Minkowski Space

Section 11.1 Minkowski Space

As we have seen, one way to obtain a non-Euclidean geometry is to modify the distance function. Nonetheless, all such geometries that we have considered so far do involve a notion of distance that assigns a positive number to every line segment. We consider here a model that does not have this property. This model, known as Minkowski space, turns out to be much more than a mathematical curiosity, as it describes the geometry of special relativity, and hence the world around us. ¹

A more extensive treatment of the geometry of special relativty can be found in Dray (2012,2021).

Consider the Euclidean plane, using rectangular coordinates \((x,t)\text{,}\) and define the squared distance between two points \(A=(x_A,t_A)\) and \(B=(x_B,t_B)\) to be

\begin{equation} d_M(A,B)^2 = (x_B-x_A)^2 - (t_B-t_A)^2\tag{11.1.1} \end{equation}

which differs from (squared) Euclidean distance by a crucial minus sign. ²

The use of \(d_M\) to denote “distance” should be interpreted as a notational convention, as \(d_M^2\) can be positive, negative, or zero. Nonetheless, we will never consider \(d_M\) to be a complex number; if necessary, we will instead consider \(\sqrt{|d_M^2|}\text{.}\)

What do circles look like in this geometry?

That's easy – they're hyperbolas!

Several Minkowski circles are shown in Figure 11.1.1. As can be inferred from the figure, there are three different types of “distance”, depending on whether the squared distance is positive, negative, or zero. In the first case, the line between the two points are said to be spacelike, in the second case timelike, and in the third case lightlike or null. These names reflect the role these distances play in special relativity, a theory of spacetime. ³ Light is special in special relativity; lightlike lines are special in Minkowski space – they have zero length!

Our notion of “squared distance” is more commonly referred to as the (squared) spacetime interval, with “time” measuring timelike intervals, and “distance” referring only to spacelike intervals.

Figure 11.1.1. Circles in hyperbola geometry.

That's not the only surprising feature of this geometry. Can you draw a 3–4–5 triangle? In Euclidean geometry, the hypotenuse is always the longest side of a right triangle. Not so in Minkowski space! Two such triangles are shown in Figure 11.1.2, again using the convention that the “length” of a timelike side is the square root of the absolute value of the squared distance (see Footnote 2).

Figure 11.1.2. Two 3–4–5 triangles in Minkowski space.

OK, so when are two lines perpendicular in this geometry? Recall that a circle is perpendicular to its radii. That principle holds here as well. ⁴ This construction appears in Figure 11.1.3; the two vectors shown are perpendicular. The somewhat surprising conclusions is that each of the triangles shown in Figure 11.1.4 is in fact a right triangle.

One way to show that radii are perpendicular to circles is to use the dot product. As the tip of the vector \(\vv\) from the center of the circle moves along the circle, its magnitude \(|\vv|\) remains constant. But \(|\vv|^2=\vv\cdot\vv\text{,}\) and differentiating this equation implies that \(\vv\) is perpendicular to its derivative – which is tangent to the circle.

Figure 11.1.3. The radial and tangent vector fields to a hyperbola, which are perpendicular in hyperbola geometry.