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


Solution 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.

Solution 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!


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Updated on February 27, 2020


  • Qmechanic
    Qmechanic over 2 years

    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.

    • Florin Andrei
      Florin Andrei over 10 years
      You can't transfer that image to actual space because then you'd have to visualize curvature in 4 dimensions, which is impossible. The purpose of what the teacher did was to alleviate that problem.
    • Qmechanic
      Qmechanic over 9 years
    • Calmarius
      Calmarius about 9 years
      @FlorinAndrei It's possible to visualize 4 dimensions as an animation. It's possible to visualize curved 4D space-time by rendering the animation from multiple observer's viewpoints.
  • resgh
    resgh over 9 years
    By the way psychology research suggests that the humans' vision perception of the world (3 Dimensional vision) completely depends on external stimuli we were all exposed to during youth. So perhaps with appropriate training we humans just might be able to visualize 4D space!