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I am learning Reissner-Nordstrom black holes and I have learnt that the black hole contains a net charges. The static field due to it ( through the Energy Momentum tensor) exists even outside the event horizon. But I also know that nothing can escape out of an event horizon. So I have two questions-:

  1. Why does the electric field ( static ) of a charge on the Reissner-Nordstrom black hole exists even outside the event horizon of the black hole?

  2. If the black hole gains a some more charge ( say a charged particle does fall in the time like singularity) then the net charge on the black hole increases and a static condition is again reached. The electric field of the new static hole should also now increase. Will the change in electric field propagate outside the horizon also increase? If so how? This increase in the electric field could act as a causal influence which has now propagated outside the horizon but that should not have happened. ( The old hole had a horizon so nothing should pierce through it, and the new hole again reaches a static condition; so the change in the electric field has pierced through the horizon of the previous hole which shouldn’t have happened)

Qmechanic
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Shashaank
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1 Answers1

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The trick is, it always existed there, outside. Even before the black hole formed, it probably was a massive star, and had its static electromagnetic field (this is what you are asking about). When the black hole forms, it just happens that the event horizon (which is not a physical object, it is just a boundary) gets situated as a boundary so that the EM field exist even outside of it. No information, no particles need to travel from inside the event horizon.

As for how the electric field gets out of the horizon, the best answer is that it doesn't: it was never in the horizon to begin with! A charged black hole formed out of charged matter. Before the black hole formed, the matter that would eventually form it had its own electric field lines. Even after the material collapses to form a black hole, the field lines are still there, a relic of the material that formed the black hole.

Detection of the Electric Charge of a Black Hole

Your second question is even more trickier. What we are talking about is a static EM field. Again, the event horizon is a boundary, not a physical object. Where does the charged particle in your question come from? It comes from the outside world. When does it start to add to the strength of the static EM field of the black hole? Just when it enters the horizon? Nope. The charged particle in your question already strengthens the black hole's static EM field outside the horizon. This is the answer to your question. No need for any particle or information to travel from inside the horizon. When the charged particle enters the horizon, the static EM field of the black hole is already stronger.

Árpád Szendrei
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  • So to give a contribution to the Energy Momentum tensor it's only necessary for the charge to be somewhere not necessarily at r=0. It's field lines will be both inside the horizon and outside the horizon when it's outside the horizon so it will give a contributions in both the areas. Is that right....I was thinking that to give a contribution to the energy momentum tensor the charge should be at r=0. – Shashaank Apr 05 '21 at 06:00
  • But hold on. When the charge is outside the horizon it's moving and its field will not be static. The field of the hole is static. Even if this charge contributes to the energy momentum tensor both inside and outside the horizon, the contribution won't be of a static field. So the whole energy momentum tensor would be a sum of a static field and a dynamic field. So the whole situation won't be static anymore and the RN metric won't be valid. Is that right. – Shashaank Apr 05 '21 at 06:04
  • You are correct that the particle doesn't need to be at r=0 (doesn't even need to be inside the horizon) to contribute to the static EM field, and the field lines are both inside and outside. Now the static EM field is just a distinction from EM waves (radiation). There are near fields (static EM field), and far fields (radiation). We call the former static EM field but this does not mean that it is static. It just means it is distinguished from EM waves (radiation). https://en.wikipedia.org/wiki/Near_and_far_field – Árpád Szendrei Apr 05 '21 at 15:52
  • "Thus, EMR is sometimes referred to as the far field. In this language, the near field refers to EM fields near the charges and current that directly produced them, specifically electromagnetic induction and electrostatic induction phenomena." https://en.wikipedia.org/wiki/Electromagnetic_radiation – Árpád Szendrei Apr 05 '21 at 16:16
  • ok I get that. But I am saying when the charge is falling in the black hole,its field is not a static field. It will radiate too. So I am just saying that the whole picture will become dynamic and the static Reissner Nordstrom will not hold true. Is that right? – Shashaank Apr 05 '21 at 18:13
  • @Shashaank If it will radiate then you are correct. I do not know whether the conditions are met for it to radiate. Just because an electron falls in, it might not radiate. Usually you need accelerated charges to radiate. The change in the static field itself might not cause EM radiation. – Árpád Szendrei Apr 05 '21 at 18:49
  • Electrostatics only deals with charges at rest. Moving charges (even with a constant velocity) don’t have static field. It need not accelerate just moving with a constant velocity is a departure from Electrostatics to Electrodynamics. So as the electron falls it is surely moving. So it won’t produce the static field Q/r^2 which goes into the stress energy tensor in this static scenario. A moving charge ( need not be accelerating) should render the RN solution invalid. That’s what I was trying to say. Do you not agree that just a moving charge should render the RN soln invalid. – Shashaank Apr 05 '21 at 19:12
  • @I think you are correct. Though, the name static here refers to the near field. Electrostatics and electrodynamics is not the same as static field and EM radiation I believe. A moving electron should have a static EM field around it. The name static here just refers to the near field. It does not mean that it cannot change at all. – Árpád Szendrei Apr 05 '21 at 20:20