# Why is space and time spoken of as one thing

I've never understood why it's spacetime and not space and time. In other words why aren't space and time spoken of as two different things. Can someone explain to me why this is

There are different words for different aspects of space. For example, consider: length, width, and height. Other words include depth and breadth. We can speak of them as different things if we choose to, but we generally consider them to be part of unified concept of space. Why?

It's because we understand that these words just pick out measurements along specific directions in space set by context or the speaker's viewpoint, and that there is no intrinsic difference between those directions. Depending on which way they are facing, what's length to one person is width to another.

More formally, there is a symmetry between directions in space. We can rotate freely and the intrinsic properties of space look just the same, e.g. the Euclidean metric (infinitesimal distance formula) $$\mathrm{d}s^2 = \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2$$ is invariant under the orthogonal group of rotations, $\mathrm{O}(3)$. Together with spatial translations (which are like rotations about a point at infinity), this forms the Euclidean group of isometries, $\mathrm{ISO}(3)$. Because of this symmetry that "intermixes" directions of space, we think of those different directions as being unified one into one thing: space.

In special relativity, spacetime has a Minkowski metric (in units of $c=1$) $$\mathrm{d}s^2 = -\mathrm{d}t^2 + \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2\text{,}$$ and it is invariant under the Lorentz group $\mathrm{O}(1,3)$ that act like rotations in spacetime; together with the translations (in space and in time), the full isometry group is the Poincaré group. In fact, the usual Lorentz transformation (say, boosting along the $x$-axis) is exactly a rotation with by a hyperbolic angle in the $tx$ plane.

That's why people began to think of spacetime as one unified whole. The reasons are essentially the same as why people think of space as one unified whole despite it having different directions: there is a symmetry that "intermixes" those directions, albeit slightly differently in spacetime because of the different sign of the temporal and spatial directions in the metric.

In general relativity, this is generalized further, but the details aren't immediately relevant here.

Why is space and time spoken of as one thing?

Because it's how Minkowski presented it in 1908. See Space and Time where you can read this: "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".

I've never understood why it's spacetime and not space and time.

You're right to do so. Because spacetime is actually a mathematical abstraction that people mistake for reality. The Earth is surrounded by space, not spacetime. Light moves through space, not spacetime. See this physics stack exchange answer where relativist Ben Crowell says "Objects don't move through spacetime. Objects move through space". When you move fast through space your measurements of space and time change, but it's you that's changed, not space, and not the things in space. For example a star doesn't actually length-contract to a discoid shape just because you step on the gas.

In other words why aren't space and time spoken of as two different things.

They are. The thing to appreciate is that whilst a lot of popscience articles blithely talk about spacetime, people like Einstein were very particular about saying space when they meant space. See for example Einstein's 1920 Leyden Address where he referred to the continuum of space-time and yet described a gravitational field as space that's neither homogeneous nor isotropic. The distinction between the two tends to get lost. In a gravitational field space is inhomogeneous, and the inhomogeneity diminishes with distance in a non-linear fashion. When you plot it using say light-clocks at different elevations, you see a curve on your plot. Your plot depicts curved spacetime. That's the map. But inhomogeneous space is the territory. See Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime.

Can someone explain to me why this is?

By and large it's because people don't have a clear understanding of relativity. Space and time are not on an equal footing. I can hop forward a metre but you can't hop forward a second. There is no real rotation when you accelerate, it's just an abstract mathematical rotation, wherein your "local motion" is of necessity reduced by your macroscopic motion through space because the total rate of motion can't exceed c, because of the wave nature of matter. See my time dilation answer here on stack exchange. Since when you look at what clocks do you appreciate that the time is a cumulative measure of local motion, you're actually better off thinking in terms of space and motion rather than space and time.

• Objects move through time as well as space. – Rob Jeffries Mar 15 '16 at 22:20
• @Rob Jeffries : no, they don't. That's a figure of speech. See for example this answer. You don't literally climb to a higher temperature, and in similar vein you don't literally move through time. You can appreciate this via the gedanken stasis box. No motion of any kind occurs in the stasis box. So when I shut you inside for five years then open the door, you think I opened it immediately. You "travelled to the future" by not moving at all whilst everything else did. – John Duffield Mar 15 '16 at 22:44
• I'm doing it right now. x,y,z and t coordinates are always relative to something else. – Rob Jeffries Mar 15 '16 at 23:04
• If a particle has a spatial position $\mathbf{x}(t)$, we can say that the particle moves in space, even though it also has some curve as its trajectory over its entire history (e.g., an ellipse for a Keplerian orbit). Similarly, in spacetime, we can have $x^\mu(\lambda) = (t(\lambda),x(\lambda),\ldots)$, so what's the problem in saying it moves in spacetime, even though it also has a worldline in spacetime over its entire history? (I agree with Ben Crowell regarding speed, but this more general point seems to needlessly insisting on forbidding something totally harmless and straightforward.) – Stan Liou Mar 16 '16 at 0:02
• @Rob Jeffries : you are not moving through time. When you measure the motion of something you can see that it gradually changes position in space, you can see that it moves through x y z space. Motion through space is evidential and empirical. But when you measure time t you refer to a clock. And a clock is a device that features something else moving through space, such as a pendulum, in a regular cyclical fashion. There is no scientific basis for the notion of moving through time. Objects don't move through spacetime. Objects move through space. And they don't move through time either. – John Duffield Mar 16 '16 at 0:03