Reference: Hobson, M.P., Efstathiou, G. P. & Lasenby, A. N. (2006), General Relativity: An Introduction for Physicists; Cambridge University Press. Problem 2.1.
Having worked through Moore’s General Relativity Workbook up to the point where we’ve obtained the Einstein equations and worked through a few solutions to obtain some metric tensors in several situations, I think it’s time to back up and have another look at general relativity from a slightly more mathematical viewpoint. I say ‘slightly’ as I don’t want to delve into the rigorous mathematics of differential geometry (although I’m sure that’s a rewarding pastime), but I found Moore’s derivations to rely a fair bit on hand waving to justify the results.
There are many books on general relativity that use a more mathematical approach than Moore’s (in fact, pretty well all of them do), so we’re spoiled for choice. I had a look at the immense tome Gravitation by Misner, Thorne and Wheeler but, although it’s elegantly written I found it a bit hard to follow (as well as being just toooo big) as it tends to jump around a lot between the main text and numerous ‘boxes’ and sidebars. There aren’t that many problems to work out either. Weinberg’s Gravitation and Cosmology is too advanced (and doesn’t have any problems to work through). The books by d’Inverno and Schutz that I started to work through a couple of years ago are also rather difficult to follow.
The book I’ve chosen in the one by Hobson et al referenced above. It received good reviews (well, except for the price) on Amazon and having read the first couple of chapters, it does seem to explain things clearly for the most part and strikes a good balance between a proper mathematical treatment and physical intuition. So I’ll go with it for now and see how we do.
To begin, we’ll review the concept of a manifold. What I wrote in the earlier post is still a good summary of manifolds and the curves and surfaces that can be embedded in them, so I’ll just add a few comments here.
A manifold is actually a more general mathematical object than my previous post may have implied. A completely general manifold is any set of points that can be parameterized. By ‘parameterized’, I mean that it is possible to define a finite set of parameters whose values uniquely determine each of the points in the manifold. Presumably we could pick, say, 10 points in 3-d space at random and parameterize them by assigning the values 1 through 10, in some order, to the points. If we call the parameter then the value of specifies which point we’re referring to. Since only one parameter is needed, the manifold is one-dimensional. The parameter(s) used to locate the points in the manifold are known as coordinates.
Most manifolds in physics are continuous and differentiable. These terms mean much the same thing as when they are applied to mathematical functions. A manifold is continuous if, for every point if by varying the coordinates that specify by an infinitesimal amount, we find other points that are infinitesimally distant from . Thus our “10 random points” manifold above is not continuous. A straight line is a continuous manifold since the location on the line can be specified by one parameter, and if we vary this parameter slightly starting at a point , we move slightly away from to one side or the other.
A differentiable manifold is one where it is possible to define a scalar field (that is, a scalar function of the points in the manifold) at each point that can be differentiated with respect to each of the coordinates.
The number of coordinates required to locate a point within the manifold is the dimension of the manifold. We can define a submanifold (or surface) with dimensions by specifying constraints of the form
for . If then there is only one constraint and the submanifold so defined is known as a hypersurface. For example, a sphere is a hypersurface within 3-d space, a line is a hypersurface within a plane, and so on. [See the earlier post for a few examples.]
The choice of coordinates for a given manifold is completely arbitrary. As an example, suppose we’re working in 3-d Euclidean space (denoted ). We can use rectangular coordinates or spherical coordinates . The standard transformation equations between the two systems are
To transform differentials we can use the chain rule
In matrix form, we have the Jacobian matrix
Hobson calls this the transformation matrix but doesn’t make clear what it transforms. It does not transform coordinates; for that we must use the equations 2. To see what does transform, suppose we have a curve in 3-d space defined in cartesian coordinates by
where is a parameter which traces out the curve as it varies. The tangent vector to the curve is defined as the vector joining two adjacent points on the curve, in the limit as the separation tends to zero, or:
where a dot denotes a derivative with respect to . From 6 we can divide both sides by to get
That is, provided the Jacobian matrix is invertible (non-singular), we can convert the tangent vector back and forth between the two coordinate systems. Thus the Jacobian matrix transforms derivatives of the coordinates, not the coordinates themselves.
From linear algebra, a square matrix is invertible if its determinant is non-zero, so these coordinate transformations will work at all points where . This determinant is known as the Jacobian determinant or sometimes just the Jacobian. [In a previous post, we looked at the role of the Jacobian in transforming volume elements in integration.] The determinant is most easily calculated by expanding about the bottom row in 7:
This is the familiar factor multiplied into the spherical coordinate volume element.
For 2 general coordinate systems and the Jacobian matrix is
where . The inverse transformation swaps primed and unprimed coordinates to give
The product (using the summation convention and the chain rule to condense line 1 to line 2) is
where the last line follows because and are independent coordinates if so the derivative of one with respect to the other is zero. Therefore is the identity matrix and , so . For our example the Jacobian of the inverse transformation is therefore
The transformation is invertible except when or . When , and are indeterminate (they can be anything) and similarly, when or , we’re at the poles and is indeterminate. However, there’s nothing singular about the actual geometry of the sphere at these points; the problem is purely in our choice of coordinates.
Example As a specific example of a coordinate transformation using the Jacobian, we’ll look at the 2-d transformation between rectangular and polar coordinates. Suppose we have a curve defined parametrically in polar coordinates by
Then the tangent to the curve is given by
The curve is a spiral, as shown in blue in the plot, where the tangent at (where and ) is shown in red:
The tangent vector in polar coordinates is
since at points to the left and points downwards.
Now we can convert this tangent vector to rectangular coordinates by using the Jacobian matrix. The transformation between polar and rectangular coordinates is
so we have
The transformation of the tangent vector is
At , and so the tangent vector in rectangular coordinates is