Re: curvature of spacetimew

JanPB wrote:
> [... sectional curvatures of submanifolds]

For certain manifolds, such as Schwarzschild spacetime, there are
specific symmetries that permit one to select a clear and obvious set of
2-d submanifolds for which to compute sectional (Gaussian) curvatures.
Specifically, the existence of a timelike Killing vector makes this
_seem_ quite natural. But in the absence of such a Killing vector this
approach is useless, as one can obtain _any_ sectional curvatures by
suitable choice of the 2-d submanifold (that is equivalent to selecting
coordinates, but phrased differently).

As you say, the sectional curvature is independent of coordinate choice
ON THE SUBMANIFOLD. But it is inherently dependent on the selection of
the 2-d submanifold. In most cases (such as the examples in this
thread), the 2-d submanifold is specified by setting 2 of the 4
coordinates ON 4-D SPACETIME to constants, so the sectional curvature is
_not_ independent of the coordinates on spacetime used to select the
submanifold itself. So your statement of coordinate independence
involves a pun on "coordinates" -- these curvatures are independent of the coordinates used to compute the curvature, but they are NOT independent of the coordinates used to specify the 2-d submanifold by setting 2 coordinates to a constant.

Here's a simple example that shows how unrelated to the full manifold these sectional curvatures of 2-d surfaces can be: In 3-d Euclidean space, using Cartesian coordinates the 2-d submanifold x=constant has zero Gaussian curvature. But in the same manifold, using spherical coordinates the 2-d submanifold r=constant has positive Gaussian curvature. The 3-d manifold itself has no curvature, and Riemann=0.


Tom Roberts tjroberts@xxxxxxxxxx
.

Relevant Pages

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    ... They contain an boundaryless submanifold ... whose curvature is everywhere minimal. ... The submanifold can sweep out the manifold, ...
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  • Re: curvature of spacetimew
    ... as one can obtain _any_ sectional curvatures by ... | suitable choice of the 2-d submanifold (that is equivalent to selecting ... | coordinates ON 4-D SPACETIME to constants, ... But in the same manifold, ...
    (sci.physics.relativity)
  • Re: curvature of spacetimew
    ... For certain manifolds, such as Schwarzschild spacetime, there are ... as one can obtain _any_ sectional curvatures by ... ON THE SUBMANIFOLD. ...
    (sci.physics.relativity)