Which basis-forms are pseudo-tensors?

SiennaTheGr8 · Jul 31, 2024

My understanding is that the Hodge dual of a pseudo-form is always a "true" pseudo-form, and vice versa. However, I'm a little confused about how this applies to basis-forms in general.

I believe I understand how it works for the ##0##-form case: the basis ##0##-form is the scalar ##1## (obviously a true tensor), and its Hodge dual is the basis ##n##-form (for an ##n##-dimensional manifold), which then must be a pseudo-tensor (equal to the volume form in an orthonormal system, if I'm not mistaken).

But what about for other situations? For example, here is an excerpt from Frankel's The Geometry of Physics (3rd edition, p. 363), where the context is ##4##-dimensional Minkowski spacetime:

##*## takes ##p##-forms into pseudo (##4 - p##)-forms. [...] Since the coordinates are orthonormal and ##\sqrt{ | g | } = 1##, we can probably avoid the use of (14.3) [Frankel's general definition for the Hodge dual]. ##*(dx^2 \wedge dx^3)## has the property that ##(dx^2 \wedge dx^3) \wedge *(dx^2 \wedge dx^3) = \| dx^2 \wedge dx^3 \|^2 \, dt \wedge dx^1 \wedge dx^2 \wedge dx^3##. Since the ##dx^\alpha## are orthonormal and ##\| dx^\alpha \|^2 \, = +1## for ##\alpha = 1, 2, 3##, we see that ##\| dx^2 \wedge dx^3 \|^2 \, = \| dx^2 \|^2 \, \| dx^3 \|^2 \, = +1##, and so ##*(dx^2 \wedge dx^3) = dt \wedge dx^1##.

What confuses me is: if ##*(dx^2 \wedge dx^3) = dt \wedge dx^1##, then either ##dx^2 \wedge dx^3## or its Hodge dual ##dt \wedge dx^1##—both of which are basis ##2##-forms—must be a pseudo-form, and the other must be a true tensor. How would one know which is which? And this is of course just an example; my question is more general. I feel that something simple is going right over my head!

SiennaTheGr8 · Jul 31, 2024

I should add that in the case of Minkowski spacetime, I'd guess that the basis ##2##-forms involving ##dt## are all of the same "type" (either the pseudo-tensors or the true tensors). But again, my question is more general: how to tell in the general case which basis forms are pseudo-tensors and which are true tensors.

SiennaTheGr8 · Aug 4, 2024

I'm wondering if ultimately it boils down to the chosen "canonical" ordering of the basis ##n##-form (for an ##n##-dimensional space).

For example, in ##4##-dimensional Euclidean space (so we don't get side-tracked by the "different" nature of time), half of the basis ##2##-forms must be true tensors, and the other half must be pseudo-tensors (because the Hodge dual takes ##2##-forms to ##2##-forms). Say we have ##*(dx \wedge dy) = dz \wedge dw##. One of these basis ##2##-forms must be a true tensor and the other a pseudo-tensor, and it seems to me that it can only be a matter of convention which is which. This must be connected with how one defines the basis ##4##-form, right? If, for instance, the basis ##4##-form is defined like ##dx \wedge dy \wedge dz \wedge dw##, and if in that case ##dx \wedge dy## is a true tensor so that ##*(dx \wedge dy) = dz \wedge dw## is a pseudo-tensor, then my suspicion is that changing the definition of the basis ##4##-form to ##dx \wedge dz \wedge dy \wedge dw## would render ##dx \wedge dy## a pseudo-tensor and ##dz \wedge dw## a true tensor. Or something like that.

Does anyone know whether I'm on the right track?

robphy · Aug 13, 2024

I don't have the full story... I think the following points are relevant.

With pseudo-forms (odd-forms, twisted-forms), there is an orientation in its definition (see p.86),
that is not captured by the notation you are using [not using the general definition used by Frankel].

Definition: A pseudo-p-form ##\alpha## on a vector space ##E## assigns,
for each orientation ##o## of ##E##,
an exterior p-form ##\alpha_0## such that if the orientation is reversed the exterior form is replaced by its negative
\alpha_{-o}=-\alpha_o

Have a look at Burke's unfinished document
https://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf
See

18 Twisted Differential Forms, especially the notations at the top of 18.2
and the "##\widehat{dx} dy## hat notation for missing basis forms" at the bottom of 18.2, spilling into 18.3
27. The Star Operator, especially the notation in 27.2

Do a Google-Books search for
"twisted differential forms" "28.10" burke
https://www.google.com/search?q="twisted+differential+forms"+"28.10"+burke&tbm=bks
and notice the orientation that is included geometrical figure.

and
"twisted differential forms" "hodge star" burke

The figures with orientations are likely based on figures in

On ordinary quantities and W-quantities. Classification and geometrical applications
J. A. Schouten; D. Van Dantzig
Compositio Mathematica (1940)
Volume: 7, page 447-473
ISSN: 0010-437X
https://eudml.org/doc/88752
http://archive.numdam.org/article/CM_1940__7__447_0.pdf