Difference between proper time and coordinate time

[Silviu]

Hello! I found this questions in several places, but no answer made me fully understand it, so I decided to give it one more try here. I am not sure I understand the difference between them in GR. I have a feeling of the proper time as the time measured by the clock of someone moving with a certain speed and/or around a massive object. But if we have 2 random events (no motion between them) what is the proper time between them? I know we have ##\frac{d\tau}{dt}=\sqrt{g_{00}}##, but this doesn't really help me to get a physical interpretation of proper time. Also, if the events are far away from each other, such that ##g_{00}## can take different values, which one should you use? Now for the coordinate time, I found some definitions in terms of inertial frames, but for example in the Godel metric, everything is spinning, so I am not sure I can see how you can have an inertial frame there. So any explanation would be greatly appreciated. Thank you!

 

[phinds]

I have a feeling of the proper time as the time measured by the clock of someone moving with a certain speed and/or around a massive object.

Much too limited a statement. Proper time is the time measured by any clock in its own frame of reference, regardless of its motion relative to any other object and regardless of its depth in a gravity well.

 

[Silviu]

Much too limited a statement. Proper time is the time measured by any clock in its own frame of reference, regardless of its motion relative to any other object and regardless of its depth in a gravity well.

So if let's say 2 stars explode at random places in the universe and I am at a place ##x## moving with speed ##v##, the proper time I measure between the explosions, would be what I measure on my own clock, regardless of anything else that happens in the Universe?

 

[A.T.]

But if we have 2 random events (no motion between them) what is the proper time between them?

There is no unique proper time between two events. There can be different worldlines with different proper time intervals between them, in both SR and GR.

 

[Orodruin]

Proper time is the time measured by a clock following a given world line. It is an invariant and is a property of the world line itself, it has no dependence on whatever coordinate system you are using.

But if we have 2 random events (no motion between them) what is the proper time between them?

Your parenthesis here points to a fundamental misunderstanding of what an event is. An event is just a point in space-time, it has no meaningful concept of "motion".

So if let's say 2 stars explode at random places in the universe and I am at a place ##x## moving with speed ##v##, the proper time I measure between the explosions, would be what I measure on my own clock, regardless of anything else that happens in the Universe?

You do not measure any proper time between explosions, you only measure the proper time that is on your clock and you have to make a meaningful question based on that. For example, "what is the proper time difference I measure between me seeing the first explosion and me seeing the second?" would be an appropriate question that has a definitive answer.

 

[Silviu]

Proper time is the time measured by a clock following a given world line. It is an invariant and is a property of the world line itself, it has no dependence on whatever coordinate system you are using.Your parenthesis here points to a fundamental misunderstanding of what an event is. An event is just a point in space-time, it has no meaningful concept of "motion".

You do not measure any proper time between explosions, you only measure the proper time that is on your clock and you have to make a meaningful question based on that. For example, "what is the proper time difference I measure between me seeing the first explosion and me seeing the second?" would be an appropriate question that has a definitive answer.

So you mean if I measure ##\tau_1## as the first explosion and ##\tau_2## as the second one, I conclude that the proper time (as measured by me) between them is ##d\tau=\tau_2-\tau_1##? And being an invariant, someone else measuring the same quantity but from its own frame, would get the same results?

 

[Orodruin]

So you mean if I measure ##\tau_1## as the first explosion and ##\tau_2## as the second one, I conclude that the proper time (as measured by me) between them is ##d\tau=\tau_2-\tau_1##? And being an invariant, someone else measuring the same quantity but from its own frame, would get the same results?

No. This is your proper time along your world line. Anyone else computing your time will obtain that you measure the same proper time difference, regardless of what coordinates they are using.

 

[phinds]

So you mean if I measure ##\tau_1## as the first explosion and ##\tau_2## as the second one, I conclude that the proper time (as measured by me) between them is ##d\tau=\tau_2-\tau_1##? And being an invariant, someone else measuring the same quantity but from its own frame, would get the same results?

Go back and read post #4

 

[Silviu]

Go back and read post #4

But in #5 it is said that proper time is an invariant. Doesn't this mean it is the same for all observers i.e. a scalar?

 

[Mister T]

So if let's say 2 stars explode at random places in the universe and I am at a place ##x## moving with speed ##v##, the proper time I measure between the explosions, would be what I measure on my own clock, regardless of anything else that happens in the Universe?

You would have to be present at both events. Proper time is the time that elapses between two events that occur at the same place.

 

[Silviu]

No. This is your proper time along your world line. Anyone else computing your time will obtain that you measure the same proper time difference, regardless of what coordinates they are using.

So what is the coordinate time then? Who measures it?

 

[Orodruin]

But in #5 it is said that proper time is an invariant. Doesn't this mean it is the same for all observers i.e. a scalar?

It is an invariant for a given world line.

 

[Orodruin]

So what is the coordinate time then? Who measures it?

Nobody. It is just a coordinate. In some situations, it is set up in such a way that it corresponds to the proper time of a distant observer (such as in the Schwarzschild metric) along their world line and with an additional simultaneity convention using hypersurfaces orthogonal to the time-like Killing field. Other times, such as in cosmology, it is made to correspond to the proper time of a comoving observer along their world line.

 

[Silviu]

It is an invariant for a given world line.

So in special relativity, if I stay still relative to an inertial frame I measure a ##ds=d\tau##. If someone else moves with constant speed v and hence has a different world line than me, he would still measure the same ##ds##. Doesn't this mean that ##ds## and hence the proper time is an invariant for different world lines, too? Or this doesn't hold in GR?

 

[Orodruin]

There is not one proper time. There is a proper time for every possible world line. They are generally different. You have to specify what world line you are referring to when you talk about a proper time.

 

[Silviu]

There is not one proper time. There is a proper time for every possible world line. They are generally different. You have to specify what world line you are referring to when you talk about a proper time.

So ##ds^2## is not a Lorentz scalar in general, but just for a given world line?

 

[Ibix]

3d analogy: proper time is the distance your fitbit (other step counting devices are available) measures. Events are places.

"The distance I walked between places A and B" makes no sense unless you were at A and B. Similarly "the proper time between events A and B" makes no sense unless you were at A and B. And even then it's not unique - it depends what route you took. In SR there is an obvious route to assume - the straight line. But not so in curved spacetime.

My phone's almost out of battery- hope that helps a bit!

 

[Ibix]

Plugged in again.

So ##ds^2## is the distance-squared along an infinitesimal part of your (or whoever's) personal path through spacetime. The integral of ##\sqrt {ds^2}## is the distance along your personal path.

So there's no "proper time between events". There is a proper time along a particular path between two events on that path.

What is coordinate time? All you do is choose a set of worldlines and use systematically spaced events along those worldlines as reference events. In SR, you pick a family of objects that are mutually at rest, and choose coordinate time to be their time. In Schwarzschild coordinates in Schwarzschild spacetime we pick observers who are hovering at constant radius and not rotating. We adjust their clocks to tick at slightly odd rates, which vary systematically with altitude, and the ticks of these clocks mark out a systematic way of "naming" a time.

Going back to a simpler analogy, start with a piece of blank paper. You can draw lines on it and mark off distances along those lines. That's useful if you want to know how long it would take you to trace along those lines. But if we're clever and use lines that are arranged systematically, now we've created graph paper - coordinates.

Hope that helps.

 

[SiennaTheGr8]

"The distance I walked between places A and B" makes no sense unless you were at A and B. Similarly "the proper time between events A and B" makes no sense unless you were at A and B. And even then it's not unique - it depends what route you took. In SR there is an obvious route to assume - the straight line.

Yes—unfortunately for the beginner, there's a slight abuse of terminology that's very common in SR, where people will casually refer to "the proper time between events," or "the proper time interval between events," when what they really mean is "the spacetime interval between timelike-separated events" (perhaps divided by ##c##), or equivalently, "the proper time that would elapse during an inertial journey between the events." Also fine (I think) is "the inertial proper time interval between events."

The reason it's an abuse of terminology is that "proper time" is just the wristwatch time of any traveler. There are an infinite number of possible paths through spacetime a traveler can take between timelike-separated events, and there's a "proper time interval" for each of them.

 

[Dale]

But if we have 2 random events (no motion between them) what is the proper time between them?

There isn’t one. Proper time is defined along a timelike worldline, not between two events.

However, you may be able to construct all possible timelike worldlines between the two events, and find the one that extremizes the proper time. That path will be a geodesic.

So you mean if I measure ##\tau_1## as the first explosion and ##\tau_2## as the second one, I conclude that the proper time (as measured by me) between them is ##d\tau=\tau_2-\tau_1##? And being an invariant, someone else measuring the same quantity but from its own frame, would get the same results?

Proper time is only defined for events on a worldline, not for events off the worldline. The clock measuring the proper time would have to be present at both explosions so that it’s worldline would include both explosions. No worldline, no proper time.

 

[Orodruin]

However, you may be able to construct all possible timelike worldlines between the two events, and find the one that extremizes the proper time. That path will be a geodesic.

In this context it might also be mentioned that there is no guarantee that this worldline will be unique.

 

[Mister T]

So in special relativity, if I stay still relative to an inertial frame I measure a ##ds=d\tau##.

If you're measuring the interval between two events that occur in the same place, yes. Note that you need only one clock to perform your measurement because that one clock can be present at both events.

If someone else moves with constant speed v and hence has a different world line than me, he would still measure the same ##ds##.

Yes. But, that person will need two clocks because no one single clock (that's at rest relative to him) can be present at both events. He will therefore need to synchronize his two clocks, using some simultaneity convention. His value of ##ds## will be ##\sqrt{dt^2-dx^2}## where ##dt## is coordinate time.

Note that ##dt## is always bigger than ##d\tau##. This is called time dilation.

 

[cianfa72]

What is coordinate time? All you do is choose a set of worldlines and use systematically spaced events along those worldlines as reference events.

Just to check my understanding of coordinate time: does it exist a specific meaning attached to "systematically spaced events" ? Thanks

 

[Heikki Tuuri]

Just to check my understanding of coordinate time: does it exist a specific meaning attached to "systematically spaced events" ? Thanks

Coordinates mean a systematic way to specify the points on a manifold.

For example, the latitude and the longitude are a systematic way to specify the points on the surface of Earth.

In General relativity, the Schwarzschild coordinates are one systematic way to specify the points (which are also called events) in the spacetime which surrounds a spherically symmetric static mass.

Coordinates can often be chosen such that they have some connection to what a real observer (a human) would measure. In the Schwarzschild case, far away from the mass, a static observer will measure the proper length (= real, measured length) to be almost equal to the length given in the Schwarzschild coordinates.

But since you are free to choose any coordinate system, there may be no connection at all to what a real observer would measure.

 

[cianfa72]

What is coordinate time? All you do is choose a set of worldlines and use systematically spaced events along those worldlines as reference events. In SR, you pick a family of objects that are mutually at rest, and choose coordinate time to be their time. In Schwarzschild coordinates in Schwarzschild spacetime we pick observers who are hovering at constant radius and not rotating. We adjust their clocks to tick at slightly odd rates, which vary systematically with altitude, and the ticks of these clocks mark out a systematic way of "naming" a time.

We always claim as a mantra that coordinates in GR are just labels for spacetime events. That’s true of course. Nevertheless I believe it is important to highlight the following: from a physical operative point of view we have to establish how to assign such coordinates to events in order to map them.

Here @Ibix describes how observers hovering at fixed radius in Schwarzschild spacetime systematically assign coordinate time to events that happen at their fixed spatial location.

That’s really interesting !

 

[Dale]

from a physical operative point of view we have to establish how to assign such coordinates to events in order to map them

Yes, definitely. But it is important to know that the mapping does not need to be any specific relationship to clock and ruler measurements. All that is necessary is that each event in a region of spacetime map to a unique coordinate and that the mapping be continuous.

 

[cianfa72]

Yes, definitely. But it is important to know that the mapping does not need to be any specific relationship to clock and ruler measurements. All that is necessary is that each event in a region of spacetime map to a unique coordinate and that the mapping be continuous.

Yes, then starting from a coordinate chart for which a physical operative "procedure" to assign events to such coordinates has been specified, we can move to other coordinate charts just by smooth mathematical transformations from it.

Take for example Gullstrand-Painlevé coordinates for Schwarzschild geometry. Start from Schwarzschild coordinates and get the GP coordinates just by a suitable transformation from it.

 

[Ibix]

We always claim as a mantra that coordinates in GR are just labels for spacetime events. That’s true of course. Nevertheless I believe it is important to highlight the following: from a physical operative point of view we have to establish how to assign such coordinates to events in order to map them.

Yes - that's the difference between physics and differential geometry. At some point we need to state that some of the numbers in our model correspond to readings on some instrument in the real world. Then we can go about checking if those readings evolve as the theory predicts.

 

[cianfa72]

Yes - that's the difference between physics and differential geometry. At some point we need to state that some of the numbers in our model correspond to readings on some instrument in the real world.

Actually, to the extent of coordinate chart, there is no requirement for direct/immediate corrispondence between events and "numeric labels" assigned to them. Indeed we need a metric to calculate stuff and check their corrispondence with reading of some real-world instrument.

 

[Dale]

to the extent of coordinate chart, there is no requirement for direct/immediate corrispondence between events and "numeric labels" assigned to them.

Yes, that is the definition of a coordinate chart. There must be a one-to-one and continuous correspondence between the events in an open subset of spacetime and points in an open subset of R4.

 

[PeterDonis]

At some point we need to state that some of the numbers in our model correspond to readings on some instrument in the real world.

Yes, but if we are doing things properly, none of those numbers will be simple coordinate values or simple coordinate components of tensors. Every number that corresponds to readings on some instrument in the real world will be expressed as an invariant--a number that is independent of any choice of coordinates.

Unfortunately, many sources gloss over this fact and focus on examples (such as an inertial frame in SR, realized by sets of clocks and rulers as Einstein, for example, did) where what we would normally call coordinate values or component values are numerically equal to relevant invariants for expressing distances and times. This unfortunately invites the mistaken belief that the coordinate or component values themselves are physically meaningful. But this conflation of the two distinct concepts only works for particular choices of coordinates in particular highly special spacetimes, and needs to be unlearned as soon as you go beyond those special cases.

 

[cianfa72]

Yes, but if we are doing things properly, none of those numbers will be simple coordinate values or simple coordinate components of tensors.

Yes of course. Nevertheless, as said before, I believe it is really important to have --at least in principle-- a physical operative procedure to assign such coordinates to events in spacetime.

 

[PeterDonis]

I believe it is really important to have --at least in principle-- a physical operative procedure to assign such coordinates to events in spacetime.

For making actual measurements, yes, you have to have some way of doing this. A good example would be the way barycentric coordinates for the solar system are defined and how they are matched up with actual observational data.

However, if we are talking about theoretical physics, often it is not even possible to define such a procedure, at least not for an entire spacetime. For example, consider the interior of a black hole: nobody who falls in can send any measurement data back out, so there is no way for anyone outside to have a physical procedure to assign coordinates to any events inside the hole. But that doesn't mean we can't build theoretical models of the interiors of black holes, and use coordinates in those models.

 

[cianfa72]

However, if we are talking about theoretical physics, often it is not even possible to define such a procedure, at least not for an entire spacetime. For example, consider the interior of a black hole: nobody who falls in can send any measurement data back out, so there is no way for anyone outside to have a physical procedure to assign coordinates to any events inside the hole.

In BH case, what we get is basically an "extension" for the BH interior of the exterior solution given in coordinates for which we know in advance their physical interpretation (at least for some of them).

As explained in Carroll we start assuming a spherical symmetry for the solution we're looking for. This implies since the beginning the enforcement of a metric with the symmetry of ##S^2## sphere -- i.e. the solution spacetime is foliated by 2-spheres. Each of them is parametrized/labeled by parameters ##t,r##.

Now my point is that from a physical viewpoint we know what a 2-sphere is. For a fixed ##t## we don't know in advance whether the geometry of the ##t= \text{const}## hypersurface will or will not be Euclidean, however in principle/imagination we know that we can build in the BH exterior region a family of concentric shells (2-spheres) parametrized by ##r## foliating that spacelike hypersurface (even though we do not know in advance the physical interpretation/properties of ##r## parameter).

Note indeed that Carroll in section 7 of his Lecture Notes on GR assumes since the beginning the following metric for each 2-sphere
$$d\Omega^2 = d\theta^2 + sin^2\theta d\phi^2$$

 

[PeterDonis]

In BH case, what we get is basically an "extension" for the BH interior of the exterior solution given in coordinates for which we know in advance their physical interpretation (at least for some of them).

Yes, but that's still not the same as having "a physical operative procedure" for assigning coordinates in the interior. All of the things you discuss are theoretical items; they're not the same as having actual, physical observers that can exchange information in order to physically assign coordinates.