Physics 2212, Dr' Holm

1
Special Theory of Relativity

• Principle of Relativity (from Newton in 1600s)
• The motion of bodies included in a given space
  are the same among themselves whether that space
  is at rest or moves uniformly forward in a
  straight line.
• Impact Experiments (e.g., dropping a ball,
  period of a pendulum, two objects colliding) in a
  boxcar moving with constant velocity will have
  the same results as the same experiments in a
  boxcar at rest.

2
Newtons Principle of Relativity(restated)

• The laws of mechanics (Newtons Laws) are
  invariant (the same) in all inertial
  (non-accelerating) frames of reference.
• OR
• Absolute uniform motion (or absolute rest)
  cannot be detected.

3
Galilean Transformation(relating frames of
reference)
4
Principle of Relativity and the Galilean
Transformation
Check!
5
Newtons Principle of Relativity(restated again!)

• Newtons Laws are invariant under a Galilean
  Transformation.

6
Difficulties Arose by the Late 19th Century

• The new Laws of Electromagnetism (Maxwells
  Equations)
• (1) were NOT invariant under a Galilean
  transformation
• (2) predicted electromagnetic waves (e.g.,
  light) moving at an absolute speed (c 3 x 108
  m/s).
• (Maybe Maxwells Equations were wrong?
• NO - predictions agreed with experiment)

7
Initial Reasoning/Theory(to eliminate
difficulties)

• (1) Newtons Principle of Relativity does
  NOT hold for electromagnetism.
• (2) There is an absolute rest frame (ether) in
  which the speed of light is c 3 x 108 m/s.

8
Testing to Validate Initial Reasoning/Theory

• Experiments were conducted to detect absolute
  rest frame (ether).
• Michelson-Morley Experiment
• Results Negative - absolute rest frame was not
  detected.

9
Meanwhile ...

• H. A. Lorentz asked (and answered) the following
  academic mathematical question
• Does there exist a set of transformation
  equations for which Maxwells Equations are
  invariant?
• YES!
• Lorentz Transformation

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Lorentz Transformation
Galilean Transformation
Lorentz Transformation
11
Lorentz Transformation

• Lorentz transformation was considered
  non-physical since
• (1) it linked time to position, and
• (2) Newtons Laws were not invariant under a
  Lorentz transformation.

12
Summary of Difficulties

• Newtons Laws and Maxwells Equations are both
  correct.
• Newtons Laws obey Newtons Principle of
  Relativity
• invariant under a Galilean Transformation
• no absolute rest frame
• Maxwells Equations do not obey Principle of
  Relativity
• not invariant under a Galilean Transformation
• predicts absolute speed of light, implying
  absolute rest frame
• Absolute rest frame cannot be detected
  experimentally
• Aside Maxwell Equations are invariant under a
  Lorentz Transformation, but Newtons Laws are
  not.
• Besides, Lorentz Transformation yields non
  physical results.

13
Enter Einstein (in 1905)
Two postulates of Special Theory of Relativity

• The Principle of Relativity holds for ALL laws
  of physics.

(including Maxwells Equations!)
Then what about that absolute value for the speed
of light?

• The speed of light is the same for ALL inertial
• frames of reference.

(c is just not the speed of light in some
absolute rest frame!)
Okay, then what are the correct transformation
equations? (One set wont work for BOTH Newtons
Laws and Maxwells Equations!)
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What are the Correct Transformation Equations?
But, according to Einstein (2nd postulate)
and
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What are the Correct Transformation Equations?
Assume transformation equations of the general
form
A to be determine
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What are the Correct Transformation Equations?
Solve for t
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Summary/Conclusion

• The correct set of transformation equations is
  the Lorentz Transformation.
• 1st postulate states that Principle of Relativity
  holds for ALL laws of physics.
• Maxwells Equations are invariant under a Lorentz
  Transformation while Newtons Laws are not.
• Therefore, Newtons Laws are wrong!

18
Newtons 2nd Law Revisited
Relativistically-correct form of Newtons 2nd Law
rest mass
relativistic mass
19
Impact of Relativistic Mass
Recall that power, P, is the time rate of change
of energy, E
relativistic mass
Newtons 2nd Law
energy and mass are equivalent!
20
Example
Let m 1 gram
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Time Dilation
Determine the relationship of the time
intervals between two events as measured in each
frame.
22
Time Dilation
time dilation
proper time is the minimum time between events.
23
Causality
Can the sequence of events as observed in one
frame be reversed as observed in another frame?
If so, what if one event causes the other event
(causality) (e.g., the throwing of a rock and
the breaking of a window)?
Example
Event A firing of gun (x0 m, t0 s)
Event B electron hitting screen (x1 m, t0.1
ns)
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Causality
Can this happen?
Check the speed of the electron
NO!
Why?
Check the mass of the electron
Therefore, c is the speed limit of the universe.
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Causality
When can the sequence of two events be different
in another reference frame?
Stated another way, given
when can
or
or
Thus,
Therefore, if a signal moving at c is sent at the
time, tA, and from the location, xA, of Event A
toward the location, xB, of Event B, it will not
arrive at xB, before time tB (when Event B
occurs).
Event A and Event B cannot be casually linked.
26
Length Contraction
Consider a meter stick at rest along the x-axis
in S.
Let Event A be the measurement of the left end of
the stick.
Let Event B be the measurement of the right end
of the stick.
L0 is independent of the times (tA tB) of the
measurements and is called the proper length.
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Length Contraction
Classically (Galilean transformation)
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Length Contraction
Relativistically (Lorentz transformation)
length contraction
Proper length is the length of an object as
measured in a reference frame in which the object
is at rest, or
the distance between two events as measured in a
reference frame in which the spatial coordinates
of those events are constant.
29
Example Space Travel
Event A Leaving the earth
Event B Arriving at Planet X
In S (observer on earth)
(they agree on the speed)
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Addition of Velocities
What speed does observer in S measure?
(Einsteins 2nd postulate says u c.)
31
Kinetic Energy
Classically
Relativistically
Binomial expansion (theorem)
32
Space Travel Example

• In observers A frame of reference, Alpha
  Centauri is 4 light-years (ly) from earth and
  spaceship C
• is 2 light-years behind spaceship B.

• At the instant that spaceship B passes earth,
  observer A and an observer (observer B) aboard
• spaceship B set their respective clocks to
  zero.

• When spaceship B reaches Alpha Centauri,
  observer B sends a signal (at the speed of light)
• back to earth and continues on with the same
  velocity.

Event 1 spaceship C arriving at earth
Event 2 spaceship B arriving at Alpha Centauri
( sending signal back)
Event 3 signal arriving at earth
When do these events occur according to observer
A and observer B?
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Space Travel Example
According to observer A (in S)
Event 1
?
Event 2
Event 3
Event 1 spaceship C arriving at earth
Event 2 spaceship B arriving at Alpha Centauri
( sending signal back)
Event 3 signal arriving at earth
When do these events occur according to observer
A and observer B?
34
Space Travel Example
Distance , L, from spaceship B to spaceship C is
proper length (contracted in S)
Distance , D, from spaceship B to Alpha Centauri
is contracted relative to S (proper length in
S)
Event 1 spaceship C arriving at earth
Event 2 spaceship B arriving at Alpha Centauri
( sending signal back)
Event 3 signal arriving at earth
When do these events occur according to observer
A and observer B?
35
Space Travel Example
Event 1
Event 2
Event 3
Event 1 spaceship C arriving at earth
Event 2 spaceship B arriving at Alpha Centauri
( sending signal back)
Event 3 signal arriving at earth
When do these events occur according to observer
A and observer B?
36
Time for Signal to Return to Earth
Event 1 spaceship C arriving at earth
Event 2 spaceship B arriving at Alpha Centauri
( sending signal back)
Event 3 signal arriving at earth
When do these events occur according to observer
A and observer B?
37
Time for Signal to Return to Earth(alternate
solution)
Which event, 1 or 2, occurs first?
FINAL QUESTIONS
According to A, Bs clock runs fast or slow ?
According to B, As clock runs fast or slow ?
Event 1
Event 2
Event 3
Event 1 spaceship C arriving at earth
Event 2 spaceship B arriving at Alpha Centauri
( sending signal back)
Event 3 signal arriving at earth
When do these events occur according to observer
A and observer B?