Gravity bends space, so how can space have only 3 dimensions? [duplicate]

Question

Due to the laws of gravity and electromagnetic attraction (decreases with $\frac{1}{r^2}$) we know that space should be limited to 3 dimensions.

At the same time we know that gravity bends space. All visualization of that are basically elastic 2D membranes where heavy balls are placed to create "gravity wells" of various sizes. The watcher is left to imagine all that happening with one more dimension and then you've got the way how gravity bends space.

However, the membrane visualization works by distorting the 2D plane into the 3rd dimension. If I go 1 dimension higher, logically 3D space should be distorted into the 4th dimension.

Basically, gravity distorts space into a dimension that shouldn't exist according to the laws of gravity?

Question: where does this logical chain of arguments have a hole?

(Assume I have a lot of popular science knowledge and a background in Mathematics and computer science, but no physics)

Answer 1

What you're asking is actually a very important topic in general relativity. One of the most important features of the mathematical surfaces used in general relativity is that their curvature can be defined without making reference to an "external" space into which we have to distort the surface.

In the popular visual presentation of general relativity we take a 2-dimensional surface and we "press" into it to cause it to bend. This seems to require a 3rd dimension in which to press the surface. However, one of the most important discoveries in the study of curved surfaces is "Gauss' theorema egregium", which essentially states that we can fully describe the curvature of a surface without needing to make any reference to a surrounding, higher-dimensional space in which it is embedded.

This ability to describe curvature without referencing the surrounding space is called intrinsic curvature (as opposed to extrinsic) if you want the fancy terminology. It also makes sense that we be able to describe our universe without needing to make reference to an "external space" around the universe into which it curves.

Just as an additional note: spacetime, which is the surface on which general relativity is formulated is a four-dimensional surface, and so the presence of mass and energy curves spacetime rather than just space alone. Unfortunately there is no way to nicely draw 4-dimensional space like you can with 3-dimensional space.

Answer 2

The membrane analogy works great, but only if you don't think about it too much. The only reason it works (balls orbit around the depression) is because there is a real gravitational field that no one ever mentions, the one on the planet where the experiment is being done. It wouldn't work at all in free-fall in space.

Instead of thinking of it as bending, it would be better to think of it as distorting. Consider the part of a balloon opposite the open end. The rubber there tends to be tougher than elsewhere, even when the balloon is inflated, so much so that one can insert a pin through it without causing it to pop.

The analogy is that just as there is more rubber concentrated in that one small area, there is much more space concentrated near large masses. Measuring the distance across one of these concentrated areas produces a value that is larger than what d = c÷𝜋 would predict. No extra dimension is needed.

Answer 3

Space is curved if you don't come back to your starting point when you walk around a square. Or equivalently, you wind up at different points if you walk east-then-north vs north-then-east.

The surface of the Earth is curved in this sense. It doesn't show for a small square. But try a really large square. Start on the equator.

• Walk 1/4 of the way around the world to the east. Turn left and walk 1/4 of the way around the world to the north. You are at the north pole.

• Walk 1/4 of the way around the world to the north. Turn right, and walk 1/4 of the way around the world. (OK, it isn't east because coordinates are weird at the north pole.) But you are on the equator.

In GR, a mass causes distortions of distance and time. If you are in orbit, the distance to the center of a star is deeper than you would expect from dividing the circumference traced out by the orbit by $2\pi$. Time runs slower at the surface than in orbit.

Space-time is 4 dimensional, so you get an extra direction you can walk around the block. You can also wait a while.

So trace out this "square" where one side is distance, and the other time. Start at a point above the star.

• Have a person at the top wait a bit, then find the point/time a distance X below him at that time.
• Find the point/time a distance X below the top person right now. Have someone at that bottom point wait a bit.

Time is slower at the bottom. In his travel through time, the bottom person passes through the the point/time the top person picks out. But when he does, he isn't done waiting yet.