Bohr Modelof Particle MotionIn the Schwarzschild Metric

Weldon J. Wilson

Department of Physics

University of Central Oklahoma

Edmond, Oklahoma

Email: wwilson@ucok.edu

WWW: http://www.physics.ucok.edu/~wwilson

OUTLINE

Schwarzschild Metric Effective Potential Bound States - Circular Orbits Bohr Quantization Summary

SCHWARZSCHILD METRIC2222221222 sin drdrdrdtcds

2SS 2

1cGM

rr

r

where

dtc

rc

rcr

mc

drdrdrdtcmc

dsmcS

2

222

2

22

2

212

222222122

sin

sin

Leads to the action

And corresponding Lagrangian

2

222

2

22

2

212 sin

cr

cr

cr

mcL

HAMILTONIAN FORMULATIONUsing the standard procedure, the Lagrangian

With

Yields the Hamiltonian

2

222

2

22

2

212 sin

cr

cr

cr

mcL

L

pL

prL

pr ,,

LpprpH r

22

22

2

2222242

sinr

pc

r

pccpcmH r

ORBITAL MOTIONThe Hamiltonian

Leads to planar orbits with conserved angular momentum

Using

22

22

2

2222242

sinr

pc

r

pccpcmH r

2

2222242

rcL

cpcmH r

Lpp constant,2

,0

CIRCULAR ORBITSFor circular orbits

And the Hamiltonian

becomes

2

2222242

rcL

cpcmH r

0constant rpRr

2

2242

2222

2242

RcL

cm

cpRcL

cmH r

0

EFFECTIVE POTENTIALThe Hamiltonian for circular orbits

is the total energy (rest energy + effective potential energy) of the mass m in a circular orbit of radius R in the “field” of the mass M.

2S2

2242S 2

,1cGM

rRcL

cmR

rH

2

2242S1

RcL

cmR

rE

0.5

0.6

0.7

0.8

0.9

1

1.1

1 10 100

R/Rs

E/m

c2

2

2242S

eff 1RcL

cmR

rVE

EFFECTIVE POTENTIAL

RADIAL FORCE EQUATIONThe radial force equation can be obtained from

Differentiation gives

2/1

3

22

2

224242

4

22

3

22

2

42

23

2

rcLr

rcL

rrcm

cm

rcLr

rcL

rrcm

Fss

ss

r

Which must vanish for the circular orbit ( )

2

2242Seff 1,

rcL

cmr

rE

rE

r

VFr

0rp

ALLOWED RADII OF ORBITSSetting

For the circular orbits produces the quadratic

Which can be solved for the allowed radii

02

3

22/1

3

22

2

224242

4

22

3

22

2

42

R

cLr

RcL

R

rcmcm

R

cLr

RcL

R

rcm

Fss

ss

r

032 2222242 cLrRcLRrcm ss

22

2

222

2

311

cmLr

rcm

LR s

s

0.5

0.6

0.7

0.8

0.9

1

1.1

1 10 100

R/Rs

E/m

c2

ALLOWED RADII

R+R-

BOHR QUANTIZATIONUsing the Bohr quantization condition

One obtains from

The quantized allowed radii

22

2

222

2

311

cmLr

rcm

LR s

s

,3,2,1 nnL

,3,2,1n

mcn

rr

r

nRn

C2

C2

2S

S2S

2C

2

,3

11

ENERGY – CIRCULAR ORBITSFrom the quadratic resulting from the radial force equation

One obtains

Putting this into

032 2222242 cLrRcLRrcm ss

S

2S

222

32 rR

RrcmL

2

2242S1

RcL

cmR

rE

Results in

)(

)(

S23

22

rRR

rRmcE S

ENERGY QUANTIZATIONFrom the energy

One obtains the quantized energy levels

where

)(

)(

S23

22

rRR

rRmcE S

,3,2,1,)(

)(

S23

2S2

n

rRR

rRmcE

nn

nn

mcn

rr

r

nRn

C2

C2

2S

S2S

2C

2

,3

11

References

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 slideshttp://www.physics.ucok.edu/~wwilson