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RELATIVITY
Updated April 28, 2008
Corrected version
Dr. Bill Pezzaglia
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I. Introduction
II. Is Everything Relative?
III. Mass
IV. General Relativity
V. Summary
I. Introduction
A. Outline of Talk
B. MetaPhysics (about
Physics)
C. Light: The Starry
Messenger
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I B1. Metaphysics (About Physics)
Why is this called the
“year of physics”?
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I B2. MetaThemes
How do I talk about Einstein’s
theory without mathematics?
Fortunately,
the
principles of
physics are
not
mathematical
in origin.
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I C1. Light: the Starry Messenger
All we know
about the universe
comes from light
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I C2. Review: Spectra
•If you spread
starlight out with a
prism, you find
“spectral lines”
contributed by the
atoms.
•Each element
has unique lines
in a precise
location
(wavelength)
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I C3. Review: Doppler Effect
Example:
Binary Star
Redshift:
when moving
away from us,
the spectral
lines are
shifted to the
red.
Shift is
proportional
to speed
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II. Is Everything Relative?
A. Mach and Rotational
Relativity
B. Relative Velocity
C. Time and Space Distortions
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A1. Rotational Relativity
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Newton argued that water in a rotating bucket will make
the shape of a parabola due to centrifugal force.
The
presence of
centrifugal
and coriolis
forces
confirm
that you are
in a noninertial
rotating
frame of
reference
A2. Mach’s Principle
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1883 Mach argued that one can not tell if
the bucket is rotating, or instead
The stars
are
rotating
around
the
bucket
A3. Frame Dragging
Mach argued a centrifugal force will appear in both
cases, indicating only a relative rotational motion
1896 Friedlander attempted (failed) to
measure if there is a centrifugal force
introduced inside of a big rotating flywheel.
Frame Dragging:
Recently Gravity
Probe B has been
orbiting the earth to
see if there is a
similar effect: that the
rotating earth pulls
the space around
with it
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(B1). Special Relativity
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1905 Einstein (26 years old) publishes theory of
special relativity
Motion is relative (Galileo)
there is no experiment one can do to determine
absolute motion relative to “space”.
Laws of physics
must hold in all
reference
frames which
differ only by a
constant velocity
(B2). Speed of Light
• Einstein Questioned: If you were moving at
the speed of light, could you see yourself in a
mirror held out in front of you?
• He concludes:
Speed of light is the
same for all observers
• How is this possible?
Nature conspires to distort
space and time
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(B3). Addition of Velocities
• Speeds add:
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V=
• Adding anything to speed of light gives speed of light
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B4.
c – Speed of Light
• Why is c c?
– Einstein used first used V, the convention
at the time
– c first used in 1856 by Weber, who meant
it to represent a “constant”
– Possibly: c as generic velocity comes from
Latin “celeritas” – “speed”
– The first measurement of c was
made by Reomer in 1675
C1. Lorentz-FitzGerald Contraction
1889 FitzGerald, 1892 Lorentz
•Propose a moving meter stick will
appear to shrink in length
L’ = L [1-v2/c2]1/2
•1905 Einstein deduces this from his
postulates of relativity.
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C2. Ehrenfest Paradox 1909
Ehrenfest proposes puzzle about a rotating disk.
•FRAME OF DISK
•Radius “R”
•Circumference C = 2pR
•FRAME OF Lab
•Rotating Disk’s Radius will be unchanged
(motion is perpendicular to radius)
•Circumference is moving so is it shrunk?
C’ = C [1-v2/c2]1/2
How can a rotating circle have a circumference
smaller than 2pR?
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C3. Time Dilation
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(Einstein)
•A moving clock will appear to run
slower
t 
t
2
v
1 2
c
Paradoxically, two observers each moving with a
clock, sees the OTHER clock running slowly
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C4.
Muon Decay
• Muons don’t “live” long: half of any group decay after
about a millionth of a second
• Frisch (one of my professors!) and Smith, 1963, on Mt.
Washington in NH: measure rate of cosmic muons at the
top and bottom of the mountain: Far more survive from
top to bottom than
lifetime should allow
• Traveling at close to c,
the muon’s “internal
clock”: runs slowly
compared to the
physicist’s clock – its
time is stretched out
III. Relativity of Mass
A. Equation E=mc2
B. Verification
C. Where to find it
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A1. Mass
(Einstein)
•A moving mass will appear more
massive
m0
m
v2
1 2
c
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A2. Famous Equation: E=mc2
• "It followed from the special theory of
relativity that mass and energy are both…
are but different manifestations of the same thing –
a somewhat unfamiliar conception for the average
mind. Furthermore, the equation E is equal (to)
m c squared, in which energy is put equal to mass,
multiplied with the square of the velocity of light, showed that
very small amounts of mass may be converted into a very large
amount of energy and vice versa. The mass and energy were in
fact equivalent, according to the formula mentioned above. This
was demonstrated by Cockcroft and Walton in 1932,
experimentally."
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A3. Making E=mc2 “More Correct”
• E2 = p2c2 + m2c4
• Objects without mass have energy…
and momentum (p)
– Light hitting a surface can make it move
• Negative energy solutions possible!
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B1.
E=mc2: Proven!
• Cockroft and Walton,
1932: smashed
accelerated protons into
a piece of Lithium,
showing for the first time
the conversion of mass
into energy
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B2. Is Mass Equivalent to Energy?
• Consider a deuterium nucleus,
composed of a proton and a neutron:
Md =
2.01355 amu
• The proton and neutron have mass
Mp
=
1.00728 amu
Mn
=
1.00866 amu
• Something’s funny! …
B3. It Doesn’t Add Up!
• Mp + Mn:
1.00728 amu
+
1.00866 amu
=
2.01594 amu
• That’s MORE than the mass of the deuterium
nucleus Md = 2.01355 amu! The proton and
neutron are heavier separately than when
they’re together!
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B4. Mass IS Equivalent to Energy!
• 2.01594 amu - 2.01355 amu = .00239 amu
• Energy equivalent to the mass difference
is that energy required to bind the proton
and neutron together.
• The equivalent energy is 20 times higher
than that of x-rays used by your dentist
C1. Where to find E = mc2?
• Our Sun, in which fusion reactions
exchange the mass of Hydrogen for
energy in the form of light and solar
wind
• The conversion of food mass into
energy to make your body run
• The conversion of the mass of gasoline
into energy to make your car run
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C2.
GPS: Relativity and You
• 24 Global Positioning Satellites, each with an atomic clock
– Each always transmitting its precise clock time
– Position found by times and known satellite positions
• Times must be known to within 20 to 30 billionths of a
second.
• Viewed from Earth: moving clocks run slowly, losing about
7 millionths of a seconds per day.
• General Relativity: clocks run slowly near a massive
object. Farther from Earth, satellite clocks run faster,
gaining about 45 millionths of a seconds a day.
• Without accounting for Relativity, GPS would not work!
IV. General Relativity
A. Equivalence Principle
B. Curved Space
C. Cosmology
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A1. Definition of Mass
There are 3 ways to think about mass
1. Inertial Mass
F=ma
2. Passive Gravitational Mass
F=mg
3. Active Gravitational Mass
GM
g 2
r
The “Weak Equivalence principle” says that
inertial mass equals passive gravitational mass
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A2. Galileo’s Experiment at Pisa
• 1590 Galileo’s Principle:
All bodies fall at the same rate,
regardless of mass
• 1907 Weak EEP
(Einstein Equivalence Principle)
All bodies will follow same path,
independent of internal structure
(e.g. mass or composition)
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A3. The Equivalence Principle
Reference at rest with Gravity is indistinguishable to a reference
frame which is accelerating upward in gravity free environment.
The apple accelerating downward due to gravity looks the same as an
apple at rest in space, with the floor accelerating upward towards it.
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A4. Gravitational Redshift
Time is distorted by gravity
1. Time runs slower in a gravity field (your feet
are aging slower than your head)
2. Photons leaving the sun lose energy pulling
 GM
away and are “redshifted”


rc
2
3. If the mass is big enough, the escape speed
becomes the speed of light, which means
light cannot escape, i.e. it is redshifted out of
existence (“black hole”)
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b1. Bending of Starlight (Date?)
• Newton: Light is NOT affected by gravity
• Einstein: Elevator example shows light must be
affected by gravity.
• Predicts starlight will be bent around sun!
• 1919 Measured by Eddington!
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b2. Curved Space
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Einstein argued that mass curves space, and “gravity” is
simply particles following the curves of space
b3. The Holonomy of Curvature
Levi-Civita (1917)
[two years after Einstein’s General Relativity]
showed that a vector “parallel
transported” around a closed
loop will be rotated due to
curvature.
We experience the rotation
(holonomy angle)
as gravitational force
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b4. Curvature and Black Holes
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Einstein’s General
Theory of Relativity
Gravity is “curved space”
Big curvature makes a
Black Hole that you can
fall in and never get out
For example, when people
throw things into my tuba,
they are never seen again.
b5. Schwarzschild Radius
If any mass is compressed into a size smaller than the
“Schwarzschild Radius”, it will become a black hole
This can happen during
a supernova explosion,
or later by additional mass
falling on a neutron star.
Anything that comes closer
than the Schwarzschild
Radius, will fall in and
never escape.
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b6. Observing Black Holes
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They are black, how do we see them?
Find one in a binary system. As mass falls into the black
hole there will be some radiation released
b7. Jets
Not all the material falls into
the hole. Some is ejected
at very high energies
out “jets” along the
axis of the black hole.
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b8. Galactic Jets
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Chandra X-Ray Telescope
Sees jets coming out of
galaxy M87, suggesting
there is a BIG black hole at
the center.
b9. Radio Lobes from galaxy Centaurus A
Again, suspect a big black
hole in the center
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C.1 Static Cosmology
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Copernicus Cosmological Principle:
• Universe is isotropic & Homogeneous
• No Preferred Center
(leads to conservation of momentum)
• Verified by Hubble’s Galaxy Survey
C.2 Newtonian Cosmology
Newton proposed that the
universe must be infinite to be
balanced; a finite universe would
collapse due to gravity
But:
• 1823 Olber shows universe is not infinite
•
1920 Eddington shows Newton’s Universe
is Unstable and would collapse
•
1929 Inconsistent with Hubble’s Law (expanding universe)
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C.3 Olber’s Paradox (1823?)
Heinrich Olber 1758-1840
discoverer of the minor planets Pallas and Vesta
If the universe is infinite in extent, and filled with
stars, Why is the sky dark at night?
For example, in this cactus forest, if you
look between two cacti, you just see
another cactus further away. In some
places you see blue sky because the
forest is not infinite, but if it was, you’d
just see cacti everywhere, and so the
universe would be “green”.
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C.4 Olber’s Paradox
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It’s the same with stars. If you look between two stars, you will
just see another.
Aren’t more distant stars fainter due to the inverse square law?
Yes, but, at a greater distance, the number of stars per angle will
increase by the distance squared!
The two effects cancel out! Everywhere the sky should be as
bright as the sun!
C5 Curved Finite Universe
1917 Einstein proposes universe:
•Is finite, curved like a ball
(this fixes Olber’s Paradox)
•But gravity would still collapse it
•Proposes negative pressure
(cosmological constant) prevents
collapse
•Later calls this his “biggest blunder”
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C6 Curved Space
If we live in a (positively)
curved space, then no matter
what direction we look, we
might see that back of our
head!
Its like living on the surface of
a big ball.
This would explain why sky is
dark at night.
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C7. Galactic Recession
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•1912 Vesto Melvin Slipher first measures “doppler redshifts” of
galaxies
•1917 Willem de Sitter interpreted Slipher’s red shifts as a Doppler
effect, meaning that the galaxies are moving away from us (known for
a time as the “de-Sitter effect). [He perhaps is the first to propose
that the universe is expanding]
•1923 Carl Wirtz combined Slipher’s red shift measurements with
rough estimates of the distances to galaxies based on their apparent
size, and proposed a velocity-distance law, i.e. that redshift is
proportional to distance. [This result has been historically ignored, and
Hubble got most of the credit]
C8
Big Bang Theory
•1922 Friedmann, and 1927
Father Lemaitre show that
another solution to
Einstein’s equation would
be that the universe is
expanding from kinetic
energy leftover from a “big
bang” creation. No need for
negative pressure.
•This would also explain the
observed “redshifts”
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C9 Hubble’s Law
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To visualize the expansion of our three-dimensional universe, imagine a
two-dimensional universe crisscrossed by a grid of parallel lines (like on
a piece of graph paper). The animation shows five galaxies that happen to
lie where gridlines cross. As the universe expands in all directions, the
gridlines and the attached galaxies spread apart. From the viewpoint of
any one of the galaxies, all the other galaxies appear to be moving away.
The more distant a galaxy is, the more rapidly it appears to be receding.
•1929 Hubble’s Law
C10 Hubble’s Law (1929)
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(recessional speed) =(const)x(Distance)
Or:
v=Hd
H=Hubble’s Constant, which will be the
“slope” of the line. The approximate
value is 75 km/(sec-MPC)
MPC=megaparsec (a million parsecs).
•The velocity is measured using the Doppler redshift (Z)
•Redshift Defined: Z = /
–
Example: if wavelength is 520 nm, but you measure it to be 572 nm,
then =52 nm, and Z=52/520=0.1
•Doppler Formula relates velocity to redshift
–
–
v/c = Z (approximately, valid for v<<c)
c=speed of light
•Example: Z=0.1 means the speed is 10% the speed of light.
C11 Doppler Formula
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Relativistic Doppler Formula
If we have: Z=v/c, then what about
distant quasars that have Z=3. Does
this mean its traveling 3 times the
speed of light?
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3.5
3
No
The true “relativistic” doppler formula:
(Z+1)2 = (c+v)/(c-v)
Redshift Z
2.5
2
1.5
As you approach the speed of light,
the redshift goes to infinity.
1
0.5
Redshift of Z=3 is speed of 88.25%
speed of light.
0
0
0.2
0.4
0.6
speed (v/c)
0.8
1
C12 Hubble Time: 13 billion years
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If the universe has
been expanding at
a constant rate,
then by “playing
the movie
backwards”, 13
billion years ago
the universe was a
single point (big
bang!)
C13 Decelerating Universe
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We expect the
universe is
decelerating due
to all the
gravitational
attraction between
galaxies. Hence
the age of the
universe is
somewhat less
than the Hubble
Time (13 billion
years)
C14 Open vs Closed Universe
CLOSED UNIVERSE: Just like a ball
thrown upward will fall back to earth
due to gravity, we might expect the
universe will slow down, and collapse.
OPEN UNIVERSE: If you throw a ball
upward fast enough, it won’t fall back,
but it certainly will slow down due to
gravity
We expect that the universe
must be decelerating due to
gravity. Whether the universe
is open or closed is only a
question of how fast it is
slowing down.
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C15. Hubble Ultra Deep Field
•March 2004 -- the Hubble
Ultra Deep Field survey
probed an area about three
minutes of arc square,
shows 10,000 galaxies.
•Mass is too small to make
universe closed (and explain
observed gravitational
forces)
•Must propose universe is
mostly made of “dark
matter”.
•This makes the entire
universe one gigantic “black
hole” (perhaps inside of
someone else’s universe?
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C16 Measuring Deceleration
Redshift is
proportional to
expansion rate
Brightness of cluster
is a measure of
distance, which is a
measure of lookback
time into the past.
Expect to see
deviations from
Hubble’s law in the
past, when universe
was expanding faster.
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C17 The Universe is Accelerating?
1998 Measurements of distant supernova (i.e. in
the distant past) were 20% fainter than expected.
Interpretation:
universe WAS
slowing down for first
half of lifetime
BUT, since then, it
has been
ACCELERATING
This is a big surprise
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V. Summary
A. Special Relativity
B. General Relativity
C. The End of Physics
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Special relativity
• Physics the same in all frames
• Speed of light – c – independent of
source, observer, and frame
• Energy and mass are equivalent:
E = mc2
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General relativity
• Equivalence Principle: gravity is
indistinguishable from an accelerating
reference frame
• Inertial and Gravitational mass are the
same
• “Gravity” is the curvature of space
Gravity Waves?
• Not discovered yet, but we are looking for them.
They would distort space as they travel by!
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V1. The End of Physics?
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The following statement made by a Nobel prize winning physicist:
“The most important fundamental laws and facts of
physical science have all been discovered, and these
are now so firmly established that the possibility of
their ever being supplemented in consequence of
new discoveries is exceedingly remote.”
Albert Abraham Michelson
1903
(before relativity and quantum mechanics were invented)
References
• Some material “borrowed” from Phil Kesten, Santa
Clara University
• Bassett & Edney, “Introducing Relativity”, Icon Books
(2002), p. 56
• Hartle, “Gravity”, Addison Wesley (2003)
• D’Inverno, “Introducing Einstein’s Relativity”,
Clarendon Press (1992)
• Misner, Thorne & Wheeler, “Gravitation”, Freeman &
Company (1973)
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