Regular negative mass black holes under time transformations

[muruep01]

17 hours ago, Markus Hanke said:

Any Lorentz transformation is simply a hyperbolic rotation (+boost) of your coordinate system - you are merely labelling the same physical events in your spacetime in a different way. You are always free to do this, since it has no physical consequences in the classical world - the energy-momentum tensor is the conserved Noether current associated with time translation invariance, so you can choose to do this either in the future direction, or into the past, the difference just being a sign convention. The actual dynamics of the system are the exact same, so no laws of physics change form.

So of course you can describe classical anti-particles as propagating backwards in time (relative to their positive-energy counterparts), but that just means you’ve chosen a different sign convention in your description of the system. And again, all Lorentz transformations (antichronous and orthochronous) are diffeomorphisms, so applying them to a given metric does not change anything about the geometry of that spacetime.

There is no change in particle trajectories, such a spacetime has the exact same geodesic structure, only future-directed time-like unit vectors are now sign-inverted, so all processes “run backwards”, and all energies are negative wrt to the ordinary case. All curvature tensors and their invariants remain unaffected, so this is the same geometry described simply in a sign-inverted way.

Negative energies imply negative masses, which have gravitational repulsive interactions between them whether you introduce them in Netwonian gravity together with the equivalence principle, so that inertial mass is also negative, or into GR in the mass term of the Schwarzschild solution.

I quote: "Equation (14.67) shows that time reversal It changes the sign of the energy and thus the sign of the mass (14.4<»). Consequently it transforms every motion of a particle of mass m into a motion of a particle of mass -m." https://ayuba.fr/pdf/souriau1997-chapter14.pdf

 

17 hours ago, Markus Hanke said:

Sure. However, this isn’t the only thing it does - it also changes the spatial components of the 4-vector such that its overall norm remains conserved. This is why 4-vectors (more generally: objects that are representations of the Lorentz group) are covariant under all Lorentz transformations, and laws formulated with them retain their form.

That is why I insist that a parity transformation may take place together with a time transformation, so that the spatial components of the 4-vector remains the same.

[Markus Hanke]

5 hours ago, muruep01 said:

Negative energies imply negative masses, which have gravitational repulsive interactions between them whether you introduce them in Netwonian gravity together with the equivalence principle, so that inertial mass is also negative, or into GR in the mass term of the Schwarzschild solution.

Yes, but this isn’t the point here. Stellar black holes start off with ordinary stars, which are described by energy-momentum tensor fields that always satisfy the positive energy condition. Then they undergo collapse, which can be described by an appropriate interior solution to the EFE. None of these solutions lead to a geometry below the horizon such as the one you describe, at least not as far as I am aware. You might be able to manually construct such a geometry be glueing together patches of suitable spacetimes (though I doubt even that is possible), but whether such a construct is a valid solution to the field equations for a physically reasonable collapse process is an entirely different matter. I very much doubt this, but I would also encourage you to actually go and try to derive such a geometry - it would be very instructive. If you do find something, then please present it here, I would be curious to take a closer look.

5 hours ago, muruep01 said:

That is why I insist that a parity transformation may take place together with a time transformation, so that the spatial components of the 4-vector remains the same.

I don’t quite understand - you said you want to apply a full Lorentz transformation to the 4-vector, so this already affects all four components, and guarantees that the vector norm is preserved.

[muruep01]

On 10/17/2023 at 6:10 AM, Markus Hanke said:

I don’t quite understand - you said you want to apply a full Lorentz transformation to the 4-vector, so this already affects all four components, and guarantees that the vector norm is preserved.

A time transformation T = (diag(-1, 1, 1, 1)) and a parity transformation P = (diag(1, -1, -1, -1)) together lead to a unitary proper antichronous transformation PT = (-diag(1, 1, 1, 1))

Edited October 25, 2023 by muruep01