A variation on the twin paradox

[pervect]

That is, of course, exactly what normally I understand!

However, in this case, where A is the COM observer and B the 'bobbing' observer and using a notation in which the time elapse between encounters is written, we have:

\Delta \tau_{A,A} being A's time elapse as measured by A,
\Delta \tau_{A,B} being A's time elapse as computed by B,
\Delta \tau_{B,B} being B's time elapse as measured by B,
\Delta \tau_{B,A} being B's time elapse as computed by A,

so \Delta \tau_{A,A} & \Delta \tau_{B,B} are measured by the identical clocks carried by A and B in their inertial frames of reference respectively.

We have said that:
\Delta \tau_{A,A} = \Delta \tau_{A,B} and
\Delta \tau_{B,B} = \Delta \tau_{B,A} as they are geometric objects independent of the coordinate systems in which they may be calculated.

Now either:
\Delta \tau_{A,A} > \Delta \tau_{B,B} or
\Delta \tau_{A,A} < \Delta \tau_{B,B} or
\Delta \tau_{A,A} = \Delta \tau_{B,B} and intuition tells you it is the first of these options that would be proved correct in an actual experiment.

Intuition isn't terribly reliable - I intuitively guessed the third option a while back, so my intuition was different than yours. And it was just as wrong, because after doing the calculation I'm convinced any of the above cases can be true depending on the density profile of the planet.

However that would also mean \Delta \tau_{A,B} > \Delta \tau_{B,B} as \Delta \tau_{A,A} = \Delta \tau_{A,B} would it not?

In which case in B's inertial frame of reference, A's space-time interval is computed by B to be greater than B's space-time interval, even though it is A that is moving and B that is stationary in that inertial frame of reference, against all space-time intuition about relative moving clock time dilation.The moving clock is 'ticking' faster than the stationary one, in other words, the moving observer A is aging more quickly than the stationary B!

This is perfectly possible. I'm having a bit of a problem untangling your notation or why you think there is a difficulty.

For any observer, using a +--- sign convention, we can write

d\tau^2 = g_{00} dt^2 + 2 g_{01} dt dx + g_{11} dx^2

which can be re-written as

d\tau = \sqrt{g_{00} + 2 g_{01} \frac{dx}{dt} + g_{11} (\frac{dx}{dt})^2 } dtThe numerical values of the g_ij will depend on the coordinate system used.

Let's look at it from the viewpoint of the stationary observer at the center of the planet, which we will call obsever A.

You are assuming, I believe, that we adopt coordinates so that g_00=1, g_01=0, and g_11 = 1 for the stationary observer. (We don't have to adopt such coordinates, but that's what I'm getting from what you wrote). Then the above intergal reduces to \tau_{A,A} = t.

Now let's consider the calculation of \tau_{B,A}

The moving obserer will have g_00, g_01, and g_11 as functions of time. We do not expect the metric coefficeints for observer B to represent a Minkowskian metric, because observer B is far away from A in a gravity field.

g_00 may be greater than 1 for the moving obsever. In fact, we expect g_00 to be greater than 1. Well, I expect it to be greater than 1.

Therfore it is possible in principle for dtau > dt.

The moving observer will have g_11 approximately equal to -1. Thus it is possible in principle for dtau < dt.

To decide which is the case, we need more information.

In english:

The moving observer is higher in a gravity well. Thus it's clock is ticking faster because of it's height. The moving obsever is also moving. Thus it's clock is ticking slower due to relativistic time dilation. Which effect dominates is not clear - it could be either one.

 

[Garth]

Yes thank you, I simply took the first option as an example to chew on!

And I certainly agree with your analysis from A's POV, I would be interested if it could also be reworked from B's POV, with the metric expressed in B's coordinate system.

What I am chewing over is the comparison of two inertial observers traveling between two close encounter events via separate geodesics, seeing the situation from each POV.

In fact that these two inertial observer's are not equivalent because the gravitational field is stationary for A but not for B.

In other words any difference between them is imparted by the distribution of the matter in the rest of the universe, which in this case is the planet Earth.

I suppose the significance of any issues raised simply depends on whether you think that difference is itself significant or not.

I'll revert back to thinking of the comsological closed space scenario!

Garth

 

[yogi]

Something i am not relating to - why is the bobbing clock a better inertial system than a free-floating clock in circular orbit about a spherically symmetrical mass? The latter is much easier to deal with since from the perspective of a clock at the center of mass the relative velocity and the gravitational potential of the orbiting clock is constant.

 

[Garth]

It was simply that 'relative simultaneity' does not present itself as a problem at the close encounters at the centre of the Earth. If the two clocks are close enough then there is no ambiguity in their meaurements of t₁, t'₁ and t₂, t'₂. the two geometric 'objects' are uniquely determined as
\Delta \tau_{A,A} = t_2 - t_1 and
\Delta \tau_{B,B} = t&#039;_2 - t&#039;_1.

Garth

 

[yogi]

Grath ..So basically, you were contriving a situation that allowed the clocks to obtain nearly instantaneous measurement of the other during passby. What if the Earth centered clock is synchronized with a clock on top of a large tower, e.g. at the North Pole, which has a height equal to the altitude of a polar orbiting clock - except for the difference in gravitational potential (for which we can adjust) we now have one clock fixed in the Earth centered frame, a second clock in the same frame but at different but constant gravitational potential (The clock on the North Pole tower) and an orbiting clock. Once the North Pole clock is adjusted to account for the difference in the G potential, it will remain in sync with the Earth centered clock - each time the satellite clock (SC) passes by the North Pole clock (NPC) they will read each other. The result is that the NPC will be found to accumulate more time between orbit encounters than the SC clock. In this experiment, which I see as analogous to your oscillating clock, the measurements will confirm that the proper rate of the orbiting clock (SC) is less than than the proper rate of the NPC. I also see it as a local model of the cosmological twin paradox - but it is not a paradox for the same reason, namely since you should not apply the mutually contradictory statement that "two clocks in relative motion each measure the other clock to be running slow" to a situation involving actual time dilation - that statement is only true when the apparent times are measured using the standard two clock technique in one frame. In the case of the oscillating clock and the orbiting clock, the experiment measures actual time difference rather than apparent time dilation - so the commonly heard statement about reciprocal observations of clocks in relative motion show the other to be slow, while true in the sense that that the other clock appears to be running slow, is not true in the sense that the other clock is actually running slow - the SC will be slower than the NPC because your experiment (and my alternative) measure the difference in the proper rate of the clocks in question

 

[Hurkyl]

The point of the bobbing observer is that we don't have to contrive some means of comparing their clocks -- when two clocks have a flyby, there's a "god-given" way to compare them.

What if the Earth centered clock is synchronized with a clock on top of a large tower,

We've discussed before that the word "synchronized" by itself is entirely meaningless in SR.

In GR, the case is far, far worse. In SR, you at least had the notion of global inertial coordinate charts -- in GR, you don't even have that.

For example, if two observers, A and B, have worldlines that never intersect, or only intersect once, then there is a coordinate chart in which A's clock is always running faster than B's clock, and there is another coordinate chart in which B's clock is always running faster than A's clock.

You seem to be doing the same thing you've done in SR: you pick your favorite coordinate chart, and decree that everything is supposed to be done relative to that.

But this obscures one of the most important facts about GR: that any coordinate chart is good enough to do physics! (meaning that all of the laws of physics take an identical form in any coordinate chart)

is not true in the sense that the other clock is actually running slow

In particular, that this sentence has absolutely no physical meaning. To say that a clock is running "slow" at any point in time, you need to have some "standard" time to which you can compare it and take derivatives.

When two clocks are co-located, we have two physical notions of time we can use, one for each clock. It then makes sense to ask about the rate of one clock with respect to the other clock. Mathematically, this question is nothing more than asking for the inner product of their velocity vectors.

When two clocks are not co-located, you cannot physically compare them. You have to invent some sufficiently global notion of time coordinate, and compare your clocks to that time coordinate. Since this requires the invention of some notion of coordinates, it's an unphysical concept.

If the clocks have meetings at which their clock readings can be compared, it does make sense to say that one clock accumulated more time than the other between meetings, but that still isn't enough to give any meaning to the differential rate of one clock with respect to another.

The result is that the NPC will be found to accumulate more time between orbit encounters than the SC clock.

All that being said, I thought the exact opposite would be true in the coordinate chart I think you're using.

Your "clock" at the NPC isn't a clock at all: it doesn't measure proper time. Because you've somehow "synchronized" it with the clock at the Earth's center (ECC), that means that your NPC will be running slower than any ordinary clock that is also located in the tower at the north pole.

I think that this further means that your NPC will accumulate less time between flybys with the SC.

 

[yogi]

Hurkyl - what was intended - perhaps not explained clearly enough - is we have an Earth centered clock (ECC) which we synchronze with another clock. The latter is moved to a tower on the North pole. If no correction is put in, this NPC will run faster than the Earth centered clock. Or if you introduce a scaling factor you can make sure it runs at the same rate as the Earth centered clock. Either way, the clock in orbit (SC) will run slower than the NPC if the NPC and SC have been synchronized at the top of the tower and the SC then put into circular polar orbit.

I am saying the utilization of a simple geometry created by a circular orbit makes it clear there is no paradox - as long as their is an unbroken chronological chain of synchronization between the ECC, NPC and SC, then each time the SC passes by the NPC and the readings are exchanged, they can be related to the ECC.

You state: "If the clocks have meetings at which their clock readings can be compared, it does make sense to say that one clock accumulated more time than the other between meetings, but that still isn't enough to give any meaning to the differential rate of one clock with respect to another."

And I say it does - if at every meeting the SC clock is 7ns less than the reading on the NPC clock - and the NPC clock is always at the same distance from the ECC we have a direct way to compare the SC clock with the ECC because the NPC can send a radio signal to the ECC and the ECC can send a radio signal to the NPC.

Now its true that if the NPC is height corrected so that it runs in sync with the ECC, it will not read the proper time for a clock in that position - but it is a clock - running at a constant offset rate - but that does not negate the impact of what it measures - we know how to adjust for the offset in arriving at the result

 

[yogi]

Hurkyl- one more point which we discussed in another thread - in actuality, we need not even use a clock at the top of the tower - we can simply have the SC clock transmit a signal indicative of the SC clock reading whenever it passes w/i ten meters of the tower - this is picked up by the ECC and compared to the ECC time - the distance is always the same - when you subtract out the difference produced by the different G potential between the ECC and the SC, what is left is an ongoing confirmation updated each time the SC passes the tower, of the actual difference in the proper rate of the ECC and the SC clocks.

 

[Hurkyl]

Or if you introduce a scaling factor you can make sure it runs at the same rate as the Earth centered clock. Either way, the clock in orbit (SC) will run slower than the NPC if the NPC and SC have been synchronized at the top of the tower and the SC then put into circular polar orbit.

In the usual coordinates we use for near-Earth, I'm quite sure that this is wrong. Because you've tuned the NPC to run at the rate of the ECC (according to the specified coordinate chart), the NPC should accumulate less time than the SC between flybys.

And I say it does - if at every meeting the SC clock is 7ns less than the reading on the NPC clock - and the NPC clock is always at the same distance from the ECC we have a direct way to compare the SC clock with the ECC because the NPC can send a radio signal to the ECC and the ECC can send a radio signal to the NPC.

Ok fine.

To the best of my knowledge, to compute a differential rate, you need to take a derivative of something with respect to something else.

So, I don't know how I can take a derivative when all I have is a discrete sequence of values.

I know how to take the derivative of 2t with respect to t.

I do not know how to take the derivative of {2, 4, 6, 8, ...} with respect to {1, 2, 3, 4, ...}. I don't even know what that would mean.

So how do you plan on defining the rate of a clock?

 

[yogi]

If we assume a transmission is sent once each time the SC clock passes by the tower in my post 128, the rate of the SC clock relative to the ECC clock is the time between consecutive transmissions sent by the SC clock as recorded on the SC clock divided by the time between two consecutive signals arriving at the ECC clock. If there is no altitude correction, the time between SC transmissions as measured on the SC clock will be less than the time recorded between receptions as measured by the ECC clock - but since we know the altitude and it is constant, we also know how fast the SC will run if it is corrected to account for the difference in G potential - in fact we can always put two clocks in orbit - one uncorrected for altitude, the other not. Depending upon your objective you can use one or the other.

Note - I do not claim to define a universal clock rate - I only conclude from this simple arrangement that one clock runs fast and the other slow -and there is no ambiguity about which is which - and I claim this is no different than the cosmological twin paradox in compact space - there is no uncertainty as to which clock is running slower. Here we have a well defined set of initial conditions - a central ECC and a way to adjust other clocks so that we can keep track of what has taken place vis a vis synchronization, off setting rates to compensate for either motion of G potential - whatever we want - we actually have a do-able experiment and a lot of confirming date from GPS. In the cosmological twin paradox, an ambiguity is created because it is assumed the two clocks in passing one another are each running at the same proper rate - but that is not the general case. As Garth observes - two clocks following the same geodesic would not in general log the same amount of time between successive passes - one needs to know more about how they got in relative motion to explain why they run at different proper rates - and once this is known, there is no paradox.

 

[Garth]

My musings above were over the ramifications of Hurkl's statement:

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.

I am now satisified, thank you!

Is not the difference between the bobbing/COM (or ECC and SC examples) and the cosmological twin case that of the difference between local and global symmetries?

i.e. The local (non-global) symmetry of the Earth's gravitaitonal field removes the ambiguity (described above #130) whereas in a totally homogeneous and isotropic FRW universe - or even a flat one with a non-trivial topology - that ambiguity can only be resolved by the global topology?

Of course - wearing my Machian hat - I have to ask: "What is it that determines that topology?"

Garth

 

[George Jones]

Is not the difference between the bobbing/COM (or ECC and SC examples) and the cosmological twin case that of the difference between local and global symmetries? i.e. The local (non-global) symmetry of the Earth's gravitaitonal field removes the ambiguity (described above #130) whereas in a totally homogeneous and isotropic FRW universe - or even a flat one with a non-trivial topology - that ambiguity can only be resolved by the global topology?

I wouldn't phrase it this way. All the cases that you mention are examples of a more general situation.

Two obsersvers are coincident at distinct spacetime events A and B, and, between A and B, the worldlines of the 2 observers are different timelike geodesics. To parametrize each worlline by proper time, an integral involving the metric along each worldlne must be performed. The metric is a tensor field, and, in general, assumes different values at the events along the different worldlines. This is the fundamental asymmetry that causes the 2 elapsed proper times to differ. It may sometimes be the case that topological quantities can easily be used to point out the asymmetry in worldlines, but it's the differing set of events, and the thus differing values of the metric tensor that is at the heart of the issue.

General statements like "moving clocks run slow" and "clocks in a gravitational field ..." are often misleading, because they don't always apply, and because it is so easy to apply them incorrectly. These statements can sometimes be useful if applied very, very carefully, but I find that I am too easily led astray by them.

The metric tells all!

Of course - wearing my Machian hat - I have to ask: "What is it that determines that topology?"

And what determines the metric.

What I know (just recently learned!) is that it's possible for there to be two different connections with the property that a curve is a geodesic of the first connection if and only if it is a geodesic of the second connection.

Is this also possible when both connections are metric-compatible and torsion-free?

I really have no idea whatsoever if the Einstein field equations would prevent this phenomenon.

Here's my thinking that caused my initial objection. If the distribution of energy, matter, and momentum (i.e, T) in the universe is specified, then, in a chart, Einstein's equation is a set of 10 coupled partial differential equations for the components of the metric. I wondered whether existence and uniquess theorems applied, thus pinning down the metric.

I worried, though, about stuff like: a single chart doesn't cover the entire universe; the "hole argument; etc.

Hawking and Ellis says "Thus the field equations really provide only six independent differential equations for the metric. This is in fact the correct number of equations to determine the spacetime, since four of the ten components of the metric can be givem arbitrary values by use of the four degrees of freedom to make coordinate transformations. ... Therefore the field equations should define the metric only up to am equivalence class under diffeomorphisms, and there are four degrees of freedom to make diffemorphisms."

Regards,
George

 

[Hurkyl]

Is this also possible when both connections are metric-compatible and torsion-free?

First off, I was under the impression that for any connection, there is a metric with which it is compatable. Is that true?

Anyways, according to Spivak, if you don't allow the geodesics to be reparametrized, then there is a unique torsion-free connection with those geodesics.

If you allow geodesics to be reparametrized, then for any connection \nabla and one form \omega, the connection defined by \bar{\nabla}_X Y := \nabla_X Y + \omega(X) Y + \omega(Y) X is a connection with the same torsion and geodesics as \nabla. (And all such connections are of this form)

 

[selfAdjoint]

First off, I was under the impression that for any connection, there is a metric with which it is compatable. Is that true?

Connections are more general than metrics. You can have a principle bundle over a non-metric space and define a connection on it. This in fact happens in many approaches to LQG.