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I recently learnt that the conservation laws are a consequence of the symmetries of space and time (the Lagrangian in Newton mechanics). Since space-time change in a black hole wouldn't quantum mechanics also change?

Let me give an example by what I mean:
In classical space:
$$x' = x + e \\ L(x') = L(x)$$

But the above transformation is not true in relativistic space. Hence, the conservation of momentum changes in special relativity (i.e momentum is not $mv$). So similarly shouldn't all the conservation laws change for a black hole and therefore it's action?

Side note: the reason I was thinking of a black-hole was that the incoming particle of mass will not drastically affect the space-time (due to the heavy centre)

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1 Answer 1

You don't need black holes to break spacetime symmetries. In general relativity, the Poincaré symmetries (rotations, boosts and translations) are routinely broken, in some cases a spacetime may have no symmetries at all (a WILD SPACETIME).

In that case, yes, energy and momentum conservation are violated. You can get such things as particle pairs being spontaneously generated by the gravitational field.

Of course you can recover those symmetries by pretending that spacetime is flat and that the curvature is actually another field.

Also, in spacetime, you always have a local Poincaré symmetry. Meaning basically that the differential version of those laws are still correct, but not the integral form.

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Wait wouldn't that mean quantum mechanics needs to be changed (because energy and momentum conservation are violated)? whereas quantum mechanics is restricted it's conservation by the dirac equation? –  Anant Saxena Jan 10 at 16:38
    
Also can't an equation of quantum mechanics be created according to transformations of the blackhole? –  Anant Saxena Jan 10 at 16:41
    
You do have to change the equations of quantum mechanics when you're dealing with spacetime, and it is not so easy. In the case of black holes, it even has some unfortunate effects, such as an electron orbiting a black hole not having a probability going up to 1 (corresponding to the effect of possibly crashing in the singularity). –  Slereah Jan 10 at 17:42
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