Black holes: charge, spin, mass

 Great question—this gets right to the heart of how black holes are described in general relativity.

1. Where does the charge come from?

A black hole’s charge is simply the net electric charge of whatever formed it.

• If a collapsing star (or matter falling in) has more protons than electrons, the black hole will end up positively charged.
  
• If it has excess electrons, it becomes negatively charged.
  

In principle, charge is just another conserved quantity like mass–energy or angular momentum. When matter falls in, its charge contributes to the total.

👉 However, in practice:

• Astrophysical black holes are expected to be nearly neutral.
  
• Any significant charge would attract opposite charges from surrounding plasma and quickly neutralize.
  

2. Where is the charge “located”?

This is subtle—and depends on how you think about spacetime physics.

Short answer:

• The charge is not localized at a definite place like a classical object.
  
• To an outside observer, it behaves as if it is inside the event horizon, but its effects are encoded in the entire spacetime.
  

3. What an outside observer sees

From outside the black hole, the electromagnetic field behaves exactly like that of a charged object:

• The solution of Einstein’s equations for a charged black hole is the Reissner–Nordström metric.
  
• It produces an external electric field:\[
  E(r) = \frac{Q}{4\pi \epsilon_0 r^2}
  \]
  

This field exists outside the horizon, so:

✅ The charge is measurable from outside
✅ It contributes to the gravitational field
✅ It is part of the black hole’s “no-hair” properties (mass, charge, spin)

4. Is the charge on the event horizon?

There are two useful ways to think about this:

(A) Membrane paradigm (intuitive picture)

In this effective model:

• The event horizon behaves like a conducting surface
  
• The charge acts as if it is spread over the horizon
  

So in this picture: 👉 The charge is “on” the event horizon (like charge on a metal sphere)

This is not fundamental—it’s a useful analogy for external observers.

(B) Full general relativity (more precise view)

In the exact theory:

• The charge is associated with the entire spacetime solution, not a localized object
  
• Field lines extend from infinity down toward the black hole
  
• The “source” of the field is effectively hidden behind the horizon
  

Key point: 👉 Nothing inside the horizon can be directly probed, but its total charge influences the exterior field.

5. What about the singularity?

This is tricky because physics breaks down there.

• In classical solutions, you might say the charge is “at” the singularity
  
• But:
• The singularity is not a normal physical location
  
• It’s a place where the theory itself stops being valid
  

So: ❗ It’s not meaningful (in current physics) to say the charge is literally sitting at the singularity.

6. A deeper way to think about it

In modern terms, charge is defined by a surface integral at infinity:

\[
Q = \frac{1}{4\pi} \int \mathbf{E} \cdot d\mathbf{A}
\]

Meaning:

• The charge is determined by the behavior of the electric field far away
  
• You don’t need to know where the charge “is” inside
  

👉 This is very different from classical intuition: The charge is a global property of the spacetime, not a localized object you can point to.

7. Summary

• Origin: Comes from the net charge of collapsing matter
  
• Location (intuitively):
• Membrane picture → spread over the event horizon
  
• Location (physically):
• Not localized; encoded in the spacetime geometry
  
• Determined by field behavior outside
  
• Singularity:
• Not a meaningful place to assign charge in a physical sense
  

✅ Big takeaway:
For black holes, “charge” is best thought of as something defined by the external electromagnetic field rather than something sitting at a specific internal location.

If you want, I can compare this with how charge works for ordinary objects or explain how this ties into the “no-hair theorem” more deeply.

Where is the mass in a black hole?

 This is one of the deepest conceptual points in general relativity—you’re asking exactly the right question.

Let’s answer it carefully, because the intuitive “mass sits somewhere” picture breaks down.

🧭 Short answer

After a black hole forms:

The mass is not sitting at a place like ordinary matter.
Instead, it is encoded in the entire spacetime geometry outside the horizon.

🪐 Before collapse: easy

For a star:

• Matter is distributed throughout a volume
  
• The gravitational field comes from that matter
  

So you can say:
👉 “The mass is where the matter is.”

🕳️ After collapse: the picture changes

Once the star collapses into a black hole:

• All matter is hidden behind the event horizon
  
• Classical general relativity says it ends up at (or near) the singularity
  

But here’s the key twist:

🔑 The external gravitational field no longer “points” to a location of matter.

🌌 Where is the mass really?

In GR, mass isn’t like a lump sitting somewhere.

Instead:

• Mass is defined by how spacetime behaves far away
  
• This is called the ADM mass (or similar global quantities)
  

So:

✅ The mass is a global property of spacetime, not a local object.

🧠 Crucial idea: gravity without matter

After formation:

• The outside region is vacuum (no matter present)
  
• Yet gravity is still there
  

This is not a bug—it’s a feature of GR:

Spacetime curvature can exist without local matter, as long as it’s sourced somewhere in the past.

🧲 Analogy (imperfect but helpful)

Imagine:

• You create a whirlpool in water using a paddle
  
• Then remove the paddle
  

The water keeps swirling.

Where is the cause of the swirl now?

• Not at a point
  
• It’s in the motion of the fluid itself
  

Similarly:

The black hole’s mass is encoded in the curvature pattern of spacetime.

🕳️ What about the singularity?

You might think:

“Okay, so the mass must be at the singularity.”

But this is tricky:

• The singularity is not part of spacetime (it’s where the theory breaks down)
  
• GR does not describe it physically
  

So we can’t really say:

✅ “The mass is located there” in any normal sense

Instead:

The singularity is a sign that our description has failed, not a well-behaved object holding mass.

🧩 The key shift in thinking

In Newtonian physics:

• Matter → causes gravity
  
• Gravity → points back to matter
  

In GR:

• Matter helps determine spacetime
  
• But once formed, spacetime can carry:
• Mass
  
• Energy
  
• Angular momentum
  

Even in regions with no matter present

🌀 Back to your rotation question

Now this ties directly to your earlier question:

• The black hole’s angular momentum is not stored in spinning matter
  
• The mass is not sitting somewhere like a lump
  

Both are:

✅ Encoded in the geometry of spacetime itself

🧠 A sharper statement

The most precise answer is:

A black hole’s mass is defined by how spacetime curves at infinity, not by a localized distribution of matter inside.

⚠️ Important limitation

All of this is within classical general relativity.

We expect:

• A future theory of quantum gravity may give a clearer answer
  
• Possibly replacing the singularity with something physical
  

But right now:

“Where is the mass?”
has the answer:
👉 Not anywhere local you can point to.

✅ Bottom line

• The matter that formed the black hole has fallen behind the horizon
  
• It no longer acts like a normal mass distribution
  
• The mass lives in the gravitational field (spacetime curvature) itself
  
• The singularity is not a usable physical “location” of mass
  

If you want, the next natural step is exploring how mass can be defined purely from geometry (e.g., ADM mass or Komar mass), which makes this idea fully precise.