36.8.2 Length

We’ve already introduced the idea of length contraction, but we didn’t precisely define just what we mean by the length of a moving object. The length of an object at rest is clear because we can take all the time we need to measure it with meter sticks, surveying tools, or whatever we need. But how can we give clear meaning to the length of a moving object?

A reasonable definition of an object’s length is the distance $L = Δ x = x_{R} - x_{L}$ between the right and left ends when the positions $x_{R}$ and $x_{L}$ are measured at the same time t. In other words, length is the distance spanned by the object at one instant of time. Measuring an object’s length requires simultaneous measurements of two positions (i.e., two events are required); hence the result won’t be known until the information from two spatially separated measurements can be brought together.

FIGURE 36.32 shows an object traveling through reference frame S with velocity v. The object is at rest in reference frame $S'$ that travels with the object at velocity v; hence the length in frame $S'$ is the proper length $ℓ$ . That is, $Δ x' = x_{R}^{'} - x_{L}^{'} = ℓ$ in frame $S'$ .

The figure illustrates the length of an object moving with a velocity with respect to two reference frames.

FIGURE 36.32 Full Alternative Text

At time t, an experimenter (and his or her assistants) in frame S makes simultaneous measurements of the positions $x_{R}$ and $x_{L}$ of the ends of the object. The difference $Δ x = x_{R} - x_{L} = L$ is the length in frame S. The Lorentz transformations of $x_{R}$ and $x_{L}$ are

\begin{array}{l} x_{R}^{'} = γ (x_{R} - v t) \\ x_{L}^{'} = γ (x_{L} - v t) \end{array}

(36.24)

where, it is important to note, t is the same for both because the measurements are simultaneous.

Subtracting the second equation from the first, we find

x_{R}^{'} - x_{L}^{'} = ℓ = γ (x_{R} - x_{L}) = γ L = \frac{L}{\sqrt{1 - β^{2}}}

Solving for L, we find, in agreement with Equation 36.15, that

L = \sqrt{1 - β^{2}} ℓ

(36.25)

This analysis has accomplished two things. First, by giving a precise definition of length, we’ve put our length-contraction result on a firmer footing. Second, we’ve had good practice at relativistic reasoning using the Lorentz transformation.

NOTE Length contraction does not tell us how an object would look. The visual appearance of an object is determined by light waves that arrive simultaneously at the eye. These waves left points on the object at different times (i.e., not simultaneously) because they had to travel different distances to the eye. The analysis needed to determine an object’s visual appearance is considerably more complex. Length and length contraction are concerned only with the actual length of the object at one instant of time.