This document discusses the quantum mechanics of black holes. It notes that quantum mechanics allows for time-reversed transitions between states, which contradicts classical general relativity descriptions of black holes. Specifically, general relativity says that a black hole absorbing another object can occur, but the reverse process of emission cannot. However, quantum mechanics predicts both absorption and emission are possible. Hawking resolved this contradiction by showing that black holes can spontaneously emit elementary particles via a quantum process, allowing the time-reversed transition and upholding the second law of thermodynamics at the quantum scale.

1. Quantum
Mechanics of
Black Holes
Muhammad Zeeshan Khalid

2. Black Holes
• A black hole is a region of space-time from which gravity
prevents anything, including light, from escaping.
• The boundary of the region from which no escape is possible
is called the event horizon.

3. QM of BH
• In QM if time dependent transition is possible from an initial
state |i> to final state |f> , then it is also possible to have
transition from |f> to |i>.
• This contradicts to Black holes.
Let, B=Macroscopic Black Hole
A=Macroscopic body
B*=Heavier BH combining B & A
• GR Tells us this reaction will occur when body(A) and Black
hole (B) will come closer
B+AB*

4. Cont..
• QM tells us Reverse reaction is also possible with an
equivalent amplitude.
B*A+B
• That reverse reaction is exactly what does not happen,
according to classical general relativity.
• Another physical principle that is also seemingly violated by
the existence of a black hole is time-reversal symmetry which
says, “if a physical process is possible, then the time-reversed
process is also possible.”

5. Time Reversal
• The important problem with time reversal is that in everyday
life, it simply does not appear to be valid. We can spill a cup of
water on to the ground, but the water never spontaneously
jumps up into the cup.
• Entropy at atomic level Explain this concept why time
reversible is not possible in everyday life.
 Increase in Number of states.
• The water could jump back up into the cup if the initial
conditions are just right at the atomic level, but this is
prohibitively unlikely.

6. Black Hole Entropy and Hawking Radiation
• A black hole should be understood as a complex system with
an entropy that increases as it grows.
• When a black hole B absorbs some other system A in the
process A+BB*, its entropy increases along with its mass
accordance with 2nd law of thermodynamics.
• The reverse reaction B*A+B diminishes the entropy hence
violating the 2nd law of thermodynamics.

7. Cont.
• If the irreversibility found in black hole physics is really the
sort of irreversibility found in thermodynamics, then it should
break down if A is not a macroscopic system but a single
elementary particle.
• This is what Hawking found in a calculation. A Black hole
spontaneously emits elementary particles.
• The typical energy of these particles is proportional to Planck’s
constant, so the effect is purely quantum mechanical in nature,
and the rate of particle emission by a black hole of
astronomical size is extraordinarily small, far too small to be
detected.

8. Conclusion
• So According to Hawking insight black hole is potentially no
different from any other quantum system, with reactions
A+BB* & B*A+B occurring in both direction at
microscopic level