Length Contraction Special Relativity - fsu.edu
Transcript of Length Contraction Special Relativity - fsu.edu
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
College Physics B - PHY2054C
Special Relativity
11/10/2014
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CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Outline
1 Einstein’s Theories of Relativity
Special Relativity
2 Time Dilation
3 Length Contraction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Special Relativity
1 The speed of light is the maximum possiblespeed, and it is always measured to have thesame value by all observers.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Speed of Light
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Special Relativity
1 The speed of light is the maximum possiblespeed, and it is always measured to have thesame value by all observers.
2 There is no absolute frame of reference, and noabsolute state of rest.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Reference Frames
A reference frame can be thought of as a set of coordinate axes.
Inertial reference frames move with a constant velocity.
The principle of Galilean relativity is the idea that the laws
of motion should be the same in all inertial frames.
• For example, adding or subtracting a constant velocity
does not change the acceleration of an object and if
Newton’s Second Law (∑ ~F = m~a ) is obeyed in one
inertial frame, it is obeyed in all inertial frames.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Relativity
The term relativity arises when a situation is described from
two different points of view.
When the railroad car moves with a constant velocity, Ted and
Alice see different motions of the ball.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 1
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Alice, what is the ball’s velocity along x just after
the ball is released?
A zero
B v in +x direction
C v in −x direction
D v/2 in +x direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 1
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Alice, what is the ball’s velocity along x just after
the ball is released?
A zero
B v in +x direction
C v in −x direction
D v/2 in +x direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 2
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Alice, what is the ball’s velocity along x just before
the ball lands at Ted’s feet?
A zero
B v in +x direction
C v in −x direction
D v/2 in +x direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 2
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Alice, what is the ball’s velocity along x just before
the ball lands at Ted’s feet?
A zero
B v in +x direction
C v in −x direction
D v/2 in +x direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 3
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Alice, what is the acceleration of the ball along x?
A zero
B g in +y direction
C g in −y direction
D g/2 in −y direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 3
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Alice, what is the acceleration of the ball along x?
A zero
B g in +y direction
C g in −y direction
D g/2 in −y direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 4
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Ted, what is the force Fx on the ball (m = 0.6) kg
along x?
A zero
B mg in +y direction
C mg in −y direction
D mg/2 in −y direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 4
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
According to Ted, what is the force Fx on the ball (m = 0.6) kg
along x?
A zero
B mg in +y direction
C mg in −y direction
D mg/2 in −y direction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 5
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
Do Ted and Alice agree on the value of Fx?
A yes
B no
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Question 5
Ted travels in a railroad car at constant velocity while his motion
is watched by Alice, who is at rest on the ground. Ted’s speed
v is much less than the speed of light. Ted releases a ball from
his hand and observes that in his reference frame the ball falls
directly downward. Hence, according to Ted, the component of
the ball’s velocity along the horizontal direction is zero.
Do Ted and Alice agree on the value of Fx?
A yes
B no
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Relativity
• Ted observes the ball’s motion purely along the vertical.
• Alice sees projectile motion in both the x- and y-directions.
• Both agree that ay = g (due to gravity) and ax = 0.
➜ Newton’s Second Law is obeyed.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Galilean Relativity and Light
According to Maxwell’s equations, the speed of light, c, has a
constant value:
• He also showed that the speed of light is independent of
the motion of both the source and the observer.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Galilean Relativity and Light
According to Maxwell’s equations, the speed of light, c, has a
constant value:
• He also showed that the speed of light is independent of
the motion of both the source and the observer.
1. Newton’s mechanics predict that the speed of the light
wave relative to Alice should be c + v .
2. According to Maxwell’s theory, Ted and Alice should both
observe the light wave to move with speed c.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Michelson-Morley Experiment
1887: Michelson and Morley attempted to determine
Earth’s motion relative to the “absolute” space through
which light supposedly moved by measuring the speed of
light at different times of the day and on different days of
the year.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Michelson-Morley Experiment
1887: Michelson and Morley attempted to determine
Earth’s motion relative to the “absolute” space through
which light supposedly moved by measuring the speed of
light at different times of the day and on different days of
the year.
Far from measuring the
properties of absolute
space, the experiment
demolished the entire
concept.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Galilean Relativity and Light
Galilean Relativity and electromagnetism do predict different
results for observers in different inertial frames:
• Experiments showed that Maxwell’s theory was correct.
• The speed of light in the vacuum is always c.
• Galilean relativity for how the speed of light depends on
the motion of the source is wrong.
➜ Einstein developed theory of relativity: Special Relativity.
Two Postulates
1 All laws of physics are
the same in all inertial
reference frames.
2 The speed of light in the
vacuum is a constant.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Inertial Reference Frames
The modern definition of an inertial reference is one inwhich Newton’s First Law holds:
If a particle moves with a constant velocity, then thereference frame is inertial.
➜ Earth’s acceleration is small enough that it can beignored (can be considered an inertial system).
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Outline
1 Einstein’s Theories of Relativity
Special Relativity
2 Time Dilation
3 Length Contraction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Light Clock
The two postulates lead to a surprising result concerning the
nature of time.
A light clock keeps time by using a pulse
of light that travels back and forth between
two mirrors:
• The time for the clock to “tick” once is
the time needed for one round trip:
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Moving Light Clock
The clock moves with a constant velocity v relative to the
ground:
• From Ted’s reference frame, the light pulse travels up and
down between the two mirrors: ∆t 0 = 2l/c.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Moving Light Clock
The clock moves with a constant velocity v relative to the
ground:
• From Ted’s reference frame, the light pulse travels up and
down between the two mirrors: ∆t 0 = 2l/c.
• Alice sees the light pulse travel a longer distance, but the
speed of light is the same for Alice as for Ted.
➜ Because of the longer distance, according to Alice the
light will take longer to travel between the mirrors.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Moving Light Clock
For Alice, the time for one tick of the clock is:
∆t =∆t 0
√
1 − v2
c2
➜ The time for Ted is different from the time for Alice.
The operation of the clock depends on the relative motion.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Time Dilation
Special relativity predicts that moving clocks run slow.
This effect is called Time Dilation.
For typical terrestrial speeds, the difference between ∆t and ∆t 0
is negligible. ∆t 0 is called the proper time:
∆t =∆t 0
√
1 − v2
c2
=∆t 0
√
1 −(100 mph)2
c2
≈∆t 0
1
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Time Dilation
When the speed v is small compared to c, the factor√
1 − v2/c2 is very close to 1:
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Time Dilation
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Outline
1 Einstein’s Theories of Relativity
Special Relativity
2 Time Dilation
3 Length Contraction
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Lorentz Contraction
γ =1
√
1 − v2/c2:
L0 = v∆t 6= L = v∆t 0
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Lorentz Contraction
When measuring the length of the moving meterstick,you do so by noting the positions of the two ends atthe same time, according to your clock.
However, those two events – the two measurementsyou make – do not occur at the same time as seen bythe moving observer. In relativity, time is relative, andsimultaneity (the idea that two events happen “at thesame time”) is no longer a well-defined concept.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Special Relativity
1 The speed of light is the maximum possiblespeed, and it is always measured to have thesame value by all observers.
2 There is no absolute frame of reference, and noabsolute state of rest.
3 Space and time are not independent, but areunified as spacetime.
CollegePhysics B
Einstein’sTheories ofRelativity
Special Relativity
Time Dilation
LengthContraction
Special Relativity
Relativistic Addition of Velocities:
v ′ =v1 + v2
1 + v1 v2
c2
1 When two velocities are much less than the speed of light,
the relativistic addition of velocities gives nearly the same
result as the Newtonian equation.
➜ Okay for speeds less than ∼ 10% of the speed of light!
2 Experiments with particles moving at very high speeds
show that the relativistic result is correct.