Physics Asked on January 2, 2021
In Edmund Bertschinger’s lecture notes Introduction to Tensor Calculus for General Relativity, following equation (14) he tells us:
The component $P_mu$ of the one-form $overset{sim}{P}$ is often called the covariant component to distinguish it from the contravariant component $P^mu$ of the vector $vec{P}$ . In fact, because we have consistently treated vectors and one-forms as distinct, we should not think of these as being distinct ”components” of the same entity at all.
I really don’t understand some of the distinctions Dr. Bertschinger is making. The above statement, in particular seems contrary to how I have learned tensor analysis (from many sources, including books in his bibliography). Am I correct in understanding that there is no mathematical distinction between vectors and one-forms on a (pseudo-)Riemannian manifold endowed with a metric? That is, Bertschinger is suggesting certain geometric objects are natively vectors, while others are natively one-forms, but each can be easily converted to the other.
Edit to add protest:
To paraphrase Evar Nering’s Linear Algebra and Matrix Theory: A homomorphism or isomorphism defined uniquely by intrinsic properties, independent of the choice of basis, is said to be natural or canonical.
Am I correct in understanding that there is no mathematical distinction between vectors and one-forms on a (pseudo-)Riemannian manifold endowed with a metric?
No. Vectors and one-forms are distinct mathematical objects with distinct transformation behavior under changes of coordinate chart. There are also subtleties when defining pushforwards and pullbacks which distinguish the two; for a generic smooth map $Phi:Mrightarrow N$, one can push vectors forward from $M$ to $N$ and pull one-forms back from $N$ to $M$, but the reverse operations are not always well-defined. The fact that the space of vectors and the space of one-forms are isomorphic does not mean that the elements of those spaces are the same.
For a generic smooth manifold, there is no canonical isomorphism that naturally maps the space of vectors to the space of one-forms. Such isomorphisms exist, but there's no particular reason why we need to choose one over the other - any nondegenerate bilinear form is sufficient to do the job.
A metric is such a form, so on metric manifolds we typically choose the metric itself to define the partnership between a vector and its one-form dual. Even in GR, though, this is not always the case; when working with linearized gravity, it is conventional to use the Minkowski background metric, rather than the full metric, to map vectors to their one-form partners. On symplectic manifolds, where there is generally no metric at all, one uses the native symplectic form.
Vectors and one-forms are intimately related, insofar as they can be put into one-to-one correspondence. However, the modern perspective treats vectors and covectors as distinct objects, and to view them that way is conceptually far cleaner than the "different components of the same object" idea that is a bit more old school.
Answered by J. Murray on January 2, 2021
There are (at least) two occasions when it is useful to distinguish contravariant and covariant tensors from each other.
Answered by Bence Racskó on January 2, 2021
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