# Simultaneity

#### Alan and the spark

Alan stands by a railroad track. There is a small electrical wire sticking out of the ground. Every now and then, it emits a spark.

Light waves emitted by the spark radiate outwards from the wire in a sphere which expands at the speed of light.

#### Alan, Beth and the spark

Beth stands on a stationary railroad car, right next to Alan.

A spark appears.

Light waves radiate outwards at the speed of light.

Both Alan and Beth agree that the waves propagate at the same speed: c = 300,000 km/sec.

#### Alan, moving Beth, and the spark

In Alan's frame of reference ...

Alan stands by the railroad track. Beth stands on a railroad car which moves at speed v to the right relative to Alan.

As she passes Alan, a spark occurs at their mutual location.

Beth and the railroad car continue to roll onwards to the right. The wave front from the spark expands outwards.

Note that Beth is moving at a pretty heft fraction of the speed of light! (Note that she must also have pretty stiff hair gel.)

Alan notes that the waves propagate at speed c = 300,000 km/sec.

In Beth's frame of reference ...

Alan, the tracks, and the wire are moving to the left at speed v.

As Alan passes her, a spark occurs at their mutual location.

Alan, the tracks and the wire continue to move to the left. The wave front from the spark expands outwards.

Beth notes that the waves propagate at speed c = 300,000 km/sec.

#### Alan, moving Beth, and two sparks

Beth has a really cool tape deck sitting on the railroad car at her feet.

It has a cassette of Beethoven's Fifth Symphony sitting in it, rewound to the very beginning. The deck has two light sensors: one at the front end, and one at the back end. It is programmed to start playing the tape if a light wave strikes both sensors at the same time.

In Alan's frame of reference ...

Alan stands by the railroad track. There are two loose wires on the track. Beth stands on a railroad car which moves at speed v to the right relative to Alan.

As Beth passes Alan, both of the wires spark simultaneously.

Light radiates outwards from the two sparks at the speed of light.

Both of the wavefronts reach Alan at the same time.

Questions:

1. Does the tape deck start to play?

In Beth's frame of reference ...

Beth (and her tape deck) are at the center of the railroad car. Alan, the tracks, and the landscape whizz by to the left at speed v. Alan stands by the railroad track. There are two loose wires on the track. Beth stands on a railroad car which moves at speed v to the right relative to Alan.

Just as Alan passes Beth, sparks occur simultaneously at both ends of the railroad car.

Questions:

1. Draw the wave fronts radiating from the sparks in Beth's frame of reference.

Small version of events in Beth's frame
2. Does the tape deck start to play?

The railroad car comes to a halt a short distance down the track. Beth grabs her tape deck and hops off the car onto the ground. She and Alan walk towards each other. They meet.

Big Question:

1. Is the tape deck playing?