GR vs quantum vacuum Lorentz invariance

[mieral]

is spacetime Lorentz invariant like the quantum vacuum?

They say the quantum vacuum is Lorentz invariant.. you can't locate it at any place.. but if spacetime manifold is also Lorentz invariant and you can't locate it at any place.. how come the Earth can curve the spacetime around the Earth and not some exoplanet light years away.

And if spacetime is not Lorentz invariant like vacuum, then what are the limited invariance it obeys and how do you make it Lorentz invariant?

 

[PeterDonis]

is spacetime Lorentz invariant like the quantum vacuum?

Spacetime is locally Lorentz invariant like the quantum vacuum, yes.

They say the quantum vacuum is Lorentz invariant.. you can't locate it at any place

That's not local Lorentz invariance, that's global translation invariance. Whether global translation invariance holds depends on the spacetime geometry; it's not an intrinsic property of the quantum vacuum.

how come the Earth can curve the spacetime around the Earth and not some exoplanet light years away

Because the spacetime geometry due to the Earth is not globally translation invariant; it looks different at different places.

 

[mieral]

Spacetime is locally Lorentz invariant like the quantum vacuum, yes.
That's not local Lorentz invariance, that's global translation invariance. Whether global translation invariance holds depends on the spacetime geometry; it's not an intrinsic property of the quantum vacuum.

a) What kind of spacetime geometry has global translation invariance?

b) Is the quantum vacuum inside spacetime? Or is spacetime inside the quantum vacuum?

c) One has global translation invariance.. the other hasn't. Shouldn't it be consistent they should have a symmetry much like electric/magnetic field, space/time, etc.?

d) Is there a transformation that can transform spacetime to quantum vacuum and quantum vacuum into spacetime.. is this one of the goals of quantum gravity?

Because the spacetime geometry due to the Earth is not globally translation invariant; it looks different at different places.

 

[PeterDonis]

What kind of spacetime geometry has global translation invariance?

Minkowski spacetime is the simplest example. If we limit it to space translations, any homogeneous spacetime, i.e., all of the FRW spacetimes used in cosmology.

Is the quantum vacuum inside spacetime? Or is spacetime inside the quantum vacuum?

Mu. (The question is not well posed so it doesn't have a well-defined answer.)

One has global translation invariance.. the other hasn't.

Where are you getting that from?

Is there a transformation that can transform spacetime to quantum vacuum and quantum vacuum into spacetime

I don't understand what such a thing would mean.