The Einstein for Everyone website was a great eye opener for me.

Read the lectures in the "Non-euclidean geometry" and "General relativity" sections. It explains all without demotivating you with the hard math.

The key idea is don't try to imagine the curved space-time wrapped in higher dimensions it won't work. Just think in terms of converging and diverging "parallel" lines.

On Earth's surface (a sphere) moving along initially parallel lines eventually intersect (positive curvature). On a saddle like surface the initially parallel lines diverge (negative curvature).

Now let's put gravitation into the picture. Drop two bodies, that have vertical separation between them. As they fall the vertical distance between them increases. If you plot the action on a space-time diagram you see their world lines diverge, this means on the vertical direction the curvature is negative.

Now drop two bodies that have horizontal separation between them. They fall towards the Earth's center, so their separation will decrease, if you plot the action on a space-time diagram you will see their world-lines converge, so in horizontal directions there is positive curvature. If you sum up the curvatures you get 0, because there is no matter density outside the earth.

Now if you would drill hole through the Earth, and drop balls with vertical separations between them you will see that their wordlines will converge (as gravation is weaker inside Earth), so the curvature is positive, in all the 3 directions (since it doesn't matter where do you drill the holes). So if you sum it up you get a positive number, because matter density inside the Earth is positive.

What Einstein's equations describe is that the net space-time curvature at a point is proportional with the matter density at that point. It's easy to say but solving these equations are incredibly hard.

The main idea is that it follows the path of least resistance. The light "wants" to travel the shortest path, which is (almost) always a straight line. However, since spacetime is curved, this straight line happens to be curved as well! Shown in the image are two hypothetical light paths, one that follows the curvature of spacetime and one that doesn't.

Consider the drawing on the left first. Initially, it might appear that the one that goes straight (the yellow line) would be travelling the shorter distance, because it looks like a straight line on the chalk board. However, it isn't a straight line, because the spacetime it's in is curved. Likewise, while the red line appears to be longer, it's not, because it's actually following the straight line in the spacetime that it's in.

If we "uncurve" the spacetime, we can clearly see that the yellow line is actually the longer of the two. Since the light "wants" to minimize the distance it travels, it'll end up taking the red line's path. The reason that it "wants" to take the shortest path is beyond my ability to explain, but it can be solved using the Variational Principle.

## Best Answer

As you travel through the warped spacetime, you would not notice much difference. This is because any spacetime regions you are likely to ever encounter look exactly the same as flat spacetime locally. This is great news for us because it means we'd always be able to assume that propelling ourselves forward will actually make us go forward and not to the side or something weird like that.

As a distant observer, we can actually see what warped space looks like. A phenomenon called gravitational lensing allows us to see the warping of spacetime as light passes by objects of large mass. Take a look at the picture below. That ring around the star is the light from a galaxy behind the star that has passed through warped spacetime.

Practised observers can actually see a few more examples of lensing in that picture (I can see 5 others, but I'm not an astronomer. How many do other people see?), but you get the idea. Pretty cool, eh?

Understandably, you may feel that this visualization does not give you an intuitive understanding about how things should behave in warped spacetime. That is why you often see the 2D simplification; it's easier to get the point across. But if you want a visual image of how gravity actually warps/curves spacetime, you'll find nothing better (at least, nothing I can think of) than a nice picture of an Einstein ring, like the one above.

_{Picture taken from Wikipedia}