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In Schwarzschild coordinates the line element of the Schwarzschild metric is given by:

$$ds^2=\Big(1-\frac{r_s}{r}\Big)\ c^2dt^2-\Big(1-\frac{r_s}{r}\Big)^{-1}dr^2-r^2(d\theta^2+\sin^2\theta\ d\phi^2).$$

In the asymptotic limit where $r>>r_s$ the Schwartzschild metric becomes:

$$ds^2=c^2dt^2-dr^2-r^2(d\theta^2+\sin^2\theta\ d\phi^2),$$

which is the Minkowski metric of flat spacetime.

But observations show that real astronomical objects are embedded in an expanding spatially flat FRW metric given in polar co-ordinates by:

$$ds^2=c^2dt^2-a^2(t)\ dr^2-a^2(t)\ r^2(d\theta^2+\sin^2\theta\ d\phi^2).$$

Therefore maybe the Schwarzschild metric should be given by:

$$ds^2=\Big(1-\frac{r_s}{r}\Big)\ c^2dt^2-a^2(t)\Big(1-\frac{r_s}{r}\Big)^{-1}dr^2-a^2(t)\ r^2(d\theta^2+\sin^2\theta\ d\phi^2).$$

Perhaps this metric would only be useful to describe a gravitational system whose size is comparable to the Universe itself?

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    $\begingroup$ FWIW, the Schwarzschild metric is the vacuum solution for a static, spherically symmetric spacetime. Think about that. $\endgroup$ – Alfred Centauri Sep 24 '16 at 23:02
  • $\begingroup$ Related: physics.stackexchange.com/q/2110/2451 and links therein. $\endgroup$ – Qmechanic Sep 24 '16 at 23:16
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    $\begingroup$ Have you tried plugging that into the Einstein tensor and figuring out what would be the stress energy tensor and the equations for a(t)? $\endgroup$ – Bob Bee Sep 25 '16 at 2:28
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Actually, if you write (1-2M/ra(t)) in your coefficients for the dt and dr terms in your metric, you have the McVittie Metric, which was proposed in 1933. Although it is an exact solution to the Einstein equations, there has been a lot of confusion in the literature about what it represents. See:

https://arxiv.org/abs/1003.4777

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Matching the Schwarzschild metric onto the metric of an expanding universe is not trivial. Einstein and Straus tried it in the 1940s but their paper, as I recall, has a mistake. A solution was given in 1956 by C. Gilbert in the MNRAS:

http://adsabs.harvard.edu/full/1956MNRAS.116..678G

As I said, the solution is not trivial.

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