# Is the spacetime generated by isolated system always asymptotically flat?

I read a saying in wiki of asymptotically flat spacetime http://en.wikipedia.org/wiki/Asymptotically_flat_spacetime

"In general relativity, an asymptotically flat vacuum solution models the exterior gravitational field of an isolated massive object. Therefore, such a spacetime can be considered as an isolated system: a system in which exterior influences can be neglected."

So my questions are :

1.Given cosmological constant $\Lambda=0$, is the spacetime generated by isolated system always asymptotic flat? " I remember that there exist vaccum solutions that are not asymptotically flat(see below). So can an isolated system generate a vaccum solution outside the system that is not asymptotically flat? Is it possible that although two objects are far enough away from each other, the gravitational effect cannot be neglected?

For example, $$ds^2=-2xydt^2+2dtdz+dx^2+dy^2$$ the determinant is $-1$ for all $t,x,y,z$, the Riemann Tensor is constant but not zero in all spacetime. And the Ricci tensor is zero in all spacetime. So this is a vaccum solution even in global spacetime, while it is still not a asymptotically flat spacetime.

• The gravity of a limited amount of mass confined to a finite volume is, except for possible gravitational waves that are being emitted by the system, essentially indistinguishable from Newtonian gravity, which, as we know, vanishes very quickly with distance. If you have a counterexample, I would love to see it. – CuriousOne Sep 6 '14 at 7:26
• @CuriousOne You can see the example in my text. – 346699 Sep 6 '14 at 7:55
• What about the presence of a CC that makes it asymptotically dS or AdS? – TwoBs Sep 6 '14 at 8:32
• @TwoBs What's the meaning of "CC"? – 346699 Sep 6 '14 at 8:52
• CC is the cosmological constant. Even with no matter at all the space would not be asymptotically flat – TwoBs Sep 6 '14 at 8:57

The confusion is purely an issue of language about what is meant by "vacuum solution" and whether a cosmological constant (cc) $\Lambda$ is included or not. You can usually tell by the context which is the case, but I can see how this can be confusing.