# Why spacetime?

by Daniel Ranard

Last week's Nobel prize in physics was awarded for the observation of gravitational waves, the famous ripples in spacetime. You can read about these waves first predicted by Einstein, but I want to talk about a more basic idea: that of spacetime itself. Why do physicists insist on "spacetime"—why can't we content ourselves with just space and time?

Many thinkers before Einstein pondered the connection between time and space. Medieval timeline makers must have understood the analogy between points and lengths of space, on the one hand, and instants and durations of time, on the other. It's an analogy rendered physical by the timeline itself. Still earlier, sundials mapped temporal durations to spatial intervals. In Edgar Allen Poe's book-length "Eureka: A Prose Poem," he concludes that "Space and Duration are one." But contrary to Poe, modern physicists do not contend space and time have an identical character. Indeed, the differences would appear obvious: for instance, we always move forward in time, while in space we may remain still.

Though spatial and temporal directions may differ, Einstein and his contemporaries realized they must be considered together, part of a geometrical whole. In one limited sense, space and time had already been considered together for centuries. A graphical timeline emphasizes time as a dimension; if you add a dimension of space to your timeline, you create the spacetime arena. More quantitatively, if you make a graph of an object's position over time, then the background of the graph – the two-dimensional plane, with axes of both space and time – suggests a notion of spacetime. Such graphs predate even Descartes, who's most often credited with the invention of Cartesian coordinates; Oresme and others drew similar figures long before.

Modern illustrations of spacetime take the same form: the Cartesian plane, with axes of time and space. But the modern marriage of space and time entails far more than just a nailing together of the axes. Though many thinkers drew time and space together, it was the insights of Einstein in 1905 that bound them inextricably. In fact, it was Einstein's old teacher, the mathematician Hermann Minkowski, who most clearly cast Einstein's results in the language of spacetime.

Minkowski began his 1908 lecture Space and Time by declaring that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." What could that mean? We often label the three axes of space as x,y,z, measuring width, height, and depth, while t labels time. The four dimensions of x, y, z, and t together constitute spacetime. Why put them together? That is, why does their union constitute a four-dimensional arena whose components demand joint consideration?

To understand why x, y, z, and t belong together, let's first understand why, at the least, x, y, and z belong together. That is, why do we usually conceive of space itself as a unified whole, rather than the amalgamation of three independent dimensions? Our conception of unified space comes so naturally that it's hard to pin down our reasoning. It's the decomposition of space into three components that seems more unnatural, not the unification. But let's try to pinpoint why x, y, and z belong together.

Imagine a friend who never turns his body, always staring straight ahead. He notes how far objects are to the left or right, a number he calls x. He also notes how far objects are in the forward-backward direction, a number he calls y. (We'll ignore z here.) He has poor peripheral vision, and though he happily walks about – front-back, sideways, diagonally, however he pleases – all the while, he never turns. Our friend conceives of two distinct axes, x and y. And although he concedes that the axes are best drawn together, he contends that all points in space have two separate, fundamental properties: their x-position and their y-position. Likewise, all objects have two fundamental properties: their width and depth.

Now imagine that one day, our friend manages to turns his head a bit. His notions of left-right and front-back suddenly shift. As he peers out from his new vantage point, those two fundamental properties – width and depth – are now changed, intermingled. A bit of depth becomes width; a bit of width becomes depth. He's also struck that after a while, his new perspective doesn't feel so different. Although width and depth are mixed, the laws of physics appear the same. In fact, as he walks and turns, eventually he even forgets which way he started; he discerns no privileged perspective. His old notion of x-position and y-position remain useful, but they also appear somewhat arbitrary. All directions mix among themselves: he now knows a unified space.

Space and time deserve a similar unification. Before Einstein, however, we were in the position of our friend with the rigid gaze. Space and time do intermingle, but only when you "turn your head." And the analog of rotating your gaze, in the case of space and time, is to travel at enormous speeds.

At everyday speeds, we are like our friend turning his head a mere fraction of a degree. The mixing is imperceptible. But at higher speeds, durations in time mix with extensions in space. Just like the notion of "width" and "depth" are relative to the rotation of our friend's gaze, mixing as he swivels his head, likewise the notions of width and depth mix with temporal duration. And just as our friend eventually forgets he's turned his head because the world proceeds in a sensible way, so does a passenger on a train eventually forget the motion of the train.

The mixing of x and y when our friend swivels his gaze does have a different character than the mixing of x, y, and z with t when we travel at high speeds. Einstein and Minkowski explained this difference using a new sort of geometry, unifying space and time in a different way than space itself is unified.

Perhaps most impressively, Einstein didn't even need to "turn his head" to understand the intermingling of space and time – he just managed to properly imagine traveling at high speeds. (If only we could all properly imagine the shifting of our gaze, even when, for whatever reason, we must stare straight ahead!) And so with the help of Minkowski, he invented spacetime.

P.S. One of the wonderful features of Einstein's relativity and Minkowski's spacetime is that the subject requires only high school algebra and focused thought to learn in a precise way. You can even find it rendered clearly in Einstein's own words. Or hear it from 3QD's own S. Abbas Raza, here.

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