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22. Black Holes
• Einstein’s Special Theory of Relativity
• Einstein’s General Theory of Relativity
• Black holes exist in some binary star systems
• Supermassive black holes at of galaxy centers
• Two
properties of nonrotating black holes
• Three properties of
rotating
black holes
• Falling into a black hole is an infinite voyage
• Black holes evaporate
Einstein’s Special Theory of Relativity
• Published in 1905
– “Special” ⇒ It applies only in special circumstances
• No acceleration or deceleration
– “Relative” in contrast to “absolute”
• Neither distance nor time are absolute
• Founded on two basic principles
– Description of physical reality is always the same
• Presumes no acceleration (constant velocity)
• Gravity causes acceleration (gravity is ignored)
– Speed of light in vacuum
is always the same
• Independent of speed & direction of motion
• The concept of spacetime
– By definition, speed = distance per unit time
– By implication, spacetime is a fourth dimension
Changeability & Constancy
• Things that are changeable
– Length contraction
• Length approaches zero
– Time
• Time
– Mass
as c is approached
dilation
nearly stands still
as c is approached
dilation
• Mass approaches infinity as c is approached
• Things that are
constant
– The speed of light in vacuum
• Independent of any two observers
• All matter must move less than speed of light
– Two mutually contradictory phenomena
• Length would contract to zero
• Mass would expand to infinity
Relativistic Length Contraction
Relativistic Time Dilation
Einstein’s General Theory of Relativity
• Einstein’s “geometric theory of gravity”
– Gravity affects the shape of space & the flow of time
• Spacetime becomes distorted
• Principle of equivalence
– All accelerations appear identical
• Things accelerating relative to a stationary massive object
• Things accelerating relative to a
moving spacecraft
Tests of General Relativity
• Gravitational bending of light
– Caused by distortions in spacetime
– First tested in the 1919 total solar eclipse
• Precession of Mercury’s orbit
– Explained 531 arc-sec / century due to planets
– Unexplained 43 arc-sec / century due to relativity
• Gravitational redshift & time dilation
– Light slowed in gravitational fields
– Time slowed in gravitational fields
• Gravitational waves
– Caused by rapidly oscillating massive objects
– Not confirmed as of 29 May 2014
Acceleration Equivalence Principle
Spacetime Gravitational Curvature
A Normal Star
The “well” is closed
Gravitational Deflection of Light
Precession of Mercury’s Orbit
One Way to Form a Black Hole
• Life of a high-mass star
– Gravity is normal
• Light escapes the star surface in nearly straight lines
• Core collapse to a large-diameter neutron star
– Gravity begins to increase as diameter decreases
• Light escapes the star surface in slightly curving lines
• Core collapse to a small-diameter neutron star
– Gravity increases as diameter decreases
• Light escapes the star surface in strongly curving lines
• Core collapse to a
black hole
– Gravity almost infinitely strong as diameter nears 0
• Light curves back to the surface, unable to escape
– The matter continues to collapse forever
• No kind of pressure can ever stop it
The Formation of a Black Hole
Curved Spacetime Near a Black Hole
A Black Hole
The “well” is open
Black Holes in Binary Star Systems
• The nature of the search
– Black holes cannot be seen directly
• No emitted or reflected light can escape
– Black holes can be detected indirectly
• Binary star systems are very common
• A small number of binary systems contain a black hole
• Kepler’s third law tells us the mass of the black hole
• Some recent successes
– Cygnus X-1
• X-ray flickering as quickly as 0.01 seconds
• Necessarily < 3,000 km in diameter
• Possibly gas concentrations in an accretion disk
– V 404 Cygni
• Doppler shift confirms orbital period of 6.47 days
• Unseen companion > 6.26 M☉
A Model of the Cygnus X-1 System
Supermassive Black Holes in Galaxies
• Galactic mass
– A typical galaxy contains 1011 M☉
– A tiny fraction of this = A supermassive black hole
• One example: M87
– High-resolution HST Doppler shift measurements
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Stars orbiting galaxy center at 100’s of km / sec
Kepler’s third law tells us the mass
The galactic center contains ~ 3.0 . 109 M☉
The galactic center is the diameter of the Solar System
– Dozens of other supermassive black holes identified
• Analysis of surrounding gas & dust
– The Milky Way galaxy
• Our galaxy apparently also has a supermassive black hole
Four Kinds of Black Holes
• Primordial
black holes
– Hypothesized by Stephen Hawking
• May have formed as part of the Big Bang event
• Masses ranging from Mcloud droplet to MEarth
• No evidence that they actually exist
• Stellar-mass
black holes
– Produced during supernova events
• > 4 M☉
• Intermediate-mass black holes
– Produced by
accretion
• > 103 M☉
• Supermassive
– Produced during
• > 106 M☉
black holes
galaxy birth
Properties of Nonrotating Black Holes
• These are theoretical objects
– Virtually all celestial objects rotate
– They rotate faster as they become smaller
• Properties of nonrotating black holes
– A center
Singularity
• Degenerate electron & neutron pressure cannot support it
• The mass continues to collapse forever
• It becomes infinitely dense yet ever increasing density
– A surface
Event horizon
• Maximum distance at which even light cannot escape
The Schwarzchild Radius
• Property of nonrotating black holes
– Distance from the singularity to the event horizon
• Considered to be the “surface” of a black hole
– Directly proportional to the mass of the black hole
2×G×M
RSch =
2
c
Geometry of Non-Rotating Black Holes
3 Numbers Describe Black Holes
• Mass
– This exerts force that acts over long distances
– Kepler’s third law tells us the mass of the black hole
• Observe stars orbiting very close to the black hole
• Electric charge
– This exerts force that acts over long distances
• Appropriate instruments on a hypothetical spacecraft
– Expected to be very nearly zero
• Number of p+ & e– in any vicinity is normally about equal
• Angular momentum
– This distorts spacetime over long distances
• Observe gas spiraling into a black hole
– Virtually all black holes are expected to rotate
• Everything we observe has some angular momentum
Rotating (Kerr) Black Holes
• Predicted by Roy Kerr
1963
– Known angular momentum of high-mass stars
• Most rotating black holes spin thousands of times per sec.
• Much faster than the fastest neutron stars
• Effects of rotating black holes
– The singularity becomes an infinitely thin ring
• It is centered on the geometric center of the black hole
– Spacetime is dragged along
• Ellipsoidal ergoregion surrounds spherical event horizon
• An unusual property of rotating black holes
– The Penrose process
• Objects skipping off the ergoregion can steal momentum
Geometry of a Rotating Black Hole
One Concept of a Black Hole
Wormholes & Time Machines
• Einstein & Rosen discover passageways
– Einstein-Rosen bridges
• Connect two parallel universes
– Wormholes
• Connect two parts of our Universe
• Problems
– Powerful gravity causes the passageway to collapse
• Travel faster than light required to traverse the bridge
– Pressure may generate antigravity
• This may support the bridge long enough for travel
• Time machines
– Move one end of the wormhole very quickly
• Generate a time differential between the two ends
– The problem of violating causality
Einstein-Rosen Bridge (Another Universe)
A Wormhole (Within Our Universe)
Fall Infinitely Into a Black Hole
• Playing with black holes
– Seems like a normal star at ~ 1,000 RSch
• ~ 1,500 km for a 5.0 . M☉ black hole
– Allow a probe to fall into the black hole
• Acceleratesinitially when gravitational time slowing small
• Decelerates later when gravitational time slowing large
• Seems to take infinitely long to enter the event horizon
– Tidal forces become overwhelming
• Probe is stretched out by
differential gravity
– The probe & its atoms are torn apart
• Probe is squeezed in by force directed to the singularity
• Strange phenomenon
– Blurred distinction between space & time
• Limited ability to move backward in time
• Limited ability to move backward in space
Distortion Falling Into a Black Hole
Black Holes Evaporate
• Basic physical processes
– Heisenberg uncertainty principle
• Both location & velocity cannot be known with certainty
• If one is well known, the other is poorly known
– Virtual pairs form just outside the event horizon
• One particle escapes & the other is captured
• Mass is lost by the black hole
• Basic properties
– Black holes have a “temperature”
• Measure of the amount of mass lost per unit time
• This temperature is inversely proportional to mass
– A 1015 kg black hole is effectively ~ 10+9 K
– A 1031 kg black hole is effectively ~ 10–7 K
– Very different results
0.5 . MDeimos
5.0 . M☉
• Primordial
black holes evaporate very quickly
• Supermassive black holes evaporate very slowly
Mass Evaporation from a Black Hole
Important Concepts
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Special theory of relativity
– Attempt to understand EMR
• Acceleration must be zero
– Foundational principles
• Description of reality constant
• Speed of light in vacuum constant
– Famous predictions
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E = m . c2
Length contraction
Time & mass dilation
Concept of spacetime
General theory of relativity
– Attempt to understand gravity
• Acceleration may be non-zero
– Famous predictions
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Equivalence of all acceleration types
Bending of light by massive objects
Gravitational redshift
Existence of gravity waves
Existence of black holes
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Black holes
– Primordial, stellar mass, supermassive
– Non-rotating black holes
• Point singularity & event horizon
– Rotating black holes
• Ring singularity & event horizon
• Also an ellipsoidal ergoregion
– Passageways
• Wormholes to our Universe
• Einstein-Rosen bridges to other univ.
– Falling into black holes
• Tidal stretching toward singularity
• Squeezing perpendicular to singularity
– Evaporation of black holes
• Equivalent of black hole temperature
• Inversely proportional to mass
• Primordial bl. holes evaporate quickly