Limits to the Equivalency Principle?

[whabbear]

Physicists: I'm struggling with some issues stemming from jointly considering General Relativity and Einstein's Equivalency Principle. Consider, first, two objects occupying two distinct frames of reference: An apple hanging from a tree on the Earth’s surface, and a geostationary satellite far (but directly) above the apple. Both objects have a clock affixed to them. There is no relative motion between the two: Measure the distance between them at any time, and the distance measurement would not vary. Yet, if the clock aboard the geosynchronous satellite and a clock affixed to the apple try to measure the duration of any arbitrary event in the vicinity of the apple, they won’t agree. The clock affixed to the apple will record a shorter duration for the event than the clock affixed to the satellite, because the curvature of spacetime is different in the apple's frame of reference than in the satellite's frame of reference.

Now, consider two identical spacecraft located adjacent to each other somewhere out in space, each one containing a clock. Initially, the two spacecraft start out in the same reference frame, as they are motionless relative to each other. If an event happens inside one of the spacecraft, both clocks will record the same duration for it. However, if one of the pair then fires up its engines and starts to accelerate away from the other, time dilation effects will occur, and the clocks onboard the two spacecraft will no longer record equal durations for events occurring in their twin.

First question: Is the time dilation that accompanies the fact that there's now time dilation between the two spacecraft due entirely to the fact that there is now relative motion between them (as opposed to the fact that the relative motion is of an accelerating form), and is therefore a phenomenon of Special Relativity, rather than General Relativity? Now suppose, as a thought experiment, one of the twin spacecraft fires up its engines, but can't actually move relative to the inert spacecraft because the active vehicle is butting up against some “immovable object” (meaning that, in some sense, the active spacecraft is experiencing a force “trying” to accelerate it, even though the immovable object prevents it from actually moving). Is the fact that there's a force being applied to the active spacecraft sufficient to induce a local spacetime curvature, and hence, despite the fact that there is no relative motion between the two spacecraft, time dilation effects kick in?

If the answer to my second question is "No", then it seems to me that there's a fundamental limitation to the Equivalency Principle. The apple on the Earth's surface is constantly "trying" to follow its shortest path through gravity-induced curved spacetime, but is prevented from doing so by the stronger force that keeps it attached to the apple tree. If this countervailing force is removed, say by a strong wind the severs the apple from the tree, the apple immediately begins to accelerate toward the Earth's surface with dynamics that are equivalent to the dynamics of an unimpeded accelerating spacecraft. But before that, while the apple is still connected to the tree, the mere "attempt to accelerate" manifests itself in the form of a time dilation effect. Again, would the same be true for the two spacecraft in my thought experiment?

[studiot]

A good length description setting out your stall.

 

Well done, most seem to make their OP too long or too short.

 

However I fear you have misunderstood frames of reference.

 

Why do you say that the apple and the satellite are in different frames of reference?

 

You haven't explicityl mentioned the centre of the Earth, but by your reasoning this should be in yet another frame.

 

Further you state that the 'distance between them is not changing.

 

How are you reckoning this distance? In which frame of reference?

 

Please explain.

Edited July 5, 2017 by studiot

[Janus]

Physicists: I'm struggling with some issues stemming from jointly considering General Relativity and Einstein's Equivalency Principle. Consider, first, two objects occupying two distinct frames of reference: An apple hanging from a tree on the Earths surface, and a geostationary satellite far (but directly) above the apple. Both objects have a clock affixed to them. There is no relative motion between the two: Measure the distance between them at any time, and the distance measurement would not vary. Yet, if the clock aboard the geosynchronous satellite and a clock affixed to the apple try to measure the duration of any arbitrary event in the vicinity of the apple, they wont agree. The clock affixed to the apple will record a shorter duration for the event than the clock affixed to the satellite, because the curvature of spacetime is different in the apple's frame of reference than in the satellite's frame of reference.

 

Now, consider two identical spacecraft located adjacent to each other somewhere out in space, each one containing a clock. Initially, the two spacecraft start out in the same reference frame, as they are motionless relative to each other. If an event happens inside one of the spacecraft, both clocks will record the same duration for it. However, if one of the pair then fires up its engines and starts to accelerate away from the other, time dilation effects will occur, and the clocks onboard the two spacecraft will no longer record equal durations for events occurring in their twin.

 

First question: Is the time dilation that accompanies the fact that there's now time dilation between the two spacecraft due entirely to the fact that there is now relative motion between them (as opposed to the fact that the relative motion is of an accelerating form), and is therefore a phenomenon of Special Relativity, rather than General Relativity?

Whether or not the time dilation measured is entirely due to the relative motion depends on which spacecraft is making the measurement. If it is the spacecraft that is not firing its engines, then this true. However if the measurement is made from the accelerating spacecraft, then the answer is no. From according to this spacecraft there is an additional time dilation component in addition to that due to the velocity difference. This factor is determined by the magnitude of the acceleration, the distance between the two craft, and the relative position of the craft with respect to each other as measured along the acceleration vector. If the spacecraft is accelerating away from the other craft, this will add to the time dilation measured by the relative motion. If it is accelerating towards the other craft, it will act to decrease the time dilation. This can bring up the following interesting situation: Both spacecraft are accelerating in the same direction so that, according to each craft, the distance between them remains the same. In such a case, the leading craft will measure a clock in the trailing craft as running slow, and the trailing space craft will measure the clock in the leading craft as running fast. This is despite the fact that they are at rest with respect to each other in their own frames and both experience the same magnitude of acceleration. This all still falls under the purview of Special Relativity as there is no gravity and no curvature of space-time.

Now suppose, as a thought experiment, one of the twin spacecraft fires up its engines, but can't actually move relative to the inert spacecraft because the active vehicle is butting up against some immovable object (meaning that, in some sense, the active spacecraft is experiencing a force trying to accelerate it, even though the immovable object prevents it from actually moving). Is the fact that there's a force being applied to the active spacecraft sufficient to induce a local spacetime curvature, and hence, despite the fact that there is no relative motion between the two spacecraft, time dilation effects kick in?

The answer is no.

 

If the answer to my second question is "No", then it seems to me that there's a fundamental limitation to the Equivalency Principle. The apple on the Earth's surface is constantly "trying" to follow its shortest path through gravity-induced curved spacetime, but is prevented from doing so by the stronger force that keeps it attached to the apple tree. If this countervailing force is removed, say by a strong wind the severs the apple from the tree, the apple immediately begins to accelerate toward the Earth's surface with dynamics that are equivalent to the dynamics of an unimpeded accelerating spacecraft. But before that, while the apple is still connected to the tree, the mere "attempt to accelerate" manifests itself in the form of a time dilation effect. Again, would the same be true for the two spacecraft in my thought experiment?

The gravitational time dilation component does not depend on whether or not the apple is resisting the force of gravity. At the instant the apple is severed by from the tree, it is still motionless and it still has the same time dilation with respect to the orbiting clock. As is begins to fall and change speed, this relative speed difference will add an additional time dilation component.

 

To go back to the two accelerating spacecraft scenario. We will attach the apple to the tail of our lead rocket. This will be the equivalent of hanging from the tree. When it detaches, in the frame of the two craft, it "falls" towards the trailing spacecraft. (in a frame which measures the craft as accelerating, it stops accelerating and the trailing spacecraft accelerates towards it.)

Now we are back to a situation like we had when only one of the craft was accelerating. The lead spacecraft will measure a clock attached to the apple to run slow due to its increasing relative velocity compounded by the increasing distance between them and direction and magnitude of the acceleration. The trailing spacecraft will measure it running slow due to the relative velocity, but there will be a countering effect due to the acceleration towards the apple.

 

The thing to understand is the gravitational time dilation is not related to the difference in forces experienced by the clocks, but to the relative positions in the field. The apple clock does not run slower because it experiences a stronger gravity than the orbiting clock, but because because it is lower in the gravity field. (and this would be true even if there were no difference in the strength of gravity at the two altitudes.)

Edited July 5, 2017 by Janus

[Mordred]

Dang I was going to respond after work but I like your response better +1.

 

Looks like you covered all the essentials

Edited July 6, 2017 by Mordred

[whabbear]

Janus, thank you so much! Your answers and accompanying descriptions were right on!

[Tim88]

Overall Janus already gave a very good reply; however there's an issue with phrasing that may be worth mentioning. SR is formulated with respect to the same reference frames as classical physics ("inertial frames"), and the standard, so-called "clock hypothesis" of relativity is that acceleration has no effect on the rate of perfect clocks.

In other words: clock rate is a function of speed and not of acceleration.

 

The equivalence principle of GR tells us that the observed phenomena must be the same as when the craft is at rest in a uniform gravitational field. One can mimic a gravitational field by means of an accelerating frame. And it is only from such an "accelerating spacecraft view" -according to which the accelerating craft is not accelerating but at rest in a fictitious gravitational field- that one obtains the apparent differences in clock rates that Janus provided for the two accelerating spacecrafts.

The Doppler effect is ignored if one pretends that the craft is constantly in rest, and this leads to the apparent difference in clock rates.

 

From an "instantly co-moving frame" view, the observed frequency difference is fully due to the Doppler effect if the spacecraft undergo identical acceleration from rest (BTW that's how Einstein calculated it first, assuming zero difference in clock rates).

 

There is another issue with "constant distance", as already hinted at by Studiot: with identical acceleration from rest as measured with an inertial frame, the distance will remain constant according to that frame's reckoning. However, the distance between the craft as measured with instantly co-moving frames will slowly appear to increase. To avoid that (and obtain a constant "proper distance"), the rear craft should have a very slightly greater acceleration. That leads to a small correction in the calculation of observed frequencies.

 

And of course, all this is still SR. Predicting that the same phenomena will be observed in a real gravitational field belongs to GR.

Edited July 10, 2017 by Tim88

[J.C.MacSwell]

 

 

There is another issue with "constant distance", as already hinted at by Studiot: with identical acceleration from rest as measured with an inertial frame, the distance will remain constant according to that frame's reckoning. However, the distance between the craft as measured with instantly co-moving frames will slowly appear to increase. To avoid that (and obtain a constant "proper distance"), the rear craft should have a very slightly greater acceleration. That leads to a small correction in the calculation of observed frequencies.

 

Right. Or in the case of the two craft actually being 2 distant parts of the same larger craft, the back better do a little more accelerating to avoid the larger craft being stretched apart. "Avoiding" the dilation in the original frame would be catastrophic.

[Tim88]

Right. Or in the case of the two craft actually being 2 distant parts of the same larger craft, the back better do a little more accelerating to avoid the larger craft being stretched apart. "Avoiding" the dilation in the original frame would be catastrophic.

 

Maybe you meant that '"avoiding" the contraction [of the accelerating craft] in the original frame' would be catastrophic. Yes indeed; and common rockets take care of that length contraction themselves - that's the essence of Lorenz contraction.

[J.C.MacSwell]

 

Maybe you meant that '"avoiding" the contraction [of the accelerating craft] in the original frame' would be catastrophic. Yes indeed; and common rockets take care of that length contraction themselves - that's the essence of Lorenz contraction.

Thanks T88. That is what I meant.