How can we use distances between points to define geometry?

[pervect]

Taylor and Wheeler frequently mention the following exercise: (see for instance MTW, or online their excerpt from "Black Holes" http://www.eftaylor.com/pub/chapter2.pdf)

Consider turning a rowboat upside down, and driving nails into it's surface. Create a table of distances between "nearby nails", where said distance is measured along the surface of the rowboat, as with a string for example.

This statement is usually presented as a way of intuitively defining a geometry. I've always found it very helpful, though when I mention it to others puzzled about the issue of what curvature and geometry "means", it doesn't always seem to satisfy them.

I was thinking about this statement, and I realized that it would be very interesting, and perhaps enlightening, as to exactly HOW one could go from this notion of geometry as a table of distances to one of the usual representations in GR. The first choice would be to go from the nail-data table to information on the curvature itself - which in this simple case is just a scalar at every point. Computing some sort of metric information would also be another approach, this would also require some sort of induced coordinate system, I suppose.

Perhaps one of the weakness of the above statement is that it's just made, without actually being demonstrated.

So, what sort of algorithm could get the curvature scalar information (or something from which this could be calculated, a metric tensor perhaps) for this simple 2d case?

It would also be interesting as to what additions to the algorithm would be needed to extend it for the case where the rowboat is replaced with 4d space-time, and the "nails" by events in said space-time.

 

[Mentz114]

I also have some unease about this. To get the distances presumably some sort of geometry must be assumed so it sounds tautologous. If there was a table of distances and coordinates, the metric could be found, I think.

Maybe the nails are vectors normal to the surface in which case it is possibe.

 

[DrGreg]

The "table of distances between nearby nails" effectively defines the metric (in the limit as the nails get closer together).

The table tells you what g(dx, dx) is, and then you can recover the full metric via

g(dx + dy, dx + dy) = g(dx, dx) + 2 g(dx, dy) + g(dy, dy)​

(the metric being bilinear). Note I've formulated this in a coordinate-independent way, considering the metric as a 2-form, a bilinear scalar-valued map acting on the tangent space.

 

[pervect]

OK, thanks - I was hoping for something less abstract, but an infinite table giving the distances between "nearby" points is, as you point out, just what the metric does. Usually one replaces the infinite table of distances into an easy-to-manage formula, but the end result is the same.

I'm not sure if this will make the semi-philosophical point that "distances define geometry" any easier to appreciate - which is what I was initially hoping to accomplish, since we get so many questions about curvature and geometry.

I think perhaps one of the obstacles is convincing oneself that one doesn't really need angles to define geometry. The abstract argument is that the same metric that defines distances defines angles as well ..[edit - just had a thought]

Is this in general true (that distances define angles), or does this come from assuming we have a "metric space" to start with?

 

[Dale]

I think perhaps one of the obstacles is convincing oneself that one doesn't really need angles to define geometry. The abstract argument is that the same metric that defines distances defines angles as well

That actually was my first thought as well. I guess that as long as you have sufficient information to make an inner product of every possible pair of tangent vectors then you have both distances and angles.

 

[A.T.]

I also have some unease about this. To get the distances presumably some sort of geometry must be assumed so it sounds tautologous.

Maybe because "geometry" is an ambiguous word. It can refer to a general concept which defines how distances are calculated. Or it can refer to a property of a specific observed object (it's metric, or "shape").

 

[bcrowell]

OK, thanks - I was hoping for something less abstract

Here's a more concrete treatment: http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.3

I think perhaps one of the obstacles is convincing oneself that one doesn't really need angles to define geometry. The abstract argument is that the same metric that defines distances defines angles as well ..
[...]
Is this in general true (that distances define angles), or does this come from assuming we have a "metric space" to start with?

A metric space is a very general thing.

I think the relevant fact is that the space is locally Euclidean.

For example, Taxi-cab geometry is a metric space: http://en.wikipedia.org/wiki/Taxicab_geometry It may be possible to define angular measure in Taxi-cab geometry, but it wouldn't be the same as Euclidean angle measure.

Another good example is affine geometry, which gives you as much of a system of distance measurement as you can have without also allowing Euclidean angle measurement.

 

[atyy]

I wonder if the key to converting the Taylor and Wheeler illustration to geometry is that these distances are not any old distances, but are geodesic or extremal distances (the string must be taut).

So if one lays out a cartesian grid (coorindinates) on a piece of (flat!) paper, we can tell whether the surface represented is flat or curved by the extremal distances (taut string) assigned to nearby points.

 

[pervect]

A lot of good points here.

It seems pretty clear that using "taxi-cab" distances would mess up the geometry in a major way from what's intended. So a bit more specification is needed, I'm not sure how to word it all rigorously...

The reading in "TIme and Matter" on the classical results was interesting, but a lot of the results were expressed in terms of angle deficits. It seemed to me there should be a more direct way to figure out curvature, without measuring angles at all.

After some thought, I realized that on a sphere, one can detect curvature by measuring the distance between four points. For instance, the diagonals of a "square", i.e. a quadrilateral with four equal sides, will be longer than they will be in a plane.

I'm pretty sure that one can't detect curvature measuring only the distances between three points - one will get three numbers, but they should be possible to embed in 2d space.

The easy way to figure out that the length of diagonals on a quadrilateral on a sphere is to use the relation between arc length and chord length. The sides of the square and the diagonals of the square will all be chords of the "great circle length" on the sphere.

 

[PAllen]

I'll repeat a point made in some other threads here. J. L. Synge, in his classic GR book, shows that distances measured between 5 spacetime points are necessary and sufficient to determine presence of curvature in 4-D spacetime. He has a section on a "5 point curvature detector" based on this theory.

 

[atyy]

After some thought, I realized that on a sphere, one can detect curvature by measuring the distance between four points. For instance, the diagonals of a "square", i.e. a quadrilateral with four equal sides, will be longer than they will be in a plane.

I'm pretty sure that one can't detect curvature measuring only the distances between three points - one will get three numbers, but they should be possible to embed in 2d space.

I believe Weinberg gives exactly this example and the generalization to N points in the first chapter of his old GR textbook.