World Science Scholars
2.2 The Mathematics of Black Holes
summary
It is possible for black holes to exist even without any matter.
• Chandrasekhar called black holes “the most perfect macroscopic objects there are” due to their elegance and simplicity.

Black holes can be described by three important characteristics.
• The first of these is mass.
• The second is horizon area.
• The third is surface gravity, which can be thought of as the force exerted at infinity to keep a unit mass suspended at the horizon.
• A more familiar system, such as a fluid, can also be easily described using three quantities: entropy, energy, and temperature.

Black holes possess a set of laws analogous to the laws of thermodynamics.
• The temperature of a system or surface gravity of a black hole must be the same throughout a system in equilibrium.
• The change in the energy of a fluid is proportional to the temperature and the change in entropy. The change in mass of a black hole is proportional to the surface gravity and the change in area.
• The entropy of a fluid and the area of a black hole must increase in any physical process.

From this fact, physicists were able to gradually infer that black holes are thermodynamic objects.
• This relation is governed by the equation $S_{BH} = \frac{k_{B}c^3 A}{4G\hbar}$
• From the constants in this equation, we see that understanding this relationship required understanding statistical mechanics (Boltzmann’s constant), relativity (the speed of light), gravity (Newton’s constant), and quantum mechanics (Planck’s constant).

In statistical mechanics, the number of microstates of a system scales exponentially with the entropy of the system.
• An solar mass black hole would have an entropy of 1077.
• This implies that it would have exponentially more microstates—so many that it would take more atoms than there are in the universe to write the number down.

The Black Hole Information Paradox points to a clash between general relativity and quantum mechanics.
• According to general relativity, information which falls into a black hole is lost forever.
• According to quantum mechanics, information can never be lost.

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