Group ResearchRelativistic Motions around a Black Hole

2010 KIAS-SNU Physics Winter CampDate : February 7, 2010Talk : 주부경, 조창우, 김은찬, 서윤지, 고성문, 노대호

Table

• What is space-time ?

• How a particle moves?

- As a geometry

• The black-hole

• ?

What is space-time ?Space + Time (additional dim)

성기오빠!한화리조트

215호에서 만나

215호? 알았어 !!

< 2134ft, N37, E128 >

한화리조트 215호

< t, 2134ft, N37, E128 >

Additional Dimension

t = t성기 연아 t t성기 연아≠

How a particle moves?As a Geometry

Action - Euclidian Space

2 2 2

2 2 2

ds dx dy dz

dx dy dz dtdt dt dt

= + +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∫ ∫

∫

Action - Euclidian Space

• Euler – Lagrange Equation

0d L Ldt xx

•

⎛ ⎞∂ ∂− =⎜ ⎟

⎜ ⎟ ∂∂⎝ ⎠

0v•

→ =

Action – In special relativity

2 2 2 2 2 2d dt dx dy dz dSτ− = − + + + =

2d dS dx dxμ νμντ τ η= = − = −∫ ∫ ∫

1 0 0 00 1 0 00 0 1 00 0 0 1

μνη

−⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

dx dx d Ldd d

μ ν

μνη σ σσ σ

→ − =∫ ∫

Action - In special relativity

• Euler – Lagrange Equation

0d L Ld xx

μμσ •

⎛ ⎞∂ ∂⎜ ⎟ − =⎜ ⎟ ∂∂⎝ ⎠

⇒

Action – In general relativity

2d dS g dx dxμ νμντ τ= = − = −∫ ∫ ∫

( )?gμν =dx dxg d Ldd d

μ ν

μν σ σσ σ

→ − =∫ ∫

The black holeExtremely curved space-time

Schwarzschild metric

22 2 2 2 2 2 2

2

2

2(1 ) ( sin )21

GM drds c dt r d dGMc rc r

θ θ ϕ= − − + + +−

Q=0, S=0, Massive

Constant of motion

2

2 2 2

2 2 22

2 3

2(1 )( )

( )

1 ( ) ( ) ( )

1 1 ( )2 2 2

ttt

t rtt rr

M dte u g ur d

dl u g u rd

u u g u g u g u

e dr M l Mlm d r r r

φφφ

φφφ

ξτ

φητ

ετ

= − ⋅ = − = −

= ⋅ = =

⋅ = − = + +

−→ = = − + −

Radial motion

2

2 2

0, 1

10 ( )2

1 ( ) ( )t rtt rr

l e

dr Md r

u u g u g u

τ

= =

= −

⋅ = − = +

Near the horizon

12(1 )

2 2(1 )

dt Md rdr M Mdr dt

d ddt r r

τ

τ τ

−= −

= = − −

2dr Md rτ

= −

Event horizon

Eddington-Finkelstein coordinates

2 lo g 12

rt r MM

υ= − − −

2 2 2 2 2 22(1 ) 2 ( sin )Mds d d dr r d dr

υ υ θ θ φ= − − + + +

22(1 ) 2 0M d d drr

υ υ− − + =

( )const ingoing radial light raysυ =

2(1 ) 2 0M d drr

υ− − + =

2( 2 log 1)2rr M constM

υ − + − =

•

1/23/2 1/2

* 1/2

2 ( 2 ) 12 [ ( ) 2( ) log ]3 2 2 ( 2 ) 1

r r r Mt t MM M r M

+= + − − +

−

22 2 2 2 2 2 2

2(1 ) ( sin )

(1 )

M drds dt r d dMrr

θ θ ϕ= − − + + +−

Reissner–Nordström metric

Q≠0, S=0, Massive

Eddington again…

2~2

2~ ~2 2

~ ~

2

22

~

ln( )

1

(1 ) 2 (1 ) 0

1 , ( ) ( )

111

Mt t M r Mr M

h f

ds h d t hd t dr h dr

t r const d t drM dr dtdt dr Mr M dr fr

r

d t hdr hdtdr f

= + − −−

≡ −

= − − + + + =

+ = → = −

→ = − = − = −− −

+=

−

=

We need elevator

Black Hole

Reference

• Wikipedia– Key Word

Black hole, Action Principle, General Relativity, Metric, Lagrangian, etc

• Google– Key Word

Black hole & charged, space-time

• Book– Gravity (J.B.Hartle)– Introducing Einstein`s Relativity (Ray D’Inverno)