Spacetime diagrams can help you to visualize relativity

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 A B C D x

Which line represents light? t A B C D x

Which line represents an impossible motion? t A B C D x Do question #1.

Draw a line for an object moving at 3/5 c. This will be the t’ axis for the ‘moving’ frame. t x

Which one is correct? A t B C t t’ x x

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

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

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

The speed of light is one light-year per year in all frames, so c = Dx/Dt. t t’ x’ rise = Dt run = Dx x

Similarly, c = Dx’/Dt’. t t’ rise = Dt’ x’ run = Dx’ x Do question #2.

The Cosmic Speed Limit The cyclotron at TRIUMF can form charged pions moving at 0. 96 c. These decay by emitting muons and neutrinos at high speeds. The emitted particles go faster than 0. 96 c but they never go faster than c. Suppose you launch a pod at ½ c from a rocket traveling at ½ c relative to the Earth. How fast will it go?

The pod travels at ½ c relative to the rocket. Which line is the pod’s? A B t t’ C D x’ x

The pod must cover one unit of space in two units of time. A B t t’ C D x’ x

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

It travels at 4/5 c t t’ x’ x Do question #3.

Time Slows Down You are in a rocket ship moving at 3/5 c past the Earth and send a signal to Earth every time your heart beats. Doctors on Earth say that your pulse is slow. If they sent a signal to you every time their heart beat, you would say that their heart was beating A) fast B) normal C) slow How is this possible?

This line marks all the places with simultaneous later times, t. t t’ x’ The x axis marks all the places where x t = 0.

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

What is g, if the relative speed of the two frames is 3/5 c? g= 1/ 1 – v 2/c 2 g= 5/4 t t” t’ x’ x

What is t’, if t = 10? t 10 t’ 8 x’ x Frame F sees time slowed in F’.

Both frames can What is t, if t’ = 8? see each other’s time slowed t t’ because they disagree on 10 8 simultaneity. 6. 4 x’ There are three spacetime points, not two. x Frame F’ sees time slowed in F.

If the black event caused another event then a signal had to travel at c, or a slower speed to the second event. In what order do the three explosions happen? t t’ x’ Which events(s) could the black x explosion cause?

Muons are formed at the top of the atmosphere by cosmic rays and race toward the Earth at 0. 995 c. What is gamma at this speed?

The atmosphere is 10 km think. How long does it take for the muons to pass through the atmosphere? t = 10, 000 m/0. 995 x 3. 00 x 108 m/s = 3. 35 x 10 -5 s

The muons time has been slowed down. How much time passes for them? 3. 3 x 10 -5 s/10 = 3. 3 x 10 -6 s

The muons say that the time is shorter, but they agree on the speed of 0. 995 c. How is this possible? They say that the atmosphere is only 1 km thick. The atmosphere is moving relative to them and is contracted by gamma.

This a muon going at 3/5 c through the atmosphere. The muons measure a shorter time and they also measure a shorter atmosphere. t’ t 10 8 x’ 8 10 x

The muons measure the t shortest time for any frame and this is called the ‘proper’ time. 10 The Earth measures the longest distance and this is called the ‘proper’ distance t’ 8 x’ 8 10 x