G. Srinivasan

Stability of Stars

• For a star to be stable, , the ratio of specific

heats, should be greater than 4/3.

• If < 4/3, the compressibility would be negative.

• Since the pressure varies over the star, the relevant parameter is the pressure-averaged value of over the star.

2

0

2

0

ln

ln

Pr

Pr

ad

s

R

R

P

dr

dr

M, R M’, R’

The two configurations are said to be “homologous” if the Homologous mass shells are located at homologous points.

' '

' '

r r m m

R R M M

Pressure = weight of unit cylinder

2

' '

4

M

m

Gm dmP

r

• Compress sphere adiabatically and homologously.

• Will the star still be in hydrostatic equilibrium?

• RHS varies as

• LHS varies as

• If ad > 4/3, LHS increases STRONGER with compression then the weight on RHS. Star will BOUNCE BACK!

4'R

R

3' 'R

R

1

2

4/3

6 8 10 12 14

log

Stability of Matter

1

2

4/3

6 8 10 12 14

log

Stable

Unstable

Stability of Matter

• In the 1960s, Chandrasekhar made a definitive study of stability of stars in GTR. As is to be expected, there is an important correction to Newtonian criterion for stability.

• In the post-Newtonian limit,

• Gravity is stronger in GTR and can render a star UNSTABLE to radial pulsation when they are nearly Newtonian.

Stability of Stars in General Relativity

2

4

3crit

GM

Rc is of order unity

Stable White Dwarfs

Stable neutron stars

OV: Oppenheimer – Volkoff Equation of State

0.2

0.6

1.0

1.4

M

M

107 109 1011 1013 1015 1017

Central Density ( g cm-3 )

Mmax

1

4/3

6 8 10 12 14

log

Stable

Unstable

Stability of Stars in GTR

2

4

3crit

GM

Rc

2

4

3crit

GM

Rc 4

3

g

crit

RR

• For example, a white dwarf becomes unstable in GTR when its radius is ~ 5000 Rg . (Chandrasekhar and Tooper, 1964).

• The early models for Quasars invoked “Supermassive stars”. These would be supported by radiation pressure for which = 4/3.

• Chandrasekhar pointed out (1964) that such stars would be unstable in GTR. (Fowler & Feynman).

General Relativistic Instability

• Oppenheimer and Volkoff (1938) derived the maximum mass of neutron stars using a “turning point criterion”.

• After Chandra’s paper of 1964, Misner & Zapolsky found that the neutron star was unstable to radial pulsations precisely at the extremum.

• Similarly, the upper mass limit of a white dwarf coincides with the point of instability to collapse.

• But this coincidence arises only because the EOS used for equilibrium analysis is the same one used to model radial pulsations.

• The above digression sets the stage for our discussion of black holes of general relativity.

ij ij ij

4

1 8 GR g R T

2 c

“Scarcely anyone who fully understands

this theory can escape from its magic.”

25 November 1915

In formulating the laws of gravitation, one may ask three

questions.

1) When can we say that there is no gravitational field?

2) What are the equations which determine the gravitational

field in vacuum, outside material bodies?

3) What are the equations that obtain in regions of space

where matter is present?

Laws of gravitation

2U 0

2U 4 G

Newtonian Gravity

• Gravitational field is described by a scalar potential U.

(1) In the absence of gravitational field,

(2) In regions of space outside matter,

(3) In regions where matter density is ,

U 0

Einstein’s generalizations

• In the absence of gravity, particles experience no acceleration.

• In a curvilinear coordinate system particles will experience “inertial acceleration” due to “fictitious forces”.

• If there is no gravity, we should be able to find a transformation such that the Christoffel symbols vanish identically.

2 i

2

d x0.

ds

j

2 ki ji

2 k

dx dx

ds

d x

s0

ds d

Einstein’s Zeroth Law of Gravity

• According to Riemannian geometry, the condition that

the Christoffel symbols vanish identically is that the

Riemann-Christoffel tensor vanishes.

• This is a necessary and sufficient condition.

• This is Einstein’s zeroth law of gravity.

i

jklR 0. U 0

In regions where no matter is present

• Einstein guessed that the generalization of Laplace’s equation is the vanishing of the Ricci tensor.

• The above equation is, like Laplace’s eqn., a second order differential equation. There are 10 equations for the 10 coefficients of the metric tensor

• The above equation reduces to Laplace’s equation in the limit

l

jkl jkR R 0. 2U 0

ijg

c

• In Newtonian theory one relates the gravitational potential to the material density .

• In relativity, density alone will not suffice to specify the distribution of matter and its motions.

• In special relativity one introduces the energy-momentum tensor, Tij, with ten components. The (0,0) component of this tensor is c2.

• One would therefore be inclined to relate the Ricci tensor (the generalization of ) to the energy-momentum tensor Tij .

In regions where matter is present

2U 4 G

ij ijR T 2U 4 G

?

2U

Conservation Laws

• In special relativity, conservation laws follow from the relation that the ‘divergence of the energy-momentum tensor vanishes’.

• In a non-Euclidean geometry, this should be replaced by

• The “guess” will not work because

ijdiv(T ) 0

covariant divergence ijT 0

ij ijR T

Covariant divergence of ijR 0

Einstein’s field equations

• However,

• Therefore, the generalization of Poisson’s equation is

• The coupling constant has to be determined.

Covariant divergence of ij ij1R g R

20

ij ij ij1R g R T .

2

2U 4 G

Newtonian limit

• In the limit of ‘weak gravity’, the metric is given by,

• Using this metric, one can find the limiting form of Einstein’s equations. In the limit Einstein’s equations reduce to

2 2 2 2 2 2

2

2U1

cds c dt dx dy dz

c

2 41U c

2

2U 4 G 4

8 G

c

ij ij ij1R g R T .

2

Einstein’s field equations

• These equations determine both the geometry (LHS) and the states of motion (RHS).

• The conservation laws are built into them.

ij ij ij

4

1 8 GR g R T

2 c

Karl Schwarzschild

2 i j

ijds g dx dx

Within a month of the announcement of Einstein’s discovery, Karl Schwarzschild found an EXACT solution to Einstein’s equations. Einstein was astonished by this! Schwarzschild’s solution described the curvature of space and time near a spherically symmetric massive body.

Schwarzschild metric

• Schwarzschild’s solution of Einstein’s equations for the gravitational field describes the curvature of space and time near a spherically symmetric massive body.

2 2 2 2 2 2 2 2

2

2

2 11 sin

21

GMds c dt dr r d d

GMrc

rc

“warpage of time” “warpage of space”

• This describes not only the “curvature of space” but

also the “warpage of time” near the star. This is just

dilatation of time due to gravity.

• This leads to gravitational redshift of light emitted

from the stars surface. Near the star, time flows more

slowly than far away. Therefore, the frequency of light

will be lowered by the same amount.

• Radiation from the star will be shifted to longer

wavelengths.

The Schwarzschild Radius

• Exact result in General Relativity,

• At the critical radius Rg = 2GM / c2 , the time dilatation

becomes infinite. Time does not flow at all!

• As seen by an outside observer, frequency goes to zero.

The wavelength of radiation becomes infinite.

• The star will appear BLACK.

1

2

2

2 /1

GM c

R

• The surface of the black hole is known as the EVENT HORIZON

• It is a semi permeable surface. Things can only go into the horizon. Nothing can come out.

• Inside the event horizon, there are no allowed trajectories with a constant distance from the centre.

• The infalling matter cannot arrest its collapse.

• The ONLY allowed trajectories are those along which all matter will fall to the centre of the black hole.

Schwarzschild metric

2 2 2 2 2 2 2 2

2

2

2 11 sin

21

GMds c dt dr r d d

GMrc

rc

Circumference = 2r. Distance between r1 and r2 is

2

12 1

1

r

rg

drr r

R

r

1g

R

r

is the curvature factor.

Approximate metric at large distances from the centre.

2 2 2 2 2

0 2

2GMds ds dr c dt

rc

Small correction

At large distance from the masses, every field appears centrally symmetric. Therefore, the above metric is the metric at large distances from ANY system of bodies. The above metric can also be written as

2 2 2 2 2 2

2 2

2 21 1

GM GMds c dt dx dy dz

rc rc

Effective potential in Newtonian theory

2 21v ;

2

dL m

GMmE m

rr

dt

2 2 2 22

2v

dr d dr Lr

dt dt dt mr

2 2

2

1

2 2

dr E GM L

dt m r mr

V(r)

Effective potential in Newtonian theory

Circular orbit Elliptical orbit

Effective potential in Einstein’s theory

Constants of motion: 2 const. 1

grd L E dt

rd m m r d

2 2 2 2 2 2 211

1

g

g

rc d c dt dr r d

rr

r

Substitute for dt and d from the constants of motion. Simplify.

1

2 22

2 21 1

grE L

dr cdm r m r

2 2 2

2 2 2

11 1

grdr E L

c d m r m r

Effective potential in Einstein’s theory

2 2 2

2 2 2

11 1

grdr E L

c d m r m r

2 2E V

m m

1

2 2

2 21 1

grV L

m r m r

2

2 2

Newton2

V GM L

m r m r

Effective potential in Einstein’s theory

1

2 2

2 21 1

grV L

m r m r

2

2 2

Newton2

V GM L

m r m r

Einstein Newton

ORBITS in Einstein’s theory

g

r

R

3

r =

Rg

Last stable circular orbit

3g

r R

v = 0.5 c

The last “unstable” circular orbit is at a radius of 1.5 Rg . The velocity of the particle will be ‘c’.

2 1

An

gula

r m

om

entu

m

3g

L mcr

g

r

R

3

r =

Rg

Last stable circular orbit

3g

r R

v = 0.5 c

2 1

An

gula

r m

om

entu

m

3g

L mcr28

9E mc

Important new result

In GTR, there exists a minimum radius for stable circular orbits, and a corresponding minimum energy for circular motion.

3g

r R 3g

L mcr 28

9E mc

An observer at infinity will see the particle orbiting with a period T.

1

2 (3 ) 121

1 2 / 31 2

g

g g

g

rd dt

r

r rT

r ccr

Important new result

Orbiting particle emits frequency 0. The received frequency at infinity is given by:

1 1

2 2

0

1 v/c1

1 v/c

gr

r

02

0

2

3

0

2

3

g

r

R

3

r =

Rg

2 1

An

gula

r m

om

entu

m

Gravitational Capture

2

E

mc

1

2g

L

mcr

3g

L

mcr

If L < 2mcrg , then E=1 does NOT intersect the potential curve V(r, L). Therefore there is no turning point. The particle will be gravitationally captured by the body.

The particle is at rest at infinity. E= rest mass energy

Cross Section for Gravitational capture

b

Angular momentum: . L < 2mcrg corresponds to . All particles with impact parameter smaller than this will be captured.

vL m b

2

v

gcr

b

2

2 24

vg

cb r

The capture cross section is

Gravitational Capture

2g

L mcr

2g

L mcr

Gravitational capture is not possible in Newtonian mechanics.

Radial Motion of Particles

• Let the particle be at rest at infinity. E=mc2 is a constant.

• General expression for the energy is

• Since energy is a constant,

• Plug this into Schwarzschild metric (radial motion: d=d=0)

1g

rE dt

m r d

2

2 21

gr

d dtr

1 g g

r rdrc

dt r r

Falling towards a black hole

1

2

2 2

2 21

dr GM GM

dt rc rc

The “velocity” of a particle as seen by distant observer

“Physical time” at a given point r: d 0, as r rg . “Proper radial distance” measured by observer at position r.

1

2

001

gr

d g dt dtr

shell 1

2

1g

drdr

r

r

shell

1

1

g

g

rdr drc

rdt dt r

r

“Physical time” at a given point r: d 0, as r rg . “Proper radial distance” measured by observer at position r.

1

2

001

gr

d g dt dtr

2

1, as

1

g

g

GMa r r

rr

r

Acceleration:

shell 1

2

1g

drdr

r

r

Falling towards a black hole

The velocity as seen by an observer on a stationary infinitely rigid shell

1

2

2 2

2 21

dr GM GM

dt rc rc

The “velocity” of a particle as seen by distant observer

1

2

2

shell

2dr GM

dt rc

Falling towards a black hole

gR 4

gR

0.5 c

c

0

dr

dt

shell

dr

dt

Time to reach Rg from r = r1

• The free fall time can be obtained by integrating dr/dt.

• This will diverge. As seen by the distant observer, the free fall time to Rg is infinite.

• The “proper time interval” for free fall is given by:

1

2

2 2

2 21

dr GM GM

dt rc rc

3 3

2 2

12

3

g

g g

r r r

c r r

Time to reach Rg from r = r1

• The “proper time interval” for free fall is given by:

• This is finite due to two reasons.

• Time is “slowed down” in a gravitational field.

• The “physical velocity” of the free falling clock tends to c, as r tends to rg . There is a consequent “Lorentz contraction”.

3 3

2 2

12

3

g

g g

r r r

c r r

Falling into a black hole

• Let a body, at rest at infinity, fall towards the BH. The time taken to fall from the event horizon (Rg ) to the centre is

as measured by the infalling clock.

2

3

gR

c

gR

Radial motion of photons

• For light, ds = 0. For radial motion, d = d = 0.

• Schwarzschild metric gives us:

• The distant observer would find that light has slowed down!

• But the “physical velocity” is given by

• Locally, Special Relativity must always prevail!

1g

rdrc

dt r

11

00

gdx drc

d dtg

Motion of light around a Schwarzschild BH

Critical “impact parameter”

3 3

2g

R3 3

2g

R

3

2g

R

Inside the horizon

• The Schw. Metric cannot be extended inside the horizon.

• The dt 2 term becomes NEGATIVE, and dr2 term becomes

POSITIVE.

• t-coordinate becomes SPACELIKE. r-cord becomes TIMELIKE.

• r = constant line cannot serve as the ‘radial coordinate’.

• The only TIMELIKE trajectories are those with decreasing r.

2

2GM1 0

rc

The Lemaitre coordinates (1933)

21

33

g

3r R c r

2

Choose co-ords { R, , , φ }

422

32 2 2 2 2 23

g2

3

g

dR 3ds c d (R c ) r (d sin d )

23

(R c )2r

In these coords, there is no singularity at rg :

is timelike and R is spacelike everywhere. The metric is not

stationary; enters explicitly.

g

3r (R c )

2

. .

.

.

.

.

.

At large distances, the point of emission lies at the centre of the expanding spherical wavefront surface.

At short distances from the event horizon, the wavefront surface is displaced by the strong gravity. At precisely the horizon, the spherical wavefront surface touches the horizon INTERNALLY. Inside, the expanding wavefront detaches itself from the point of emission. Light emitted in any direction is pulled towards the central singularity.

R

Light signals: g

1

3

g

rd 1c

dR r2 1

(R c )3 r

+ -

dc 1

dR

dc 1

dR

dc 1

dR

R = co

nstan

t

Roger Penrose