Note on a common visualisation of spacetime curvature.

A common visualisation of spacetime curvature is presented below:

Mass curves spacetime

A mass is depicted curving a two-dimensional sheet, which changes the paths of surrounding matter. This visualization is utilized in this very popular science video:

As you can see in the video, the mass bends the sheet of cloth. The bending alters the paths of surrounding rolling matter.

The ‘mistake’ in the analogy:

An often and understably weird concept is that mass curves the sheet of cloth because of the actual ‘gravity’ acting on the ball. This makes the analogy seem to break down, because it uses the very thing it tries to explain. It’s like trying to explain what apples are using apples while describing it. It feels circular.

XKCD even made a comic about this problematic analogy:

 

I argue however that this analogy is not perfect, but not defective because of the reason depicted in the xkcd. To explain that reason, I first want to specify the difference between “How?” and “Why?”-questions.

“How?”-questions versus “Why?”-questions. 

Physics tries to answer ‘how?’-questions and not often you can answer the ‘why?’-questions. For example, Newton found the mathemetical way ‘how’ gravity works. But he realized it does not even remotely begin to answer the ‘why?’-question. He famously said: “I feign no hypotheses”. He had no clue why gravity happens.

Now, it is very important to understand that we still have no clue why gravity happens! Einstein answered the ‘How?’-question even more accurate than Newton, but it is a misconception to think that he answered the ‘Why?’-question.

Newton answered the ‘how?’-question with his inverse square law and Einstein answered the ‘how?’-question with saying that matter curves spacetime and spacetime tells matter how to move.

But why does matter curve spacetime? No one knows, although there are good candidates. The reason we cannot answer ‘why?’ mass warps spacetime is the reason why the analogy actually works. Because the reason why the mass curves the sheet of cloth is obscured, the reason why mass curves spacetime is also unknown.

I do think the analogy of the curving of sheets of cloth does not highlight inertial trajectories along geodesics enough, which is actually what Einstein found out. There are other visualisations that represent this fact better, I presented them below.

Why does mass curve spacetime?

The answer to the ‘why?’-question can probably be found when we completely and accurately answered the ‘how?’-question. Since there is no quantum-mechanical formulation of gravity, the ‘how?’-question is still not fully answered.

Resources for other good visualizations of spacetime curvature: 

http://www.relativitet.se/spacetime1.html

http://demoweb.physics.ucla.edu/content/10-curved-spacetime

http://www.adamtoons.de/physics/gravitation.swf