Understanding Proper Time in General Relativity & Special Relativity

In summary: This is similar to the situation in Euclidean spaces, where there are different global topologies possible, but locally (at least in regions that are simply-connected) all such spaces are indistinguishable from one another in terms of their geometry.
  • #1
binbagsss
1,254
11
I'm reading An Intro to GR, Hughston and Tod, it says that in GR the idea is that the geometry of st varies from point to point, represented by allowing the metric to vary over space-time.

It uses a (+,-,-,-) signature and so ##proper time=ds^{2}##.

It then makes the comment that proper time depends on the the observers location (compared to SR where it doesn't).

My question:

I believe it is always the case that the comtponents of a metric in GR are functions of space-time, and so I understand that proper time will vary for observers dependent upon location, but,

In flat space am I correct in thinking that the components also can depend upon space-time, e.g use of polar coordinates, and the only case where it doesn't is cartesian coordinates?

So from the above statement I would conclude that proper time also varies from location to location in SR. So shouldn't the statement instead be that if there does not exist coordinate system in which the components of the metric are not functions of space-time, the space-time is curved from which it follows that proper time is location dependent.

Thanks
 
Last edited:
Physics news on Phys.org
  • #2
binbagsss said:
In flat space am I correct in thinking that the components also can depend upon space-time, e.g use of polar coordinates, and the only case where it doesn't is cartesian coordinates?
Yes, if you have curvilinear coordinates, your metric will in general depend on position. In SR, the coordinates where this is not happening are Minkowski coordinates. Cartesian coordinates are for Euclidean geometry.
 
  • #3
Orodruin said:
Yes, if you have curvilinear coordinates, your metric will in general depend on position. In SR, the coordinates where this is not happening are Minkowski coordinates. Cartesian coordinates are for Euclidean geometry.

Oh right, thanks. And what's the difference? They are both to describe flat space-time right?

I know that there's a metric with (+,+,+,+) signture where you can't categorize the vectors into tl,null,sl.
And that the Minkowki metric, signature (+,-,-,-) or (-,+,+,+), allows categorization,

But I thought that the difference lies in the metric signature, not the coordinate system.

Thanks.
 
  • #4
Minkowski coordinates are the Minkowski space equivalent of Cartesian coordinates in Euclidean spaces. The transformations between different sets of Minkowski coordinates (ie, Lorentz transformations) are different than the relations between different Cartesian coordinate systems (rotations).
 
  • #5
Am I correct when I say that if there does not exist coordinate system in which the components of the metric are not functions of space-time, the space-time is curved?
 
  • #6
binbagsss said:
It then makes the comment that proper time depends on the the observers location (compared to SR where it doesn't).

Can you give an exact quote? As you state this, it doesn't seem right, because proper time elapsed for a given observer always depends on that observer's path through spacetime. But I suspect the book actually meant something else (possibly they meant what you suggest as an alternate interpretation, about whether or not the metric coefficients vary from event to event).
 
  • #7
binbagsss said:
Am I correct when I say that if there does not exist coordinate system in which the components of the metric are not functions of space-time, the space-time is curved?

The converse is certainly true: if there does exist a coordinate chart in which the metric coefficients are constant, then spacetime is flat (since obviously all derivatives of the metric coefficients are zero, and therefore the Riemann tensor vanishes identically).

Knowing that, we can easily see that your proposition is true as well, as follows: for the converse to be true, as above, but your proposition to be false, there would have to be at least two flat spacetimes, because we know Minkowski spacetime does have a coordinate chart with constant metric coefficients, and is therefore flat by the converse proposition above (as of course we know it is). In other words, there would have to be some other spacetime that does not admit a coordinate chart in which the metric coefficients are constant, but still has a vanishing Riemann tensor (which is what "flat spacetime" means). But this is not possible; the easiest way to see this is to observe that all the invariants that determine the spacetime geometry are functions of the Riemann tensor, so if the Riemann tensor vanishes identically, all the invariants must also vanish. So any spacetime with vanishing Riemann tensor must have the same geometry as Minkowski spacetime, and must therefore admit a coordinate chart with constant metric coefficients. So any spacetime that does not have a chart in which the metric coefficients are constant must be curved.
 
  • #8
PeterDonis said:
So any spacetime with vanishing Riemann tensor must have the same geometry as Minkowski spacetime, and must therefore admit a coordinate chart with constant metric coefficients.
I'm just going to add that this is a local statement. There are flat space-times which are not globally equivalent to Minkowski space.
 
  • #9
Orodruin said:
There are flat space-times which are not globally equivalent to Minkowski space.

Yes, this is a good point, saying that a spacetime is "flat" does not tell you the global topology. However, any flat spacetime, regardless of global topology, must have the same local geometry as Minkowski spacetime, and therefore any open subregion of such a spacetime must admit a chart with constant metric coefficients. The only difference with a global topology other than that of Minkowski spacetime is that you may not be able to cover the entire spacetime with one such chart.
 

1. What is proper time in relativity?

Proper time is the time measured by a clock that is moving along the same worldline as the observer. It is the time experienced by an object or observer, and it will always be the shortest possible time interval between two events in spacetime.

2. How is proper time related to special relativity?

In special relativity, proper time is used to describe the time experienced by an object or observer in a specific reference frame. It is a fundamental concept in understanding the effects of time dilation and length contraction in moving frames of reference.

3. How is proper time related to general relativity?

In general relativity, proper time is used to describe the time experienced by an object or observer in a curved spacetime. It takes into account the effects of gravity, which can cause time to pass at different rates in different locations.

4. Can proper time be measured?

Yes, proper time can be measured using a clock that is stationary in the same reference frame as the object or observer. This clock will measure the proper time experienced by the object or observer.

5. What is the significance of proper time in relativity?

Proper time is a crucial concept in relativity as it allows us to understand the effects of time dilation and length contraction in moving frames of reference, as well as the effects of gravity on the passage of time. It also helps us to make accurate predictions and calculations in both special and general relativity.

Similar threads

  • Special and General Relativity
Replies
17
Views
2K
  • Special and General Relativity
Replies
14
Views
532
  • Special and General Relativity
Replies
11
Views
1K
  • Special and General Relativity
2
Replies
58
Views
4K
  • Special and General Relativity
Replies
17
Views
391
  • Special and General Relativity
Replies
16
Views
589
  • Special and General Relativity
Replies
3
Views
1K
  • Special and General Relativity
Replies
20
Views
729
  • Special and General Relativity
Replies
10
Views
670
  • Special and General Relativity
Replies
19
Views
1K
Back
Top