This post introduces “densities”. There is a very nice book, available on arxiv, by Sean M. Carroll: “Lecture Notes on General Relativity“. There, on p. 52, Carroll writes “… we don’t like tensor densities, we like tensors.” Well, I do like tensor densities, I consider them important. They exist, and they have their independent life. While it is true that once you have a nondegenerate metric on your manifold, you can avoid using tensor densities. But what if you want to contemplate “pre-metric electrodynamics,” or “pre-metric General Relativity”? I see no reason for prohibiting such ideas. In fact, I very much like them. Therefore, here, we do like tensor densities.
We are discussing geometric objects. Geometric objects, and we are discussing only linear objects of the first order, are associated with linear representations of 
in vector spaces. So far we have discussed the defining representation (the natural action of on 
) and the contragradient representation. We have also discussed the “identity” representation, without specifying the space on which the representing transformations act. But, for the identity transformation, it does not really matter: the identity transformation of , on any 
, is a direct sum of identity transformations on 
so we can think that the identity transformation associates with each matrix 
simply number 
![\[\text{Id}(A)\xi = \xi\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-d7d015468f5bf383d26d3196198d6a30_l3.png "Rendered by QuickLaTeX.com")
for all 
.
This way we have defined, using transformation properties of their components, tensors (contravariant and covariant), and scalars. But there are two other important representations of on 
. They are used for defining even and odd tensors, and for defining tensor densities of different weights.
First, let us recall the general basic construction. We have introduced this construction for the group acting on the set of all bases in 
. But it works as well for any group 
that acts from the right on a set 
in a simply transitive (that is the action is transitive and free) way. We then form the Cartesian product 
Whenever we have a left action 
of on a set 
( representation of in 
), we have an equivalence relation in 
![\[ (b,x)\sim (b',x')\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-62381ee8f91eeda616460fc3798622d6_l3.png "Rendered by QuickLaTeX.com")
if and only if there exists 
such that 
, and 
Note: is the set of all bijective maps 
.
We then denote by 
the set of equivalence classes, and call it the space of geometrical objects of type .
Remark In physics, selecting 
can often be interpreted as selecting a “gauge”, 
is “another gauge”, while 
is interpreted as the corresponding “gauge transformation” (of some scalar, or vector-valued, wave function etc.). Usually we have a different at different points, so we deal we a “principal fibre bundle”, and we construct this way an “associated fiber bundle”. But here we are playing with just one point. Once we learn how to deal with just one point, having several, or a continuum, of points, is not a problem. Only when we want to examine what changes when we move from one point to another – we need extra work – we have to develop the concepts of “path-dependent parallel transport” and “covariant differentiation”.
Let us now return to the case of . There are two important one-dimensional representations of this group. They will be used to define even and odd tensors (or tensors and pseudo-tensors), and to define tensor densities.
The orientation line.
Here we take
![\[ \rho(A)\xi = \text{sgn}(\det(A))\xi,\, A\in GL(n),\,\xi\in \RR.\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-99b20f35d53203dfeda0b1b397686950_l3.png "Rendered by QuickLaTeX.com")
We have a linear representation of on 
(Why?) Combining this representation with the defining and contagradient representation we obtain new tensor transformation laws that involve sign of the determinant. They are called “odd” or “twisted”, or pseudo-” tensors (covariant, contravariant, mixed). Usually the “orientation line”, i.e. the space 
is forgotten, and only the “twisted” transformation law is written. Exceptions are publications using explicitly “orientation line bundle”. Moreover, different authors may use somewhat different terminology here. For instance Schouten [3, p. 12] instead of “odd” or “twisted” talks about “
-tensors”.
A curious reader may note that using here is, perhaps, unnecessary. It would be enough to take for the two-point set 
It can be done in this way, for sure, but we want stay on the category of general real vector spaces, and their tensor products, and also there is no harm in using the full rather than only two pints there, once we understand what it is for – wa want to distinguish between positive and negative numbers.
Tensor densities (or “relative tensors”, as some authors call them [1]).
The second important representation (or, better, a whole class of representations) of on us defined as follows: for eny real number 
let 
be defined as
![\[\rho_w(A)\xi=|\det(A)|^{-w}\xi,\, A\in GL(n),\,\xi\in\RR.\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-a8d0725eed6c4d0b35a20fc2840cae03_l3.png "Rendered by QuickLaTeX.com")
Geometric objects of type are called scalar densities of weight . We can then, as before, combine this transformation rule with the standard tensor transformation rules to obtain tensor densities (untwisted or twisted) of weight 
Some authors do not use the absolute value of the determinant in the definition of 
they use just the determinant [1],[2]. But then they require 
to be an integer. We use the absolute value, and we have an extra option of twisting, if needed.
Exercise 1. Verify that is indeed a representation of 
References
[1] Lovelock, D., Rund, H., Tensors, Differential Forms, and Variational Principles, Dover 1989.
[2] Synge, J.L, Schild, A., Tensor Calculus, Dover 1978.
[3] Schouten, J.A., Ricci Calculus, Springer 1954.
Afternotes:
21-12-25 16:20
Just while ago I had a chat with Perplexity AI concerning a related topic. In particular it tells you why you should never rely on AI:
[latexepage]
Ark, let me post here again about my own interpretation of gravity. I promise not to take advantage of your kindness anymore.
————————
A localization in a linear space $V$ of the algebra of linear vector fields $\mathcal{A}_{\nabla}(V)$ means constructing (using the Jacobian of coordinate transformations of the space $V$) a morphism
\begin{equation}
V\to \mathrm{Aut}(V)\to \mathrm{Aut}\mathcal{A}_{\nabla}(V)
\end{equation}
which leads to the localization of its Lie algebra $\mathcal{A}_{T}(V)$
\begin{equation}
V\to \mathrm{Aut}(V)\to \mathrm{Aut}\mathcal{A}_{T}(V)
\end{equation}
In turn, if $V$ is the Minkowski space $\mathcal{M}$, then the streamlines of the vector field dual to the differential 1-form of the time flow
\begin{equation}
\tau=\tau_{t}\mathrm{d}t + \tau_{x}\mathrm{d}x + \tau_{y}\mathrm{d}y + \tau_{z}\mathrm{d}z
\end{equation}
which form the time coordinate, and the coordinates of the foliation whose hypersurfaces are orthogonal to the vector field dual to the 1-form of the vacuum flow $\mathrm{d}t$, together form the curvilinear local-time coordinates of the Minkowski space $\mathcal{M}$, with the Jacobian of the transformation of curvilinear coordinates $\mathrm{J}$, where $\mathrm{d}t’=\tau\,,\mathrm{d}x’=\mathrm{d}x\,,\mathrm{d}y’=\mathrm{d}y\,,\mathrm{d}z’=\mathrm{d}z$, belongs to the morphism $\mathcal{M}\to \mathrm{Aut}(\mathcal{M})$. Note that the matrix of the metric tensor in curvilinear coordinates $\mathrm{g}$ is calculated using the formula
\begin{equation}
\mathrm{g}=\mathrm{J}^{T}\mathrm{J}\,\eta
\end{equation}
where $\eta$ is the matrix of the metric tensor of flat Minkowski space. Moreover, if the algebra $\mathrm{Aut}\mathcal{A}_{\nabla}(V)$ is isomorphic to $M(2,\mathbb{C})$, then we have a chain of mappings
\begin{equation}
\mathcal{M}\to\bigwedge^{1}T^{*}\mathcal{M} \to \mathrm{Aut}\, M(2,\mathbb{C})
\end{equation}
and if the algebra $\mathrm{Aut}\mathcal{A}_{\nabla}(V)$ is isomorphic to $M(4,\mathbb{C})$, that is, to the Clifford algebra $\mathrm{Cl}(\mathcal{M})$, into which the Minkowski space is embedded, then we have a chain of mappings
\begin{equation}
\mathcal{M}\to\bigwedge^{1}T^{*}\mathcal{M} \to \mathrm{Aut}\,\mathrm{Cl}(\mathcal{M})
\end{equation}
Thus, the differential 1-form of the time flow defines a localization in Minkowski space of an isomorphic $sl(2,\mathbb{C})$ (or $sl(4,\mathbb{C})$) Lie algebra of linear vector fields of the 4-dimensional (or 8-dimensional) real space. It is expected that the local Lie algebras generated by harmonic (minimal) 1-forms correspond to the Lie algebras of the Killing vector fields of a pseudo-Riemannian manifold with a metric that minimizes the Hilbert–Einstein action
\begin{equation}
\int \mathrm{R}\,\mathrm{d}V = \int \left(\tau^{*},\mathrm{d}t^{*}\right)\,\mathrm{d}t\wedge\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z
\end{equation}
where $\mathrm{R}\,\mathrm{d}V$ is the product of the scalar curvature and the differential volume element of the pseudo-Riemannian manifold, and the integrand $\left(\tau^{*},\mathrm{d}t^{*}\right)$ denotes the scalar product of the vector fields dual to the differentials $\tau$ and $\mathrm{d}t$. In other words, we have two different ways of describing gravity: internal (using the metric tensor $\mathrm{g}$, which obeys Einstein’s equation) and external (using the 1-form $\tau$, which obeys the harmonic condition $\Delta\tau=0$). Note also that the description of gravity using the minimal flow in Minkowski space returns us to historical attempts to justify gravity with ether.
Igor, you wrote [latexepage]. There should be no e between x and p.
Additionally I am not sure if the interpreter understands \begin{equation}. It understands $$ and \[ \]. If you want to number equations, you need to do it by hand like
\[ x=y \tag{(1)}\]
Оказывается, понимает и \begin{equation
Well, try again.
Ark, let me post here again about my own interpretation of gravity. I promise not to take advantage of your kindness anymore.
————————
A localization in a linear space
of the algebra of linear vector fields 
means constructing (using the Jacobian of coordinate transformations of the space 
) a morphism
(1)
which leads to the localization of its Lie algebra
(2)
In turn, if
is the Minkowski space 
, then the streamlines of the vector field dual to the differential 1-form of the time flow
(3)
which form the time coordinate, and the coordinates of the foliation whose hypersurfaces are orthogonal to the vector field dual to the 1-form of the vacuum flow
, together form the curvilinear local-time coordinates of the Minkowski space 
, with the Jacobian of the transformation of curvilinear coordinates 
, where 
, belongs to the morphism 
. Note that the matrix of the metric tensor in curvilinear coordinates 
is calculated using the formula
(4)
where
is the matrix of the metric tensor of flat Minkowski space. Moreover, if the algebra 
is isomorphic to 
, then we have a chain of mappings
(5)
and if the algebra
is isomorphic to 
, that is, to the Clifford algebra 
, into which the Minkowski space is embedded, then we have a chain of mappings
(6)
Thus, the differential 1-form of the time flow defines a localization in Minkowski space of an isomorphic
(or 
) Lie algebra of linear vector fields of the 4-dimensional (or 8-dimensional) real space. It is expected that the local Lie algebras generated by harmonic (minimal) 1-forms correspond to the Lie algebras of the Killing vector fields of a pseudo-Riemannian manifold with a metric that minimizes the Hilbert–Einstein action
(7)
where
is the product of the scalar curvature and the differential volume element of the pseudo-Riemannian manifold, and the integrand 
denotes the scalar product of the vector fields dual to the differentials 
and 
. In other words, we have two different ways of describing gravity: internal (using the metric tensor 
, which obeys Einstein’s equation) and external (using the 1-form 
, which obeys the harmonic condition 
). Note also that the description of gravity using the minimal flow in Minkowski space returns us to historical attempts to justify gravity with ether.
So, it works. I mean latex. I still do not see where your gravity is. In the standard formulation you can derive geodesic equations from general covariance of the action principle. What is exactly your idea is not clear to me from what you wrote. I do not understand what is your gravity? What you require from tau apart of it being a 1-form? How it relates to gravity?
The whole trick is in equation 7. Variation of the integral on the left leads to the Einstein equations, and variation of the integral on the right leads to zero Laplacian of the differential 1-form of the time flow.
R is undefined. Scalar curvature of what? Of the metric? Which metric? dV is undefined. If dV is dx dy dz dt , then the integral of RdV is not an ,invariant. Usually one is using sqrt{|det g|}dx dy dz dt to make the integral invariant.
The integral on the left pertains to a pseudo-Riemannian manifold, so its volume element is expressed in terms of the metric, just as you wrote. The scalar curvature is also expressed in terms of the manifold’s metric. But that’s not the point of the concept; the point is that the metric is generated by the flow of time. Also, I made a mistake there: before equation 4,
should be replaced with 
.
Igor, I realized I do not really understand. You wrote
But that would imply
Equality here means that instead of $dt’$, we take $\tau$. Not $\tau$, but $\tau/|\tau|$.
Equality here means that instead of
, we take 
. Not 
, but 
.
Igor, do I understand correctly that your J is the matrix
Then
Is that what you mean?
Ark, I’m not good at calculations, I could have messed up. Concepts are my thing.
You do not need to calculate. Just give me your explicit formula for your J. Then I will be able to do the calculations. Without explicit formula formula for J no meaningful discussion is possible. With the explicite formula – we can go somewhere. You said J is Jacobian, But now it is clear that it is not Jacobian. It is something “similar” to Jacobian, but not Jacobian. So: what is it? Write down just this one matrix.
If the Jacobian assumes that the variable $t’$ must necessarily be a function, then yes it is not a Jacobian.
What is it then? Can you down an explicit matrix for J? It is not a calculation. It is a definition. A thing. And you say you are good with things. Do it should be easy for you. You wrote a whole page of formulas before. Now write just one 4×4 matrix, please.
Хорошо, соглашусь с вами – это матрица похожая на якобиан.
@Igor
“соглашусь с вами – это матрица похожая на якобиан.”
Except that it is impossible to define it. Jacobian of coordinate transformations can be easilt defined, but your J is impossible to define. Like in potery. So you are only pretending doing math. In fact, you are writing math-looking poetry. Or, you are playing games with us? Or, perhaps, both?
To prove that I am wrong write an explicit expression for J in terms of 1-form tau.
In turn, if
is the Minkowski space 
, then the streamlines of the vector field dual to the differential 1-form of the time flow
(1)
which form the time integral lines, and the coordinates of the foliation whose hypersurfaces are orthogonal to the vector field dual to the 1-form of the vacuum flow
, together form a curvilinear local-time integral network of the Minkowski space 
, with the Jacobian-like transformation matrix of differentials 
, belongs to the morphism 
. Note that, as in curvilinear coordinates, the matrix of the metric tensor 
in a curvilinear local-time network is calculated by the formula
(2)
where
is the matrix of the metric tensor of flat Minkowski space.
Igor, you can’t keep yourself from teasing!
Why can’t you write J explicitly, in a matrix form? Just do it. Or is it forbidden by Law in your country? What penalty awaits those who write explicitly matrix expression for J? Five years? 25 years? Life sentence?
The initial stage of reconciling the two approaches will be considered using the example of a spherically symmetric metric with a singularity at a point mass of unit mass. Let
be an orthogonal matrix (
) of order three such that 
, where 
. Then the Jacobian corresponding to the quadratic differential form ?? in a non-orthogonal coordinate system has the form
(1)![\begin{equation*} J= \left(\begin{array}[pos]{cccc} \exp(\frac{1}{\gamma})\cosh(\frac{1}{\gamma}) & \exp(\frac{1}{\gamma})\sinh(\frac{1}{\gamma}) & 0 & 0\\ 0 & \exp(\frac{-1}{\gamma}) &0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)\operatorname{diag}(1,A) \end{equation*}](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-c6988bec8e3eac5a6f745d5d7036e204_l3.png "Rendered by QuickLaTeX.com")
where
, and in the orthogonal coordinate system it is reduced to the form
(2)![\begin{equation*} J=\left(\begin{array}[pos]{cccc} \exp(\frac{1}{\gamma})\cosh(\frac{1}{\gamma}) & \exp(\frac{1}{\gamma})\sinh(\frac{1}{\gamma}) & 0 & 0\\ \exp(\frac{-1}{\gamma})\sinh(\frac{1}{\gamma}) & \exp(\frac{-1}{\gamma})\cosh(\frac{1}{\gamma}) & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)\operatorname{diag}(1,A) \end{equation*}](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-a72c14b76687f2e734bc9872d3986f60_l3.png "Rendered by QuickLaTeX.com")
It only remains to check the differential form
for harmonicity, which is almost obvious due to the fact that 
.
@Igor:
“the metric is generated by the flow of time”
How?
The metric is induced by the flow through the curvature of the time coordinate. Or are you confused by the fact that (according to equation 4) an off-diagonal matrix is obtained. Well, we have non-orthogonal coordinates. It is enough to go to the orthogonal basis and we get the diagonal matrix of the metric tensor.
“The metric is induced by the flow through the curvature of the time coordinate. ”
How?
1) What is primary? \tau^\mu or \tau_\mu?
2a) If you choose \tau^\mu, what are the conditions that \tau^\mu is assumed to satisfy?
2b) If you choose \tau_\mu, what are the conditions that \tau_\mu is assumed to satisfy?
What is primary – the velocity of the particle or the measurement result in the form of a traveled path? The question is philosophical, but in our case, the time flow is caused by the movement of particles of matter.
Igor, it looks like you are avoiding my questions. If there is no metric yet, there is an essential mathematical difference between vector fields and 1-forms. You are not allowed to replace one by another, as it would lead to a nonsense. So, do not replace mathematics by a philosophy. So, is tau_\mu or tau^\mu the object that somehow, in a mysterious way, in your mind, “defines” some metric. So, would you be so kind and answer 1,2a,2b?
Why is there no metric? By definition, there is a metric in Minkowski space. Therefore, raising (lowering) indexes is a well-defined operation. Another question is how the Lorentz metric is induced, but that’s not the point here.
OK. So you are using flat Minkowski metric. For this metric the curvature is zero, in every coordinate system. But you also have, as you say, another metric, evidently with not necessarily zero curvature. It is somehow, defined, as you say, by tau. How?
For the sake of completeness, I’ll add it anyway. The Lorentz metric is generated by the topology of the manifold in which the time stream moves, namely $S^{3}\times S^{1}$.
How is your non-Minkowski metric defined by tau? Can you answer this question?
The scalar curvature must be related to the modulus of the 1-form $\tau$. I can’t immediately answer how exactly.
Может быть, $R=|\tau|$
So, you have some other, non-Minkowski metric, and you have action principle, which is not generally covariant. Right? What are your dynemical variables? Only the metric? Or also tau?
Or, perhaps, only tau???
The dynamic variable is the 1-form of the flow in Minkowski space, and the metric and scalar curvature are derivatives. Alternatively,
@Igor
“Может быть, $R=|\tau|$”
How you define $|\tau|$?
So, you assume that
Do you also assume, as a constraint, that
?
Sorry. It should be
, not 
.
[latex]
““Может быть, $R=|\tau|$””
But what about $g$?
Может быть WHAT?
Equation 4
Now, I think, I understand. Thank you.
Did you calculate the scalar curvature of this g in terms of tau?
No, I haven’t calculated it. It looks like second or even third derivatives will be needed.
I calculated. Got 1660 terms that need now to be simplified into a compact form. Yes, there are derivatives of the third order there, and also high nonlinearity.
Sorry, it has 1954 terms.
A conformal metric that could be Minkowski in places sounds OK but it can be something else in places too like degenerate at infinity. Something extra like a tau seems needed if only poetically. I used to think it was just a volume form but now I think it’s a symplectic form. I like that the intersection of SO(4,4) and Sp(8,R) is U(2,2).
Riemann mentioned
and Vladimirov’s school is currently looking into such extensions. Also Finsler geometries.
Yes, the time flow always remains timelike. As for the condition of flow integrability, it is not assumed in advance, but follows from the condition of flow harmonicity, which is a consequence of the variational equation.
@Igor
“integrability (…) follows from the condition of flow harmonicity, which is a consequence of the variational equation.”
Can you show me a proof of this whole statement?
I childishly “proved” this in a note about parallelepipeds in the section on minimal flows in the book you know
You can’t “prove it”, even childishly, if the action principle is not precisely defined. So, first define precisely your action principle. Then provide a proof, so that everybody can see it. You wrote a long comment that is hard to follow. Write now something that is clear – your “proof”.
I’ll have to see what I was proving there.
Repeat post with truncated matrices
We will consider the initial stage of reconciling the two approaches using the example of a spherically symmetric metric with a singularity on the three-dimensional unit sphere. Let
be an orthogonal matrix (
) of order three such that 
, where 
. Then the Jacobian corresponding to the quadratic differential form ?? in a non-orthogonal coordinate system has the form
(1)![\begin{equation*} J= \left(\begin{array}[pos]{cccc} \exp(\frac{1}{\gamma})\cosh(\frac{1}{\gamma}) & \exp(\frac{1}{\gamma})\sinh(\frac{1}{\gamma}) & 0 & 0\\ 0 & \exp(\frac{-1}{\gamma}) &0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)\operatorname{diag}(1,A) \end{equation*}](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-c6988bec8e3eac5a6f745d5d7036e204_l3.png "Rendered by QuickLaTeX.com")
where
, and in the orthogonal coordinate system it is reduced to the form
(2)![\begin{equation*} J=\left(\begin{array}[pos]{cccc} \exp(\frac{1}{\gamma})\cosh(\frac{1}{\gamma}) & \exp(\frac{1}{\gamma})\sinh(\frac{1}{\gamma}) & 0 & 0\\ \exp(\frac{-1}{\gamma})\sinh(\frac{1}{\gamma}) & \exp(\frac{-1}{\gamma})\cosh(\frac{1}{\gamma}) & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)\operatorname{diag}(1,A) \end{equation*}](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-a72c14b76687f2e734bc9872d3986f60_l3.png "Rendered by QuickLaTeX.com")
It only remains to check the differential form
for harmonicity, which is almost obvious due to the fact that
(3)