Chapter 37
Special Relativity
37.2: The postulates:
Validity of Maxwell’s equations
The Michelson-Morley experiment
37.2: The postulates, The Ultimate Speed:
The speed of light in
vacuum has been defined
to be exactly
c =299 792 458 m/s.
It is found to be the
ultimate speed.
37.3: Measuring an Event:
An event is something that happens, and every event can be assigned three space coordinates
and one time coordinate.
1.The Space Coordinates. We imagine the observer’s coordinate system fitted with a closepacked, three-dimensional array of measuring rods, one set of rods parallel to each of the three
coordinate axes. These rods provide a way to determine coordinates along the axes.
2.The Time Coordinate. For the time coordinate, we imagine that every point of intersection
in the array of measuring rods includes a tiny clock, which the observer can read because the
clock is illuminated by the light generated by the event.
3.The Spacetime Coordinates. The observer can now assign spacetime coordinates to an
event by simply recording the time on the clock nearest the event and the position as
measured on the nearest measuring rods. If there are two events, the observer computes their
separation in time as the difference in the times on clocks near each and their separation in
space from the differences in coordinates on rods near each.
37.3: Measuring an Event:
# Synchronization of all clocks
is needed and achievable.
37.4: The Relativity of Simultaneity:
37.4: The Relativity of Simultaneity, A Closer Look:
Fig. 37-4 The spaceships of Sally and Sam and the occurrences of events from Sam’s
view. Sally’s ship moves rightward with velocity, v . (a) Event Red occurs at positions
RR’ and event Blue occurs at positions BB’; each event sends out a wave of light. (b)
Sam simultaneously detects the waves from event Red and event Blue. (c) Sally detects
the wave from event Red. (d) Sally detects the wave from event Blue. With the postulate
of constant speed of light, Sally would say event Red happens earlier.
37.5: The Relativity of Time:
37.5: The Relativity of Time:
In the previous case, Sally measures a proper
time interval, and Sam measures a greater
time interval. The amount by which a
measured time interval is greater than the
corresponding proper time interval is called
time dilation.
g is called the Lorentz factor.
Example, Time dilation of spacecraft which returns to Earth:
Example, Time dilation and travel
distance for a relativistic particle:
37.6: The Relativity of Length:
37.6: The Relativity of Length:
Let L0 be the length of a rod that you measure when the rod is stationary (meaning
you and it are in the same reference frame, the rod’s rest frame). If, instead, there is
relative motion at speed v between you and the rod along the length of the rod, then
with simultaneous measurements you obtain a length L given by
Since the Lorentz factor g is always greater than unity if there is relative motion, L
is less than L0.
37.6: The Relativity of Length, Proof:
Consider that both Sally, seated on a train moving through a station, and Sam,
again on the station platform, want to measure the length of the platform.
Sam, using a tape measure, finds the length to be L0, a proper length, because the
platform is at rest with respect to him. Sam also notes that Sally, on the train,
moves through this length in a time Dt =L0/v, where v is the speed of the train.
Therefore,
For Sally, however, the platform is moving past her. She finds that the two events
measured by Sam occur at the same place in her reference frame. She can time
them with a single stationary clock, and so the interval t0 that she measures is a
proper time interval. To her, the length L of the platform is given by
Therefore,
and finally,
Example, Time dilation and length contraction:
Example, Time dilation and length contraction as seen in outrunning a supernova :
37.7: The Galilean Transformation:
37.7: The Lorentz Transformation Equations:
Based on the
constancy of the
speed of light in
all inertial
frames
The Lorentz transformation equations are:
(The equations are written with the assumption that t =t’= 0
when the origins of S and S’ coincide.)
37.7: The Lorentz Transformation Equations:
37.8: Some Consequences of Lorentz Transformation Equations:
Simultaneity:
37.8: Some Consequences of Lorentz Transformation Equations:
Time Dilation:
37.8: Some Consequences of Lorentz Transformation Equations:
Length Contraction:
Example, Time dilation and reversing the sequence of events:
Example, Time dilation and reversing the sequence of events, cont.:
37.9: The Relativity of Velocities:
37.10: The Doppler Effect of Light:
The Doppler effect is a shift in detected frequency for traveling waves.
The Doppler effect for light waves depends on only one velocity, the relative
velocity, v, between source and detector, as measured from the reference frame of
either.
Let f0 represent the proper frequency of the source—that is, the frequency that is
measured by an observer in the rest frame of the source. Let f represent the
frequency detected by an observer moving with velocity relative to that rest frame.
Then, when the direction of is directly away from the source,
For low speeds (b « 1), this equation can be expanded in a power series in b and
approximated as
37.10: The Doppler Effect of Light, Astronomical Effect:
# Red shift and blue shift
37.10: The Doppler Effect of Light, Transverse Doppler Effect:
The transverse Doppler effect.
In terms of period of oscillations,
Where To is the proper period.
The Doppler effect of light in general:
1- b 2
f = f0
1+ br
37.11: A New Look at Momentum:
# momentum conservation in danger with the old definition p =mv
# concept of four vectors, proper time as a scalar
37.11: A New Look at Energy, Total Energy:
E = g mc
2
In a system undergoing a chemical or nuclear reaction, a change in the total mass energy of
the system due to the reaction is often given as a Q value. The Q value for a reaction is
obtained from the relation
37.11: A New Look at Energy, Kinetic Energy:
For v << c
37.11: A New Look at Energy, Momentum and Kinetic Energy:
E = g mc
p = g mv
2
37.11: A New Look at Energy, Mass Energy:
An object’s mass m and the equivalent energy E0 are related by:
This energy that is associated with the mass of an object is called mass energy or rest
energy.
Masses are usually measured in atomic mass units, where
1 u =1.660 538 86 10-27 kg,
and energies are usually measured in electron-volts or multiples of it, where
1 eV=1.602 176 462 10-19 J.
Example, Energy and momentum of a relativistic electron:
Example, Energy and discrepancy in travel time:
Homework:
Problems 14, 20, 32, 38, 58