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Lecture 27:
Black Holes
Stellar Corpses:

white dwarfs
collapsed cores of low-mass stars
 supported by electron degeneracy
 white dwarf limit 1.4 Msun


neutron stars
collapsed cores of high-mass stars
 supported by neutron degeneracy
 neutron star limit about 3 Msun


black holes

collapse to a singularity
General Relativity

the equivalence principle:
gravity and acceleration are equivalent –
i.e., one cannot discriminate between
being at rest in a gravitational field and
being accelerated in the absence of
gravity
gravity
=
acceleration
General Relativity

mass causes space-time to curve:
Imagine space-time as a fourdimensional rubber sheet. Any object
with mass causes this sheet to become
deformed.
General Relativity

the curvature of space-time tells
matter how to move:
What we perceive as gravity arises from
the curvature of space-time. Masses
follow the ‘straightest possible paths’
possible given the curvature.
Strange consequences of the
Equivalence Principle:

gravitational time dilation:
time runs slower near a
massive object
flashes take a longer
time to reach
flashes take a
shorter time to
reach
Strange consequences of the
Equivalence Principle:


gravitational time dilation:
time runs slower near a
massive object
gravitational redshifting:
light escaping from a massive
object is shifted towards lower
frequencies/longer wavelengths
Strange but true: observations
confirming the predictions of
general relativity

gravitational lensing (bending of light by
gravity) confirmed during a solar eclipse
in 1919
Strange but true: observations
confirming the predictions of
general relativity


gravitational lensing (bending of light by
gravity) confirmed during a solar eclipse
in 1919
precession of the perihelion of Mercury:
general relativity predicts a correction
to Newton’s Law, which fits the
observations
574 arcsec per century
Newtonian theory
predicted 531 arcsec
per century
Strange but true: observations
confirming the predictions of
general relativity



gravitational lensing (bending of light by
gravity) confirmed during a solar eclipse
in 1919
precession of the perihelion of Mercury:
general relativity predicts a correction
to Newton’s Law, which fits the
observations
gravitational redshifting: spectral lines
from white dwarfs are shifted; direct
confirmation in 1960
Black Holes


general relativity predicts that there
can be singularities in space-time,
places where the density of matter
becomes infinite
‘black holes’ are the name for one
kind of ‘singular solution’ in the
equations.
Formation of a Black Hole
The paths of photons
in curved space-time
Escape velocity from a black hole



remember (from Chapter 5) the
escape velocity is given by
vesc = [2GM/R]1/2
what if the escape velocity was equal
to the speed of light?
this would set a maximum radius for
which light could escape from an
object with a given mass
The Schwarzschild radius
v2esc = [2GM/R] = c2
 RS = 2GM/c2
or RS = [3.0 x M/Msun] km
The Schwarzschild radius


the larger the mass of a black
hole, the larger the Schwarzschild
radius
once light or any object has
crossed the Schwarzschild radius
(or event horizon), it can never
escape the force of gravity of the
black hole.
Black holes have no hair

all information about the material
that is inside the event horizon of a
black hole is lost, except
mass
 charge
 angular momentum

Black hole Entropy Theorem


The total amount of information
(entropy) in the Universe cannot
decrease (second law of
thermodynamics)
this is what lead Bekenstein and
Hawking to the idea that Black holes
must radiate
Falling into a black hole



stretched by tidal forces
time slows down
radiation is redshifted
Observational Evidence


there is evidence that black holes
formed from collapsed stars exist in
some X-ray binaries
most promising candidate:
Cygnus X-1: 18 Msun star orbiting an
unseen companion with a mass of 10
Msun
 too massive to be a neutron star and
too small to be an ordinary star

Cygnus X-1
Supermassive Black Holes


there is very good evidence from the
motions of stars and gas near the
centers of galaxies that most galaxies
(including our own) contain
‘supermassive black holes’ – black
holes weighing millions to billions of
solar masses
how these objects formed is still
something of a mystery…
M87
White holes, Wormholes, and
tunnels through hyperspace



black holes are only one of the
several kinds of singularities in the
equations of general relativity
white holes are sort of like the
opposite of black holes
a wormhole is a black hole
connecting to a white hole
Einstein-Rosen bridge
Wormhole
the end