solutions of general relativity · solutions of general relativity with singularities 9.5.03 p....

Black Holes

Solutions of General Relativity with Singularities 9.5.03 p.

Solutions of General Relativity with Singularities

by Mathias Garny

Black Holes

Solutions of GR with Singularities

• Introduction to the types of Black Holes• The static, spherically symmetric Black Hole

(Schwarzschild - Black Hole)• The stationary, rotating Black Hole

(Kerr – Black Hole)• Some dynamical phenomena involving Black Holes

Black Holes

The types of Black Holes

three stationary types:

• Mass M• not rotating• uncharged

• Mass M• Ang. mom. J• uncharged

• Mass M• Ang. mom. J• Charge Q

Schwarzschild Kerr Kerr-Newman

astrophysically irrelevantmathematically analogous

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End-state-solutions

• gravitational collapse• dynamic evolution• asymptotic approach of stationary solutions⇒ astrophysically relevantTypes

gravitational collapsestationary

Black Hole

Black Holes

Schwarzschild Black Holes

• Search for solution – of the Einstein-Equation

– in vacuum – spherically symmetric

• Schwarzschild-metric (G = 1, c = 1, M¯ ≅ 1.5km)

Black Holes

Properties of the SS-metric

asymptotically flat for r → ∞ flat Minkowski-space for M = 0

spherically symmetric

singular metric elements for

r = 2Mr = 0

singular metric

element forθ = 0, π θ = 0

θ = πcoordinate singularity

singularities of spacetime ?

Black Holes

Free fall into a black hole

all geodesics end at r = 0

geodesics „well-behaved“ at r = 2M

test mass reaches r = 0 in a finite amount of time

Black Holes

Free fall into a black hole

film A: recorded in the spaceshipfilm B: recorded on earth

The astronomers salute

Black Holes

Spaghettification

singularity

Guess:Tidal forces• finite at r = 2M• infinite at r = 0

r = 2M coordinate singularity

r = 0 real phys. singularity

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„Intrinsic“ singularity of spacetime

• Exact math. definition is not easy• Some simple criteria:

– Endpoint of geodesics– Curvature →∞

If any of scalars →∞ ⇒ SINGULARITY

Problem:components of

Riemann Tensorcoordinate-dependant

Solution:Riemann Scalars

gµνRµν

RµναβRµναβ

Rµναβ gαβRµνδεgδε

Black Holes

Singularities of SS-solution

r = 2M r = 0

all curvature-scalars stay finite

intrinsic singularitycoordinate singularity

all geodesics end at r = 0geodesics well-behaved

Black Holes

Finkelstein-coordinates

• Aim: Find coordinates regular at r = 2M• One Possibility: Use proper time of infalling photons

time →

well defined for all values 0 < r < ∞

Black Holes

SS → Finkelstein

cyan: singularity at r = 0red: horizon at r = 2Myellow: worldlines of infalling lightraysochre: worldlines of outgoing lightrayspurple: lines of constant radiusblue: lines of constant SS-time

Black Holes

SS → Finkelstein

geodesics seem to be irregular at r = 2M

it is obvious that geodesics are regular

at r = 2M

Black Holes

Causal structure & lightcones

Trajectories of any objects only inside lightcones !(otherwise velocity exeeds c)

• flat spacetime: All lightcones have identical slopeand orientation

• curved spacetime: slope and orientation can change from point to point

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Causal structure of a black hole

For r < 2M onlyinward-directed

motion Lightcones tilt more and more inwardTypes

„Top-View“

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Horizon

light emitted at r < 2M never gets out

light emitted at r = 2M gets infinitly redshifted

Horizon at r =2 M

whatever forces may act: nothing can avoid the crash

no escape possible

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Gravitational collapse

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Kruskal-coordinates – Why?

• gµν neither 0 nor ∞ at r = 2M• infalling and outgoing lightrays travel at ±45°⇒ causal structure easy to see

• analytic expansion of SS-coordinatesTypes

Kruskal coordinates

Schwarzschild coordinates

Black Holes

Finkelstein → Kruskal

cyan: singularity at r = 0red: horizon at r = 2Myellow: worldlines of infalling lightraysochre: worldlines of outgoing lightrayspurple: lines of constant radiusblue: lines of constant SS-time

Black Holes

Kruskal - coordinates

Black Holes

Kruskal - coordinates

horizon

singularity r = 0

r < 2Mr > 2M

our universe

Black Holes

Kruskal – causal structure

singularity r = 0

r > 2Mour

universe

lightflash emitted outside horizon:outgoing ray r →∞ingoing ray r → 0

lightflash emitted inside horizon:„outgoing“ ray r → 0ingoing ray r → 0

Black Holes

Kruskal – analytic expansion

e-Types

Time-reversed universe should be also a solution⇒ supplement the Kruskal-diagram

Black Holes

• When a star collapses, no white hole is produced• White holes violate 2nd law of thermodynamics• White holes are unstable configurations

2 asym. flat regions not causally connected

white hole

Black Holes

• When a star collapses, no white hole is produced• White holes violate 2nd law of thermodynamics• White holes are unstable configurations

physically unrealisticTypes

Black Holes

Kruskal – realistic scenario

physically realisticTypes

DynsThe metric inside the star is not a vacuum solution⇒ Kruskal only outside the star

Black Holes

Significance of singularities in GR

Newton Einstein

Solve Rµν = 0 ⇒ SS-solution

• singularity at r = 0• unique spherical symm.

vacuum solution (Birkhoff-theorem)

Solve � φ = 0 ⇒• singularity at r = 0• unique spherical symm.

vacuum solutionTypes

potentialspace

curvaturespace

Difference between Newton-singularity and Einstein-singularity ?

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Significance of singularities in GR II

∼ 1/r ∼ 1/r6Types

potential of a point-charge→ no point-charge in class. phy.

Black Holes

∼ 1/r

∼ 1/r ∼ 1/r6Types

potential of a point-charge→ no point-charge in class. phy.⇒ replace by small homogenous

sphere of Radius R⇒ stable solution for all R⇒ no singularity

Black Holes

∼ 1/r

∼ 1/r ∼ 1/r6

sphere of Radius R⇒ stable solution for all R⇒

connect homogenous sphere⇒ stable solution only for R > 2M

no singularity22

Black Holes

∼ 1/r

∼ 1/r ∼ 1/r6

R < 2M

sphere of Radius R⇒ stable solution for all R⇒

connect homogenous sphere⇒ stable solution only for R > 2M

For R < 2M no stable solution⇒ collapse to a point

(as long as GR is valid)⇒ no singularity real phys. singularity

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Summary of SSBH

• SS-solution – unique sph. symm. vacuum solution of GR – singularity at r = 0– one parameter M

• mass M compressed in a volume with radius

⇒ matter must collapse to a point

• Horizon

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Rotating Kerr – Black Hole

• axially symmetric, stationary solution (Kerr, 1963)

• stationary spacetime ( , dtdφ - terms)• axial φ - symmetry ( )• discrete symmetries

• for a = 0 ⇒ SSBH• asymptotically flat for r →∞⇒ describes rotating BH of mass M and ang. mom. J =aM

t → –t, φ→ –φ ↔ rotation t → –t, a→ –a ↔ ang. momentum ∼ a

Black Holes

Horizon and static limitSSBH KBH

ring singularity

static limit

horizon

axis of rotation

ergosphere

spacespace

horizon

singularity

Black Holes

Horizon and static limitSSBH KBH

ring singularity

static limit

horizon

axis of rotation

ergosphere

spacespace

horizon

singularity

Dynsspace

horizon static limit

ergosphere ring singularity

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Frame dragging II

trajectories of infalling photons in a not rotating black hole

Black Holes

Frame dragging II

trajectories of infalling photons in a rotating black hole

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Frame dragging II

trajectories of infalling photons in a faster rotating black hole

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Frame dragging III

trajectories of infalling photons in a faster rotating black hole

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Extracting energy from a BH

• SSBH:

• KBH:

• Penrose process: Deceleration of rotating BH

not extractable by classical means

Mirr = irreducable mass < M

extractable by classical means

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Penrose process

Nakdpower plant based on the penrose-process

Projectile disintegrates into 2 parts:• 1 fragment decelerates BH • 1 fragment escapes and gains

energy29

Black Holes

No-hair theorem

gravitational collapse formation of a Kerr-BH only described by M, J

gravitational collapse

What happens?

No-hair theorem:After the gravitational collapse of any arbitrary mass distribution the endstate is described by the Schwarzschild, Kerr or Kerr-Newman solution with only 3 free parameters M, J, Q

Black Holes

Conclusions from no-hair-theorem

• The horizon has always the shape of a sphere(Because thats true for SSBH, KBH, KNBH)

magnetized star implodes

BH it forms has no magnetic field

square star implodes

BH it forms is round

star with mountain implodes

BH it forms has nomountain

Black Holes

Gravitational waves

• BH keeps only 0th multipole moment of initial mass distribution

• all higher multipole moments are radiated away by gravitational waves

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Summary

• Only three stationary vacuum solutions M, J, Q

• singularity encapsulated in a horizon

• inside the horizon, the distance to the singularity is getting smaller for any object „as inevitably as time goes on“

• any gravitational collapse with generic initial data achieves one of the three stationary solutions

Black Holes

Prospect of naked singularities

• KBH: horizon at ⇒ horizon vanishes if a/M > 1 ( (a/M)¯ = 0.185 )⇒ naked singularity⇒ orbits from singularity to ∞

• „cosmic censorship“:

„All singularities are enclosed in a horizon and therefore causally encapsulated“ (Penrose)

Black Holes

Prospect of naked singularities

parameter space of initial mass distributions

below critical surface implosion

above critical surface explosion

on critical surface formation of naked

singularity

solutions of general relativity · solutions of general relativity with singularities 9.5.03 p....

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