Do black holes exist in 1+1 dimensional spacetime? I'm currently working in 1+1 dimensional spacetime and would like to know if black holes can exist in such a manifold? I think they can because the Schwarzchild metric has the coordinate singularity, not to mention the actual singularity at $r=0$, attached to the $g_{tt}$ and the $g_{rr}$ metric components. However, as the radius is usually oriented in 3 dimensional space, I'm unsure...
 A: Black holes are solutions of vacuum EFE on a space-time with singularities.
EFE in the vacuum are:
$$
G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 0.
$$
These are sometimes written as
$$
R_{\mu \nu} = 0,
$$
which is completely equivalent in spacetimes of all dimensionalities except for $d = 2$.
This is easy to see. Write down the contraction of the EFE:
$$
0 = G_{\mu \nu} g^{\mu \nu} = R - \frac{1}{2} R d = R \left( 1 - \frac{d}{2} \right)
$$
implies that either $R = R_{\mu \nu} = 0$, or $d = 2$. In the latter case there are no restrictions on $R$ whatsoever.
Any 2-dimensional spacetime is a solution of vacuum EFE in 2 dimensions, because that vacuum EFE is an identity, not an equation.
This can be also seen by noticing that by Gauss-Bonnet theorem, the Einstein-Hilbert action in 2 dimensions is a topological invariant proportional to the Euler characterestic
$$
S_{EH} = \frac{1}{16 \pi G} \intop_{\mathcal{M}} d^2 x \sqrt{| \det g |} R = \frac{1}{8 G} \chi (\mathcal{M})
$$
that doesn't depend on the metric at all. Its variations w.r.t. the metric are all trivial and no constraining equation on the metric can be obtained via the least action principle.
The conclusion is that in 1+1 dimensions, any Lorentzian metric is a solution to the EFE. You can write down metrics that look like black holes, but those won't be predictions of the theory, rather stuff you put in by hand.
GR looks very different in $d = 2$.
Interestingly, $d = 3$ is also a very special case for GR (ask a separate question about this if you want an expanded answer).
Only starting with $d = 4$ does GR look like a theory of the gravitational interaction.
