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.