How to compare the proper times of two spatially distant mass points?

[Peter Strohmayer]

TL;DR Summary

Is a comparison of the proper times of two mass points resting relative to each other possible if their two world lines never intersect and the intersection at the first event a and the intersection at the second event b are replaced by pairs of events a' and b' which - from the point of view of a reference frame in which both mass points rest - are simultaneous with the first pair of events?

Do all synchronized clocks in a reference system always show the same time?

Is this part of the definition of a frame of reference?

Have the clocks always passed the same proper time from zero (see below)?

Would the knowledge of the proper time of a clock between two events lead to the knowledge of the proper times of all clocks between simultaneous pairs of events?

If no: Is this conclusion impossible because the simultaneous events are not simultaneous from the point of view of other reference frames?

Thus, would the spatiotemporal intervals between pairs of events no longer be the same from the point of view of each reference frame?

 

[Dale]

Is a comparison of the proper times of two mass points resting relative to each other possible if their two world lines never intersect … from the point of view of a reference frame in which …

Not in a frame invariant way.

are simultaneous with the first pair of events?

This is the key issue. Other frames will not agree that a pair of such events is simultaneous.

 

[Nugatory]

Is a comparison of the proper times of two mass points resting relative to each other possible if their two world lines never intersect and the intersection at the first event a and the intersection at the second event b are replaced by pairs of events a' and b' which - from the point of view of a reference frame in which both mass points rest - are simultaneous with the first pair of events?

We can measure or calculate the proper time between event a and b and the proper time between a' and b', but that doesn't tell us much. It's like measuring or calculating the distance between London and Moscow on the one hand, and between Cairo and Durban on the other - they are both perfectly fine and well-defined quantities but they aren't particularly related to one another.

Do all synchronized clocks in a reference system always show the same time?

Not necessarily. The clocks on the wall in the Dallas airport are synchronized with the clocks on the wall in the New York airport but the Dallas clocks consistently read one hour behind the New York clocks. That's because they were zeroed at different times (using the frame in which both are at rest to define "different times").

Is this part of the definition of a frame of reference?

No. A frame of reference is a convention for assigning x, y, z, and t coordinates to events in spacetime. Some frames of reference are more convenient for some problems than others; frames of reference in which the Dallas and New York airports are at rest so that their clocks can be synchronized are especially convenient when we're planning travel between those cities.

Have the clocks always passed the same proper time from zero (see below)?

Because their worldlines have never crossed there is no single zero event to start measuring from. We can choose the four events a, b, a', b' such that the proper time between the pairs a,b and a',b' is the same; that's what we're doing when we say that there's one hour between the events "clock in New York reads 3:00 PM" and "clock in New York reads 4:00 PM" and also between the events "clock in Dallas reads 2:00 PM" and "clock in Dallas reads 3:00 PM"

Would the knowledge of the proper time of a clock between two events lead to the knowledge of the proper times of all clocks between simultaneous pairs of events?

Yes, provided that we also have the position information and - crucially - have specified what we mean by "simultaneous", which means specifying our simultaneity convention. Without that, we haven't uniquely specified the two remote events, let alone the proper time between them.

Thus, would the spatiotemporal intervals between pairs of events no longer be the same from the point of view of each reference frame?

The interval between pairs of events is the same in all frames, always. It's analogous to the way that the ruler distance between two points on a sheet of paper is the same and equal to $\sqrt{(\Delta x)^2 + (\Delta y)^2}$ no matter where we put the origin and how we rotate our x-y grid.

 

[Ibix]

Do all synchronized clocks in a reference system always show the same time?

There are two asects to "synchronised", which is the zeroing and the tock rate. Two clocks that are zeroed simultaneously according to some inertial frame, and have zero motion relative to each other will always show the same time according to that frame. They will always show different times according to any other frame. But this is vacuous, because that is the definition of "same time" in such reference frames. So the only interesting physics here is whether it is possible or not to define such frames.

Have the clocks always passed the same proper time from zero (see below)?

According to the frame in which they were zeroed, yes. According to other frames, no.

Would the knowledge of the proper time of a clock between two events lead to the knowledge of the proper times of all clocks between simultaneous pairs of events?

Not as you state it. With additional restrictions (which include picking a frame), possibly.

Thus, would the spatiotemporal intervals between pairs of events no longer be the same from the point of view of each reference frame?

The interval between a pair of events is invariant. That doesn't mean you always have enough information to determine it.

 

[PeterDonis]

Is a comparison of the proper times of two mass points resting relative to each other possible if their two world lines never intersect and the intersection at the first event a and the intersection at the second event b are replaced by pairs of events a' and b' which - from the point of view of a reference frame in which both mass points rest - are simultaneous with the first pair of events?

You can always compare the proper times of two objects if you have already determined which pair of events on each worldline you will use for the comparison. Once you have determined events a and b on one worldline and events a' and b' on the other, the proper time of one object between a and b is an invariant, and so is the proper time of the other between a' and b'. So you are comparing two invariants.

The issue that got you all tangled up in your previous thread was, how do you determine the pairs of events? And the answer to that is that, unless both pairs are the same, a' = a and b' = b, as is the case in the standard "twin paradox" where the twins meet twice, there is no frame invariant way of picking corresponding pairs of events. You can still pick pairs of events, but whatever way you do it will be frame dependent. For example, your method of picking the pairs of events here only works for a specific frame, the frame in which both objects are at rest (which, in turn, requires that there is such a frame, which is a very special condition that in most scenarios of interest will not be true). Applying the rule "use events on the two worldlines which are simultaneous" gives different events for different frames, and so the comparison will give different results.

 

[jbriggs444]

TL;DR Summary: Is a comparison of the proper times of two mass points resting relative to each other possible if their two world lines never intersect and the intersection at the first event a and the intersection at the second event b are replaced by pairs of events a' and b' which - from the point of view of a reference frame in which both mass points rest - are simultaneous with the first pair of events?

What is the "proper time of a mass point"? I know how to measure the proper time elapsed on a trajectory. Do you, perhaps, mean the elapsed time from a starting event to an ending event on a timelike world line?

Do all synchronized clocks in a reference system always show the same time?

What does this even mean? A clock that always shows the same time is right at most twice a day.

Is this part of the definition of a frame of reference?

Is what part of the definition of a frame of reference?

Have the clocks always passed the same proper time from zero (see below)?

At any given time coordinate in a reference frame, all of the clocks that were zeroed at time zero in that frame and have been running correctly since will be displaying that time coordinate. Yes. That is by definition.

Would the knowledge of the proper time of a clock between two events lead to the knowledge of the proper times of all clocks between simultaneous pairs of events?

What is "the proper time between simultaneous pairs of events"? There can be no timelike trajectory between such events. So no defined proper time between them.

If no: Is this conclusion impossible because the simultaneous events are not simultaneous from the point of view of other reference frames?

No. It is impossible for other reasons.

Thus, would the spatiotemporal intervals between pairs of events no longer be the same from the point of view of each reference frame?

The interval between a pair of events along a given trajectory is an invariant. In the flat space time of special relativity, we usually ignore the requirement for a trajectory and assume that we are using the [unique] geodesic path between the two events.

 

[Nugatory]

@Peter Strohmayer you have been asking the same question and getting the same answers over and over again in multiple threads - clearly the way we’re explaining this stuff isn’t helping you see or clear up your misunderstanding.

You might be better served by a different approach: get hold of a copy of Taylor and Wheeler’s “Spacetime Physics” and work through it from the beginning. Do not speed through the early chapters thinking you understand that basic stuff - you don’t. The first edition is legal and free on the internet, and we can help over any hard spots if you get stuck.

Please don’t just post the same old stuff here without having gone through that exercise or something similar. This thread is already on the wrong side of the forum rule about restarting closed threads and another like it won’t help anyone.

This thread is closed.