Talk:Tensor density

3771:

conventions are clearly used by notable sources, you can proceed from the assumption that they should both be mentioned in the article. For internal consistency, however, the article does need to select one only of the prevailing conventions for its presentation of the details. A further principle is not to change the convention in use in an article without consensus. This essentially means that arguments about the best convention have little point other than to sway the consensus prior to any change. JRSpriggs and Sławomir have both expressed their preference, YohanN7 is essentially saying the same as I've said now, and as you will gather from what I've said above, I fail to grasp the utility of the concept of tensor densities, nor do I see a natural preference for either convention. Simply put, if you wish to change the article, you will need to persuade the other involved editors to agree to the changes that they are presently opposed to, and taking a combative stance is not going to help with that. Editing works best through mutual understanding and cooperation. As you settle in as a Knowledge editor, I expect you'll learn to enjoy being a part of a mutual, cooperative endeavour that we all find rewarding. Your attention to detail and energy for researching specifics are a plus here. You may find

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coordinate-dependent, and thus (quite unnecessarily) undermines the fundamental motivation of tensors: coordinate-independence. It leads to defining densities in the sense of this article (of weight +1?) as entities that have been scaled to allow integration (using what I would call a flawed form of integration, or more specifically, volume element) over a hypervolume region to yield an invariant quantity. This article describes tensor densities, but fails to explain why they have any value, or what motivates them. As far as I can tell, the motivation will relate to a specific form of integration, where it was presumably decided to retain a form that rescaled the units with a change of coordinates, and dumped the responsibility for compensating this on the quantity being integrated. I would encourage a knowledgeable editor to insert such a motivation into the article; surely it must be sourceable. —

4625:

unfortunately. On oriented manifolds, the bundle of densities in naturally isomorphic to the bundle of n-forms, where n is the dimension of the base manifold. In general, a non-oriented manifold M has an oriented double cover, N→M, together with an action of the group with two elements Z_2. The bundle of densities is naturally isomorphic the bundle of Z_2 equivariant maps from N to the bundle of n-forms on M. Such a thing has an invariant integral because it can be pulled back to an n-form on the oriented manifold N, integrated there, and then divided by two. Notice that everything in this construction is tensorial: there are no absolute values appearing. But the densities are tensors

195: 5072:(arbitrary) choice to coordinatize the manifold, and then vary the metric on the manifold without "changing the shape" of the manifold under the chosen coordinatization. This allows one to vary and determine the components of the metric in terms of the chosen "fixed" coordinatization at each point. Other (non-"fixed") coordinatizations would be possible. So in this case, it has a particular calculational convenience based on a simplifying assumption on the choice of coordinates. I this picture it helps to group the "variable" expressions together. I still don't see that this motivates the concept of a standalone definition of a tensor density, though. — 2910: 2214: 1807: 2273:

manifold that is integrable. Examples are charge densities, mass densities, population densities, (even "volume densities", although if we think of this, we are likely to confuse ourselves). The main point is that a density is something that can be integrated over small pieces of the manifold (in a coordinate-independent manner), producing a single number (scalar). The Jacobian change of variables formula dictates what the transformation law for a density must be under a change in coordinates. It must vary by the inverse of the Jacobian of the coordinate change. That is,

5868:-dimensional submanifold. It is therefore nothing to do with a weight, in the sense used here. You would want to be able to integrate the density of mass (a 3-density) in 3-space, for example, or a flux density (a 2-density) over a 2-surface in 3-space (for example, electric displacement field). Something that can be integrated along a curve (e.g. along a world line to yield proper time) is a 1-density. A scalar, by analogy, is a 0-density, and it can only be integrated on a 0-d manifold (i.e. summed over discrete points of the manifold). These 410:

to compensate for that and make the integral invariant, we must multiply the scalar density by 1/16. Under this doubling of coordinates, an arrow (contravariant vector) would have its components double. So the metric tensor (a covariant matrix) would have its components multiplied by 1/4. The determinant of the metric would be multiplied by 1/256. So the square-root of its absolute value would be multiplied by 1/16. This is just the factor needed to change an ordinary scalar into a scalar density. So that square-root should have weight +1.

84: 2515: 1819: 1412: 74: 53: 4706:@Q Lee's density measures volumes spanned by n vectors in the tangent space at each point of the manifold. That's, for instance, what every n-form does, some of them with a sign, this one is unsigned. These things can be integrated and that's the point. (The same is true for volume forms on orientable manifolds.) This article addresses what happens when the coordinates are changed. Edit: What I'm trying to say is that there aren't two meanings of density in Lee's density. 4658:) to a publicly viewable page in his book. It should be clear that he is defining it in a coordinate-independent fashion. In a nutshell, Lee's density is defined to scale its result according to the determinant of a transformation applied to its input vectors, but this article's density is defined to scale its result according to the Jacobian determinant of the transformation of the basis (potentially in addition to the scaling as per Lee's density, if it is an 5256:-form is far simpler theoretically, and does not need new rules to deal with the construction of densities. Use of the Hodge dual in the Maxwell's equations is natural; one needs both the quantity and its Hodge dual at various places. Any "density" other than the Hodge dual equivalent I'm guessing does not have a natural use anywhere. The article also makes no reference to the Hodge dual, but should since it is the natural interpretation for your example. — 5531: 2905:{\displaystyle {\mathfrak {T}}_{\nu '_{1}\cdots \nu '_{m}}^{\mu '_{1}\cdots \mu '_{n}}=\left|{\frac {\partial x'}{\partial x}}\right|^{w}{\frac {\partial x^{\mu '_{1}}}{\partial x^{\mu _{1}}}}\cdots {\frac {\partial x^{\mu '_{n}}}{\partial x^{\mu _{n}}}}{\frac {\partial x^{\nu _{1}}}{\partial x^{\nu '_{1}}}}\cdots {\frac {\partial x^{\nu _{m}}}{\partial x^{\nu '_{m}}}}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}} 2209:{\displaystyle {\mathfrak {T}}_{\nu '_{1}\cdots \nu '_{m}}^{\mu '_{1}\cdots \mu '_{n}}=\left|{\frac {\partial x}{\partial x'}}\right|^{w}{\frac {\partial x^{\mu '_{1}}}{\partial x^{\mu _{1}}}}\cdots {\frac {\partial x^{\mu '_{n}}}{\partial x^{\mu _{n}}}}{\frac {\partial x^{\nu _{1}}}{\partial x^{\nu '_{1}}}}\cdots {\frac {\partial x^{\nu _{m}}}{\partial x^{\nu '_{m}}}}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}} 1802:{\displaystyle {\mathfrak {T}}_{\nu '_{1}\cdots \nu '_{m}}^{\mu '_{1}\cdots \mu '_{n}}=\left|{\frac {\partial x'}{\partial x}}\right|^{w}{\frac {\partial x^{\mu '_{1}}}{\partial x^{\mu _{1}}}}\cdots {\frac {\partial x^{\mu '_{n}}}{\partial x^{\mu _{n}}}}{\frac {\partial x^{\nu _{1}}}{\partial x^{\nu '_{1}}}}\cdots {\frac {\partial x^{\nu _{m}}}{\partial x^{\nu '_{m}}}}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}} 185: 158: 3239:

reflects what it looks like in (good) textbooks as well. Alternatives are mentioned. But don't misunderstand me as proposing that every fringe convention should be mentioned. Is Weinberg's authority enough to make his choice of convention get mention in this case? Don't know. I haven't read his work on GR, just his QFT vol 1,2,3. (Yes, these were totally unconventional - and great.) Just trying to reason in general terms here.

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symbol is non-tensorial but can be defined without a volume form or metric, whereas the volume form is tensorial but is best defined in terms of the metric. The definition of a density weighting is the function giving the proportionality between them, which is consequently only defined when both a basis is specified and there is a volume form. This severely limits the definition of densities in metric-independent contexts. —

4598:-form. So yes, I'm saying that "that kind of density" (Lee's) is entirely different from "this kind of density" (this article's), but that only "that kind" can naturally be called a "density", that "this kind" should have a different name, and that there is thus no natural selection between the two weight conventions, and indeed no need for "this kind" at all (a statement for which I'll probably be taken to task). — 3203:

think is fairly limited as far as it goes). Maybe Weinberg did have good reasons, but Weinberg is somewhat unconventional in his entire approach to GR, and he often uses the opposite conventions of others working in that area (cf. Penrose, Hawking, for example). I have no doubt that he had his reasons for choosing these conventions, but so far no one has been able to give a compelling account of those reasons.

22: 6264:-form has all of the dx's in it, so it's actually coordinate-invariant. That is what (in physics) distinguishes a scalar from a non-scalar. It's not a question of whether it is a "number" or "non-number" so much, which I think is how most mathematicians read the word "scalar". In particular, contrary to Quondum's interpretation, this does not require any kind of duality to be on hand. 3952:, it seems to me that the concept is best understood in the absence of a metric tensor, and the crucial feature appears to be that a density is a quantity that can be integrated (intrinsically) over a region of the manifold. I deduce that this is a coordinate-independent concept, which puts it at odds with this article. If one defines a volume form in terms of the coordinates as say 5062:, even here the use of tensor densities vanishes. I'm going to (not for the first time) claim that in a properly formulated basis-independent treatment where this simple point is kept in mind, the entire tensor density treatment is superfluous. Okay, lotsa people are going to turn in their graves and it will offend many others, but please prove me wrong with a counterexample. — 5362: 354:, where what is meant is "the determinant of the metric tensor coefficient matrix ". By way of establishing the context in which a reader is to interpret this article, it should at least point out at the start that it is using the formalism in which the matrix of coefficients representing a tensor with respect a basis, and that here this matrix is referred to as 6441: 3924: 3881: 3116: 4253: 3356:

Essentially, since the word "density" is used, it will trigger certain natural associations as its selection was presumably intended to do. Thus, I would expect the components a tensor density of weight +1 to scale exactly as the components of something that I would call a density, and hopefully this would suggest a convention.

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If a tensor density is used in the place of the Hodge dual, what is really happening is that the dual was found using the Levi-Civita symbol instead of the volume form of the manifold. The volume form depends upon the metric, but the Levi-Civita symbol depends upon a choice of basis. The Levi-Civita

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Your presumptions are somewhat naive. The crux here is that in GR the metric is a dynamical quantity (i.e. you are dealing with a family of mathematically distinct (pseudo)-Riemannian. In order to make sense of the variational principle it is necessary to introduce an external frame of reference that

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Correct me if I'm wrong, but there is little conceptual difference and no substantial difference at all between the two conventions. The conceptual difference is that "ordinary" densities get weight +1 with our convention, and -1 with the other. Both are equally invariantly integrable since while our

3325:

Consider the sign convention for the metric (-,+,+,+) vs (+,-,-,-). I personaly use the mostly plus convention, but as long as people are consistant they are both right. By saying that "Often it is said that definitions and conventions are arbitrary and that it does not matter which version one uses.

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I think it is misleading to think of the volume form as a density in this context, since the "volume form" might refer to a scalar or a density, depending on what "volume form" means. If "volume form" means the wedge product of the differentials of the coordinate functions, then it is not coordinate

1357:

It is useful to realize that there are different conventions for the definition of weight of a tensor density that differ by a sign. Carroll (like for example Weinberg) use a convention in which the Levi-Civita symbol with lower indices has weight +1. Consequently, the square root of g has weight -1

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ii) The reference I used were from a textbook called 'Spacetime and Geometry: An Introducton to General Relativity' by Sean Carroll which is a graduate level textbook and widely accepted in physics today. It so happens that Carroll made lecture notes as well (which I referenced). However his 'lecture

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instead. How does this scalar density transform? For simplicity, let us consider a change of coordinate system in which each of the coordinates is doubled. Then the hyper-volume of the region over which one is integrating would seem to increase by a factor of 16 (if one ignores the metric tensor). So

366:. A coordinate-based approach still makes sense with a nonholonomic basis, and thus tensor densities can defined in the same way in terms of the determinant of the transformation matrix, but the transformation matrix cannot be expressed as the matrix of partial derivatives as assumed in this article. 341:

of the space of basis vectors, thus formally capturing the transformation requirement. In this context (whether formally or in the less formal physicists' approach), tensor densities are easily defined. In a coordinate-free approach to tensors, or even with wording that is agnostic to which approach

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This article chooses to define a volume element that depends on the choice of coordinatization with a holonomic basis, and is much like the Hodge dual. It non-tensorial, and unless an invariant structural volume element (what we've been calling the volume form) is also defined, other weightings are

5526:{\displaystyle {\begin{aligned}&\Omega =fdx^{1}\land dx^{2}\land \cdots \land dx^{n}&{\text{is a tensor (density of weight 0)}}\\&f&{\text{is a scalar density with weight 1}}\\&dx^{1}\land dx^{2}\land \cdots \land dx^{n}&{\text{is a tensor density of weight -1}}\end{aligned}}} 4830:. The whole, complex treatment of weights appears to be an artefact of failing to identify an invariant volume element at the start. (I've said this before above.) Furthermore, the nontensorial even/odd/pseudo sign shebang can be abolished by using a double cover of the manifold, as per Sławomir. — 3370:

Maybe I am interpreting JRSpriggs incorrectly, but it sounds like he is trying to state that there is the one and only convention. (Whatever that convention is) and all other conventions and by extention people who use these "wrong" conventions are stupid/wrong... (In analogy it sounds like religion

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I agree with Yohan completely. As an aside: I have been thinking and (I will probably be criticised severely for this), I think (personally) the name 'Tensor density' is a misnomer. The reason being is, when one hears the word 'density' or the words '____ density', our minds automatically think that

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In the end, any vocabulary is a matter of convention. But not all vocabularies are equally useful. In this case, calling something a "density" should agree with the premathematical use of the term, in my opinion. The convention should be chosen so that mass, charge, population, etc, are densities.

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Although there are several things I dislike about his edit, the main problem is that he is getting the sign of the weight of the determinant of the metric tensor wrong. Essentially he has reversed the definition of weight given in the first section of the article (without changing it there). I think

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are three distinct things with distinct names and distinct articles even if many conflate them more or less on purpose because the conflation has become second nature with time, like has been done here in this thread. Just as long as the articles are kept straight, there shouldn't be a problem. But

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Right. There are three ways of thinking of a skew (0,n) form. One is as a tensor (weight zero), one is as a scalar (a differential n-form of weight zero), and one is as a density (a scalar with weight 1, in our convention). All of these perspectives are connected by a "natural isomorphism" (that

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on the part of other editors. I've given an interpretation that differs from yours and is compatible with none of these authors, conventions or you being slighted. In any event, Knowledge does not go by what is "right" or "true", but rather by what is published by notable sources. Given that both

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I don't mean to say that Weinberg is mathematically wrong in his choice. I have been using "wrong" in quotes, because it is after all only my opinion. But no one so far seems to have produced a clear motivation for the opposite convention, aside from just appeal to authority (an authority which I

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because the convention used there seemed to me to be wrong (or at least inconsistent with most conventions). Let me explain why JRSpriggs is "right" and other conventions are "wrong". I will just discuss scalar densities. (Here by "density" I just mean "1-density"). A density is a quantity on a

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FYI, Personally I will keep believing what I believe but you probably don't care about that. I will continue physics and continue with my convention and if I fail university and life and become a hobo on the streets because of it, then SO BE IT, (You can come by and laugh at my stuipidity then). By

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This is what I think too, but the third item is shaky. It relies on the split I made (the only LaTex ITT) above of a 4-form into factors. It is rather inelegant and I doubt that a mathematician would call it "natural". They have bundles and smooth maps in mind, not a chop-up in coordinates. I have

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There are three ways of thinking of a skew (0,n) form. One is as a tensor (weight zero), one is as a scalar (a differential n-form of weight zero), and one is as a density (a scalar with weight 1, in our convention). All of these perspectives are connected by a "natural isomorphism" (that is, an

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The Hodge dual is not defined on a manifold without a volume form, which one can derive from the metric tensor and an orientation. You don't need the Hodge dual in circumstances in which it is not defined ;-) It only arose due to the mention of Maxwell's equations, where the Levi-Civita symbol is

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Hm. I think I understand parts of what you mean, but not all. I don't see what the Hodge dual has to do with this, I'll take it as homework. I though the point with tensor densities is that we know what they are already and that those with density 1 can be used with the "usual" volume element (not

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pushes one into the concept of densities as defined in this article. I would personally work with the true Hodge dual, not the scaled version (being a purist), but from a practical perspective, simply rearranging the coordinates into a vector (or scalar etc.) form with a mental tag that these are

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Now all these physicists understand and know GR and (more to the point) come to the same conclusions. I highly doubt anyone would say that any of these people are wrong simply because they use a convention that someone dislikes. Yet by JRSpriggs argument he is basically saying that people like me,

3321:

Sorry but I disagree with you JRSpriggs: 1) Conventions ARE arbitrary and just because person A uses a different convention from person B doesn't make it wrong. As long as they are consistent with their conventions the physics that the two are describing are COMPLETELY EQUIVALENT. The math will be

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Tensor densities (of weight +1) are called "densities" precisely because they are used as the mathematical representation of things which are called densities in ordinary life (classical physics). Sławomir Biały is right about that. Action (actually the Lagrangian density, as anonymous pointed out

939:

It looks like things are going to work out just fine. I'm not an expert on the subject, so I'll leave it to you guys who are to sort it out. But I have a hunch that it is just a matter of definition or convention. If this is the case, then both conventions can be mentioned in the article, with the

4679:

Thanks for the link. The usual bare word "density" refers to an even 1-density (in the terminology of this article). That is, one for which the Jacobian appears in the absolute value. This is probably worth mentioning. Also, I wonder whether the distinction between even and odd densities is a

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At the risk of making myself more unpopular than I already am, I'll (hesitantly) say that there seems to be very little real substance in what has been said (with close to religious conviction) in this thread. What it seemingly boils down to is where to put minus-signs and inverses. I feel little

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Often it is said that definitions and conventions are arbitrary and that it does not matter which version one uses. This is wrong. It is important to make them as intuitive as possible, especially for people just learning the subject. Just imagine if when you swept your finger to the right over a

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choice. At that point a motivation could well be given if there (like in our case) is a compelling reason for making the choice. Then, in the interest of completeness, list some other common choices (naturally without motivation unless there is one), and then get on with the article. This pattern

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is not the same as a scalar density if I get it right. The "1" refers to the weight. Such tensor densities can be integrated invariantly using the "usual" volume element because it has weight -1, making the whole package invariant. I can only assume that the "n-forms are 1-densities" is standard

5176:

A 4-form IS a scalar density. It is just a different way of talking about the same thing. Personally, I prefer to talk about tensor densities than about differential forms. It makes the transformation properties clearer. And while you think that my method is unnecessarily coordinate-dependent, I

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I don't think it is clear cut in our discussion what is required for an invariant integral to exists. It is hinted that objects with weight +1 are invariantly integrable, no other. I'd say that it depends on what pre-exists in the integrand. If there is an invariant volume element in place, then

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the other convention. I can't promise it wont be reverted, but you might as well try. But don't get the idea that there should be a parallel development of the two conventions (I'd be quick to revert) or anything like it. I'd try to describe it in words in one sentence with one reference (hint:

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Perhaps we should avoid the use of the terms "right" and "wrong" in this discussion. And I interpret JRSpriggs differently – he seems to be saying that we do ourselves a disservice if we choose to completely sweep the benefits of selecting certain conventions under the rug, with which I agree.

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below) is a scalar density. The electric charge density and current density form a vector density. It could be represented either as a contravariant vector density with 4 components or equivalently as an anti-symmetric three-times-covariant ordinary tensor with 64 (mostly redundant) components.

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I find the double cover of a manifold idea a rather beautiful way of dealing with the integrability orientation issue – I was toying with it but was unsure; I'm glad you mentioned it. In any event, Lee's use of densities appears to be a way of dealing with the orientability problem. Here is a

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The premathematical (in the sense of me not properly understanding the detail) use would lead me to expect that a scalar density (which I'd assume would be said to have weight +1) would be one that when integrated over a region (using the volume element), yields a true scalar. Charge and mass

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So I again apoligize. Truthfully, looking back, I am appalled at how stupid and mentally challenged I am. I really should be in a mental institution come to think of it. I know I graduated with honors but, honestly, I am SHOCKED that I passed elementry school with my mentally challanged brain.

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a tensor, but I assume that this is because of the nonlinearity of the absolute value. There are (IMO) cleaner ways to deal with non-orientability than via the absolute value. The sign of the Jacobian is not of concern to the weight mentioned above, only its scaling. If one uses the Jacobian

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This is where I fall off the wagon, though, since it seems to me that scalar densities (in the everyday sense) are invariant, and all tensors transform with weight 0. Thus, the concept of a "tensor density" as described in this article is an artefact of retaining a form of integration that is

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Well, actually, I haven't addressed your comment and appear to be stating simply the contrary (I actually do not know how to deal with the variational principle), so I'll try to rephrase what I suspect underlies your comment. Presumably, for your statement to be true, one must have made the

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On the utility of tensor densities. The usefulness of tensor densities becomes apparent when you apply the variational principle to a gravitational action. The invariant (weight 0) volume form is not invariant under variations of the metric, the coordinate 4-form is. Consequently, it is the

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6445:) (I know, British English...) applies to true volume forms as well, apart from the absolute value. But I agree, the lead can be misinterpreted as saying that the integration isn't invariant. It is. The point is that, just like volume forms, there are two parts, one of which behaves like a 4624:

If a density in Lee's sense is something that can be integrated, it has weight +1 in our convention (reams of text explain this above). I only have the old electronic version of his book on me. I'm away from my dead tree copies at the moment. He doesn't discuss densities in that version

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Ironically, there is one possible convention that seems to me to be better than the present choice of ours. That would be to use the other convention, but with an additional flip in the sign of the weight. That way Jacobian determinants of the inverse transformation are kept out of sight,

2957:. This is NOT WRONG. It is just a convention. So by saying (and I quote) "Your teacher was mistaken, if he equated what Carroll called the "naive volume element" with a density of weight +1. ...That is, your basic assumption is wrong." your simply telling everyone that you are arrogant! 5120:

which have zero or two (or more) metric tensors, this is not possible. In those cases, tensor densities are essential because Lagrangian densities must be scalar densities of weight +1 and there is no unique metric which one can use to convert them to the "equivalent" ordinary scalar.

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as our volume element, the whole point of multiplying by the Jacobian determinant elsewhere disappears. The volume element becomes tensorial, and we need only tensors, never "weighted densities" in the sense of this article. In short, we never need to work with anything not of weight

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arbitrary. In the latter case there is a natural choice. The other choice isn't wrong because it will produce identical results (apart from a sign switch for weights). But it goes counter the intuitive when it comes to the word "density". The convention doesn't match the terminology.

5836:-density" and "volume element" to mean something incompatible with this usage. His is the abstract picture: that of differential geometry. I contend that Sławomir has used the term "density" in both (incompatible) senses without highlighting this, though I'd have to hunt out where. — 5177:

think that your method is unnecessarily dependent on abstract nonsense (category theory) and does not allow you the flexibility to easily mix tensor indices with densities (i.e. differential forms). Also, Maxwell's equations are more naturally represented using densities for the ρ,

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that is a coordinate-independent "density of volume" (though one needs an initial frame for a density reference). Now I am guessing here, but this invariant "density of volume" is a 3-form, and thus an ordinary tensor; this is deserving of the name "density". This is, however,

5831:

I have little issue with what Sławomir says: it makes sense, in the third instance provided that one accepts the definition of this article for "density", and that one must interpret a "represented as" as having been elided. My problem is that Penrose uses the terms "density",

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you keep fixed while varying the metric. Of course, a coordinate frame is not the only way to do this (you could try it using non-coordinate frames). It is however it is by far the most intuitive because it allows you to use the ordinary Euler-Lagrange equations for fields.

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is a +1 tensor density and the wedge products of coordinate differentials is a -1 tensor density. The difference is that the density is endowed with an absolute value (making part of it an even tensor density, not an authentic tensor density) and the volume form not.

1362:. The definition currently appearing in the article uses the opposite convention, where the square root of g has weight +1. This is not a matter of right or wrong, but a matter of convention, and whatever convention is used in the article should be used consistently. 5674:

Above, JRSpriggs said "A 4-form IS a scalar density", and Sławomir has made a similar-sounding assertion. In a context such as this, such statements become very confusing. With suitable nomenclature, in my mind such a statement should be impossible. A 4-form is a

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form) right away. Needless to say, there seems to be little point in manufacturing a tensor density if a true tensor is available, except under unusual circumstances (e.g. variation of action). (The Levi-Civita symbol, b t w, is a tensor density, that much I know.)

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But Tensors are purely abstract and mathematical concepts which can be written on paper and were INVENTED to understand GR yes, but one can't see tensors, pick them up, or interact with them at all in any way. (Like numbers for example...you cannot pick up 2)...

5098:

In my naïvity, I cannot distinguish what you describe from what I was picturing. I can see that before a metric is determined, a coordinate-determined volume element has utility. But the utility of the whole theory of tensor densities is clearly eluding me.

5583:, why would one want to consider the other quantities? This perspective generalizes to pretty much every case. Also on terminology, let's distinguish between natural tensorial densities and the densities as defined in this article. I propose calling them " 4893:

context other than as a calculation tool, some time ago in another thread somewhere. At first blush your (and JSpriggs's) claim seems to be a claim of a substantive (i.e. with mathematically significant) meaning to tensor densities. As I commented

4434:. That is perfectly adequate for a convention that the article doesn't make use of. Secondly, and more importantly, your attempt is a one page long continuation of your angered argumentation on this talk page. Did you honestly expect it to stay? 446:

It does not seem necessary or useful to apply page protection. I support JRSpriggs's perspective, and hope that the IP will choose to discuss here. Using informative edit summaries and the opening of this thread may be helpful in this direction.

511:

Hi, This is the unregistered user with the changing IP address. I only found the 'editing talk' section now and I do apologize for not writing here sooner. There are two main resons why (as you said) I 'reverse the sign of the weight'.

5964:

This perspective assumes that there is a natural duality, which is the case if there is a volume form as part of the defined structure of the manifold (what Penrose calls the volume element). I think of the implied equivalence between

5845:

If you understand, then you can perhaps explain to me what he means? First problem: A n-form is a tensor (weight zero according to this article). This makes it impossible for it to have weight 1 as a density according to this article.

3268:

For example mass density can be integrated to give a mass of a object that we can pick up...charge density can be integrated to give charge of an object which we can see (some of it) for example electric charge and electricity, etc.

3329:

Math IS arbitrary, numbers are arbitrary...Just look at complex numbers for example. You won't see a '2+i' walking down the street and smiling at you now would you? But we use numbers to describe the world, so how can it be wrong?

1217: 5660:-densities you mean tensor densities? why not simply tensors and tensor densities (occasionally scalar densities)? But then we really need to know what JRSpriggs means by 4-form = scalar density or misunderstandings will result. 402:

that changing that definition would be wrong as it would not be the usual definition and would conflict with the usage of the word "density" in ordinary speech. A weight of +1 should be used for "density" in the ordinary sense.

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is to be used, tensor densities don't work. For example, the concept of a determinant of a tensor in the coordinate-free formulation only makes sense in the case of a linear transformation (necessarily a type-(1,1) tensor; see

4732:

has weight -1. When multiplied with something with weight +1, these weights add and the net result is a true scalar with weight 0 that is immune to coordinate changes. Also, Lee's density appears to be just an even 1-density.

3850:

convention assigns the weight -1 to the "coordinate volume element" (wedge of coordinate differentials), the other assigns +1; hence the net result is a true scalar (vector, tensor) when multiplied together with the density.

2409:

article and the present article follow the mnemonic "new variables downstairs in weighting factor" and they are presumably using the "correct" convention. Your change of variables formula above have "new variables upstairs".

3831:

Actually, upon closer inspection, the article was already mentioning the other convention before the definitions. All that was needed was a reference. I added Weinberg (and got rid of a reference needed tag in the process).

3919:

Ah, but that is far away from intuitive notions about classical densities and, furthermore, already has a choice of convention (ours) built in. The motivation would then be commonality, not naturality. But let us not argue

3141:

briefly our choice of definition (so that it goes into the top part of the article), and there also mention the alternative, with pop-up citations. This would increase the quality of the article, and keep the peace longer.

4471:

I'm not saying you are wrong (or right). As I suspected, this is much a matter of definition and convention. Even so, since you have been edit warring, the original version should stand as long as no consensus is reached.

5733:, which says: "... a density is a spatially varying quantity on a differentiable manifold which can be integrated in an intrinsic manner". Let's also be clear on something: in the sense of this definition, multiplying by 4975: 361:

A second point, related in that it suffers from the same encyclopaedic style problem, namely that it makes the implicit assumptions of a certain approach without stating the assumptions, is that the definitions assume a

6116: 5231:

Ah, well, this is moderately a subtle point, and a case where normal tensors do exactly the right thing in terms of transformations, without the need to introduce additional mathematical machinery. It is called the

6346:

to it yields a density. At least the formulas in Lee's book all indicate this, without exception, though I'm too lazy to bring out pencil and paper to compute myself so I may be wrong (and might go wrong even if I

5679:-density and a scalar density is a 1-density. They are not the same mathematical object. By "natural tensorial densities" I mean tensors that may be intrinsically integrated over a region. Sorry, scratch the term 5565:, (or the metric volume form for fields) and without inner product there is no Hodge dual, at least not from what I can glean from the Hodge dual article. I still don't see what Hodge duals have to do with this. 519:) In that book (and course) we went over tensor densities. One thing we showed/learned was that the square-root of the determinant of the metric is a scalar density of weight -1. (Sean Carroll shows it on p.52). 2509:

For your information I spoke to THREE different university professors who all have a PhD IN THEORETICAL PHYSICS specializing in GRAVITY. They ALL said that the transformation of tensor densities defined as:

662:

As further proof that this is correct, consider a famous action, say the Einstein-Hilbert (EH) action. For simplicity lets look at the EH-action with zero cosmological constant and no matter. The EH action is

4248:{\displaystyle d{\bar {x}}^{1}\land \ldots \land d{\bar {x}}^{n}=\operatorname {det} \left({\frac {\partial {\bar {x}}^{j}}{\partial x^{i}}}\right)dx^{1}\land \ldots \land dx^{n}\qquad {\text{Lee (12.13)}},} 350:), in its wording implicitly denies the alternative abstract definition of tensors. For example, the phrase "the determinant of the metric tensor" does not make sense using the abstract geometric concept of 2367: 823: 3371:- there is only one God and all other Gods are false Gods and if you belive in them then you go to hell). There are MANY physicists who work in the same feild (eg Gravity) who use different conventions. 297:

relate to transformation properties? Physicists tend to dislike to formally define the concepts they are using, and when they (most reluctantly) do define them, they don't always agree among themselves.

5953:

components in all but only one real degree of freedom, and satisfying the tensor transformation rules of such a tensor. This perspective will make sense those who like to think of tensors as a bunch of

464:

The IP(s) doesn't seem to be interested in discussion. This has been going on for some time. Therefore page protection is both necessary and useful (useful in that he/she(/they) is forced to discuss).

6468:

I give up. This is taking too many words to get anywhere. Already you are heading of in a direction on which there is no contention, suggesting that you've failed to realize where the problem is. —

6530: 4050:(it has weight zero and no indices). Now there is also a canonical identification of 3-forms on the manifold with the bundle of skew-symmetric tensors of type (0,3). Skew forms of the top degree 1054: 5147:

That is an interesting complication. However, in such theories one probably cannot transfer the theory of tensor densities directly without some complications. In nonmetric and other cases, would

5367: 5249:(rather than differing by the determinant of the basis transformation (or some mix), not the Jacobian of the coordinate transformation – the distinction is significant for non-holomorphic bases) 713: 398:

I have been in an edit-war with an unregistered user with a constantly changing IP address. This makes for a very difficult situation as it is almost impossible to communicate with such people.

337:

the matrix of coefficients with respect to a chosen basis, and it excludes the coordinate-free concept of a tensor. I've seen this concept mathematically formalized somewhere in WP as a matrix

6138: 4551:

Do you still have access to Lee's book? You can find that kind of density in chapter 14 (old ed). It allows you to do invariant integration on non-orientable manifolds. Sorry for interrupting.

4364:

Do you still have access to Lee's book? You can find that kind of density in chapter 14 (old ed). It allows you to do invariant integration on non-orientable manifolds. Sorry for interrupting.

5025: 4662:-form density), and cannot be made tensorial via your double cover construction. These are two very different scaling dependencies. Thus, the two meanings of "density" are entirely distinct. — 605:

is some constant to be determined. The scalar is a type (0,0) tensor density of weight 0 (i.e. a scalar), so that is good. Since the volume element is a tensor density of weight +1 we require

1263: 140: 4430:

No, JRSpriggs won't delete it. I will. First off, I told you the article is updated already with enough on other conventions already by me. You can find it as the last paragraph of section

581: 3276:

So by calling something a 'tensor density' one assumes that one can get a tensor which is a physical thing. But it's NOT. It makes more sense for it to be called just a 'weighted tensor'

2379:

the wedge product of the coordinate functions. Thus if the wedge product of the coordinate functions has units of volume, densities have units of 1/volume (or (whatever unit)/volume).

3967:, one has an inherently coordinate-dependent volume form, but if volume is to be a coordinate-independent concept, one is forced to multiply this expression by the Jacobian determinant 5151:-forms not be the equivalent of choice, thus not using scalar densities at all? I've got a feeling (quite possibly wrong) that without at least this, integration becomes meaningless. — 1133: 653: 6572: 1378:

Ok, this lasted too long and I am sick of it. You know what? I give up. I raise the white flag. YOUR RIGHT AND I (with my tiny miniscule insignificant pee brian) AM WRONG!!!! HAPPY?

4454:

Do you remember my item 3 above? You wrote "That I did not know and I apologize!" in response too it. You seem to understand the principle, but just don't care about following it.

3874:

about having +1 for weight for an ordinary density though I prefer it before -1. A "weight" is anyhow something that is over, above and beyond the concept of a classical density.

3223:

through. I am not defending unconventional choices, nor do I say every choice is as good as any other (though I wrote something that could be misinterpreted to that effect above).

5060: 3846:

I also added a brief motivation for the choice of convention in the article along with the difference in the other convention. The latter is contained in the popup reference.

3002:

To be honest, I'm reacting a bit against the right-or-wrong formulation. Chose convention as suits the occasion. It is little more (actually no more) than choosing a basis in

1071:

Also, density is used in lieu of four anti-symmetric covariant indices. So you should not combine both a notion of weighted density and tensor indices in your volume element.

6372:

Oh yes, here's a real problem. "1-density" links to the density article. The tensor densities in the present article have (I think) also been referred to as 1-densities ITT.

5086:

I also do not buy your "it is only a computational tool" argument. This is true for many mathematical concepts. You could argue that derivatives are only computational tools.

263: 168: 5708:

Best is to avoid inventing terminology, "may intrinsically be integrated" only adds to the confusion. What exists suffices, and not only tensors can be integrated. B t w, a

740: 5244:

a 1-density as defined in this article) as its Hodge dual (also a true tensor) so that one is working with a vector for convenience, then only the desire to use the same

3719: 3644: 3571: 3498: 3425: 769: 4386:

Ok I will put at the bottom of the article a side note about the other convention and *hope* that it doesn't get deleted. (Though I suspect it will) Does that sound ok?

1385:

At the beginning you said there were other things you didn't like about my post. I was going to ask you what but I don't need to because I can guess...it is EVERYTHING.

3087:@Sławomir, I agree of course. I was speaking in general terms. I don't have a problem at all with the Knowledge articles making internal choices that could be said are 1298: 4025:

as a density of weight 0. Ergo: the word "density" is being used with two distinct meanings. In light of this, I do not follow you when you say "the rest follows". —

1330: 6011:

Penroses book b t w, so it's ok with me to refer to it. (I started working again on that thing for classical groups. Feel free to drop by, link is on my user page.)

5737:

before integrating does not fit the definition. You can integrate a volume form over a region of a manifold. You don't first multiply it with a volume element. No

3265:

this is something that (once integrated) is a physical thing, e.g. can be visualized or picked up...in other words one can interact with it, be it in implicit way.

5797:-densities. He does not once refer to a metric, a weight, coordinates, or transformation properties. WP and Penrose could just as well be from different planets. — 3742: 3667: 3594: 3521: 3448: 2955: 3177:

do so in a way that results in an invariant integral. For that, one needs an actual scalar. This scalar is obtained by taking the product of this "volume form"

2932: 623: 603: 369:

Am I being reasonable in saying that this article simply does not adequately outline the context and assumptions sufficiently before launching into the detail? —

1135:

is a scalar density of -1!!! Furthermore (Carroll also shows this) the volume element transforms under a co-ordinate transformation with a determinant as such:

251: 1138: 5817:

It would be nice to see this detailed, both in terms of those natural isomorphisms and in terms of a breakdown of a 1-form into factors like I did above.

5027:(with a fillip to deal with sign on non-orientable manifolds) is the invariant quantity, and that as long as one does not try to work separately with the 6557: 4493:

Now you screw up pretty badly. I had written "I'm not saying you are wrong (or right)." You changed that to "I'm not saying you are right." Here's diff:

3326:

This is wrong." you are saying that either ALL particle physicists are wrong or ALL relativity physicists are wrong, which (if you ask me) is arrogant.

3091:. But other choices in common existence could (and should imo) be mentioned, and not be called wrong, and, likewise, the Wiki-choice shouldn't be called 130: 4898:, this formulation appears to be a peculiarity of the formulation, and cannot have mathematical significance. In particular, if one looks at the action 6552: 771:. But it is a known fact that the EH-action is a true scalar. Or if you insist that it has weight 0 then you are saying that the EH-action should be 6577: 6567: 241: 405:

The integral of a scalar field over a hyper-volume of space-time is not an invariant. To make an invariant integral, one has to integrate a scalar

4904: 6226:

In the notation system I prefer, only the coefficients are used, the basis vectors are discarded as redundant (and coordinate dependent) trash.

5608:

sometimes used as a sloppy Hodge dual (4-current density is a 3-form, but it is often expressed as its Hodge dual for simplicity). Ignore it. —

3023:

Wouldn't you agree? Choosing the opposite convention therefore seems arbitrary and unmotivated. Also it is not a common one in my experience.

3899:

Not religious conviction, but usually on the literature the bare word "density" means "1-density". From this basic premise, the rest follows.

106: 6521:

the same as understanding them. To understand them it is necessary to grasp their mathematical underpinnings. To define a tensor by referring

194: 6562: 4415: 3751: 1342: 924: 5116:

If there is exactly one (nonsingular) metric tensor field, then one can avoid using tensor densities (with some inconvenience). However, in

4054:

have a transformation law under change of coordinates, one that naturally identifies them as densities (at least on an oriented manifold).

6042: 4608:

I think Lee's density is a density in the sense of the present article with weight -1 (our convention). Haven't checked thoroughly though.

4524: 3340: 3286: 2967: 2474: 2232: 835: 4408:

I added a secton titled "Note about conventions" Is this fine? I hope it won't get deleted...though I bet JRSpriggs will delete it...lol

961:

Your teacher was mistaken, if he equated what Carroll called the "naive volume element" with a density of weight +1. In ordinary language

4843:

But identifying a volume form at the start is not a canonical choice. The bundle of densities is trivial, but not canonically trivial.

6534: 5117: 4726:

The article ought to address why 1-densities are invariantly integrable. As far as I can see it is because the "usual" volume element

217: 4489:

Yes and I am sorry about that. This did last too long and should have stopped....I am sorry and I'll stop contributing to this page.

3948:

Sławomir, I can't shake the feeling that different concepts are being conflated under a shared name, causing a confusion. Looking at

5285:

Why would there be any particular problems with tensor densities with metric gone? Their definition doesn't make use of the metric.

1395: 97: 58: 1335:

Also I highly doubt that both Carroll and my professors are wrong...espcially that they have a PhD in physics, specilizing in GR.

5600:-density, the sign depending on whether the indices are raised or lowered. To facilitate clarity, I'd suggest moving away from a 2279: 774: 4889:

JRSpriggs did mention this example before when I tried to claim that tensor densities have no intrinsic mathematical value in

3103:. It is hard to find even the slightest typo is his books, errors in equations are almost unheard of, and would he make gross 331:

I have a possibly different, but essentially related issue – the chosen wording. The concept of a tensor used in this article

5617:

Okay, but the question remains. Why would there be any particular problems with tensor densities with metric gone? These are

6271: 6212:{\displaystyle f={\frac {1}{24}}\epsilon ^{\alpha \beta \gamma \delta }\omega _{\alpha \beta \gamma \delta }=\omega _{1234}} 4850: 4687: 4636: 4336: 4281: 4061: 3906: 3210: 3030: 2989: 2496: 2457: 2431: 2386: 2982:

Fine. If three PhDs agree that a "mass density" or "charge density" is not a density at all, clearly they must be right.

282: 208: 163: 970: 6582: 6437:

to a volume form. The same description as in the second paragraph in the lead (my browser tells me you spell lead wrong

5936:), it is the same object that is being represented in each case. Sławomir, please feel free to shred my interpretation. 3816:

Weinberg has higher weight than Carroll), the more economical, the bigger chance it lasts, but that's really up to you.

33: 6328:

of Lee. These are ready to be invariantly integrated and are as such not tensor densities which need a volume element

4274:

You misunderstand me. A skew symmetric tensor of type (0,n) has n downstairs indices. It transforms as a 1-density.

666: 4986: 5789:-density is not what is defined in this article, it is something that can (on its own) be integrated. He says that 876:

It is customary here at Knowledge that when an edit war occurs, then the matter should be settled on the talk page

2253:

so that you do not make life difficult for other editors. I know this because I found out the hard way! — Cheers,

1225: 2467:

The convention where the weight of g is -2 is NOT I repeat NOT wrong...it is just a convention (Which I prefer)

293:

Is there a definition of the thingies in the article coming from a pure math source, like a definition that does

6265: 4844: 4681: 4630: 4330: 4275: 4055: 3900: 3799:@Anonymous: The choice of signature is AFAIK arbitrary, while the choice of convention for tensor density isn't 3204: 3053: 3024: 2983: 2490: 2451: 2425: 2400: 2380: 531: 6240:

Thank you for the concise answer. Now it all fits together as it should without any more guesswork on my part.

4419: 3755: 3344: 3290: 1382:

the looks of it, my miniscule insignificant pee brain and I deserve it. (and according to you it will happen)

1346: 928: 4528: 2971: 2478: 2236: 839: 21: 5764:, I need language to draw attention to the distinction. Without that, this discussion is not worth pursuing. 1108: 628: 5876:-forms, and hence all tensors. Weight is not a concept that enters into the picture in Penrose's version. — 4594:-linear, and hence a tensor. Aside from the issue with the sign, Lee's definition is compatible that of an 1399: 3862:

our convention, but with the defining formula reformulated in a more natural way (Jacobian determinant of

3772: 6449:

of weight +1 and one of which behaves as a tensor density of weight -1. Compare the LaTeX ITT. There the

6339:

tensor density of weight +1. Conversely, if you have an even tensor density of weight +1, then attaching

2268:

I'm not sure what the current state of this discussion is. I only just became aware of it after editing

867:

You need to update the article with references supporting your claims (or you are likely to be reverted).

742:

has weight +1 which (since the volume element has weight +1) means that the intergral has overall weight

6420: 6325: 5754: 5729: 5090: 4878: 3949: 3789: 2258: 1366: 1089: 385: 39: 83: 5945:

I interpret this to mean that one can represent it as a fully antisymmetric covariant tensor of order

5033: 2489:

Fine. Except in that convention, the word "density" is unrelated to its usual mathematical meaning.

625:

to be a object of weight -1 so that the combination is a (0,0) tensor of weight 0. It so happens that

4520: 4411: 3747: 3336: 3282: 3052:

densities do not qualify, since they are components of tensors, not scalars, unlike the trace of the

2963: 2470: 2228: 1391: 1338: 920: 831: 6282:

So – do you agree that we have two colliding sets of terminology? And that we need to distinguish? —

5758:(an article which unfortunately then proceeds to confuse the meanings), and a density as defined in 656: 523: 6231: 5769: 5202: 5126: 4754:

objects of weight 0 will be invariantly integrable with respect to it. If, on the other hand, only

4000:

the concept represented by a "tensor density" of this article; this article considers the quantity

3333:

2) Action is not a density but rather a functional. You probably meant to say Lagrangian density.

3311: 3064:" means, but I guess this'd be the most natural. Is this the sense in which it is currently used? — 2249: 1076: 415: 381:

See Spivak's book listed in the references. (It should be easy to find a digital copy using google)

216:

on Knowledge. If you would like to participate, please visit the project page, where you can join

105:

on Knowledge. If you would like to participate, please visit the project page, where you can join

5781:-surfaces, as well as defining a volume element. It may be noted that his "volume element" is an 718: 89: 6256:

I just want to add to what JRSpriggs said. It is in this sense that I mean that a differential

73: 52: 3099:

or whatever, preferably with an explanation. It is a potential blunder to call, say, Weinberg,

6469: 6459: 6425: 6377: 6315: 6283: 6245: 6016: 5999: 5890: 5877: 5851: 5837: 5822: 5798: 5718: 5700: 5665: 5639: 5609: 5604:. The Jacobian of the change of coordinates and the tensor transformation matrix then diverge. 5570: 5542: 5321: 5290: 5257: 5152: 5100: 5073: 5063: 4867:(again splitting out what seems like a separate topic, by copying a post from a thread above) 4831: 4800: 4738: 4711: 4663: 4651: 4613: 4599: 4556: 4507: 4477: 4462: 4439: 4395: 4369: 4319: 4026: 3933: 3889: 3837: 3821: 3776: 3680: 3605: 3532: 3459: 3386: 3361: 3244: 3147: 3065: 3013: 2415: 945: 889: 855: 745: 495: 469: 448: 433: 370: 321: 303: 6033:

To YohanN7, in response to your question above about Ω: Yes, you are essentially correct. If

4655: 3302:

cell phone, the image moved upward or spun around or shrank, rather than moving to the right.

5087: 4875: 4587:| (as one can do on an orientable manifold, or locally on any manifold), then Lee's density 3786: 2254: 1363: 1086: 912: 382: 6335:

attached before integration. But they are related. Densities contain a "factor" that is an

4874:

Lagrangian density (weight +/- 1) that naturally appears in your Euler-Lagrange equations.

4329:

is, an isomorphism that does not require any choices to be made or additional structure).

4310:

below. What do you and what do you not include when you "measure" the weight. If we have a

3785:

Lagrangian density (weight +/- 1) that naturally appears in your Euler-Lagrange equations.

1268: 5601: 3381:

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity

3234:

be in the literature. When there are multiple conventions around, the article should make

1409:

Ok I just realized that my argument was valid if you defined the tensor transformation as

1303: 515:

1) When I took a General Relativity (GR) course we used the GR textbook by Sean Carroll. (

363: 2446:

The long-term edit war (which I think justifies semiprotection), has spilled over to the

3008:. Sometimes you don't even want it to be orthogonal because it would complicate things. 2372:

This is the convention adopted in the present article, and in JRSpriggs' comment above.

1388:

Have a good life, this future stupid IDIOTIC loser (aka me) won't bother you again!!!!

6227: 5760: 5198: 5122: 5028: 4431: 3724: 3649: 3576: 3503: 3430: 3307: 2937: 1072: 411: 200: 6529:

that does the transforming shows a complete lack of understanding of what tensors are.

2917: 1212:{\displaystyle d^{n}x'=\left\vert {\frac {\partial x'}{\partial x}}\right\vert d^{n}x} 608: 588: 6546: 5648:@Quondum, may I ask what you mean by "natural tensorial densities"? You exemplify by 4810:

Yes, and you're hopefully starting to get my point. If we use the (invariant) weight-

3808:. The reason this is so is made clear in this thread, especially in JRSprigg's posts. 3767: 4602: 4560: 4543:(one post copied from above thread, separating as above thread is getting unwieldy) 828:

P.s. I use absolute value on g to ensure positivity even in a Riemannian manifold.

6455: 6373: 6311: 6241: 6012: 5886: 5847: 5818: 5714: 5695:, and I'm guessing that the natural dimensionality of the region of integration is 5661: 5635: 5566: 5538: 5286: 4796: 4734: 4707: 4609: 4552: 4503: 4473: 4458: 4435: 4391: 4365: 4315: 3929: 3885: 3833: 3817: 3240: 3143: 3057: 3009: 2411: 941: 885: 851: 491: 465: 429: 317: 299: 5811:

isomorphism that does not require any choices to be made or additional structure).

5683:-form; I'm too unsure of what objects qualify, though I suspect that they are all 5621:

to us, e.g. like in the case of EM if I understand what JRSpriggs means correctly.

4046:

that we would call the density (a "volume density"). The three-form itself is a

4981: 4895: 343: 184: 157: 102: 5357:

A bit on terminology. Consider a n-form Ω, a true tensor. Then, in coordinates

2375:

Put another way, a density is something that defines a scalar field when it is

5552: 5233: 190: 79: 6424:

and see which of these each of the two paragraphs appears to be describing. —

5806:

Yes, you are right. There is confusion. Here is what Sławomir wrote earlier:

4306:

I think I understand what you mean. This here is surely related to the topic

4970:{\displaystyle S={1 \over 2\kappa }\int R{\sqrt {-g}}\,\mathrm {d} ^{4}x\;,} 3766:

I suggest you mellow your interpretation; it is a principle in Knowledge to

6503:

section profoundly fail to explain what a tensor density is to the reader.

4457:

Please don't do any more edits to the article before consensus is reached.

516: 6291:

Actually no. I have changed my mind for the tenth time in a few days now.

4785:(including everything!), then it can be multiplied by an object of weight 4760:

is in place, then an object of weight +1 will be invariantly integrable.

3181:! This is what the slogan "densities have a well-defined integral" means. 5981:-form; indeed, there is a direct mapping between the two: the Hodge dual. 1085:

Saying that people adhering to a different convention are wrong is silly.

6495:

Can someone who understands the subject please define it in the article?

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Hi, we all make mistakes. An important thing on Knowledge is to follow

2240: 1403: 1369: 1350: 1092: 1080: 962: 949: 932: 893: 859: 843: 499: 473: 451: 437: 419: 388: 375: 325: 307: 213: 6111:{\displaystyle \Omega =f\,dx^{1}\land dx^{2}\land \cdots \land dx^{n}} 3775:

and the numerous articles and essays that it links to illuminating. —

3173:

invariant (it picks up a factor of the Jacobian). But of course, it

2447: 2406: 2269: 428:

It would force him/her to either quit or get an account and discuss.

347: 5977:. There is a natural isomorphism between this scalar field and the 5727:

I am not inventing that particular terminology, I am taking it from

4314:, then the total weight of the thing must be 0 as far as I can see. 940:

most common one being the "main" definition to be used throughout.

526:, equation 2.44). Now, in curved space-time we want the expression 5748:

Confusion is being caused by allowing sloppy abuse of terminology

870:

Lecture notes do generally not constitute acceptable references.

657:

http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll2.html

524:

http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll2.html

5987:

one is as a density (a scalar with weight 1, in our convention)

5975:

a scalar field that must be multiplied by the structural n-form

5864:-density" is a field that can be integrated over a region of a 5752:. When I choose to distinguish between a density as defined in 5579:

Looks good. In this example, as you say, if one has the 4-form

4500:

Do not ever edit what other people have written on a talk page!

522:

2) The volume element is a (0,n) tensor density of weight +1. (

2424:

yes, you're right. It should be the inverse of the jacobian.

907:

i) I did support my claims both on the talk and the article.

15: 6410:

Ah – a form of concession at last, at least from one person:

5785:-form that is part of the structure of the manifold, and his 4769:

is in place in the integrand, no invariant element, i.e. not

3858:

ordinary densities are assigned weight +1. (Admittedly, this

3226:

The point is this: The articles are supposed to reflect what

2362:{\displaystyle f(y)=f(x)|\det(\partial y/\partial x)|^{-1}.} 490:

He reverted. This guy is not here to discuss. Semi-protect!

5624:@JRSpriggs, I'd like to know too what you mean by a 4-form 3322:

different but the answer/interpretation will be the same.

818:{\displaystyle S\propto \int {\frac {R}{\sqrt {-g}}}d^{4}x} 316:

be defined without reference to transformation properties.

4574:. I don't have complete access. He claims "a density is 3811:

There's nothing stopping you from editing the article and

394:

Sign of the weight of the determinant of the metric tensor

5960:

one is as a scalar (a differential n-form of weight zero)

3527:

Introduction to General Relativity Spacetime and Geometry

913:

http://preposterousuniverse(dot)com/spacetimeandgeometry/

585:

to be a intergral of a scalar over a proper n-form. Here

911:

notes' are identical to his textbook. Here is his blog:

6133:

is an anti-symmetric covariant tensor (weight 0), then

5773:§12.5 and §12.7 gives an encapsulation of integration, 4494: 872:

Nor does your own arguments - even if they are correct.

312:

For example, a tensor (w/o all pseudo, etc qualifiers)

6260:-form (of weight zero) is a "scalar". A differential 5885:

I actually had Sławomir's quote in mind, not Penrose.

4307: 6141: 6126:-form. Otherwise, it is coordinate-dependent garbage. 6045: 5934:

There are three ways of thinking of a skew (0,n) form

5555:

properly without a metric? An inner product will do,

5365: 5036: 4989: 4907: 4086: 3727: 3683: 3652: 3608: 3579: 3535: 3506: 3462: 3433: 3389: 2940: 2920: 2518: 2282: 1822: 1415: 1306: 1271: 1228: 1141: 1111: 973: 777: 748: 721: 669: 631: 611: 591: 534: 3721:

signature and uses the definition that g has weight

1049:{\displaystyle Density={\frac {d\,Mass}{d\,Volume}}} 212:, a collaborative effort to improve the coverage of 101:, a collaborative effort to improve the coverage of 1068:

to volume. That is, your basic assumption is wrong.

346:). This article (to some extent unlike the article 6211: 6110: 5525: 5054: 5019: 4969: 4247: 3736: 3713: 3661: 3638: 3588: 3565: 3515: 3492: 3442: 3419: 2949: 2926: 2914:is a convention that is RIGHT. By THIS convention 2904: 2361: 2208: 1801: 1324: 1292: 1257: 1211: 1127: 1048: 817: 763: 734: 707: 647: 617: 597: 575: 3279:What do you guys think? Or am I off my rockers? 708:{\displaystyle S\propto \int R{\sqrt {-g}}d^{4}x} 3060:. Here there is room to define what "of weight 2315: 715:. Do you deny this is correct? According to you 6525:to "how it transforms" and without ever saying 5020:{\displaystyle {\sqrt {-g}}\,\mathrm {d} ^{4}x} 3107:? (Having said that, his mathematical taste is 6573:Start-Class physics articles of Low-importance 5628:a scalar density. Do you identify it with the 3928:. Article ok (last sentence in lead + popup)? 6508:someone who actually understands this concept 1258:{\displaystyle {\sqrt {\vert g\vert }}d^{n}x} 864:Three things you need to know (for a start): 8: 5767:At this juncture I'll note that Penrose, in 3137:So, I suggest, in the present case, that we 1237: 1231: 1120: 1114: 640: 634: 4570:I presume you're referring to John M Lee's 917:iii) That I did not know and I apologize! 576:{\displaystyle \int \phi (x^{\mu })Ad^{n}x} 19: 6310:I'm tired and can't think straight a t m. 4517:Fine, I am sorry I'll stop contributing. 152: 47: 6203: 6181: 6162: 6148: 6140: 6102: 6080: 6064: 6044: 6037:is a scalar density (of weight +1), then 5596:The Levi-Civita symbol is, as you say, a 5514: 5506: 5484: 5468: 5450: 5434: 5426: 5404: 5388: 5366: 5364: 5043: 5038: 5035: 5008: 5003: 4990: 4988: 4953: 4948: 4935: 4914: 4906: 4779:By extension if the integrand has weight 4237: 4229: 4207: 4187: 4172: 4161: 4160: 4153: 4134: 4123: 4122: 4103: 4092: 4091: 4085: 3726: 3682: 3651: 3607: 3578: 3534: 3505: 3461: 3432: 3388: 2939: 2919: 2894: 2881: 2876: 2869: 2856: 2851: 2845: 2844: 2829: 2824: 2807: 2802: 2792: 2775: 2770: 2753: 2748: 2738: 2727: 2722: 2702: 2697: 2687: 2673: 2668: 2648: 2643: 2633: 2627: 2598: 2579: 2563: 2558: 2548: 2532: 2527: 2521: 2520: 2517: 2347: 2342: 2327: 2310: 2281: 2198: 2185: 2180: 2173: 2160: 2155: 2149: 2148: 2133: 2128: 2111: 2106: 2096: 2079: 2074: 2057: 2052: 2042: 2031: 2026: 2006: 2001: 1991: 1977: 1972: 1952: 1947: 1937: 1931: 1902: 1883: 1867: 1862: 1852: 1836: 1831: 1825: 1824: 1821: 1791: 1778: 1773: 1766: 1753: 1748: 1742: 1741: 1726: 1721: 1704: 1699: 1689: 1672: 1667: 1650: 1645: 1635: 1624: 1619: 1599: 1594: 1584: 1570: 1565: 1545: 1540: 1530: 1524: 1495: 1476: 1460: 1455: 1445: 1429: 1424: 1418: 1417: 1414: 1305: 1270: 1246: 1229: 1227: 1200: 1167: 1146: 1140: 1112: 1110: 998: 972: 806: 787: 776: 747: 722: 720: 696: 682: 668: 632: 630: 610: 590: 564: 548: 533: 4962: 517:http://arxiv.org/pdf/gr-qc/9712019v1.pdf 6531:2600:1700:E1C0:F340:21E2:140D:C505:D61A 6055: 5000: 4945: 1128:{\displaystyle {\sqrt {\vert g\vert }}} 1023: 1004: 648:{\displaystyle {\sqrt {\vert g\vert }}} 289:Definition from elsewhere than physics? 154: 49: 5559:? The Hodge dual uses the Levi-Civita 3452:Misner, Thorne, Wheeler in their book 655:fulfills this requirement as seen by ( 344:Determinant#Abstract_algebraic_aspects 6433:On an oriented manifold a density is 3646:signature and says that g has weight 3573:signature and says that g has weight 3500:signature and says that g has weight 3427:signature and says that g has weight 3230:in the literature, not what we think 7: 6438: 5971:a differential n-form of weight zero 5777:-forms and their integrability over 3921: 3878: 3113: 904:In reference to your three items... 901:Hi Yohan, thank you for your reply. 206:This article is within the scope of 95:This article is within the scope of 5029:inherently basis-dependent quantity 2846: 2522: 2150: 1826: 1743: 1419: 38:It is of interest to the following 6324:The trickiest point may by be the 6046: 5372: 5118:alternatives to general relativity 5039: 5004: 4949: 4863:On the utility of tensor densities 4792:and then invariantly integrated. 4308:#Invariantly integrable quantities 4180: 4156: 3600:Introducing Einsteins's Relativity 2817: 2795: 2763: 2741: 2715: 2690: 2661: 2636: 2614: 2601: 2332: 2321: 2121: 2099: 2067: 2045: 2019: 1994: 1965: 1940: 1913: 1905: 1714: 1692: 1660: 1638: 1612: 1587: 1558: 1533: 1511: 1498: 1183: 1170: 850:Hi, and welcome to the talk page. 14: 6558:Low-priority mathematics articles 6499:The introductory section and the 5557:but then you have a metric, right 5452:is a scalar density with weight 1 5436:is a tensor (density of weight 0) 5055:{\displaystyle \mathrm {d} ^{4}x} 4749:Invariantly integrable quantities 3973:, thus arriving at an expression 3677:Weinburg and Carroll who use the 115:Knowledge:WikiProject Mathematics 6553:Start-Class mathematics articles 6439: 6223:is a scalar density (weight +1). 5941:One is as a tensor (weight zero) 5551:More hm. How do you compute the 5516:is a tensor density of weight -1 5236:. If one wants to represent an 4795:Are these observations correct? 4572:Introduction to Smooth Manifolds 4390:The article is updated already. 3922: 3879: 3114: 2405:I'm slightly confused. Both the 193: 183: 156: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 6578:Start-Class relativity articles 6568:Low-importance physics articles 6510:please rewrite these sections? 3877:So now I have screwed up badly 825:which, obviously is wrong. QED 246:This article has been rated as 135:This article has been rated as 4166: 4128: 4097: 3708: 3684: 3633: 3609: 3560: 3536: 3487: 3463: 3414: 3390: 2343: 2338: 2318: 2311: 2307: 2301: 2292: 2286: 1319: 1307: 554: 541: 1: 6539:00:45, 12 November 2017 (UTC) 4763:The default seems to be that 4539:Comparing to Lee's definition 4255:so it is a density of weight 4235: 1219:Thus it is a density of +1. 326:19:23, 28 November 2013 (UTC) 308:14:06, 29 November 2013 (UTC) 283:Talk:Tensor density/Archive 1 261:This article is supported by 226:Knowledge:WikiProject Physics 220:and see a list of open tasks. 109:and see a list of open tasks. 6563:Start-Class physics articles 3379:Steven Weinburg in his book 735:{\displaystyle {\sqrt {-g}}} 229:Template:WikiProject Physics 6276:20:56, 14 August 2014 (UTC) 6599: 4538: 4080:The transformation law is 3806:That is all there is to it 3598:Ray D'Inverno in his book 252:project's importance scale 6473:13:13, 10 July 2014 (UTC) 6464:06:32, 10 July 2014 (UTC) 6429:04:02, 10 July 2014 (UTC) 6382:01:21, 10 July 2014 (UTC) 6320:00:45, 10 July 2014 (UTC) 5745:≠ 0 fits this definition. 5593:-densities" respectively. 4432:Tensor density#Definition 3744:are wrong/stupid/etc... 3714:{\displaystyle (-,+,+,+)} 3639:{\displaystyle (+,-,-,-)} 3566:{\displaystyle (-,+,+,+)} 3525:Sean Carroll in his book 3493:{\displaystyle (-,+,+,+)} 3420:{\displaystyle (-,+,+,+)} 764:{\displaystyle +2\not =0} 376:14:28, 14 June 2014 (UTC) 264:the relativity task force 260: 245: 178: 134: 67: 46: 6287:23:41, 9 July 2014 (UTC) 6250:19:50, 9 July 2014 (UTC) 6236:19:27, 9 July 2014 (UTC) 6021:17:43, 9 July 2014 (UTC) 6003:17:10, 9 July 2014 (UTC) 5895:15:34, 9 July 2014 (UTC) 5881:15:18, 9 July 2014 (UTC) 5856:14:12, 9 July 2014 (UTC) 5841:13:22, 9 July 2014 (UTC) 5827:10:46, 9 July 2014 (UTC) 5802:06:13, 9 July 2014 (UTC) 5723:05:11, 9 July 2014 (UTC) 5713:jargon/notational abuse. 5704:22:44, 8 July 2014 (UTC) 5670:19:31, 8 July 2014 (UTC) 5644:19:06, 8 July 2014 (UTC) 5613:18:44, 8 July 2014 (UTC) 5575:18:12, 8 July 2014 (UTC) 5547:16:19, 8 July 2014 (UTC) 5325:15:27, 8 July 2014 (UTC) 5295:11:34, 8 July 2014 (UTC) 5261:13:28, 8 July 2014 (UTC) 5207:10:30, 8 July 2014 (UTC) 5156:05:29, 8 July 2014 (UTC) 5131:03:40, 8 July 2014 (UTC) 5104:22:30, 7 July 2014 (UTC) 5094:19:50, 7 July 2014 (UTC) 5077:14:30, 7 July 2014 (UTC) 5067:13:39, 7 July 2014 (UTC) 4882:08:34, 7 July 2014 (UTC) 4855:13:30, 6 July 2014 (UTC) 4835:05:34, 6 July 2014 (UTC) 4805:04:31, 6 July 2014 (UTC) 4743:03:09, 6 July 2014 (UTC) 4716:03:35, 6 July 2014 (UTC) 4692:02:41, 6 July 2014 (UTC) 4667:00:55, 6 July 2014 (UTC) 4641:23:09, 5 July 2014 (UTC) 4618:21:57, 5 July 2014 (UTC) 4603:21:14, 5 July 2014 (UTC) 4561:19:06, 5 July 2014 (UTC) 4533:21:15, 2 July 2014 (UTC) 4512:19:18, 2 July 2014 (UTC) 4482:14:43, 2 July 2014 (UTC) 4467:13:53, 2 July 2014 (UTC) 4444:20:32, 5 July 2014 (UTC) 4424:19:47, 5 July 2014 (UTC) 4400:19:20, 5 July 2014 (UTC) 4374:19:06, 5 July 2014 (UTC) 4341:18:02, 6 July 2014 (UTC) 4324:17:36, 6 July 2014 (UTC) 4286:15:39, 6 July 2014 (UTC) 4066:14:02, 6 July 2014 (UTC) 4030:18:54, 5 July 2014 (UTC) 3938:16:41, 5 July 2014 (UTC) 3911:16:21, 5 July 2014 (UTC) 3894:14:41, 5 July 2014 (UTC) 3842:11:45, 5 July 2014 (UTC) 3826:09:56, 5 July 2014 (UTC) 3793:08:34, 7 July 2014 (UTC) 3780:04:17, 5 July 2014 (UTC) 3760:02:32, 5 July 2014 (UTC) 3365:18:48, 4 July 2014 (UTC) 3349:05:07, 4 July 2014 (UTC) 3316:04:36, 4 July 2014 (UTC) 3295:04:13, 4 July 2014 (UTC) 3249:15:31, 4 July 2014 (UTC) 3215:12:45, 4 July 2014 (UTC) 3152:03:26, 4 July 2014 (UTC) 3069:02:14, 4 July 2014 (UTC) 3035:01:41, 4 July 2014 (UTC) 3018:00:41, 4 July 2014 (UTC) 2994:23:22, 3 July 2014 (UTC) 2976:23:11, 3 July 2014 (UTC) 2501:22:35, 3 July 2014 (UTC) 2483:21:29, 3 July 2014 (UTC) 2462:19:38, 3 July 2014 (UTC) 2450:article as well, fwiw. 2436:16:28, 3 July 2014 (UTC) 2420:15:50, 3 July 2014 (UTC) 2391:13:32, 3 July 2014 (UTC) 2263:08:29, 3 July 2014 (UTC) 2241:02:30, 3 July 2014 (UTC) 1404:19:08, 2 July 2014 (UTC) 1370:10:49, 2 July 2014 (UTC) 1351:08:06, 2 July 2014 (UTC) 1093:10:49, 2 July 2014 (UTC) 1081:06:22, 2 July 2014 (UTC) 950:01:54, 2 July 2014 (UTC) 933:01:00, 2 July 2014 (UTC) 894:00:09, 2 July 2014 (UTC) 860:22:42, 1 July 2014 (UTC) 844:21:58, 1 July 2014 (UTC) 500:20:41, 1 July 2014 (UTC) 474:15:52, 1 July 2014 (UTC) 452:14:32, 1 July 2014 (UTC) 438:13:35, 1 July 2014 (UTC) 420:05:54, 1 July 2014 (UTC) 389:09:00, 3 July 2014 (UTC) 141:project's priority scale 5860:Penrose's meaning for " 4042:In your example, it is 4015:as a density of weight 2934:is a scalar density of 2270:Tensor#Tensor densities 2218:(then g has weight +2) 1811:(then g has weight -2) 1064:related to volume, not 98:WikiProject Mathematics 6213: 6112: 5527: 5056: 5021: 4980:one sees that the the 4971: 4249: 3773:Knowledge:Five pillars 3738: 3715: 3663: 3640: 3590: 3567: 3517: 3494: 3444: 3421: 2951: 2928: 2906: 2363: 2210: 1814:BUT you defined it as 1803: 1326: 1300:. So it is a ordinary 1294: 1293:{\displaystyle -1+1=0} 1259: 1213: 1129: 1050: 819: 765: 736: 709: 649: 619: 599: 577: 28:This article is rated 6421:Density on a manifold 6214: 6113: 5755:Density on a manifold 5730:Density on a manifold 5528: 5057: 5022: 4972: 4250: 3950:Density on a manifold 3739: 3716: 3664: 3641: 3591: 3568: 3518: 3495: 3445: 3422: 2952: 2929: 2907: 2364: 2211: 1804: 1327: 1325:{\displaystyle (0,0)} 1295: 1260: 1214: 1130: 1051: 880:any changes are made. 820: 766: 737: 710: 650: 620: 600: 578: 6139: 6043: 5363: 5034: 4987: 4905: 4680:useful one at all. 4084: 3725: 3681: 3650: 3606: 3577: 3533: 3504: 3460: 3431: 3387: 3054:stress–energy tensor 2938: 2918: 2516: 2280: 1820: 1413: 1304: 1269: 1226: 1139: 1109: 971: 965:is characterized by 775: 746: 719: 667: 629: 609: 589: 532: 121:mathematics articles 6583:Relativity articles 6418:. Read the lede of 5872:-densities are all 5770:The Road to Reality 3219:I'm not getting my 2901: 2837: 2783: 2710: 2656: 2589: 2587: 2571: 2556: 2540: 2225:Have a good life! 2205: 2141: 2087: 2014: 1960: 1893: 1891: 1875: 1860: 1844: 1798: 1734: 1680: 1607: 1553: 1486: 1484: 1468: 1453: 1437: 1222:So the combination 1105:Carroll SHOWS that 209:WikiProject Physics 6209: 6108: 6056: 5750:in this discussion 5523: 5521: 5052: 5017: 5001: 4967: 4963: 4946: 4245: 4236: 3866:transformation.)) 3737:{\displaystyle -2} 3734: 3711: 3662:{\displaystyle +2} 3659: 3636: 3589:{\displaystyle -2} 3586: 3563: 3516:{\displaystyle +2} 3513: 3490: 3443:{\displaystyle -2} 3440: 3417: 3095:, just natural or 2950:{\displaystyle -2} 2947: 2924: 2902: 2843: 2825: 2771: 2698: 2644: 2575: 2559: 2544: 2528: 2519: 2359: 2206: 2147: 2129: 2075: 2002: 1948: 1879: 1863: 1848: 1832: 1823: 1799: 1740: 1722: 1668: 1595: 1541: 1472: 1456: 1441: 1425: 1416: 1360:in that convention 1322: 1290: 1255: 1209: 1125: 1046: 1024: 1005: 815: 761: 732: 705: 659:, equation 2.41). 645: 615: 595: 573: 90:Mathematics portal 34:content assessment 6156: 5517: 5453: 5437: 4998: 4943: 4927: 4523:comment added by 4414:comment added by 4240: 4194: 4169: 4131: 4100: 3768:assume good faith 3750:comment added by 3339:comment added by 3285:comment added by 3105:conceptual errors 2966:comment added by 2927:{\displaystyle g} 2841: 2787: 2736: 2682: 2621: 2473:comment added by 2231:comment added by 2145: 2091: 2040: 1986: 1925: 1738: 1684: 1633: 1579: 1518: 1394:comment added by 1341:comment added by 1265:has total weight 1240: 1190: 1123: 1044: 923:comment added by 834:comment added by 800: 799: 730: 690: 643: 618:{\displaystyle A} 598:{\displaystyle A} 279: 278: 275: 274: 271: 270: 151: 150: 147: 146: 6590: 6517:the concepts is 6444: 6443: 6442: 6345: 6334: 6300:tensor densities 6268: 6218: 6216: 6215: 6210: 6208: 6207: 6195: 6194: 6176: 6175: 6157: 6149: 6122:is an invariant 6117: 6115: 6114: 6109: 6107: 6106: 6085: 6084: 6069: 6068: 5932:By implication ( 5711: 5678: 5659: 5653: 5633: 5599: 5592: 5582: 5532: 5530: 5529: 5524: 5522: 5518: 5515: 5511: 5510: 5489: 5488: 5473: 5472: 5454: 5451: 5438: 5435: 5431: 5430: 5409: 5408: 5393: 5392: 5246:component values 5240:-form (which is 5061: 5059: 5058: 5053: 5048: 5047: 5042: 5026: 5024: 5023: 5018: 5013: 5012: 5007: 4999: 4991: 4976: 4974: 4973: 4968: 4958: 4957: 4952: 4944: 4936: 4928: 4926: 4915: 4847: 4829: 4824: 4820: 4819: 4813: 4791: 4784: 4775: 4768: 4759: 4731: 4684: 4633: 4535: 4426: 4333: 4278: 4258: 4254: 4252: 4251: 4246: 4241: 4238: 4234: 4233: 4212: 4211: 4199: 4195: 4193: 4192: 4191: 4178: 4177: 4176: 4171: 4170: 4162: 4154: 4139: 4138: 4133: 4132: 4124: 4108: 4107: 4102: 4101: 4093: 4058: 4024: 4018: 4014: 3994: 3972: 3966: 3927: 3926: 3925: 3903: 3884: 3883: 3882: 3762: 3743: 3741: 3740: 3735: 3720: 3718: 3717: 3712: 3668: 3666: 3665: 3660: 3645: 3643: 3642: 3637: 3595: 3593: 3592: 3587: 3572: 3570: 3569: 3564: 3522: 3520: 3519: 3514: 3499: 3497: 3496: 3491: 3449: 3447: 3446: 3441: 3426: 3424: 3423: 3418: 3351: 3297: 3207: 3119: 3118: 3117: 3027: 3007: 2986: 2978: 2956: 2954: 2953: 2948: 2933: 2931: 2930: 2925: 2911: 2909: 2908: 2903: 2900: 2899: 2898: 2886: 2885: 2875: 2874: 2873: 2861: 2860: 2850: 2849: 2842: 2840: 2839: 2838: 2833: 2815: 2814: 2813: 2812: 2811: 2793: 2788: 2786: 2785: 2784: 2779: 2761: 2760: 2759: 2758: 2757: 2739: 2737: 2735: 2734: 2733: 2732: 2731: 2713: 2712: 2711: 2706: 2688: 2683: 2681: 2680: 2679: 2678: 2677: 2659: 2658: 2657: 2652: 2634: 2632: 2631: 2626: 2622: 2620: 2612: 2611: 2599: 2588: 2583: 2567: 2557: 2552: 2536: 2526: 2525: 2493: 2485: 2454: 2428: 2404: 2383: 2368: 2366: 2365: 2360: 2355: 2354: 2346: 2331: 2314: 2243: 2215: 2213: 2212: 2207: 2204: 2203: 2202: 2190: 2189: 2179: 2178: 2177: 2165: 2164: 2154: 2153: 2146: 2144: 2143: 2142: 2137: 2119: 2118: 2117: 2116: 2115: 2097: 2092: 2090: 2089: 2088: 2083: 2065: 2064: 2063: 2062: 2061: 2043: 2041: 2039: 2038: 2037: 2036: 2035: 2017: 2016: 2015: 2010: 1992: 1987: 1985: 1984: 1983: 1982: 1981: 1963: 1962: 1961: 1956: 1938: 1936: 1935: 1930: 1926: 1924: 1923: 1911: 1903: 1892: 1887: 1871: 1861: 1856: 1840: 1830: 1829: 1808: 1806: 1805: 1800: 1797: 1796: 1795: 1783: 1782: 1772: 1771: 1770: 1758: 1757: 1747: 1746: 1739: 1737: 1736: 1735: 1730: 1712: 1711: 1710: 1709: 1708: 1690: 1685: 1683: 1682: 1681: 1676: 1658: 1657: 1656: 1655: 1654: 1636: 1634: 1632: 1631: 1630: 1629: 1628: 1610: 1609: 1608: 1603: 1585: 1580: 1578: 1577: 1576: 1575: 1574: 1556: 1555: 1554: 1549: 1531: 1529: 1528: 1523: 1519: 1517: 1509: 1508: 1496: 1485: 1480: 1464: 1454: 1449: 1433: 1423: 1422: 1406: 1353: 1331: 1329: 1328: 1323: 1299: 1297: 1296: 1291: 1264: 1262: 1261: 1256: 1251: 1250: 1241: 1230: 1218: 1216: 1215: 1210: 1205: 1204: 1195: 1191: 1189: 1181: 1180: 1168: 1159: 1151: 1150: 1134: 1132: 1131: 1126: 1124: 1113: 1055: 1053: 1052: 1047: 1045: 1043: 1018: 999: 935: 846: 824: 822: 821: 816: 811: 810: 801: 792: 788: 770: 768: 767: 762: 741: 739: 738: 733: 731: 723: 714: 712: 711: 706: 701: 700: 691: 683: 654: 652: 651: 646: 644: 633: 624: 622: 621: 616: 604: 602: 601: 596: 582: 580: 579: 574: 569: 568: 553: 552: 425:Page protection? 234: 233: 232:physics articles 230: 227: 224: 203: 198: 197: 187: 180: 179: 174: 171: 160: 153: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 6598: 6597: 6593: 6592: 6591: 6589: 6588: 6587: 6543: 6542: 6497: 6440: 6340: 6329: 6266: 6199: 6177: 6158: 6137: 6136: 6132: 6098: 6076: 6060: 6041: 6040: 5709: 5676: 5655: 5654:-forms. And by 5649: 5629: 5602:holonomic basis 5597: 5588: 5580: 5520: 5519: 5512: 5502: 5480: 5464: 5459: 5456: 5455: 5448: 5443: 5440: 5439: 5432: 5422: 5400: 5384: 5370: 5361: 5360: 5037: 5032: 5031: 5002: 4985: 4984: 4947: 4919: 4903: 4902: 4865: 4845: 4827: 4817: 4815: 4814: 4811: 4786: 4780: 4770: 4764: 4755: 4751: 4727: 4682: 4631: 4541: 4518: 4452: 4416:207.161.101.195 4409: 4331: 4276: 4256: 4225: 4203: 4183: 4179: 4159: 4155: 4149: 4121: 4090: 4082: 4081: 4056: 4020: 4016: 4001: 3974: 3968: 3953: 3923: 3901: 3880: 3752:207.161.172.197 3745: 3723: 3722: 3679: 3678: 3648: 3647: 3604: 3603: 3575: 3574: 3531: 3530: 3502: 3501: 3458: 3457: 3429: 3428: 3385: 3384: 3334: 3280: 3205: 3115: 3025: 3003: 2984: 2961: 2936: 2935: 2916: 2915: 2890: 2877: 2865: 2852: 2820: 2816: 2803: 2798: 2794: 2766: 2762: 2749: 2744: 2740: 2723: 2718: 2714: 2693: 2689: 2669: 2664: 2660: 2639: 2635: 2613: 2604: 2600: 2594: 2593: 2514: 2513: 2491: 2468: 2452: 2426: 2398: 2381: 2341: 2278: 2277: 2226: 2194: 2181: 2169: 2156: 2124: 2120: 2107: 2102: 2098: 2070: 2066: 2053: 2048: 2044: 2027: 2022: 2018: 1997: 1993: 1973: 1968: 1964: 1943: 1939: 1916: 1912: 1904: 1898: 1897: 1818: 1817: 1787: 1774: 1762: 1749: 1717: 1713: 1700: 1695: 1691: 1663: 1659: 1646: 1641: 1637: 1620: 1615: 1611: 1590: 1586: 1566: 1561: 1557: 1536: 1532: 1510: 1501: 1497: 1491: 1490: 1411: 1410: 1389: 1343:205.200.233.195 1336: 1302: 1301: 1267: 1266: 1242: 1224: 1223: 1196: 1182: 1173: 1169: 1163: 1152: 1142: 1137: 1136: 1107: 1106: 1019: 1000: 969: 968: 925:205.200.233.195 918: 829: 802: 773: 772: 744: 743: 717: 716: 692: 665: 664: 627: 626: 607: 606: 587: 586: 560: 544: 530: 529: 396: 364:holonomic basis 291: 231: 228: 225: 222: 221: 199: 192: 172: 166: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 6596: 6594: 6586: 6585: 6580: 6575: 6570: 6565: 6560: 6555: 6545: 6544: 6496: 6493: 6492: 6491: 6490: 6489: 6488: 6487: 6486: 6485: 6484: 6483: 6482: 6481: 6480: 6479: 6478: 6477: 6476: 6475: 6447:tensor density 6416:tensor density 6395: 6394: 6393: 6392: 6391: 6390: 6389: 6388: 6387: 6386: 6385: 6384: 6359: 6358: 6357: 6356: 6355: 6354: 6353: 6352: 6351: 6350: 6349: 6348: 6322: 6280: 6279: 6278: 6267:Sławomir Biały 6224: 6221: 6220: 6219: 6206: 6202: 6198: 6193: 6190: 6187: 6184: 6180: 6174: 6171: 6168: 6165: 6161: 6155: 6152: 6147: 6144: 6130: 6127: 6120: 6119: 6118: 6105: 6101: 6097: 6094: 6091: 6088: 6083: 6079: 6075: 6072: 6067: 6063: 6059: 6054: 6051: 6048: 6026: 6025: 6024: 6023: 5996: 5995: 5994: 5993: 5984: 5983: 5982: 5957: 5956: 5955: 5930: 5929: 5928: 5927: 5926: 5925: 5924: 5923: 5922: 5921: 5920: 5919: 5918: 5917: 5916: 5915: 5914: 5913: 5912: 5911: 5910: 5909: 5908: 5907: 5906: 5905: 5904: 5903: 5902: 5901: 5900: 5899: 5898: 5897: 5815: 5814: 5813: 5765: 5761:Tensor density 5746: 5741:-density with 5646: 5622: 5605: 5594: 5587:-forms" and " 5549: 5535: 5534: 5533: 5513: 5509: 5505: 5501: 5498: 5495: 5492: 5487: 5483: 5479: 5476: 5471: 5467: 5463: 5460: 5458: 5457: 5449: 5447: 5444: 5442: 5441: 5433: 5429: 5425: 5421: 5418: 5415: 5412: 5407: 5403: 5399: 5396: 5391: 5387: 5383: 5380: 5377: 5374: 5371: 5369: 5368: 5355: 5338: 5337: 5336: 5335: 5334: 5333: 5332: 5331: 5330: 5329: 5328: 5327: 5306: 5305: 5304: 5303: 5302: 5301: 5300: 5299: 5298: 5297: 5274: 5273: 5272: 5271: 5270: 5269: 5268: 5267: 5266: 5265: 5264: 5263: 5218: 5217: 5216: 5215: 5214: 5213: 5212: 5211: 5210: 5209: 5165: 5164: 5163: 5162: 5161: 5160: 5159: 5158: 5138: 5137: 5136: 5135: 5134: 5133: 5109: 5108: 5107: 5106: 5084: 5051: 5046: 5041: 5016: 5011: 5006: 4997: 4994: 4978: 4977: 4966: 4961: 4956: 4951: 4942: 4939: 4934: 4931: 4925: 4922: 4918: 4913: 4910: 4887: 4886: 4885: 4884: 4864: 4861: 4860: 4859: 4858: 4857: 4846:Sławomir Biały 4838: 4837: 4750: 4747: 4746: 4745: 4723: 4722: 4721: 4720: 4719: 4718: 4699: 4698: 4697: 4696: 4695: 4694: 4683:Sławomir Biały 4672: 4671: 4670: 4669: 4644: 4643: 4632:Sławomir Biały 4621: 4620: 4568: 4567: 4566: 4565: 4564: 4563: 4540: 4537: 4525:206.45.181.160 4515: 4514: 4496: 4451: 4448: 4447: 4446: 4403: 4402: 4381: 4380: 4379: 4378: 4377: 4376: 4354: 4353: 4352: 4351: 4350: 4349: 4348: 4347: 4346: 4345: 4344: 4343: 4332:Sławomir Biały 4295: 4294: 4293: 4292: 4291: 4290: 4289: 4288: 4277:Sławomir Biały 4265: 4264: 4263: 4262: 4261: 4260: 4244: 4232: 4228: 4224: 4221: 4218: 4215: 4210: 4206: 4202: 4198: 4190: 4186: 4182: 4175: 4168: 4165: 4158: 4152: 4148: 4145: 4142: 4137: 4130: 4127: 4120: 4117: 4114: 4111: 4106: 4099: 4096: 4089: 4073: 4072: 4071: 4070: 4069: 4068: 4057:Sławomir Biały 4035: 4034: 4033: 4032: 3943: 3942: 3941: 3940: 3914: 3913: 3902:Sławomir Biały 3829: 3828: 3809: 3797: 3796: 3795: 3733: 3730: 3710: 3707: 3704: 3701: 3698: 3695: 3692: 3689: 3686: 3674: 3673: 3672: 3671: 3670: 3669: 3658: 3655: 3635: 3632: 3629: 3626: 3623: 3620: 3617: 3614: 3611: 3596: 3585: 3582: 3562: 3559: 3556: 3553: 3550: 3547: 3544: 3541: 3538: 3523: 3512: 3509: 3489: 3486: 3483: 3480: 3477: 3474: 3471: 3468: 3465: 3450: 3439: 3436: 3416: 3413: 3410: 3407: 3404: 3401: 3398: 3395: 3392: 3368: 3367: 3357: 3341:206.45.181.185 3319: 3318: 3303: 3287:206.45.181.185 3262: 3261: 3260: 3259: 3258: 3257: 3256: 3255: 3254: 3253: 3252: 3251: 3224: 3206:Sławomir Biały 3191: 3190: 3189: 3188: 3187: 3186: 3185: 3184: 3183: 3182: 3179:with a density 3161: 3160: 3159: 3158: 3157: 3156: 3155: 3154: 3128: 3127: 3126: 3125: 3124: 3123: 3122: 3121: 3078: 3077: 3076: 3075: 3074: 3073: 3072: 3071: 3042: 3041: 3040: 3039: 3038: 3037: 3026:Sławomir Biały 2997: 2996: 2985:Sławomir Biały 2968:207.161.69.254 2946: 2943: 2923: 2897: 2893: 2889: 2884: 2880: 2872: 2868: 2864: 2859: 2855: 2848: 2836: 2832: 2828: 2823: 2819: 2810: 2806: 2801: 2797: 2791: 2782: 2778: 2774: 2769: 2765: 2756: 2752: 2747: 2743: 2730: 2726: 2721: 2717: 2709: 2705: 2701: 2696: 2692: 2686: 2676: 2672: 2667: 2663: 2655: 2651: 2647: 2642: 2638: 2630: 2625: 2619: 2616: 2610: 2607: 2603: 2597: 2592: 2586: 2582: 2578: 2574: 2570: 2566: 2562: 2555: 2551: 2547: 2543: 2539: 2535: 2531: 2524: 2504: 2503: 2492:Sławomir Biały 2475:142.132.71.246 2465: 2464: 2453:Sławomir Biały 2443: 2442: 2441: 2440: 2439: 2438: 2427:Sławomir Biały 2401:Sławomir Biały 2382:Sławomir Biały 2370: 2369: 2358: 2353: 2350: 2345: 2340: 2337: 2334: 2330: 2326: 2323: 2320: 2317: 2313: 2309: 2306: 2303: 2300: 2297: 2294: 2291: 2288: 2285: 2266: 2265: 2233:206.45.181.160 2201: 2197: 2193: 2188: 2184: 2176: 2172: 2168: 2163: 2159: 2152: 2140: 2136: 2132: 2127: 2123: 2114: 2110: 2105: 2101: 2095: 2086: 2082: 2078: 2073: 2069: 2060: 2056: 2051: 2047: 2034: 2030: 2025: 2021: 2013: 2009: 2005: 2000: 1996: 1990: 1980: 1976: 1971: 1967: 1959: 1955: 1951: 1946: 1942: 1934: 1929: 1922: 1919: 1915: 1910: 1907: 1901: 1896: 1890: 1886: 1882: 1878: 1874: 1870: 1866: 1859: 1855: 1851: 1847: 1843: 1839: 1835: 1828: 1794: 1790: 1786: 1781: 1777: 1769: 1765: 1761: 1756: 1752: 1745: 1733: 1729: 1725: 1720: 1716: 1707: 1703: 1698: 1694: 1688: 1679: 1675: 1671: 1666: 1662: 1653: 1649: 1644: 1640: 1627: 1623: 1618: 1614: 1606: 1602: 1598: 1593: 1589: 1583: 1573: 1569: 1564: 1560: 1552: 1548: 1544: 1539: 1535: 1527: 1522: 1516: 1513: 1507: 1504: 1500: 1494: 1489: 1483: 1479: 1475: 1471: 1467: 1463: 1459: 1452: 1448: 1444: 1440: 1436: 1432: 1428: 1421: 1373: 1372: 1321: 1318: 1315: 1312: 1309: 1289: 1286: 1283: 1280: 1277: 1274: 1254: 1249: 1245: 1239: 1236: 1233: 1208: 1203: 1199: 1194: 1188: 1185: 1179: 1176: 1172: 1166: 1162: 1158: 1155: 1149: 1145: 1122: 1119: 1116: 1100: 1099: 1098: 1097: 1096: 1095: 1069: 1060:So density is 1058: 1057: 1056: 1042: 1039: 1036: 1033: 1030: 1027: 1022: 1017: 1014: 1011: 1008: 1003: 997: 994: 991: 988: 985: 982: 979: 976: 959: 953: 952: 899: 898: 897: 896: 883: 882: 881: 874: 868: 836:206.45.187.125 814: 809: 805: 798: 795: 791: 786: 783: 780: 760: 757: 754: 751: 729: 726: 704: 699: 695: 689: 686: 681: 678: 675: 672: 642: 639: 636: 614: 594: 572: 567: 563: 559: 556: 551: 547: 543: 540: 537: 509: 508: 507: 506: 505: 504: 503: 502: 481: 480: 479: 478: 477: 476: 457: 456: 455: 454: 441: 440: 426: 395: 392: 379: 378: 367: 359: 290: 287: 286: 285: 277: 276: 273: 272: 269: 268: 259: 256: 255: 248:Low-importance 244: 238: 237: 235: 218:the discussion 205: 204: 201:Physics portal 188: 176: 175: 173:Low‑importance 161: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 6595: 6584: 6581: 6579: 6576: 6574: 6571: 6569: 6566: 6564: 6561: 6559: 6556: 6554: 6551: 6550: 6548: 6541: 6540: 6536: 6532: 6528: 6524: 6520: 6516: 6513:Note: Merely 6511: 6509: 6504: 6502: 6494: 6474: 6471: 6467: 6466: 6465: 6461: 6457: 6452: 6448: 6436: 6432: 6431: 6430: 6427: 6423: 6422: 6417: 6413: 6409: 6408: 6407: 6406: 6405: 6404: 6403: 6402: 6401: 6400: 6399: 6398: 6397: 6396: 6383: 6379: 6375: 6371: 6370: 6369: 6368: 6367: 6366: 6365: 6364: 6363: 6362: 6361: 6360: 6344: 6338: 6333: 6327: 6323: 6321: 6317: 6313: 6308: 6307: 6302: 6301: 6296: 6295: 6290: 6289: 6288: 6285: 6281: 6277: 6273: 6269: 6263: 6259: 6255: 6254: 6253: 6252: 6251: 6247: 6243: 6239: 6238: 6237: 6233: 6229: 6225: 6222: 6204: 6200: 6196: 6191: 6188: 6185: 6182: 6178: 6172: 6169: 6166: 6163: 6159: 6153: 6150: 6145: 6142: 6135: 6134: 6128: 6125: 6121: 6103: 6099: 6095: 6092: 6089: 6086: 6081: 6077: 6073: 6070: 6065: 6061: 6057: 6052: 6049: 6039: 6038: 6036: 6032: 6031: 6030: 6029: 6028: 6027: 6022: 6018: 6014: 6009: 6008: 6007: 6006: 6005: 6004: 6001: 5990: 5989: 5988: 5985: 5980: 5976: 5972: 5968: 5963: 5962: 5961: 5958: 5952: 5948: 5944: 5943: 5942: 5939: 5938: 5937: 5935: 5896: 5892: 5888: 5884: 5883: 5882: 5879: 5875: 5871: 5867: 5863: 5859: 5858: 5857: 5853: 5849: 5844: 5843: 5842: 5839: 5835: 5830: 5829: 5828: 5824: 5820: 5816: 5812: 5808: 5807: 5805: 5804: 5803: 5800: 5796: 5792: 5788: 5784: 5780: 5776: 5772: 5771: 5766: 5763: 5762: 5757: 5756: 5751: 5747: 5744: 5740: 5736: 5732: 5731: 5726: 5725: 5724: 5720: 5716: 5707: 5706: 5705: 5702: 5698: 5694: 5690: 5686: 5682: 5673: 5672: 5671: 5667: 5663: 5658: 5652: 5647: 5645: 5641: 5637: 5632: 5627: 5623: 5620: 5616: 5615: 5614: 5611: 5606: 5603: 5595: 5591: 5586: 5578: 5577: 5576: 5572: 5568: 5564: 5563: 5558: 5554: 5550: 5548: 5544: 5540: 5536: 5507: 5503: 5499: 5496: 5493: 5490: 5485: 5481: 5477: 5474: 5469: 5465: 5461: 5445: 5427: 5423: 5419: 5416: 5413: 5410: 5405: 5401: 5397: 5394: 5389: 5385: 5381: 5378: 5375: 5359: 5358: 5356: 5352: 5351: 5350: 5349: 5348: 5347: 5346: 5345: 5344: 5343: 5342: 5341: 5340: 5339: 5326: 5323: 5318: 5317: 5316: 5315: 5314: 5313: 5312: 5311: 5310: 5309: 5308: 5307: 5296: 5292: 5288: 5284: 5283: 5282: 5281: 5280: 5279: 5278: 5277: 5276: 5275: 5262: 5259: 5255: 5250: 5247: 5243: 5239: 5235: 5230: 5229: 5228: 5227: 5226: 5225: 5224: 5223: 5222: 5221: 5220: 5219: 5208: 5204: 5200: 5196: 5192: 5188: 5184: 5180: 5175: 5174: 5173: 5172: 5171: 5170: 5169: 5168: 5167: 5166: 5157: 5154: 5150: 5146: 5145: 5144: 5143: 5142: 5141: 5140: 5139: 5132: 5128: 5124: 5119: 5115: 5114: 5113: 5112: 5111: 5110: 5105: 5102: 5097: 5096: 5095: 5092: 5089: 5085: 5081: 5080: 5079: 5078: 5075: 5069: 5068: 5065: 5049: 5044: 5030: 5014: 5009: 4995: 4992: 4983: 4964: 4959: 4954: 4940: 4937: 4932: 4929: 4923: 4920: 4916: 4911: 4908: 4901: 4900: 4899: 4897: 4892: 4883: 4880: 4877: 4872: 4871: 4870: 4869: 4868: 4862: 4856: 4852: 4848: 4842: 4841: 4840: 4839: 4836: 4833: 4823: 4809: 4808: 4807: 4806: 4802: 4798: 4793: 4790: 4783: 4777: 4774: 4767: 4761: 4758: 4748: 4744: 4740: 4736: 4730: 4725: 4724: 4717: 4713: 4709: 4705: 4704: 4703: 4702: 4701: 4700: 4693: 4689: 4685: 4678: 4677: 4676: 4675: 4674: 4673: 4668: 4665: 4661: 4657: 4656:newer edition 4653: 4648: 4647: 4646: 4645: 4642: 4638: 4634: 4628: 4627:on the bundle 4623: 4622: 4619: 4615: 4611: 4607: 4606: 4605: 4604: 4601: 4597: 4593: 4590: 4586: 4582: 4577: 4573: 4562: 4558: 4554: 4550: 4549: 4548: 4547: 4546: 4545: 4544: 4536: 4534: 4530: 4526: 4522: 4513: 4509: 4505: 4502: 4501: 4497: 4495: 4492: 4491: 4490: 4487: 4484: 4483: 4479: 4475: 4469: 4468: 4464: 4460: 4455: 4449: 4445: 4441: 4437: 4433: 4429: 4428: 4427: 4425: 4421: 4417: 4413: 4406: 4401: 4397: 4393: 4389: 4388: 4387: 4384: 4375: 4371: 4367: 4363: 4362: 4361: 4360: 4359: 4358: 4357: 4342: 4338: 4334: 4327: 4326: 4325: 4321: 4317: 4313: 4309: 4305: 4304: 4303: 4302: 4301: 4300: 4299: 4298: 4297: 4296: 4287: 4283: 4279: 4273: 4272: 4271: 4270: 4269: 4268: 4267: 4266: 4242: 4230: 4226: 4222: 4219: 4216: 4213: 4208: 4204: 4200: 4196: 4188: 4184: 4173: 4163: 4150: 4146: 4143: 4140: 4135: 4125: 4118: 4115: 4112: 4109: 4104: 4094: 4087: 4079: 4078: 4077: 4076: 4075: 4074: 4067: 4063: 4059: 4053: 4049: 4045: 4041: 4040: 4039: 4038: 4037: 4036: 4031: 4028: 4023: 4013: 4009: 4005: 3999: 3993: 3989: 3985: 3981: 3977: 3971: 3965: 3961: 3957: 3951: 3947: 3946: 3945: 3944: 3939: 3935: 3931: 3918: 3917: 3916: 3915: 3912: 3908: 3904: 3898: 3897: 3896: 3895: 3891: 3887: 3875: 3873: 3867: 3865: 3861: 3857: 3851: 3847: 3844: 3843: 3839: 3835: 3827: 3823: 3819: 3814: 3810: 3807: 3802: 3798: 3794: 3791: 3788: 3783: 3782: 3781: 3778: 3774: 3769: 3765: 3764: 3763: 3761: 3757: 3753: 3749: 3731: 3728: 3705: 3702: 3699: 3696: 3693: 3690: 3687: 3656: 3653: 3630: 3627: 3624: 3621: 3618: 3615: 3612: 3601: 3597: 3583: 3580: 3557: 3554: 3551: 3548: 3545: 3542: 3539: 3528: 3524: 3510: 3507: 3484: 3481: 3478: 3475: 3472: 3469: 3466: 3455: 3451: 3437: 3434: 3411: 3408: 3405: 3402: 3399: 3396: 3393: 3382: 3378: 3377: 3376: 3375: 3374: 3373: 3372: 3366: 3363: 3358: 3354: 3353: 3352: 3350: 3346: 3342: 3338: 3331: 3327: 3323: 3317: 3313: 3309: 3304: 3300: 3299: 3298: 3296: 3292: 3288: 3284: 3277: 3274: 3270: 3266: 3250: 3246: 3242: 3237: 3233: 3229: 3225: 3222: 3218: 3217: 3216: 3212: 3208: 3201: 3200: 3199: 3198: 3197: 3196: 3195: 3194: 3193: 3192: 3180: 3176: 3171: 3170: 3169: 3168: 3167: 3166: 3165: 3164: 3163: 3162: 3153: 3149: 3145: 3140: 3136: 3135: 3134: 3133: 3132: 3131: 3130: 3129: 3112: 3111: 3106: 3102: 3098: 3094: 3090: 3086: 3085: 3084: 3083: 3082: 3081: 3080: 3079: 3070: 3067: 3063: 3059: 3055: 3050: 3049: 3048: 3047: 3046: 3045: 3044: 3043: 3036: 3032: 3028: 3021: 3020: 3019: 3015: 3011: 3006: 3001: 3000: 2999: 2998: 2995: 2991: 2987: 2981: 2980: 2979: 2977: 2973: 2969: 2965: 2958: 2944: 2941: 2921: 2912: 2895: 2891: 2887: 2882: 2878: 2870: 2866: 2862: 2857: 2853: 2834: 2830: 2826: 2821: 2808: 2804: 2799: 2789: 2780: 2776: 2772: 2767: 2754: 2750: 2745: 2728: 2724: 2719: 2707: 2703: 2699: 2694: 2684: 2674: 2670: 2665: 2653: 2649: 2645: 2640: 2628: 2623: 2617: 2608: 2605: 2595: 2590: 2584: 2580: 2576: 2572: 2568: 2564: 2560: 2553: 2549: 2545: 2541: 2537: 2533: 2529: 2511: 2507: 2506:To JRSpriggs 2502: 2498: 2494: 2488: 2487: 2486: 2484: 2480: 2476: 2472: 2463: 2459: 2455: 2449: 2445: 2444: 2437: 2433: 2429: 2423: 2422: 2421: 2417: 2413: 2408: 2402: 2397: 2396: 2395: 2394: 2393: 2392: 2388: 2384: 2378: 2377:multiplied by 2373: 2356: 2351: 2348: 2335: 2328: 2324: 2304: 2298: 2295: 2289: 2283: 2276: 2275: 2274: 2271: 2264: 2260: 2256: 2252: 2251: 2246: 2245: 2244: 2242: 2238: 2234: 2230: 2223: 2219: 2216: 2199: 2195: 2191: 2186: 2182: 2174: 2170: 2166: 2161: 2157: 2138: 2134: 2130: 2125: 2112: 2108: 2103: 2093: 2084: 2080: 2076: 2071: 2058: 2054: 2049: 2032: 2028: 2023: 2011: 2007: 2003: 1998: 1988: 1978: 1974: 1969: 1957: 1953: 1949: 1944: 1932: 1927: 1920: 1917: 1908: 1899: 1894: 1888: 1884: 1880: 1876: 1872: 1868: 1864: 1857: 1853: 1849: 1845: 1841: 1837: 1833: 1815: 1812: 1809: 1792: 1788: 1784: 1779: 1775: 1767: 1763: 1759: 1754: 1750: 1731: 1727: 1723: 1718: 1705: 1701: 1696: 1686: 1677: 1673: 1669: 1664: 1651: 1647: 1642: 1625: 1621: 1616: 1604: 1600: 1596: 1591: 1581: 1571: 1567: 1562: 1550: 1546: 1542: 1537: 1525: 1520: 1514: 1505: 1502: 1492: 1487: 1481: 1477: 1473: 1469: 1465: 1461: 1457: 1450: 1446: 1442: 1438: 1434: 1430: 1426: 1407: 1405: 1401: 1397: 1393: 1386: 1383: 1379: 1376: 1375:To JPSpriggs 1371: 1368: 1365: 1361: 1356: 1355: 1354: 1352: 1348: 1344: 1340: 1333: 1316: 1313: 1310: 1287: 1284: 1281: 1278: 1275: 1272: 1252: 1247: 1243: 1234: 1220: 1206: 1201: 1197: 1192: 1186: 1177: 1174: 1164: 1160: 1156: 1153: 1147: 1143: 1117: 1103: 1102:To JRSpriggs 1094: 1091: 1088: 1084: 1083: 1082: 1078: 1074: 1070: 1067: 1063: 1059: 1040: 1037: 1034: 1031: 1028: 1025: 1020: 1015: 1012: 1009: 1006: 1001: 995: 992: 989: 986: 983: 980: 977: 974: 967: 966: 964: 960: 958:To anonymous: 957: 956: 955: 954: 951: 947: 943: 938: 937: 936: 934: 930: 926: 922: 915: 914: 908: 905: 902: 895: 891: 887: 884: 879: 875: 873: 869: 866: 865: 863: 862: 861: 857: 853: 849: 848: 847: 845: 841: 837: 833: 826: 812: 807: 803: 796: 793: 789: 784: 781: 778: 758: 755: 752: 749: 727: 724: 702: 697: 693: 687: 684: 679: 676: 673: 670: 660: 658: 637: 612: 592: 583: 570: 565: 561: 557: 549: 545: 538: 535: 527: 525: 520: 518: 513: 501: 497: 493: 489: 488: 487: 486: 485: 484: 483: 482: 475: 471: 467: 463: 462: 461: 460: 459: 458: 453: 450: 445: 444: 443: 442: 439: 435: 431: 427: 424: 423: 422: 421: 417: 413: 408: 403: 399: 393: 391: 390: 387: 384: 377: 374: 373: 368: 365: 360: 357: 353: 349: 345: 340: 336: 335: 330: 329: 328: 327: 323: 319: 315: 310: 309: 305: 301: 296: 288: 284: 281: 280: 266: 265: 258: 257: 253: 249: 243: 240: 239: 236: 219: 215: 211: 210: 202: 196: 191: 189: 186: 182: 181: 177: 170: 165: 162: 159: 155: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 6526: 6522: 6518: 6514: 6512: 6507: 6505: 6500: 6498: 6450: 6446: 6434: 6419: 6415: 6411: 6342: 6336: 6331: 6305: 6304: 6299: 6298: 6293: 6292: 6261: 6257: 6123: 6034: 5997: 5992:not defined. 5986: 5978: 5974: 5970: 5966: 5959: 5950: 5946: 5940: 5933: 5931: 5873: 5869: 5865: 5861: 5833: 5809: 5794: 5790: 5786: 5782: 5778: 5774: 5768: 5759: 5753: 5749: 5742: 5738: 5734: 5728: 5696: 5692: 5688: 5687:-forms, 0 ≤ 5684: 5680: 5656: 5650: 5630: 5625: 5618: 5589: 5584: 5561: 5560: 5556: 5537:Yay or nay? 5253: 5248: 5245: 5241: 5237: 5194: 5190: 5186: 5182: 5178: 5148: 5070: 4979: 4890: 4888: 4866: 4821: 4794: 4788: 4781: 4778: 4772: 4765: 4762: 4756: 4752: 4728: 4659: 4626: 4595: 4591: 4588: 4584: 4583:instead of | 4580: 4575: 4571: 4569: 4542: 4519:— Preceding 4516: 4499: 4498: 4488: 4485: 4470: 4456: 4453: 4410:— Preceding 4407: 4404: 4385: 4382: 4355: 4311: 4051: 4047: 4043: 4021: 4011: 4007: 4003: 3997: 3991: 3987: 3983: 3979: 3975: 3969: 3963: 3959: 3955: 3876: 3871: 3868: 3863: 3859: 3855: 3852: 3848: 3845: 3830: 3812: 3805: 3800: 3746:— Preceding 3675: 3599: 3526: 3453: 3380: 3369: 3335:— Preceding 3332: 3328: 3324: 3320: 3281:— Preceding 3278: 3275: 3271: 3267: 3263: 3235: 3231: 3227: 3220: 3178: 3174: 3138: 3109: 3108: 3104: 3100: 3096: 3092: 3088: 3061: 3058:Ricci scalar 3004: 2962:— Preceding 2959: 2913: 2512: 2508: 2505: 2469:— Preceding 2466: 2376: 2374: 2371: 2267: 2248: 2227:— Preceding 2224: 2220: 2217: 1816: 1813: 1810: 1408: 1390:— Preceding 1387: 1384: 1380: 1377: 1374: 1359: 1337:— Preceding 1334: 1221: 1104: 1101: 1066:proportional 1065: 1061: 919:— Preceding 916: 909: 906: 903: 900: 877: 871: 830:— Preceding 827: 661: 584: 528: 521: 514: 510: 406: 404: 400: 397: 380: 371: 355: 351: 338: 333: 332: 313: 311: 294: 292: 262: 247: 207: 137:Low-priority 136: 96: 62:Low‑priority 40:WikiProjects 5973:as meaning 5954:components. 5793:-forms are 5197:variables. 4982:volume form 4239:Lee (12.13) 3872:religiosity 3454:Gravitation 3097:appropriate 2960:Good day! 2255:Steelpillow 1396:206.45.29.3 112:Mathematics 103:mathematics 59:Mathematics 30:Start-class 6547:Categories 6527:what it is 6501:Definition 5634:of above? 5553:Hodge dual 5234:Hodge dual 4486:To Yohan 3813:mentioning 3801:completely 3110:horrendous 356:the tensor 169:Relativity 6435:identical 6347:compute). 6306:densities 6228:JRSpriggs 5710:1-density 5252:from the 5199:JRSpriggs 5123:JRSpriggs 4896:elsewhere 4654:(or this 4383:Hi Yohan 3602:uses the 3529:uses the 3456:uses the 3383:uses the 3308:JRSpriggs 2250:etiquette 1073:JRSpriggs 1062:inversely 412:JRSpriggs 5967:a scalar 5735:anything 4521:unsigned 4412:unsigned 3748:unsigned 3337:unsigned 3283:unsigned 3232:ought to 3175:does not 3139:motivate 2964:unsigned 2471:unsigned 2229:unsigned 1392:unsigned 1339:unsigned 1332:tensor. 921:unsigned 832:unsigned 339:function 6470:Quondum 6456:YohanN7 6426:Quondum 6412:density 6374:YohanN7 6326:density 6312:YohanN7 6294:Tensors 6284:Quondum 6242:YohanN7 6013:YohanN7 6000:Quondum 5949:, with 5887:YohanN7 5878:Quondum 5848:YohanN7 5838:Quondum 5819:YohanN7 5799:Quondum 5715:YohanN7 5701:Quondum 5662:YohanN7 5636:YohanN7 5610:Quondum 5567:YohanN7 5539:YohanN7 5322:Quondum 5287:YohanN7 5258:Quondum 5153:Quondum 5101:Quondum 5074:Quondum 5064:Quondum 4832:Quondum 4797:YohanN7 4735:YohanN7 4708:YohanN7 4664:Quondum 4610:YohanN7 4600:Quondum 4553:YohanN7 4504:YohanN7 4474:YohanN7 4459:YohanN7 4450:To Anon 4436:YohanN7 4405:Yohan 4392:YohanN7 4366:YohanN7 4316:YohanN7 4027:Quondum 3930:YohanN7 3886:YohanN7 3864:forward 3834:YohanN7 3818:YohanN7 3777:Quondum 3362:Quondum 3241:YohanN7 3144:YohanN7 3089:natural 3066:Quondum 3056:or the 3010:YohanN7 2412:YohanN7 963:density 942:YohanN7 886:YohanN7 852:YohanN7 492:YohanN7 466:YohanN7 449:Quondum 430:YohanN7 407:density 372:Quondum 318:YohanN7 300:YohanN7 250:on the 223:Physics 214:Physics 164:Physics 139:on the 6303:, and 5562:tensor 4312:tensor 4048:scalar 4019:, and 2448:Tensor 2407:Tensor 878:before 352:tensor 348:Tensor 36:scale. 6515:using 5626:being 5619:given 4771:√(−g) 3221:point 3101:wrong 3093:right 6535:talk 6523:only 6506:Can 6460:talk 6378:talk 6337:even 6316:talk 6272:talk 6246:talk 6232:talk 6205:1234 6131:αβγδ 6129:If ω 6017:talk 5969:and 5891:talk 5852:talk 5823:talk 5719:talk 5666:talk 5640:talk 5571:talk 5543:talk 5291:talk 5203:talk 5193:and 5127:talk 4851:talk 4801:talk 4739:talk 4712:talk 4688:talk 4652:link 4637:talk 4629:N. 4614:talk 4557:talk 4529:talk 4508:talk 4478:talk 4463:talk 4440:talk 4420:talk 4396:talk 4370:talk 4356:--- 4337:talk 4320:talk 4282:talk 4062:talk 3934:talk 3907:talk 3890:talk 3838:talk 3822:talk 3756:talk 3345:talk 3312:talk 3291:talk 3245:talk 3211:talk 3148:talk 3031:talk 3014:talk 2990:talk 2972:talk 2497:talk 2479:talk 2458:talk 2432:talk 2416:talk 2387:talk 2259:Talk 2237:talk 1400:talk 1347:talk 1077:talk 946:talk 929:talk 890:talk 856:talk 840:talk 496:talk 470:talk 434:talk 416:talk 322:talk 304:talk 6519:not 5699:. — 5242:not 4891:any 4776:. 4576:not 4144:det 4010:∧ d 4006:∧ d 3998:not 3990:∧ d 3986:∧ d 3962:∧ d 3958:∧ d 3856:and 3236:its 2316:det 314:may 295:not 242:Low 131:Low 6549:: 6537:) 6462:) 6414:≠ 6380:) 6318:) 6297:, 6274:) 6248:) 6234:) 6201:ω 6192:δ 6189:γ 6186:β 6183:α 6179:ω 6173:δ 6170:γ 6167:β 6164:α 6160:ϵ 6154:24 6093:∧ 6090:⋯ 6087:∧ 6071:∧ 6047:Ω 6019:) 5893:) 5854:) 5825:) 5721:) 5691:≤ 5668:) 5642:) 5598:±1 5573:) 5545:) 5497:∧ 5494:⋯ 5491:∧ 5475:∧ 5417:∧ 5414:⋯ 5411:∧ 5395:∧ 5373:Ω 5293:) 5205:) 5189:, 5185:, 5181:, 5129:) 4993:− 4938:− 4930:∫ 4924:κ 4853:) 4822:dx 4818:−g 4803:) 4773:dx 4766:dx 4757:dx 4741:) 4729:dx 4714:) 4690:) 4639:) 4616:) 4589:is 4559:) 4531:) 4510:) 4480:) 4465:) 4442:) 4422:) 4398:) 4372:) 4339:) 4322:) 4284:) 4257:−1 4220:∧ 4217:… 4214:∧ 4181:∂ 4167:¯ 4157:∂ 4147:⁡ 4129:¯ 4116:∧ 4113:… 4110:∧ 4098:¯ 4064:) 4052:do 4017:±1 3978:= 3936:) 3909:) 3892:) 3860:is 3840:) 3824:) 3758:) 3729:− 3688:− 3631:− 3625:− 3619:− 3581:− 3540:− 3467:− 3435:− 3394:− 3347:) 3314:) 3293:) 3247:) 3228:is 3213:) 3150:) 3033:) 3016:) 2992:) 2974:) 2942:− 2892:μ 2888:⋯ 2879:μ 2867:ν 2863:⋯ 2854:ν 2827:ν 2818:∂ 2805:ν 2796:∂ 2790:⋯ 2773:ν 2764:∂ 2751:ν 2742:∂ 2725:μ 2716:∂ 2700:μ 2691:∂ 2685:⋯ 2671:μ 2662:∂ 2646:μ 2637:∂ 2615:∂ 2602:∂ 2577:μ 2573:⋯ 2561:μ 2546:ν 2542:⋯ 2530:ν 2499:) 2481:) 2460:) 2434:) 2418:) 2389:) 2349:− 2333:∂ 2322:∂ 2261:) 2239:) 2196:μ 2192:⋯ 2183:μ 2171:ν 2167:⋯ 2158:ν 2131:ν 2122:∂ 2109:ν 2100:∂ 2094:⋯ 2077:ν 2068:∂ 2055:ν 2046:∂ 2029:μ 2020:∂ 2004:μ 1995:∂ 1989:⋯ 1975:μ 1966:∂ 1950:μ 1941:∂ 1914:∂ 1906:∂ 1881:μ 1877:⋯ 1865:μ 1850:ν 1846:⋯ 1834:ν 1789:μ 1785:⋯ 1776:μ 1764:ν 1760:⋯ 1751:ν 1724:ν 1715:∂ 1702:ν 1693:∂ 1687:⋯ 1670:ν 1661:∂ 1648:ν 1639:∂ 1622:μ 1613:∂ 1597:μ 1588:∂ 1582:⋯ 1568:μ 1559:∂ 1543:μ 1534:∂ 1512:∂ 1499:∂ 1474:μ 1470:⋯ 1458:μ 1443:ν 1439:⋯ 1427:ν 1402:) 1349:) 1273:− 1184:∂ 1171:∂ 1079:) 948:) 931:) 892:) 858:) 842:) 794:− 785:∫ 782:∝ 725:− 685:− 677:∫ 674:∝ 550:μ 539:ϕ 536:∫ 498:) 472:) 436:) 418:) 334:is 324:) 306:) 167:: 6533:( 6458:( 6451:f 6376:( 6343:x 6341:d 6332:x 6330:d 6314:( 6270:( 6262:n 6258:n 6244:( 6230:( 6197:= 6151:1 6146:= 6143:f 6124:n 6104:n 6100:x 6096:d 6082:2 6078:x 6074:d 6066:1 6062:x 6058:d 6053:f 6050:= 6035:f 6015:( 5998:— 5979:n 5951:n 5947:n 5889:( 5874:p 5870:p 5866:p 5862:p 5850:( 5834:p 5832:" 5821:( 5795:p 5791:p 5787:p 5783:n 5779:p 5775:p 5743:w 5739:w 5717:( 5697:k 5693:n 5689:k 5685:k 5681:n 5677:0 5664:( 5657:w 5651:n 5638:( 5631:f 5590:w 5585:n 5581:Ω 5569:( 5541:( 5508:n 5504:x 5500:d 5486:2 5482:x 5478:d 5470:1 5466:x 5462:d 5446:f 5428:n 5424:x 5420:d 5406:2 5402:x 5398:d 5390:1 5386:x 5382:d 5379:f 5376:= 5289:( 5254:n 5238:n 5201:( 5195:M 5191:P 5187:H 5183:D 5179:J 5149:n 5125:( 5099:— 5091:R 5088:T 5050:x 5045:4 5040:d 5015:x 5010:4 5005:d 4996:g 4965:, 4960:x 4955:4 4950:d 4941:g 4933:R 4921:2 4917:1 4912:= 4909:S 4879:R 4876:T 4849:( 4828:0 4816:√ 4812:0 4799:( 4789:N 4787:− 4782:N 4737:( 4710:( 4686:( 4660:n 4635:( 4612:( 4596:n 4592:R 4585:J 4581:J 4555:( 4527:( 4506:( 4476:( 4461:( 4438:( 4418:( 4394:( 4368:( 4335:( 4318:( 4280:( 4259:. 4243:, 4231:n 4227:x 4223:d 4209:1 4205:x 4201:d 4197:) 4189:i 4185:x 4174:j 4164:x 4151:( 4141:= 4136:n 4126:x 4119:d 4105:1 4095:x 4088:d 4060:( 4044:J 4022:v 4012:z 4008:y 4004:x 4002:d 3992:z 3988:y 3984:x 3982:d 3980:J 3976:v 3970:J 3964:z 3960:y 3956:x 3954:d 3932:( 3905:( 3888:( 3836:( 3820:( 3790:R 3787:T 3754:( 3732:2 3709:) 3706:+ 3703:, 3700:+ 3697:, 3694:+ 3691:, 3685:( 3657:2 3654:+ 3634:) 3628:, 3622:, 3616:, 3613:+ 3610:( 3584:2 3561:) 3558:+ 3555:, 3552:+ 3549:, 3546:+ 3543:, 3537:( 3511:2 3508:+ 3488:) 3485:+ 3482:, 3479:+ 3476:, 3473:+ 3470:, 3464:( 3438:2 3415:) 3412:+ 3409:, 3406:+ 3403:, 3400:+ 3397:, 3391:( 3343:( 3310:( 3289:( 3243:( 3209:( 3146:( 3120:) 3062:w 3029:( 3012:( 3005:R 2988:( 2970:( 2945:2 2922:g 2896:n 2883:1 2871:m 2858:1 2847:T 2835:′ 2831:m 2822:x 2809:m 2800:x 2781:′ 2777:1 2768:x 2755:1 2746:x 2729:n 2720:x 2708:′ 2704:n 2695:x 2675:1 2666:x 2654:′ 2650:1 2641:x 2629:w 2624:| 2618:x 2609:′ 2606:x 2596:| 2591:= 2585:′ 2581:n 2569:′ 2565:1 2554:′ 2550:m 2538:′ 2534:1 2523:T 2495:( 2477:( 2456:( 2430:( 2414:( 2403:: 2399:@ 2385:( 2357:. 2352:1 2344:| 2339:) 2336:x 2329:/ 2325:y 2319:( 2312:| 2308:) 2305:x 2302:( 2299:f 2296:= 2293:) 2290:y 2287:( 2284:f 2257:( 2235:( 2200:n 2187:1 2175:m 2162:1 2151:T 2139:′ 2135:m 2126:x 2113:m 2104:x 2085:′ 2081:1 2072:x 2059:1 2050:x 2033:n 2024:x 2012:′ 2008:n 1999:x 1979:1 1970:x 1958:′ 1954:1 1945:x 1933:w 1928:| 1921:′ 1918:x 1909:x 1900:| 1895:= 1889:′ 1885:n 1873:′ 1869:1 1858:′ 1854:m 1842:′ 1838:1 1827:T 1793:n 1780:1 1768:m 1755:1 1744:T 1732:′ 1728:m 1719:x 1706:m 1697:x 1678:′ 1674:1 1665:x 1652:1 1643:x 1626:n 1617:x 1605:′ 1601:n 1592:x 1572:1 1563:x 1551:′ 1547:1 1538:x 1526:w 1521:| 1515:x 1506:′ 1503:x 1493:| 1488:= 1482:′ 1478:n 1466:′ 1462:1 1451:′ 1447:m 1435:′ 1431:1 1420:T 1398:( 1367:R 1364:T 1345:( 1320:) 1317:0 1314:, 1311:0 1308:( 1288:0 1285:= 1282:1 1279:+ 1276:1 1253:x 1248:n 1244:d 1238:| 1235:g 1232:| 1207:x 1202:n 1198:d 1193:| 1187:x 1178:′ 1175:x 1165:| 1161:= 1157:′ 1154:x 1148:n 1144:d 1121:| 1118:g 1115:| 1090:R 1087:T 1075:( 1041:e 1038:m 1035:u 1032:l 1029:o 1026:V 1021:d 1016:s 1013:s 1010:a 1007:M 1002:d 996:= 993:y 990:t 987:i 984:s 981:n 978:e 975:D 944:( 927:( 888:( 854:( 838:( 813:x 808:4 804:d 797:g 790:R 779:S 759:0 756:≠ 753:2 750:+ 728:g 703:x 698:4 694:d 688:g 680:R 671:S 641:| 638:g 635:| 613:A 593:A 571:x 566:n 562:d 558:A 555:) 546:x 542:( 494:( 468:( 447:— 432:( 414:( 386:R 383:T 358:. 320:( 302:( 267:. 254:. 143:. 42::

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Index