Romy asks, “What’s the relationship between space and time?”
When you’re planning to meet up with someone, what do you specify? Just saying Grand Central isn’t enough, and just saying noon isn’t enough either. You need both a place and a time.
The same is true in physics. Physics explains the world of matter, moving and changing and interacting with other matter. All these events — motion, change, interaction — happen somewhere in space and somewhen in time.
It’s not too hard for us to think of space as a big, concrete whole and ignore our place in it. It’s very hard to do this for time. We think of “now” as absolute in a way that we don’t think of “here” as absolute. We look at events like this:
But why take time-slices? Why not look at it like this?
Because, you might say, that’s weird. A particle — or a ball, or a person — has a certain shape that stays mostly the same as it travels through time. Why would we think of it as a time-worm sitting in a pre-made spacetime block universe?
As it turns out, a quick trip into the world of large-scale physics makes it very hard to consider space and time separately. The reason is that space and time aren’t quite independent. Things that are farther away are also, in a sense, earlier. And this relationship bends and twists based on how heavy stuff is. Don’t worry, I’ll explain.
There’s a cosmic speed limit in our universe. We call it c, short for celeritas, which means “swiftness” in Latin — quite the understatement. If you were driving at c, you could get from New York to San Francisco in one one-hundredth of a second. Swift indeed!
This speed limit is totally and completely unbreakable. Light can’t break it. Information can’t break it. If something happens twenty light-years away, it is literally impossible for us to know for the next twenty years if it happened or not. If the sun flicked off, it would take eight minutes for us to find out (and then die instantly). Many of the stars you see at night are already dead.
So there’s a very real sense in which stuff going on here is separated from stuff going on very far away. If an event takes place today on Earth, in the next year it can only affect stuff within a light-year of Earth — a pretty cozy area, compared to the vastness of all space! The reverse is also true: an event on Earth can only be affected by stuff that was, in the last year, within a light-year of Earth. Everything further away is “time-like separated” from that event.
For an Earthbound mortal like you or me, the total extent of all things we can know about are things that are “close enough” in spacetime. The past, as far as we’re concerned, is just what’s inside our measly light-cone. The future too.
This is the Universe that we see. It’s not the actual full Universe, which is four-dimensional (three space dimensions, one time dimension). It’s a projection of the Universe onto a three-dimensional space. It’s like watching shadow puppets: you’re not seeing the actual three-dimensional puppets, just the two-dimensional projection.
Projections are simple enough when you’re projecting onto a flat thing like a wall. But our Universe isn’t flat.
Standing on the surface of the Earth and looking around, you might think the world was flat. The curvature isn’t obvious unless you take a lot of very careful measurements. At a small, human scale, the surface of a sphere is indistinguishable from a flat plane.
That’s not the only time something like that can happen. Two things with totally different overall “topologies” — like a plane and a sphere — can look exactly the same when you’re only looking at a little piece.
We have a fancy-sounding name for stuff like that: something is a 2-manifold if it looks like 2-space (a.k.a. a flat plane) when you hop into Google Street View. The surface of a donut is also a 2-manifold, because if you were teensy-tiny and standing anywhere on it, you’d think you were on a flat plane.
Our Universe seems to be 3-space (a.k.a. normal three-dimensional space). But we can’t send satellites outside the Universe to snap a picture. So all we really know is that it’s a 3-manifold. There are infinitely many 3-manifolds with different topologies, and without any extra hints we can’t tell which one we’re in.
But we do have some extra hints.
If an object has no forces acting on it, it’ll move at a constant speed in a straight line. Of course, it could also stand still, but that’s just a special case where constant speed is zero. Think of a hockey puck sliding on ice, or an astronaut floating through space. The only reason we don’t see things gliding like that normally is because of Earthly forces like gravity, friction, air resistance, etc.
But a “straight line” only makes sense if space isn’t curved, like on a flat plane or in normal three-dimensional space. If space were the surface of a sphere or donut, how would a freely moving object glide?
The more general version of a “straight line” is a geodesic, which just means the shortest path between two points. If space is curved, the geodesic can end up looking pretty weird. It can look even weirder if all you see is a projection of the path onto a non-curvy space.
For example, if you fly from New York to Hong Kong, the plane will take the quickest path — the geodesic. This path goes over the North Pole. But your in-flight map will project the curved surface of a sphere onto the flat surface of your screen. And that makes the path look very silly.
How does this give us any hints about the shape of the Universe? It all comes back to the universal speed limit.
When Einstein proposed in 1905 that there was a maximum speed for anything to travel, there was one problem: gravity. Gravity acts instantly across huge distances. The gravitational pull from the sun affects the Earth now, not eight minutes from now. The electromagnetic force has a speed-of-light delay, but not gravity. What gives?
This puzzle was solved ten years later, by Einstein himself. As it turns out, gravity is a sham force. It looks like a force from where we’re standing, but really it’s just the curvature of spacetime. The path of the Earth around the sun is just a free-moving object following a geodesic in curved four-dimensional spacetime. It’s like the puck gliding on ice.
The general theory of relativity tells us that matter doesn’t just sit idle in space — it bends spacetime around it, like a metal ball tugging on stretched fabric. Since we can only see time-slices of our Universe — projections onto our light-cone — the path of a free-moving object looks as silly as the Hong Kong flight on a map.
The math works out so that gravity looks just like a force if you’re inside the Universe it’s bending. This is one of the challenges of science: the world we’re trying to explain is the very same one we’re trapped inside. And being pinned to a tiny speck of space dust for a blink of a cosmic eye gives us a pretty weird perspective on reality.
It even makes us think — silly us! — that space and time are two separate things.
More on this: Stephen Hawking, A Brief History of Time, Ch. 2
More from Romy Asks:
Originally published at milobeckman.com on May 7, 2015.