We need to check if there is a proof that Maxwell's equations can be satisfied also under a dynamic gravitational field.
Gauss's law does not hold for a global observer far away, who measures the field strength by the energy he can receive if a test charge is moved a vector ds. The energy is transmitted to the global observer as light, and that light suffers a redshift if it has to climb from a gravitational potential well. The global observer sees the field E weaker than the local observer. The global observer may interpret this that the electric field has polarized space. Some electric lines of force end at opposite charges which polarization has brought there.
The global observer probably sees the zone close to the horizon of a black hole as an equipotential surface, and that the electric lines of force all end before reaching the horizon. For him, the space enclosed by the horizon is an equipotential area where the electric field is zero.
On the other hand, local observers see the lines of force continue unbroken to the horizon. If an observer jumps into the black hole, then the no drama hypothesis implies that he will see electric lines of force continue smoothly also behind the horizon.
Electric field lines which enter a black hole and come out of a white hole
Let us have a positive electric charge close to a black hole.
We may ask what happens if a black hole is connected through a wormhole to a white hole. What kind of an electric field E does an observer measure if he jumps in, during his journey into another universe?
Let us have two observers who jump. Observer A jumps from the side of a positive charge, and B from the opposite side.
A will probably see himself fall in the direction of the field line throughout his journey and then pop up from a white hole.
B will probably see himself falling in the opposite direction of a field line.
Maybe the white hole will appear to be an electric dipole, when A and B observe it in the universe where they fell? The white hole seems to have an induced positive charge on the side of A and an opposite negative charge on the side of B. The universe on the white hole side will contain a dipole field which is extending at the speed of light into the universe.
For a global observer on the black hole side, there seems to be induced negative charge close to the horizon on the side where A jumped, and induced positive charge on the side of B.
An electric charge which falls through to a white hole
Suppose that A throws a positive charge to a black hole and A jumps in after the charge. B jumps in from the other side.
Both A and B will probably observe themselves falling against the direction of electric field lines.
If the field lines do not break for a local observer, then the field lines probably continue unbroken out of the white hole up to the charge which already popped out of the white hole. Then a global observer on the white hole side probably will see an induced negative charge around the white hole horizon. The field lines which start from the positive charge end up at the negative charge close to the horizon.
But how much negative charge does the global observer see in the white hole?
What is the electric field like close to a white hole?
A global observer sees the electric field weak close to a black hole, because the energy that a local observer may get by moving a test charge, is transmitted as light to the global observer, and there is a redshift in the light.
If there is a local observer close to a white hole and he sends light to a global observer, then there is a blueshift. The global observer will see the electric field E stronger than a local observer.
Suppose that we have a positive charge which did not pop out of the white hole but has come to the universe some other way.
A black hole attracts field lines. It is logical that a white hole repels them. Let us assume that local observers see every line avoid the white hole. They may interpret this that behind the horizon there are induced charges which prevent field lines from entering. It would be like what happens to magnetic field lines at a superconductor.
The local observers see E very weak close to the white hole horizon. Maybe a global observer sees it as moderate strength?
The global observer must be able to define a global electric potential. If the local observer sees a very weak field E close to the horizon, then the global one might see a potential which looks reasonable.
Imagine a "neutron star" which is made of negative mass. It would repel most of the electric field lines, but allow a few of them pass through. To enforce energy conservation and avoid a perpetuum mobile, the global observer must see the field strength E roughly the same inside the star and around it. Then the local observer must see E very weak inside the star. This example suggests that the local observer, indeed, sees E weak close to the white hole horizon.
But now we have a dilemma. If a positive charge pops out of the white hole, how can its field lines go inside the hole, as seen by a local observer? A local observer is supposed to see a moderate electric field E all the way as he goes down the wormhole and comes out. A moderate field E for a local observer at the horizon is an infinite field for the global observer.
Suppose that we have a positive charge which did not pop out of the white hole but has come to the universe some other way.
A black hole attracts field lines. It is logical that a white hole repels them. Let us assume that local observers see every line avoid the white hole. They may interpret this that behind the horizon there are induced charges which prevent field lines from entering. It would be like what happens to magnetic field lines at a superconductor.
The local observers see E very weak close to the white hole horizon. Maybe a global observer sees it as moderate strength?
The global observer must be able to define a global electric potential. If the local observer sees a very weak field E close to the horizon, then the global one might see a potential which looks reasonable.
Imagine a "neutron star" which is made of negative mass. It would repel most of the electric field lines, but allow a few of them pass through. To enforce energy conservation and avoid a perpetuum mobile, the global observer must see the field strength E roughly the same inside the star and around it. Then the local observer must see E very weak inside the star. This example suggests that the local observer, indeed, sees E weak close to the white hole horizon.
But now we have a dilemma. If a positive charge pops out of the white hole, how can its field lines go inside the hole, as seen by a local observer? A local observer is supposed to see a moderate electric field E all the way as he goes down the wormhole and comes out. A moderate field E for a local observer at the horizon is an infinite field for the global observer.
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