Group Research Relativistic Motions around a Black Hole 2010 - - PowerPoint PPT Presentation

Group Research

Relativistic Motions around a Black Hole

2010 KIAS-SNU Physics Winter Camp Date : February 7, 2010 Talk : 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 = t

성기 연아

t t

성기 연아

≠

How a particle moves?

As a Geometry

Action - Euclidian Space

2 2 2 2 2 2

ds dx dy dz dx dy dz dt dt dt dt = + + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∫ ∫ ∫

Action - Euclidian Space

• Euler – Lagrange Equation

d L L dt x x

• ⎛

⎞ ∂ ∂ − = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠

v

• →

=

Action – In special relativity

2 2 2 2 2 2

d dt dx dy dz dS τ − = − + + + =

2

d dS dx dx

μ ν μν

τ τ η = = − = −

∫ ∫ ∫

1 1 1 1

μν

η − ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

dx dx d Ld d d

μ ν μν

η σ σ σ σ → − =

∫ ∫

Action - In special relativity

• Euler – Lagrange Equation

d L L d x x

μ μ

σ

• ⎛

⎞ ∂ ∂ ⎜ ⎟ − = ⎜ ⎟ ∂ ∂ ⎝ ⎠

⇒

Action – In general relativity

2

d dS g dx dx

μ ν μν

τ τ = = − = −

∫ ∫ ∫

( )

? gμν =

dx dx g d Ld d d

μ ν μν

σ σ σ σ → − =

∫ ∫

The black hole

Extremely curved space-time

Schwarzschild metric

2 2 2 2 2 2 2 2 2 2

2 (1 ) ( sin ) 2 1 GM dr ds c dt r d d GM c r c r θ θ ϕ = − − + + + −

Q=0, S=0, Massive

Constant of motion

2 2 2 2 2 2 2 2 2 3

2 (1 )( ) ( ) 1 ( ) ( ) ( ) 1 1 ( ) 2 2 2

t tt t r tt rr

M dt e u g u r d d l u g u r d u u g u g u g u e dr M l Ml m d r r r

φ φφ φ φφ

ξ τ φ η τ ε τ = − ⋅ = − = − = ⋅ = = ⋅ = − = + + − → = = − + −

Radial motion

2 2 2

0, 1 1 ( ) 2 1 ( ) ( )

t r tt rr

l e dr M d r u u g u g u τ = = = − ⋅ = − = +

Near the horizon

1

2 (1 ) 2 2 (1 ) dt M d r dr M M dr dt d d dt r r τ τ τ

−

= − = = − −

2 dr M d r τ = −

Event horizon

Eddington-Finkelstein coordinates

2 lo g 1 2 r t r M M υ = − − −

2 2 2 2 2 2

2 (1 ) 2 ( sin ) M ds d d dr r d d r υ υ θ θ φ = − − + + +

2

2 (1 ) 2 M d d dr r υ υ − − + =

( ) const ingoing radial light rays υ = 2 (1 ) 2 M d dr r υ − − + =

2( 2 log 1) 2 r r M const M υ − + − =

• 1/2

3/2 1/2 * 1/2

2 ( 2 ) 1 2 [ ( ) 2( ) log ] 3 2 2 ( 2 ) 1 r r r M t t M M M r M + = + − − + −

2 2 2 2 2 2 2 2 2

(1 ) ( sin ) (1 ) M dr ds dt r d d M r r θ θ ϕ = − − + + + −

Reissner–Nordström metric Q≠0, S=0, Massive

Eddington again…

2 ~ 2 2 ~ ~ 2 2 ~ ~ 2 2 2 ~

ln( ) 1 (1 ) 2 (1 ) 1 , ( ) ( ) 1 1 1 M t t M r M r M h f ds h d t hd t dr h dr t r const d t dr M dr dt dt dr M r M dr f r r d t h dr h dt dr 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)