Gravity from the Ground Up by Bernard Schutz

Author:Bernard Schutz
Language: eng
Format: epub
Publisher: Cambridge University Press

When the triangle’s two sides are perpendicular, then θ = 90°, so that the last term is zero, leaving the usual Pythagorean theorem. But in general one needs the cosine term to compensate for the fact that the distances x and y are measured in directions that are not perpendicular to each other. We see that the cosine formula is a special case of the formula Equation 18.5 with C = −2 cos θ and A = B = 1. The cosine formula can be thought of as the distance formula in flat space with straight coordinates that are skewed, so that there is an angle θ between the coordinate axes, an angle that is not necessarily a right angle.

Of course, the cosine rule is a formula that is correct only for a triangle in a flat two-dimensional plane. But every smooth curved space is locally flat (i.e. flat if we look at a small enough piece of it), so if the differences Δx and Δy are small enough, we can interpret the distance formula Equation 18.5 exactly as a version of the cosine rule. Therefore, C at any point just measures the angle θ between the directions of the coordinates at that point: cos θ = −C/2AB.

This shows that an ordinary space (not spacetime), where squared distances must be positive, cannot have a distance formula with arbitrary coefficients: C2 must be smaller than or equal to 4AB (in order that cos θ should be less than or equal to one) and A and B themselves must be positive.

Figure 18.3 illustrates a coordinate system for a two-dimensional surface that is curved. It shows that, even when the coordinates are drawn in a very regular and smooth way, they stretch and turn to follow the surface. If we choose any grid line in the diagram and move along it, we see that the distances (measured along the surface, of course) between successive intersections with other grid lines do not remain a constant length: the grid is stretched and compressed. We also see that grid lines do not always intersect at right angles. The distance formula for this surface, in this coordinate system, will have non-zero functions A, B, and C.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.