Schwarzschild metric in expanding Universe

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?

• FWIW, the Schwarzschild metric is the vacuum solution for a static, spherically symmetric spacetime. Think about that. – Alfred Centauri Sep 24 '16 at 23:02
• Related: physics.stackexchange.com/q/2110/2451 and links therein. – Qmechanic Sep 24 '16 at 23:16
• 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)? – Bob Bee Sep 25 '16 at 2:28