B Spatial curvature around a spherically symmetric mass

[cianfa72]

Yes. The point is, so what? Why would you even consider those curves in the first place if you're interested in the spatial curvature of a 3-surface of constant $t$?

Ok yes, got it.

 

[PAllen]

A geodesic can be a geodesic both manifolds and sub-manifolds. We can describe a geodesic by giving a starting point, and a direction at that starting point. Then for an example of what I mean, consider, a geodesic in the 4d Schwarzschild space-time, starting at the point r=2, theta=0, phi=0, t=0, with an inital velocity/direction given by d/dt = d/dr = d/d theta=0 and d/ d phi nonzero. This curve is a geodesic in the 4d Schwarzschild space-time, the 3d spatial surface of the 4d spacetime (a submanifold of the Schwarzschild spacetime with t=0), and the 2d manifold of the r-theta equatorial plane (a 2d submanifold of the Schwarzschild metric with t=0 and theta=0).

While your first statement above is correct as a possibility, in the particular case it is not. To make matters easy,
reference:

www.physics.usu.edu/Wheeler/GenRel2015/Notes/GRSchwarzschildGeodesics.pdf

and putting in an initial tangent of (0,0,0,1) , one finds the geodesic equations do not vanish for dr’/ds. Thus even though the starting tangent is only in the phi direction, the radial coordinate will change along the geodesic of the spacetime. Thus your proposed curve is definitely not a geodesic of the spacetime. Whether it is a geodesic of the 3-d spatial slice requires additional analysis, which I will make if I have time and interest.

 

[A.T.]

A geodesic can be a geodesic both manifolds and sub-manifolds. We can describe a geodesic by giving a starting point, and a direction at that starting point. Then for an example of what I mean, consider, a geodesic in the 4d Schwarzschild space-time, starting at the point r=2, theta=0, phi=0, t=0, with an inital velocity/direction given by d/dt = d/dr = d/d theta=0 and d/ d phi nonzero. This curve is a geodesic in the 4d Schwarzschild space-time, the 3d spatial surface of the 4d spacetime (a submanifold of the Schwarzschild spacetime with t=0), and the 2d manifold of the r-theta equatorial plane (a 2d submanifold of the Schwarzschild metric with t=0 and theta=0).

While your first statement above is correct as a possibility, in the particular case it is not.

A trivial example of a geodesic in spacetime and space is radial free fall.

 

[PAllen]

A trivial example of a geodesic in spacetime and space is radial free fall.

Not really. For radial free fall geodesic, both t coordinate and r coordinate change as a function of an affine parameter. For a radial spatial geodesic, the t coordinate does’t change at all. Thus, they are completely different curves in spacetime. What is actually true is that a radial spatial geodesic in Schwarzschild geometry is both a geodesic of spacetime and a geodesic of a constant SC time slice.

 

[A.T.]

A trivial example of a geodesic in spacetime and space is radial free fall.

Not really.

- The world-line of a radial free fall is a geodesic in space-time
- The spatial-path of a radial free fall is a geodesic in space

 

[PAllen]

Not really. For radial free fall geodesic, both t coordinate and r coordinate change as a function of an affine parameter. For a radial spatial geodesic, the t coordinate does’t change at all. Thus, they are completely curves in spacetime.

- The world-line of a radial free fall is a geodesic in space-time
- The spatial-path of a radial free fall is a geodesic in space

But they are two different curves. The latter is not even well defined as such - you have to specify that you mean radial path in an SC coordinate constant time slice. Then, the spatial the spatial path is a geodesic of both the slice and the spacetime. For other coordinates, a radial spatial path need not be a geodesic of the spacetime, though it would normally be a geodesic of the slice.

The claim under discussion was a specific path being a geodesic of the spacetime and the submanifold. Your example is simply not relevant in that you are talking about two different paths.

 

[pervect]

While your first statement above is correct as a possibility, in the particular case it is not. To make matters easy,
reference:

www.physics.usu.edu/Wheeler/GenRel2015/Notes/GRSchwarzschildGeodesics.pdf

and putting in an initial tangent of (0,0,0,1) , one finds the geodesic equations do not vanish for dr’/ds. Thus even though the starting tangent is only in the phi direction, the radial coordinate will change along the geodesic of the spacetime. Thus your proposed curve is definitely not a geodesic of the spacetime. Whether it is a geodesic of the 3-d spatial slice requires additional analysis, which I will make if I have time and interest.

I'd agree that dr/ds is not zero. But I wouldn't expect it to be zero.

When I talk about the Scwharzschild space-time, formally I'm talking about coordinate [t,r,theta,phi], geometric units, and a metric with the line element $-(1-r_s/r) dt^2 + dr^2/(1-r_s/r) + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2$

When I talk about the 3d surface of the Schwarzschild metric, I'm talking about projecting the Schwarzschild metric to 3d space. There are of course many possible projections, so this is vague. Specifically, then, it's a metric with coordinates ($r, \theta, \phi$)) and a line element of $dr^2/(1-r_s/r) + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2$. Then the main goal is to discuss it's curvature.

One can think of this as the 3d hypersurface of constant t in the Schwarzschild metric. This is the easiest to graps, but it's a coordinate-dependent definition. I had previously presented a coordinate independent definition, but it was admittadely a bit abstract. It was based on the idea of considering a subspace of the tangent vector space at some point on the manifold, then mapping this subspace to a submanifold of the Schwarzschild manifold itself, using the exponential map between the tangent space and the manifold. Probably for my purposes it would be better to stick with the coordinate dependent defintion.

When I talk about the 2d equatorial "r-theta" plane, and it's curvature, I'm talking about the plane with coordinates $(r, \phi)$ and the line element $dr^2/(1-r_s/r) + r^2 d\phi^2$, which can be thought of as the surface of Flamm's paraboloid, which is presented in Wiki and other places as a way to visualize (part of) the Schwarzschild curvature in the r-phi plane. So I expect dr/ds to be nonzero, because the space in question is curved.

 

[PAllen]

Ah, ok @pervect, I thought you were saying that a curve constant r, t, and theta with varying phi was a geodesic of spacetime. Now I see you meant that a spacetime geodesic tangent to this curve (which will have r varying) is also a geodesic of the indicated 3d and 2d submanifolds (metric induced from the 4d metric). That seems right. I did not understand what you were proposing.

 

[PeterDonis]

I had previously presented a coordinate independent definition

The usual coordinate-independent definition of the spacelike 3-surfaces in question is that they are the 3-surfaces that are orthogonal to the integral curves of the timelike Killing vector field of the spacetime.

 

[PAllen]

Ah, ok @pervect, I thought you were saying that a curve constant r, t, and theta with varying phi was a geodesic of spacetime. Now I see you meant that a spacetime geodesic tangent to this curve (which will have r varying) is also a geodesic of the indicated 3d and 2d submanifolds (metric induced from the 4d metric). That seems right. I did not understand what you were proposing.

I verified explicitly from the general geodesic equations that any spacelike geodesic with initial tangent having no timelike component, and no theta component, and a starting position in the equatorial plane (theta = pi/2), will be fully contained in the equatorial plane. It will thus be a geodesic of the 4-manifold, a constant SC time spatial slice, as well as the equatorial (r,phi) 2-manifold. This includes a spatial radial geodesic as a subcase, as well any initial (r,phi) direction within the equatorial surface.

Then, by rotational symmetry arguments, one can say:

Any geodesic of a constant time SC slice is also a geodesic of the spacetime as well as of a particular 2-manifold subset of this slice containing this geodesic. In the case of a purely radial geodesic, there will be infinite such 2-manifolds, instead of just one.