Why Must Time Be A Dimension?

November 28, 2016

When we think about how we can move through the Universe, we immediately think of three different directions. Left-or-right, forwards-or-backwards, and upwards-or-downwards: the three independent directions of a Cartesian grid. All three of those count as dimensions, and specifically, as spatial dimensions. But we commonly talk about a fourth dimension of a very different type: time. But what makes time a dimension at all? That’s this week’s Ask Ethan question from Thomas Anderson, who wants to know:

I have always been a little perplexed about the continuum of 3+1 dimensional Space-time. Why is it always 3 [spatial] dimensions plus Time?

Let’s start by looking at the three dimensions of space you’re familiar with.

Here on the surface of the Earth, we normally only need two coordinates to pinpoint our location: latitude and longitude, or where you are along the north-south and east-west axes of Earth. If you’re willing to go underground or above the Earth’s surface, you need a third coordinate — altitude/depth, or where you are along the up-down axis — to describe your location.

After all, someone at your exact two-dimensional, latitude-and-longitude location but in a tunnel beneath your feet or in a helicopter overhead isn’t truly at the same location as you. It takes three independent pieces of information to describe your location in space.

But spacetime is even more complicated than space, and it’s easy to see why. The chair you’re sitting in right now can have its location described by those three coordinates: x, y and z. But it’s also occupied by you right now, as opposed to an hour ago, yesterday or ten years from now. In order to describe an event, knowing where it occurs isn’t enough; you also need to know when, which means you need to know the time coordinate, t. This played a big deal for the first time in relativity, when we were thinking about the issue of simultaneity. Start by thinking of two separate locations connected by a path, with two people walking from each location to the other one.

You can visualize their paths by putting two fingers, one from each hand, at the two starting locations and “walking” them towards their destinations. At some point, they’re going to need to pass by one another, meaning your two fingers are going to have to be in the same spot at the same time. In relativity, this is what’s known as a simultaneous event, and it can only occur when all the space components and all the time components of two different physical objects align.

This is supremely non-controversial, and explains why time needs to be considered as a dimension that we “move” through, the same as any of the spatial dimensions. But it was Einstein’s special theory of relativity that led his former professor, Hermann Minkowski, to devise a formulation that put the three space dimensions and the one time dimension together.

We all realize that to move through space requires motion through time; if you’re here, now, you cannot be somewhere else now as well, you can only get there later. In 1905, Einstein’s special relativity taught us that the speed of light is a universal speed limit, and that as you approach it you experience the strange phenomena of time dilation and length contraction.

But perhaps the biggest breakthrough came in 1907, when Minkowski realized that Einstein’s relativity had an extraordinary implication: mathematically, time behaves exactly the same as space does, except with a factor of c, the speed of light in vacuum, and a factor of i, the imaginary number √(-1).

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