ELi5: Physicists say space is ‚flat‘, but how can 3D space be flat?

Question

My assumption is that the short answer is that light travels in a 45 degree angle in a space-time diagram?

In: Physics

Answer 1

It depends a bit on the context – what sort of physics we are looking at and whether we are looking at space locally or globally.

“Flat” and “curved” in this sort of area don’t have their normal, every-day meanings, but have more generalised, mathematical meanings – i.e. mathematicians and physicists started working with these terms when dealing with 2d surfaces, and generalised them to 3d or 4d.

The simplest way of looking at this is about how distances work.

You walk 5m in a line. You walk another 2m in a line. How far are you from where you started? Somewhere between 3m and 7m (depending on which direction you walked the second time), and we can work that out with a bit of trig, or using pythagoras’s theorem and some geometry.

But that only applies in “flat” space. If the space you are in is “curved” inwards, maybe you are 2m-6m away from where you started. If the space “curves” outwards maybe you could be 8m or more away from where you started.

In “curved” space, distances don’t work the way we normally treat them as working.

Mathematically this can get pretty messy, and we sort of need differential geometry. At the risk of going too deep, in “flat” space we have Pythagoras’s Theorem for measuring the distance (*ds*) between two points:

> *ds*^2 = *dx*^2 + *dy*^2 + *dx*^2

or if we are in 4-spacetime we get something like (up to convention):

> *ds*^2 = *dt*^2 – *dx*^2 – *dy*^2 – *dx*^2

These are our “metrics”, our way of describing how distances work, and importantly they should be the same no matter out perspective (within the rules we are using).

“Curved” space (or spacetime) would have a different metric. Which can look like all kinds of things. Maybe there are extra constants, maybe there are “cross” terms (like a *dx*.*dy* term), all sorts of fun things can happen.

But ultimately it means that distances don’t work the way they “should.”

And this is true locally with General Relativity; space gets bunched up around stuff with mass/energy, so the distance in a “straight line” past something with energy is longer than it “should” be.