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Section 11.2 Light Boxes

Remarkably, the notion of area in special relativity is exactly the same as in ordinary Euclidean geometry! Even more remarkably, Minkowskian area can be used to measure distance and time!  1 

The presentation in this section was motivated by the discussion of causal domains in Salgado (2016), a preprint of which is available here.

Figure 11.2.1 shows a hyperbola connected to the origin by means of a double-null parallelogram, that is, a parallelogram all of whose sides are at \(45^\circ\text{.}\) Since all of its sides are lightlike, such parallelograms are called light boxes. Furthermore, the area of this parallelogram remains constant as the upper vertex moves along the hyperbola!

Figure 11.2.1. Using light boxes to measure spacetime interval.

It is straightforward to work out the coordinates of the vertices \(X\) and \(Y\) at the sides of the parallelogram, which are precisely (up to sign) half the difference and sum, respectively, of the components of point \(Q\) at the tip of the parallelogram. That is, if \(P=(0,0)\) and \(Q=(x,t)\text{,}\) then

\begin{equation} X=\left(-\frac{t-x}{2},\frac{t-x}{2}\right) , \qquad Y=\left(\frac{t+x}{2},\frac{t+x}{2}\right) .\tag{11.2.1} \end{equation}

First using the Pythagorean theorem to determine the lengths of the sides \(a\) and \(b\text{,}\) the (Euclidean) area \(A\) of the box is therefore given by

\begin{equation} A^2 = a^2 b^2 = \left(2\frac{(t-x)^2}{4}\right)\left(2\frac{(t+x)^2}{4}\right)\tag{11.2.2} \end{equation}

so that \(A=\frac12(t^2-x^2)\text{.}\) The area of the parallelogram is (proportional to) the squared distance from \(P\) to \(Q\text{!}\)

Physically, such double-null parallelograms correspond to bouncing a beam of light off of a mirror that remains at fixed distance from the worldline connecting \(P\) and \(Q\text{.}\) Such mirrors can be used as clocks along this worldline, with times expressed in multiples of the round-trip travel time for the beam of light – one “tick” of the clock. But now we can measure time by counting ticks. A similar argument enables the use of double-null parallelograms to measure (spacelike) distance.

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