A variation on the twin paradox

[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.