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

Types

SSBH

KBH

Dyns

2

Black Holes


The types of Black Holes

three stationary types:

Types

SSBH

KBH

Dyns

• 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

M M

J

3

Black Holes


End-state-solutions

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

SSBH

KBH

Dyns

gravitational collapsestationary

Black Hole

4

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)

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

5

Black Holes


Properties of the SS-metric

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

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

spherically symmetric

singular metric elements for

r = 2Mr = 0

singular metric

element forθ = 0, π θ = 0

θ = πcoordinate singularity

singularities of spacetime ?

6

Black Holes


Free fall into a black hole

all geodesics end at r = 0

geodesics „well-behaved“ at r = 2M

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

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

7

Black Holes


Free fall into a black hole

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

The astronomers salute

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

7

Black Holes


Spaghettification

singularity

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

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

r = 2M coordinate singularity

r = 0 real phys. singularity

8

Black Holes


„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

9

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

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

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

10

Black Holes


Finkelstein-coordinates

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

Schw

arzs

child

tim

e →

Fink

elst

ein

time →

well defined for all values 0 < r < ∞

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

11

Black Holes


SS → Finkelstein

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

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

12

Black Holes


SS → Finkelstein

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

geodesics seem to be irregular at r = 2M

it is obvious that geodesics are regular

at r = 2M

12

Black Holes


Causal structure & lightcones

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

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

• flat spacetime: All lightcones have identical slopeand orientation

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

13

Black Holes


Causal structure of a black hole

For r < 2M onlyinward-directed

motion Lightcones tilt more and more inwardTypes

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

space

space

„Top-View“

14

Black Holes


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

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

no escape possible

14

Black Holes


Gravitational collapse

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

15

Black Holes


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

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

Kruskal coordinates

Schwarzschild coordinates

16

Black Holes


Finkelstein → Kruskal

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

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

17

Black Holes


Kruskal - coordinates

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

18

Black Holes


Kruskal - coordinates

horizon

horiz

on

singularity r = 0

r < 2Mr > 2M

our universe

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

18

Black Holes


Kruskal – causal structure

horiz

on

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

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

18

Black Holes


Kruskal – analytic expansion

Kru

skal

tim

e →

e-Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

e+

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

19

Black Holes



Kru

skal

tim

e →

• 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

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

19

Black Holes



Kru

skal

tim

e →

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

physically unrealisticTypes

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

19

Black Holes


Kruskal – realistic scenario

Kru

skal

tim

e →

physically realisticTypes

SSBH

Sing

Fink

Krus

Sum

KBH

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

20

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

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

potentialspace

space

curvaturespace

space

Difference between Newton-singularity and Einstein-singularity ?

21

Black Holes


Significance of singularities in GR II

∼ 1/r ∼ 1/r6Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

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

22

Black Holes



∼ 1/r

R

∼ 1/r ∼ 1/r6Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

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

22

Black Holes



∼ 1/r

R

∼ 1/r ∼ 1/r6

R>2M

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns


sphere of Radius R⇒ stable solution for all R⇒

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

no singularity22

Black Holes



∼ 1/r

R

∼ 1/r ∼ 1/r6

R < 2M

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns


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

22

Black Holes


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

Types

SSBH

Sing

Fink

Krus

Sum

KBH

Dyns

23

Black Holes


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

24

Types

SSBH

KBH

Dyns

Black Holes


Horizon and static limitSSBH KBH

ring singularity

static limit

horizon

axis of rotation

ergosphere

spacespace

space

horizon

singularity

Types

SSBH

KBH

Dyns

25

Black Holes


Horizon and static limitSSBH KBH

ring singularity

static limit

horizon

axis of rotation

ergosphere

spacespace

space

horizon

singularity

Types

SSBH

KBH

Dynsspace

space

horizon static limit

ergosphere ring singularity

φ

25

Black Holes


Frame dragging II

Types

SSBH

KBH

Dyns

trajectories of infalling photons in a not rotating black hole

26

Black Holes


Frame dragging II

Types

SSBH

KBH

Dyns

trajectories of infalling photons in a rotating black hole

26

Black Holes


Frame dragging II

Types

SSBH

KBH

Dyns

trajectories of infalling photons in a faster rotating black hole

26

Black Holes


Frame dragging III

Types

SSBH

KBH

Dyns

trajectories of infalling photons in a faster rotating black hole

27

Black Holes


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

28

Types

SSBH

KBH

Dyns

Pen

Hair

Nakd

Black Holes


Penrose process

Types

SSBH

KBH

Dyns

Pen

Hair

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

Types

SSBH

KBH

Dyns

Types

SSBH

KBH

Dyns

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

Pen

Hair

Nakd

30

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

31

Types

SSBH

KBH

Dyns

Pen

Hair

Nakd

Black Holes


Gravitational waves

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

• all higher multipole moments are radiated away by gravitational waves

Types

SSBH

KBH

Dyns

Pen

Hair

Nakd

32

Black Holes


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

Types

SSBH

KBH

Dyns

33

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)

Types

SSBH

KBH

Dyns

Types

SSBH

KBH

Dyns

Pen

Hair

Nakd

34

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

35

Types

SSBH

KBH

Dyns

Pen

Hair

Nakd

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

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