Selected Questions -Answers From All Experts Astronomy Forum (Black Hole Astrophysics)

Question: Hello. I want to know about the basic astrophysics, etc.  laws for black holes, especially the key parameters that define them, including how calculated.  I am also a 'big boy' and can handle any kind of math or physics involved- don't need to be spoon fed!  - Uber Genius, Cambridge, MA

Answer:   A good place to start is by first noting that only three properties are needed to describe most black holes: the mass M, angular momentum L and the electric charge Q. For Kerr black holes, see e.g.

http://www.daviddarling.info/encyclopedia/K/Kerr_black_hole.html

the properties are only two: Mass M and angular momentum L.  They are electrically uncharged.

In greater generality, let's be clear the same laws for standard physics, e.g. general relativity, Maxwell's electrodynamics, quantum mechanics and laws applicable to matter and radiation, and thermodynamics, also apply to black holes.  For example, in regard to thermodynamics black holes must also conform to the 2nd law which defines entropy, in this case:

S _BH  = kA/ 4 L _p ²

where k defines a constant and A is the surface area of the event horizon with the Planck length:

 L _p    =   Ö G ħ/  c³   

The laws of thermodynamics applied to black holes are also called the "laws of black hole dynamics".  Thus, the first law of black hole dynamics - by analogy to the first law of thermodynamics - is just the standard law of conservation of total energy, now supplemented by the laws of conservation of total momentum, angular momentum and electric charge. (Note: Beware in many texts angular momentum is denoted by 'S'. I denote it by L so as not to confuse it with the entropy, S).

In the case of an infalling electric charge, the first law states that the total charge Q of a black hole - as measured by the electric flux emerging from it - changes by an amount equal to the total charge that falls down the hole, i.e.

D Q = q _in


Now, consider in terms of the black hole dynamics what transpires when two holes collide and coalesce.  Begin by first defining P1 and P2, the respective  4-momenta of the two black holes as measured gravitationally or when they are sufficiently separated they have negligible influence on each other. Then:

P1  =  m u ⁱ


P2 = mv ⁱ

The invariant mass is m,  and respective 4 velocities are:


u   ⁱ   = (u ¹ , u  ²  , u   ³   , u  ⁴   )   

v  ⁱ  =    (v ¹  , v ²  , v  ³   , v ⁴ ) 


OR:

 u  ⁱ  =   d xi / dt 

  v ⁱ   = d x ' i / dt      (i = 1....4)

(P1 and P2 are 4-vectors in the surrounding asymptotically flat spacetime)

Similarly, we let J1 and J2 be their respective total angular momentum tensors (not intrinsic angular momentum vectors) relative to an arbitrarily chosen origin of coordinates, say  C _o   in the surrounding flat spacetime.  Similarly, we let P3 and J3 be the total 4-momentum and total angular momentum of the final (e.g. coalesced) black hole.  And let  P _r   and J _r   be the total 4-momentum and total angular momentum radiated as gravitational waves e.g.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102

Then:

P3 =   P1  + P2 -   P _r      and J3 =  J1   + J2   -   J _r

Limiting cases:

We already saw the limiting case for the Kerr black hole, e.g. for which Q = 0

The limiting case of L = 0 applies to the Reissner -Nordstrom geometry

The limiting case of Q = 0 = L  applies to the Schwarzschild geometry.

The last entity is regarded as "dead" in the sense that it is impossible to extract any of its mass-energy. This is because it is neither rotating (L = 0) or charged (Q = 0)

The preceding are all grounded in one fundamental proposition, that a "black hole can have no hair".  I.e. there are no other independent characteristics or physical properties apart from M, Q and L to specify it.