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Ranku
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When an object is in freefall, it is in a locally inertial frame of reference. If two objects are in freefall, can their locally inertial frames of reference have different relative velocities?
Yes, and in the presence of tidal gravity their relative velocity can change over time.Ranku said:When an object is in freefall, it is in a locally inertial frame of reference. If two objects are in freefall, can their locally inertial frames of reference have different relative velocities?
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.
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?
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?
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?
I don’t understand this question.Ranku said:So the object closer to the gravitational source will have greater relative velocity than the object further from it?
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:Two objects in freefall will see the same acceleration and can have different velocities if they do not fall at the same time or if they are in different gravitational fields.
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.
No, relative velocities between objects give rise to apparent time dilation and length contraction.Ranku said:relative velocities between two inertial frames of reference give rise to time dilation and length contraction
Relative velocities between objects that are moving on different inertial worldlines that, because of tidal gravity, separate or converge, can give rise to apparent time dilation and length contraction, yes. But typically such relative velocities are way too small for such phenomena to be observable.Ranku said:does relative velocities between two locally inertial frames of reference within freefall also give rise to the same phenomena?
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?
What do you mean by apparent time dilation?PeterDonis said:relative velocities between objects give rise to apparent time dilation and length contraction
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.
I mean that the terms "time dilation" and "length contraction" in SR refer to observer-dependent appearances: clocks moving relative to an observer appear to run slow to that observer, and rulers moving relative to an observer appear to be shorter to that observer. But those appearances are symmetric; a second observer moving along with the clock and ruler will not find the clock to run slow or the ruler to be shorter, but will find clocks and rulers moving along with the first observer to run slow/be shorter.Dale said:What do you mean by apparent time dilation?
So is that because the local inertial frame of reference of the observer has a higher velocity?TonyStewart said:I think ... When an observer is closer to a gravitational field, time runs more slowly compared to an observer farther away from the field. Similarly, lengths in the direction of the gravitational field appear contracted.
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?
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?
No, time dilation and length contraction are entirely due to relative motion in SR.TonyStewart 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.
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.
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.
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?
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}##.
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?
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?
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 the only symmetrical possibility. If the relative gravitational time dilation between the observers is not zero, it will not be symmetrical either; both observers will agree on which one's clock is running slower.A.T. said:If the scenario is symmetrical then then both time dilations are symmetrical.
A locally inertial frame is a coordinate system in which the laws of physics take on their simplest form. In other words, the effects of gravity and acceleration are negligible in this frame, making it an ideal reference frame for studying motion and other physical phenomena.
Freefall is the motion of an object under the sole influence of gravity. In a locally inertial frame, an object in freefall will appear to be at rest and not experiencing any acceleration. This is because the frame itself is accelerating at the same rate as the object due to the effects of gravity.
The principle of equivalence states that the effects of gravity are indistinguishable from the effects of acceleration. This means that an observer in a uniformly accelerating frame will experience the same physical laws as an observer in a gravitational field. This principle is the basis for understanding the relationship between freefall and locally inertial frames.
In a locally inertial frame, relative velocities can be calculated using the laws of special relativity. This means that the velocity of an object relative to an observer in a locally inertial frame will be the same as the velocity of the object relative to an observer in an inertial frame. However, if the observer is also accelerating, the relative velocity will appear to be different due to the effects of acceleration.
Locally inertial frames allow us to simplify complex physical systems and accurately describe the behavior of objects in motion. They also help us understand the fundamental principles of relativity and gravity. Additionally, many physical laws and equations are derived and tested in locally inertial frames, making them an essential tool in the study of physics.