Is the spacetime generated by isolated system always asymptotically flat?

Question

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.

Answer 1

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.

In the wikipedia article: "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." it is implied that there is no cc.

In the statement: "There exist vaccum solutions that are not asymptotically flat." a cc is allowed. The example you provide in the question is also of this type, ie it only satisfies the Einstein equations if the cc term is included.

The difference in language is a matter of perspective. You can either treat the cc as an inherent property of spacetime and hence allow it in "vacuum solutions" or think about it some kind of "dark energy" which is then an added form of energy and should not be allowed in vacuum solutions.

Both options are fine as long as you know what you are talking about. In that sense, we can say that an object far away from another object can be treated as an isolated system, i.e. we can neglect the influence by others, but if there is cc we cannot neglect the interaction of the object and the cc.