text:
In this section, we fix some basic setting-up to prove Theorem~\ref{ThmASUnivAbVar}. Let $M$ be a connected mixed Shimura variety of Kuga type associated with $(P,\cX^+)$. Let $\bu \colon \cX^+ \rightarrow M$ be the uniformization. Use $\Delta \subseteq \cX^+ \times M$ to denote $\mathrm{graph}(\bu)$. Let $\tilde{\pi} \colon (P,\cX^+) \rightarrow (G,\cX^+_G)$ and $\pi \colon M \rightarrow M_G$ be as in Notation~\ref{NotationMSV}.

Let $\mathscr{Z} = \mathrm{graph}(\tilde{Z} \rightarrow \bu(\tilde{Z}))$ be a complex analytic irreducible subset of $\mathrm{graph}(\bu)$. Let $\mathscr{Z}^{\Zar}$ be the smallest algebraic subvariety of $\cX^+ \times M$ containing $\mathscr{Z}$. We wish to prove
\[
\dim \mathscr{Z}^{\Zar} - \dim \mathscr{Z} \ge \dim \tilde{Z}^{\biZar}.
\]
It is clear that we may replace $\mathscr{Z}$ by a complex analytic irreducible component of $\mathscr{Z}^{\Zar} \cap \Delta$. Hence Theorem~\ref{ThmASUnivAbVar} is equivalent to the following statement.
\begin{thm}\label{ThmEquivalentFormOfAS}
Let $\mathscr{B}$ be an irreducible algebraic subvariety of $\cX^+ \times M$, and let $\mathscr{Z}$ be a complex analytic irreducible component of $\mathscr{B} \cap \Delta$. Assume $\mathscr{B} = \mathscr{Z}^{\Zar}$. Then
\[
\dim \mathscr{B} - \dim \mathscr{Z} \ge \dim \tilde{Z}^{\biZar},
\]
where $\tilde{Z}$ is the image of $\mathscr{Z}$ under the natural projection $\cX^+ \times M \rightarrow \cX^+$.
\end{thm}
We shall prove the Ax-Schanuel theorem in the form of Theorem~\ref{ThmEquivalentFormOfAS}. Since any connected mixed Shimura subvariety of $M$ is bi-algebraic, we may replace $(P,\cX^+)$ by the smallest connected mixed Shimura datum such that $\tilde{Z} \subseteq \cX^+$ and replace $M$ accordingly. Then we still have $\mathscr{B} \subseteq \cX^+ \times M$. Use
\[
\mathrm{pr}_{\cX^+} \colon \cX^+ \times M \rightarrow \cX^+, \quad \mathrm{pr}_M \colon \cX^+ \times M \rightarrow M
\]
to denote the natural projections. They are clearly algebraic. We still have $\tilde{Z} = \mathrm{pr}_{\cX^+}(\mathscr{Z})$.

Consider the action of $\Gamma$ on $\cX^+ \times M$ via its action on the first factor. Then $\Delta$ is $\Gamma$-invariant.

\subsection{Fundamental Set}\label{SubsectionFundamentalSet}

Fix a Shimura embedding $(P,\cX^+) \hookrightarrow (G_0,\cD^+) \times (P_{2g,D,\mathrm{a}}, \cX_{2g,\mathrm{a}}^+)$. Such an embedding exists by Theorem~\ref{ThmReductionLemma}.

Let $\mathfrak{F}_{\Sp_{2g}}$ be a Siegel fundamental set for the action of $\Sp_{2g,D}(\Z)$ on $\mathfrak{H}_g^+$. Let $\mathfrak{F}_{P_{2g,\mathrm{a}}} \subseteq \cX_{2g,\mathrm{a}}^+$ be defined as $i_{\mathfrak{H}_g^+}\left( (-k,k)^{2g} \times \mathfrak{F}_{\Sp_{2g}} \right)$ for some $k \ge 1$, where
\[
i_{\mathfrak{H}_g^+} \colon V_{2g}(\R) \times \mathfrak{H}_g^+ \cong \cX_{2g,\mathrm{a}}^+
\]
is the real-algebraic map defined in \eqref{EqTwoComplexStrOfX2g}. Then $\mathfrak{F}_{P_{2g,\mathrm{a}}}$ is a fundamental set for the action of $\Z^{2g} \rtimes \Sp_{2g,D}(\Z)$ on $\cX_{2g,\mathrm{a}}^+$.

Let $\mathfrak{F}_{\cD^+}$ be a fundamental set for the action of $\Gamma_0$ on $\cD^+$ as in \cite[Theorem~1.9]{KlinglerThe-Hyperbolic-}. Then

$\mathfrak{F} = \mathfrak{F}_{\cD^+} \times \mathfrak{F}_{P_{2g,\mathrm{a}}}$

is a fundamental set for $\bu \colon \cD^+ \times \cX_{2g,\mathrm{a}}^+ \rightarrow S_0 \times \mathfrak{A}_g$. Moreover $\bu|_{\mathfrak{F}}$ is definable in $\R_{\mathrm{an},\exp}$ by \cite{PeterzilDefinability-of}.

It is possible to choose $\mathfrak{F}_{\Sp_{2g}}$, $k$ and $\mathfrak{F}_{\cD^+}$ such that $\mathfrak{F}\cap \cX^+$ is a fundamental set for $\bu \colon \cX^+ \rightarrow M$. Replace $\mathfrak{F}$ by $\mathfrak{F} \cap \cX^+$, then $\bu|_{\mathfrak{F}}$ is definable in $\R_{\mathrm{an},\exp}$. We may furthermore enlarge $\mathfrak{F}$ such that $\mathfrak{F}$, resp. $\mathfrak{F}_G:= \tilde{\pi}(\mathfrak{F})$, is open in $\cX^+$, resp. in $\cX_G^+$, in the usual topology.

\subsection{Some results on the pure part}
Let $\tilde{Z}$ be as in Theorem~\ref{ThmEquivalentFormOfAS}. In this subsection we summarize some known results on the pure part, which will be used in the proof of Theorem~\ref{ThmBignessOfQStab}. Assume $\dim \tilde{\pi}(\tilde{Z}) > 0$. All distances and norms $|| \cdot ||_{\infty}$ below are as in \cite[$\mathsection$5.1]{KlinglerThe-Hyperbolic-}.

Fix $\tilde{z}_{0,G} \in \tilde{\pi}(\tilde{Z})$. All constants below depend only on $\cX^+_G$, $\mathfrak{F}_G$, $\tilde{\pi}(\tilde{Z})$, and $\tilde{z}_{0,G}$.

For any $T > 0$, define
\begin{equation}\label{EqGeodesicBallPurePart}
B^{\mathrm{horz}}(\tilde{z}_{0,G}, T) = \{ \tilde{z}_G \in \cX^+_G : d^{\mathrm{horz}}(\tilde{z}_G, \tilde{z}_{0,G}) < \log T \},
\end{equation}
where $d^{\mathrm{horz}}(\cdot, \cdot)$ is the $G^{\mathrm{der}}(\R)^+$-invariant hyperbolic metric on $\cX^+_G$.

\begin{lemma}\label{LemmaInftyNormPointInTheBase}
For each $\tilde{z}_G \in B^{\mathrm{horz}}(\tilde{z}_{0,G}, T) \subseteq \cX^+_G$, we have
\begin{equation*}\label{EqInftyNormPointInTheBase}
|| \tilde{z}_G ||_{\infty} \le c_0 T.
\end{equation*}
\end{lemma}
\begin{proof}
For $\tilde{z}_G \in B^{\mathrm{horz}}(\tilde{z}_{0,G}, T) \subseteq \cX^+_G$, there exists $g \in G(\R)^+$ such that $g\cdot \tilde{z}_{0,G} = \tilde{z}_G$. Then by \cite[Lemma~5.4]{KlinglerThe-Hyperbolic-}, we have

\[
\log || g ||_{\infty} \le d^{\mathrm{horz}}(\tilde{z}_G, \tilde{z}_{0,G}) < \log T.
\]
Hence we are done.

\end{proof}

Now let us consider
\begin{equation}\label{EqGammaGIntersectionNonEmpty}
L_G(T):=\{ \gamma_G \in \Gamma_G : \gamma_G \mathfrak{F}_G \cap B^{\mathrm{horz}}(\tilde{z}_{0,G}, T) \not= \emptyset \}.
\end{equation}

\begin{lemma}\label{LemmaGammaGUpperBound}
There exist two constants $c_1, c_2 >0$ such that
\begin{equation*}\label{EqGammaGUpperBound}
H(\gamma_G) \le c_1 T^{c_2}\quad \text{ for any }\quad \gamma_G \in L_G(T).
\end{equation*}
\end{lemma}
\begin{proof}
For any $\gamma_G \in L_G(T)$, we have $\gamma_G^{-1}\tilde{z}_G \in \mathfrak{F}_G$ for some $\tilde{z}_G \in B^{\mathrm{horz}}(\tilde{z}_{0,G},T)$.
Thus the result follows from Lemma~\ref{LemmaInftyNormPointInTheBase} and \cite[Lemma~5.5]{KlinglerThe-Hyperbolic-}.
\end{proof}

All volumes below are hyperbolic volumes.

\begin{lemma}\label{LemmaVolumeLowerBoundPurePart}
There exist two constants $c_3 >0$ and $c_4 > 0$ such that
\begin{equation*}\label{EqVolumeLowerBoundPurePart}
\mathrm{vol}(\tilde{\pi}(\tilde{Z}) \cap B^{\mathrm{horz}}(\tilde{z}_{0,G},T)) \ge c_3 T^{c_4 \dim \tilde{\pi}(\tilde{Z})}, \quad \forall T \gg 0.
\end{equation*}
\end{lemma}
\begin{proof}
This is \cite[Corollary~3]{HwangVolumes-of-comp}.
\end{proof}

\begin{lemma}\label{LemmaVolumeUpperBoundPurePart}
There exists a constant $c_5 > 0$ such that
\begin{equation*}\label{EqVolumeUpperBoundPurePart}
\mathrm{vol}(\tilde{\pi}(\tilde{Z}) \cap \gamma_G\mathfrak{F}_G) \le c_5
\end{equation*}
for all $\gamma_G \in \Gamma_G$.
\end{lemma}
\begin{proof}
This follows from \cite[Proposition~3.2]{BakkerThe-Ax-Schanuel}; our $\tilde{\pi}(\tilde{Z}) \cap \gamma_G\mathfrak{F}_G$ is the image of the $Z \cap \gamma \Phi$ in \textit{loc.cit.} under the projection $\cX^+_G \times M_G \rightarrow \cX^+_G$. As our notation is somewhat different from \textit{loc.cit.}, we briefly recall the proof. Denote by $\tilde{X} = \tilde{Z}^{\Zar}$ and $Y = \bu(\tilde{Z})^{\Zar}$. The assumption of Theorem~\ref{ThmEquivalentFormOfAS} ($\mathscr{Z}$ is a complex analytic irreducible component of $\mathscr{B} \cap \Delta$) implies that $\tilde{Z}$ is a complex analytic irreducible component of $\tilde{X} \cap \bu^{-1}(Y)$. Hence $\tilde{\pi}(\tilde{Z})$ is a complex analytic irreducible component of $\tilde{\pi}(\tilde{X}) \cap \bu_G^{-1}(\pi(Y))$, where $\bu_G \colon \cX^+_G \rightarrow M_G$ is the uniformizing map. It is possible to cover $\mathfrak{F}_G$ with finitely many semi-algebraic subsets $\{\Sigma_i\}$ such that each $\Sigma_i$ can be written in terms of (Siegel) coordinates; see \cite[Lemma~5.8]{KlinglerThe-Hyperbolic-} or \cite[Proposition~3.2]{BakkerThe-Ax-Schanuel}. For each dominant projection $p$ from $\gamma_G^{-1}\tilde{\pi}(\tilde{Z}) \cap \Sigma_i$ to $\dim \tilde{\pi}(\tilde{Z})$ coordinates, it can be computed that $p(\gamma_G^{-1}\tilde{\pi}(\tilde{Z}) \cap \mathfrak{F}_G \cap \Sigma_i)$ has finite volume. As the K\"{a}hler form with respect to which we compute the volume is $G^{\der}(\R)^+$-invariant, it suffices to bound the degree of the projections uniformly for $\gamma_G \in \Gamma_G$. But the function $G(\R)^+ \rightarrow \R$, $g^{-1}\tilde{\pi}(\tilde{X}) \cap (\bu_G^{-1}(\pi(Y)) \cap \mathfrak{F}_G) \cap \Sigma_i \mapsto \deg(p)$ is a definable function with value in $\Z$. Hence the image must be a finite set, meaning that the degree is uniformly bounded for $g \in G(\R)^+$, and in particular for $\gamma_G \in \Gamma_G$. Hence we are done.
\end{proof}

config: arxiv


TITLE: Find the solution to this 4-by-4 matrix ODE
QUESTION [0 upvotes]: Find the solution to the system $y'=Ay$, where A is given by:
$A=
  \begin{pmatrix}
    0 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 \\
    0 & 0 & -1 & 3 \\
    0 & 0 & 0 & -1 \\
  \end{pmatrix}
$
I have found the eigenvalues to be $\lambda_{1,2}=0, \lambda_{3,4}=-1$. Note that I am personally only interested in the contribution I will get from the zero eigenvalues, because I know the ones for -1 will give me solutions that tend to zero. So this is actually a part of a different problem I have to solve.
I found the eigenvector for $\lambda_{1,2}=0$ to be $\begin{pmatrix}
a\\ b\\ 0\\
0\\\end{pmatrix}$ (can someone double check that I've done this right?).
But here I've only gone to the null space of $(\lambda I - A)$, and I'm not sure whether I can argue that the contribution from the zero eigenvalues will be of the form:
$\begin{pmatrix}
a\\ 0\\ 0\\
0\\\end{pmatrix} + \begin{pmatrix}
0\\ b\\ 0\\
0\\\end{pmatrix}$
Or do I need to proceed in a different way that would result in, say, finding a solution of the form $t \bf{v}$ for some eigenvector v?

REPLY [2 votes]: You should have a solution that looks like a linear combination of the following:
$$\left(\begin{array}{rrr}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & e^{-t} & 3 e^{-t} t\\
0 & 0 & 0 & e^{-t}
\end{array}\right).$$


TITLE: Binomial coefficient; substitution?
QUESTION [1 upvotes]: How do we get this equality?
\begin{eqnarray*}
\binom{n+l-1}{l}q^l=\binom{-n}{l}(-q)^l
\end{eqnarray*}
I haven't worked a lot with the binomial coefficient, so I don't really have a feel for these things... I really hope someone would help me with this, even though I haven't shown what I've tried, because I haven't been able to come up with much.

REPLY [2 votes]: Why, that's simple.
$$\binom{n+l-1}{l}={\overbrace{(n+l-1)(n+l-2)\dots n}^{\color{red}{l\;terms}}\over l!}$$
On the other hand,
$$\binom{-n}{l}={\overbrace{(-n)(-n-1)\dots(-n-l+1)}^{\color{red}{l\;terms}}\over l!}={n(n+1)\dots(n+l-1)\over l!}\cdot(-1)^l$$


TITLE: Little-o proof for $3n^2 = o(n^4)$
QUESTION [0 upvotes]: Logically, I know this affirmation to be true. For all $c$, there is a $n_0>0$ that will make $3n^2 < c n^4$.
But when I try to prove it, I come to the resolution that it's false.
Here is how I get this conclusion:
$$f(n) \leq 3n^2 < n^4.$$
So we have $\forall c$, a $n_0 > n \in \mathbb{N}$ where:
$3n^4 < c n^4$  (This is a step I don't fully grasp, but one that seems to be used a lot in asymptotic notations proof)
Which is false in the case of $c = 1$, $\forall n>0 \in \mathbb{N}$.
What would be another way to go about this?

REPLY [1 votes]: Let's rewrite $3n^2 < c \cdot n^4$ as $\frac{3n^2}{n^4}=\frac{3}{n^2}<c$. Left side of last inequality tends to $0$, which fulfills requirements for $n$ and $c$.
As to your last question, then let me ask how you get $n^4$ in left hand of inequality?


text:
\subsection{Generalization}\label{sec:generalization}

We follow the problem setting of~\cite{el2009transductive}.
For fixed $M\in\mathbb{N}_+$, we create a training set by uniformly randomly drawing $M$ sample points \textit{without} replacement from $V$ and treating the remaining $U$ sample points as a test set.
We think of training and test errors as random variables with respect to the random partition of $V$.
Reference \cite{el2009transductive} introduced the Rademacher complexity for transductive learning and derived the generalization gap bounds.
We obtain the following proposition by applying it to our setting.
For a hypothesis space $\mathcal{F}\subset \{\mathcal{X}\to \widehat{\mathcal{Y}}\}$, we denote its transductive Rademacher complexity by $\mathfrak{R}(\mathcal{F})$.
We define $S:=\frac{4(M+U)(M\wedge U)}{(2(M+U)-1)(2(M\wedge U)-1)}$, which is close to $1$ when $M$ and $U$ are sufficiently large.
See Section~\ref{sec:proof-of-generalization} for the definition of $\mathfrak{R}(\cdot)$ and the proof of the proposition.
\begin{proposition}\label{prop:generalization-gap}
    There exists a universal constant $c_0 > 0$ such that for any $\delta' > 0$, with a probability of at least $1-\delta'$ over the random partition of samples, the output $\hat{Y}$ of Algorithm~\ref{alg:train} satisfies
    \begin{align*}
        \mathcal{R}(\widehat{Y})
        \leq \widehat{\mathcal{R}}(\widehat{Y}) + \sum_{t=1}^T \eta^{(t)}\mathfrak{R}(\mathcal{F}^{(t)})
        + c_0Q\sqrt{M\wedge U} + \sqrt{\frac{SQ}{2}\log \frac{1}{\delta'}}.
    \end{align*}
\end{proposition}


TITLE: Every bounded subset of $R^k$ lies in a compact subset of $R^k$
QUESTION [2 upvotes]: P: Every bounded subset of R$^k$ lies in a compact subset of R$^k$ (p52, Rudin)

How is the above true?
I know that, by theorem, a set $E$ in $R^k$ is closed and bounded if and only if $E$ is compact. We don't know if the subset is closed so I think we can't apply the theorem to conclude P.

REPLY [2 votes]: If $A$ is bounded, then $A\subseteq \overline{B_r(0)}$ for some $r>0$.
Exercise 1: Prove or disprove: every closed subset of $\mathbb{R}^k$ is contained in a compact subset of $\mathbb{R}^k$.
Exercise 2: Prove that every closed subset of $\mathbb{R}^k$ is the union of countably many compact subsets of $\mathbb{R}^k$. (Hint: prove that $\mathbb{R}^k$ is the union of compact subsets of $\mathbb{R}^k$ and note that the intersection of a closed set and a compact set in $\mathbb{R}^k$ is compact.)
I hope this helps! (Also, thanks very much to Ted Shifrin below who pointed out improvements!)


TITLE: Polynomials that vanish on an arbitrary set of elements of a field
QUESTION [1 upvotes]: Can someone give me a hint how to solve the following (rather vague) exercise?

Let $S$ be a subset of $F^n$, where $F$ is a field and $F^n$ is the $n$-dimensional linear space over itself. Describe the set $L=\{p\in F[X_1,\ldots,X_n] \mid p \text{ vanishes on } S\}$ and give a geometric interpretation of $F[X_1,\ldots,X_n] /L$.

I looked in the Wikipedia page on formal polynomials in the hope to find something useful and it seemed to me that the Hilbert Nullstellensatz might be what is hiding behind this question, but couldn't adapt it, or even find a clear analogy (probably because I have absolutely no knowledge of the Nullstellensatz).
(Please bare also in mind, that my knowledge of algebra is limited to the basic definition of rings, ideals etc. plus some rather easy theorem about them - so nothing very deep.)

REPLY [3 votes]: You could say that  $L\subset F[X_1,\ldots,X_n]$  is a reduced ideal, reduced meaning that if $p^r\in L$ for some positive integer $r$, then $p\in L$. Can you see why this is true?
Now call $regular$ any function $S\to F$ given by a polynomial $p\in F[X_1,\ldots,X_n]$, that is a function of the form $s\mapsto f(s_1,...,s_n)$.
Try  to prove that your mysterious quotient  $F[X_1,\ldots,X_n] /L$ is isomorphic to the ring of regular functions on $S$.
Don't worry about the Nullstellensatz: it can't be applied anyway because $F$ is not assumed algebraically closed.
And also: there are zillions of  explicit, elementary and well-formulated exercises for beginners in algebraic geometry. I 'm not sure  that this is one of them. So don't be discouraged if you couldn't solve it!


TITLE: prove that the finite group G has a fixed point in S if |G|=$p^r$, p prime, |S|=N, (p,N)=1
QUESTION [0 upvotes]: I am asked to solve the following
Let G be a finite group with $p^r$ elements, where p is prime. if G acts on the finite set S with N elements and $(p,N)=1$, prove that there exists $s\in S$ such that $g*s=s,\forall g\in G$
I am able to get that $\sum_{g\in G}I(g)>0$ but I don't see how to convert that into $I(g)>0, \forall g\in G$

REPLY [1 votes]: Let $G_{s}$ and $O_{s}$ be the stabilizer and orbit of an element $s \in S$.
Then $p^{r} = |G| = |G_{s}||O_{s}|$, Hence $|O_{s}| = p^{k}$.
On another hand we number the orbits that make up $S$ arbitrarily, as $O_{1}, ..., O_{k}$. Then $|S| = \sum_{i=1}^{k}|O_{i}|$ with $|O_{i}| = p^{e_{i}}$. If $e_{i} \geq 1$ for all $i$, then we get that $p$ divide $|S|$, which contradicts the hypothesis. Hence $e^{j} = 0$ for some $j$, so that $|O_{j}| = 1$, but this implies the result and we are done as it was promised.


text:
\begin{document}

\title{Gauge Fixing in the Maxwell Like Gravitational Theory in Minkowski Spacetime
and in the Equivalent Lorentzian Spacetime}
\author{Rold\~{a}o da Rocha$^{(1)}$ and Waldyr A. Rodrigues Jr.$^{(2)}$\\$^{(1)}\hspace{-0.05cm}${\footnotesize Centro de Matem\'{a}tica,
Computa\c{c}\~{a}o e Cogni\c{c}\~{a}o}\\{\footnotesize Universidade Federal do ABC, 09210-170, Santo Andr\'e, SP,
Brazil}\\{\small roldao.rocha@ufabc.edu.br}\\$^{(2)}\,\hspace{-0.1cm}${\footnotesize Institute of Mathematics, Statistics
and Scientific Computation}\\{\footnotesize IMECC-UNICAMP CP 6065}\\{\footnotesize 13083-859 Campinas, SP, Brazil}\\{\small walrod@ime.unicamp.br or walrod@mpc.com.br}}
\date{11 September 2009}
\maketitle

\begin{abstract}
In \ a previous paper we investigate a Lagrangian field theory for the
gravitational field, which is there represented by a section $\{\mathfrak{g}
^{\alpha}\}$ of the coframe bundle\ over Minkowski spacetime $(M\simeq
\mathbb{R}^{4},\overset{\circ}{

\slg

},\mathring{D}\mathbf{,\tau}_{\overset{\circ}{

\sslg

}},\mathbf{\uparrow})$. Such theory, under appropriate conditions, has been
proved to be equivalent to a Lorentzian spacetime structure $(M\simeq
\mathbb{R}^{4},

\slg

,D,\mathbf{\tau}_{

\sslg

},\uparrow)$ where the metric tensor $

\slg

$ satisfies the Einstein field equation. Here, we first recall that according
to quantum field theory ideas gravitation is described by a Lagrangian theory
of a possible \textit{massive graviton field} (generated by matter fields and
coupling also to itself) living in Minkowski spacetime. The graviton field is
moreover supposed to be represented by a symmetric tensor field\textbf{
}$\mathbf{h}$ carrying the representations of spin two and zero of the Lorentz
group. Such a field, then (as it is well known) must necessarily satisfy the
gauge condition given by Eq.(\ref{RESTRICTION}) below. Next, we introduce an
\textit{ansatz} relating $\mathbf{h}$ with the $1$-form fields $\{\mathfrak{g}
^{\alpha}\}$. Then, using the Clifford bundle formalism we derive from our
Lagrangian theory the exact wave equation for the graviton and investigate the
role of the gauge condition given by Eq.(\ref{RESTRICTION}) \ by asking the
question: does Eq.(\ref{RESTRICTION}) fix any gauge condition for the field $

\slg

$ of the \textit{effective} Lorentzian spacetime structure ($M\simeq
\mathbb{R}^{4},

\slg

,D,\tau_{

\sslg

},\uparrow$) that represents the field \ $\mathbf{h}$ in our theory? We show
that no gauge condition is fixed a priory, as it is the case in General
Relativity. Moreover we investigate under which conditions we may fix Logunov
gauge condition $\mathring{D}_{\gamma}\left(  \sqrt{-\left\vert g_{\mu\nu
}\right\vert }g^{\gamma\kappa}\right)  =0$.

\end{abstract}

\section{Introduction}

In a previous paper\footnote{Please, consult the arXiv version of \cite{edrod}
which corrects an error of the printed version.}, using the Clifford bundle
formalism, a Lagrangian theory of the gravitational field as field in the
Faraday sense, i.e., an object of the same ontology as the electromagnetic
field living on a Minkowski spacetime structure $\mathbf{M=}$ $(M\simeq
\mathbb{R}^{4},\overset{\circ}{

\slg

},\mathring{D}\mathbf{,\tau}_{\overset{\circ}{

\sslg

}}$ $,\mathbf{\uparrow})$ has been formulated\footnote{Minkowski spacetime
will be called Lorentz vacuum, in what follows. Moreover in the $5$-uple
$(M\simeq\mathbb{R}^{4},\overset{\circ}{

\slg

},\mathring{D}\mathbf{,\tau}_{\overset{\circ}{

\sslg

}},\mathbf{\uparrow)}$, $\overset{\circ}{

\slg

}$ is a Minkowski metric, $\mathring{D}$ is its Levi-Civita connection,
$\mathbf{\tau}_{\overset{\circ}{

\sslg

}}$ is the volume element defining a global orientation and $\uparrow$ refers
to a time orientation. The objects in the Lorentzian spacetime structure
$\mathbf{L}=(M\simeq\mathbb{R}^{4},

\slg

,D,\tau_{\mathtt{

\sslg

}},\uparrow)$\ have similar meanings. In what follows $\mathtt{\mathring{g}}$
denotes the metric of the cotangent bundle relative to the structure
$\mathbf{M}$. If $\mathtt{\mathring{g}}=\mathring{g}^{\kappa\iota}
\partial_{\kappa}\otimes\partial_{\iota}$ and $\overset{\circ}{

\slg

}=\mathring{g}_{\kappa\iota}dx^{\kappa}\otimes dx^{\iota}$ then $\mathring
{g}^{\kappa\iota}\mathring{g}_{\iota\xi}=\delta_{\xi}^{\kappa}$. Also
\texttt{g }denotes the metric of the cotangent bundle relative to the
structure $\mathbf{L}$ and if $\mathtt{g}=g^{\kappa\iota}\partial_{\kappa
}\otimes\partial_{\iota}$ and $

\slg

=g_{\kappa\iota}dx^{\kappa}\otimes dx^{\iota}$, then $g^{\kappa\iota}
g_{\iota\xi}=\delta_{\xi}^{\kappa}$ More details, if needed are given, e.g.,
in \cite{rodoliv2006}.}. The theory has been constructed on two assumptions.
The \textit{first} one is that the gravitational field is represented by a
\textit{coframe} $\{\mathfrak{g}^{\alpha}\}$, with \ $\mathfrak{g}^{\alpha}
\in\sec\bigwedge\nolimits^{1}T^{\ast}M\hookrightarrow\sec\mathcal{C}
\ell(M,\mathtt{\mathring{g}})$ whose \textit{dynamics} is encoded in
Lagrangian density $\mathcal{L}_{m}^{\mathbf{M}}$ (see Eq.(\ref{10hyp2})
below) which is of the Yang-Mills type (containing moreover a gauge fixing
term and an auto interaction term related to the \textquotedblleft
vorticity\textquotedblright\ of the fields). The theory is invariant under
diffeomorphisms and under local Lorentz transformations of the coframe
$\{\mathfrak{g}^{\alpha}\}$. The gravitational field couples universally with
the matter fields and in such a way that \ the coupling resulting from the
presence of energy-momentum due to matter fields in some region of Minkowski
spacetime distorts the Lorentz vacuum\footnote{A region of Minkowski spacetime
void of matter fields will be called Lorentz vacuum.} in much the same way
that stresses in an elastic body produces \textit{plastic} deformations in it
\cite{zorawski}. To present some additional details we need to introduce some
notation. So, let $\{\mathrm{x}^{\mu}\}$ be a set of global
coordinates\footnote{If \ $\{\mathbf{x}^{\mu}\}$ and $\{\mathbf{x}^{\prime\mu
}\}$ are global coordinate functions \ in the Einstein-Lorentz-Poincar\'{e}
gauge, the coordinates of $\mathfrak{e}\in M$ in are $\{\mathrm{x}^{\mu
}\}:=\{\mathbf{x}^{\mu}(\mathfrak{e})\}$, $\{\mathrm{x}^{\prime\mu
}\}:=\{\mathbf{x}^{\prime\mu}(\mathfrak{e})\}$ and \textrm{x}$^{\prime\mu
}=\Lambda_{\nu}^{\mu}$\textrm{x}$^{\nu}$, with $\Lambda_{\nu}^{\mu}$ a proper
and orthochronous Lorentz transformation.} for $M$ in the
Einstein-Lorentz-Poincar\'{e} gauge associated to arbitrary inertial reference
frame\footnote{An inertial reference frame satisfies $\mathring{D}I=0$. See
\cite{rodoliv2006} for details.} $I=\partial/\partial\mathrm{x}^{0}\in\sec
TM$. Let $\{\partial/\partial\mathrm{x}^{\mu}\}$ be orthonormal basis for $TM$
and $\{\gamma^{\mu}=d\mathrm{x}^{\mu}\}$ the corresponding dual basis for
$T^{\ast}M$. We take\footnote{$\bigwedge\nolimits^{p}T^{\ast}M$ denotes the
bundle of $p$-forms, $\bigwedge T^{\ast}M=

{\displaystyle\bigoplus\nolimits_{p=0}^{4}}

\bigwedge\nolimits^{p}T^{\ast}M$ is the bundle of multiform fields,
$\mathcal{C}\ell(M,\mathtt{\mathring{g}})$ denotes the Clifford bundle of
differential forms. The symbol $\sec$ means section. All \ `tricks of the
trade' necessary for performing the calculations of the present paper are
described in \cite{rodoliv2006}.} $\gamma^{\mu}\in\sec\bigwedge\nolimits^{1}
T^{\ast}M\hookrightarrow\sec\mathcal{C}\ell(M,\mathtt{\mathring{g}})$. Of
course, we have
\begin{align}
\overset{\circ}{

\slg

}\text{ }  &  =\eta_{\alpha\beta}\gamma^{a}\otimes\gamma^{\beta}\label{zero}\\
\mathtt{\mathring{g}}  &  =\eta^{\alpha\beta}\frac{\partial}{\partial
\mathrm{x}^{\alpha}}\otimes\frac{\partial}{\partial\mathrm{x}^{\beta}
},\nonumber
\end{align}
and we recall that to each (non degenerated) metric tensor, say
$\mathtt{\mathring{g}}\in\sec T_{0}^{2}M$ there corresponds an unique
invertible metric extensor field $\mathring{g}:\sec

{\displaystyle\bigwedge\nolimits^{1}}

T^{\ast}M\rightarrow\sec

{\displaystyle\bigwedge\nolimits^{1}}

T^{\ast}M$, while the metric tensor $\overset{\circ}{

\slg

}\in\sec T_{2}^{0}M$ is represented by the extensor field $\mathring{g}^{-1}$.
Our second assumption is that there are a \textit{plastic distortion field}
described by an extensor field $h:\sec

{\displaystyle\bigwedge\nolimits^{1}}

T^{\ast}M\rightarrow\sec

{\displaystyle\bigwedge\nolimits^{1}}

T^{\ast}M$ that distorts the cosmic lattice represented by the $\gamma^{a}$
producing the fields $\mathfrak{g}^{\alpha}$ which may be used to introduce on
on $T^{\ast}M$ an effective Lorentzian metric field, respectively
\texttt{g}$\in\sec T_{0}^{2}M$ and $

\slg

\in\sec T_{2}^{0}M$\ and such that the associated extensor fields are
\begin{equation}
g=h^{\dagger}h,\text{...}g^{-1}=h^{-1}h^{\dagger-1}. \label{01}
\end{equation}
It is useful for application of the formalism we have in mind to suppose that
the metric field $

\slg

$ is the pullback under a diffeomorphism of a metric field $

\mbox{\boldmath{$\eta$}}

\in\sec T_{2}^{0}M$ according to the following scheme:
\begin{equation}

\slg

=\mathtt{h}^{\ast}

\mbox{\boldmath{$\eta$}}

=\eta_{\alpha\beta}\theta^{\mathbf{\alpha}}\otimes\theta^{\beta} \label{02}
\end{equation}
where
\begin{equation}
\theta^{\mathbf{\alpha}}:=\mathtt{h}^{\ast}\mathfrak{g}^{\mathbf{\alpha}
}=h^{-1}(\gamma^{\alpha}) \label{teta}
\end{equation}
\begin{equation}

\mbox{\boldmath{$\eta$}}

:=\eta_{\alpha\beta}\mathfrak{g}^{\mathbf{\alpha}}\otimes\mathfrak{g}^{\beta}
\label{new met}
\end{equation}
\begin{equation}
g(\theta^{\alpha})\underset{\mathtt{\mathring{g}}}{\cdot}\theta^{\beta
}=h(\theta^{\alpha})\underset{\mathtt{\mathring{g}}}{\cdot}h(\theta^{\beta
})=\mathfrak{\gamma}^{\alpha}\underset{\mathtt{\mathring{g}}}{\cdot
}\mathfrak{\gamma}^{\beta}=\eta^{\alpha\beta} \label{03}
\end{equation}
Once the Levi-Civita connection $D$ of $

\slg

$ is introduced in the game, it is possible to show that the deformed
gravitational fields $\theta^{\alpha}=\mathtt{h}^{\ast}\mathfrak{g}^{\alpha}$
satisfy Maxwell like field equations (which follows from the variational
principle \cite{fr}). Moreover, in our theory each nontrivial gravitational
field configuration, i.e., one from which not\textit{ all} the $\mathfrak{g}
^{\alpha}$ are exact differentials can be interpreted as generating an
effective Lorentzian spacetime $(M\simeq\mathbb{R}^{4},

\slg

,D,\mathbf{\tau}_{

\sslg

},\uparrow)\simeq(M\simeq\mathbb{R}^{4},\eta,\overset{\eta}{D}=\mathtt{h}
^{\ast-1}D,\mathbf{\tau}_{\eta},\uparrow)$ where $

\slg

$ satisfies Einstein equation\footnote{Or by an effective teleparallel
spacetime, see \cite{edrod}, the arXiv version.
\par
{}}.

Recall next that it is a physicist dream to construct a quantum theory for the
gravitational field, where the quanta of the field are the so called
gravitons. In such (yet to be constructed) theory the gravitational field is
supposed to be represented by a distribution valued symmetric field operator
acting on the Hilbert space of the system. Classically that field is
represented by a symmetric tensor (distribution)
\begin{equation}
\mathbf{h}=\mathbf{h}_{\alpha\beta}\vartheta^{a}\otimes\vartheta^{\beta}
\in\sec T_{0}^{2}M, \label{h field}
\end{equation}
where\footnote{Given arbitrary coordinate functions $\{

\slx

^{\mu}\}$ covering $U\subset M$ \ with coordinates $\{x^{\mu}\}$ such that $

\slx

^{\mu}(\mathfrak{e})=x^{\mu}$, we write, as usual, $\{\frac{\partial}{\partial
x^{\mu}}\}$ for the coordinate tangent vector fields and $\{dx^{\mu}\}$ for
the coordinate cotangent covectors.} $\vartheta^{\mu}:=dx^{\mu}$ $\in
\sec\bigwedge\nolimits^{1}T^{\ast}M\hookrightarrow\sec\mathcal{C}
\ell(M,\mathtt{\mathring{g}})$, with $\{x^{\mu}\}$ arbitrary coordinates
covering $U\subset M$. Such a general field, as it is well known
\cite{barnes,fronsdal} carries a direct sum of irreducible representations of
the Lorentz group, one carrying spin two, one carrying spin one and two
carrying spin zero. Now, consider the tensor field $\mathbf{h}^{\prime}\in\sec
T_{1}^{1}M$
\begin{equation}
\mathbf{h}^{\prime}=\mathbf{h}_{\beta}^{\alpha}\frac{\partial}{\partial
x^{\alpha}}\otimes dx^{\beta}
\end{equation}
If we impose that \textrm{div}$\mathbf{h}^{\prime}=0$, i.e.,\textrm{ }the
restriction\footnote{In \cite{edrod} we show explicitly how to determine the
extensor field $h$ once $

\slg

$ is known in a given basis.}
\begin{equation}
\mathring{D}_{\alpha}\mathbf{h}_{\beta}^{\alpha}=0, \label{RESTRICTION}
\end{equation}
(where $\mathbf{h}_{\beta}^{\alpha}:=\mathring{g}^{\alpha\kappa}
\mathbf{h}_{\kappa\beta},\overset{\circ}{

\slg

\text{ }}=\mathring{g}_{\alpha\beta}\vartheta^{a}\otimes\vartheta^{\beta}$ )
then the field $\mathbf{h}$ carries only the irreducible representations with
spin 2 and one with spin zero of the Lorentz group. This restriction is the
one appropriate for the description of gravitons with non null mass $m$. Next
we introduce the main purpose of this paper, which is to investigate (using
the Clifford bundle formalism) the consequences of the \textit{ansatz}

\begin{align}
\mathbf{h}^{\prime}  &  =h=\mathbf{h}_{\beta}^{\alpha}\frac{\partial}{\partial
x^{\alpha}}\otimes dx^{\beta}=h_{\beta}^{\alpha}\frac{\partial}{\partial
\mathrm{x}^{\alpha}}\otimes d\mathrm{x}^{\beta}\nonumber\\
\mathfrak{g}^{\alpha}  &  :=h_{\beta}^{\alpha}\gamma^{\beta}. \label{def g}
\end{align}

We then show how to derive from our Lagrangian theory the \textit{exact} wave
equation for the graviton field and we obtain a reliable conservation law for
the energy-momentum tensor of the gravitational plus the matter fields in
Minkowski spacetime.\

We also ask the question: does Eq.(\ref{RESTRICTION}) fix any gauge condition
for the field $

\slg

$ of the \textit{effective} Lorentzian spacetime structure ($M\simeq
\mathbb{R}^{4},

\slg

,D,\tau_{

\sslg

},\uparrow$) that is a well defined functional of the field $\mathbf{h}$ in
our theory? We show that no gauge condition is fixed a priory, as it is the
case in General Relativity (GR). Thus, writing $\mathtt{g}=g^{\alpha\beta
}\frac{\partial}{\partial x^{\alpha}}\otimes\frac{\partial}{\partial x^{\beta
}}$ we do not need, e.g., to fix in our theory Logunov gauge condition
\begin{equation}
\mathring{D}_{\gamma}\left(  \sqrt{-\det(g_{\mu\nu})}g^{\gamma\kappa}\right)
=0,\label{logunov}
\end{equation}
which, is indeed a result of a postulate in Logunov's theory
\cite{logunov1,logunov2}. Since Logunov thinks that Eq.(\ref{logunov}) \ is
very important, since according to him it fixes a \textit{unique} solution of
Einstein equations\footnote{Even for the case of a zero mass graviton.} once a
matter distribution and a \textit{coordinate chart} are given, thus
eliminating (possible) ambiguities in predictions of experiments, we discuss
briefly this issue.

\section{The Wave Equation for the\textbf{ }$\mathfrak{g}^{\alpha}$}

We recall that the dynamics of the fields $\mathfrak{g}^{\alpha}$ in a region
of $M$ is given by
\begin{equation}
\mathcal{L=L}_{g}^{\mathbf{M}}+\mathcal{L}_{m}^{\mathbf{M}}, \label{8.5neww}
\end{equation}
where $\mathcal{L}_{m}^{\mathbf{M}}$ is the Lagrangian density of the matter
fields and
\begin{equation}
\mathcal{L}_{g}^{\mathbf{M}}=-\frac{1}{2}d\mathfrak{g}^{\alpha}\wedge
\underset{\eta}{\star}d\mathfrak{g}_{\alpha}+\frac{1}{2}\underset{\eta}
{\delta}\mathfrak{g}^{\alpha}\wedge\underset{\eta}{\star}\underset{\eta
}{\delta}\mathfrak{g}_{\alpha}+\frac{1}{4}d\mathfrak{g}^{\alpha}
\wedge\mathfrak{g}_{\alpha}\wedge\underset{\eta}{\star}(d\mathfrak{g}^{\alpha
}\wedge\mathfrak{g}_{\alpha})+\frac{1}{4}m^{2}\mathfrak{g}_{\alpha}
\wedge\underset{\eta}{\star}\mathfrak{g}^{\alpha}, \label{10hyp2}
\end{equation}
is invariant under local Lorentz transformations\footnote{We observe that the
various coefficients in Eq.(\ref{10hyp2}) have been selected in order for
$\mathcal{L}_{g}^{M}$ to be invariant under arbitrary local Lorentz
transformations. This means, as the reader may verify that under the
transformation \ $\mathfrak{g}^{\mathbf{a}}\mapsto u\mathfrak{g}^{\mathbf{a}
}u^{-1}$, $u\in\sec\mathrm{Spin}_{1,3}^{e}(M,\mathtt{g})\hookrightarrow
\sec\mathcal{C}\ell(M,\mathtt{g})$, $\mathcal{L}_{g}^{M}$ is invariant modulo
an exact form.}, which is a kind of gauge freedom, a crucial ingredient of our
theory, as showed in \cite{edrod}.

The $\mathfrak{g}^{\alpha}$ couple universally to the matter fields in such a
way that the energy momentum $1$-form of the matter fields are given by
\begin{equation}
\underset{\eta}{\star}\mathfrak{T}_{\mathbf{M}}^{\alpha}=\frac{\partial
\mathcal{L}_{m}^{\mathbf{M}}}{\partial\mathfrak{g}_{\alpha}}. \label{10hyp2'}
\end{equation}
\medskip

Each one of the fields $\mathfrak{g}^{\alpha}$ in Eq.(\ref{10hyp2}) resembles
a potential of an electromagnetic field. Indeed, the first term is of the
Yang-Mills type, the second term is a kind of gauge fixing term (analogous to
the \textit{Lorenz} condition for the gauge potential of the electromagnetic
potential), and more important, the condition given by Eq.(\ref{RESTRICTION})
is equivalent to
\begin{equation}
\underset{\overset{\circ}{

\sslg

}}{\delta}\mathfrak{g}^{\alpha}=0. \label{EQUIV}
\end{equation}
Indeed, given the coordinates functions $\{

\slx

^{\mu}\}$ for $U\subset M$, $

\slx

^{\mu}(\mathfrak{e})=\mathrm{x}^{\mu}$ ($\mathfrak{g}^{\alpha}=h_{\beta
}^{\alpha}\gamma^{\beta}$ and $\overset{\circ}{

\slg

}=\eta_{\alpha\beta}\gamma^{\alpha}\otimes\gamma^{\beta}$) it is\footnote{As
usual we put $\mathring{D}_{\frac{\partial}{\partial x^{\kappa}}}\left(
\mathbf{h}_{\beta}^{\alpha}dx^{\beta}\otimes\frac{\partial}{\partial
x^{\alpha}}\right)  :=(\mathring{D}_{\kappa}\mathbf{h}_{\beta}^{\alpha
})dx^{\beta}\otimes\frac{\partial}{\partial x^{\alpha}}$.}
\begin{align}
\underset{\overset{\circ}{

\sslg

}}{\delta}\mathfrak{g}^{\alpha}  &  =-

\bpartial

\underset{\overset{\circ}{

\sslg

}}{\lrcorner}\mathfrak{g}^{\alpha}=-\gamma^{\kappa}\underset{\overset{\circ}{

\sslg

}}{\lrcorner}(\mathring{D}_{\frac{\partial}{\partial\mathrm{x}^{\kappa}}
}\left(  h_{\beta}^{\alpha}\gamma^{\beta}\right)  )\nonumber\\
&  =-\left(  \mathring{D}_{\kappa}h_{\beta}^{\alpha}\right)  \eta^{\kappa
\beta}=\partial_{\kappa}h_{\lambda}^{\kappa}=0
\end{align}
Moreover, in \textit{general} we also have
\begin{equation}
\underset{\eta}{\delta}\mathfrak{g}^{\alpha}\neq0, \label{no gaugef}
\end{equation}
where for any $A_{p}\in\sec\bigwedge\nolimits^{p}T^{\ast}M\hookrightarrow
\sec\mathcal{C}\ell(M,\mathtt{\mathring{g}})$, it is $\underset{\eta}{\delta
}A_{p}:=-\gamma^{\kappa}\underset{\eta}{\lrcorner}D_{\frac{\partial}{\partial
x^{\kappa}}}A_{p}$.

Also, the third term in the Lagrangian density is a self-interacting term,
which is proportional to the square of the total\ `vorticity' $\Omega
=d\mathfrak{g}^{\alpha}\wedge\mathfrak{g}_{\alpha}$ associated to the $1$-form
fields $\mathfrak{g}^{\alpha}$. This shows that in the Lagrangian density the
$\mathfrak{g}^{\alpha}$ does \textit{not} couple with the energy-momentum
tensor of the gravitational field\footnote{On this respect see the discussion
of \cite{pada}.}, which according to the Lagrangian formalism is given by
$\frac{\partial\mathcal{L}_{g}^{\mathbf{M}}}{\partial\mathfrak{g}^{\alpha}}$.
We finally recall that $\ $as showed in details in \cite{edrod} $\mathcal{L}
_{g}:=\mathtt{h}^{\ast-1}\mathcal{L}_{g}^{\mathbf{M}}$ has the same form of
$\mathcal{L}_{g}^{\mathbf{M}}$ with the substitutions \ $\mathfrak{g}^{\alpha
}\mapsto\theta^{\alpha}$ and $\underset{\eta}{\star}\mapsto\underset{

\sslg

}{\star}$ and differs (when the graviton mass is null) from the
Einstein-Hilbert Lagrangian by an exact differential.

Also, as showed in details. e.g., in \cite{fr} variation of $\int
\mathcal{L}_{g}^{\mathbf{M}}$ produces the following equations of motions
\begin{equation}
d\underset{\eta}{\star}\mathcal{S}^{\alpha}+\text{ }\underset{\eta}{\star
}\mathfrak{t}^{\alpha}+\frac{1}{2}m^{2}\underset{\eta}{\star}\mathfrak{g}
^{\alpha}=-\text{ }\underset{\eta}{\star}\mathfrak{T}^{\alpha} \label{8.27}
\end{equation}
with $\underset{\eta}{\star}\mathfrak{t}_{\mathbf{\ }}^{\kappa}\in
\sec\bigwedge\nolimits^{3}T^{\ast}M\hookrightarrow\mathcal{C\ell}\left(
T^{\ast}M,\mathtt{\mathring{g}}\right)  $ and $\underset{\eta}{\star
}\mathcal{S}^{\kappa}\in\sec\bigwedge\nolimits^{2}T^{\ast}M\hookrightarrow
\mathcal{C\ell}\left(  T^{\ast}M,\mathtt{\mathring{g}}\right)  $ given by
\begin{align}
\underset{\eta}{\star}\mathfrak{t}_{\mathbf{\ }}^{\kappa}  &  =\frac
{\partial\mathcal{L}_{g}^{\mathbf{M}}}{\partial\mathfrak{g}_{\kappa}}=\frac
{1}{2}[(\mathfrak{g}_{\kappa}\underset{\eta}{\lrcorner}d\mathfrak{g}^{\alpha
})\wedge\underset{\eta}{\star}d\mathfrak{g}_{\alpha}-d\mathfrak{g}^{\alpha
}\wedge(\mathfrak{g}_{\kappa}\underset{\eta}{\lrcorner}\underset{\eta}{\star
}d\mathfrak{g}_{\alpha})]\nonumber\\
&  +\frac{1}{2}d\left(  \mathfrak{g}_{\kappa}\underset{\eta}{\lrcorner
}\underset{\overset{\circ}{

\sslg

}}{\star}\mathfrak{g}^{\alpha}\right)  \wedge\underset{\eta}{\star}
d\underset{\eta}{\star}\mathfrak{g}_{\alpha}+\frac{1}{2}\left(  \mathfrak{g}
_{\kappa}\underset{\eta}{\lrcorner}d\underset{\eta}{\star}\mathfrak{g}
^{\alpha}\right)  \wedge\underset{\eta}{\star}d\underset{\eta}{\star
}\mathfrak{g}_{\alpha}+\frac{1}{2}d\mathfrak{g}_{\kappa}\wedge\underset{\eta
}{\star}\left(  d\mathfrak{g}^{\alpha}\wedge\mathfrak{g}_{\alpha}\right)
\nonumber\\
&  -\frac{1}{4}d\mathfrak{g}^{\alpha}\wedge\mathfrak{g}_{\alpha}\wedge\left[
\mathfrak{g}_{\kappa}\underset{\eta}{\lrcorner}\underset{\eta}{\star}\left(
d\mathfrak{g}^{\iota}\wedge\mathfrak{g}_{\iota}\right)  \right]  -\frac{1}
{4}\left[  \mathfrak{g}_{\kappa}\underset{\eta}{\lrcorner}\left(
d\mathfrak{g}^{^{\iota}}\wedge\mathfrak{g}_{\iota}\right)  \right]
\wedge\underset{\eta}{\star}\left(  d\mathfrak{g}^{\alpha}\wedge
\mathfrak{g}_{\alpha}\right)  , \label{7.1016}
\end{align}

\begin{equation}
\underset{\eta}{\star}\mathcal{S}_{M}^{\kappa}=\frac{\partial\mathcal{L}
_{g}^{\mathbf{M}}}{\partial d\mathfrak{g}_{\kappa}}=-\mathfrak{g}^{\alpha
}\wedge\underset{\eta}{\star}(d\mathfrak{g}_{\alpha}\wedge\mathfrak{g}
_{\kappa})+\frac{1}{2}\mathfrak{g}_{\kappa}\wedge\underset{\eta}{\star
}(d\mathfrak{g}^{\alpha}\wedge\mathfrak{g}_{\alpha}). \label{7.10.17}
\end{equation}

For what follows we need also the following equivalent expression for the
$\underset{\eta}{\star}\mathcal{S}^{\kappa}$ obtained, e.g., in
\cite{edrod2008},

\begin{equation}
\underset{\eta}{\star}\mathcal{S}^{\kappa}=\frac{1}{2}\underset{\eta}{\star
}\left[  -(\mathfrak{g}_{\alpha}\underset{\eta}{\lrcorner}d\mathfrak{g}
^{\alpha})\wedge\mathfrak{g}^{\kappa}-(\mathfrak{g}^{\alpha}\underset{\eta
}{\lrcorner}d\mathfrak{g}_{\alpha})\wedge\mathfrak{g}^{\kappa}+(\mathfrak{g}
^{\kappa}\underset{\eta}{\lrcorner}d\mathfrak{g}^{\alpha})\wedge
\mathfrak{g}_{\alpha}-d\mathfrak{g}^{\kappa}\right]  . \label{S}
\end{equation}
We write moreover
\begin{equation}
\underset{\eta}{\star}\mathcal{S}^{\kappa}=-\frac{1}{2}\underset{\eta}{\star
}d\mathfrak{g}^{\kappa}+\underset{\eta}{\star}\mathfrak{K}^{\kappa} \label{SS}
\end{equation}
and insert this result in Eq.(\ref{8.27}) obtaining:
\begin{equation}
-\frac{1}{2}d\underset{\eta}{\star}d\mathfrak{g}^{\kappa}+\frac{1}{2}
m^{2}\underset{\eta}{\star}\mathfrak{g}^{\kappa}=-\underset{\eta}{\star
}\left(  \mathfrak{t}^{\kappa}+\mathfrak{T}^{\kappa}+\underset{\eta}{\star
}^{-1}d\underset{\eta}{\star}\mathfrak{K}^{\kappa}\right)  \label{SSS}
\end{equation}

Before proceeding we recall that we have the conservation law
\begin{equation}
d\underset{\eta}{\star}\left(  \mathfrak{t}^{\kappa}+\mathfrak{T}^{\kappa
}+\underset{\eta}{\star}^{-1}d\underset{\eta}{\star}\mathfrak{K}^{\kappa
}-\frac{1}{2}m^{2}\underset{\eta}{\star}\mathfrak{g}^{\kappa}\right)  =0,
\label{cl1}
\end{equation}

We now add the term $-\frac{1}{2}d\underset{\eta}{\delta}\mathfrak{g}^{\kappa
}$ to both members Eq.(\ref{SSS}) and next apply the operator $\underset{\eta
}{\star}^{-1}$ to both sides of that equation, thus obtaining the equivalent
equation:
\begin{equation}
-\frac{1}{2}\underset{\eta}{\delta}d\mathfrak{g}^{\kappa}-\frac{1}
{2}d\underset{\eta}{\delta}\mathfrak{g}^{\kappa}+\frac{1}{2}m^{2}
\mathfrak{g}^{\kappa}=-\left(  \mathfrak{t}^{\kappa}+\mathfrak{T}^{\kappa
}+\underset{\underset{\eta}{\star}}{\delta}\mathfrak{K}^{\kappa}+\frac{1}
{2}d\underset{\underset{\eta}{\star}}{\delta}\mathfrak{g}^{\kappa}\right)
\label{cl2}
\end{equation}
We now recall the definition of the Hodge D'Alembertian, which in the Clifford
bundle formalism is the square of the Dirac operator $

\bpartial

:=\vartheta^{\alpha}\overset{\eta}{D}_{e_{\alpha}}$ acting on sections of the
Clifford bundle \cite{rodoliv2006}, i.e.,
\begin{equation}
\overset{\eta}{\lozenge}\mathfrak{g}^{\kappa}:=(-\underset{\eta}{\delta
}d-d\underset{\eta}{\delta})\mathfrak{g}^{\kappa}=

\bpartial

\,^{2}\mathfrak{g}^{\kappa} \label{cl3}
\end{equation}
and recall moreover the following nontrivial decomposition \cite{rodoliv2006}
of $

\bpartial

^{2}$,
\begin{equation}

\bpartial

\,^{2}\mathfrak{g}^{\kappa}=

\bpartial

\underset{\eta}{\cdot}

\bpartial

\mathfrak{g}^{\kappa}+

\bpartial

\wedge

\bpartial

\mathfrak{g}^{\kappa}, \label{cl4}
\end{equation}
where $\overset{\eta}{\square}:=

\bpartial

\underset{\eta}{\cdot}

\bpartial

$ is the \textit{covariant D'Alembertian} and $

\bpartial

\wedge

\bpartial

$ is the \textit{Ricci operator associated to the Levi-Civita connection
}$\overset{\eta}{D}$ \textit{of }$

\mbox{\boldmath{$\eta$}}

$ . Moreover, we have
\begin{equation}

\bpartial

\wedge

\bpartial

\mathfrak{g}_{\kappa}=\mathcal{R}_{\kappa}=R_{\kappa\iota}\vartheta^{\iota},
\label{cl5}
\end{equation}
where $\mathcal{R}_{\kappa}\in\sec\bigwedge\nolimits^{1}T^{\ast}
M\hookrightarrow\mathcal{C\ell}\left(  T^{\ast}M,\mathtt{\mathring{g}}\right)
$ are the Ricci $1$-form fields and $R_{\kappa\iota}$ are the components of
the Ricci tensor.

This permits us to rewrite Eq.(\ref{cl2}) as
\begin{equation}
\frac{1}{2}\overset{\eta}{\square}\mathfrak{g}^{\kappa}+\frac{1}{2}
m^{2}\mathfrak{g}^{\kappa}=-\mathfrak{T}^{\kappa}-\mathfrak{t}^{\kappa
}-\underset{\eta}{\delta}\mathfrak{K}^{\kappa}-\frac{1}{2}d\underset
{\underset{\eta}{\star}}{\delta}\mathfrak{g}^{\kappa}-\mathcal{R}^{\kappa
}=-\mathbf{T}^{\kappa}. \label{cl6}
\end{equation}
Thus, writing
\begin{equation}
\phi_{\iota}^{\kappa}:=h_{\mu}^{\kappa}\frac{\partial\mathrm{x}^{\mu}
}{\partial x^{\iota}},
\end{equation}
\begin{equation}
\mathfrak{g}^{\kappa}=\phi_{\iota}^{\kappa}dx^{\iota},\text{ }\mathbf{T}
^{\kappa}=\mathbf{T}_{\iota}^{\kappa}dx^{\iota},\text{ } \label{cl7}
\end{equation}
and taking into account that \cite{rodoliv2006}
\begin{equation}
\overset{\eta}{\square}\mathfrak{g}^{\kappa}=(g^{\alpha\beta}\overset{\eta}
{D}_{\alpha}\overset{\eta}{D}_{\beta}\phi_{\iota}^{\kappa})dx^{\iota}
\label{cl8}
\end{equation}
we get from Eq.(\ref{cl6})
\begin{equation}
g^{\alpha\beta}\overset{\eta}{D}_{\alpha}\overset{\eta}{D}_{\beta}\phi_{\iota
}^{\kappa}+m^{2}\phi_{\iota}^{\kappa}=-2\mathbf{T}_{\iota}^{\kappa},
\label{cl9}
\end{equation}
which is in our theory a possible form for the (covariant) equation for the
graviton field.in Minkowski spacetime. The last statement follows because
$\overset{\eta}{D}_{\alpha}$ can be easily be expressed in terms of the
$\mathring{D}_{\alpha}$ using the formulas of the Appendix.

\begin{remark}
If we recall that \emph{\cite{rodoliv2006}} for an arbitrary $L\in
\sec\bigwedge\nolimits^{p}T^{\ast}M,J\in\sec\bigwedge\nolimits^{r}T^{\ast}M$,
$p\leq r,$ we have:
\begin{align}
d\mathtt{h}^{\ast}L &  =\mathtt{h}^{\ast}dL\nonumber\\
\underset{

\sslg

}{\star}d\underset{

\sslg

}{\star}\mathtt{h}^{\ast}L &  =\text{ }\mathtt{h}^{\ast}\underset{\eta}{\star
}d\underset{\eta}{\star}L,\nonumber\\
\mathtt{h}^{\ast}L\underset{\eta}{\lrcorner}\mathtt{h}^{\ast}J &
=\mathtt{h}^{\ast}(L\underset{\eta}{\lrcorner}J),\label{ndif4}
\end{align}
we can immediately write from \emph{Eq.(\ref{SSS})} that
\begin{equation}
-\frac{1}{2}d\underset{

\sslg

}{\star}d\theta^{\kappa}+\frac{1}{2}m^{2}\underset{

\sslg

}{\star}\theta^{\kappa}=-\underset{

\sslg

}{\star}\left(  t^{\kappa}+T^{\kappa}+\underset{

\sslg

}{\star}^{-1}d\underset{

\sslg

}{\star}K^{\kappa}\right)  \label{lorentzian equiv}
\end{equation}
and also

\begin{equation}
\underset{

\sslg

}{\delta}\left(  T^{\kappa}+t^{\kappa}+\underset{

\sslg

}{\delta}K^{\kappa}-\frac{1}{2}m^{2}\theta^{\kappa}\right)  =0,
\label{clawlorentziann}
\end{equation}
where
\begin{equation}
\theta^{\kappa}=\mathtt{h}^{\ast}\mathfrak{g}^{\kappa},\text{ }t_{g}^{\alpha
}=\mathtt{h}^{\ast}(\mathfrak{t}^{\kappa}+\underset{\eta}{\delta}
\mathfrak{K}^{\kappa})=t^{\kappa}+\underset{\eta}{\delta}K^{\kappa},\text{
}T^{\alpha}=\mathtt{h}^{\ast}\mathfrak{T}^{\alpha}. \label{auto}
\end{equation}

\end{remark}

\begin{remark}
\emph{Eq.(\ref{clawlorentziann}) }express as we already anticipated a reliable
conservation law for the total energy-momentum of the matter plus the
gravitational field. However, take notice that in GR this result depends on
the fixing of a cotetrad basis and changing it by a local Lorentz
transformation changes accordingly the energy-momentum tensor of the
gravitational field. In fact this last result has already been known since the
work\footnote{Which however did not use the present crystal clear formalism.}
of M\o ller \emph{\cite{moller}}. Moreover, we see that imposing the Lorenz
type gauge $\underset{

\sslg

}{\delta}\theta^{\kappa}=0=\underset{\eta}{\delta}\mathfrak{g}^{\kappa}$ to
the dynamic gravitational fields amounts to exclude the graviton energy
density from the \ conservation law.
\end{remark}

\section{Which Gauge to Use for $

\slg

$ in the Effective Lorentzian Spacetime?}

We already recalled that our Lagrangian density differs for the
Einstein-Hilbert Lagrangian by an exact form. But it can also be written as:
\begin{equation}
\mathcal{L}_{g}=-\frac{1}{2}(d\mathfrak{g}_{\alpha}\wedge\mathfrak{g}
^{\mathbf{\beta}})\wedge\underset{

\sslg

}{\star}(d\mathfrak{g}_{\mathbf{\beta}}\wedge\mathfrak{g}^{\mathbf{\alpha}
})+\frac{1}{4}d\mathfrak{g}_{\mathbf{\alpha}}\wedge\mathfrak{g}
^{\mathbf{\alpha}}\wedge\underset{

\sslg

}{\star}(d\mathfrak{g}_{\mathbf{\beta}}\wedge\mathfrak{g}^{\mathbf{\beta}
})\label{Lb}
\end{equation}
which can be shown to be equivalent (modulus an exact differential) to

\begin{equation}
\mathcal{L}_{g}=-\frac{1}{2}(d\vartheta_{\alpha}\wedge\vartheta^{\beta}
)\wedge\underset{

\sslg

}{\star}(d\vartheta_{\beta}\wedge\mathfrak{\vartheta}^{\alpha})+\frac{1}
{4}d\vartheta_{\alpha}\wedge\vartheta^{\alpha}\wedge\underset{

\sslg

}{\star}(d\vartheta_{\beta}\wedge\vartheta^{\beta})\label{Lb1}
\end{equation}
where $\{\vartheta^{\alpha}\}$ is an  an \textit{arbitrary }coframe basis,
\textit{not} necessaritly a $

\slg

$ orthonormal one.

This pemits us \cite{fr} to obtain an equation analogous to Eq.(\ref{8.27}), i.e.,

\begin{equation}
d\underset{

\sslg

}{\star}S^{\alpha}+\underset{

\sslg

}{\star}t^{\alpha}+\frac{1}{2}m^{2}\underset{

\sslg

}{\star}\vartheta^{\alpha}=-\underset{

\sslg

}{\star}T^{\alpha}.\label{b2}
\end{equation}

So, let us examine the structure of Eq.(\ref{b2}) in a coordinate basis
$\{\vartheta^{\mu}=dx^{\mu}\}$\footnote{Keep in mind that $\vartheta
_{\mathbf{\alpha}}=g_{\mathbf{\alpha\beta}}dx^{\mathbf{\beta}}$.}. We
immediately see that a conservation law (distinct from the previous one
established above) \footnote{Recall that in GR Eq.(\ref{b2}) \ implies in a
pseudo conservation law because in that theory (without a Minkowski spacetime
interpretation, as here) $S^{\alpha}$ are expressed in terms of connection
-forms of the Levi-Civita connection of $

\slg

$ and thus are not \textit{indexed }forms. Details may be found in
\cite{edrod2008}.} in the effective Lorentzian spacetime structure
(\textit{excluding} the energy associated with the graviton mass) exists for
$\underset{

\sslg

}{\star}(T^{\mu}+t^{\mu})$ if
\begin{equation}
\underset{

\sslg

}{\delta}\vartheta^{\mu}=0.\label{b4a}
\end{equation}
This of course, implies that
\begin{equation}
\lozenge x^{\mu}=-d\underset{

\sslg

}{\delta}x^{\mu}-\underset{

\sslg

}{\delta}dx^{\mu}=0,\label{b5a}
\end{equation}
i.e., the coordinates must be \textit{harmonic}.

Now, if
\begin{equation}
D_{\frac{\partial}{\partial x^{\nu}}}\frac{\partial}{\partial x^{\mu}}
=\Gamma_{\nu\mu}^{\rho}\frac{\partial}{\partial x^{\rho}},\text{ \ \ }
D_{\frac{\partial}{\partial x^{\nu}}}\vartheta^{\mu}=-\Gamma_{\nu\rho}^{\mu
}\vartheta^{\rho}
\end{equation}
we have
\begin{align}
\underset{

\sslg

}{\delta}\vartheta^{\mu}  &  =-\vartheta^{\nu}\underset{

\sslg

}{\lrcorner}D_{\frac{\partial}{\partial\mathrm{x}^{\nu}}}\vartheta^{\mu
}=\vartheta^{\nu}\underset{

\sslg

}{\lrcorner}\left(  \Gamma_{\nu\alpha}^{\mu}\vartheta^{\alpha}\right)
=\Gamma_{\nu\alpha}^{\mu}g^{\nu\alpha}=0,\label{b7a}\\
\lozenge x^{\mu}  &  =0\Rightarrow\Gamma_{\nu\alpha}^{\mu}g^{\nu\alpha}=0
\end{align}

Now, if $\Gamma_{\nu\alpha}^{\mu}g^{\nu\alpha}=0$ we have
\begin{align*}
D_{\mu}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})  &  =\frac{\partial}{\partial
x^{\mu}}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})+\Gamma_{\mu\alpha}^{\nu}
\sqrt{-\det(g_{\mu\nu})}g^{\mu\alpha}\\
&  =\frac{\partial}{\partial x^{\mu}}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})\\
&  =\frac{\partial}{\partial x^{\mu}}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu
})+\mathring{\Gamma}_{\mu\alpha}^{\nu}\sqrt{-\det(g_{\mu\nu})}g^{\mu\alpha
}-\mathring{\Gamma}_{\mu\alpha}^{\nu}\sqrt{-\det(g_{\mu\nu})}g^{\mu\alpha}\\
&  =\mathring{D}_{\mu}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})-\mathring{\Gamma
}_{\mu\alpha}^{\nu}\sqrt{-\det(g_{\mu\nu})}g^{\mu\alpha}
\end{align*}
and since $D_{\mu}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})=0$ we get that

\begin{equation}
\lozenge x^{\mu}=0\Rightarrow\mathring{D}_{\mu}(\sqrt{-\det(g_{\mu\nu})}
g^{\mu\nu})=\mathring{\Gamma}_{\mu\alpha}^{\nu}\sqrt{-\det(g_{\mu\nu})}
g^{\mu\alpha} \label{harm x Do}
\end{equation}

In particular in a coordinate basis $\{\gamma^{\mu}=d\mathrm{x}^{\mu}\}$,
where $\{\mathrm{x}^{\mu}\}$ are global coordinates for $M$ in
Einstein-Lorentz-Poincar\'{e} gauge\footnote{Recall that the energy-momentum
conservation law of any \textit{Lorentz invariant} field theory is
unambiguously formulated global coordinates in Einstein-Lorentz-Poincar\'{e}
gauge.} where the connection coefficients are $\mathtt{\mathring{\Gamma}}
_{\mu\alpha}^{\nu}=0$ ($\mathtt{\mathring{D}}_{\frac{\partial}{\partial
\mathrm{x}^{\nu}}}\frac{\partial}{\partial\mathrm{x}^{\mu}}=\mathtt{\mathring
{\Gamma}}_{\nu\mu}^{\rho}\frac{\partial}{\partial\mathrm{x}^{\rho}}$, $

\slg

=\mathtt{g}_{\mu\nu}d\mathrm{x}^{\mu}\otimes d\mathrm{x}^{\nu}$) we have
\[
\lozenge\mathrm{x}^{\mu}=0\Rightarrow\mathtt{\mathring{D}}_{\mu}(\sqrt
{-\det(\mathtt{g}_{\mu\nu})}\mathtt{g}^{\mu\nu})=\frac{\partial}
{\partial\mathrm{x}^{\mu}}(\sqrt{-\det(\mathtt{g}_{\mu\nu})}\mathtt{g}^{\mu
\nu})=0.
\]
Also, for arbitrary \textit{non} harmonic coordinates functions $\{x^{\mu}\}$
we get ($\mathring{D}_{\frac{\partial}{\partial x^{\nu}}}\frac{\partial
}{\partial x^{\mu}}=\mathring{\Gamma}_{\mu\nu}^{\kappa}\frac{\partial
}{\partial x^{\kappa}},D_{\frac{\partial}{\partial x^{\nu}}}\frac{\partial
}{\partial x^{\mu}}=\Gamma_{\mu\nu}^{\kappa}\frac{\partial}{\partial
x^{\kappa}}$) that
\begin{equation}
\lozenge x^{\mu}=-\Gamma_{\alpha\nu}^{\mu}g^{\alpha\nu}=-\mathring{\Gamma
}_{\alpha\nu}^{\mu}g^{\alpha\nu}-K_{\alpha\nu}^{\mu}g^{\alpha\nu}. \label{b8}
\end{equation}

Now, observe that the so called \textit{Logunov gauge condition }
(Eq.(\ref{logunov}))
\[
\mathring{D}_{\mu}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})=0.
\]
does \textit{not} imply that the coordinates are harmonic ones, for we have
(using the formulas in the Appendix)
\begin{align}
\mathring{D}_{\mu}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})  &  =D_{\mu}
(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})+K_{\mu\kappa}^{\nu}\sqrt{-\det(g_{\mu\nu
})}g^{\mu\kappa}\nonumber\\
&  =K_{\mu\kappa}^{\nu}\sqrt{-\det(g_{\mu\nu})}g^{\mu\kappa}
\end{align}
and thus
\begin{equation}
\mathring{D}_{\mu}(\sqrt{-\det(g_{\mu\nu})}g^{\mu\nu})=0\Rightarrow
K_{\mu\kappa}^{\nu}\sqrt{-\det(g_{\mu\nu})}g^{\mu\kappa}=0, \label{bb100}
\end{equation}
and we see that under those conditions\footnote{The ones in Logunov theory.}
the \textit{allowed} coordinate functions must always satisfy the constraint:
\begin{equation}
\lozenge x^{\mu}=-\Gamma_{\alpha\nu}^{\mu}g^{\alpha\nu}=-\mathring{\Gamma
}_{\alpha\nu}^{\mu}g^{\alpha\nu}. \label{B10''}
\end{equation}

Finally we comment that Logunov \cite{logunov1,logunov2} imposed
Eq.(\ref{logunov}) as a gauge condition in his theory because he
\textit{postulated} (differently from what is the case here, see below)) that
the relation between $\underset{L}{g}^{\alpha\beta}$ and the gravitational
field $\mathbf{h}^{\alpha\beta}$ is given by
\begin{equation}
\underset{L}{\varkappa}\underset{L}{g}^{\alpha\beta}:=\mathring{g}
^{\alpha\beta}+\mathbf{h}^{\alpha\beta}. \label{logunov postulate}
\end{equation}
\ \

Now, the second member of Eq.(\ref{logunov postulate}) implies immediately
taking into account Eq.(\ref{RESTRICTION}) that $\mathring{D}_{\alpha
}(\underset{L}{\varkappa}\underset{L}{g}^{\alpha\beta})=0$ and thus from the
second line in Eq.(\ref{shazam}) (in Appendix) we get that $\mathring
{D}_{\alpha}\left(  \sqrt{-\det(\underset{L}{g}\text{ }_{\alpha\beta}
)}\underset{L}{g}^{\alpha\beta}\right)  =0.$

In our theory, defining $\{\vartheta_{\mu}\}$ as the \textit{reciprocal} basis
of $\{\vartheta^{\mu}\}$ relative to \texttt{g}, i.e., \texttt{g}
$(\vartheta_{\mu},\vartheta^{\nu})=\delta_{\mu}^{\nu}$ we can write directly
from Eq.(\ref{the teta}) that

\begin{align}

\slg

&  =\eta_{\alpha\beta}h^{-1}(\gamma_{\alpha})\otimes h^{-1}(\gamma_{\beta
})\nonumber\\
&  =g^{\kappa\iota}\vartheta_{\kappa}\otimes\vartheta_{\iota},
\end{align}
and clearly, differently from Logunov's theory we have:
\begin{equation}
\varkappa g^{\kappa\iota}\neq\underset{L}{\varkappa}\underset{L}{g}
^{\alpha\beta} \label{WTHEORY}
\end{equation}
from where it follows that there is no need to impose a priory any gauge
condition (as it is the case in GR) for the "metric" field $

\slg

$ of the effective Lorentzian spacetime.

\section{Conclusions}

We showed that if gravitation is to be described by a massive graviton field
living in Minkowski spacetime which is represented by a symmetric tensor
field\textbf{ }$\mathbf{h}$ carrying the representations of spin \textit{two}
and \textit{zero} of the Lorentz group and thus satisfying the gauge condition
given by Eq.(\ref{RESTRICTION}) then the \textit{effective} Lorentzian
spacetime structure that represents the gravitational field (under the ansatz
given by Eq.(\ref{def g})) of a given energy-momentum distribution is such
that the field $

\slg

$ solving the effective Einstein-Hilbert equation (with cosmological
constant)---as it is the case in GR--- does \textit{not} need to satisfy a
priory any fixed gauge. We showed moreover that the Logunov gauge condition
$\mathring{D}_{\gamma}\left(  \sqrt{-\det(g_{\mu\nu})}g^{\gamma\kappa}\right)
=0$ ( which in his theory is indeed a postulate) does not hold\ in general in
our theory without ad hoc hypothesis. If such a gauge is postulated it implies
that the allowed coordinate functions to the ones satisfying Eq.(\ref{B10''}).
Moreover, we proved that the imposition of the Lorenz type gauge gauge
$\underset{

\sslg

}{\delta}\theta^{\kappa}=0=$ $\underset{\eta}{\delta}\mathfrak{g}^{\kappa}$ to
the dynamic gravitational fields amounts to exclude the graviton energy
density from the energy-momentum\ conservation law.

Logunov thought that the importance of the gauge condition $\mathring
{D}_{\gamma}\left(  \sqrt{-\det(g_{\mu\nu})}g^{\gamma\kappa}\right)  =0$ in
determining the effective Lorentzian spacetime generated by an energy-momentum
distribution can be seem from the following example \cite{logunov1,logunov2}.
Let $\{t,r,\theta,\varphi\}$ be the usual spherical coordinates in Minkowski spacetime.

If we try to solve the (effective) Einstein-Hilbert equations (in the zero
mass graviton case) for the field generated by a point mass at the origin of
the coordinate system we get immediately that the following \textquotedblleft
metric\textquotedblright\ fields are solutions of those equations,
\begin{equation}

\slg

_{s}=\left(  1-\frac{2m}{r}\right)  dt\otimes dt-\left(  1-\frac{2m}
{r}\right)  ^{-1}dr\otimes dr-r^{2}(d\theta\otimes d\theta+\sin^{2}\theta
d\varphi\otimes d\varphi), \label{digr2}
\end{equation}
and
\begin{align}

\slg

_{i}  &  =\left(  \frac{r+\lambda-m}{r+\lambda+m}\right)  dt\otimes dt-\left(
\frac{r+\lambda+m}{r+\lambda-m}\right)  dr\otimes dr\nonumber\\
&  -(r+\lambda+m)^{2}(d\theta\otimes d\theta+\sin^{2}\theta d\varphi\otimes
d\varphi), \label{digr3}
\end{align}
with $\lambda$ an arbitrary real parameter. Now, both solutions have the same
asymptotic behavior when $r\rightarrow\infty$. Which one should we use for the
descriptions of physical processes? It is important to emphasize that both
metrics even if expressed in the same coordinates \textit{are
diffeomorphically equivalent} since.it is possible to perform a coordinate
trasformation in Eq.(\ref{digr2}) which makes it in the new variables to have
the apperance of Eq.(\ref{digr3}). If we take into account that the
\textit{meaning} of the coordinates in each one are \textit{different} because
we can know what the spacetime labels mean, only after we fix a metric on it.
Specifically this statement means that those labels are associated with
physical distances and time lapses measured by ideal rods and clocks in
different ways.

But Logunov thinks that $

\slg

_{s}$ and $

\slg

_{i}$\ given in the \textit{same} coordinate basis even if diffeomorphically
equivalent are physically distinguished through experiments and so fixing one
of them as the correct one implies in the existence of an additional
theoretical criterion and such a criterion does not exists in GR. He claims
that the metric $

\slg

_{i}$ when $\lambda=0$ that satisfies the condition $\mathring{D}_{\gamma
}\left(  \sqrt{-\det(g_{\mu\nu})}g^{\gamma\kappa}\right)  =0$ is the
\textit{only} one that fits correctly all known data on solar system
experiments.\ Does the method used by astronomers methods for determining the
coordinates of their probes always fix those coordinates as being the
spherical coordinates of Minkowski spacetime and fix the metric to be $

\slg

_{i}$?. It is hard to believe in that possibility...

A last comment is in order. We start our considerations by postulating that
the distortion field $h$ is symmetric since it has been constructed from the
symmetric tensor field \textbf{h}. However, from the general theory of plastic
deformations of the Lorentz vacuum presented in \cite{fr} it is quite clear
that we can construct symmetric metric tensor fields associated to non
symmetric $h$ fields\footnote{In \cite{fr} it is directly derived from the
variational principle and an appropriate Lagrangian the field equations for
the plastic extensor field $h$.}. This observation shows that the quantum
theory of the gravitational field must be more complex than one where the $

\slg

$ field is supposed to arise from the existence of a symmetric graviton field.
We will return to this issue in another publication.\medskip

\appendix\textbf{Appendix\medskip}

Let $(M,\overset{\circ}{

\slg

},\mathring{D}\mathbf{)}$ and $(M,

\slg

,D)$ be the two Lorentzian structures\footnote{More general formulas relating
two arbitrary general connections may be found, e.g., in
\cite{rodoliv2006,quinrod1995}.} on the same manifold $M$ such that
\begin{equation}
\mathring{D}\overset{\circ}{

\slg

}=0,D

\slg

=0,
\end{equation}
with the \textit{non-metricity} of $D$ relative to $\overset{\circ}{

\slg

}$ being given by:
\[
\mathbf{Q:=-}D\overset{\circ}{

\slg

}.
\]
Let moreover the connection coefficients of $\mathring{D}$ and $D$ in
arbitrary coordinates $\{x^{\mu}\}$ covering $U\subset M$ be:

\begin{equation}
\mathring{D}_{\partial_{\alpha}}dx^{\rho}=-\mathring{\Gamma}_{\alpha\beta
}^{\rho}dx^{\beta},D_{\partial_{\alpha}}dx^{\rho}=-\Gamma_{\alpha\beta}^{\rho
}dx^{\beta},
\end{equation}
and
\begin{equation}
Q_{\alpha\beta\sigma}=-D_{\alpha}\mathring{g}_{\beta\sigma},
\end{equation}
Define the components of the \textit{strain tensor} of the connection $D$ by:
\begin{equation}
S_{\alpha\beta}^{\rho}=2\Gamma_{\alpha\beta}^{\rho}-2\mathring{\Gamma}
_{\alpha\beta}^{\rho}.
\end{equation}

Then
\begin{align}
Q_{\alpha\beta\sigma}  &  =\frac{1}{2}(\mathring{g}_{\mu\sigma}S_{\alpha\beta
}^{\mu}+\mathring{g}_{\beta\mu}S_{\alpha\sigma}^{\mu}),\label{1077}\\
S_{\alpha\beta}^{\rho}  &  =\mathring{g}^{\rho\sigma}(Q_{\alpha\beta\sigma
}+Q_{\beta\sigma\alpha}-Q_{\sigma\alpha\beta}).
\end{align}
Also,
\begin{equation}
Q_{\alpha\beta\sigma}+Q_{\sigma\alpha\beta}+Q_{\beta\sigma\alpha}
=S_{\alpha\beta\sigma}+S_{\sigma\alpha\beta}+S_{\beta\sigma\alpha},
\end{equation}
where $S_{\alpha\beta\sigma}=\mathring{g}_{\rho\sigma}S_{\alpha\beta}^{\rho}$.

Putting
\begin{equation}
K_{\alpha\beta}^{\!\rho}=\frac{1}{2}S_{\alpha\beta}^{\rho}. \label{1168}
\end{equation}
we have
\begin{equation}
K_{\alpha\beta}^{\!\rho}=-\frac{1}{2}\mathring{g}^{\rho\sigma}(D_{\!\alpha
}\mathring{g}_{\beta\sigma}+D_{\!\beta}\mathring{g}_{\sigma\alpha}
-D_{\!\sigma}\mathring{g}_{\alpha\beta}) \label{1130}
\end{equation}

The relation between the curvature tensor $R_{\mu}{}^{\rho}{}_{\!\alpha\beta}$
associated with the connection $D$ and the Riemann curvature tensor
$\mathring{R}{}_{\mu}{}^{\rho}{}_{\!\alpha\beta}$ of the Levi-Civita
connection $\mathring{D}$ associated with the metric\texttt{ }$\overset{\circ
}{

\slg

}$ \ are given by:
\begin{equation}
R_{\mu}{}^{\rho}{}_{\!\alpha\beta}=\mathring{R}_{\mu}{}^{\rho}{}
_{\!\alpha\beta}+J_{\mu}{}^{\rho}{}_{\![\alpha\beta]}, \label{1070}
\end{equation}
where:
\begin{equation}
J_{\!\mu}{}^{\rho}{}_{\!\alpha\beta}=\mathring{D}_{\!\alpha}K_{\beta\mu
}^{\!\rho}-K_{\beta\sigma}^{\!\rho}K_{\alpha\mu}^{\!\sigma}=D_{\!\alpha
}K_{\beta\mu}^{\!\rho}-K_{\alpha\sigma}^{\!\rho}K_{\beta\mu}^{\!\sigma
}+K_{\alpha\beta}^{\!\sigma}K_{\sigma\mu}^{\!\rho}. \label{1070a}
\end{equation}
Multiplying both sides of Eq.(\ref{1070}) by $\frac{1}{2}\theta^{\alpha}
\wedge\theta^{\beta}$ we get for the curvature $2$-forms of the two
connections $D$ and $\mathring{D}$:
\begin{equation}
\mathcal{R}_{\mu}^{\rho}=\mathcal{\mathring{R}}_{\mu}^{\rho}+\mathfrak{J}
_{\mu}^{\rho}, \label{1208}
\end{equation}
where we have written:
\begin{equation}
\mathfrak{J}_{\mu}^{\rho}=\frac{1}{2}J_{\!\mu}{}^{\rho}{}_{\![\alpha\beta
]}\theta^{\alpha}\wedge\theta^{\beta}.
\end{equation}

The relation between the Ricci tensors\footnote{For the Ricci tensor
$Ricci=R_{\mu\alpha}dx^{\mu}\otimes dx^{\nu}$, we use the convention
$R_{\mu\alpha}:=R_{\mu}{}^{\rho}{}_{\!\alpha\rho}$.} of the connections $D$
and $\mathring{D}$ is:
\begin{equation}
R_{\mu\alpha}=\mathring{R}_{\mu\alpha}+J_{\mu\alpha}, \label{1174}
\end{equation}
with
\begin{align}
J_{\mu\alpha}  &  =\mathring{D}_{\!\alpha}K_{\rho\mu}^{\!\rho}-\mathring
{D}_{\!\rho}K_{\alpha\mu}^{\!\rho}+K_{\alpha\sigma}^{\!\rho}K_{\rho\mu
}^{\!\sigma}-K_{\rho\sigma}^{\!\rho}K_{\alpha\mu}^{\!\sigma}\nonumber\\
&  =D_{\!\alpha}K_{\rho\mu}^{\!\rho}-D_{\!\rho}K_{\alpha\mu}^{\!\rho
}-K_{\sigma\alpha}^{\!\rho}K_{\rho\mu}^{\!\sigma}+K_{\rho\sigma}^{\!\rho
}K_{\alpha\mu}^{\!\sigma}.
\end{align}

Recall that the connection $\mathring{D}$ plays with respect to the tensor
field $

\slg

$ a role analogous to that played by the connection $D$ with respect to the
metric tensor $\overset{\circ}{

\slg

}$ and in consequence we shall have similar equations relating these two pairs
of objects. In particular, the strain of $\mathring{D}$ with respect to $

\slg

$ equals the negative of the strain of $D$ with respect to $\overset{\circ}{

\slg

}$, since we have:
\[
S_{\alpha\beta}^{\rho}=\Gamma_{\alpha\beta}^{\rho}+\Gamma_{\beta\alpha}^{\rho
}-\mathring{b}_{\alpha\beta}^{\rho}=-(\mathring{\Gamma}_{\alpha\beta}^{\rho
}+\mathring{\Gamma}_{\beta\alpha}^{\rho}-b_{\alpha\beta}^{\rho}),
\]
where $b_{\alpha\beta}^{\rho}=\mathring{\Gamma}_{\alpha\beta}^{\rho}
+\mathring{\Gamma}_{\beta\alpha}^{\rho}$ and $b_{\alpha\beta}^{\rho}
=\Gamma_{\alpha\beta}^{\rho}+\Gamma_{\beta\alpha}^{\rho}$ . Furthermore, we
have that:
\begin{align}
K_{\alpha\beta}^{\!\rho}  &  =-\frac{1}{2}\mathring{g}^{\rho\sigma
}(D_{\!\alpha}\mathring{g}_{\beta\sigma}+D_{\!\beta}\mathring{g}_{\alpha
\sigma}-D_{\!\sigma}\mathring{g}_{\alpha\beta})\nonumber\\
&  =\frac{1}{2}g^{\rho\sigma}(\mathring{D}_{\!\alpha}g_{\beta\sigma}
+\mathring{D}_{\!\beta}g_{\alpha\sigma}-\mathring{D}_{\!\sigma}g_{\alpha\beta
}).
\end{align}
Now, recall that given arbitrary coordinates $\{x^{\alpha}\}$ covering
$U\subset$ $M$ \ and $\{x^{\prime\alpha}\}$ covering covering $V\subset$ $M$
($U\cap V\neq\varnothing$)\ a set of functions $A_{\nu_{1}...\nu_{s}}^{\mu
_{1}...\mu_{r}}(x^{\alpha})$ is said to be the components of a relative tensor
field of type$\left(
\begin{array}
[c]{c}
r\\
s
\end{array}
\right)  $ and weight $w$ if under the coordinate transformation $x^{\alpha
}\mapsto x^{\prime\beta}$ with Jacobian $J=\det\left(  \frac{\partial
x^{\prime\alpha}}{\partial x^{\beta}}\right)  $these functions transform as
\cite{lovrund}
\begin{equation}
A_{\kappa_{1}...\kappa_{s}}^{\prime\lambda_{1}...\lambda_{r}}(x^{\prime\beta
})=J^{w}\frac{\partial x^{\prime\lambda_{1}}}{\partial x^{\mu_{1}}}
...\frac{\partial x^{\prime\lambda_{1}}}{\partial x^{\mu_{1}}}\frac{\partial
x^{\nu_{1}}}{\partial x^{\kappa_{1}}}...\frac{\partial x^{\nu_{s}}}{\partial
x^{\kappa_{s}}}A_{\nu_{1}...\nu_{s}}^{\mu_{1}...\mu_{r}}(x^{\alpha}).
\label{relative tensor}
\end{equation}
The covariant derivative of a relative tensor field relative to a given
arbitrary connection $\nabla$ defined in $M$ \ such that $\nabla
_{\frac{\partial}{x^{\nu}}}dx^{\mu}=-\mathbf{L}_{\nu\alpha}^{\mu}dx^{\alpha}$
is given by
\begin{align}
\nabla_{\kappa}A_{\nu_{1}...\nu_{s}}^{\mu_{1}...\mu_{r}}  &  =\frac{\partial
}{\partial x^{\kappa}}A_{\nu_{1}...\nu_{s}}^{\mu_{1}...\mu_{r}}+\mathbf{L}
_{\iota\kappa}^{\mu_{p}}A_{\nu_{1}....................\nu_{s}}^{\mu_{1}
...\mu_{p-1}\iota\mu_{p+1}...\mu_{r}}\nonumber\\
&  -\mathbf{L}_{\nu_{q}\kappa}^{\iota}A_{\nu_{1}...\nu_{q-1}\iota\nu
_{q+1}...\nu_{s}}^{\mu_{1}....................\mu_{r}}-w\mathbf{L}
_{\kappa\sigma}^{\sigma}A_{\nu_{1}...\nu_{s}}^{\mu_{1}...\mu_{r}}.
\label{cdrt}
\end{align}
In particular we have
\begin{align}
\mathring{D}_{\alpha}(\sqrt{-\det(\mathring{g}_{\mu\nu})})  &  =\partial
_{\gamma}\left(  \sqrt{-\det(\mathring{g}_{\mu\nu})}\right)  -\mathring
{\Gamma}_{\gamma\rho}^{\rho}\sqrt{-\det(\mathring{g}_{\mu\nu})}=0,\nonumber\\
\mathring{D}_{\alpha}\left(  \frac{1}{\sqrt{-\det(\mathring{g}_{\mu\nu})}
}\right)   &  =\partial_{\gamma}\left(  \frac{1}{\sqrt{-\det(\mathring{g}
_{\mu\nu})}}\right)  +\mathring{\Gamma}_{\gamma\rho}^{\rho}\frac{1}
{\sqrt{-\det(\mathring{g}_{\mu\nu})}}=0,\nonumber\\
D_{\alpha}(\sqrt{-\det(g_{\mu\nu})})  &  =\partial_{\gamma}\left(  \sqrt
{-\det(g_{\mu\nu})}\right)  -\Gamma_{\gamma\rho}^{\rho}\sqrt{-\det(g_{\mu\nu
})}=0,\nonumber\\
\mathring{D}_{\alpha}\left(  \frac{1}{\sqrt{-\det(g_{\mu\nu})}}\right)   &
=\partial_{\gamma}\left(  \frac{1}{\sqrt{-\det(g_{\mu\nu})}}\right)
+\Gamma_{\gamma\rho}^{\rho}\frac{1}{\sqrt{-\det(g_{\mu\nu})}}=0.
\label{particular}
\end{align}

Now, if we define
\begin{equation}
\varkappa:=\sqrt{\frac{\det(g_{\mu\nu})}{\det(\mathring{g}_{\mu\nu})}},
\end{equation}
we can easily prove the the following relations:
\begin{align}
&  K_{\rho\sigma}^{\!\rho}=-\frac{1}{2}\mathring{g}^{\alpha\beta}D_{\!\sigma
}\mathring{g}_{\alpha\beta}=\frac{1}{2}g^{\alpha\beta}\mathring{D}_{\!\sigma
}g_{\alpha\beta}=\frac{1}{\varkappa}\partial_{\sigma}(\varkappa)\nonumber\\
&  g^{\alpha\beta}K_{\alpha\beta}^{\!\rho}=-\frac{1}{\varkappa}\mathring
{D}_{\!\sigma}(\varkappa g^{\rho\sigma})=-\frac{1}{\sqrt{-\det(g_{\mu\nu})}
}\mathring{D}_{\!\sigma}(\sqrt{-\det(g_{\mu\nu})}g^{\rho\sigma})
\label{shazam}\\
&  \mathring{g}^{\alpha\beta}K_{\alpha\beta}^{\!\rho}=\frac{1}{\varkappa^{-1}
}D_{\!\sigma}(\varkappa^{-1}\mathring{g}^{\rho\sigma}).\nonumber
\end{align}

Another useful formulas valid for our \textit{particular} connections
$\mathring{D}$ and $D$ are:
\begin{equation}
\begin{array}
[c]{l}
\mathring{D}_{\!\alpha}K_{\rho\beta}^{\!\rho}=\mathring{D}_{\!\beta}
K_{\rho\alpha}^{\!\rho}\\
D_{\!\alpha}K_{\rho\beta}^{\!\rho}=D_{\!\beta}K_{\rho\alpha}^{\!\rho}.
\end{array}
\end{equation}


TITLE: What does $^33$ mean?
QUESTION [1 upvotes]: I came across a type of function/notation on a test that I'd never seen before, and for formatting reasons, I haven't been able to find any answers online. Can anyone help me with what this means/how to solve it? Thanks in advance.
the question:

REPLY [1 votes]: The notation $^n a$ is usually used to denoted tetration. Tetration is the next hyperoperation after exponentiation.
So to build up through the various lower level hyperoperations from  successor, addition, multiplication, and exponentiation to your particular example of tetration we have:
\begin{align*}
3 + 3 &= 3 + \underbrace{(1 + 1 + 1)}_{\text{3 copies of 1 added to 3}} = 6\\
3 \times 3 &= \underbrace{3 + 3 + 3}_{\text{3 copies of 3 combined by addition}} = 9\\
3^3 &= \underbrace{3 \times 3 \times 3}_{\text{3 copies of 3 combined by multiplication}} = 27\\
^3 3 &= \underbrace{3^{3^3}}_{\text{3 copies of 3 combined by exponentiation}} = 7 \, 625 \, 597 \, 484 \, 987
\end{align*}


TITLE: Is there a theory for cellular automata propagating signals in straight lines?
QUESTION [0 upvotes]: Is there a theory explaining how a cellular automata can propagate signals in straight lines?
For example, this video shows how some "signals" travel down at a diagonal, even though they are composed of 5 white cells.

What I'm wondering is, has any work been done to describe the different cases or general models for how different numbers of "white cells" can travel in different angles in a grid? So in the video, there are 5 white cells, traveling at roughly a 45 degree angle. But what about 6 white cells, what about 20, or even just 1? And what about a 44 degree angle, or a 44.995 degree (some very precise) angle? What are the general patterns for how the cells move at each step?
Basically, what would the rules be for those types of "signal propagations", or general formulas based on angle and number of cells, how to make it move along a path. That sort of thing. Has any work been done in this area?

REPLY [1 votes]: From the computer science point of view, there is a survey by Marianne Delorme and Jacques Mazoyer: Signals on Cellular Automata.
From the dynamical system point of view, people sometimes talk about defects or particles or kinks, which are similar to what you would call a signal.  See for example the relevant section of this survey by Marcus Pivato and the references therein.
You could also check out this (and related) articles by James Hanson and James Crutchfield on what they call the computational mechanics of cellular automata.