Lecture 27: Black Holes

Stellar Corpses: white dwarfs white dwarfs  collapsed cores of low-mass stars  supported by electron degeneracy  white dwarf limit 1.4 M sun neutron stars neutron stars  collapsed cores of high-mass stars  supported by neutron degeneracy  neutron star limit about 3 M sun black holes black holes  collapse to a singularity

General Relativity the equivalence principle: 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: mass causes space-time to curve: Imagine space-time as a four- dimensional rubber sheet. Any object with mass causes this sheet to become deformed.

General Relativity the curvature of space-time tells matter how to move: 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 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 gravitational time dilation: time runs slower near a massive object gravitational redshifting 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 v esc = [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 v 2 esc = [2GM/R] = c 2  R S = 2GM/c 2 or R S = [3.0 x M/M sun ] 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.

Example: Find the Schwarzschild radius of a black hole with the mass of the Earth (6 x kg).

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 M sun star orbiting an unseen companion with a mass of 10 M sun  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