some height, well above the Earth's surface if necessary. They do not have to be synchronised, just working accurately.

They each record t₁ & t'₁ at the first encounter and t₂ & t'₂ at the second. Once the results have been exchanged, by radio, then they can each compare t₂ - t₁ against t'₂ - t'₁ and see which is greater.

I agree that if the Earth determines which inertial frame of reference records the longer duration then that would not be GR. But how else would the result be determined? My POV is that GR needs to fully include Mach's Principle which is what I have done in http://en.wikipedia.org/wiki/Self_creation_cosmology at this moment.

Garth

 

[Garth]

Garth - to further embellish on what I think I am trying to say, consider a non rotating Earth centered clock A and a GPS satellite clock B that is compensated only for its height. If A sends out radar pulses every 10 microseconds according to A's measurement of time 10 and B sends out radar pulses every 10 microsceonds according to B's measurment of time, and both A and B measure the times of arrival of the return pulses - then since the distance between A and B is always constant for a circular orbit, A will be able to determine that the proper rate of B clock is always less than the proper rate of A clock. In other words, this geometry provides a convenient method of determining the difference in the proper rate of both clocks. Real time dilation involves a difference between proper rates - there is no paradox here. I do not see why the same reasoning cannot be applied to the cosmological case i.e., assuming the universe to be a Hubble sphere - put a clock at some point C and establish two other clocks J and K at R = c/H that pass each other as they circumscribe the universe following a geodesic - although we would have to wait a long time for the signals to be returned - in our imagination we could surmise that the C clock would reveal the difference between the proper rates of J and K

Our posts crossed.

IMHO the difference in this example is the issue of a local laboratory. It is because the two inertial observers pass arbitrarily close to each other that the paradox arises, both should be equivalent, yet they are not.

This seems to raise questions about the Equivalence Principle.

If as you correctly suggest the difference in clock rate is due to the geometry of space-time, or topology in the cosmological case, what is it that determines that geometry if is not the mass of the Earth?

Garth

 

[Hurkyl]

It is because the two inertial observers pass arbitrarily close to each other that the paradox arises, both should be equivalent, yet they are not.

They are equivalent.

Equivalent things are allowed to give different answers when they're asked different questions.

It almost seems like you're suggesting "they travel inertially" is a complete specification of the two problems -- which would explain why you think there's a paradox.

 

[Garth]

They are equivalent.

Equivalent things are allowed to give different answers when they're asked different questions.

It almost seems like you're suggesting "they travel inertially" is a complete specification of the two problems -- which would explain why you think there's a paradox.

Hi Hurkyl!
Do you think they are equivalent in the cosmological case of a closed 'compact' space?

And what are the "different questions" you refer to in the Earth centred case? The only question I have asked each observer is: "How much time has elapsed between encounters?"

Garth

 

[Hurkyl]

Hi Hurkyl!
Do you think they are equivalent in the cosmological case of a closed 'compact' space?

Yes. But again, you're asking two different questions, so it is not a paradox that you get two different answers.

The whole thing seems trivial (to me) if you ask it in geometric terms:

(1) We draw two non-parallel straight lines L and M on a cylinder.
(2) The lines intersect multiple times. Pick two consecutive intersections and call them P and Q.
(3) Compute the length of the line segment PQ along L.
(4) Compute the length of the line segment PQ along M.
(5) Gasp in confusion when the two lengths aren't equal!

Of course, to really be an exact description of the cosmological twin paradox, we need to integrate the Minowski metric along the line segments instead of the Euclidean metric.

The only question I have asked each observer is: "How much time has elapsed between encounters?"

Yes: you've asked two different questions.

"Observer 1: how long was it?"
"Observer 2: how long was it?"

These questions are not identical -- there's no reason to think they should have the same answer.

 

[Garth]

Yes. But again, you're asking two different questions, so it is not a paradox that you get two different answers.

The whole thing seems trivial (to me) if you ask it in geometric terms:

(1) We draw two non-parallel straight lines L and M on a cylinder.
(2) The lines intersect multiple times. Pick two consecutive intersections and call them P and Q.
(3) Compute the length of the line segment PQ along L.
(4) Compute the length of the line segment PQ along M.
(5) Gasp in confusion when the two lengths aren't equal!

Of course, to really be an exact description of the cosmological twin paradox, we need to integrate the Minowski metric along the line segments instead of the Euclidean metric.

In agreement with your example let us take the compact space to be Einstein's static cylindrical model for the sake of the argument.

There is no surprise in the fact that the two lengths/space-time intervals are different, the problem is: "In whose frame of reference is the diagram drawn?"

You can draw it in either observer's frame, they both think they are the one that is stationary and the other is the one that is moving.

So which one actually does have the longer duration, and how is that observer to be selected?

Yes: you've asked two different questions.

"Observer 1: how long was it?"
"Observer 2: how long was it?"

These questions are not identical -- there's no reason to think they should have the same answer.

Well I call that asking the same question to two different observers.

I do not think they do have the same answer!

That is the problem, again if, as you say they are equivalent, whose answer gives the longer duration and how is that observer selected?

GArth

 

[pervect]

In agreement with your example let us take the compact space to be Einstein's static cylindrical model for the sake of the argument.

There is no surprise in the fact that the two lengths/space-time intervals are different, the problem is: "In whose frame of reference is the diagram drawn?"

My understanding is this:

The winding number is a topological quantity that is not dependent on any particular choice of reference frame (i.e. choice of coordinates).

I'm not quite sure of the mathematical details.

Given that we assume that the above statement is true, the answer to the question becomes clear - the winding number distinguishes the obsevers.

 

[Hurkyl]

So which one actually does have the longer duration, and how is that observer to be selected?

That is the problem, again if, as you say they are equivalent, whose answer gives the longer duration and how is that observer selected?

By computation! You integrate the metric along the worldline.

In the SR analysis of the classic twin paradox, we have a very clever way of avoiding direct computation: we can invoke the Minowski version of the triangle inequality.

There isn't a general purpose shortcut, though. Unless you have a specialized theorem to invoke for the situation at hand, you have to compute.


Unfortunately, I don't know of a good way of actually computing things without picking a coordinate chart. But the process -- and the answer -- is the same no matter what chart we use.

My understanding is this:

The winding number is a topological quantity that is not dependent on any particular choice of reference frame (i.e. choice of coordinates).

I'm not quite sure of the mathematical details.

Given that we assume that the above statement is true, the answer to the question becomes clear - the winding number distinguishes the obsevers.

The winding number will allow you to make statements such as:

"He's gone around the universe one more time than me in that direction!"

which is, of course, equivalent to "I've gone around the universe one more time than him in the opposite direction!"

But this is still not a complete description of their paths on the cylindrical space-time: it still contains insufficient information to figure out which one measures more time between meetings.

 

[Garth]

By computation! You integrate the metric along the worldline.

In the SR analysis of the classic twin paradox, we have a very clever way of avoiding direct computation: we can invoke the Minowski version of the triangle inequality.

There isn't a general purpose shortcut, though. Unless you have a specialized theorem to invoke for the situation at hand, you have to compute.


Unfortunately, I don't know of a good way of actually computing things without picking a coordinate chart. But the process -- and the answer -- is the same no matter what chart we use.

The problem is that the oscillating observer thinks that as she suffers no inertial forces she can take herself to be stationary. In a totally equivalent calculation you have to integrate the metric along her worldline using the Schwarzschild metric transformed into her system of coordinates in which she remains at the centre and it is the other observer, and the Earth that is moving.

In that case her duration is easy: as dx' = dy' = dz' = 0 then

\int d\tau&#039; = \int dt&#039;

the problem is working it out for the other observer in these coordinates.

The winding number will allow you to make statements such as:

"He's gone around the universe one more time than me in that direction!"

which is, of course, equivalent to "I've gone around the universe one more time than him in the opposite direction!"

But this is still not a complete description of their paths on the cylindrical space-time: it still contains insufficient information to figure out which one measures more time between meetings.

I concur, I think this problem exists in both problems, the answer to the conundrum in my understanding must be that the extra required information is given by the distribution of the other mass in the universe/Earth. i.e. It is a paradox that is only resolved by application of Machian principles.

Garth

 

[yogi]

That is all that they can ever measure. However they can each make the two measurements of time at consequtine encounters. And then radio each other the results. One result will definitely be a longer duration than the other, but which one and how is that to be determined.
Garth

When measurements of the clocks are made during close fly-by by reading the other clock - you are actually obtaining the actual time dilation (the difference between the proper time accumulated by one clock in comparison to the proper time accumulated by the other clock - the accumulated proper times in general will not be equal - even though both clocks are following cosmic geodesics (or a local orbit). But that is not paradoxical - it only becomes paradoxical if one attempts to incorporate w/i the same argument, the apparent slowing of the other clock that would be observed by each observer when he considers himself stationary. This latter measurement is a distortion of reality - in actuality we never make this experiment - we simply take it as a true. Moreover it is tacitly assumed that since both clocks are in inertial frames, each clock is equally capable of making a measurement and that the measurment would be the same if the roles are reversed - but where is it written that both clocks will be running at the same proper rate - and if they are not then they would not make identical measurements of the slowing of the other clock

While the twin problem can be analysed using apparent observations - and this methodology leads to the same time loss as that obtained directly by differencing the accumulated proper time logged by each clock, the mathematical statements cannot logically be both true at the same time.

 

[yogi]

Garth - i was wondering if Mach's principle were applied to a universe where all matter exists in a concentrated lump ..e.g., at the center of a Hubble sphere - the metric for the surrounding space is spherically symmetrical as determined by the central mass - Would a clock orbiting the central mass and a clock following a geodesic determined by the central mass be governed by the same relativistic relationships?

 

[Garth]

When measurements of the clocks are made during close fly-by by reading the other clock - you are actually obtaining the actual time dilation (the difference between the proper time accumulated by one clock in comparison to the proper time accumulated by the other clock - the accumulated proper times in general will not be equal - even though both clocks are following cosmic geodesics (or a local orbit). But that is not paradoxical - it only becomes paradoxical if one attempts to incorporate w/i the same argument, the apparent slowing of the other clock that would be observed by each observer when he considers himself stationary. This latter measurement is a distortion of reality - in actuality we never make this experiment - we simply take it as a true. Moreover it is tacitly assumed that since both clocks are in inertial frames, each clock is equally capable of making a measurement and that the measurment would be the same if the roles are reversed - but where is it written that both clocks will be running at the same proper rate - and if they are not then they would not make identical measurements of the slowing of the other clock

While the twin problem can be analysed using apparent observations - and this methodology leads to the same time loss as that obtained directly by differencing the accumulated proper time logged by each clock, the mathematical statements cannot logically be both true at the same time.

I concur yogi, although we cannot actually make this measurement, though I suppose in future you might use an asteroid and its field as the base for an experiment, it is a useful and instructive scenario for a 'gedanken'.

If the mathematical statements cannot be true at the same time, which of course I agree with, it is where I come in, then what needs to be changed, the fact that equivalent inertial clocks are not equally capable of making a measurement? If that is the case, what principle do you use to decide between them?

As far as a 'lump' universe is concerned both inertial clocks would be telling their own time, and it would be possible to transform from one time scale to the other. A third clock that might be considered to be in a privileged position would be the one at the Centre of Mass and the one 'at infinity' from the mass, and comoving with it. That clock in my understanding would be recording the greatest proper time between any contrived inertial encounters.

Garth

 

[Hurkyl]

The problem is that the oscillating observer thinks that as she suffers no inertial forces she can take herself to be stationary.

Why is that a problem?

I concur, I think this problem exists in both problems, the answer to the conundrum in my understanding must be that the extra required information is given by the distribution of the other mass in the universe/Earth.

Well, it's wrong. The mass distribution doesn't contribute anything to the problem. The metric is all that matters.


Now, it might be possible, by watching all matter get pushed around for all time, to solve for the metric and then have enough information to work out the proper time along a path. I don't know enough about GR to know if the Einstein field equations are that strong.

However, I do know it is possible for two different metrics to push all matter around in the exact same way. So knowing the mass simply might not be enough to work out the time everybody experiences.

However this does lead to a measurement problem (but not a paradox): the difference between idealized and physical clocks. If two different metrics (meaning different readouts for idealized clocks, because the proper time is different!) have identical action on matter, that should extend to physical clocks.

 

[George Jones]

Up until now I have agreed with what Hurkyl has said in this thread, but now he's lost me somewhat.

The mass distribution doesn't contribute anything to the problem. The metric is all that matters. ... However, I do know it is possible for two different metrics to push all matter around in the exact same way.

I don't understand this. A solution to Einstein's equation includes a pair (g, t), where g is the metric and T is the energy-momentum tensor.

Are you saying that it's possible for (g, T) and (h, T), where g =/= h (but T is the same), to both be solutions to Einstein's equation, including the same boundary conditions?

Could you give an example?

Regards,
George

 

[Garth]

Why is that a problem?

Because, in their own inertial coordinate systems, they both can take themselves to be stationary. Therefore, if as you say the two observers are equivalent, why should one have the 'right' answer and the other the 'wrong' one?

To reiterate, they both think that their clock, and not the other clock, should have recorded the greater time elapse. Yet obviously if the two time elapses are not equal only one obsever will be correct, but which one?

On what basis do you select between the two?

Well, it's wrong. The mass distribution doesn't contribute anything to the problem. The metric is all that matters.

I disagree.

In agreement with what Bernard said later, what is it that determines the metric if it is not the distribution of mass and energy?

However this does lead to a measurement problem (but not a paradox): the difference between idealized and physical clocks. If two different metrics (meaning different readouts for idealized clocks, because the proper time is different!) have identical action on matter, that should extend to physical clocks.

I am not sure what you mean here. By "identical action on matter" being extended to physical clocks, do you mean the two clocks should record identical time durations between consecutive encounters? Surely this is not the situation we are discussing in this paradox.

In fact, as we have established, in the situation posited above the clocks record different time durations; the paradox lies in the fact that at the second encounter, in an arbitrarily small enough region, both the two clocks have remained in inertial frames of reference, on geodesics, throughout, and therefore, as you said, are equivalent.

So on what basis do you choose between them?

Garth

 

[JesseM]

Because, in their own inertial coordinate systems, they both can take themselves to be stationary. Therefore, if as you say the two observers are equivalent, why should one have the 'right' answer and the other the 'wrong' one?

Are you talking about the situation of one observer at the center of the Earth and the other oscillating up and down? How can either have a non-local "inertial coordinate system" when they are in curved spacetime?

 

[Garth]

Are you talking about the situation of one observer at the center of the Earth and the other oscillating up and down?

Yes

How can either have a non-local "inertial coordinate system" when they are in curved spacetime?

Let A be the observer at the center of the Earth be A
and the other oscillating up and down be B

They are both inertial observers and can base a coordinate system with themsleves as the origin. Times and distances are measured by standard clocks and rulers kept in inertial frames of reference at their respective origins, and radar may be used to measure distance.

A's metric is that of the Schwarzschild solution. B's is very complicated and I don't know of anyone who has looked at it, but perhaps others do.

Nevertheless, although we cannot easily calculate the exact time durations between consecutive encounters, measured by A as \Delta \tau_A and B as \Delta \tau_B, as the signature of the metric is (-+++) , d\tau^2 = - g_{\mu\nu}dx^{\mu}dx^{\nu}, we can be sure that:

A thinks that \Delta \tau_A &gt; \Delta \tau_B
and B thinks that \Delta \tau_B &gt; \Delta \tau_A

So, which is correct when \Delta \tau_A and \Delta \tau_B are compared at the second encounter, and how is that observer selected by the physical setup?

Garth

 

[JesseM]

Let A be the observer at the center of the Earth be A
and the other oscillating up and down be B

They are both inertial observers and can base a coordinate system with themsleves as the origin. Times and distances are measured by standard clocks and rulers kept in inertial frames of reference at their respective origins, and radar may be used to measure distance.

A's metric is that of the Schwarzschild solution.

What do you mean when you say the clocks and rulers would be "kept in inertial frames of reference at their respective origins"? If a given clock or ruler was at a constant distance from A's position the center of the earth, then it would not be following a geodesic path and would therefore be moving non-inertially, right?