Time must be relative

If the speed of light is the same even if the light source itself is moving at some speed relative to the person doing the measuring, then how does this affect the way different observers measure time and space and combine them into spacetime?

Let's take the red and blue cars from the previous example, and let's mount a laser on top of the blue car. We'll put a mirror on the blue car as shown in the figures, and aim the laser at the mirror perpendicular to the direction the blue car is travelling.

Here is the laser pulse viewed in the frame of reference of the blue car.

Within the time interval T' measured by the driver of the blue car operating the laser, the laser light travels a distance 2 L from the laser to the mirror and back.

Meanwhile, the driver of the red car sees the blue car go whizzing by. According to the red driver, the laser took a total time T to hit the mirror and return. (She's willing to agree with the driver of the blue car that the distance between the laser and the mirror is L, since neither driver has any velocity in that direction.) She measures the blue car to have travelled a distance X = U T and the laser pulse to have travelled a total distance of 2 D = c T in time T.

Here is the laser pulse viewed in the frame of reference of the red car.

Here is where the problem arises. Since we just deduced above that c T' = 2 L and c T = 2 D, the only way we could have T' = T would be if D = L.

Therefore, the red and blue drivers do not measure time equally..

Now how can we sort this out and find out how their measurements of time should differ in order to account for the observed constancy of the speed of light?

What both drivers agree on is the distance L perpendicular to the motion of the cars. Using the Pythagorean Rule on the laser pulse's path in space, we get

L² + (X/2)² = (c T/2)² .

From the blue driver's point of view L² = (c T'/2)² , and putting these together and using X = U T we get: (c T')² + (X)² = (c T)² or

T = T' / (1 - (U/c)²)^1/2

This is stupendous! The drivers of the blue and red cars don't measure the same time for the laser to hit the mirror and come back! We are forced to conclude this if we want to be consistent with the observed constancy of the speed of light (so far) in Nature.

This has profound implications for the mathematical modelling of space and time as observed in Nature. This means we have to expand the idea of Euclidean analytic geometry to include the observer-dependent relativity of measurements of time and space. This opens up a gigantic can of mathematical worms, eventually bringing us to black holes, wormholes and at least the abstract mathematical possibility of time travel.

What we've learned here is called Relativistic time dilation. The process that occurred in the blue driver's rest frame with in time T' was perceived by the red driver to have occurred in time T = T' / (1 - (U/c)²)^1/2, which can be much much greater than T' if U is close to the speed of light c.