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Effects like gravitational waves and the curvature surrounding black holes do not occur in spacetimes with one time-like coordinate and two space-like coordinates. This is because the Einstein Field Equations fully constrain the Riemann tensor in fewer than four dimensions.

Are there other interesting effects that exist in four spacial dimensions plus a time-like dimension that vanish in our (3,1) universe? Or is the qualitative difference between (2,1) and (3,1) solely due to the transition between a fully-constrained Riemann tensor and one with freedom?

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  • $\begingroup$ I am not qualified to write a comprehensive answer, but from what I know the (2,1) and (3,1) cases are different because of the absence of dynamic degrees of freedom in (2,1). That is, higher-dimensional GR is pretty similar to the (3,1) case except that the number of degrees of freedom (graviton polarizations if you wish) is greater than 2. $\endgroup$ – Prof. Legolasov Nov 12 '16 at 7:21
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I) For starters, for higher-dimensional GR with $n\geq 5$ spacetime dimensions, an event horizon (which always has codimension 2) needs not be homotopic to a sphere $S^{n-2}$. E.g. for $n=5$, there are also black rings.

II) On the other hand, low-dimensional GR with $n\leq 3$ spacetime dimensions

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    $\begingroup$ And the topology of these black rings is fixed: there are no knots in 4+ dimensional space. Just an observation. $\endgroup$ – Prof. Legolasov Nov 12 '16 at 11:38
  • $\begingroup$ It also appears that black rings are unstable and could result in naked singularities, which the universe really dislikes. $\endgroup$ – Draco18s May 11 '17 at 18:12
  • $\begingroup$ Correction to the answer (v2): The word event horizon above should be replaced by a spatial slice of the event horizon. $\endgroup$ – Qmechanic May 20 '18 at 13:37
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A physical way to see that there are no gravitational waves in spacetime dimension $d~=~3$ is with the degrees of freedom of a wave. An electromagnetic or other gauge field has an electric $\vec E$ and magnetic field $\vec B$. In a sourceless region These fields are orthogonal in an electromagnetic or gauge field wave. Yet if you have only two spatial dimensions you do not have enough dimensions for the wave to propagate. The old right hand rule for the $\vec E$ and $\vec B$ fields and the direction or propagation $\vec k$ means you have insufficient number of spatial dimensions for waves. This means you have static field configurations, which is a topological field theory.

The extends to gravitational waves as well which require in a weak field limit the perturbing metric elements $h_{++}$ and $h_{\times\times}$ for two polarization directions. This means you have traceless field tensors $E_{ij}$ and $B_{ij}$ for the electric and magnetic analogues of the gravitational field in the spatial manifold embeded in spacetime. Again if you have only two spatial dimensions you "run out of dimensions" for wave progagation. This is manifested more mathematically in the vanishing of the Weyl tensor. There is though a parallel tensor with conformal structure called the Cotton tensor. The Weyl tensor defines the conformal properties of spacetime, and the Cotton tensor does the same for lower dimensional spacetimes.

For dimensions higher than $4$ we can think about black holes with Gauss theory in a Newtonian setting. If we have any spatial surface $\Sigma^n$ with dimension $n$ larger than $2$ the gravitational field inside a Gaussian surface $S$ enclosing a region $V~\subset~\Sigma^n$ we have $$ \int_S {\bf F}\cdot da~=~\int_V\nabla\cdot {\bf F}dV~=~4\pi G\rho, $$ for $\rho$ the mass density. For three dimension the area enclosing the source is a $2$-sphere and we get Newtonian gravity. In general we get a form of this field that appears as $$ {\bf F}~=~-\frac{k}{r^{n-1}}. $$ and the potential that gives the field ${\bf F}~=~-\nabla U$ has a form $$ U~=~-\frac{k}{r^{n-2}}. $$ The metric elements of the Schwarzschild metric $g_{tt}~=$ $g_{rr}^{-1}~=~(1~-~2m/r$ become modified accordingly with $g_{tt}~=~(1~-~c/r^{n-2})$. This is admittedly rather heuristic and a more rigorous derivation is needed.

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  • $\begingroup$ Just to make sure I understand what you're doing, that's a derivation of what Newtonian gravity would look like followed by the assumption that GR would look somewhat similar? $\endgroup$ – NoethersOneRing Nov 12 '16 at 13:40
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    $\begingroup$ Right. GR in higher dimensions in a weak field limit will be a higher dimension form of Newtonian gravity. To make this more airtight requires a more complete analysis, but the results are much the same. $\endgroup$ – Lawrence B. Crowell Nov 12 '16 at 13:56
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There are lots and lots of qualitatively new things that emerge in higher dimensions. As QMechanic mentions, event horizons don't need to be topologically spherical. It's also known that generic initial conditions can lead to naked singularities, so the cosmic censorship hypothesis fails. I believe that black holes with hair also exist in higher dimensions.

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  • $\begingroup$ What sort of hair? Do you have a source? $\endgroup$ – NoethersOneRing Jan 12 '17 at 15:02
  • $\begingroup$ @NoethersOneRing Here's one: arxiv.org/abs/1311.6065 $\endgroup$ – tparker Jan 12 '17 at 22:49
  • $\begingroup$ Very cool! Does it matter that the spacetimes are asymptotically AdS, or would they exist in asymptotically flat spacetimes too? $\endgroup$ – NoethersOneRing Jan 13 '17 at 0:09
  • $\begingroup$ @NoethersOneRing No idea, that's above my pay grade. (PS don't forget to upvote ;-) ) $\endgroup$ – tparker Jan 13 '17 at 1:27

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