Bowling ball on a mattress

Published on March 10, 2015

Bowling ball on a mattress

When explaining general relativity, this image is often coming up: imagine a mattress, and a bowling ball on it. The bowling ball is like a planet, and the mattress is the space. It’s visible that the bowling is curving the mattress: any marble launched nearby will roll toward the ball. This image has its benefits, but it’s misleading for several reasons:

  1. The mattress shouldn’t even exist before the marble is launched.
  2. The gravity shouldn’t deform the mattress/spacetime: the gravity is the mattress/spacetime.
  3. The mattress is a spatial object but what is really deformed is not only space but spacetime.
  4. Finally, this analogy is involving gravity to explain gravity… Doesn’t feel right.

Points 4 is more a pedagogical problems, but 1, 2 and 3 are more fundamental. Indeed our old “Newtonian” vision was that the spacetime pre-exists to things: the space and the time was like an immutable arena. The objects exist and move in this arena. the space and the time was outside the Universe and was used to measure the position and the moment of events, without being affected by them. The gravity was a force, and with its little arms it would pull objects together.

There is no mattress

Einstein’s theories broke this vision: the space and the time are now part of the world and are affected by the things in it. However the mattress vision is not radical enough. Indeed it’s not a big mental step to say: “OK, the gravity is not anymore pulling objects directly together with its little arms, but now it is bending an intermediate thing which is the spacetime”. In fact, as stated by 1. and 2., the mattress/spacetime is not an intermediate pre-existing thing that would just be deformed by the Gravity: the spacetime is the gravity. They are the same thing. The spacetime is not pre-existing to the objects, it is created by them. In term of fields, the spacetime is the field of the gravity, just like the electromagnetic field is the field associated with electricity.

Spacetime matters

Point 3 is another important problem. The mattress example makes us think that gravity deforms only space. But this would not explain free-fall: if the earth was deforming only space, I wouldn’t necessarily fall toward it. The moon could be stationary, even if earth is deforming space around it. However if I am stationary, I still have a “speed vector” in the spacetime: my vector is aligned only on the time axis. But since earth is deforming the spacetime, my speed vector will soon hit some space axis: I am falling.