Twins Paradox: Exploring Clocks in a Closed Room

[m4r35n357]

Yogi

Serious answer this time. Non-geodesic paths encounter more proper time than geodesic ones, even in GR. GR just allows more than one geodesic path between two events. What more must GR provide? I don't see the problem . . .

 

[PeterDonis]

Non-geodesic paths encounter more proper time than geodesic ones, even in GR.

This is wrong in two ways. First, in SR, non-geodesic paths between the same pair of events have less elapsed proper time, not more. Second, in GR, both cases are possible: non-geodesic paths that have less proper time than geodesic ones between the same two events, and non-geodesic paths that have more.

For example, consider the following scenario: one astronaut hovers at a constant altitude over a non-rotating planet. A second astronaut is in a circular orbit about the planet at the same altitude. A third astronaut launches himself upward from the first's position at the same instant that the second one passes, in such a way that he free-falls upward and then free-falls back down so that he arrives back at the first astronaut at the same instant that the second one passes again in his orbit.

The three astronauts follow three different paths between the same pair of events. Astronaut #1's path is non-geodesic; the other two are geodesic. Astronaut #1 has more elapsed proper time than #2 between the pair of shared events, and less than #3.

 

[m4r35n357]

This is wrong in two ways. First, in SR, non-geodesic paths between the same pair of events have less elapsed proper time, not more. Second, in GR, both cases are possible: non-geodesic paths that have less proper time than geodesic ones between the same two events, and non-geodesic paths that have more.

Yeah, got that the wrong way round, oops! My head hurts a bit thinking about the scenarios, I'll need a bit of time to think about them. Still, my point was that there is no misunderstanding or mystery about "twin paradoxes" in GR; they can all be calculated and resolved.

 

[PeterDonis]

my point was that there is no misunderstanding or mystery about "twin paradoxes" in GR; they can all be calculated and resolved.

Yes, definitely.

 

[yogi]

How so? The Schwarzschild solution, for example, is a vacuum solution--no stress-energy anywhere

The Schwarzschild solution is a solution for the space and time outside a mass based upon the mass ...look at the coefficient of the time increment

ds^2 - (1-2M/r)dt^2 ...

the fact there is no mass embedded in the solution for the exterior space is not the same as saying "no stress energy anywhere"

While reasonable minds may differ upon interpretation, the formula for time dilation in GR is totally based upon energy - the escape velocity that determines the gravitational potential - If you believe there are any experiments validating the fact that an object traveling a curved path at constant velocity (as described by Einstein in his 1905 paper, will incur a time dilation component greater than that predicted by SR, I would like to see the proof of such.

Look at the elements that make up the time dilation in GR The time dilation is given by dt* = dt(1-2GM/rc^2)^1/2 = dt(1- v^2/c^2)^1/2 where v is the escape velocity

There is no change in time dilation unless acceleration results in a change in height (PE) or a change in velocity KE) and therefore I stand on my statement. A clock traveling a curved path at constant velocity keeps the same time as a clock on straight track traveling at constant velocity

Regards

Yogi

 

[PeterDonis]

the fact there is no mass embedded in the solution for the exterior space is not the same as saying "no stress energy anywhere"

It does if the solution is describing a black hole.

the formula for time dilation in GR is totally based upon energy - the escape velocity that determines the gravitational potential

First of all, you have this backwards; the gravitational potential determines the escape velocity, not the other way around.

Second, how is any of this "based upon energy"? The gravitational potential is a geometric feature of the spacetime (more precisely, it is directly definable in terms of a geometric feature, namely the spacetime's timelike Killing vector field).

If you believe there are any experiments validating the fact that an object traveling a curved path at constant velocity (as described by Einstein in his 1905 paper, will incur a time dilation component greater than that predicted by SR, I would like to see the proof of such.

You're going to have to be more specific about what you mean by "an object traveling a curved path at constant velocity". Do you mean an object traveling around in a circle?

You're also going to have to be more specific about what you mean by "time dilation component". Do you just mean the "tick rate" of the object's clock as compared to coordinate time in a fixed inertial frame?

There is no change in time dilation unless acceleration results in a change in height (PE) or a change in velocity KE) and therefore I stand on my statement. A clock traveling a curved path at constant velocity keeps the same time as a clock on straight track traveling at constant velocity

How does the comparison between two objects at different heights in a gravitational field have anything to do with "a clock traveling a curved path at constant velocity"?

 

[yogi]

It does if the solution is describing a black hole.
First of all, you have this backwards; the gravitational potential determines the escape velocity, not the other way around.

Second, how is any of this "based upon energy"? The gravitational potential is a geometric feature of the spacetime (more precisely, it is directly definable in terms of a geometric feature, namely the spacetime's timelike Killing vector field).
You're going to have to be more specific about what you mean by "an object traveling a curved path at constant velocity". Do you mean an object traveling around in a circle?

You're also going to have to be more specific about what you mean by "time dilation component". Do you just mean the "tick rate" of the object's clock as compared to coordinate time in a fixed inertial frame?
How does the comparison between two objects at different heights in a gravitational field have anything to do with "a clock traveling a curved path at constant velocity"?

The Black Hole is what generally comes to mind in the Schwarzschild solution - the mass is concentrated in a black hole and the exterior space is curved. This is a different subject than the curvature experienced by a spaceship tethered to a pole so that it travels at constant velocity and therefore constant energy with constant centripetal acceleration, ergo, there is no change in energy, time dilation relative to the Earth is uniform through-out the entire trip. That was my original analogy - in such a case, a round a trip voyage is the same as a one way voyage - there is no special turnaround acceleration involved since the curvature of the path is constant during the entire journey - that was Einstein's example in part IV of his "Electrodynamics of Moving bodies. As far as time dilation, we are talking about comparing the total time of a circular path which begins on Earth and ends at the same point - that was Einstein's example - it was perfectly correct as originally presented - there is NO correction imposed by GR for a circular path in free space - contrary to many misstatements on these and other forums where a curved path is immediately relegated to a problem requiring GR - it doesn't require GR. GR is not involved.

Finally to clarify the height energy - a clock traveling on Earth or in any other "g" field, if subjected to different heights, would to that extent, experience time changes due to GR since the PE is a function of the height in determining the time dilation in a gravitational field - but that potential energy is immediately seen as the KE that corresponds to the velocity acquired to leave the Earth and never return (7 mi/sec) which in this sense, points in the direction of an absolute reference frame rather than a relative one. My point in saying that time differences depend upon energy differences follows from the fact, that in all experiments to date (except one), are based upon adding energy and measuring a slower clock rate for the object put into motion wrt the earth.

 

[PeterDonis]

This is a different subject than the curvature experienced by a spaceship tethered to a pole so that it travels at constant velocity

Do you mean in flat spacetime? If so, there is no "curvature"; spacetime is flat. The spaceship's worldline is curved, but spacetime itself is not. Or...

so that it travels at constant velocity and therefore constant energy with constant centripetal acceleration, ergo, there is no change in energy, time dilation relative to the Earth is uniform through-out the entire trip.

...do you mean, for example, a spaceship tethered on a pole attached to the surface of the Earth, so it goes around in a circle at a constant altitude? If so, yes, spacetime is curved, but it's curved the same for this spaceship as it is for a spaceship "hovering" at the same altitude but at rest relative to the Earth. The only difference is in the curvature of the worldlines of the two ships.

in such a case, a round a trip voyage is the same as a one way voyage - there is no special turnaround acceleration involved since the curvature of the path is constant during the entire journey

I'm still confused as to whether you mean the flat spacetime or the curved spacetime case; but taking the flat spacetime case for discussion, since that's the one Einstein was talking about, yes, an inertial observer at rest at one point on the circular path of the observer going around in a circle will have more elapsed time between two successive meetings of the two. I'm still not sure what this has to do with comparing two observers at rest at different altitudes in a gravitational field, though.

As far as time dilation, we are talking about comparing the total time of a circular path which begins on Earth and ends at the same point - that was Einstein's example - it was perfectly correct as originally presented - there is NO correction imposed by GR for a circular path in free space

Assuming that by "free space" you mean "flat spacetime", yes, this is correct; SR is sufficient to analyze any scenario in flat spacetime, even if some of the worldlines involved are curved (i.e., accelerated). GR is only necessary if spacetime is curved.

However, if the scenario is set on Earth, then it is not in flat spacetime. If everything happens at exactly the same altitude, you can finesse that, which is basically what Einstein did. But why do that when you can just as easily set the scenario in flat spacetime to begin with?

a clock traveling on Earth or in any other "g" field, if subjected to different heights, would to that extent, experience time changes due to GR since the PE is a function of the height in determining the time dilation in a gravitational field

The PE does not determine the time dilation; the position of the observer with respect to the timelike Killing vector field of the spacetime determines both the PE and the time dilation.

but that potential energy is immediately seen as the KE that corresponds to the velocity acquired to leave the Earth and never return (7 mi/sec)

Again, this is determined by position relative to the timelike Killing vector field; it is that position which is the fundamental quantity, and it is a geometric quantity, not an "energy" quantity.

which in this sense, points in the direction of an absolute reference frame rather than a relative one.

There is a unique reference frame picked out by the timelike Killing vector field of the spacetime, yes. (In the case of Schwarzschild spacetime, this is Schwarzschild coordinates.) Put another way, the spacetime we are talking about has a particular symmetry, and any spacetime with a particular symmetry will have a particular reference frame picked out that matches up with that symmetry. Again, the fundamental fact is the symmetry, and that is a geometric fact.

all experiments to date (except one), are based upon adding energy and measuring a slower clock rate for the object put into motion wrt the earth.

What is the one exception?

 

[georgir]

http://www.convertalot.com/relativistic_star_ship_calculator.html
It can't take distances over 15 digits normally, but if you know your way around a browser's HTML inspector you can change that.
Then, assuming the formula it uses is correct, you can see that accelerating 1g for 95 on-board years can take you almost 1885540714000000000000 light years away :) Yes, that is ten orders of magnitude more than the observable universe.