Black Holes, Gravity to the Max

By Dr. Harold Williams

of Montgomery College Planetariumhttp://montgomerycollege.edu/Departments/planet/

Black HolesJust like white dwarfs (Chandrasekhar limit: 1.4 Msun),

there is a mass limit for neutron stars:

Neutron stars can not exist with masses > 3 Msun

We know of no mechanism to halt the collapse of a compact object with > 3 Msun.

It will collapse into a single point – a singularity:

But only at the end of time relative to an outside observer. => A black hole!

Escape VelocityVelocity needed to

escape Earth’s gravity from the surface: vesc

≈ 11.6 km/s.

vesc

Now, gravitational force decreases with distance (~ 1/d2) => Starting out high

above the surface => lower escape velocity.

vesc

vescIf you could compress

Earth to a smaller radius => higher escape velocity

from the surface.

Escape Velocity Equation

• Newtonian gravity

• Ves=√(2GM/R)

• Ves, escape velocity in m/s

• G, Newtonian universal gravitational constant, 6.67259x10-11m3/(kg s2)

• M, mass of object in kg

• R, radius of object in m

The Schwarzschild Radius

=> There is a limiting radius where the escape velocity

reaches the speed of light, c:

Vesc = cRs = 2GM ____ c2

Rs is called the Schwarzschild radius.

G = gravitational constant

M = mass

General Relativity

• Extension of special relativity to accelerations• Free-fall is the “natural” state of motion• Space+time (spacetime) is warped by gravity

Black Holes• John Michell, 1783:

would most massive things be dark?

• Modern view based on general relativity

• Event horizon: point of no return

• Near BH, strong distortions of spacetime

Schwarzschild Radius and Event Horizon

No object can travel faster than the speed of light

We have no way of finding out what’s

happening inside the Schwarzschild radius.

=> nothing (not even light) can escape from inside

the Schwarzschild radius

“Event horizon”

“Black Holes Have No Hair”Matter forming a black hole is losing

almost all of its properties.

black holes are completely determined by 3 quantities:

mass

angular momentum

(electric charge)

The electric charge is most likely near zero

The Gravitational Field of a Black Hole

Distance from central mass

Gra

vita

tion

al

Po

ten

tial

The gravitational potential (and gravitational attraction force) at the Schwarzschild

radius of a black hole becomes infinite.

General Relativity Effects Near Black Holes

An astronaut descending down towards the event horizon of

the black hole will be stretched vertically (tidal effects) and

squeezed laterally unless the black hole is very large like thousands of solar masses.

General Relativity Effects Near Black Holes (II)

Time dilation

Event horizon

Clocks starting at 12:00 at each point.

After 3 hours (for an observer far away

from the black hole): Clocks closer to the black hole run more slowly.

Time dilation becomes infinite at the event horizon.

Observing Black HolesNo light can escape a black hole

=> Black holes can not be observed directly.

If an invisible compact object is part of a binary,

we can estimate its mass from the orbital

period and radial velocity.

Mass > 3 Msun

=> Black hole!

Detecting Black Holes

• Problem: what goes down doesn’t come back up

• Need to detect effect on surrounding stuff Hot gas in accretion disks Orbiting stars Maybe gravitational lensing

Compact object with > 3 Msun must be a

black hole!

Stellar-Mass Black Holes• To be convincing, must

show that invisible thing is more massive than NS

• First example: Cyg X-1• Now 17 clear cases• Still a small number!

Jets of Energy from Compact Objects

Some X-ray binaries show jets perpendicular

to the accretion disk

Model of the X-Ray Binary SS 433

Optical spectrum shows spectral lines from material

in the jet.

Two sets of lines: one blue-shifted, one red-shifted

Line systems shift back and forth across each other due to jet

precession

Black Hole X-Ray Binaries

Strong X-ray sources

Rapidly, erratically variable (with flickering on time scales of less than a second)

Sometimes: Quasi-periodic oscillations (QPOs)

Sometimes: Radio-emitting jets

Accretion disks around black holes

Gamma-Ray Bursts (GRBs)Short (~ a few s), bright bursts of gamma-rays

Later discovered with X-ray and optical afterglows lasting several hours – a few days

GRB of May 10, 1999: 1 day after the GRB 2 days after the GRB

Many have now been associated with host galaxies at large (cosmological) distances.

Probably related to the deaths of very massive (> 25 Msun) stars.

Black-Hole vs. Neutron-Star Binaries

Black Holes: Accreted matter disappears beyond the event horizon without a trace.

Neutron Stars: Accreted matter produces an X-ray flash as it impacts on the

neutron star surface.

Stars at the Galactic Center

Fig. 14-23, p. 301

Fig. 14-23a, p. 301

Gamma-Ray Bursts (GRBs)

Short (~ a few s), bright bursts of gamma-rays

Later discovered with X-ray and optical afterglows lasting several hours – a few days

GRB of May 10, 1999: 1 day after the GRB 2 days after the GRB

Many have now been associated with host galaxies at large (cosmological) distances.

Probably related to the deaths of very massive (> 25 Msun) stars.

Black Holes and their Galaxies

Gravitational Waves

• Back to rubber sheet• Moving objects

produce ripples in spacetime

• Close binary BH or NS are examples

• Very weak!

Gravitational Wave Detectors

Numerical Relativity

• For colliding BH, equations can’t be solved analytically Coupled, nonlinear, second-order PDE!

• Even numerically, extremely challenging Major breakthroughs in 2006-2007

• Now many groups have stable, accurate codes

• Can compute waveforms and even kicks

Colliding BH on a Computer: From NASA/Goddard Group

What Lies Ahead

• Numerical relativity continues to develop Compare with post-Newtonian analyses

• Initial LIGO is complete and taking data

• Detections expected with next generation, in less than a decade

• In space: LISA, focusing on bigger BH Assembly of structure in early universe?

Mass – Inertial vs. Gravitational

• Mass has a gravitational attraction for other masses

• Mass has an inertial property that resists acceleration Fi = mi a

• The value of G was chosen to make the values of mg and mi equal

'

2g g

g

m mF G

r

Einstein’s Reasoning Concerning Mass

• That mg and mi were directly proportional was evidence for a basic connection between them

• No mechanical experiment could distinguish between the two

• He extended the idea to no experiment of any type could distinguish the two masses

Postulates of General Relativity

• All laws of nature must have the same form for observers in any frame of reference, whether accelerated or not

• In the vicinity of any given point, a gravitational field is equivalent to an accelerated frame of reference without a gravitational field– This is the principle of equivalence

Implications of General Relativity

• Gravitational mass and inertial mass are not just proportional, but completely equivalent

• A clock in the presence of gravity runs more slowly than one where gravity is negligible

• The frequencies of radiation emitted by atoms in a strong gravitational field are shifted to lower frequencies– This has been detected in the spectral lines emitted by

atoms in massive stars

More Implications of General Relativity

• A gravitational field may be “transformed away” at any point if we choose an appropriate accelerated frame of reference – a freely falling frame

• Einstein specified a certain quantity, the curvature of spacetime, that describes the gravitational effect at every point

Curvature of Spacetime

• There is no such thing as a gravitational force– According to Einstein

• Instead, the presence of a mass causes a curvature of spacetime in the vicinity of the mass– This curvature dictates the path that all freely

moving objects must follow

General Relativity Summary

• Mass one tells spacetime how to curve; curved spacetime tells mass two how to move– John Wheeler’s summary, 1979

• The equation of general relativity is roughly a proportion:

Average curvature of spacetime energy density– The actual equation can be solved for the metric which

can be used to measure lengths and compute trajectories

Testing General Relativity

• General Relativity predicts that a light ray passing near the Sun should be deflected by the curved spacetime created by the Sun’s mass

• The prediction was confirmed by astronomers during a total solar eclipse

Other Verifications of General Relativity

• Explanation of Mercury’s orbit– Explained the discrepancy between observation

and Newton’s theory

• Time delay of radar bounced off Venus• Gradual lengthening of the period of binary

pulsars (a neutron star) due to emission of gravitational radiation

Black Holes

• If the concentration of mass becomes great enough, a black hole is believed to be formed

• In a black hole, the curvature of space-time is so great that, within a certain distance from its center, all light and matter become trapped

Black Holes, cont

• The radius is called the Schwarzschild radius– Also called the event horizon

– It would be about 3 km for a star the size of our Sun

• At the center of the black hole is a singularity– It is a point of infinite density and curvature where

spacetime comes to an end