Black Hole Physics

1
Black Hole Physics
Schwarzschild metric
What happens at
Can it happen?
White Dwarf
Neutron Star
Black Hole
2
Free FallRadial (Radical?) Plunge
Wristwatch time
Far away coordinates
Rule 1 Go straight in local coordinate system.
Rule 2 Actual trajectory is the one that
maximizes wristwatch time.
r
Dr1
Procedure Vary tM at fixed rM such that total
wristwatch time of trip is maximized.
Dr2
t
T
0
3
Free Fall (continued)
r
Dr1
Dr2
t
T
0
4
Energy Conservation
Far away
Constant of Motion
Potential Energy
Kinetic Energy
5
Raindrop in Far Away Time
Definition
For far away observer, object never reaches black
hole!
6
Raindrop in Shell Time
Shell observer Observer stationary in
gravitational field who has set up his/her
coordinate system there.
Time dilation
Length contraction
Raindrop sees shell go by at same velocity that
shell sees raindrop go by.
7
Velocity Comparison
Shell velocity
Faraway velocity
8
Time Comparison
Faraway time
Wristwatch time
9
Orbiting
Wristwatch time
Far away coordinates
Rule 1 Go straight in local coordinate system.
Rule 2 Actual trajectory is the one that
maximizes wristwatch time.
Procedure Vary fM at fixed rM such that total
wristwatch time of trip is maximized.
10
Computing the Orbit
Energy Conservation
Angular Momentum
Schwarzschild metric
Strategy Integrate r(t), then integrate f(t) and
t(t).
11
Newtonian Effective Potential
Newton radial equation
Effective Potential/Unit Mass
E/m
Angular Momentum effect
Effective Potential
Orbit Excursions
r
Gravitational potential
12
Relativistic Effective Potential
(Effective Potential/Unit Mass)2
13
Relativistic Effective Potential
Plunger
E/mc2
Bound Orbit
Circular Orbit
r/R
14
Circular Orbit
No Stable orbits
One stable and one unstable orbit
15
Light Orbit
Equations for orbits of massive particles
Problems

• m0
• t0

Procedure Write in terms of far away time, and
let m go to zero.
16
Light Orbit (continued)
Impact parameter
Note
Shell velocities
Note
17
Effective Potential for Light
Problem both terms have position dependence
Problem initial conditional dependence in
potential term
Effective potential
Kinetic type term
Total energy term
18
Effective Potential
All photons have the same effective potential.
Decreasing blight
V(r)
Constant blight
r/Rsch