Why do we know space-time is locally flat?

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Why do we know space-time is locally flat,then we can use differentiable 4-manifold to describe the space-time?Do the Equivalence Principle say this?Classically space-time in GR is smooth,but what about if we consider the quantum effect?
 

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PeterDonis
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Why do we know space-time is locally flat,then we can use differentiable 4-manifold to describe the space-time?
Because doing so gives us a theory, General Relativity, that matches the results of experiments to very high accuracy.

Do the Equivalence Principle say this?
The EP says so, yes, but that's not why we know it's true, because it just shifts the question to, why do we know that the EP is true? The answer to that is the same answer I gave above.

Classically space-time in GR is smooth,but what about if we consider the quantum effect?
No one has ever observed any signs of spacetime itself being quantized. According to most physicists, this is to be expected since if spacetime is quantized, the length scale on which we would expect to see such effects is roughly the Planck length, which is 20 orders of magnitude smaller than the smallest length scale we can probe experimentally (roughly the size of an atomic nucleus).
 
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Then what is Quantum Gravity?What is quantized in this theory?
 
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PeterDonis
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Then what is Quantum Gravity?What is quantized in this theory?
We don't know because we don't have a theory of quantum gravity. Many physicists believe that spacetime will end up being quantized in such a theory, but that's only a speculation at this point.
 
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Why do we know space-time is locally flat
Empirically speaking - because every experiment we perform in a small enough local region matches the predictions of Special Relativity. Since that model is based on flat Minkowski spacetime, we can deduce that spacetime is locally flat. There are of course also more theoretical reasons, but I think you can get my drift.
 
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