Map analogy for how spatial curvature causes gravity

[md65536]

In various topics on this forum, I get the idea that it is still a mystery how or "why" spacetime curvature causes gravitational acceleration.

Am I mistaken? Did Einstein and others who understand GR also understand that gravity immediately follows from spacetime curvature?

It seems intuitive to me that it does (thus "why"), however the details are not at all intuitive, and I wouldn't even begin to know where to begin with the math.

 

My basic "understanding" of how spatial curvature causes gravitation is as follows:

1. Movement through curved space means that the distances between remote points changes as you move through space. I think this is because distances in curved space are defined in terms of local flat spatial curvature. For example, if you go to google maps, you get a flat representation of the curved Earth; if you scroll the page north or south, the map scale ruler will change in size, meaning that the relationship between pixels and meters changes. In real life, the scale ruler would stay the same size while the "map" (ie. your measurements of the universe) changed and scaled or warped.

2. Uniform oscillatory motion of particles in a curved mapping of the curvature of space will tend to be non-uniform in a flatter mapping of the same space.

3. Repeatedly moving according to one mapping of spatial curvature (eg flat), and then changing the mapping to fit the different curvature of the new location in curved space, will tend to accelerate you towards greater spatial curvature or whatever.

 

 

As an analogy of this, consider a typical world map (Mercator projection). Uniform distances on these maps do not map to uniform distances on the Earth. A very small area around the poles is expanded to take up the entire width of the map.

 

Imagine someone doing a random walk on the flat map, and then following that path on a globe. They could do this by picking a random direction to move, and moving by a fixed (or random) distance on the map. Clearly they would tend toward the poles more than toward any other specific point on the globe, because the poles take up so much more room on the map.

 

An observer on the globe who is unaware of the map but saw the walker tending towards a pole might think that some force is drawing them there, but there is no force... just a uniform movement in one mapping that corresponds to non-uniform movement in another.

 

Obviously this isn't exactly how gravity works because it's still a random walk, and the person will still wander away from the pole, and gravity is not evidently random. This map example ignores point 3, which would involve something like the walker creating a new and different map at each step, or perhaps changing step size with each step; so acceleration isn't demonstrated. The details of what such new maps (or step size) would look like is a complication that eludes me. They would certainly not be Mercator projections.

 

 

 

Does this make sense?

Is this a good analogy for how gravity works? Are there already similar analogies (and better)?

Edited August 16, 2011 by md65536

[ajb]

In various topics on this forum, I get the idea that it is still a mystery how or "why" spacetime curvature causes gravitational acceleration.

Am I mistaken? Did Einstein and others who understand GR also understand that gravity immediately follows from spacetime curvature?

 

My understanding is that the equivalence principle strongly suggests that we live on a Lorentzian manifold.

 

Section 4 of Carroll's Lecture Notes on General Relativity (arXiv:gr-qc/9712019v1) gives arguments as to why the equivalence principle implies curvature.

[md65536]

My understanding is that the equivalence principle strongly suggests that we live on a Lorentzian manifold.

 

Section 4 of Carroll's Lecture Notes on General Relativity (arXiv:gr-qc/9712019v1) gives arguments as to why the equivalence principle implies curvature.

 

 

That looks like an excellent resource, thanks.

After a quick glance at the section it looks like his explanation works with simpler requirements (his works with slow-moving objects but mine requires high-speed oscillation of all masses).

I'll have to spend some time trying to figure out the lecture notes.

[md65536]

That looks like an excellent resource, thanks.

After a quick glance at the section it looks like his explanation works with simpler requirements (his works with slow-moving objects but mine requires high-speed oscillation of all masses).

I think I judged these explanations with the wrong criteria. I'd assumed that explanations involving the fewest requirements are superior, because if another explanation requires some other precondition, then the former explanation must be more general and account for a greater range of phenomena.

 

But I think instead that all that matters is whether or not the required precondition is observed in all cases that the various explanations cover.

 

In this case, the precondition is the requirement that "all mass must oscillate at the speed of light." So the only question is whether or not that's true, which I think it is (based just on a vague understanding or misunderstanding of matter and energy).

 

If it's true that all mass oscillates, then an explanation of gravity that makes use of that could actually be simpler or perhaps more intuitive, because it may correspond more closely with reality.

 

 

 

However, correspondence with reality means correspondence with observations and thus also with theories (and their explanations) that correspond well with observations, and the judgment of that is how well the math corresponds. I've been trying to convince myself that this speculation "works", while avoiding figuring out the math, which is probably foolish.

 

 

 

 

Since we already have equations that correspond well with observations involving gravity, I think the way to turn an idea into a viable explanation is to "massage" the idea and the math to fit each other, while ensuring that one has a reasonable explanation for why each step is being done. Ideally, I'd like to take some already accepted ideas about mass (such as ubiquitous oscillation) and spacetime curvature, and show that existing equations can be derived directly from that.