Proper Time in Special vs. General Relativity

[schaefera]

We know that the proper time between two events is the shortest possible time between those two events that can be measured in any frame. This follows from the idea that moving clocks run slow-- a stationary clock at rest in S' which moves relative to S at a constant speed v will be time dilated to run slow, and from the Lorentz transformation it follows immediately that if two events happen at the same location in S', all other frames require MORE time than the proper time measured in S' between those two events (which is, quantitatively γ*Δt').

Now, I've also heard that in general relativity, the proper time of a clock is the time measured by a clock along a geodesic-- that is, the time measured by a clock which has MAXIMIZED its proper time.

Is it strange that in SR, proper time is the smallest possible time between events while in GR the proper time is maximized by a clock which experiences no non-gravitational forces?

How are these two ideas compatible? Why does one minimize while the other maximizes proper time?

 

[Dale]

We know that the proper time between two events is the shortest possible time between those two events that can be measured in any frame. This follows from the idea that moving clocks run slow-- a stationary clock at rest in S' which moves relative to S at a constant speed v will be time dilated to run slow, and from the Lorentz transformation it follows immediately that if two events happen at the same location in S', all other frames require MORE time than the proper time measured in S' between those two events (which is, quantitatively γ*Δt').

Hi schaefera, you have this backwards, the proper time is the LONGEST possible time. Think about the twins "paradox", the inertial twin may age 20 years while the traveling twin only ages 1, 1 year is LESS time than 20.

Technically the proper time is the time along any path. That means that the proper time is path dependent in both GR and SR. What you are actually describing is a geodesic path, which is a path that maximizes the proper time with respect to local variations in the path, again in both GR and SR.

 

[schaefera]

But I thought the shortest time between two events is the time measured in the rest frame... so the frame in which an observer sees them both happening at the same location? My textbook, at least, claims that it is this frame in which proper time is defined. It also says that all other frames measure times longer than the proper time.

EDIT: Textbook is Modern Physics by Krane.

 

[Dale]

My apologies, I misunderstood your question. I hope I don't confuse you.

The proper time, in both SR and GR, is the time actually measured by a single clock. There is no such thing as "the" proper time between two events since the time measured by the clock between the two events depends on the path it takes. However, there are paths which maximizes that time, and those paths are called geodesics.

There is another thing called coordinate time. In both SR and GR coordinate time is measured by a system of synchronized clocks, and the coordinate time between two events depends on the synchronization convention for the pair of clocks at the two events. In SR inertial frames (i.e. using Einstein's synchronization convention) the coordinate time is indeed minimized in the frame where the two events occur at the same location.

The two ideas are compatible because they are talking about two very different things. One is coordinate time (two clocks, one at each event, and some synchronization convention) and the other is proper time (one clock which moves along some path between both events).

 

[schaefera]

So, just to make sure I understand, the proper time is dependent on which clock and which path we choose to consider... meaning that every clock reads its own proper time?

Does that mean that in the twin paradox, both twins could claim their own time is their measured proper time?

The proper time my textbook talks about is really a special one in which the only possible path is a straight line (because we aren't considering acceleration) and therefore my book is actually applying the name proper time to this straight line path between the two events.

But in GR, with acceleration considered, the proper time is maximized-- but the proper time is still the amount of time measured by a clock in its own rest frame?

 

[Dale]

So, just to make sure I understand, the proper time is dependent on which clock and which path we choose to consider... meaning that every clock reads its own proper time?

Yes, exactly.

Does that mean that in the twin paradox, both twins could claim their own time is their measured proper time?

Yes. Your age is just your own proper time since birth.

The proper time my textbook talks about is really a special one in which the only possible path is a straight line (because we aren't considering acceleration) and therefore my book is actually applying the name proper time to this straight line path between the two events.

that is a little unfortunate that it is using non standard terminology. Such a path is usually called a geodesic, but the time along a geodesic is not usually given a special name.

in GR, with acceleration considered, the proper time is maximized-- but the proper time is still the amount of time measured by a clock in its own rest frame?

You can do acceleration in SR (just not gravity), but otherwise, yes.

 

[schaefera]

Thank you very much for clearing that up!

My one last question now has to do with the spacetime interval, s. If s^2=r^2-(ct)^2, where r is separation in space and t is the time difference between events, what does it mean when we solve for the t between events if we let r=0... So I'm asking about what the time in a time-like separation means. That's what we've called proper time in class, but it shouldn't really be called that?

 

[Dale]

My one last question now has to do with the spacetime interval, s. If s^2=r^2-(ct)^2, where r is separation in space and t is the time difference between events, what does it mean when we solve for the t between events if we let r=0... So I'm asking about what the time in a time-like separation means. That's what we've called proper time in class, but it shouldn't really be called that?

I wouldn't call it that. t is a coordinate time. It is the coordinate time between the two events in the inertial frame where the two events are co-located.

However, please, if you are taking a class then make sure that you can give the professor the answer that he wants to hear in the vocabulary that he wants to use. I cannot save your grade!

 

[schaefera]

Got it, thank you so much for clarifying!

 

[A.T.]

Your age is just your own proper time since birth.

Biologically, but legally? Would the traveling twin, after leaving as a child and returning only slightly aged, have to ask his brother to buy beer for him, even tough they have the same birth date in the id cart?

 

[Dale]

Biologically, but legally?

Hehe, I just had a vision of trying to explain relativity to a bunch of politicians.

 

[Parlyne]

Hehe, I just had a vision of trying to explain relativity to a bunch of politicians.

Careful. They might start pushing for schools to teach Intelligent Aging.

 

[Nugatory]

Hehe, I just had a vision of trying to explain relativity to a bunch of politicians.

This problem shows up, almost as an afterthought, near the end of Heinlein's "Tunnel in the Sky". Different physics, because Heinlein needed the kids to return from summer camp :) having aged more than their parents, hence closer to legal adulthood.