I have been attempting to find an answer to this for a few weeks, and decided to finally ask.
I’ll ask the questions at the beginning and then give the necessary background below:
(1) Are volume forms dx1∧…∧dxn n-forms and tensors?
(2) How do I integrate on Lorentzian manifolds?
(3) Is some pull-back on a manifold related to the metric of that manifold?
I’m a novice to general relativity, and I am teaching myself through Sean Carroll’s An Introduction to General Relativity: Spacetime and Geometry (see here for essentially a pre-print version). Simultaneously, I am looking at Jon Pierre Fortney’s A Visual Introduction to Differential Forms and Calculus on Manifolds in order to try to understand the mathematical infrastructure of GR in a somewhat visual way.
I am very much confused by how the two authors handle integration on manifolds. I understand that Fortney is trying to deal with somewhat more primitive analysis using Rn manifolds, while Carroll tries to deal more specifically with Lorentzian manifolds, but there is no indication in the text of either that the formulas derived for integration are not general.
Carroll (in 89-90 of his book; or page 53-54 of the linked pre-print) argues that volume forms dnx=dx0∧…∧dxn−1 are not tensors, i.e. they are not an n-forms. He then goes through to sort of prove that the proper, coordinate-invariant method of integration of a scalar function ϕ on a (Lorentzian??) manifold is given by:
where |g| is the determinate of the metric on the manifold.
Fortney (in chapter 3 and Appendix A) indicates that forms such as dxi∧dxj are n-forms and are a subset of all tensors on a manifold (specifically, the set of skew-symmetric (n,0) tensors). In Chapter 7, he derives the integration formula for integration under coordinate change θ:Rn→Rnas:
where [θ1,…,θn] are the transformed coordinates and T∗θ−1 is the pullback induced by θ on T∗.
I realize, in a very real sense, these expressions are addressing two different aspects of integration. In particular, Fortney is specifically interested in change in coordinates while Carroll is interested in a coordinate-invariant expression for integration. However, I can’t help but think these ideas should be related. In particular, it seems like the pull-back of some transformation is related to the metric on the manifold.
I’m particularly concerned by the discrepancy between Carrol and Fortney. Carroll argues volume forms are not tensors, while Fortney argues they are. Either one of them is wrong or I’m misunderstanding the objects they are talking about.
I’d say volume forms are tensors, though I’m a mathematician so I’m not sure what a tensor means:)
The two chunks of integration you are describing are related in the following way (and note that this is just an outline): In general, you have the differential forms of degree k (which you might call (k,1) tensor fields or something similar), which is a relatively abstract object, and may be defined in a coordinate-free way, and we also have their integrals. If your manifold is orientable, there are also volume forms, which are simply defined as nonzero differentiable form of top degree. For instance, the manifold Rn has the form dx1∧⋯∧dxn. (Their existance is equivalent to orientability). Given a volume form, its integral over your manifold is called the volume of the manifold. Try to convince yourself of this in the case where your manifold is some open bounded set in Rn
Next, consider a metric g on your orientable manifold. then one can construct a distinguished volume form using this metric. The idea is to use the metric to specify orthonormal bases, which have nice properties. For instance the form on Rn from above is precisely the form obtained from the standard metric on Rn.
Finally, in a chart of your manifold with coordinates x1,...,xn, you can calculate the volume form, and it turns out to be √|g|dx1∧⋯∧dxn.