Gravitational time dilation, proper time and spacetime interval

[haushofer]

Dear all,

I'm having confusion about the standard derivation of Schwarzschild's gravitational time dilation. For concreteness I'll follow the explanation of Schutz' "gravity from the ground up", but other texts argue the same. So let me rephrase Schutz's explanation (I surpress factors of c in it):

"We know that the time part of the spacetime interval gives the rate at which clocks at a fixed place in space run. If a clock is at rest, so that along its word line [itex]dr=d\theta=d\phi = 0 [/itex], then the Schwarzschild spacetime interval tells us that its proper time lapse [itex]d\tau [/itex] associated with a coordinate time lapse of [itex]dt [/itex] is given by

[tex]
(d\tau)^2 = - (ds)^2 = (1-\frac{2GM}{r}) (dt)^2
[/tex]

This determines etc. etc."

My first confusion is this: to my understanding, the spacetime interval between A and B only equals the elapsed proper time if the corresponding observer traveling between A and B is inertial. But the chosen observer here is clearly not, he's standing still ! So I'd say the equality between the spacetime interval and proper time above does not hold.

My second confusion is that if we use the standard rule for calculating the elapsed proper time for an observer traveling along a path [itex]x^{\mu}(\lambda)[/itex],

[tex]
\Delta \tau = \int \sqrt{-g_{\mu\nu} \frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}} d\lambda
[/tex]

and evaluate for our observer earlier (his t is proper time),

[tex]
\frac{dx^{\mu}}{d\tau} = (1,0,0,0)
[/tex]

then

[tex]
\Delta \tau = \int \sqrt{-g_{00} } dt = \sqrt{-g_{00} } \Delta t
[/tex]

which coincides with the result Schutz is given. And of course I'm thoroughly familiar with it. Because I'm suspicious about this "coincidence", my question to you is: what am I missing?

 

[Orodruin]

But the two are the same, the first is just the differential form of the other and when you integrate it you get the second. Note that ##dt## a priori is not the time difference between two finitely separated events although that is the result you get after the integration as you have done.

 

[haushofer]

But the two are the same, the first is just the differential form of the other and when you integrate it you get the second. Note that ##dt## a priori is not the time difference between two finitely separated events although that is the result you get after the integration as you have done.

I see that the two results are the same, but am I right in that one cannot just put [itex](d\tau)^2 = - (ds)^2[/itex] for our observer at fixed position? My last equation between the finite coordinate and proper time differences follows because the metric components do not depend on time, but my point is really that I cannot reconcile this result with the fact that apparently [itex](d\tau)^2 = - (ds)^2[/itex] for our observer at fixed position, even if he is not inertial.

 

[Orodruin]

but am I right in that one cannot just put [itex](d\tau)^2 = - (ds)^2[/tex] for our observer at fixed position?

No, this is true by definition. Just like path lengths in Riemannian geometry. It does not matter if the curve is not a geodesic, ##ds^2 = g_{ab}dx^a dx^b## describes the curve length.

 

[vanhees71]

Why not? ##\tau## is a functional of the worldline of your (pointlike) observer, and if you have ##\mathrm{d} r =\mathrm{d} \vartheta=\mathrm{d} \varphi=0##, then you can choose ##t## as the parameter, and then you immediately get ##\mathrm{d} \tau=-g_00 \mathrm{d} t^2## (east-coast convention, as you've obviously used in your postings above).

 

[haushofer]

No, this is true by definition. Just like path lengths in Riemannian geometry. It does not matter if the curve is not a geodesic, ##ds^2 = g_{ab}dx^a dx^b## describes the curve length.

But s^2 does not describe the curve length; it describes the distance, and this distance only equals the curve length for straight curves (inertial observers). I think my confusion lies in infinitesimal vs. finite differences; infinitesimally, a curve looks flat (just like observers are inertial at short time periods), and that's why [itex](d\tau)^2 = - (ds)^2[/itex], although for an non-inertial observer [itex](\Delta \tau)^2 \neq - (\Delta s)^2[/itex]. But this still keeps me with my problem: why then do we have [itex](\Delta \tau)^2 = - (\Delta s)^2[/itex] for our non-inertial observer?

 

[haushofer]

Why not? ##\tau## is a functional of the worldline of your (pointlike) observer, and if you have ##\mathrm{d} r =\mathrm{d} \vartheta=\mathrm{d} \varphi=0##, then you can choose ##t## as the parameter, and then you immediately get ##\mathrm{d} \tau=-g_00 \mathrm{d} t^2## (east-coast convention, as you've obviously used in your postings above).

Yes, I see that now, thanks! But my problem still remains.

Let me make a simple comparison to SR: if an observer travels non-inertially between the finitely (as opposed to infinitesimally) separated evens A and B using a rocket or whatever, his elapsed proper time will not be equal to the spacetime interval between A and B, right? It would be quite a coincidence if it would. So in general I'd say [itex](\Delta \tau)^2 \neq - (\Delta s)^2[/itex] between A and B.

Well, here we have a non-inertial observer who in the same spirit nevertheless measures the same amount of proper time elapsed as the spacetime interval, [itex](\Delta \tau)^2 = - (\Delta s)^2[/itex]. Considering the above that's weird to me.

 

[Orodruin]

But s^2 does not describe the curve length; it describes the distance, and this distance only equals the curve length for straight curves (inertial observers).

No, again you are thinking about finite differences. The relation is infinitesimal. Think polygon train approximations of curve length.

why then do we have [itex](\Delta \tau)^2 = - (\Delta s)^2[/itex] for our non-inertial observer?

Because, as you noted when actually doing the integration, your choice of coordinates is such that the relation does not depend on ##\tau## nor ##t##. Compare to the length of the curves of constant ##\theta## on a sphere. These curves are not geodesics, but their length is given by ##ds^2 = \sin^2(\theta_0) d\varphi^2## and so ##\Delta s = \left|\sin(\theta_0)\right| \Delta \varphi##, where ##\theta_0## is the constant value of ##\theta##.

 

[vanhees71]

The finite length is of course
$$\mathrm{d} \tau=\int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda \sqrt{-g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}},$$
where ##\dot{x}^{\mu}=\mathrm{d} x^{\mu}/\mathrm{d} \lambda##, and the world line has to be timelike, i.e., the expression under the root must be positive everywhere.

 

[haushofer]

No, again you are thinking about finite differences. The relation is infinitesimal. Think polygon train approximations of curve length.Because, as you noted when actually doing the integration, your choice of coordinates is such that the relation does not depend on ##\tau## nor ##t##. Compare to the length of the curves of constant ##\theta## on a sphere. These curves are not geodesics, but their length is given by ##ds^2 = \sin^2(\theta_0) d\varphi^2## and so ##\Delta s = \left|\sin(\theta_0)\right| \Delta \varphi##, where ##\theta_0## is the constant value of ##\theta##.

Yes, I was confused about finite v.s. infinitesimal differences. And I like your example, that's exactly what's going in in my example. So at the calculational level, I'm good. But I'm still a bit puzzled why a non-inertial observer measures a proper time between two events equal to the spacetime interval. As I see it now, it is a happy coincidence because the metric component does not depend on ##t##.

 

[Orodruin]

Yes, I was confused about finite v.s. infinitesimal differences. And I like your example, that's exactly what's going in in my example. So at the calculational level, I'm good. But I'm still a bit puzzled why a non-inertial observer measures a proper time between two events equal to the spacetime interval. As I see it now, it is a happy coincidence because the metric component does not depend on ##t##.

There is no such thing as a "finite space-time interval" in GR - only finite coordinate differences, which are arbitrary. The "coincidence" you are referring to is just the requirement of the space-time being stationary and your curve being an integral curve of the corresponding Killing field. (You will find that the same is true of my example on the sphere with ##\partial_\varphi## being the Killing field).

 

[haushofer]

There is no such thing as a "finite space-time interval" in GR - only finite coordinate differences, which are arbitrary. The "coincidence" you are referring to is just the requirement of the space-time being stationary and your curve being an integral curve of the corresponding Killing field. (You will find that the same is true of my example on the sphere with ##\partial_\varphi## being the Killing field).

Yes, I do understand that this integral curve is the "coincidence". Let me ask another question. Imagine I have two events A and B in an arbitrary curved spacetime (a solution of the Einstein equations, for concreteness). These events are not infinitesimally separated. Also imagine I let an arbitrary observer travel between these two events. Under which conditions on the spacetime and/or observer does his proper time equal the spacetime interval?

Many thanks by the way for your efforts to clear up my confusion. Maybe it's the summer holidays :P

 

[pervect]

The way I think about it is the following. Space-time intervals are a property not of a pair of points, but of a curve with endpoints. Usually in SR the curve in question is the unique geodesic between two points, but probably it'd be wrong to say this is always true, for instance in dealing with rotating frames using SR, one might compute the space-time interval of a clock that is accelerating in such a way as to have a fixed position in the rotating frame.

If one specifies for a particular problem that the space-time interval in question is to be measured along a geodesic curve, one can ask whether specifying two points uniquely determines a geodesic. The answer is "maybe", and I believe that in the neighborhood of any point in a manifold, there's a region, called the local convex region IIRC in which geodesics are unique.

 

[PeterDonis]

to my understanding, the spacetime interval between A and B only equals the elapsed proper time if the corresponding observer traveling between A and B is inertial

Schutz's use of the term "spacetime interval" appears to be somewhat nonstandard. At least in texts that focus on special relativity, the term "spacetime interval" usually has a very specific meaning: it is the "distance" between two events in flat spacetime, along the unique geodesic that connects them. Under this definition, strictly speaking there is no such thing as "the spacetime interval" in a curved spacetime at all.

Schutz is using the term "spacetime interval" to mean "the arc length along a particular curve between two events", which is a much more general concept; even in flat spacetime there will be an infinite number of "spacetime intervals" in this sense between any two events, since there are an infinite number of possible curves that connect them. In flat spacetime, only one of these curves will be a geodesic, but in curved spacetime, even that is no longer true in general.

 

[haushofer]

But isn't that arclength just the proper time along the curve? To me Schutz's use then seems like confusing lengths and distances in Euclidean space.

 

[haushofer]

I think my confusion arises because I thought the spacetime interval only equals the proper time of a curve if that curve us a geodesic and hence the observer is inertial in the GR-sense.

 

[haushofer]

The way I think about it is the following. Space-time intervals are a property not of a pair of points, but of a curve with endpoints. Usually in SR the curve in question is the unique geodesic between two points, but probably it'd be wrong to say this is always true, for instance in dealing with rotating frames using SR, one might compute the space-time interval of a clock that is accelerating in such a way as to have a fixed position in the rotating frame.

But isn't this the proper time (=length of the curve)? In the Euclidean plane we also make a Sharp distinction between distances between A and B and lengths of curves having A and B as endpoints.

 

[PeterDonis]

But isn't that arclength just the proper time along the curve?

If the curve is timelike, yes. But you can also have null and spacelike curves; they have arc lengths too.

In the Euclidean plane we also make a Sharp distinction between distances between A and B and lengths of curves having A and B as endpoints.

Yes, but if you unpack that distinction, you realize that the "distance" between A and B is just the length along one particular curve having A and B as endpoints--the unique geodesic between those points. But in a non-Euclidean geometry, there can be multiple geodesics connecting the same pair of points, so the Euclidean definition of "distance" is no longer well-defined. Similarly, in a curved spacetime, there is no unique "spacetime interval" between two events, the way there is in the flat spacetime of special relativity.

 

[SiennaTheGr8]

Let me make a simple comparison to SR: if an observer travels non-inertially between the finitely (as opposed to infinitesimally) separated evens A and B using a rocket or whatever, his elapsed proper time will not be equal to the spacetime interval between A and B, right? It would be quite a coincidence if it would. So in general I'd say [itex](\Delta \tau)^2 \neq - (\Delta s)^2[/itex] between A and B.

There's some inconsistency in how the term proper time is used. Does it refer to the wristwatch time of a (potentially non-inertial) traveler? Or does it refer to the spacetime interval expressed in units of time?

I say it's the former, and that the latter is informal or sloppy usage. I think a better term for the latter is the inertial proper time interval between two events: ##\Delta \tau | _{\vec a = \vec 0} = \Delta s / c##. (It's the total time logged by the wristwatch of an inertial traveler who journeys from one event to the other.)

I regard the proper time interval ##\Delta \tau## as a more general concept, applicable to any journey through spacetime (inertial or not).

 

[PeterDonis]

There's some inconsistency in how the term proper time is used.

I'm not sure there is any inconsistency in actual textbooks or peer-reviewed papers. The literature seems pretty consistent to me: "proper time" means arc length along a timelike curve. That would correspond to this:

Does it refer to the wristwatch time of a (potentially non-inertial) traveler?

 

[SiennaTheGr8]

I'm not sure there is any inconsistency in actual textbooks or peer-reviewed papers. The literature seems pretty consistent to me: "proper time" means arc length along a timelike curve. That would correspond to this:

Here are some examples of what I mean:

https://www.google.com/search?tbm=bks&q="proper+time+between+events"
https://www.google.com/search?tbm=bks&q="proper+time+interval+between+events"
https://www.google.com/search?tbm=bks&q="proper+time+between+two+events"
https://www.google.com/search?tbm=bks&q="proper+time+interval+between+two+events"

 

[PeterDonis]

Here are some examples of what I mean

Perhaps I'm more used to reading GR literature, where you can't get away with the special assumptions that you can in SR texts. All of these look like SR texts, and they are using the term "proper time" to mean, in the terms I would expect to see in a GR text, "the arc length along the unique timelike geodesic between two timelike separated events". But as I've said previously in this thread, that only works in SR, where spacetime is flat and there is a unique timelike geodesic between any two timelike separated events. Similarly, thinking of this proper time as the unique "spacetime interval" between those events only works in SR, in flat spacetime.

Since these are all SR texts, I would not necessarily expect them to explicitly discuss the fact that their definition of "proper time", strictly speaking, only works in flat spacetime. But in the context of this thread, that point is of course essential to recognize.

 

[PAllen]

Here are some examples of what I mean:

https://www.google.com/search?tbm=bks&q="proper+time+between+events"
https://www.google.com/search?tbm=bks&q="proper+time+interval+between+events"
https://www.google.com/search?tbm=bks&q="proper+time+between+two+events"
https://www.google.com/search?tbm=bks&q="proper+time+interval+between+two+events"

These all seem to refer to SR, where proper time between events is short hand for proper time along the unique geodesic connecting them. I don't see this as conflicting meaning, more a shorthand - 'between two events' can only be well defined if there is a unique geodesic between them, and then it is just shorthand. The general case is proper time along a world line segment.

Similarly, spacetime interval between events is shorthand for spacetime interval along the unique geodesic between them (now not restricted to being timelike), and again this is specific to SR. Spacetime interval along a curve is the more general usage.

In general, I have never seen the 'two events' language used in GR, for the reasons many have mentioned here - non-uniqueness of geodesic. However, I have seen a different usage by a small number of authors in GR. That is, the 'world function' of two events. This is a function of any pair of events that belong to some convex normal neighborhood, such that there is a unique geodesic between them among all convex normal neighborhoods containing them; then the value of the function is the spacetime interval along this geodesic. The domain of the world function is all pairs of sufficiently close events, in the sense just described.

 

[haushofer]

If the curve is timelike, yes. But you can also have null and spacelike curves; they have arc lengths too.
Yes, but if you unpack that distinction, you realize that the "distance" between A and B is just the length along one particular curve having A and B as endpoints--the unique geodesic between those points. But in a non-Euclidean geometry, there can be multiple geodesics connecting the same pair of points, so the Euclidean definition of "distance" is no longer well-defined. Similarly, in a curved spacetime, there is no unique "spacetime interval" between two events, the way there is in the flat spacetime of special relativity.

Yes, you're absolutely right. That's one important clue I was missing. Thanks.

 

[sweet springs]

Hi.

I think my confusion arises because I thought the spacetime interval only equals the proper time of a curve if that curve us a geodesic and hence the observer is inertial in the GR-sense.

Forget about ##d\tau^2## and use ##ds^2##. This is a simple subscription. Using ##d\tau^2## means using co-moving coordinate of the body where the one stays still at its origin of space coordinate and the rest of the world move. I am afraid that co-moving coordinate of the body is variable and not easy to handle in GR. Best.

 

[haushofer]

Hi.

Forget about ##d\tau^2## and use ##ds^2##. This is a simple subscription. Using ##d\tau^2## means using co-moving coordinate of the body where the one stays still at its origin of space coordinate and the rest of the world move. I am afraid that co-moving coordinate of the body is variable and not easy to handle in GR. Best.

But in the case of grav. time dilation we're interested in the proper time of an observer standing still at fixed position r.

 

[sweet springs]

Hi. In your case
[tex]ds^2=g_{00}dx_0^2[/tex]
Isn't it enuough? Why do you want to introduce [tex]d\tau=\frac{ds}{c}[/tex] ?
Which one are you interested in a dweller standing still at the fixed point or a free falling passenger?

 

[vanhees71]

Perhaps I'm more used to reading GR literature, where you can't get away with the special assumptions that you can in SR texts. All of these look like SR texts, and they are using the term "proper time" to mean, in the terms I would expect to see in a GR text, "the arc length along the unique timelike geodesic between two timelike separated events". But as I've said previously in this thread, that only works in SR, where spacetime is flat and there is a unique timelike geodesic between any two timelike separated events. Similarly, thinking of this proper time as the unique "spacetime interval" between those events only works in SR, in flat spacetime.

Since these are all SR texts, I would not necessarily expect them to explicitly discuss the fact that their definition of "proper time", strictly speaking, only works in flat spacetime. But in the context of this thread, that point is of course essential to recognize.

Proper time refers physically to the motion of a point particle. It's not necessarily along a geodesic. Thus it's defined for any time-like curve
$$\tau=\int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda \sqrt{g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}}.$$
That's it. One doesn't need to make any confusion around this really simple concept (neither in SR nor in GR).

 

[SiennaTheGr8]

Perhaps I'm more used to reading GR literature, where you can't get away with the special assumptions that you can in SR texts. All of these look like SR texts, and they are using the term "proper time" to mean, in the terms I would expect to see in a GR text, "the arc length along the unique timelike geodesic between two timelike separated events". But as I've said previously in this thread, that only works in SR, where spacetime is flat and there is a unique timelike geodesic between any two timelike separated events. Similarly, thinking of this proper time as the unique "spacetime interval" between those events only works in SR, in flat spacetime.

Since these are all SR texts, I would not necessarily expect them to explicitly discuss the fact that their definition of "proper time", strictly speaking, only works in flat spacetime. But in the context of this thread, that point is of course essential to recognize.

Yes, sorry, I was responding to @haushofer's aside about SR:

Let me make a simple comparison to SR: if an observer travels non-inertially between the finitely (as opposed to infinitesimally) separated evens A and B using a rocket or whatever, his elapsed proper time will not be equal to the spacetime interval between A and B, right? It would be quite a coincidence if it would. So in general I'd say [itex](\Delta \tau)^2 \neq - (\Delta s)^2[/itex] between A and B.

What I was getting at is that one sometimes sees talk of "the proper time between events," which doesn't quite jibe with the usual "wristwatch time of a (potentially non-inertial) traveler" definition of proper time.

Carry on!

 

[SiennaTheGr8]

These all seem to refer to SR, where proper time between events is short hand for proper time along the unique geodesic connecting them. I don't see this as conflicting meaning, more a shorthand - 'between two events' can only be well defined if there is a unique geodesic between them, and then it is just shorthand. The general case is proper time along a world line segment.

Similarly, spacetime interval between events is shorthand for spacetime interval along the unique geodesic between them (now not restricted to being timelike), and again this is specific to SR. Spacetime interval along a curve is the more general usage.

Yes, I agree that "the proper time between events" in SR is used as shorthand. It's no problem if the reader already has a good handle on things.

 

[PAllen]

Hi.

Forget about ##d\tau^2## and use ##ds^2##. This is a simple subscription. Using ##d\tau^2## means using co-moving coordinate of the body where the one stays still at its origin of space coordinate and the rest of the world move. I am afraid that co-moving coordinate of the body is variable and not easy to handle in GR. Best.

Not the way I've seen. dτ² is just -1/c² times ds², I.e. just a different metric convention, according to all GR books I own.

 

[sweet springs]

All your books allow imaginary ##\tau## for space interval?

 

[vanhees71]

No, the notion of proper time of course only makes sense for space-like trajectories in spacetime (in both SR and GR).

 

[PAllen]

All your books allow imaginary ##\tau## for space interval?

No, by pure convention, you use proper time differential for timelike or null, and for ds, any type of curve, adjusting the sign as needed. But there is no strong reason for this - it is perfectly common to use time as a unit for distance.

The main point is using proper time says nothing about coordinates in use, or even using coordinates at all. It is perfectly well defined in coordinate free conventions.

 

[vanhees71]

How do you apply it for null trajectories at all, and which physics sense should it make for space-like ones?