My assumption is that the short answer is that light travels in a 45 degree angle in a space-time diagram?
In: Physics
Flat here doesn’t mean flat like a piece of paper, it means the universe zero curvature. Flat geometry just means Euclidean. It’s the geometry we’re all most familiar with in our everyday lives, where parallel lines never meet and the sum of angles in a triangle is 180 degrees. If the universe has a curvature, those things would not be true.
Flat refers to the geometry of the universe. A flat universe is the one you always thought you lived in: Angles of a triangle sum to 180 degrees, parallel lines do not intersect, etc.
I don’t know what physicists you’re referring to, but I didn’t think anyone serious claims that “space is flat”. We acknowledge curvature in spacetime as a fundamental consequence of General Relativity. That’s how we can explain the Doppler Shifts of starlight, time dilation experienced by GPS and other high-velocity probes, etc.
What you might be confusing is the claim of space being *locally flat*. That’s more of an abstraction to explain relativistic behaviors of subatomic particles. Personally, I like to think of it as the difference between modeling terrestrial dynamic observations you can see with your eyes (microscopic) versus from a satellite (macroscopic). This “local flatness” is mathematically equivalent to creating a N-dimensional linear approximation.
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.
We say a 2d shape is flat if it doesn’t curve in a third dimension. We say that 3d space is flat if it doesn’t curve in some 4th dimension. As others say, that would show by triangle’s angles not adding up to 180 degrees.
Now, we know that space is curved locally, as heavy object’s gravity warps spacetime, but on larger scales the universe appears to be flat, at least to the extent we can measure it.
Its flat as in Euclidean geometry applies. On curved surfaces geometry works differently (parallels for example only exist in a flat space). Now of course it’s uncommon to think of 3D space as a surface but whatever you can say about a 2D flat or curved sheet you can extend things into one dimension higher.
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