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## Main Question or Discussion Point

Hello guys. Is this even a valid question? Just curious.

Thanks,

Thanks,

- Thread starter squidsoft
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- #1

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Hello guys. Is this even a valid question? Just curious.

Thanks,

Thanks,

- #2

Dale

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Well, a geodesic is an extremal (minimal or maximal) path between two events.

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In 4-D space-time one can define 3D and 2D 'slices' which may have interesting properties, but I don't know if these properties can be varied to give the correct field equations.

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The second condition is a bit more subtle. Without going into details, it can be shown that if (2) is satisfied, then the condition for minimality is that the mean curvature identically vanish. Depending on your familiarity with surface theory, you may or may not know that the mean curvature is also referred to as the "extrinsic" curvature (as opposed to the Gaussian, or "intrinsic," curvature), because it is dependent on the particular embedding chosen (rather than depending entirely on the metric structure of the surface, which is what "intrinsic" means). In other words, two surfaces that are isometric (i.e., there exists a smooth map between them that preserves the lengths of curves) need not have the same mean curvature. In GR, the spaces of interest are completely devoid of any embedding in some higher-dimensional space; the only properties that matter are the intrinsic, metric-dependent ones. Thus, the statement that the mean curvature vanish identically is completely meaningless in GR. This is not to say that spacetime manifolds

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