Why time DILATION?

Nugatory

By "fixed to a point on the equator", you mean that there is a point on the equator that is at rest in the frame, right? No such frame can be inertial because it is accelerating, and I see no way to discuss it without considering the acceleration.

(We need not bring in GR, of course)

Mentz114

You're ignoring what I said. There is no need to 'establish' differential aging. It follows from the fact that every worldline has its own invariant proper time.

Now you're agreeing with me and contradicting yourself.

Wrong. Einstein called differential aging 'the clock paradox'.

Your thinking is very woolly. Stick to invariants. Time dilation is not an invariant.

PAllen

The case of a equatorial and polar observer is really a case of twin differential aging not just time dilation because the equatorial observer keeps returning to a fixed point in the polar observer's rest frame. In exchanging signals, the polar observer is effectively getting slightly delayed information about this 'twin' situation. As a result, the accumulated average time difference between polar and equatorial clock is invariant, not observer dependent.

If two observers simply move past each other at some relative speed, each forever considers the others clock to be slow, and other observer's have various different conclusions about which clock is faster.

Note, the non-inertial character of the equatorial clock must be considered (you can't consider them the origin of a coordinate system that can use the Lorentz transform or the Minkowski metric). However, since you are interested in invariant features (exchange of signals with information), you can do the analysis most simply in any inertial frame (e.g. the polar inertial frame). This frame is sufficient for computing any actual observation or measurement made by the equatorial observer. Note also, that a frame in which the earth was moving at .999c would be almost as simple to use, and would compute identical results for the behavior of exchanged signals and world line proper time information they carry. However, it would differ radically on 'time dilation' applicable to the polar and equatorial clocks.

arindamsinha

@Nugatory - Since in SR there is no preferred frame of reference, what is the 'point on the equator' accelerating w.r.t. to? Einstein does not refer to any acceleration at all in his example. There is a way to discuss it without considering any acceleration - a preferred IRF from which both observers' velocities are measured. The polar point has no velocity in this IRF, the equatorial point does. I believe this is implicit in Einstein's example.

@PAllen - Absolutely correct, but w.r.t. what frame is the equatorial observer returning to the same point? We are talking about a single preferred IRF w.r.t. which we are looking at both observers, I think you will agree.

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A lot of discussion in this thread seems to be about differentiating the terminologies of "time dilation" and "differential aging". My understanding is that "time dilation" is just a combination of the classical Doppler effect combined with "differential aging". Would you agree?

Nugatory

Unlike velocity, acceleration does not have to be defined with respect to something else.

In SR (and GR with a more careful definition of "acceleration") an accelerated frame and a non-inertial frame are the same thing, so if I can observe non-inertial behavior I know that I'm being accelerated - even if there's nothing else around for the acceleration to be relative to. The most natural way to for me at the equator to measure my radial acceleration towards the center of the earth due to the earth's rotation is to study the behavior of an object moving in the radial direction; it will move eastward in violation of the law of inertia and I'll know that I'm not in an inertial frame. A sensitive enough accelerometer would also do the trick; and the accelerometer reading is the position of a needle on a scale, and that has to be an invariant fact not relative to anything else.

ghwellsjr

This is not true. Here is a diagram that depicts a similar situation to the one you proposed except that it uses the same parameters as the one in post #33 but without the moving twin returning:



You are giving preference to this one frame, the Inertial Reference Frame (IRF) in which the "stationary" twin remains inertial. In this IRF, the differential aging after the moving twin comes to mutual rest with the "stationary" twin is 4.5 quarter hours (a little over an hour) with the moving twin younger. But look at this diagram in which the moving twin is inertial during his moving portion of the trip:



Here there is a different criterion for what is simultaneous and now the stationary twin is the one that is younger by slightly over 2 quarter hours (a little over a half hour).

Of course, every IRF will show the same results for whatever light signals are exchanged between the twins, but that is not sufficient to establish unambiguous simultaneity and that's what you have to do to determine differential aging. Differential aging is answering the question, between this coordinate time and that coordinate time, what is the difference in how two observers age? Even if the two observers agree with each other because they implicitly are using their mutual rest IRF, that doesn't mean the question has been answered the same for all other IRF's. The only way for all IRF's to get the same answer is if the two observers are colocated at the first coordinate time and again colocated at the last coordinate time, (not necessarily the same location, not necessarily even at rest in the same IRF nor do the observers even have to be in mutual rest at either time). All this is simply to remove any ambiguity about simultaneity issues at the start and the end of the process.
The clock moving in a circle is constantly accelerating. If it weren't, it would go in a straight line. It has to accelerate in order to move in a circle. It is not inertial. It cannot be at rest in an IRF. Einstein only considered a single reference frame when discussing the differential tick rate of the two clocks and he did not address the issue of the differential aging.
You don't have a correct idea of what Time Dilation is nor of what Differential Aging is and I suspect you don't understand Relativistic Doppler
I've never brought GR into the discussion but somehow you seem to associate acceleration with GR. I'm not sure you understand what acceleration is. The clock on the equator is accelerating in order to maintain a circular path. You could, if you wanted to (I don't) consider a non-inertial frame in which the equator clock is at rest and in which the pole clock is moving but I don't think you're ready for that.

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PAllen

The 'twin like' feature of the equatorial path is that it has periodic intersections with an inertial space time path that has zero relative velocity compared to the polar observer's (presumed) inertial space time path. This circumstance is true in every coordinate system or reference frame.

You have the concept of time dilation backwards. Differential aging and Doppler are the invariant observables. Time dilation is a feature of the how a particular clock's time relates to coordinate time; it can then be used to compute the invariants: differential aging, clock time between two physically identifiable events, and Doppler.