# Alternate layman's metaphors for illustrating curved space-time

The metaphor of a surface (typically a pool table or a trampoline) distorted by a massive object is commonly used as a metaphor for illustrating gravitationally induced space-time curvature. But as has been pointed out here and elsewhere, this explanation seems (to a layman like me, at least), to be "hopelessly circular", and in the end contributes little to an understanding of how modern theories of gravitation work.

Are there other (or additional) metaphors that might be helpful in illustrating to lay readers (a) what motivates modern gravitational theory and (b) why it has greater explanatory power than Newtonian gravitation?

• Please excuse (and feel free to correct) any misuse of terminology in my question. – orome Nov 13 '11 at 17:54
• Apologies, I propose it is a possible duplicate of How exactly does curved space-time describe the force of gravity? - vote on it - I don't really understand in what sense your question is different than the previous one and why you expect people to say something that they haven't already said. The trampoline is used because the laymen may imagine it. But the real GTR doesn't need any "gravity beneath the trampoline". Instead, it says that things are moving along geodesics. – Luboš Motl Nov 13 '11 at 18:17
• The curved trampoline isn't just a "metaphor": a curved trampoline or any curved surface is a special case of a curved manifold and it is the very same kind of curvature that is used in GTR. For this reason, there can't be any "alternative layman's explanation" of the curved space because the curved space is always the same thing. What may be underestimated in the explanations for the laymen is that we have an independent way to find the "straightest possible path on a curved manifold" which doesn't require us to talk about any extra "gravity beneath the trampoline". It should be emphasized... – Luboš Motl Nov 13 '11 at 18:21
• @LubošMotl: This is a distinct question, partly in the way you've indicated in your comment above (which could be an answer): "no not really, but as generally presented it is misleading ... here's what's wrong with how it is typically presented...". It is also distinct in the specific sense asked above: is there a metaphor (or an aspect of the existing "metaphor") that illustrates why it is a better explanation than Newtonian gravitation (again, your comment points to an answer). – orome Nov 13 '11 at 18:47
• Probably the trampoline is not good because there should be many trampolines (each particular motion needs its own geodesic, they are different, not common to all particles). Maybe the best metaphor is motion of a particle under forces in a flat space with motion-dependent clocks? – Vladimir Kalitvianski Nov 13 '11 at 19:39

The best way to understand curved space-time is Einstein's way in 1907. Imagine all of space is filled with clocks which are held in place, but they need tick at different rates in order to stay simultaneous with each other. Near a massive object, the clocks tick slowly, away from masses, they tick faster.

Particles travel through space so that they locally take the path of maximum time between fixed endpoints, so that between endpoints which are close to a massive object, their path curves out a little, meaning that they are bent toward the massive object.

This is a statement of the Einstein 1907 theory of gravity, which he knew then would be the weak field, slow velocity approximation to Genera Relativity. It is counterintuitive for a few reasons:

• In geometry, straight line paths are minimum distance. In relativity the path is a local maximum. This is a consequence of the minus sign in the Pythagorean theorem in relativity. In relativity, unlike in geometry, the sum of the length of two legs of a triangle (when these are not imaginary) is always less than the third, so that straight lines maximize proper time.
• There is only one function which describes the curving of space time, and this is the clock rate. The curvature is determined by this clock rate, but it is purely a time curvature. Space is not curved at all.
• The geodesic motion is not trivial to see from the clock-rate description. You might naively think that to maximize the proper time you need to move away from massive objects, because time ticks slower near them. But the maximization is holding the endpoints fixed. To give an equation of motion without the concept of maximum proper time, you can just say that objects feel a force of attraction towards regions of slower clock-tick, and leave it at that. But this doesn't look like a geometrical condition (although it is).

I don't believe that there are two pictures of a phenomenon, one appropriate for laymen and a separate one for physicists. A correct picture is a correct picture, and is useful for both, and a misleading picture is misleading for both. This picture is used by all General Relativists when they are thinking about the weak field limit.

### Two dimensional relativistic gravity

For the two dimensional gravity, with point masses, there is a nice description which can be understood immediately. Two dimensional point masses are parallel strings moving perpendicular to the direction of motion in 3d plus time, but these strings are like pencils of light, not stationary line-masses, they are relativistic along their direction of motion. You need to have a relativistic momentum density on the strings for them to reduce to the simple limit of 2+1 gravity.

In this limit, the strings are described by 2+1 graity. The point masses in 2+1 gravity are described by cutting out a wedge from a two dimensional paper representing space-time, and gluing it back to form a cone. This description is exact--- this is what the space-time around a relativistic cosmic strings looks like. The space is called locally flat, because if you draw a least distance line it will be straight after unrolling the paper, so that the only curvature is that which can be seen from outside, not to a flat fellow living inside the paper. There is only intrinsic curvature at the tip of the cone, proportional to the deficit angle, the angular size of the wedge. This is is the mass of the string.

If you imagine a particle coming in from infinity, it travels in a straight line along the cone, but it comes out deflected in a certain way. This is easiest to see by taking two parallel lines coming in on opposite sides of the cone point--- they will intersect each other.

If you make a double-cone by cutting out two wedges, to make a slushie-shape after gluing. The paper is still locally flat, but if you draw two straight lines, the line passing between the cones will intersect the other lines. A collection of n stationary cone points describes an equilibrium stationary configuration of 2d gravity.

If you set the cone points in motion, and add some points with negative curvature which evolve in a specific way (their curvature in the 3d sense is still zero) you get t'Hooft's description of 2+1 gravity, which is an active research subject today.

If you take a piece of paper, it does not matter how you twist the paper, its intrinsic curvature is zero. (only extrinsic curvature may be non-zero)

If you take that piece of paper and make a cylinder, its curvature is still zero, but its topology changed (the structure of connections in the space)

if you take that piece of paper and make a (conical) water cup, its curvature is zero everywhere except at the bottom cusp of the cup.

Now imagine i zoom in the cusp, and see that its bottom end looks like a half hemisphere. The mean radius of this hemisphere is the curvature in the cusp. GR states that this curvature is proportional to the matter density in the cusp.

Now take a flat infinite paper bidimensional surface representing space, it is everywhere flat. Cut a hole in the paper and surgically attach the water cup. The curvature is non-zero only in the surgical scar that connects the cup with the surface.

Now imagine this surgically attachment is made continuous, so the curvature is "spread" in the surface. A paper where such a curvature exists is basically impossible to twist into a flat surface again without tearing the paper. (even the paper cup cannnot be twisted in a flat surface without tearing)

The cylinder is sort of a edge case; i can split the cylinder along its length in two maps (lets call it north and south section). Each of these maps is entirely mapeable to a flat surface without any tearing of paper, and i can make sure the lines connecting both maps also map without "twisting". This mapping is called an atlas of the cylinder, and the fact that these maps exist to a flat surface, plus some considerations in the boundaries, guarantees that the cylinder is flat in the diffeo morphic sense (even if it is topologically unlike the flat surface). The reason is that curvature is a local notion of the paper, while the topology is related to global properties of the paper

Now, geodesics of movement, which is what is the generalization of the first Newton law to curved space-time, says that objects will show apparent acceleration when in a piece of paper with non-zero curvature

I am not tackling how does this change when time and the other space-time dimensions are involved, but I hope this helps to bring a more realistic picture of the geometry.

• Could you provide a citation that explains the distinction between intrinsic and extrinsic curvature as it relates to gravitation? I'm having difficulty accepting that the curvature of a bent piece of paper is zero. – JxB Nov 13 '11 at 23:13
• intuitively intrinsic curvature is anything that requires stretching of the metric elements; this is why i choose paper instead of say, rubber. Paper is flexible in bending in orthogonal directions to the surface, but is very rigid regarding bending in directions parallel to itself, which amount to variations in the metric – lurscher Nov 14 '11 at 3:04
• – lurscher Nov 14 '11 at 3:06
• very good link i just found googling: laplace.phas.ubc.ca/BIRS/Talks/arnold2.pdf – lurscher Nov 14 '11 at 3:07
• I didn't mean to repeat your answer--- but it seems I did. I am sorry. +1. – Ron Maimon Nov 14 '11 at 6:40

One definition of (intrinsic) curvature is: the amount of deviation from Euclid's geometry formulas. The two geometry formulas most used here are a) sum of angles of a triangle equals 180 degrees b) area of a circle equals $\pi r^2$. this should be taken as a kind of 'percentage deviation' in that obviously for a very small triangle or circle the deviation will be small, but as a percentage of the area of the triangle or circle, it will not get smaller and smaller even if the triangle shrinks to zero.

The globe is curved but a cylinder is not. If you start at the North Pole and walk straight to the equator, then turn right 90 degrees and go one quarter way around it, then turn right again 90 degrees, you will get back to the NOrth pole eventually, and your triangle trip has three right angles so their sum is 270....way off. Now just try that on a cylinder...it can't happen. So the cylinder is not curved.

Check out this links for better visualizations of curved space-time, which explain Einstein's gravity model in a non-circular way, and actually include the time dimension (not just space like the common rubber-sheet).

http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_spacetime.html

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