Transcript Bohr Model of Particle Motion in Schwarrzschild Metric

Bohr-Sommerfeld
Quantization
In the Schwarzschild
(Reissner-Nordström) Metric
Weldon J. Wilson
Department of Physics
University of Central Oklahoma
Edmond, Oklahoma
Email: [email protected]
WWW: http://www.physics.ucok.edu/~wwilson
OUTLINE
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Physical Motivation
Charged Schwarzschild Metric
(Reissner-Nordström Metric)
Hamiltonian-Jacobi Equation
Contour Integration
Bohr-Sommerfeld Quantization
Energy Levels
Summary
PHYSICAL MOTIVATION
MQ
mq
A mass m with charge q = 0
bound gravitationally to the
mass M with charge Q ≠ 0.
Ultimate Goal - exact H-Atom
energy levels including
general relativistic correction.
REISSNER-NORDSTRÖM METRIC
ds 2  c 2 dt 2  1dr 2  r 2 d 2  r 2 sin 2  d 2
with
 rS rQ2 
2GM 2 GQ 2
  1   2 , rS  2 , rQ  4
c
c
 r r 
Leads to planar orbits with
L  constant
Choosing


2
 d  0
The metric becomes
ds 2  c 2 dt 2  1dr 2  r 2 d 2
Conserved Quantities
Time independence of ds2 means
that p0 is constant along the motion.
As customary we denote the
constant by
p0  E
Independence of ds2 of the
angle  implies that p is
constant. As customary,
p   L
MASS-ENERGY RELATION
2
 
2
2
1
2
2
2
ds

g
x
x


c
dt


dr

r
d


metric
The
yields
p 0  g 00 p0  1c 2 E
p  0
dr
p m
d
L
p  g p  2
r

r

So the mass energy relation
m 2c 2  g  p  p
 g 00 p 0 p 0  g rr p r p r  g p p  g p p
Yields
L
m 2 c 2  c 2 (1c  2 E ) 2  1 ( p r ) 2  r 2  2 
r 
or
E2
L2
1
r 2
m c   2  (p )  2
c
r
2 2
1
2
HAMILTON-JACOBI EQUATION
The mass energy relation
E2
L2
1
r 2
m c   2  (p )  2
c
r
2
2
E
L
( p r ) 2  2  m 2 c 2   2
c
r
2 2
or
1
Leads to the (separable !!) H-J
2
equation r 2 E 2
L
2 2
( S) 
c
2
 m c  
r2
And the integrable (!!!) action
integral
2
E2
L
2 2
Jr   
c
2
 m c  
r
2
dr
CONTOUR INTEGRAL
The action integral can be
evaluated using the contour
integral method of Sommerfeld.
2
E2
L
2 2
Jr   


m
c   2 dr
2
c
r
 2i  Residues
There are two poles, both of
order two - one at r = 0 and
the other at r = . Evaluation
of the residues one obtains ...
BOUND STATE ENERGY
The contour integral evaluates to
2 2
 i rS

m
c
r
1
S

J r  2i
|L|
2 2
2
2 
2i m c  E / c 
 2 rQ
Which can be solved for the
(classical) bound state energy
1/ 2




2 2 2
m c rS
1

2
E  mc 1 
2
4 J
 
r
1

S
r



|
L
|
 2 2 r
 

Q

 

QUANTIZATION
Using the Bohr-Sommerfeld
quantization condition
L  
J r  nh
One obtains
with
  0,1,2,3,
n  1,2,3,




2
rS 
1
1
2
En  mc 1 
2
2 
4

 C
r
1

S
n 






2
r
Q




1/ 2
2GM
GQ 2

rS  2 , rQ 
,


C
c
c4
mc
SUMMARY
The Reissner-Nordström metric
1
 rS r  2 2  rS r 
ds  1   c dt  1    dr 2
r r 
r r 


 r 2 d 2  r 2 sin 2  d 2
2
2
Q
2
2
Q
2
Lead to the Bohr-Sommerfeld
energy levels




2
rS 
1
1
2
En   mc 1 
2
2 
4

 C
r
1

S
n 
 



2 rQ 



1/ 2
REFERENCES
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Robert M. Wald, General
Relativity (Univ of Chicago Press,
1984) pp 136-148.
Bernard F. Schutz, A First Course
in General Relativity (Cambridge
Univ Press, 1985) pp 274-288.
These slides
http://www.physics.ucok.edu/~wwilson