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 t t ≠ 성기 연아 성기 연아

How a particle moves? As a Geometry

Action - Euclidian Space ∫ ∫ = + + 2 2 2 ds dx dy dz 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ dx dy dz ∫ = + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ dt ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ dt dt dt

Action - Euclidian Space • Euler – Lagrange Equation ⎛ ⎞ ∂ ∂ d L L − = ⎜ ⎟ 0 ⎜ ⎟ • ∂ dt x ∂ ⎝ ⎠ x • → = v 0

Action – In special relativity − τ = − + + + = 2 2 2 2 2 2 d dt dx dy dz dS ∫ ∫ ∫ μ ν τ = τ = − = − η 2 d dS dx dx μν − ⎛ ⎞ 1 0 0 0 μ ν ⎜ ⎟ dx dx d ∫ ∫ → − η σ = σ 0 1 0 0 ⎜ ⎟ Ld η = ⎜ μν σ σ μν ⎟ 0 0 1 0 d d ⎜ ⎟ ⎝ ⎠ 0 0 0 1

Action - In special relativity • Euler – Lagrange Equation ⇒ ⎛ ⎞ ∂ ∂ d L L ⎜ ⎟ − = 0 ⎜ • ⎟ μ σ ∂ d x μ ⎝ ∂ ⎠ x

Action – In general relativity ∫ ∫ ∫ μ ν τ = τ = − = − 2 d dS g dx dx μν ( ) μ ν g μν = dx dx ∫ ∫ → − σ = σ ? g d Ld μν σ σ d d

The black hole Extremely curved space-time

Schwarzschild metric 2 2 GM dr = − − + + θ + θ ϕ 2 2 2 2 2 2 2 ds (1 ) c dt r ( d sin d ) 2 2 GM c r − 1 2 c r Q=0, S=0, Massive

Constant of motion 2 M dt = − ⋅ ξ = − = − t e u g u (1 )( ) τ tt r d φ d φ = η ⋅ = = 2 l u g u r ( ) φφ τ d φ ⋅ = − = + + t 2 r 2 2 u u 1 g ( u ) g ( u ) g ( u ) φφ tt rr ε − 2 2 2 e 1 1 ( dr M l Ml → = = − + − 2 ) τ 2 3 m 2 2 d r 2 r r

Radial motion = = l 0, e 1 1 dr M = − 2 0 ( ) τ 2 d r ⋅ = − = + t 2 r 2 u u 1 g ( u ) g ( u ) tt rr

Near the horizon dt 2 M − = − 1 (1 ) τ d r dr 2 M τ = − dr dr dt 2 M 2 M d r = = − − (1 ) τ τ d d dt r r Event horizon

Eddington-Finkelstein coordinates r = υ − − − t r 2 M lo g 1 2 M 2 M = − − υ + υ + θ + θ φ 2 2 2 2 2 2 ds (1 ) d 2 d dr r ( d sin d ) r 2 M d − − υ + υ = 2 (1 ) 2 d dr 0 r υ = const ( ingoing radial light rays ) 2 M d − − υ + = (1 ) 2 dr 0 r r υ − + − = 2( r 2 M log 1) const 2 M

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

Reissner–Nordström metric 2 M dr = − − + + θ + θ ϕ 2 2 2 2 2 2 2 ds (1 ) dt r ( d sin d ) M r − 2 (1 ) r Q ≠ 0, S=0, Massive

Eddington again… 2 ~ M = + − − 2 t t M ln( r M ) − r M ≡ − h 1 f 2 ~ ~ = − − + + + = 2 2 ds (1 h d t ) 2 hd t dr (1 h dr ) 0 ~ ~ + = → = − t r const d t dr 2 M dr dt 1 → = − = − = − dt dr , − 2 M ( r M ) dr f − 2 ( r ) r ~ + d t 1 h = − dr 1 h dt 1 = 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)