Galilean Story

Galileo was familiar with several aspects of relativity.

Galilean Relativity differs from Special Relativity in that Special Relativity is built around the idea that the speed of light is constant in all inertial frames of reference.

In Galilean Relativity, if a flashlight on a train traveling 50 mph shoots a burst of light, the speed of the burst will be . No one is bothered that this is faster than the speed of light.

For our story we will use numbers that favor simple calculations. We will have an unspecified distance unit called “1 length”. Also, we will have a graph with ‘x’ and ‘t’ where ‘x’ is a line that coincides with the direction that the speed ramp travels and ‘t’ is time in seconds.

There is a speed ramp in an airport that travels at 0.25 lengths per second and you are capable of walking at a speed of 0.25 lengths per second.

In the first scenario you are walking alongside the speed ramp while your friend stands still on the speed ramp. Relative to the ground both of you are moving 0.25 and both of you are moving 0.00 relative to the speed ramp. Take some time to think about that.

In the second scenario you are walking at 0.25 on the speed ramp so relative to the ground you are moving 0.5 and now your friend, who is on the ground, has to trot/run to get the 0.5 relative to the ground needed to keep up with you.

We now consider the notion of “moving in time”. The motion you generate with your muscles can be used to move you back and forth in a way that changes the ‘x’ value. However, you can’t do anything to change the ‘t’ variable, you simply keep moving “one second per second”.

We have two coordinate systems in the illustration below. Each

Also, the direction of travel must always be in the positive ‘t’ direction. In the illustration below for the (gray, blue) coordinate system time moves you straight up (north) and for the (gray, green) coordinate system time moves you upward but slightly towards the east.

You are riding the speed ramp and you are using the (gray, green) coordinate system. The yellow line which is true for both you and your friend is coincident with your time direction. You are standing still (no change to your ‘x’) but you are steadily moving along in time at a rate of one second per second.

Your friend uses the (gray, blue) coordinate system and we see that after four seconds (4 units of up) your friend has moved one unit in the ‘x’ direction (the distance between adjacent blue lines). This is in agreement with the observation that your friend had to walk to keep up with you.

Bonus Features

On the (gray and green) coordinate system the yellow line segment went from (2,0) to (2,4). You will get extra points on the test for pointing out that as physicists we would prefer to type (0,4) to show a change of zero for ‘x’ and a change of four for ‘t’.

On the (gray and blue) coordinate system the yellow line segment went from (2,0) to (3,4) and we would prefer to type that as (1,4).

Below we show the two renditions of the same vector, one with the gray and blue basis vectors and the other with the gray and green basis vectors.