general relativity - In GR, is 'Static' the same as 'Time-symmetric'?

I recently attended a talk where the person stated a uniqueness result for static vacuum spacetimes whereby he came to a conclusion about a type of spacetime (a 4-manifold) by studying 3-manifolds which are embedded as hypersurfaces in the 4-manifold (similar to the analysis by Schoen and Yau for the positive mass theorem).

However, he made the assumption that the 3-manifold in the spacetime always has a vanishing second fundamental form (similar to Part I of the Schoen-Yau proof). I believe in the literature that this a special case known as the time-symmetric case, but when I asked if his argument could then be generalized to the case where this is not assumed (perhaps using a PDE), he stated that it could not as the spacetime being static implies that it contains a 3-manifold with zero second fundamental form.

I would like to confirm if that is true. Surely a manifold can be 'time-symmetric' in some sense without being 'static'. Time-symmetric is just talking about symmetry under reversal of time, whereas static means it does not change at all: it cannot even rotate as with stationary state metrics like the Kerr metric.

Answer

A stationary spacetime is one that has a timelike Killing vector. There is also a notion of an asymptotically stationary spacetime, which is what some authors mean by "stationary."Although a stationary spacetime does not have a uniquely pre- ferred time, it does prefer some time coordinates over others. In a stationary spacetime, it is always possible to find a “nice” t such that the metric can be expressed without any t-dependence in its components. A static spacetime is one that is not only stationary but also has the property that coordinates exist in which it is diagonal. (Coordinates will also exist in which it is not diagonal.)

GR does not have a notion of time-reversal that applies in all cases. Basically the structure of GR does not allow the concept of discrete symmetries to be applied.

Surely a manifold can be 'time-symmetric' in some sense without being 'static'.

Yes, this is certainly true. For example, the maximal extension of the Schwarzschild spacetime has a preferred notion of time reversal, under which the black hole and white hole regions are interchanged, and the time coordinate of a static observer in one of the exterior regions is reversed. However, this spacetime is not static, because the interior regions are not static.

Similarly, you can have an FLRW spacetime in which there is a preferred time (the time of an observer at rest with respect to the Hubble flow), and a time-reversal symmetry (big crunch or big bounce cosmologies), but it's not static.