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https://www.physicsforums.com/showpost.php?p=2973770&postcount=45,

some questions came up about what the conditions are for a spacetime to admit flat spatial slices, and for a spacetime to have a time-independent "scale factor" (see definition below). These questions seemed interesting enough to warrant a new thread.

One key definition:

(A) The term "scale factor" is here extended from what I believe is its normal usage, which refers to the coefficient a(t) in front of the spatial part of the FRW metric in at least one of its "standard" forms. Physically, a(t) tracks the "comoving distance" between nearby geodesics that are "at rest", meaning that they are the worldlines of observers who see the universe as homogeneous and isotropic. By analogy, we can extend the term to refer to the "comoving distance" between nearby worldlines "at rest" in other spacetimes (with some caveats to the definition of "comoving", since the "nearby worldlines at rest" won't always be geodesics), and specifically to the fact that, in the other spacetimes we will be considering, there is no time-dependence in the "scale factor", meaning, roughly speaking, that the "size" of a given region of space does not change with time.

The reason this came up in the other thread was my use of the term "spatially flat" to describe the FRW spacetime with k = 0. Passionflower pointed out that, since the scale factor in this spacetime is time-dependent, even though each hypersurface of constant FRW coordinate time is flat, the "size" of a given region of space changes from one spatial slice to another, so, for example, a geometric figure "drawn" using geodesics in one spatial slice would have "expanded" in a later slice. This raised the question of what would be required of a spacetime to ensure that this didn't happen.

Two proposed conditions, based on the discussion in the other thread, are:

(1) For a spacetime to admit a metric with a time-independent scale factor, it must be stationary. The "canonical" example here is the Kerr spacetime, which is stationary but not static, and which does not admit any metric with flat spatial slices. It does, however, have the property that none of the metric coefficients depend on the time (in any of the standard coordinate charts--this is, of course, a restatement of the fact that the spacetime is stationary), which means that the "scale factor" is constant in time; contrast this with, for example, any of the FRW spacetimes, which have a scale factor that varies with time (see above).

(2) For a spacetime to admit a metric with flat spatial slices *and* a time-independent scale factor, it must be static. The "canonical" example here is, of course, Schwarzschild spacetime, which admits a coordinate chart (the Painleve chart) with flat spatial slices, and which also shares the time-independent scale factor property with the Kerr metric (of which it is a special case). (Of course there are other charts for this spacetime as well, which do not have flat spatial slices, but that doesn't matter for our purposes here as long as there is *some* chart that does.)

Condition (1) seems straightforward: a time-independent scale factor requires a time-independent metric.

I'm not sure about condition (2), though, because I'm not sure the interior portion of Schwarzschild spacetime--with r < 2M--qualifies as "static". (This portion is covered by the Painleve chart, so we can't finesse the issue by only considering the exterior portion.) The definitions in Wald are, briefly, that "stationary" means there is a timelike Killing vector field, and "static" means that field is hypersurface orthogonal. In the exterior of Schwarzschild spacetime, [itex]\partial_{t}[/itex] meets both these conditions, but it doesn't in the interior since it's no longer timelike. As far as I know, there is no other Killing vector field that *is* timelike in the interior (let alone hypersurface orthogonal).

There is one additional condition that we didn't propose an answer for in the other thread:

(3) What is required for a spacetime to admit a metric with flat spatial slices (but not necessarily a time-independent scale factor)? The "canonical" example here would be the FRW spacetime with k = 0, which has flat spatial slices but a scale factor that varies with time.

The key property that enables the construction of the metric for FRW spacetime in simple form is that it is isotropic; however, that condition can't be the right one for admitting flat spatial slices because it is too strong--Schwarzschild spacetime is not isotropic (it can be represented in so-called "isotropic coordinates", but that's not the same thing--the radial direction is still fundamentally different from the other two spatial directions). So maybe "spherically symmetric" is the right condition? I believe FRW spacetimes meet the textbook definition for that, even though they're not usually thought of in that way.

Can any of the experts here shed any more light on conditions (1), (2), and (3)?