Conceptualizing the analogy of gravity

[tourmaline]

Hi there, I have a question about something that has been bothering me for quite some time now: Doesn't the notion of gravity being a curvature (or "warp") in the fabric of the universe created by a body of matter presuppose the idea of an already larger gravitational force/curvature acting upon that matter? What I mean is, the common analogy for conceptualizing gravity is to visualize a sheet or mattress (the fabric of the cosmos) whose straightness is warped due to the preesence of a body of matter...but doesn't that analogy already presume a force of gravity acting upon the body of matter (e.g. the only reason the body of mass is able to warp the mattress to begin with is because something would be acting upon it gravitationally ("pulling it down"), thus creating an indentation in the fabric. Any thoughts? Lately, it's very difficult for me to conceptualize gravity using the Einstein analogy of the warp/curvature, because I keep getting tripped up by the fact that a body of matter could not make an indentation without being acted upon itself by a larger gravitational field. Thanks for any thoughts on this.

All

Robin

 

[Chronos]

Hi tourmaline! Welcome to PF. The whole gravity thing is indeed difficult to understand - or even describe for that matter. Gravity, according to Einstein, is time and space. An analogy I like is that relativity is a four dimensional version of Pythagorean theorem. The sum of the squares equals unity.

 

[tourmaline]

Thanks Chronos. I like your analogy using the Pythagorean Theorem (such a versatile equation!).

Regarding the original analogy of relativity which was bothering me, I think I need to just learn to suspend disbelief and not be so literal/analytical when it comes to visualizing physical concepts. All analogies are just approximations... but my first instinct is always to pick something apart and look at how it is not feasible: I think I should just go with the flow...

 

[turbo]

Hi, Tourmaline:

That's an interesting name, since I facet gemstones as a hobby, and I have mined Maine tourmalines for that purpose!

My personal view is that Einstein's mathematical model of gravitation as expressed by the curvature of space-time is close-but-no-cigar. More o follow.

 

[pervect]

Hi there, I have a question about something that has been bothering me for quite some time now: Doesn't the notion of gravity being a curvature (or "warp") in the fabric of the universe created by a body of matter presuppose the idea of an already larger gravitational force/curvature acting upon that matter?

This has been discussed a lot in the relativity forum - basically, the "rubber sheet" analogy is OK, but is weak for the reasons you cite.

There are better analogs out there in GR textbooks, but they aren't as widely popularized. MTW's "Gravitation" has some nice fairly simple anologies, mixed together with some formidable tensor calculus. I'm not aware of any source that presents only the simple anologies without the advanced math, unfortunatly. A brave enough reader could pick up the book, read the simple parts, and ignore the parts that are too advanced (which is most of the book! not that that's a real obstacle).

There are probably better, simpler books, but unfortunately I don't know what they are.

The topic really needs pictures, but I'll do what I can in words. Imagine drawing a space-time diagram of a simple system with one spatial dimension and one time dimension on a flat sheet of paper.

If a horizontal line on the paper represents our observer, who is always at x=0 for all t, a slanted line on the paper represents an observer moving with some velocity 'v'. An observer moving at a velocity 'v' moves in a straight line - so does our observer - and both straight lines follow the rules of Euclidean geometry.

Now, we introuduce curvature into the picture. NOte that we are curving space-time, not just space - an important point, lacking from the usual "rubber sheet" anology.

Imagine drawing the same space-time graph on a curved surface, such as the surfacae of a sphere. One of the problems you will face is that the sphere is finite, while your sheet of paper is infinite. If you can't imagine wrapping an infinite sheet of paper around a sphere in both directions (not possible in 3d, I think it's possible with an additional spatial dimension or two), you might imagine wrapping a strip of paper many times around the equator of the sphere (wrapping it around only in one direction, and limiting the height of the paper). This is possible in 3d. Or you can ignore the problem completely, that's the favorite approach of most texts, because it doesn't really affect the results any.

Now you have to imagine what happens when you draw your space-time diagrams on this curved piece of paper. Objects that follow the straightest possible paths will be following "great circles" rather than straight lines. If you plot the path of an observer moving with a velocity 'v' following such a "straight line", you see an interesting effect. As he moves away from the origin, he reaches a maximum distance, and then starts to move back - because two great circles on a sphere through the same point diverge, but eventually re-converge. This is something that doesn't happen in Euclidean geometry, but it happens in spherical geometry.

A detailed analysis of the situation really requires quite a bit of math, but one finds that drawing a space-time diagram on such a curved sheet of paper is functionally equivalent to saying that there is a "force" between the observer and the moving object. Thus the idea of "force" can be and is replaced with the idea of "curvature".

Some keywords for more reading - this idea is known as "Geodesic deviation", and is a very important part of GR.

 

[tourmaline]

That's an interesting name, since I facet gemstones as a hobby, and I have mined Maine tourmalines for that purpose!

Hi there :-) That's awsome, I think tourmalines are the most beautiful of gems. I love that they come in so many striking colors .

A detailed analysis of the situation really requires quite a bit of math...Some keywords for more reading - this idea is known as "Geodesic deviation", and is a very important part of GR.

Yep, and it's the math part that is a sticking point for me because I only just finished Precalc 1, lol . Ah well, I'm extremely curious about, and interested in, physics. It's just frustrating to want to know everything (especially all of those intriguing symbols which are able to communicate complex ideas so thoroughly and compactly: I'm dying to understand what they all mean...but I guess I'll have to wait for Calc class ). I have a conceptual physics textbook, and it has a section on Geodesic Deviation, so I'm really glad you pointed me in that direction. Thanks alot!