IB Physics – Relativity

Relativity

Lesson 21. Time dilation

2. Lorentz Factor

3. Proper time

4. Lorentz contraction

5. Proper length

6. Twin paradox and symmetric situations

7. Muon decay; evidence for time dilation

IB Physics – Relativity

Time dilation

IB Physics – Relativity

Time dilation

2

2

1cv

tt

Think about what this means?

????

IB Physics – Relativity

Proof of formula

The proof of the time dilation formula is a standard requirement in the exam.

Carefully work through the proof using Pythagoras’ Theorem.

Make sure you understand each step.

Hints

The “prime” notation refers to measurements in the ‘moving’ frame

The speed of light is the same for all observers.

IB Physics – Relativity

The Lorentz factor

2

2

1

1

cv

v 0.1c 0.2c 0.3c 0.4c 0.5c 0.6c 0.7c 0.8c 0.9c c

For what values of v is significant ?

Lorentz Factor

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2

v/c

Work out below

IB Physics – Relativity

Example of time dilation

tt If the train passengers measure a time interval of Dt1 = 6 s and the train moves at a speed v = 0.80c, calculate the length of the same time interval measured by a stationary observer outside the train standing on the ground

IB Physics – Relativity

Atomic clocks prove time dilation!

IB Physics – Relativity

Proper time

A proper time interval is the time separating two events that take place at the same point in spaceobserved time interval = x proper time interval

Note that the proper time interval is the shortest possible time separating two events

IB Physics – Relativity

Examples of proper time

1. The time interval between the ticks of a clock carried on a fast rocket is half of what observers on Earth record. How fast is the rocket moving with respect to the Earth? What are the two events here?

2. A rocket moves past an observer in a laboratory with speed = 0.85c. The lab observer measures that a radioactive sample of mass 50 mg (which is at rest in the lab) has a half life of 2.0 min. What half-life do the rocket observers measure? Again, what are the two events?

3. In the year 2010, a group of astronauts embark on a journey toward Betelgeuse in a spacecraft moving at v = 0.75c with respect to the Earth. Three years after departure from the Earth (as measured by the astronaut’s clocks) one of the astronauts announces that she has given birth to a baby girl. The other astronauts immediately send a radio signal to Earth announcing the birth. When is the good news received on Earth (according to the Earth Clocks)?

In each case first suggest in which frame the proper time is directly measuredTsokos, 2005, p562

IB Physics – Relativity

Length ContractionAnother consequence of the invariance of the speed of light is that the distance between two points in space contracts according to an observer moving relative to the two points.

The contraction is in the same direction as the relative motion.

0L

L Measured by observer who is stationary with respect to the object

Measured by observer in a moving frame with respect to the object A paradox on length contraction

IB Physics – Relativity

Proper Length

The proper length of an object is the length recorded in a frame where the object is at rest

Any observers moving relative to the object measure a shorter length (Lorentz contraction);

lengthproper

length

IB Physics – Relativity

Examples

1. An unstable particle has a life time of 4.0 x 10-8 s in its own rest frame. If it is moving at 98% of the speed of light calculate;

a) Its life time in the lab frame

b) The length traveled in both frames.

2. Electrons of speed v = 0.96c move down the 3 km long SLAC linear accelerator.

a) How long does take according to lab observers?

b) How long does it take according to an observer moving along with the electrons?

c) What is the speed of the accelerator in the rest frame of the electrons?

Tsokos, 2005, p566

Kirk, 2003, p145

IB Physics – Relativity

The twin paradox

Link to twin paradox

Read about this in Kirk p 146.

•The paradox arises because both twins view a symmetrical situation. Explain why?•Explain why it is not a paradox.

IB Physics – Relativity

The muon experiment

This offers direct experimental evidence of time dilation

Key points

Muons have an average lifetime of 2.2 x 10-6 s in their own rest frame.

They are created 10 km up in the atmosphere with velocities as large as 0.99c.

Show that without special relativity muons are unlikely to be detected on Earth.

Muon decay explanation