Conserved Quantities & Angular Momentum in Schwarzschild Spacetime: Relativity & Cosmology, Study notes of Physics

Download Conserved Quantities & Angular Momentum in Schwarzschild Spacetime: Relativity & Cosmology and more Study notes Physics in PDF only on Docsity!

Relativity and Cosmology

lecture 18

Recap lecture 17

Discussed radial motion of massive and masslessbodies in Schwarschild spacetime Used several frames; global “far away” frame, local shellobserver and freely falling observer. Locally laws of SR apply - light travels at speed

c

massive bodies with

v < c

, acceleration, energy follows

Newton for small

v

r > r S

Globally things different. For BH - event horizon,singularity What about more general motion? orbits?

Angular momentum a la Newton^ For linear motion we know that in absence of externalforces acceleration is zero and hence

momentum

p

mv

is constant.

If switch on a force

F

this is not true. However, if force is

purely radial

F

x, y, z

F

r

there is still a conserved

quantity – angular momentum

L

mvr

mr 2 dθ^ dt

Why? Take cross product of

F

m dv^ dt

with

r

F

×

r

Hence

m d ( v × r ) dt

Thus

mv

×

r = constant

. In polar coords

v

r dθ dt

Relativity and Cosmology – p. 4

Example

For simple circular motion with radius

a

L

ma 2 ω

where

ω

dθ^ dt

must

be constant if radius

a

is constant.

Derivation

Consider a motion comprising two sections A and Bdelineated by three events with coordinates

r

r,

r, φ

(the A section with mean

r-coordinate

r A

and elapsed proper time

τ A

r, φ

r

r,

(the B section with mean

r-coordinate

r B

and elapsed proper time

τ B

The final angle

is considered fixed and only the

intermediate angle

φ

will be varied.

continued ...

Setting

dτ dφ

and using metric we find

r (^2) A φ τ A

r 2 B

φ τ B

Thus we see that

r 2 dφ dτ = constant

and can be identified

with a (conserved) angular momentum (per unit mass)for motion in a spherically symmetric spacetime. Notice similarity to derivation of energy ...

General motion

Starting from some initial position

r, φ

and the

constants

L

and

E

we can imagine computing changes

in

t

and

φ

using the equations

t

E/mc 2

GM/c 2 r

τ ∆ φ

L/m r 2

τ

Using the form of the Schwarzschild metric leads to anexpression for the change in r-coordinate: ∆

r

[

E/mc 2

2

GM c 2 r

L/m cr

2

)]

1 2

τ c Relativity and Cosmology – p. 10

Comments

Choose small

τ

• use equations to go from ( r, φ

r

r, φ

φ

Use these as new

r, φ

and do again. Keep iterating.

General path

r

τ

, φ

τ

Computer simulation ... Setting

r ′^

r/r S

τ ′

τ c/r S

ǫ

E/mc 2 l

L/

mcr S

find

r ′^

[

ǫ 2

(^1) r ′^

l r ′^2

]

1 2

τ ′ ∆ φ

l r ′^2

τ ′