I Gravitational time dilation, proper time and spacetime interval

[Ibix]

I agree that the coordinate time is a bookkeeping device, but you can stick an observer to it. I'd say the coordinate t is measured as the time difference between the two events by an observer very far away from the black hole where spacetime can be considered to be flat.

But that's a special case of coordinates that have a sensible physical interpretation, isn't it? You can replace $t$ with $t'=2t$ without breaking anything, and then there's no direct interpretation of the time coordinate in terms of anything much.

 

[vanhees71]

Yes, and that's why proper time has a direct physical meaning, while coordinates generally don't. It's as in electrodynamics: The four-potential is no direct physical meaning, because it's gauge-dependent. Only the field-strength tensor has direct physical meaning, i.e., is related to measurable quantities which are well defined by an equivalence class of appropriate measurement procedures.

 

[sweet springs]

Re:#47
$\tau$ is larger than t in SR but smaller than t in GR. It is an ambiguous saying but reveals my misunderstanding. Best.

 

[haushofer]

But that's a special case of coordinates that have a sensible physical interpretation, isn't it? You can replace $t$ with $t'=2t$ without breaking anything, and then there's no direct interpretation of the time coordinate in terms of anything much.

I interpret that physically as the same observer using a different time scale.

 

[PeterDonis]

I'd say the coordinate t is measured as the time difference between the two events by an observer very far away from the black hole where spacetime can be considered to be flat.

But such an observer cannot measure the time difference between two events that are not on his worldline. He can only assign such times as a convention, which is what a coordinate chart is.

 

[haushofer]

But such an observer cannot measure the time difference between two events that are not on his worldline. He can only assign such times as a convention, which is what a coordinate chart is.

Yes. That's the whole thing of coordinate time; it's not proper time :P But I'd still call it measuring with his clock.

 

[vanhees71]

Yes. That's the whole thing of coordinate time; it's not proper time :P But I'd still call it measuring with his clock.

But an ideal clock measures precisely its proper time!

 

[Orodruin]

The point is that measurements on a clock only are unambiguous for events on the world line of said clock (as already pointed out). The time coordinates of other events depend on an arbitrary foliation of space-time. Fine, the Schwarzschild coordinates are somewhat special in the sense that its time coordinate is chosen such that the corresponding basis vector field is a time-like Killing vector field showing that the space-time is stationary, but it is not much more than that.

 

[haushofer]

But an ideal clock measures precisely its proper time!

Yes, but coordinate times are not scary diseases, they 're just coordinate dependent. I agree with Orodruin that this particular case is special.

 

[vanhees71]

In any affine space, the parallel postulate holds. You don't need a metric or fundamental form. That's very easy to see. Just fix a point, and you can describe anything with vectors. A straight line is defined then by
$$g: \quad \vec{x}(t)=\vec{v} t+\vec{x}_0,$$
and two straight lines $g_1$ and $g_2$ are called parallel, iff $\vec{v}_1=\vec{v}_2$. Then either they are identical and have all points in common or they don't have any points in common. Nowhere did I need any reference to a scalar product or fundamental form.

If the affine space is two-dimensional then it's also easy to see that to any straight line and a given point not on this line there's exactly one parallel to the straight line going through the given point. Again you don't need any reference to a fundamental form.

 

[Dale]

A recent thread derail has been deleted, and the thread is reopened

 

[sweet springs]

Hi.

I agree that the coordinate time is a bookkeeping device, but you can stick an observer to it. I'd say the coordinate t is measured as the time difference between the two events by an observer very far away from the black hole where spacetime can be considered to be flat.

P.247 of the text https://archive.org/stream/TheClassicalTheoryOfFields/LandauLifshitz-TheClassicalTheoryOfFields#page/n257/mode/2up/search/world+time
calls this coordinate time under stationary gravitation the world time. I am not sure whether the naming is popular but common and universal nature of the coordinate time in the sense that everybody in everywhere can translate it to his or anybody's real time is well expressed in this naming. Best.

 

[Orodruin]

Hi.

P.247 of the text https://archive.org/stream/TheClassicalTheoryOfFields/LandauLifshitz-TheClassicalTheoryOfFields#page/n257/mode/2up/search/world+time
calls this coordinate time under stationary gravitation the world time. I am not sure whether the naming is popular but common and universal nature of the coordinate time in the sense that everybody in everywhere can translate it to his or anybody's real time is well expressed in this naming. Best.

So what? This should be true for any coordinate system and there is nothing special about Schwarzschild coordinates apart from what I mentioned in #58. If you pick a different foliation of space-time, any observer will still be able to identify what the corresponding time coordinate is.