Of black holes and electric charges

Factual Questions
1

Let’s start by taking the Earth (5.98x10[sup]24[/sup] kg) and compressing it down to a sphere of about r = 9 mm. According to the Schwartzchild formula (r = 2GM/c[sup]2[/sup]), the mass would collapse to a black hole with an event horizon of that size.

Now, let’s drop some electrons into this black hole. To be specific, let’s drop about 1.57550x10[sup]12[/sup] electrons. Black holes being what they are, it will of course devour them.

Since each electron has a charge of 1.60219x10[sup]-19[/sup] Coulombs, our black hole now has a charge of 2.52425x10[sup]-7[/sup] C. Similarly, since the mass of each electron is 9.10939x10[sup]-31[/sup] kg, we have added 1.43518x10[sup]-18[/sup] kg. The mass of the earth is now 5.98x10[sup]24[/sup] + 1.43518x10[sup]-18[/sup] kg, just to make sure we keep track. :wink:

Now let us take one more electron, name it Alice, and move it to a meter away from the hole. The force of gravitational attraction between Alice and the black hole will be:
[ul]F = G M[sub]1[/sub] M[sub]2[/sub] / r[sup]2[/sup]
= 6.67260x10[sup]-11[/sup] • 5.98x10[sup]24[/sup] • 9.10939x10[sup]-31[/sup] / 1[sup]2[/sup]
= 3.63484x10[sup]-16[/sup] Newtons[/ul]
But of course that’s not the only force. Our new black hole has a charge, so we should look at that as well. The force of repulsion due to electric charges between Alice and the black hole will be:
[ul]F = k Q[sub]1[/sub] Q[sub]2[/sub] / r[sup]2[/sup]
= 8.9875x10[sup]9[/sup] • 2.52425x10[sup]-7[/sup] • 1.60219x10[sup]-19[/sup] / 1[sup]2[/sup]
= 3.63484x10[sup]-16[/sup] Newtons[/ul]
So the net force between Alice and the black hole is zero. (pause for dramatic effect)

Note that the distance r is arbitrary - if you increase or decrease it, both forces will increase or decrease proportionally, so the net force between Alice and the black hole will continue to be 0.

So, assuming that the mass of the black hole has collapsed to one nondimentional point (aka singularity) within the event horizon, can I launch Alice on a path that will now come to within 2 mm of the singularity? All along this path, the net force on Alice should continue to be 0, so nothing will prevent her from continuing. But note that this trajectory will go into and then come out of the event horizon.

Now I’m sure I’ve made a mistake along the way. Black hole event horizons are supposed to be one-way only. So please stamp out my ignorance and let me know where I went wrong.

2

Is electric charge “visible” outside the event horizon? I thought charge was carried by photons, which would not be able to escape the BH, so no charge propagation. (Or is it that charge is carried by virtual photons, just like gravitation is supposed to be carried by virtual gravitons, meaning the photons could escape beyond the EH?)

3

Newton’s laws don’t hold in highly curved spacetime so your formulas don’t apply. In order to understand this, one would have to master quantum field theory in highly curved spacetime, which unfortunately I haven’t done, so I can’t explain it. But I do know that gravity near a black hole doesn’t increases as 1/r[sup]2[/sup], but instead by the much greater: (1/r[sup]2[/sup]) / (1-r/R), where R is the radius of the hole.

4

Yes. Black holes have three properties: mass, charge, and angular momentum. And indeed it is because the virtual photons mediating the electromagnetic interaction are not trapped the way real photons are.

5

yes indeed, virtual photons aren’t constrained by light speed. That’s one of the reasons they’re virutal and one of the reasons they’re not real.

6

Aha. Thanks, Ring, I knew I was missing something.

Just to be clear - this adjustment only affects the gravity equation? Is there a similar adjustment for the other forces? Does the force of charge bend space in its own way as well?

7

You don’t need quantum field thoery for this, just General Relativity. According to Einstein, F = G mM/r^2 is only a good approximation when gravity is weak. As you approach the black hole the 1/r^2 dependance is no longer accurate.

If you’re far enough away, tho, this should work.

8

Your last question can only be answered in the context of a theory that unifies electricity and gravity, for example in Kaluza-Klein theory. (I don’t know what the answer would be.) Or string theory.

My suggestion: find a string theorist and ask your question. Watch string theorist turn red.