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Ranku

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In summary: Two objects A and B are in freefall, and B is accelerating faster than the A, because it has been freefalling longer and is closer to the gravitational source.Yes, and in the presence of tidal gravity their relative velocity can change over time. Yes, and in the presence of tidal gravity their relative velocity can change over time.So the object closer to the gravitational source will have greater relative velocity than the object further from it?If the scenario is regarded as "local" (in all 4 dimensions), then tidal effects are neglected.

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Ranku

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Ibix

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If the two objects are at the same event (or close enough) then it is meaningful to compare their chosen frames, and they may choose inertial frames that are in relative motion, yes.

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Dale

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Yes, and in the presence of tidal gravity their relative velocity can change over time.Ranku said:

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Ranku

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So the object closer to the gravitational source will have greater relative velocity than the object further from it?Dale said:Yes, and in the presence of tidal gravity their relative velocity can change over time.

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Sagittarius A-Star

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If the scenario is regarded as "local" (in all 4 dimensions), then tidal effects are neglected.Ranku said:So the object closer to the gravitational source will have greater relative velocity than the object further from it?

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jbriggs444

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Wait, what? Exactly what two relative velocities are you comparing? And why are we using tangent inertial rest frames with different origins when our measurements are accurate enough to detect the effects of local tidal gravity?Ranku said:So the object closer to the gravitational source will have greater relative velocity than the object further from it?

You are comparing the speed of H (high object) in the tangent inertial frame of L with the speed of L in the tangent inertial frame of H?

Surely one would expect gravitational time dilation to mean that L's clocks to run slow so that H's trajectory runs fast.

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Ibix

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Velocity relative to what? And there's no general relationship between altitude and speed, not in GR nor Newtonian gravity.Ranku said:So the object closer to the gravitational source will have greater relative velocity than the object further from it?

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Dale

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I don’t understand this question.Ranku said:

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TonyStewart

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There is also an equivalence principle, in general relativity, that states that in a small region of spacetime, the effects of gravity are indistinguishable from the effects of acceleration.

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Ranku

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That is what I am trying to get at. Just like relative velocities between two inertial frames of reference give rise to time dilation and length contraction, does relative velocities between two locally inertial frames of reference within freefall also give rise to the same phenomena? Does that explain why closer to a gravitational field, time runs slowly and length contracts?TonyStewart said:

There is also an equivalence principle, in general relativity, that states that in a small region of spacetime, the effects of gravity are indistinguishable from the effects of acceleration.

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PeterDonis

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No, relative velocities betweenRanku said:relative velocities between two inertial frames of reference give rise to time dilation and length contraction

Relative velocities betweenRanku said:does relative velocities between two locally inertial frames of reference within freefall also give rise to the same phenomena?

That said:

No. Gravitational time dilation is not symmetric and doesn't work like apparent time dilation due to relative velocity in SR.Ranku said:Does that explain why closer to a gravitational field, time runs slowly

There is no such thing as "gravitational length contraction" so I don't know what you are talking about here.Ranku said:and length contracts?

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Dale

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What do you mean by apparent time dilation?PeterDonis said:relative velocities betweenobjectsgive rise toapparenttime dilation and length contraction

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Ranku

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I am trying clarify the following situation: Two objects A and B are in freefall, and B is accelerating faster than the A, because it has been freefalling longer and is closer to the gravitational source. Will the local inertial frame of reference of B have a greater relative velocity than that of A? If so, does that explain relativistic effect like time running slowly for B, because it is closer to the gravitational source?Dale said:I don’t understand this question.

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TonyStewart

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PeterDonis

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I mean that the terms "time dilation" and "length contraction" in SR refer to observer-dependent appearances: clocks moving relative to an observerDale said:What do you mean by apparent time dilation?

This is to contrast with, for example, gravitational time dilation, which is not symmetric; two observers at rest in a gravitational field will agree that the one at the lower altitude has his clock running slower.

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Ranku

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So is that because the local inertial frame of reference of the observer has a higher velocity?TonyStewart said:

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PeterDonis

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No. The whole concept of "velocity of the local inertial frame of reference" is not a good concept and you should not be trying to understand GR in terms of it.Ranku said:So is that because the local inertial frame of reference of the observer has a higher velocity?

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Sagittarius A-Star

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You could for example define locally an inertial reference-frame in which one of the objects is at rest and the other moving with constant ##v##. Locally, SR is valid. That means, you have for the moving object time-dilation and length contraction. Theses effects depend on the reference-frame. In this inertial reference-frame, you have locally no gravitational time-dilation.Ranku said:That is what I am trying to get at. Just like relative velocities between two inertial frames of reference give rise to time dilation and length contraction, does relative velocities between two locally inertial frames of reference within freefall also give rise to the same phenomena?

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TonyStewart

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PeterDonis

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No, time dilation and length contraction areTonyStewart said:I think... relative velocities can play a role in special relativity, where time dilation and length contraction are primarily caused by differences in relative motion

No. GR includes SR as a special case. Within a small enough patch of a curved spacetime, relative motion can give rise to time dilation and length contraction just as in the flat spacetime of SR.TonyStewart said:in GR, these effects arise from the curvature of spacetime due to gravity rather than differences in velocity between observers.

You can also get "gravitational" time dilation in SR with accelerated observers; for example, if there are two observers in a rocket accelerating in flat spacetime, one at the bottom and one at the top, both will agree that the clock of the one at the bottom runs slower.

Spacetime curvature is

Spacetime curvature also makes possible configurations that cannot exist in flat spacetime, for example, accelerated observers standing in a room on the surface of a planet, who locally see things the same as the two observers in the accelerating rocket, but who stay on the planet indefinitely instead of flying off into space.

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Sagittarius A-Star

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No, as @PeterDonis mentioned. Locally measured gravitational time-dilation is an SR effect in an accelerated reference-frame.TonyStewart said:in GR, these effects arise from the curvature of spacetime due to gravity rather than differences in velocity between observers.

Consider as local scenario an elevator-shaft (height ##h##, made of glass) in a tall building. When a lamp at the top of the elevator-shaft sends a light-pulse, the elevator-cabin starts free falling down from the top level. At the ground level, an observer ##A## stands near the elevator-shaft.

An observer ##B## in the falling elevator-cabin is at rest in an inertial reference frame.

- The lamp was at rest with reference to this free-falling frame, when it sent out the light pulse at ##t=0##.
- The observer ##A## is accelerating upwards with reference to this free-falling frame and is moving into the light with ##v\approx g t##, when the light pulse reaches his eyes at ##t \approx \frac{h}{c}##.

Observer ##A## would call the same blue-shift "gravitational blue shift" due to difference of gravitational potential ##\phi=gh## and related gravitational time-dilation by the factor ##1 + \frac{\phi}{c^2}##.

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Ranku

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Why is gravitational time dilation non-symmetrical between two observers, in contrast to the symmetry between two observers in SR?PeterDonis said:This is to contrast with, for example, gravitational time dilation, which is not symmetric; two observers at rest in a gravitational field will agree that the one at the lower altitude has his clock running slower.

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PeterDonis

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I'm not sure there's any useful way to even compare the two scenarios; they're just different. The fact that the term "time dilation" is used in connection with both of them does not mean there's any meaningful similarity. "Time dilation" is not a fundamental concept in relativity; it's just a name that, for historical reasons, gets used to refer to certain particular scenarios.Ranku said:Why is gravitational time dilation non-symmetrical between two observers, in contrast to the symmetry between two observers in SR?

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Ranku

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How would the above situation compare with an ascending elevator, with both observers in the elevator, with observer A at the top of the elevator and observer B on the floor of the elevator, in terms of doppler shift and time dilation?Sagittarius A-Star said:No, as @PeterDonis mentioned. Locally measured gravitational time-dilation is an SR effect in an accelerated reference-frame.

Consider as local scenario an elevator-shaft (height ##h##, made of glass) in a tall building. When a lamp at the top of the elevator-shaft sends a light-pulse, the elevator-cabin starts free falling down from the top level. At the ground level, an observer ##A## stands near the elevator-shaft.

An observer ##B## in the falling elevator-cabin is at rest in an inertial reference frame.

From observer ##B##'s viewpoint, the observer ##A## must see the light pulse Doppler-blue shifted by the factor approximately ##1+\frac{v}{c} = 1 + \frac{gh}{c^2}##.

- The lamp was at rest with reference to this free-falling frame, when it sent out the light pulse at ##t=0##.
- The observer ##A## is accelerating upwards with reference to this free-falling frame and is moving into the light with ##v\approx g t##, when the light pulse reaches his eyes at ##t \approx \frac{h}{c}##.

Observer ##A## would call the same blue-shift "gravitational blue shift" due to difference of gravitational potential ##\phi=gh## and related gravitational time-dilation by the factor ##1 + \frac{\phi}{c^2}##.

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Sagittarius A-Star

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The equivalence principle states, that measurements in an elevator cabin cannot help to distinguish betweenRanku said:How would the above situation compare with an ascending elevator, with both observers in the elevator, with observer A at the top of the elevator and observer B on the floor of the elevator, in terms of doppler shift and time dilation?

- upward acceleration of the cabin in absence of a gravitational field,
- the cabin being locally at rest on the surface of the earth with it's gravitational field.

In my above example in posting #21, observer ##A## and the lamp at the top of the elevator shaft are locally both at rest in an accelerated frame. The proper acceleration of each can be measured with accelerometers.

In your example, assuming you mean an accelerated ascending elevator, observer ##B## would see light, emitted by observer ##A##, blue-shifted by the factor ##1 + \frac{\phi}{c^2}##, with ##\phi=gh## and ##g## being the proper upward-acceleration of the bottom of the cabin and ##h## the height of the cabin.

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A.T.

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If the scenario is symmetrical then then both time dilations are symmetrical. But in GR this requires that both clocks are placed and move symmetrically relative to the gravitational sources.Ranku said:Why is gravitational time dilation non-symmetrical between two observers, in contrast to the symmetry between two observers in SR?

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PeterDonis

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If both observers are placed symmetrically relative to the source of gravity (for example, at the same altitude above a spherically symmetric planet or star), the relative time dilation between them is zero. But for gravitational time dilation, that is theA.T. said:If the scenario is symmetrical then then both time dilations are symmetrical.

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