Why can an electric field escape from a black hole? [duplicate]

Question

I know this is a duplicate question but the other answers didn't explain clearly my problem. The talks about no hair theory for explaining the communication and information problem of matter inside the black hole. But I just want to know how it is possible that a field can escape from a black hole. It is not clear where an electric field is made of and perhaps that is the lack of information which can answer why it can escape a black hole.

Answer 1

Electric fields don't start somewhere and move somewhere else. The field lines with arrows are just a picture telling you the direction (tangent to line and in the direction of the arrow) and the magnitude (stronger where field lines are closer together).

Really, there is just two vectors (or one bivector) at each point in space at each time, for the electric field and the magnetic field.

In electromagnetism you start by admitting that electromagnetic fields exist, even in vacuum, and that they evolve in time according to equations like $$\frac{\partial \vec B}{\partial t}=-\vec \nabla \times \vec E, \text{ and}$$

$$\frac{\partial \vec E}{\partial t}=-\frac{1}{\epsilon_0}\vec J+\frac{1}{\epsilon_0\mu_0}\vec \nabla \times \vec B.$$

And these equations are just Maxwell's and they are literally just how fields evolve. And there is a field everywhere (though it might be zero at a particular time and place) so knowing how it changes is the full and complete story about where it comes from.

Let's look at a simple example. A symmetric spherical shell of charge of radius $R$ with a zero magnetic field everywhere and an electric field of $\frac{Q}{4\pi\epsilon_0}\frac{x\hat x+y\hat y+z\hat z}{(x^2+y^2+z^2)^{3/2}}$ when $x^2+y^2+z^2\gt R^2$ and $\vec 0$ otherwise. Now we can describe the actual physical mechanism to create a field. If the charge on the surface of the shell moves inwards there is a current. And so right where the current is (on the surface) a new nonzero electric field is produced. If the current goes in radially, then the new elecetric field points radially outward (since there is a zero magnetic field and $$\frac{\partial \vec E}{\partial t}=-\frac{1}{\epsilon_0}\vec J+\frac{1}{\epsilon_0\mu_0}\vec \nabla \times \vec B.$$).

And since the electric fields are radial, they have zero circulation so the magnetic field doesn't change and so is still zero. Now new nonzero fields have been created in a region that previously had zero fields. They were created as the charge passed that location, and they persisted long after the charge moved past that region.

So we know how electromagnetic fields are created. Each region has electric fields and they change based on the balance between the current and the circulation of magnetic fields. And each region has magnetic fields and they change based on the circulation of electric fields. That's the complete story, the only thing to know is the current fields and the current, then you know how the fields change and so that's where the fields come from. And we didn't even need all the details of Maxwell for our super simple example. We can note that Coulomb fields of parameter $Q$ are static vacuum solutions and so they can hold away from charge and the charge is a source allow two vacuum solutions (Coulomb of parameter $Q$ and Coulomb of parameter zero) to math up. And indeed the names of parameters are related to how much charge (source) you need to connect them.

So regions without fields can get fields. As a shell of charge collapses the region is passes through has its zero field turn into a radially outwards field. The outwards is from the minus sign of $$\frac{\partial \vec E}{\partial t}=-\frac{1}{\epsilon_0}\vec J+\frac{1}{\epsilon_0\mu_0}\vec \nabla \times \vec B.$$ And it fall s off like $1/r^2$ since current densities $\vec J$ fall off that way when a spherical shell of charge compresses.

The same process explains how spacetime is curved outside a star.With Maxwell you started with initial electromagnetic vector fields, their time rates of change told them how to change and current and the current field gave those time rates and Maxwell explains the process.

With Einstein you start with an initial Cauchy slice of a manifold with a metric tensor field and their time rates of change (this is second order like Newton), their time rates of change tell them how to change and the second order rates of change tell the first order rates how to change. The second order rates are determined by the Stress-Energy Tensor (that's like the current in Maxwell) and also by the variations of the metric (like how the circulation of the electric field determined the rate of change of the magnetic field and how the circulation of the magentic field was part of how the electric field got its rate of change).

But what's the process? Just as Newton gave the process for acceleration coming from mass and force, and just as Maxwell gave the process for time rates coming from spatial variation and currents. So Einstein (through the Einstein Equation $G_{\mu\nu}=kT_{\mu\nu}$) gives the process for second order time rates of change to be determined by Stress-Energy and lower time order variations of the metric. This is literally how Numerical Relativity evolves initial configurations into entire manifolds. And it's like using an initial value formulation of Maxwell to get fields in all of time and space from the field everywhere at one time or using Newton to get the past or fututre from the present.

Let's look at a super simple example next. A symmetric spherical shell of energy at rest with a Schwarzschild metric of parameter $M$ on the outside and a flat Minkowski metric on the inside (this is like the spherical shell of charge with a Coulomb field in the outside and zero field in the inside). Now just like a radial electromagnetic field is a kind of electromagnetic field that doesn't change in time when there is zero current, so a Schwarzschild metric of parameter $M$ is a kind that doesn't change when there is zero Stress-Energy. Both are called vacuum solutions. And actually, the Minkowski metric is just a Schwarzschild metric with a parameter of zero, so it also is a vacuum solution and does not change in time when there is zero Stress-Energy. Technically we also needed the metric on a slice and to have zero first order derivatives in time and then we find that away from the shell of energy the metric has zero time second derivatives, but hopefully you get the picture that the metric is there and in some regions it isn't changing.

Now it helps to visualize the metric of the Schwarzschild solution as like a funnel. So if you put a coin in a funnel then you'd have a curved funnel part on the outside and then a flat part on the inside. That's our metric. But since there is energy where the metrics with two different parameters ($M$ and zero) the metric (and hence the curvature) does change there, just like the electric field changed where there was current.

So what happens? If the energy collapses radially, then the region below where it collapses to stays parameter zero. And the region below where it started stayed parameter $M$ (just like in the electric case the field stayed $Q$ on the outside of where the charge started and stayed zero on the inside of where the charge moved to). But in between those two surfaces, things change.

How do they change. In the electric and the gravitational case the shell is still just somewhere with one type ($Q$ or $M$) on the outside, and zero on the inside. They have to be, those types can only connect up to their own type when there is no source ($\vec J$ or $T_{\mu\nu}$).

So its like replacing the coin with a smaller coin. And here something interesting happens. When you remove a coin from a funnel and replace it with a smaller coin it is still that same type of funnel outside and still flat inside and the outside has the same circumference but the old parts of the funnel are now farther from the center of the coin, because the coin is deeper. If you replaced a 2m radius coin with a 1m radius coin the old circle where the old 2m coin rested is now more than 1m away from the location of the edge of the new coin.

Similarly when energy collapses inwards new space is created below where the energy used to be and above where it ends up. That new space is larger than the old region (it would be smaller if your shell had negative energy on it instead of positive energy) and it has a new curvature from the new metric. The new metric there is the same type as outside (Schwarzschild of parameter $M$) but just like how the Coulomb of parameter $Q$ is larger closer in, the Schwarzschild metric of parameter $M$ is more curved deeper in the well. The funnel is just more curved closer in so the smaller coin exposes more funnel of the existing type, so introduces larger curvature than previously existed (same type, but larger curvature).

This is literally the process by which large curvature forms. Small curvature turns into larger curvature as the energy on the boundary of the small curvature is concentrated into a smaller region. But extending the vacuum type of curvature into the new space exposed by the collapsing energy.

Exactly like how strong electric fields are created as the charge on the boundary of the small field region is concentrated into a smaller region. By extending the vacuum type of field into the space exposed by the collapsing charge.

A realistic star has many layers of parameter $M$ and $m$ so you put less curved funnels into more curved funnels and you shave off the outside part of the less curved funnel so it can fall deeper into the outer funnel (instead of shaving off the outside part of coin to get a smaller coin to place deeper).

Fields don't come out of sources and travel to where you feel them. Fields are everywhere, and they evolve according to the variations around them. Vacuum fields evolve (or are static, depending on the field) and sources allow different vacuum solutions in different regions to be pasted together.