Einstein’s theory of special relativity makes some very bizarre and counter-intuitive predictions. Anything that violates common sense like this must be strongly supported by logic and evidence before we are willing to accept it.

A rocket ship is heading toward you at ½ c. How fast does the light from its headlamps travel toward you? a) c b) 1.5 c c) 0.5 c d) something else The speed of light is c in all frames.

You have been placed in a rocket traveling at 99 % c and cannot look out the window. List all the tests that you could do to tell that you are traveling so fast. There are no tests that you can do. All inertial frames are equivalent.

1) The speed of light is c in all frames. 2) All inertial frames are equivalent. All the strange predictions of Special Relativity come from these two postulates and can be developed by examining the geometry of spacetime diagrams.

Spacetime diagrams can help you to visualize relativity. They are similar to scale diagrams and freebody diagrams.

A spacetime diagram has time vertical. The axes are calibrated in years and light-years. t x

Which line represents an object at rest? t x ABCDABCD

Which line represents light? t x ABCDABCD

Which line represents an impossible motion? t x ABCDABCD t x

We want a line for an object moving at 3/5 c. This will be the t’ axis for the ‘moving’ frame. Which one is correct? x t’ t x x t x t x x A B C

A speed of 3/5 c has a slope of 5/3. x t’ t

Which is the correct x’ axis? x t’ t x x x’ t x t x t x A B C t’ x’

Hint: What is true about light in all frames? x t’ t x x x’ t x t x t x A B C t’ x’

x t’ t The two axes must be symmetric about the path of light because light must have the same speed in all frames x’  x’  t’

The Cosmic Speed Limit Suppose you launch a pod at ½ c from a rocket traveling at ½ c relative to the Earth. How fast will it go?

x t’ t The pod travels at ½ c relative to the rocket. Which line is the pod’s? x’ ABCDABCD

x t’ t x’ The pod must cover one unit of space in two units of time. ABCDABCD

x t’ t How fast is it moving relative to the Earth? x’

x t’ t It travels at 4/5 c x’ How fast is it moving relative to the Earth?

Reality Check #1: The cyclotron at Triumf can form pions moving at 0.96 c which decay by emitting muons and neutrinos.

Many of these emitted particles go faster than 0.96 c, but none go faster than light.

Reality Check #2: At CERN, neutral pions were accelerated to c. When these pions decayed, they emitted light.

All the light emitted by the pions travelled at c.

Time Slows Down x t’ t x’ This line marks simultaneous later times. The x axis marks all the places where t = 0.

x t’t x’ The x’ axis marks all the points where t’ = 0. This line marks equal or simultaneous times t’, in the other frame, F’.

x x’ Suppose lightning strikes twice as shown. Did the strikes happen at the same time? t’ t Simultaneity is relative.

x x’ Consider the point where the two gridlines cross. Does t = t’? t’ t Hint: Does t’ = t”? t” They can’t be the same. Let’s say that t =  t’

We can find the formula for  by considering a ‘light’ clock, which sends light up and down between mirrors once every time t.

Suppose that this clock is on a rocket ship moving at a speed v to the right. We observe that the light travels farther in our frame. To keep c constant, it must also take more time, c =  x/  t. Therefore, our time interval is larger than the rocket’s. Their clock runs slower.

How much more slowly can be found by applying Pythagoras’ theorem to the diagram below. Write the equation and solve for t. ct ct’ vt  = 1/ 1 – v 2 /c 2 t = 1/ 1 – v 2 /c 2 t’ c 2 t 2 = v 2 t 2 + c 2 t’ 2 (c 2 - v 2 ) t 2 = c 2 t’ 2

This formula does not depend on the fact that we used a ‘light’ clock. Anything that ‘ticks’ will do. If you were on a rocket ship and sent a signal to Earth every time your heart beat, doctors on Earth would say that your pulse was slow.

If the Earth doctor sent you a signal every time that her heart beat, you would say that her pulse was a) faster b) normal c) slower

Hint: All inertial frames are equivalent. a) faster b) normal c) slower

What is , if the relative speed of the two frames is 3/5 c? x t’ t x’t”  = 1/ 1 – v 2 /c 2

x t’ t x’  = 5/4 What is t’, if t = 10? 10 8

What is t”  ? x x’ t” t”  = 4/5 * 8 =

t’ is shorter but it ‘looks’ longer. x t’ t x’ 108

Reality Check#3: Muons at CERN were accelerated to high speeds and lived 20 times longer than normal.

The Twin Paradox Brenda goes off at 4/5 c to a distant star and then returns. Her twin brother Ali stays on Earth. When she gets back she is no longer the same age as Ali.

x t The first part looks like this. During this half, Ali sees Brenda’s clock run slowly and Brenda sees Ali’s clock run slowly. t’ x’ Ali Brenda

x t The second part looks like this. During the second half they each see the other person’s time run slower. t’ x’ Ali Brenda

x t Suppose Ali ages 10 years during the trip. How much did Brenda age? t’ x’ 10 5 What is v? v is 4/5 c. What is  ?  is 5/3. What is the time in the middle of the trip? 3 Brenda only aged 6 years during the trip.

x t If Brenda always sees Ali’s time ticking more slowly and Ali always sees Brenda’s time ticking more slowly. Why don’t they age by the same amount? t’ x’ Brenda turns around. Her frame is not inertial and therefore not equivalent.

x t If Brenda always sees Ali’s time ticking more slowly and Ali always sees Brenda’s time ticking more slowly. When does Ali pick up the extra 4 years? t’ x’

x t During the turnaround. t’ x’

x t This is Brenda’s time at the start of the turn… t’ x’

x t... in the middle … t’ x’

x t … and at the end. t’ x’

x t Almost no time passed for Brenda during the turnaround, while a lot passed for Ali. t’ x’

Reality Check #4: GPS satellites are constantly turning around.

This makes their time run slower than ours by 8.99 x s every second.

If the GPS failed to adjust for the different times, then they would be out by 2.5 km after one day!

The Doppler Shift Brenda goes on another long voyage. x t’ t

Ali sends Brenda a message every year. How many messages does she receive on the way out, back? x t’ t

The frequency is greater on the way back when the ship is moving toward the signals. This is similar to the Doppler shift for sound. x t’ t

Brenda also sends Ali messages every year. How many will he receive compared to Brenda? A) the same number B) fewer C) more x

She sends fewer because only fewer years have passed for her, so Ali receives fewer. x t’ t

Reality Check #5: Physicists know that the universe is expanding because the frequency of light from galaxies is Doppler shifted.

The galaxies are moving so fast, that astronomers must use the formula for the relativistic Doppler shift, f o = f e (1-v/c)/(1+v/c)

This animation shows how things would look to an observer moving to the right. It shows the colour changes due to the relativistic Doppler shift and the distortion of space.