Space Fight

Spacecraft of equal rest length pass very, very close to each other as they travel in opposite directions at a relative speed of 3/5 c. F’ F

Ali, the captain of ship F has a laser cannon at his tail that he plans to fire at the nose of Brenda’s F’ ship when their nose is lined up with her tail. F’ F

It is only supposed to be a warning shot across nose and he figures it won’t hit because her F’ length is contracted. F’ F

However, his co-pilot says that the shot will hit because Brenda sees that the F length is shortened. F’ F

Who is right? Draw it on a spacetime diagram. Label the axes with a scale of one square to 10 ns.

  200 120 150 80 100 120 80 50 50 100 150 v = 3/5 c

The ships have a length of 100 nm. Put Ali’s nose at the origin The ships have a length of 100 nm. Put Ali’s nose at the origin. Draw the position of Ali’s ship at t = -50, 0, 50, 100 and 150 ns.

  200 120 150 100 120 50 50 100 150 v = 3/5 c

Put Brenda’s tail at the origin Put Brenda’s tail at the origin. Draw the position of her ship at t’ = -80, -40, 0, 40 and 80 ns.

  200 120 150 100 120 50 50 100 150 v = 3/5 c

You can see the nose of Ali’s ship and the tail of Brenda’s ship at the same spacetime point. This is event #1 At this time t=0, the laser cannon at Ali’s tail is fired. This happens at a different point and is event #2

  200 120 150 100 120 50 Event #1 Event #2 50 100 150 v = 3/5 c

You can’t show the path of the laser shot on the diagram because it moves too fast too slow parallel to the page perpendicular to the page

You can’t show the path of the laser shot on the diagram because it moves into another dimension that is perpendicular to the page. t F’ y F x

However, the ships are very close and the shot will cross the distance in almost no time. The important question is, is the nose of Brenda’s ship nearby when the shot is fired?

  200 120 150 100 120 This line marks all the positions ever occupied by the nose of Brenda’s ship. 50 50 100 150 v = 3/5 c

At t = 0 ns, where does Ali see the nose of Brenda’s ship?

At t = 0 ns, Ali sees the nose of Brenda’s ship at x = 80 ns. 50 100   200 120 150 100 120 50 At t = 0 ns, Ali sees the nose of Brenda’s ship at x = 80 ns. 50 100 150 v = 3/5 c

At t’ = 0 ns, where does Brenda say her nose is?

At t’ = 0 ns, Brenda says her nose is at x’ = 100 ns. 50   200 120 150 100 120 At t’ = 0 ns, Brenda says her nose is at x’ = 100 ns. 50 50 100 150 v = 3/5 c

When does Brenda say that her nose was closest to the laser cannon?

  200 120 150 100 120 50 At t’ = -32 ns, Brenda sees her nose beside the cannon, well before it is fired. 50 100 150 v = 3/5 c

Ali and Brenda disagree because of a difference in simultaneity Ali and Brenda disagree because of a difference in simultaneity. The simultaneity of two events depends on the frame of reference .

This is Ali’s view.

  200 120 150 100 120 50 The shot is fired when the nose and tail are lined up. The ships are in the position shown. 50 100 150 v = 3/5 c

This is Brenda’s view.

The two ships are in this position when the shot is fired.   200 120 150 The shot does not occur at the same time as the tail and nose being lined up. The events are not simultaneous. At t’ = 0, the two ships are in this position, with the nose and tail lined up. 100 120 50 50 100 150 The two ships are in this position when the shot is fired. v = 3/5 c

Not everything is relative. They both agree that the shot misses Not everything is relative. They both agree that the shot misses. They just don’t agree as to why.

Reality Check#6: Muons are formed at the top of the atmosphere by cosmic rays. Muons that are at rest have a half-life of 2.2 x 10-6 s.

However, when they are streaming toward the Earth at 0 However, when they are streaming toward the Earth at 0.995 c, we see them live 10 times longer!

The muons would disagree. Their time feels completely normal. They would say that the muons at rest in the Earth lab were living 10 times longer. How could the muons explain the fact that they managed to make it through the 10-kilometer thick atmosphere?

This diagram shows the problem with the speed slowed down. Event#2: Muon hits Earth This diagram shows the problem with the speed slowed down. Event #1: Muon enters atmosphere Atmosphere Earth

The times between events are different. Dt Dt’ The distances between events are also different. Dx Dx’

We measure the muons living ten times longer. Dt Dt’ The muons measure an atmosphere that is ten times shorter. Dx Dx’

The Earth measurement of length is the largest any frame will measure. We and our ruler are at rest with both positions. This is called the ‘proper’ length.

The muon’s time interval is the shortest any frame will measure. They and their clock are at rest with both events. This is called the ‘proper’ time. Dt

Not everything is ‘relative’. We agree on our relative speeds and the speed of light. c = Dx’ /Dt’ = Dx /Dt Dt Dt’ Dx Dx’

We also agree on the number of muons getting the Earth. We just don’t agree on the explanation of the two events.