Space must be relative, too

In the previous frame we learned that if we want to use geometry to model space and time together, in order to account for the observed constancy of the speed of light, observers moving at constant velocity relative to one another perceive the passage of time differently.

Now we will figure out whether these observers measure space differently as well.

Let's suppose the driver of the blue car below has measured the length of her car with a ruler to be length L'. She sees the red car driving by her at velocity -U. The blue driver sees that it takes a time T' for the front of the red car to pass from the front end to the back end, and she calculates that therefore L' = U T'.

L' is the blue car length, and T' is the time for the red car to pass, as measured by the blue driver.

The driver of the red car, on the other hand, sees the blue car rushing past her. She measures the time T it takes for the blue car to pass her front bumper. She then calculates the length L of the red car to be L = U T.

So we have L/L' = T/T'. Now we just need to know the relationship between T and T'. But we calculated that already in the previous frame.

L is the blue car length, and T is the time for the blue car to pass, as measured by the red driver.

Recall that in the previous section, the blue driver was timing a laser pulse and its reflection from a mirror. These were events that happened at the same place according to the blue driver, and their time separation was measured to be T', with only a single clock needed for the measurement.

The red driver saw the laser pulse and return happening in different places (because she saw the laser as moving with the blue car) and so the red driver's time measurement T could only be made with a minimum of two (synchronized) clocks.

Constraining the speed of light to be constant gave us this relationship between the time T' measured by the blue driver and the time T measured by the red driver:

T' = T (1 - (U/c)²)^1/2.

The blue driver with the laser is like the red driver in this example, trying to measure a time interval between two events happening at the same place. The blue driver in this example is like the red driver from the previous section, trying to measure a time interval between two events in two different places.

Therefore, the relationship T = T' (1 - (U/c)²)^1/2 must hold between T and T' in this example. (The roles of T and T' are switched compared with the last section because the roles of the red and blue drivers with respect to time measurement has switched, as explained above.)

But this means that the relationsip between the lengths L and L' must be

L = L' (1 - (U/c)²)^1/2

Therefore, if we want to make a mathematical model for spacetime that is consistent with the observed constancy of the speed of light, we have to conclude in this model that measurement of space and time is not the same for all observers.

What we've learned here is called Relativistic length contraction. The blue car measured to have length L' in the blue driver's rest frame was measured by the red driver to have have the length L = L' (1 - (U/c)²)^1/2, which can be much much smaller than L' if U is close to the speed of light c.

Is there anything that is the same for all observers? That is what we'll look at next.