Is there a geometric interpretation of the spacetime interval?

Question

In Euclidean space, the invariant $s^2 = x^2+ y^2+ z^2$ is equal to the length square of the position vector $r$. This is easily understood and can be represented geometrically in a graph.

On the other hand, in Minkowski space, the corresponding invariant quantity is defined as the square of the spacetime interval $s^2 = x^2 + y^2 + z^2 - c^2 t^2$.

Question: Is there a corresponding geometrical interpretation? The best I could come up with is to take $ict$ as the time coordinate.**

Secondarily, why do they call this quantity an interval?

Answer 1

The interpretation of $s^2$ is rather straightforward: Given two points or events in the spacetime, it is a coordinate-invariant way of ascribing a distance---in the sense determined by the Minkowski metric---between the two. In the formulas you wrote down, one of the two points is taken to be the origin. In the general case we have

$$s^2_{1,2}=(\vec x_1-\vec x_2)^2-c^2(t_1-t_2)^2$$

This is exactly analogous to the more intuitive Euclidean case where the signature of the metric is purely positive. The word "interval" is more or less synonymous with "distance between things", thus it is rather natural to call the distance between two spacetime points (or events) the interval between them.

Different types of separation

To make the geometric interpretation more explicit, one has to work with the notion of a light cone. This construction is described in any text on special relativity, so if you find yourself unsatisfied after reading this answer, I suggest you have a look at the literature on the subject.

Now, let's begin: If we fix a specific point $x$, we can classify all the other points in Minkowski space, as follows: For any $y\in M$, we examine $s^2_{x,y}$ and look at its sign. This way, we obtain three different types of points; each point in Minkowski space lies in one of the following sets:

$$T=\{y\in M \mid s^2_{x,y}<0\}\qquad S=\{y\in M \mid s^2_{x,y}>0\}\qquad L=\{y\in M \mid s^2_{x,y}=0\}$$

We say that $y$ is timelike separated from $x$ if it is an element of the first set, spacelike separated from $x$ if it is an element of $S$, and the third set is that of lightlike separated points (with respect to $x$).

To understand these naming conventions, we first note that light rays always travel at velocity $v=\frac{\mathrm{d}x}{\mathrm{d}t}=c$, so that any trajectory traveled by a light ray satisfies $\vec x^2=c^2t^2$. Consequently, $s^2=0$ along the trajectories traveled by light. This explains the last of the three names. Secondly, whenever $\Delta x_{1,2}^2 := (\vec x_1-\vec x_2)^2$ is larger than $c^2 \Delta t_{1,2}^2$, we see that $s_{1,2}^2>0$, so that it makes sense to call such points spacelike separated. A similar line of argument justifies the name "timelike separated".

I would also like to remark that, although I won't prove it here, the terms spacelike and timelike separation are further justified by the fact that there exists no (inertial) frame of reference where two timelike separated events are simultaneous, while there exists no frame of reference where two spacelike separated events occur in the same location, hence such points are truly separated in time and space, respectively. These claims follow straightforwardly from the Lorentz transformation laws.

The light cone diagram

Now, what is the geometry behind this whole setup? It's beautifully demonstrated, for example, in this picture from Wikipedia:

Here, the point $x$ is denoted by "observer" and taken as the origin. The set $L$ is depicted by the blue/green (double) cone originating at the origin. The set $T$ is the set of all points inside the light cone(s), both future and past, while the set $S$ describes all points outside of the cone.

Causality

Finally, we remark that the notion of a light cone is intimately related with the notion of causality: Since the light cone represents the maximum distance that light could have traveled from $x$ (in the future part) or the maximum distance from which light could have come to reach $x$ at the moment of time that we have chosen as the $t=0$ plane, it actually tells you that nothing outside of the light cone can be in causal contact with $x$. That is, past events that occurred outside the past light cone cannot have influenced the event $x$, and the event $x$ itself cannot influence any future events that lie outside of the future light cone.

Answer 2

Yes, there is a geometrical interpretation.

Firstly, note that you can make a rectangle whose sides are light rays and that has the two events on opposite corners.

To see that, if they are time like separated shoot a light ray from the earlier one to the later one and you get to that location too soon so let it keep going until it reach an event where a light ray going the opposite direction can reach the later event. Thats two sides. For the next side start out going that opposite direction and switch to the first direction when you have finally waited long enough. In the frame where they were at the same location and a time T apart you sent rays out in two opposite directions to travel $D=cT/2$ and then simultaneously bounce and come back.

For spacelike separated events, in the frame where they are simultaneous have the midpoint send a beam in the two opposite directions to bounce off the two events and come back.

You can even think of it as a common rectangle that has two spacelike separated events on two vertices and two timelike separated events on the other two.

The separation is equal two twice the area of that rectangle that had the rays as the sides. And of course the area is physically related to the clock readings on a radar time or radar distance measurement. All of this would be true in Galilean Relativity, but in Special Relativity this area is the same for any two inertial observers. The invariance comes from the fact that two two inertial moving frames literally see each other run at the same literal speed. I'll cite Mermin since I got the light rectangles description from Mermin.

Two inertial observers in relative motion must each see the other’s clock running at the same rate. The representation of this symmetry of the Doppler effect in a two-dimensional space–time diagram reveals an important geometrical fact: The squared interval between two events is proportional to the area of the rectangle of photon lines with the events at diagonally opposite vertices.

"Space–time intervals as light rectangles" by N. David Mermin in the American Journal of Physics, Volume 66, Issue 12, pp 1077-1080  (1998); http://dx.doi.org/10.1119/1.19047

Answer 3

My answer will inadvertently push two religious biases of mine upon you: that $w = ct$ and that the proper convention for the interval is $ds^2 = dw^2 - dx^2 - dy^2 - dz^2.$ Sorry in advance.

Common background

In relativity, we say that a 3D vector may be paired with a scalar as a four-vector if they transform according to the Lorentz boost when changing to a new inertial coordinate system moving with speed $\vec v$ relative to the old one. The Lorentz boost by $\vec \beta = \vec v/c$ (with $\gamma = 1/\sqrt{1 - \vec\beta\cdot\vec\beta}~$) is a transform on 4-vectors $$(\alpha,\; \vec a) ~\mapsto~ \left(\gamma~\left[\alpha - \vec \beta \cdot \vec a\right],\; \vec a + \vec\beta~\left[(\gamma - 1)~ \frac{\vec a\cdot\vec\beta}{\vec\beta\cdot\vec\beta} ~-~ \gamma~\alpha\right]\right),$$ which can be shown to preserve the inner product $(\alpha, \vec a) * (\beta, \vec b) = \alpha~\beta - \vec a \cdot \vec b.$ This product $(*)$ is therefore very important to the theory of relativity, and in particular lets us turn 4-vectors into quantities which are immediately relevant to us. The broader group that we want is called the "Poincaré group" and it is the group generated by rotations of the spatial subspace, 4D translations, Lorentz boosts, and parity flips $(\alpha, \vec a)\mapsto (-\alpha, \pm\vec a);$ it is all isometries of the $(*)$ product in 4D space.

A position vector $\vec r = [x, y, z]$ can be paired with a time $w$ to produce a 4-position for an "sudden event," a point in spacetime. Due to the translations in the Poincaré group, we will generally only want to form 4-vector products $(*)$ with differences in 4-position (4-displacements!), not actual 4-position vectors.

Light cones as expanding bubbles

Consider such a sudden event: since nothing can travel faster than the speed of light, you cannot know that it has truly happened until you are hit by the light from the event. This light comes out of the event like an expanding bubble moving at speed $c.$ We'll call it a "light bubble" but the technical term is a "future-pointing light cone." Stepping back and looking at the universe at any single time holistically: inside the light bubble are all of those points in space which "have seen" the event some time in their past; these points in spacetime are therefore the "relativistic future" of the event" if you extend it over all times.

Similarly, we can think about the event's past-pointing light cone, which is the set of all the light rays that could have been incident on the point of the event when it happened: this is another "expanding bubble," but expanding in the negative direction of time. Points inside this bubble are in the event's "relativistic past", the event was able to see them.

Such an expanding-at-speed-$c$ bubble is described by the coordinates $$(w - w_0)^2 = (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2,$$ so we see that the Lorentz boost preserves the structure of these light-bubbles, mapping light bubbles to other light bubbles, but possibly resizing them at some time relative to each other or shifting them around in space. It actually does something even more interesting due to its linearity: it preserves their topology. Given two events in spacetime, either one light bubble is "inside" the other one (A came objectively before B) and they do not intersect: or else they both will "collide" eventually as they expand. In the first case, there is a reference frame which visits event A and then continues inertially to visit event B, so in that frame both happen "right here" and therefore they are not objectively space separated. However, since nobody can go faster than $c$, there is no way for this spaceship to exit the light bubble, and so the "colliding" case means that A and B are objectively in different places: there is no reference frame that inertially visits both of them.

However, the Lorentz boost can resize both colliding bubbles to have the same size. In this reference frame, therefore, both events were simultaneous: the events are no longer objectively space-separated. So the events may either be objectively space-separated, objectively time-separated, or possibly "null-separated" if they're on the infinitely thin border between (one bubble is "inside" the other but they make contact at a point all the time; no real observer could have been at both; these are objectively both space- and time-separated but those separations can each be made arbitrarily small).

The spacetime interval as proper time, proper distance between events.

All particle motions are described by moving from an event into its relativistic future, and thus the 4-displacement $R$ between them satisfies $R * R > 0.$ In the special case that the particle performs this motion inertially, it has an inertial reference frame where these two points in spacetime are described as $(w_0, \vec 0)$ and $(w_1, \vec 0)$ and hence $R * R = (w_1 - w_0)^2.$ We call such a time difference $w_1 - w_0$ the proper time $\tau$ between the two events: it is the time measured by the coordinates which think that both events happened at the same place. It is the minimum such time between the two events; due to the structure of the Lorentz transform, every other reference frame will see the time become larger in order to preserve $\Delta w^2 - \Delta r^2 = \Delta w^2 (1 - \beta^2) = \tau^2$, so that generally you see $\Delta w = \gamma~\tau.$

If two events are objectively space-separated, then they have a 4-displacement $R$ satisfying $R * R < 0.$ In this case, $\ell = \sqrt{-R * R} = |\vec r_1 - \vec r_0|$ is the proper distance between the positions of the two events as measured by someone who saw them both as simultaneous; other people will in general see a larger distance between where these two events happened.

("Larger" may sound odd if you are used to length contraction, but you can also derive length contraction from the Lorentz transform. It involves the two worldlines $(w, \vec 0)$ and $(w, x~\hat \beta)$ where $\hat \beta$ is a unit vector in the direction we're about to boost. This becoming the slanted lines $(\gamma~w, -\gamma~\vec\beta~w)$ and $(\gamma (w - \beta~x),\;\hat \beta~[\gamma~x - \gamma~\beta~w]);$ forcing these both to have time-component 0 means that the first is $(0, 0)$ while the second is $(0,\hat\beta~\gamma~x[1 - \beta^2]) = (0, \hat\beta~x/\gamma).$ The key discrepancy to notice here is that in the length-contraction case we're talking about the distances between two things "at the same time", whereas when we boost the above "proper distance" the events are suddenly happening at two different times.)

Answer 4

In Euclidean space, the invariant $s^2=x^2+y^2+z^2$ is equal to the length squares of the position vector $r$.

"Length", or "distance between points", is an coordiante-independent (invariant) notion. In three-dimensional Euclidean space, the points are flat to each other; that means: considering any five points, $\mathsf A$, $\mathsf B$, $\mathsf J$, $\mathsf K$, $\mathsf Q$, and given the ten pairwise distance values between them, $d[~\mathsf A, \mathsf B~]$, $d[~\mathsf A, \mathsf J~]$ ..., $d[~\mathsf K, \mathsf Q~]$, then their (normalized) Cayley-Menger determinant vanishes:

0 = $ \begin{array}{|cccccc|} 0 & \left(\frac{d[~\mathsf A, \mathsf B~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf A, \mathsf J~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf A, \mathsf K~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf A, \mathsf Q~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 1 & \\ \left(\frac{d[~\mathsf B, \mathsf A~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 0 & \left(\frac{d[~\mathsf B, \mathsf J~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf B, \mathsf K~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf B, \mathsf Q~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 1 & \\ \left(\frac{d[~\mathsf J, \mathsf A~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf J, \mathsf B~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 0 & \left(\frac{d[~\mathsf J, \mathsf K~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf J, \mathsf Q~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 1 & \\ \left(\frac{d[~\mathsf K, \mathsf A~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf K, \mathsf B~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf K, \mathsf J~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 0 & \left(\frac{d[~\mathsf K, \mathsf Q~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 1 & \\ \left(\frac{d[~\mathsf Q, \mathsf A~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf Q, \mathsf B~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf Q, \mathsf J~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & \left(\frac{d[~\mathsf Q, \mathsf K~]}{d[~\mathsf A, \mathsf B~]}\right)^2 & 0 & 1 & \\ 1 & 1 & 1 & 1 & 1 & 0 & \end{array}$.

If coordinate tuples $\{ x, y, z \} \in \mathbf R^3$ are assigned to all points of this space such that for any two points, $\mathsf A$ and $\mathsf B$

$$s^2[~\mathsf A, \mathsf B~] := (d[~\mathsf A, \mathsf B~])^2 = (x[~\mathsf B~] - x[~\mathsf A~])^2 + (y[~\mathsf B~] - y[~\mathsf A~])^2 + (z[~\mathsf B~] - z[~\mathsf A~])^2,$$

then such a coordinate assignment is called "Cartesian coordinates (of three-dimensional Euclidean space)".

On the other hand, in Minkowski space, the corresponding invariant quantity is defined as the square of the spacetime interval $s^2=x^2+y^2+z^2 - c^2 t^2$

Well, this (or perhaps some variant involving certain differences between coordinate values) may indeed be taken as a definition, as far as Minkowski space is founded on considerations of algebraic relations between certain coordinate tuples, rather than geometric relations. Consequently we may ask about interpretations of the quantity "$s^2$" in terms of geometry and physics.

Question: Is there a corresponding geometrical interpretation?

Sure:

• a positive value of $s^2$ (between two distinct suitable events under consideration, say $\mathsf A$ and $\mathsf B$) is interpreted in terms of distance between two participants where one had taken part in event $\mathsf A$ and the other had taken part in event $\mathsf B$; specificly as the square of the minimum distance (or in case a minimum doesn't exist, the infimum of all distances) among all such pairs of participants;

• a negative value $s^2$ (between two distinct suitable events under consideration, say $\mathsf J$ and $\mathsf K$) is interpreted in terms of duration of one participant between having taken part (first) in one of these two events, and (then) in the other; specificly as ("$(-1)~c^2$" times) the square of the maximum duration (or in case a maximum doesn't exist, the supremum of all durations) among all those participants;

• a zero value $s^2$ (between two distinct suitable events under consideration, say $\mathsf P$ and $\mathsf Q$) is interpreted as the "signal front" of one event having reached the other event, and

• for any one event: $s^2[~\mathsf A, \mathsf A~] = 0$, too.

Secondarily, why do they call this quantity an interval?

The word "interval" is obviously related to "(spatial or temporal) separation". Apparently, the people who applied this name to the quantity $s^2$ (rather than instead to the quantity "$\text{sgn}[~s^2~]~\sqrt{\text{sgn}[~s^2~]~s^2}$") weren't particularly bothered by $s^2$ referring to squares of distance or duration values.