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?

We’ll draw it on a spacetime diagram. Ali’s rocket is 100 light- nanoseconds long so label the axes with a scale of five squares for fifty nanoseconds for the rest frame, F.

v = 3/5 c

Let Ali’s frame be the ‘rest’ frame F. Put Ali’s nose at the origin. Draw the tail at 100 ns on the x axis. Draw the position of Ali’s ship at t = -50, 0, 50, 100 and 150 ns.

v = 3/5 c

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

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, which is the origin of both axes. At this time, t=0, the laser cannon at Ali’s tail is fired. Mark this event with a dot.

v = 3/5 c

You can’t show the path of the laser shot on the diagram because it moves A) too fast B) too slow C) parallel to the page D) 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 x y F F’

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?

v = 3/5 c

v = 3/5 c This line marks all the positions ever occupied by the nose of Brenda’s ship. It doesn’t come near the cannon.

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

v = 3/5 c

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

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

v = 3/5 c

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

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

v = 3/5 c At t’ = -32 ns, Brenda sees her nose beside the cannon, well after the cannon was fired.

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.

v = 3/5 c

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

This is Brenda’s view.

v = 3/5 c

v = 3/5 c At t’ = 0, Ali’s nose is next to Brenda’s tail. At t’ = - 80 ns, the two ships are in this position and the shot is fired. At t’ = - 32 ns, Ali’s cannon is next to Brenda’s tail.

v = 3/5 c The shot does not occur at the same time as the tail and nose being lined up. The events are not simultaneous.

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