In considering an FLRW type universe, the scale factor $a$ is generally indicated as being strictly a function of time $a(t)$. Isn't this only true for a comoving frame?

In some other reference frame wouldn't the scale factor more generally be a function of position and time $a(x^i)$? Of course I get the reasons for choosing the comoving frame.


It has occurred to me that this question is strongly related to another question that I've asked Here

  • $\begingroup$ The scale factor is defined as a quantity in the comoving frame, so how do you propose to define it otherwise? $\endgroup$ – knzhou Oct 5 '18 at 11:28
  • $\begingroup$ @knzhou for a nonflat space, the scale factor is the radius of the sectional curvature, therefore it's transformation properties are well understood. (not sure about the flat space case however). for example the closed FLRW universe, the radius of the spatial three sphere is the scale factor $a$ and one can imagine distortions of this shape via Lorentz transformations, leaving the radius of curvature then position dependent. $\endgroup$ – R. Rankin Oct 5 '18 at 11:34

The answer is quite simple, what is meant by the symbol $t$ in the case of the Friedmann metrics is always the comoving time.

This is generally the case when presenting a metric, the coordinate statements of course relate only to that given coordinate system. Consider the Schwarzschild space-time. There we have Schwarzschild coordinates $r,t,\varphi,\vartheta$. The fact that the space-time is static, all the metric components are independent of $t$, is true only in the given coordinate system. In particular, the "$t$" in the Schwarzschild coordinates is proportional to a time measured by a very special class of observers, the static observers, and these will thus see the space-time as static.

Naturally, the Schwarzschild black hole will seem non-static to any other observer class, the black hole will generally seem to be moving with respect to the non-static observers. This would be reflected in the metric coefficients being dependent on the new "non-static" time $\tilde{t}$ if a corresponding coordinate system was to be adapted.

This is very much the case with the FLRW space-times (please keep the Lemaître if you include the Robertson and Walker). The statements about the metric coefficients being dependent or independent on coordinates are of course true only in the given coordinate system. In this case the time would be only the time of the comoving observers and the spatial directions the orthogonal directions to that.

  • $\begingroup$ Thank you, I asked the question because it seemed that if the scale factor transformed as I expected (which you seem to confirm), then conformal time, given by: $$\eta=\int\frac{dt}{a}$$ would be independent of the path taken between two spacetime events. I found that very fascinating. is that right? $\endgroup$ – R. Rankin Oct 5 '18 at 11:27

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