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In the standard study of asymptotically flat spacetimes one defines null infinity demanding that topologically ${\cal I}^\pm \simeq \mathbb{R}\times S^2$ (c.f. Definition 1 of this review by Ashtekar). The reason is clearly that this is what happens in flat spacetime and intuitively we want to define things so that asymptotically flat spacetimes at infinity look like Minkowski spacetime.

Nevertheless I imagine that from a mathematical perspective nothing would stop us from adapting the definition so that we have a larger class of Lorentzian manifolds, which are defined exactly like asymptotically flat spacetimes, with the only difference that now ${\cal I}^\pm \simeq \mathbb{R}\times \Sigma$ where $\Sigma$ is some arbitrary Riemann surface. In this case we would endow $\Sigma$ with a metric $\gamma_{AB}$ which would take the place of the usual round metric on $S^2$. In particular, in the usual retarded coordiantes $(u,r,x^A)$ we would have, near ${\cal I}^+$,

$$ds^2=-du^2+2dudr+r^2\gamma_{AB}dx^Adx^B+\text{corrections...}$$

My question here: has this class of spacetimes been studied? Are they physically reasonable for any $\Sigma$, for just a subset of all Riemann surfaces or they are simply not physically reasonable at all?

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  • $\begingroup$ This won't answer your question, but I know asymptotically AdS spacetimes are studied often, e.g. arxiv.org/abs/1211.6347 and cds.cern.ch/record/469051/files/0010138.pdf $\endgroup$
    – Eletie
    Commented Nov 28, 2020 at 14:07
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    $\begingroup$ Hi @Eletie, thanks for the references ! Indeed this is a different matter, I have edited the question to make it clearer. I'm thinking about spacetimes which obeys all conditions of asymptotic flatness, except that null infinity has a different topology ${\cal I}^\pm \simeq \mathbb{R}\times \Sigma$, so I'm relaxing the definition of asymptotic flatness just a bit, not that much so as to get to asymptotically AdS spacetimes. $\endgroup$
    – Gold
    Commented Nov 28, 2020 at 14:12
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    $\begingroup$ Ah! Okay I understand now, thanks for clarifying! Hopefully somebody else can give a good answer. $\endgroup$
    – Eletie
    Commented Nov 28, 2020 at 14:25

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Here are examples of solutions that have toroidal null infinities:

This are constructed from analogue of the Schwarzschild metric with planar symmetry (metric A3 in classification of Ehlers & Kundt, from a 1962 volume Gravitation: an Introduction to Current Research, edited by L. Witten) also known as Taub's plane symmetric solution: $$ ds^2 = −\frac1R\,dT^2+R\,dR^2+R^2(dx^2+dy^2). $$

By imposing periodicity in both $x$ and $y$ directions and suitably “deforming” the metric one could obtain a family of vacuum asymptotically A3 solutions with null infinities having topology $\mathbb{R}\times T^2$.

Note, that the A3 metric is not very physical, since it could be seen as a gravitational field of a negative mass naked singularity, see e.g. this paper for a discussion of its properties. One use for such spacetimes is as a test for numerical relativity codes since we can check results of numerical evolution against the exact solutions.


Also, celestial sphere of ordinary asymptotically flat spacetimes can be modified by the action of finite superrotation transformations, which would introduce conical defects at the celestial sphere geometry:

Another viewpoint on finite superrotations, is that they correspond to cosmic strings in the bulk:

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