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Black Holes and General Relativity
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The General Theory of Relativity
Intervals and Geodesics
Black Holes
The General Theory of Relativity
Newton
• Simple
• Works (extremely) well
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Describes how things fall
on earth
Describes motion of solar
system objects…predicted
existence of Neptune
• Unable to account for …
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for 43”/century of observed
574”/century shift of
mercury
perihelion…hmmm?
Einstein
• Elegant
• Works even better
• Alters basic view of
Universe
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Mass  space-time
• Accounts for …
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additional 43”/century of
observed 574”/century
shift of mercury
perihelion…
Shifting of star positions
during eclipse…
Advance of the Perihelion of Mercury
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Perihelion-nearest approach point of
Mercury’s orbit advances due to
effects from planets,etc…
Newtonian Gravitation can not
account for total observed shift of
574”/century….Planet Vulcan???
Einstein’s General Theory of
relativity could fully account for the
observed perihelion advance
http://en.wikipedia.org/wiki/Tests_of_general_relativity
http://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz
_vector
http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity
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Need to account for trivial precession
of equinoxes 1.5 degree/century
Advance of the Perihelion of Mercury
www.math.toronto.edu/~colliand/426_03/Papers03/C_Pollock.pdf
Curvature of Space-Time?
Salvador Dali's "Soft Watch at Moment of First Explosion"
Curvature of Space-Time
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The General Theory of Relativity is
fundamentally a geometric
description of how distances
(intervals) are measured in the
presence of mass.
Relativity deals with a unified spacetime.
Distances between points in the
space surrounding a massive object
are altered in a way that can be
interpreted as space becoming
curved through a fourth spatial
dimension
Rubber sheet analogy:
– Closer to ball more curvature
– Distance between two points
increases more
Mass acts on Spacetime, telling it
how to curve
Curved space-time acts on mass
telling it how to move
Curvature of Space-Time
• Curved space….can see that….kinda
• Curved time?
The rate of flow of time
is determined by the
strength of the
gravitational field
where it is measured…
Curvature of Space-Time
Space and Time
http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/general_relativity.htm
Gravity curves space ,eh?
Looks like balls follow different curves to me!!...
Curvature of Space-Time
Space and Time
Need to look at space and time….In space-time the
balls trajectory do indeed have the same curvature…
Curvature of Space-Time
Space and Time
Bullet and Ball initially take
same paths in space. Why
don’t they continue following
same path if gravity is curving
space?
http://curious.astro.cornell.edu/question.php?nu
mber=649
Bullet and Ball take different
paths in space-time
 experience different
curvature
Curvature of Space-Time
Space and Time
Bullet and Ball initially take
same paths in space. Why
don’t they continue following
same path if gravity is curving
space?
http://curious.astro.cornell.edu/question.php?nu
mber=649
Bullet and Ball take different
paths in space-time
 experience different
curvature
Curvature of Space-Time
Since spacetime itself is curved
even the trajectory of massless photons
deviates from a “straight” line
The Principle of Equivalence
We’re in a “no gravity” state?
The Principle of Equivalence
In any free-float situation, such as that in a freely falling spaceship, the
paths of objects will never bend in any direction when they are given a
certain speed. These objects will move in completely straight lines.
The Principle of Equivalence
When the Earth (or a rocket) pushes on the spaceship, the tracks curve
relative to the spaceship.
The Principle of Equivalence
The ball appears to move in the familiar
curved path which we have come to view
(since the time of Newton) as the effect of
a gravitational force directed towards the
center of the Earth.
With the platform severed from its
attachment to the Earth, the small house is in
a free-fall situation. This time the ball will
move in a straight line, unaffected by any socalled 'gravity' force.
The Principle of Equivalence
The Principle of Equivalence
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Weak Equivalence principle
– mg/mi is a constant
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The Principle of Equivalence: All
local, freely falling, nonrotating
laboratories are fully equivalent for
the performance of all physical
experiments
The Bending of light
The Bending of light
The Bending of light
Gravitational Redshift and Time Dilation
An outside observer, not in free-fall inside of the lab, would measure only the
gravitational redshift (blueshift if the photon were going downward)
Gravitational Redshift and Time Dilation
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Light pulse generated at instant
cable is released. In the time for the
photon to cross the elevator cabin,
the meter has attained a speed
v=gt=gh/c.
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Doppler blueshift should change
meter’s frequency measurement by
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In fact no frequency shift would be
observed in accordance with
principle of equivalence. The
gravitational redshift exactly
compensates by
An outside observer, not in free-fall inside of
the lab, would measure only the gravitational
redshift (blueshift if the photon were going
downward)
Gravitational Redshift
Calculation for a beam that escapes to infinity
Gravitational Redshift and Time Dilation
Gravitational Redshift
Time Dilation
Gravitational time dilation: Time passes more slowly as
the surrounding space-time becomes more curved
Intervals and Geodesics
• General Relativity allows one to
relate events (x,y,z,t) in
spacetime in the presence of
mass that results in the fabric of
spacetime being curved!!!
• T is the stress-energy tensor
which evaluates the effect of a
given distribution of mass and
energy on the curvature of
spacetime.
• G is the Einstein Tensor for
Gravity that mathematically
describes the curvature of
space-time.
• Note that the Gravitational
constant G and the speed of light
play a role in the gravitational
field equation.
Field Equation: for
calculating the geometry of
space-time produced by a
given distribution of mass and
energy
Worldlines and Light Cones
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Worldline: The path followed by an
object as it moves through
spacetime.
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Light Cone: worldline of photon
originating at event A. The speed of
light is taken to be 1.
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What is the worldline for a freely
falling object in response to the local
curvature of spacetime?
Spacetime Intervals, Proper Time and Proper
Distance
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What is a “distance” in spacetime?
Spatial distance between two points
(x1,y1,z1) and (x2,y2,z2) in flat space
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Spacetime interval: between two
events (xA,yA,zA,tA) & (xB,yB,zB,tB) in
flat spacetime
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s)2>0: timelike. Light has more
than enough time to travel between
the events A and B
s)2=0. Lightlike separation.
s)2<0: spacelike. Light does not
have enough time to travel between
the events
Proper time: the time between two
events that occurs at the same location.
s is invariant under Lorentz
transformations. An observer in another
inertial frame S’ would measure the
same interval between events A and B
Proper Distance: The distance
measured between two events A and B
in a reference frame for which they
occur simultaneously (tA=tB).
The Metric for Flat Spacetime
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Metric for flat 3-dimensional space
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Path length for arbitrary path
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Straightest possible line between
points minimizes length
Metric for flat spacetime
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Total interval along worldline
•The interval measured along any timelike
interval is the proper time multiplied by c.
•The proper time along any lightlike worldline
is zero
•The proper time along any spacelike
worldline is undefined
•In flat spacetime, the interval measured
along a straight timelike worldline is a
maximum!!!
The Metric for Flat Spacetime
Curved Spacetime and the Schwarzschild Metric
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Flat spacetime metric in polar
coordinates
•Mass acts on spacetime, telling it how to curve
•Spacetime in turn acts on mass, telling it how to
move
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Schwarzschild metric in space time
curved by a spherical body of mass
M where r>R
•Geodesic: “Straightest” possible
worldline. In a flat spacetime a
geodesic is a straight worldline. In a
curved spacetime a geodesic will be
curved.
•Any freely falling particle (including a photon)
follows the straightest possible worldline, a
geodesic, through spacetime. For a massive
particle the geodesic has a maximum or a
minimum interval, while for light , the geodesic
has a null interval.
Shortest distance between two points may not
be a straight line
The Schwarzschild metric
•Consider a sphere of radius R and
mass M placed at the origin of the
coordinate system. The coordinate r
does NOT represent distance from the
origin. A concentric sphere whose
surface is at r would have a surface
area 4r2.
•A Flamm paraboloid helps “visualize”
this curvature. Remember that “you”
would be contained in the curved
spacetime and you cannot directly
“view” the curvature into the 4th
dimension….
•Proper distance along a radial line
•Proper Time at radial coordinate r
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http://en.wikipedia.org/wiki/Schwarzschild_
metric
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http://casa.colorado.edu/~ajsh/schwp.html
http://people.hofstra.edu/Stefan_Waner/diff_geo
m/Sec15.html
http://channel.nationalgeographic.com/episode/journey-to-theedge-of-the-universe-3023/Overview#tab-interactive
The Orbit of a Satellite
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Starting with Schwarzschild metric
with dr=0, d=0 and d=dt where
v/r
Tests of General Relativity
http://en.wikipedia.org/wiki/Tests_of_general_relativity
• Classical Tests
– Perihelion Precession of Mercury
– Deflection of light by the Sun
– Gravitational Redshift of Light
• Modern Tests
– Gravitational Lensing
– Light travel time delay testing
– Equivalence Principle tests
• Gravitational redshift
• Lunar Ranging…
– Frame dragging tests
• Strong Field Tests (Neutron stars,Black holes)
• Gravitational Wave Detectors
• Cosmological Tests
Gravitational Lens
Gravitational Lens
“Einstein Cross”
Gravitational Lens Studies at the U
Gravitational Lens Studies at the U
Black Holes
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In 1783 John Mitchell pondered that
the escape velocity from the surface
of a star 500 times larger than the
sun with the same average density
would equal the speed of light.
v esc = 2GM /r = 2G(500MÄ ) /7.93RÄ = c
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Light would not be able to escape
from such a star!!!!
Naïve solution of Newtonian escape
velocity equation for c gives a radius
of R=2GM/c2 for a star whose
escape velocity equals the speed of
light.
R=2.95(M/M) km….kinda small!!!
In 1939 J. Robert Oppenheimer and
Hartland Snyder described the ultimate
gravitational collapse of a massive star
that has exhausted its sources of nuclear
fusion. They pondered what happened to
the cores of stars whose mass exceeded
the limit of neutron stars..
In 1967 the term “black hole” was coined
By John Archibald Wheeler
The Schwarzschild Radius
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Consider the Schwarzschild metric
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When the radial coordinate of the star’s
surface has collapsed to RS=2GM/c2 the
square roots in the metric go to zero. RS is
known as the Schwarzschild Radius.
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At r=RS the behavior of space and time is
“remarkable”….
The proper time measured by a clock here is
d=0.
Time has slowed to a complete stop! As
measured from a vantage point far away.
From this viewpoint nothing ever happens at
the Scwarzschild radius
Does this mean that even light is frozen in
time???
The speed of light by an observer
suspended above the star must always
be c. But from far away we can determine
that light is delayed as it moves through
curved spacetime…
•The apparent speed of light, the rate at
which the spatial coordinates of a photon
change, is called the coordinate speed of
light. For light ds=0.
• For dd=0, we have
•In flat spacetime dr/dt~c, however at
r=RS dr/dt=0
Light does appear frozen in time at the
Schwarzschild radius!!!