Motion in Schwarzschild Spacetime: Relativity Lecture 19, Study notes of Physics

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Relativity and Cosmology

lecture 19

Recap lecture 18

Discussed general motion in Schwarzschild spacetime. Derived conservation of angular momentum. Key tosolving for general motion.

Effective potential

Rewrite (dimensionless) r-equation

dr dτ

2

ǫ 2

V

r

2

where

V

r

2 r

l 2 r 2

Called effective potential. Different regimes; if

l

small monotonic inward moving

particle is drawn inevitably to

r

. For larger

l

other

scenarios possible.

Turning points

To understand what can happen consider theintersection of the curve

y

V

r

with

y

ǫ

Radial velocity given by difference in squares of twocurves. Motion confined to regions where

ǫ > V

When

ǫ

V v

. In general place where

sign

of

dr^ dt

changes. Maximum or minimum in distance. In general motion will be such as have

r

oscillate

between these limits elliptical orbit (with precession). If

ǫ

large – no intersections - straight capture.

If tune

ǫ

so just one intersection – circular orbit.

Relativity and Cosmology – p. 5

Initial conditions

Determining

L

and

E

. Imagine a shell observer

launching a satellite with a certain speed perpendicularto the radial direction. Can we use this info to determine L

and

E

and hence predict the motion?

E/mc 2

A

dt dτ

A

dt dt shell dt shell dτ

This is then

E/mc 2

AA

− 1 2 γ shell

. Thus,

E/mc 2

A

1 2 γ shell

Similarly,

L/mc

r 0 dt shell dτ

r 0 dφ cdt shell

. Thus

L/m

r 0 γ shell v shell

This can be trivially generalized to arbitrary initialdirections of motion.