Announcements
•
Exam 3 is Thursday April 11
Tentatively will cover Chapters 7, 8 & 9
Sample questions have been posted
The main equations of special
relativity
Time Dilation:
Length Contraction:
2
2
1
c
v
L
L
p
2
2
1
rest
t
t
v
c
A bit about terminology
Proper Time: time measured stationary relative to the “event”
Your watch always tells proper time for you
Proper length: length measured stationary relative to the “event”
A meter stick you carry always measures the proper
length of objects that are stationary with respect to you
Rest mass: mass measured stationary with respect to the object
A scale you carry always measures the rest mass of
objects that are stationary with respect to you
Boost factor (
G
) from the Lorentz transformation.
2
2
1
1
c
v
G
Check out links on Special Relativity at
http://www.apsu.edu/astronomy/cosmology

links

and

applets
Other Equations of Relativity
Relativistic Momentum
Relativistic Mass
Relativistic Energy
2
mc
E
0
0
2
2
1
m v
p mv m v
v
c
G
0
0
2
2
1
m
m m
v
c
G
2
2
0
0
2
2
1
m c
E m c
v
c
G
2
2
1
1
v
c
G
The Classic Doppler Effect
The classical Doppler Effect is only radial: only motion away
or towards the observer affect the observed frequency
The Relativistic Doppler Effect
1
1
1
1
1
2
2
z
c
u
c
u
c
u
c
u
obs
The relativistic Doppler Effect includes the stretching
or contracting of waves due to relative motion
and
time dilation due to the relative motion.
Note that the relativistic Doppler Effect occurs even
perpendicular to the direction of motion
Space

time
Diagrams
Anything inside
your light cone
could influence
you (below you)
or you could
influence it
(above you).
Anything outside
your light cone
cannot affect you
The Space

time Interval is
invariant. i.e.: it is the same for all
inertial reference frames
2
2
2
x
t
c
s
Note the minus sign
difference from the
Pythagorean
theorem.
This means
s
2
can be
negative (space

like)
Play with Applets on
Special Relativity
The Twin
Paradox. The
paradox goes
away if you
consider that the
traveler is in two
different inertial
reference frames
Terence stays at home while Stella
makes a trip to Alpha
Centauri.
See the Twin Paradox
Website
The “Special” of Special
Relativity was constant velocity
How do we deal with situations
where there isn’t constant
velocity? What about near the
surface of the Earth where there
is gravity? What about the
general case?
It took Einstein 10 years to figure
out General Relativity
The problem with Newton’s gravity:
the anomalous precession of the
perihelion of Mercury
The observed precession is
5600 arcseconds per
century. 5025 arcseconds is
due to the Precession of the
Equinoxes of Earth, 532 is
due to the effects of Venus,
Earth, Jupiter and all the
other planets on
Mercury.
That leaves 43 arcseconds
per century unaccounted for
by Newtonian
gravitation
and mechanics
.
Newton’s Universal Gravity is a force
law with no propagation speed
r
r
m
m
G
F
ˆ
2
2
1
There is nothing in the force
equation that says how fast it
is transmitted. In fact, it
assumes an instantaneous
transmission of the force.
An instantaneous transmission is in direct violation of
Special Relativity
Special Relativity was for “inertial”
reference frames. What about
accelerating reference frames?
The Weak Equivalence Principle
Any acceleration is
indistinguishable
from gravity
Is “freefall” an inertial reference frame?
On Earth’s surface we are
not
in
an inertial reference frame
The two “fictitious” forces we feel
are the Coriolis force and the
centrifugal force
The fictitious forces are invoked to
explain the observed accelerations
when we are in a non

inertial
reference frame
The Strong Equivalence Principle
All inertial and freely falling
reference frames are equivalent
Recall from Special Relativity: All laws of physics
are the same in all inertial reference frames
A consequence of
the equivalence
principle is the
bending of light by
massive objects
Time is changed by gravity
Gravity slows time. The stronger the gravity, the slower time
flows. Unlike Special Relativity, you can tell whose clock is slow.
The one that is deeper in the gravity well is the slower one.
Gravitational
Redshift is another
consequence of
equivalence
Light loses energy as it “climbs” out of a gravity well. This
results in the light being redshifted by gravity.
General Relativity deals with
geometry
Euclidian Geometry: Flat
Spherical Geometry
Hyperbolic Geometry
The Metric is the formula for the
distance between two points
2
2
2
2
y
h
y
x
gf
x
f
r
2
2
2
2
2
2
)
(cos
R
R
r
Euclidian geometry
Spherical geometry
Hyperbolic geometry
2
2
2
2
1
2
1
2
22
2
2
2
2
1
2
1
2
12
2
2
2
2
1
2
2
2
11
1
1
1
1
1
x
x
x
a
g
x
x
x
x
a
g
x
x
x
a
g
So what kind of geometry do we
use in General Relativity?
Riemannian geometries are locally flat. On a small
enough scale they are Euclidian
The Earth is
a good
example of a
Riemannian
Geometry
The mathematics of
Riemannian geometry
had been worked out
in the mid 1800’s by
Georg Friedrich
Bernhard Riemann
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