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I've been told that GR and QM are not compatible, is there an intuitive reason/thought experiment which demonstrates the issue? (Or one of the issues?)

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I've answered a similar question at physics.stackexchange.com/questions/387/… – Matt Reece Sep 2 '12 at 21:04
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The simple thought experiments are not related to the naive reason people give for the incompatibility--- people say that they are incompatible because of renormalizability. This is not easy to explain, and it is in fact false, because N=8 supergravity is with some scientific confidence renormalizable. But it's not a good theory.

The reason for the incompatibility is the behavior of black holes--- the fact that they have an entropy which scales as the area, not the volume. In quantum field theory, the fields near the horizon have infinite entropy, as shown by 't Hooft. So you need a different kind of quantum theory, one which is nonlocal enough to allow black holes to have entropy which goes as area.

This is the only incompatibility, since string theory is exactly such a quantum theory, and it is a consistent theory of gravity. So there is no further incompatibility left.

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Here's why:

There are two singularities interacting with each other and they each have their own direction of time (anti-parallel to each other at the largest enclosing brane/sphere). We've gotten entangled with both. One cannot create both a persistent and consistent model, because an individual ultimately only belongs to one of them (each having their own separate sphere of causality).

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Thanks for the answer but I fear that wont understand this until I've studied both theories... – user27182 Sep 4 '12 at 23:07
-1: singularities don't have a direction of time. This answer is ridiculous. The issues of singularities is not significant, as they are only present in extensions past horizons. – Ron Maimon Nov 4 '12 at 1:43
They don't with respect to reference frame outside them (where symmetry can't observe a privledged direction), but they do to the observer within them. You are in one now. – Mark J Nov 4 '12 at 3:33
@RonMaimon: What did you think singularities were for? Just to conform to your laws? – Mark J Nov 4 '12 at 3:41
@MarkJ: I am not in a singularity, a singularity is a limiting position in a manifold where the curvature diverges. This happens symmetrically in black hole exact solutions, to the past and to the future, and it is only semiclassical approximations that break this symmetry, or situations where you have asymmetric entropy. I don't think these are "my laws", rather they are the accepted best answer to the GR problem in light of string theory, and they are not going to change. "Singularity" is not a catch all phrase that you can use any way you like, it means something definite in GR. – Ron Maimon Nov 4 '12 at 4:50
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Who says QM and GR are incompatible? Nature has both, and presumably, nature is consistent. String theory appears to be a theory combining both of them. Just because we don't know how to quantize gravity yet doesn't mean it's impossible.

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