Talk:Pseudotensor

message: This entry needs some good examples.
Done --Michael C Price 11:34, 10 June 2006 (UTC)
replies: text: Yes!  The terminology is muddled in the literature, but pseudotensor is often used in gtr to mean a tensorial like object which is not generally covariant, but which only makes sense in certain coordinate systems, such as pseudo-cartesian systems. ---CH 20:19, 24 December 2005 (UTC)
replies: text:  The terminology is more than muddled, it's complete rubbish.  The article is talking about tensor densities not pseudotensors.  I've changed it to reflect the distinction -- at some stage the two concepts should be given their own pages. --Michael C Price 11:34, 10 June 2006 (UTC)

text: I agree.  Please do it.  (I could, but too busy trying to control bad edits in other physics/math topics.)---CH 23:14, 13 June 2006 (UTC)
replies: text:  Already done.  It was a learning experience :-(, but worth it.--Michael C Price 02:56, 14 June 2006 (UTC)
thread_title: Tensor densities?

message: Math.  BTW, Michael, WP discussion of this kind of confusion has been previously noted but none of the experts has taken the time to clear it up, so feel free to have a go.  I think it is important to take advantage of existing math articles and to distinguish between math (tensor algebra/calculus) and applications to physics.---CH 23:13, 13 June 2006 (UTC)
replies: text: I spent some time looking over the maths and physics categories and the only firm conclusion I came to was that I wasn't going to be happy with which ever one I chose! Still, I think it better to produce half-finished and inappropriate stubs than none at all....--Michael C Price 02:56, 14 June 2006 (UTC)
thread_title: Math or physics?

message: I believe that the term pseudotensor means different things in different contexts: in particle physics they may well gain a sign flip under an inversion but I don't believe this is true in general relativity.  The article is currently inconsistent on this issue: I don't think the Landau-Lifshitz pseudotensor, for example, flips signs under an inversion. --Michael C. Price talk 08:12, 11 April 2007 (UTC)
replies: text: I don't see why the LL pseudotensor flips sign under a spatial inversion.  IOW readers badly need your explanation. --Michael C. Price talk 07:48, 13 April 2007 (UTC)

text: I'm guessing here -- as you say, direct calculation would be very difficult -- but isn't there a good chance that the determinant sign flip will cancel since det appears both above and below?  I realise that it may not be this simple, since one appears inside the bracket and is twice differentiated, whilst the other is outside. --Michael C. Price talk 12:04, 13 April 2007 (UTC)

text: You're correct, we're talking about two different determinants. My error. --Michael C. Price talk 20:06, 13 April 2007 (UTC)

text: Since the LL pseudotensor vanishes locally in any frame where the first derivatives of the metric vanish (which by the equivalence principle is always possible anywhere), doesn't it also follow that the sign flip can't encapsulate the definition of a pseudotensor?  Adding a sign flip to a pseudotensor's transformation would not allow it to vaninh anywhere in anyframe.--Michael C. Price talk 06:02, 18 April 2007 (UTC)

text: You may well be right about the change in usage of the term "pseudotensor" from the days of Landau et al.  I suspect that the new usage is more prevalent in QFT/particle physics than GR.  I have never heard of quasitensors, but the distinction you make is logical.--Michael C. Price talk 09:48, 18 April 2007 (UTC)
thread_title: Sign flip

message: I read something further on quasitensors. They seem to be related to dual quasialgebras and quasi Hopf-algebras, and yes, some SEM (quasi?)tensors, eg Bergmann-Thompson quasitensor is mentioned in this respect. Need some more time to dip in this. --Lantonov 08:17, 18 April 2007 (UTC)
thread_title:  Quasitensors 

message: I changed the term covariant to invariant. Invariant is a more general term which refers to both covariant and contravariant tranformations. Although often covariant is used as a synonym of invariant this use is informal (see Covariance and contravariance of vectors#Informal usage:invariance). Lantonov 10:06, 23 July 2007 (UTC)
thread_title:  Covariant vs Invariant 

message: This article contains the statement "On non-orientable manifolds, one cannot define a volume form globally due to the non-orientability, but one can define a volume element".  However, this is a complete swizz (i.e. it is a fallacy), because the volume form can only be defined in terms of a basis, and usually a coordinate basis is assumed. No continuous basis field (whether this is a coordinate basis or not) exists globally on a non-orientable manifold, and consequently on cannot define a volume form globally on a non-orientable manifold.  I am ware that a volume form such as the Levi-Civita symbol takes the same component form regardless of the choice of basis, but this does not change the fact that it is only defined where there is something to reference it to – either a basis, or a volume form, neither of which exist globally (in a continuous form) on a non-orientable manifold.  —Quondum 04:15, 22 May 2015 (UTC)
replies: text: I think that the article is trying to indicate that, e.g., a Möbius strip has a definable total area.  Furthermore, despite the lack of a volume form, one could define a mass density on a non-orientable manifold and then integrate it to get a total mass.  Whether the article is conveying these ideas properly is a separate question.  Leegrc (talk) 17:10, 22 May 2015 (UTC)
replies: text: Understood that defining an integrable quantity has value, but I'm not convinced that this construction is mathematically consistent: the definition of integration itself needs to be modified to ignore orientation of the volume element to achieve consistency. The problem becomes more obvious if you consider something like a charge density (i.e. something that can be positive and negative), and then try to integrate it globally.  A simpler and more natural construction is to integrate an n-form defined on the double cover of the manifold.  My concern is to avoid the claim that a volume element is defined globally while a volume form is not.  The examples should also be separated into the two types (as per the lead's two definitions).  This is a tricky area, however.  —Quondum 17:38, 22 May 2015 (UTC)
replies: text: I think that charge density is not any harder than mass; I can imagine a Möbius strip with positive and negative charges on it and ask what is its net charge.  I agree that the double-cover approach allows one to side-step this business of volume form vs. volume element, but I am still hopeful that there is a textbook way to accurately describe the volume element approach too. Leegrc (talk) 19:09, 22 May 2015 (UTC)
replies: text: Sorry, I was overly cryptic.  In a non-orientable Möbius strip universe (with a time dimension), switching the orientation by traveling around the loop changes the sign of the charge from the perspective of the flipped observer, so the net charge is inherently zero once you integrate twice round the strip.  This holds for any true density, with or without a time dimension.  I don't think there is any way around the conundrum, but would be happy to see a coherent account refuting this.  Or even a well-sources claim that a volume element can be integrated globally on a non-orientable manifold (in a way that does not always give zero). —Quondum 20:40, 22 May 2015 (UTC)
thread_title:  Invalid claim about volume element? 

message: In fact, they are quite different. 
*  a pseudo-tensor gains an additional sign flip under some passive transform (e.g. a basis inversion), 
* a pseudo-vector gains an additional sign flip under some active transform (e.g.  a space inversion).
Therefore, a pseudo-scalar (e.g. a scalar product of an axial vector by a polar vector) is not a pseudo-tensor as well.
If I am right, the sentence “This is a generalization of a pseudovector.” must be deleted.
KharanteDeux (talk) 09:08, 11 May 2021 (UTC)
thread_title:  A pseudo-vector (aka axial vector) is not a pseudo-tensor