How can I vizualize and understand curved spaces in general relativity?

Question

I'm taking a basic physics class and the teacher described space with a special table that has curves and black holes etc. He would throw a metal ball down onto it and the class would watch it circle around a black hole and this showed the warping of space.

The instructor said that it is actually more like a table cloth because the ball would cause a warping as well.

The problem I have is transferring this image of 3d objects on a warped 2d plane to actual actual space.

Another way to look at my dilemma: There is space above the 3d object on the table, and on the sides of it. If we were on the 3d object and left if in any direct but down, we would be in space, but how is this space described? I can only picture the one "bottom" portion of space.

Help me understand the rest.

Answer 1

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.

Answer 2

Unfortunately Human beings haven't evolved to deal with thinking outside 3 spatial dimensions and as such any attempt to do so will require analogy or reduction.

The rubber ball on a sheet is useful in a reduced dimension problem but is more of a lesson in how geometry relates to particle motion rather than 4D General Relativity. Think of it as a stepping stone towards the maths, and once you've got that sorted you can investigate as many dimensions as you want!

Answer 3

Look at the world through the bottom of a wineglass with one eye. Then close that eye and open the other, seeing the world as normal. Switch back and forth a few times if necessary. It produces the same distortion as a gravitational lens, so you can visualize the warping of space a little better that way.

Answer 4

Another way to think of curved is distorted as in distance distortion.

A basic postulate of Euclidean geometry is "The shortest distance between two points is a straight line." That's what the top figure shows (below). That would be true if space were not curved, but it is. The second figure shows a curved line. In addition to the intuitive notion of what curvature means, it also changes distances.

It is curved spacetime and not just curved space, however. Assuming that you understand what unifying space and time means (a topic from special relativity), curved spacetime means that distances and times are distorted.

If you learn more about general relativity, you will get more to what the curvature means, but the basic concept is the distortion. Picturing curvature in your mind for dimensions greater than two is hard, but as long as you remember distortion you'll be OK.

Answer 5

Truly to visualize it is difficult, but there are a number of points that can help you think about it.

It's always good to start thinking in a space you can visualize and work from there. If you live on a curved surface, you can detect the curvature just by measuring distances and angles. For instance, on the Earth, if you measure a circle with a radius of ten thousand km, you will find that the circumference is less than $2\pi$ times the radius (where you measure both the circumference and the radius along the surface). That tells you that you are not on a flat surface. This means that, mathematically speaking, if the way you measure distances and angles changes in the right ways, or fails to follow the rules of Euclidean geometry, then you can use the mathematics for handling curvature, and the space (or space-time) is considered to be curved, even if there is no higher-dimensional space for it to be curved into.

This in turn means that, when you hear time is slower close to a black hole, or that distances have to be measured differently, that is the curvature. You can visualize it a little bit by taking a 2D slice and bending it in 3-space so the geometry works. That helps you see why it's called "curvature". But the essence of it is in the changing of time and distance.

Then you have to ask how the changing of time and distance leads to gravity. One point that it took me some time to understand is that, in space-time, you are always moving, mostly into the future. Motion in space consists of tilting your headlong path into the future so that it slopes in a spaceward direction as well.

In free fall, this path into the future will be as straight as it is possible to be. Depending on your convention for defining space-time distance, this means it should be the shortest space-time distance between two points (in space-time). This, finally, means that heading into the future without moving closer to a gravity source won't be the shortest path in space-time. (Actually, the most natural way to measure space-time distance is the amount of time experienced as you go from one event to another, and in that case the closest thing to a straight line is the one that maximizes the distance. This is surprising, but it comes from the odd way you have to measure distance in space-time in order to match the physics we observe.)