## Measurement is theory-laden part 2

In my previous post in this series I discussed the idea that all measurement is theory laden and gave an example of a bad explanation of a measurement. In this post, I will give a good example, from the special theory of relativity. The point is to illustrate  that a good explanation consists of taking an idea seriously and working out its consequences, not of sweeping apparently strange implications under the rug. Doing that is an error since either those new implications are false and your idea is a dud or they are true and you have discovered something new and important. You can’t find out which of those two possibilities is true without working out the consequences of the idea.

First, two definitions that will be relevant. An frame of reference is a set of physical systems used to measure the motion of other systems. An inertial reference frame (IRF) is a frame of reference in which objects that do not suffer some external force move in straight lines. Empty space far away from any masses is a good approximation to an inertial frame. If you measure everything relative to your car while it is accelerating that won’t be a good approximation to an inertial frame because you will see things accelerating when no forces act on them.

Special relativity uses two assumptions.

(1) All IRFs are the same with respect to the laws of physics. For example, if two people use different IRFs, they will both see that objects obey conservation of momentum.

(2) The speed of light in empty space has the same value in all IRFs, denoted by c.

The second assumption looks a bit strange. If you were to get in your car and drive down the road, the cars coming in the opposite direction would be moving at a different velocity relative to you if you speed up or slow down. If you’re driving at 30mph with respect to the pavement and the chap driving in the opposite direction is also travelling at 30mph then you will see him coming at you at 60mph. But you can consistently work out the consequences of these assumptions and that’s part of what matters when it comes to doing experimental tests. (It is not the only thing that matters since some consistent and well worked out ideas don’t have empirically testable consequences.)

I will illustrate two consequences of these assumptions to illustrate what I’m talking about. The consequences in question are that if an object is moving at constant velocity (moving at a constant speed and in a constant direction) with respect to an IRF will act as if it is shorter along the direction in which you are moving than in an IRF in which it isn’t moving and any clocks attached to that object will run slower. This happens in such a way as to make measurements of the speed of light come out the same in both IRFs.

Suppose that you have a clock X that is at rest in an IRF S. This clock consists of two mirrors A and B, a distance L apart with a pulse of light bouncing between them. Each time the light pulse bounces off one of the mirrors it causes a clock attached to mirror A to tick. In the frame S the time T between ticks is 2L/c.

Now suppose there is another IRF S‘ that is moving at a speed v with respect to S at a right angle to the pulse of light bouncing between the mirrors:

The distance between the mirrors in S isn’t going to change because that distance is at right angles to the direction in which the clock is moving. The time it takes the clock to tick in S‘ is T‘. The light takes T‘/2 to hit the mirror B in S‘ and during that time the mirror has moved vT‘/2. The light then takes another T‘/2 to hit the first mirror again at position C. So using Pythagoras theorem the total distance the light travels in S‘ is

$AB + BC = 2\sqrt{L^2+(vT'/2)^2}$.

The total distance travelled by the light pulse must be cT‘ because the speed of light doesn’t change between the two frames, so

$cT' = 2\sqrt{L^2+(vT'/2)^2}$

and a bit of rearrangement gives:

$T' = \frac{2L}{\sqrt{c^2-v^2}}$.

Since L = cT/2 we can write

$T' = \frac{T}{\sqrt{1-v^2/c^2}}$

Now,

$1 > \sqrt{1-v^2/c^2}$

because v < c so the amount of time it takes for the clock to tick increases.

Now suppose that we consider another frame S” moving  at speed parallel to the direction in which the light travels in the clock:

Since the far end of the clock is moving away from the light pulse it has to travel farther than the length of the clock in S” to get to the mirror at the far end of the clock:

$L'' + vt'' = ct''$.

When the light pulse is travelling back from the far end of the clock to where it started, it travels a shorter distance:

$L'' - vu'' = cu''$.

The total time is

$T'' = u'' + v'' = \frac{2L''c}{c^2-v^2}=\frac{2L''/c}{1-v^2/c^2}$.

The time measured in S‘ and S” will be the same so:

$T'' = \frac{T}{\sqrt{1-v^2/c^2}}$

$L'' = L\sqrt{1-v^2/c^2}$

and so moving objects will act as if they are shorter.

On the scales of speed and distance we use in everyday life these effects are very small, but they can be measured for particles travelling near the speed of light and they have been found.