Why do we need a theory of quantum gravity

[MathJakob]

I've been wondering, why does the world need a theory of quantum gravity? What questions would be solved if such a theory was discovered?

What does a theory of quantum gravity mean for black holes, if anything?

Try to be quite indepth about the questions rather than something like "It will tell us what happened at t = 0" :P

 

[rubi]

As you might know, Einsteins equations are
$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R = 8\pi T_{\mu\nu} \,\text{.}$$
$T_{\mu\nu}$ is the stress-energy tensor of the matter content of the theory (electrons, quarks, gauge fields, ...). The problem is that the matter is described by a quantum theory (the standard model) and thus $T_{\mu\nu}$ should really be an operator $\hat T_{\mu\nu}$. But if the right hand side is an operator, the left hand side should be an operator as well for the equation to make sense. Thus we try to find a quantum theory that describes the gravity part of the eqation.

You might ask: "Why don't we use $\left<\hat T_{\mu\nu}\right>$ for the right hand side?" People actually do this and it's called quantum field theory on curved spacetimes. It is expected to be a limiting case of the a full quantum theory.