I Synchronous Frames/Coordinate Charts

[PeterDonis]

[Moderator's note: Spun off from previous thread due to topic shift.]

it is possible to construct a frame where the time coordinate corresponds to the proper time of a set of timelike curves

I believe this is only possible if the set of timelike curves forms a congruence that is both geodesic and hypersurface orthogonal. (The congruence of worldlines at rest in a global inertial frame in Minkowski spacetime of course satisfies both these requirements.) We had a recent thread on this; I'll see if I can find it.

 

[Orodruin]

I believe this is only possible if the set of timelike curves forms a congruence that is both geodesic and hypersurface orthogonal. (The congruence of worldlines at rest in a global inertial frame in Minkowski spacetime of course satisfies both these requirements.) We had a recent thread on this; I'll see if I can find it.

I believe that this depends on what conditions you put on calling something a ”frame”. If you just mean ”coordinate system” I don’t think that is the conclusion, but I believe I remember the thread to which you are referring.

 

[PAllen]

I believe that this depends on what conditions you put on calling something a ”frame”. If you just mean ”coordinate system” I don’t think that is the conclusion, but I believe I remember the thread to which you are referring.

Those are the conditions for a synchronous coordinate patch, which has the feature that along lines of constant spatial coordinates, coordinate time difference equals proper time difference. But I agree that there are other coordinates possible that have this property, e.g. without hypersurface orthogonality, so that components like $g_{01}$ will not be zero.

 

[PeterDonis]

If you just mean ”coordinate system” I don’t think that is the conclusion

Yes, it is, because it is only possible at all to find any coordinate system that meets the requirement that coordinate time is the same as proper time along any worldline in some congruence of timelike worldlines, if the congruence has the properties I stated.

 

[Orodruin]

Yes, it is, because it is only possible at all to find any coordinate system that meets the requirement that coordinate time is the same as proper time along any worldline in some congruence of timelike worldlines, if the congruence has the properties I stated.

Take Schwarzschild spacetime with regular spatial coordinates and take t=0 as a spatial hypersurface in the exterior region. You can now introduce a time coordinate that is the proper time of stationary observers relative to this hypersurface.

This is not a global coordinate system and eventually the hypersurfaces of equal time coordinate will no longer be spatial, but it is a coordinate system that satisfies the requirement of the time coordinate being the proper time of the world lines of constant spatial coordinates.

 

[PeterDonis]

Take Schwarzschild spacetime with regular spatial coordinates and take t=0 as a spatial hypersurface in the exterior region. You can now introduce a time coordinate that is the proper time of stationary observers relative to this hypersurface.

This is not a global coordinate system and eventually the hypersurfaces of equal time coordinate will no longer be spatial, but it is a coordinate system that satisfies the requirement of the time coordinate being the proper time of the world lines of constant spatial coordinates.

Ah, I see where the disconnect is. In the previous thread referred to, there was an additional (if possibly implicit) requirement on the coordinate chart, that its surfaces of constant time be everywhere orthogonal to the congruence. Your chart does not have that property, but you're correct that it does have the other properties we described, at least on some open region within the exterior. (I think the "time extension" of that region goes to zero as the horizon is approached, but of course one can always restrict the region to a minimum value of $r$ that is a finite distance away from the horizon.)

 

[Orodruin]

In the previous thread referred to, there was an additional (if possibly implicit) requirement on the coordinate chart, that its surfaces of constant time be everywhere orthogonal to the congruence.

Indeed, as PAllen mentioned in #17. If I recall correctly, he was contributing significantly to that thread.

 

[cianfa72]

Take Schwarzschild spacetime with regular spatial coordinates and take t=0 as a spatial hypersurface in the exterior region. You can now introduce a time coordinate that is the proper time of stationary observers relative to this hypersurface.

ok, they are the 'hovering' observers at fixed $r,\theta,\phi$ Schwarzschild coordinates.

but it is a coordinate system that satisfies the requirement of the time coordinate being the proper time of the world lines of constant spatial coordinates.

So, in this specific case, the worldlines in the congruence that define the coordinate chart are not geodesics nor hypersurface orthogonal, however the coordinate chart being defined satisfies the requirement of the time coordinate being the proper time of the world lines of constant spatial coordinates.

 

[Orodruin]

nor hypersurface orthogonal

Indeed, not orthogonal to the hypersurfaces of constant time coordinate. They are of course still orthogonal to the hypersurfaces of the regular Schwarzschild time.

 

[cianfa72]

This is not a global coordinate system and eventually the hypersurfaces of equal time coordinate will no longer be spatial.

Morever, as you pointed out, the hypersurfaces of constant coordinate time will no longer be spacelike.

 

[Orodruin]

As you pointed out morever the hypersurfaces of constant coordinate time will no longer be spacelike.

Initially they will be, but after some time they will become null and after that there will be two timelike coordinates. As @PeterDonis noted in #21, I believe the time until this happens goes to zero as you approach the horizon, but I have not done this computation explicitly.

 

[cianfa72]

Initially they will be, but after some time they will become null and after that there will be two timelike coordinates.

Do you mean 'null hypersurface' and after some coordinate time hypersurfaces with 2 timelike + 1 spacelike directions (coordinates) ?

 

[Orodruin]

Do you mean 'null hypersurface' and after some coordinate time hypersurfaces with 2 timelike + 1 spacelike directions (coordinates) ?

No, the hypersurfaces of constant coordinate time will have 2 spacelike (angular) coordinates and one timelike coordinate - meaning 2 timelike coordinates in total.

but I have not done this computation explicitly.

I have now done the computation. Unless I did some arithmetic errors in the haste, the hypersurfaces become null at $t=\pm 2r^2/r_S$, with $t$ being the usual Schwarzschild time, which does not go to zero as $r\to r_S$. However, expressed in the new time coordinate $\tau = (1-r_S/r)^{1/2}t$, this corresponds to $\tau = 2r^2(1-r_S/r)^{1/2}/r_S$, which does go to zero.

 

[cianfa72]

No, the hypersurfaces of constant coordinate time will have 2 spacelike (angular) coordinates and one timelike coordinate - meaning 2 timelike coordinates in total.

Ah ok, you actually mean 2 spacelike + 1 timelike coordinates on the hypersurface plus 1 timelike coordinate adapted to the timelike worldlines in the congruence.

Btw, $\tau = (1-r_S/r)^{1/2}t$ should be the proper time of an 'hovering' observer at fixed Schwarzschild coordinate $r$ given in term of Schwarzschild coordinate time $t$ ($\tau$ becomes the coordinate time of the coordinate chart being defined).

 

[Orodruin]

Btw, τ=(1−rS/r)1/2t should be the proper time of an 'hovering' observer at fixed Schwarzschild coordinate r given in term of Schwarzschild coordinate time t (τ becomes the coordinate time of the coordinate chart being defined).

It is indeed, as mentioned in #29