Special and General Relativity
Inertial reference frame: a reference frame in which an acceleration is the result of a
force.
Examples of Inertial Reference Frames
1.
This room .
Experiment: Drop a ball. It accelerates downward at 9.8 m/s2 due to the
force of gravity.
2.
The inside of a car moving at constant speed along a straight road.
Repeat the experiment. Results are the same as in #1.
3.
The inside of an elevator that is moving either upward or downward at
constant speed. Results are the same as in #1 and #2.
Examples of Non-inertial Reference Frames
1.
2.
The interior of a car that is either speeding up, slowing down, or going
around a curve.
Experiment:
Drop a ball. If the car is slowing down. The ball accelerates downward and
towards the front of the car. The acceleration toward the front of the car is
not due to a force on the ball.
The inside of an elevator that is accelerating either upward or downward.
If the elevator is accelerating upward, the ball accelerates downward faster
than 9.8 m/s2. The additional downward acceleration is not due to a force on
the ball.
Fundamental Assumptions of the Special Theory of Relativity
1.
2.
The laws of physics have the same form in any inertial reference frame; in
other words, no experiment can be done in an inertial reference frame to
detect its state of motion.
The speed of light in vacuum has the same value when measured by any
observer, regardless of the observer’s state of motion.
Time Dilation as a Consequence of the Postulates
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•
Both O and O' (pronounced “O prime”) are in inertial reference frames. O
stands by the railroad track and watches O' move by at constant velocity.
Construct a simple clock in the railroad car, consisting of a mirror on the floor
and another directly above it on the ceiling. Let a pulse of light bounce back
and forth between floor and ceiling. Each round trip is a tick of the clock.
Reference Frame of Observer O'
Moving at Constant Velocity
Direction of the Railroad
Car’s Constant Velocity
Reference Frame of a Stationary Observer O
Time per Tick as Measured by O’
O' sees the light having only vertical motion.
L
t  t up  t down
tup
tdown
t 
L L

c c
c = the speed of light = 3.00×108 m/s
t 
2L
c
Time per Tick as Measured by O
O sees the light moving horizontally as well as vertically.
Use the Pythagorean theorem to calculate tup and tdown
c2 t 2up  L2  v 2 t 2up
L2
t  2
c  v2
t down  t up
2
up
t up 
L
c
ctup
1
1
v
ctdown
L
2
c2
vtup
vt
t  t up  t down
2L
t
c
1
1
v2
c
2
t
t
1
v2
c2
This is the time dilation formula of special relativity.
•The time per tick measured by the moving observer O' is shorter than the time
per tick measured by the stationary observer O.
•Any clock at rest relative to O' is slower than any clock at rest relative to O.
•Time elapses more slowly for a moving observer.
Length Contraction
Similar logic shows that when O measures the distance between two points on the train
along the direction of motion, he gets a smaller result than O' does. This is called length
contraction and its quantitative expression is the following equation.
v2
L = L¢ 1 - 2
c
Ĺ́ ́́ is the distance between two events as measured by someone at
rest relative to the events, and L is the distance measured by
someone moving relative to them.
Examples
1.
2.
Suppose that O and O' have identical meter sticks and that the speed of O'
relative to O is 0.8 times the speed of light. When O measures the length of
O'’s meter stick, he finds that it is only 0.6 m long. Likewise, when O'
measures Os meter stick, he finds that O’s meter stick is only 0.6 m long.
A and B are 30 year old twins. A stays on Earth while B travels in a
spaceship at 0.99875 times the speed of light to a star 10 light years away
and immediately returns after making some observations that take only a
few hours. How old are the twins when B arrives back at Earth? How far
did B travel?
A watches B make a round trip of L = 2 ×10 = 20 light years while moving at about
1 light year per year. For him, 20 years elapse (t = 20 years) and he is (30 + 20) = 50
years old.
On the other hand, B experiences time dilation. For her, the elapsed time is
t' = 20 × 0.05 = 1 year. She is only 31 years old when she returns.
While A watches B make a 20 light year round trip, B finds herself traveling
only L' = 20× 0.05 = 1 light year.
3. Atomic Clock Experiment by C.O. Alley in 1975
t  15 hours
v  104 m / s  313 mph
t  5.6  107 s measured.
t  5.7  107 s predicted by special relativity.
4. Muon decay
lifetime in laboratory = 2  106 s
measured lifetime of cosmic ray-generated muons = 30  106 s
5. Dependence of Mass on Speed
m
m0
v
1  
c
2
6. Mass-energy Equivalence
E  m0 c 2
Spacetime and General Relativity
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•
•
•
•
•
•
The examples on the previous slide illustrate the fact that the time between two
events and the distances between them are not absolute. They depend on the
motion of the person who measures them.
However, this doesn’t mean that everything is relative. It is important to know
what is absolute (independent of the observer). Physicists recognize that a
combination of space and time called spacetime is the same for all observers.
For both Aand B, the spacetime interval between B́́s departure and return is
zero.
The space-time of special relativity is one in which the shortest spacetime path
between two points is a straight line. In a vacuum, for example, the path of a
ray of light is a straight line
The presence of mass bends space-time. The result is what we call gravity. A
satellite in orbit, for example, follows the curvature of spacetime. Physics that
takes this into account is called general relativity.
According to general relativity, gravity causes a time dilation similar to that
caused by motion.
Gravity affects time by slowing it down; i.e., time elapses more slowly where
gravity is strong.
Gravity affects spacetime by bending it, stretching it, and compressing it.
Local Inertial Reference Frames
A local inertial reference frame is one that is falling freely and is small enough for tidal effects
to be negligible.
L2
L3
A planet or
some other
massive object
a=g
L1 is far from any massive
object and is accelerating. It is
not a local inertial reference
frame
L1
a = -g
Experiments in L2 will show the
effects of gravity. It is not a local
inertial reference frame.
L3 is falling freely. It is a local inertial
reference frame if it is small enough that no
experiment in L3 can show the effects of
gravity.
The Equivalence Principle
•
•
•
If L is at rest in a region where there is a uniform gravitational field g and L'
is far from any massive bodies but is undergoing an acceleration, – g, identical
experiments in the two laboratories will give identical results.
A locally uniform gravitational field is equivalent to a uniform acceleration
No experiment conducted inside a sufficiently small laboratory can distinguish
between a uniform gravitational field and a uniform acceleration of that
laboratory.
L'
a = -g
A
B
L
A
B
Laser beam from A to B in L'.
AB curved due to acceleration.
g
Laser beam from A to B in L.
Same result as in L'. The effect
of gravity is to bend spacetime.
In a freely falling laboratory, AB
would be a straight line.
The Bending of the Path of a Light Ray as it Passes a
Massive Object (Star, Neutron Star, Black Hole)
S
C
E
The diagram shows Earth (E), a massive object (C), and a distant star (S). C could be a star,
neutron star, or black hole.
The solid green line shows the path of a light ray from S. Instead of being a straight line as it
would if C were not present, it is curved because C bends the space-time in its neighborhood.
From Earth, the star appears to be located in the direction indicated by the dotted line.
Animations showing what we would see if we approached a neutron star or black hole can be
found at http://antwrp.gsfc.nasa.gov/htmltest/rjn_bht.html.
Double Einstein Rings
A foreground galaxy ( about 3 billion light years away) bends space around it to form a
“gravitational lens”. As light from two more remote galaxies travels toward us, it
follows the curvature of this space to form the “Einstein rings” shown here.
Deflection of starlight passing near the sun:
Prediction of general relativity: 1.75".
Best measurement: 1.66" ± 0.18"
Gravitational Time Dilation
Time elapses more slowly in regions where gravity is strong than in
regions where it is weak. Specifically, time elapses more slowly near a
compact stellar object than it does far from it. This is expressed
quantitatively by the equation at the right.
t
t
R
1 s
r
t is the time interval between two events that take place at distance r from the center of
the compact object as measured by an observer very far from the compact object.
t' is the time interval between the same events as measured by an observer O' at distance
r from the center of the compact object.
Rs = the Schwartzschild radius = (3 km) × M
M = the mass of the compact object in solar masses.
An indestructible astronaut O' falling into a black hole would find himself very quickly
crossing the event horizon (time t'). As we watched him, according to the time dilation
formula, we would never see him arrive at the event horizon because as r approaches Rs, t
approaches infinity.
Perihelion Precession
Measured Perihelion Precession
Planet
Observed Excess
Precession
Relativistic
Prediction
Mercury
43.11±0.45
43.03
Venus
8.4 ± 0.48
8.6
Earth
5.0 ± 1.2
3.8
Icarus
9.8 ± 0.8
10.3