Black Holes

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Black Holes
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Euclidean Geometry Euclidean space exists all
around you. Any and all geometry that you study
in high school is Euclidean geometry. Consider a
triangle ? ? ?
? ? ? 180?
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Riemannian Geometry Riemannian geometry is used
on curved surfaces, such as the surface of our
Earth. If a triangle is drawn by connecting
three distant cities such as Toronto, Paris, Hong
Kong, the lines would traces out arcs along the
surface of the Earth. Consider the triangle

Paris
Toronto
? ? ? ? 180?
Hong Kong
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General Relativity begins ...
After creating Special Relativity, Einstein
worked diligently to expand his theories to
include gravity and to help to predict the
observable universe. He realized that the
Special Theory of Relativity described a universe
where one event may appear to occur at different
times - Simultaneity. That meant that the
Newtonian concepts of planetary motion and
gravitational forces must be incorrect - i.e.
Newton would predict that the moon orbits the
Earth because the Earths gravitational force
pulls on the moon and the moons gravitational
force pulls on the Earth. Both of these forces
must act simultaneously, and that can no longer
be guaranteed as a result of Special
Relativity. Einstein proposed two Gedanken
experiments, one that predicted a need for
non-Euclidean geometry and the second that
predicted that energy feels the effects of
gravity!
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Gedanken Experiment 1 Spinning Space
Station A Measures Co 2? ro or 2?
Co ro Angular velocity ? rv
A Euclidean B B Measures r
ro and C Co ? 1 - v2/c2 C Co ? 1
- v2/c2 2? ? 1 - v2/c2 r ro

NOT Euclidean
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Gedanken Experiment 2
Recall Energy of a photon is E hf where E
energy h Planks constant f
frequency of light
Eblue hfblue
Ered hfred
Eblue gt Ered ? fblue gt fred
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Gedanken Experiment 2
a
At E1 mass m is at rest At E2 mass m
spontaneously changes to a photon of light
(magic!) and becomes E3 At E4 photon m
spontaneously changes back to mass m (more
magic!) E4 must equal E1, and the only way to
ensure that is to have the frequency of the
photon get less as the photon moves away from the
gravitational field - Red Shifted - which is
physically observed!
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Curved Space-Time Gravity Newtonian concepts
about gravity and simultaneous forces were
unacceptable in light of Special Relativity.
Einsteins conceptual picture of the universe
as a continuous sheet of malleable material, into
which massive bodies (planets, stars, black
holes) made depressions, provided a better
description of gravity.
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Curved Space-Time Gravity
(x2,y2)
(x1,y1)
This equation describes the distance between any
two points on the surface of Einsteins curved
space-time, otherwise known as our Universe.
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Curved Space-Time Gravity
Infinitesimal steps along space-time surface
New Look Same Equation
Infinitesimal steps along space-time converted
to spherical coordinate system
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Verification of General Relativity
? Perihelion Shift of Mercury General Relativity
precisely predicts the amount Mercurys orbit
will precess (wobble) over time. Newtonian
equations could not account for the orbital
variation.
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Verification of General Relativity
? Gravitational Bending of Light General
relativity predicts that the bending angle for a
light ray in the vicinity of a point mass to be
where ? bending angle G universal
gravitational constant M mass c speed of
light R distance from mass
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Verification of General Relativity
? Gravitational Bending of Light
Contd Observational Evidence The solar
eclipse of May 29, 1919, was to provide the stage
on which the verification of Einstein's result
was set. Eddington monitored the positions of
stars grazing the limb of the Sun during the
eclipse. Observations revealed that the
positions of the stars shifted by 1?98, 1?60
and 0?30 respectively. These results
confirm Einstein's prediction. A very definite
result has been obtained, that light is deflected
in accordance with Einstein's law of
Gravitation.' Eddington
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Verification of General Relativity
? Gravitational Bending of Light Contd
Light is bent by gravity OR light follows the
curved space-time near massive objects.
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Verification of General Relativity
? Gravitational Bending of Light Contd
Principe Island, 29 May 1919. Marked on a
negative photo of the 1919 solar eclipse are the
positions of stars examined in the historic test
of Einstein's theory of gravity (from Memoirs of
the Royal Astronomical Society LXII, Appendix
Plate 1). Library of the Royal Astronomical
Society, London.
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Verification of General Relativity
? Red Shift and Time Delay
General Relativity exactly predicts the observed
amount of redshift and time delay communication
signals sent to the surface probes on Mars when
it is on the far side of the sun.
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Black Holes
Consider a region on the space-time surface.
Arbitrarily choose a circular shape that is
static, non-rotating, and in a vacuum described
by
r
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Black Holes
To find how the space-time surface (also known as
gravity) behaves on the circle when r gt R you
must determine the line integral of the 4D
equation.
r
Time goes to zero!
Space becomes infinite
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Black Holes
The escape velocity for a planet is the minimum
speed that an object must have to be flung free
of the planets gravity.
r
Therefore at the Schwarzchild radius, the escape
velocity becomes the speed of light! For any
values of R lt r, the escape velocity becomes
greater than the speed of light nothing, not
even light can escape.
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Black Holes
Solve for the Schwarzchild radius of our sun,
which currently has a radius of 700 000 000
m. Given msun 2 x 1030 kg G 6.673 x 10-11 N
m2/kg2 c 3.0 x 108 m/s
r
rSchwarzchild 3 km