Higher Dimensional Gravity and Farkas Property in Oriented Matroid Theory

TL;DR

This paper explores a duality symmetry in higher-dimensional gravity with a split manifold, revealing an analogy with Farkas property in oriented matroid theory, and discusses potential applications in physics like cosmology.

Contribution

It introduces a duality invariance in a split higher-dimensional gravity model, connecting it to Farkas property in oriented matroid theory, and suggests broader physical implications.

Findings

Duality transformation $ ightarrow$ invariance of one part of the action

Special case $ ightarrow$ simplified form when $ ext{psi} = ext{phi}^{-1}$

Analogy established between Farkas property and gravitational duality

Abstract

We assume gravity in a $d$-dimensional manifold $M$ and consider a splitting of the form $M=M_{p}\times M_{q}$, with $d=p+q$. The most general two-block metric associated with $M_{p}$ and $M_{q}$ is used to derive the corresponding Einstein-Hilbert action $\mathcal{S}$. We focus on the special case of two distinct conformal factors $\psi $ and $\phi $ ($\psi $ for the metric in $M_{p}$ and $\phi $ for the metric in $M_{q}$), and we write the action $\mathcal{S}$ in the form $\mathcal{S=S}_{p}\mathcal{+S}%_{q} $, where $\mathcal{S}_{p}$ and $\mathcal{S}_{q}$ are actions associated with $M_{p}$ and $M_{q}$, respectively. We show that a simplified action is obtained precisely when $\psi =\phi ^{-1}$. In this case, we find that under the duality transformation $\phi \leftrightarrow \phi ^{-1}$, the action $\mathcal{S}_{p}$ for the $M_{p}$-space or the action $\mathcal{S}%_{q}$ for the $M_{q}$-space remain invariant, but not both. This result establishes an analogy between Farkas property in oriented matroid theory and duality in general relativity. Furthermore, we argue that our approach can be used in several physical scenarios such as 2t physics and cosmology.

Equations

$$\gamma_{AB}=\left(\begin{array}[]{cc}g_{\mu\nu}(x,y)&0\\ 0&g_{ij}(x,y)\end{array}\right),$$

$${\mathcal{R}}_{BCD}^A={\partial }_C{\Gamma }_{DB}^A-{\partial }_D{\Gamma }_{CB}^A+{\Gamma }_{CF}^A{\Gamma }_{DB}^F-{\Gamma }_{DF}^A{\Gamma }_{CB}^F,$$

$${\Gamma }_{BC}^A=\frac{1}{2}{g}^{AF}(g_{BF,C}+g_{CF,B}-g_{BC,F}).$$

$$\begin{array}[]{ccc}\Gamma_{\alpha\beta}^{\mu}=\left\{{}_{\alpha\beta}^{\mu}\right\},&\Gamma_{\alpha i}^{\mu}=\frac{1}{2}g^{\mu\lambda}g_{\alpha\lambda,i}\ ,&\Gamma_{ij}^{\mu}=-\frac{1}{2}g^{\mu\lambda}g_{ij,\lambda},\\ &&\\ \Gamma_{jk}^{i}=\left\{{}_{jk}^{i}\right\}&\Gamma_{j\alpha}^{i}=\frac{1}{2}g^{il}g_{jl,\alpha}\ ,&\Gamma_{\alpha\beta}^{i}=-\frac{1}{2}g^{il}g_{\alpha\beta,l}.\end{array}$$

$${\mathcal{R}}_{\nu \alpha \beta }^{\mu }=R_{\nu \alpha \beta }^{\mu }+{\Gamma }_{\alpha k}^{\mu }{\Gamma }_{\beta \nu }^k-{\Gamma }_{\beta k}^{\mu }{\Gamma }_{\alpha \nu }^k,$$

$$R_{\nu \alpha \beta }^{\mu }={\partial }_{\alpha }{\Gamma }_{\beta \nu }^{\mu }-{\partial }_{\beta }{\Gamma }_{\alpha \nu }^{\mu }+{\Gamma }_{\alpha \lambda }^{\mu }{\Gamma }_{\beta \nu }^{\lambda }-{\Gamma }_{\beta \lambda }^{\mu }{\Gamma }_{\alpha \nu }^{\lambda }.$$

$${\mathcal{R}}_{\nu \alpha \beta }^{\mu }=R_{\nu \alpha \beta }^{\mu }+\frac{1}{4}(g^{\mu \lambda }{g}^{kl}{g}_{\beta \lambda ,k}{g}_{\alpha \nu ,l}-g^{\mu \lambda }{g}^{kl}{g}_{\alpha \lambda ,k}{g}_{\beta \nu ,l}).$$

$${\mathcal{R}}_{jkl}^i=R_{jkl}^i+\frac{1}{4}(g^{\lambda \tau }{g}^{im}{g}_{lm,\lambda }{g}_{kj,\tau }-g^{\lambda \tau }{g}^{im}{g}_{km,\lambda }{g}_{lj,\tau }).$$

$${\mathcal{R}}_{i\nu j}^{\mu }={\partial }_{\nu }{\Gamma }_{ji}^{\mu }-{\partial }_j{\Gamma }_{\nu i}^{\mu }+{\Gamma }_{\nu \lambda }^{\mu }{\Gamma }_{ji}^{\lambda }-{\Gamma }_{j\lambda }^{\mu }{\Gamma }_{\nu i}^{\lambda }+{\Gamma }_{\nu k}^{\mu }{\Gamma }_{ji}^k-{\Gamma }_{jk}^{\mu }{\Gamma }_{\nu i}^k.$$

$${\mathcal{R}}_{i\nu j}^{\mu }={\mathcal{D}}_{\nu }{\Gamma }_{ji}^{\mu }-{\mathcal{D}}_j{\Gamma }_{\nu i}^{\mu }-{\Gamma }_{j\lambda }^{\mu }{\Gamma }_{\nu i}^{\lambda }-{\Gamma }_{jk}^{\mu }{\Gamma }_{\nu i}^k.$$

$${\mathcal{R}}_{i\nu j}^{\mu }=-\frac{1}{2}{\mathcal{D}}_{\nu }{g}_{ij}^{,\mu }-\frac{1}{2}{\mathcal{D}}_j(g^{\mu \lambda }{g}_{\lambda \nu ,i})-\frac{1}{4}{g}^{\mu \lambda }{g}^{\alpha \sigma }{g}_{\lambda \alpha ,j}{g}_{\sigma \nu ,i}+\frac{1}{4}{g}^{kl}{g}_{kj}^{,\mu }{g}_{li,\nu }.$$

$${\mathcal{R}}_{\mu j\nu }^i=-\frac{1}{2}{\mathcal{D}}_j{g}_{\mu \nu }^{,i}-\frac{1}{2}{\mathcal{D}}_{\nu }(g^{il}{g}_{lj,\mu })-\frac{1}{4}{g}^{il}{g}^{km}{g}_{lk,\nu }{g}_{mj,\mu }+\frac{1}{4}{g}^{\alpha \lambda }{g}_{\alpha \nu }^{,i}{g}_{\lambda \mu ,j}.$$

$${\mathcal{R}}_{AB}={\mathcal{R}}_{AKB}^K,$$

$$\mathcal{R}=g^{\mu \nu }{\mathcal{R}}_{\mu \nu }+g^{ij}{\mathcal{R}}_{ij}.$$

$$\mathcal{R}=g^{\mu \nu }({\mathcal{R}}_{\mu \alpha \nu }^{\alpha }+{\mathcal{R}}_{\mu k\nu }^k)+g^{ij}({\mathcal{R}}_{i\alpha j}^{\alpha }+{\mathcal{R}}_{ikj}^k).$$

$$\mathcal{R}=g^{\mu \nu }{\mathcal{R}}_{\mu \alpha \nu }^{\alpha }+g^{ij}{\mathcal{R}}_{ikj}^k+2g^{\mu \nu }{g}^{ij}{\mathcal{R}}_{\mu i\nu j}.$$

$$\begin{array}[]{c}\mathcal{R}=\ ^{1}R+\ ^{2}R\\ \\ -\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\nu,i}g_{\alpha\beta,j}+\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\alpha,i}g_{\nu\beta,j}\\ \\ -\frac{1}{4}g^{\mu\nu}g^{ij}g^{kl}g_{ij,\mu}g_{kl,\nu}+\frac{1}{4}g^{\mu\nu}g^{ij}g^{kl}g_{ik,\mu}g_{jl,\nu}\\ \\ -\frac{1}{2}g^{ij}\mathcal{D}_{\mu}g_{ij}^{\quad,\mu}-\frac{1}{2}\mathcal{D}_{i}(g^{\mu\nu}g_{\mu\nu}^{\quad,i})\\ \\ -\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\alpha,i}g_{\nu\beta,j}+\frac{1}{4}g^{\mu\nu}g^{ij}g^{kl}g_{ik,\mu}g_{jl,\nu}\\ \\ -\frac{1}{2}g^{\mu\nu}\mathcal{D}_{i}g_{\mu\nu}^{\quad,i}-\frac{1}{2}\mathcal{D}_{\mu}(g^{ij}g_{ij}^{\quad,\mu})\\ \\ -\frac{1}{4}g^{\mu\nu}g^{ij}g^{kl}g_{ik,\mu}g_{jl,\nu}+\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\alpha,i}^{\quad}g_{\nu\beta,j}.\end{array}$$

$$\begin{array}[]{c}\mathcal{R}=\ ^{1}R+\ ^{2}R-\mathcal{D}_{\mu}(g^{ij}g_{ij}^{\quad,\mu})-\mathcal{D}_{i}(g^{\mu\nu}g_{\mu\nu}^{\quad,i})\\ \\ -\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\nu,i}g_{\alpha\beta,j}-\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\alpha,i}g_{\nu\beta,j}\\ \\ -\frac{1}{4}g^{\mu\nu}g^{ij}g^{kl}g_{ij,\mu}g_{kl,\nu}-\frac{1}{4}g^{\mu\nu}g^{ij}g^{kl}g_{ik,\mu}g_{jl,\nu}.\end{array}$$

$$\begin{array}[]{c}\mathcal{S}=\int_{M}\sqrt{{}^{1}g}\sqrt{{}^{2}g}\mathcal{R}=\int_{M}[\sqrt{{}^{1}g}\sqrt{{}^{2}g}\ ^{1}R+\sqrt{{}^{1}g}\sqrt{{}^{2}g}\ ^{2}R\\ \\ -\sqrt{{}^{1}g}\mathcal{D}_{\mu}(\sqrt{{}^{2}g}g^{ij}g_{ij}^{\quad,\mu})-\sqrt{{}^{2}g}\mathcal{D}_{i}(\sqrt{{}^{1}g}g^{\mu\nu}g_{\mu\nu}^{\quad,i})\\ \\ +\frac{1}{4}\sqrt{{}^{1}g}\sqrt{{}^{2}g}(g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\nu,i}g_{\alpha\beta,j}-g^{\mu\nu}g^{\alpha\beta}g^{ij}g_{\mu\alpha,i}g_{\nu\beta,j}\\ \\ +g^{\mu\nu}g^{ij}g^{kl}g_{ij,\mu}g_{kl,\nu}-g^{\mu\nu}g^{ij}g^{kl}g_{ik,\mu}g_{jl,\nu})].\end{array}$$

$$\begin{array}[]{c}\mathcal{S}=\int_{M}\sqrt{{}^{1}g}\sqrt{{}^{2}g}\mathcal{R}=\int_{M}\sqrt{{}^{1}g}\sqrt{{}^{2}g}\ [^{1}R+^{2}R+\\ \\ +\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}g^{ij}(g_{\mu\nu,i}g_{\alpha\beta,j}-g_{\mu\alpha,i}g_{\nu\beta,j})+\frac{1}{4}g^{\mu\nu}g^{ij}g^{kl}(g_{ij,\mu}g_{kl,\nu}-g_{ik,\mu}g_{jl,\nu})].\end{array}$$

$$\mathcal{S}={\int }_M\sqrt{{}^{1}g}\sqrt{{}^{2}g}{(}^{1}R{+}^{2}R),$$

$$\begin{array}[]{c}\mathcal{S}=\int_{M}\sqrt{{}^{1}g}\sqrt{{}^{2}g}\ [^{1}R+^{2}R+\\ \\ +p(p-1)\psi^{-2}\varphi^{-2}\tilde{g}^{ij}\psi_{,i}\psi_{,j}+q(q-1)\psi^{-2}\varphi^{-2}\tilde{g}^{\mu\nu}\varphi_{,\mu}\varphi_{,\nu}].\end{array}$$

$${}^{1}R={\psi }^{-2}{[}^1\tilde{R}-(p-1)(p-4){\psi }^{-2}{\psi }_{,\lambda }{\psi }^{,\lambda }-2(p-1){\psi }^{-1}{\nabla }_{\lambda }{\psi }^{,\lambda }]$$

$${}^{2}R={\phi }^{-2}{[}^2\tilde{R}-(q-1)(q-4){\phi }^{-2}{\phi }_{,i}{\phi }^{,i}-2(q-1){\phi }^{-1}{\nabla }_i{\phi }^{,i}]$$

$$\begin{array}[]{c}\mathcal{S=}\int_{M}\sqrt{{}^{1}\tilde{g}}\sqrt{{}^{2}\tilde{g}}[\psi^{p-2}\varphi^{q~{}1}\tilde{R}+\psi^{p}\varphi^{q-2~{}2}\tilde{R}\\ \\ -(p-1)(p-4)\psi^{p-4}\varphi^{q}\psi_{,\lambda}\psi^{,\lambda}-2(p-1)\psi^{p-3}\varphi^{q}\nabla_{\lambda}\psi^{,\lambda}\\ \\ +p(p-1)\psi^{p-2}\varphi^{q-2}\psi_{,i}\psi^{,i}\}\\ \\ -(q-1)(q-4)\psi^{p}\varphi^{q-4}\varphi_{,i}\varphi^{,i}-2(q-1)\psi^{p}\varphi^{q-3}\nabla_{i}\varphi^{,i}\\ \\ +q(q-1)\psi^{p-2}\varphi^{q-2}\varphi_{,\lambda}\varphi^{,\lambda}\}].\end{array}$$

$$\begin{array}[]{c}\mathcal{S=}\int_{M}\sqrt{{}^{1}\tilde{g}}\sqrt{{}^{2}\tilde{g}}[(\psi^{p-2}\varphi^{q})^{~{}1}\tilde{R}+(\psi^{p}\varphi^{q-2})^{~{}2}\tilde{R}\\ \\ +(p-1)(p-2)\psi^{p-4}\varphi^{q}\psi_{,\lambda}\psi^{,\lambda}+2q(p-1)\psi^{p-3}\varphi^{q-1}\varphi_{,\lambda}\psi^{,\lambda}\\ \\ +p(p-1)\psi^{p-2}\varphi^{q-2}\psi_{,i}\psi^{,i}\\ \\ +(q-1)(q-2)\psi^{p}\varphi^{q-4}\varphi_{,i}\varphi^{,i}+2p(q-1)\psi^{p-1}\varphi^{q-3}\varphi_{,i}\psi^{,i}\\ \\ +q(q-1)\psi^{p-2}\varphi^{q-2}\varphi_{,\lambda}\varphi^{,\lambda}].\end{array}$$

$$\begin{array}[]{c}\mathcal{S=}\int_{M}\sqrt{{}^{1}\tilde{g}}\sqrt{{}^{2}\tilde{g}}[(\varphi^{q-p+2})^{~{}1}\tilde{R}+(\varphi^{q-p-2})^{~{}2}\tilde{R}\\ \\ +(p-1)(p-2)\varphi^{q-p}\varphi_{,\lambda}\varphi^{,\lambda}+2q(p-1)\varphi^{q-p}\varphi_{,\lambda}\varphi^{,\lambda}\\ \\ +p(p-1)\varphi^{q-p-4}\varphi_{,i}\varphi^{,i}\\ \\ +(q-1)(q-2)\varphi^{q-p-4}\varphi_{,i}\varphi^{,i}+2p(q-1)\varphi^{q-p-4}\varphi_{,i}\varphi^{,i}\\ \\ +q(q-1)\varphi^{q-p}\varphi_{,\lambda}\varphi^{,\lambda}],\end{array}$$

$$\begin{array}[]{c}\mathcal{S=}\int_{M}\sqrt{{}^{1}\tilde{g}}\sqrt{{}^{2}\tilde{g}}[(\varphi^{q-p+2})^{~{}1}\tilde{R}+(\varphi^{q-p-2})^{~{}2}\tilde{R}\\ \\ +[(p-1)(p-2)+2q(p-1)+q(q-1)]\varphi^{q-p}\varphi_{,\lambda}\varphi^{,\lambda}\\ \\ +[(q-1)(q-2)+2p(q-1)+p(p-1)]\varphi^{q-p-4}\varphi_{,i}\varphi^{,i}].\end{array}$$

$$(p-1)(p-2)+2q(p-1)+q(q-1)=(p+q-1)(p+q-2)$$

$$(q-1)(q-2)+2p(q-1)+p(p-1)=(p+q-1)(p+q-2),$$

Full text

HIGHER DIMENSIONAL GRAVITY AND FARKAS

PROPERTY IN ORIENTED MATROID THEORY

J. A. Nieto*⋆∗* 111nieto@uas.uasnet.mx and E. A. León*∗* 222ealeon@posgrado.cifus.uson.mx

*⋆*Facultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán Sinaloa, México.

*∗*Departameto de Investigación en Física de la Universidad de Sonora, Hermosillo Sonora , México

Abstract

We assume gravity in a $d$-dimensional manifold $M$ and consider a splitting of the form $M=M_{p}\times M_{q}$, with $d=p+q$. The most general two-block metric associated with $M_{p}$ and $M_{q}$ is used to derive the corresponding Einstein-Hilbert action $\mathcal{S}$. We focus on the special case of two distinct conformal factors $\psi$ and $\varphi$ ($\psi$ for the metric in $M_{p}$ and $\varphi$ for the metric in $M_{q}$), and we write the action $\mathcal{S}$ in the form $\mathcal{S=S}_{p}\mathcal{+S}_{q}$, where $\mathcal{S}_{p}$ and $\mathcal{S}_{q}$ are actions associated with $M_{p}$ and $M_{q}$, respectively. We show that a simplified action is obtained precisely when $\psi=\varphi^{-1}$. In this case, we find that under the duality transformation $\varphi\leftrightarrow\varphi^{-1}$, the action $\mathcal{S}_{p}$ for the $M_{p}$-space or the action $\mathcal{S}_{q}$ for the $M_{q}$-space remain invariant, but not both. This result establishes an analogy between Farkas property in oriented matroid theory and duality in general relativity. Furthermore, we argue that our approach can be used in several physical scenarios such as 2t physics and cosmology.

Keywords: Higher dimensional gravity, gravitational duality, 2t physics,cosmology

Pacs numbers: 04.60.-m, 04.65.+e, 11.15.-q, 11.30.Ly

December, 2009

Duality has been an important concept in several physical scenarios, including string theory [1], $M$-theory [2], Matroid theory [3], MacDowell-Mansouri gravity (see Refs. [4]-[5] and references therein) and cosmological models [6][7]. Here, we explore the possible relevance of duality in higher dimensional gravity. We first consider general relativity in a $d-$dimensional manifold $M$ with arbitrary space-time signature. We then proceed to split such manifold in the form $M=M_{p}\times M_{q}$, with $p+q=d$, assuming a two-block diagonal metric. Considering such splitting, we obtain the Einstein-Hilbert action $\mathcal{S}$. For the special case in which the two metrics associated with $M_{p}$ and $M_{q}$ are expressed in terms of the conformal factors $\psi$ and $\varphi$, respectively, we show that the action $\mathcal{S}$ can be written as $\mathcal{S=S}_{p}\mathcal{+S}_{q}$, where $\mathcal{S}_{p}$ and $\mathcal{S}_{q}$ are actions associated with $M_{p}$ and $M_{q}$, respectively. Taking $\psi=\varphi^{-1}$, the reduced action is analysed. The key idea is to consider the invariance of such reduced action under the duality transformation $\varphi\leftrightarrow\varphi^{-1}$. In this context, we find that the invariance of $\mathcal{S}$ does not follow, but rather we discover that, under this duality transformation, the action $\mathcal{S}_{p}$ is invariant, provided $p\rightleftarrows q+2$, while $\mathcal{S}_{q}$ is not. And conversely, the action $\mathcal{S}_{q}$ is invariant, provided $p\rightleftarrows q-2$, while $\mathcal{S}_{p}$ is not. We argue that this result establishes an analogy between Farkas property in oriented matroid theory [8] and duality in higher dimensional gravity. It is worth mentioning that the Farkas property has been used as an alternative to define the concept of oriented matroid [9]. In turn, oriented matroid theory has been proposed [10]-[11] as the appropriate mathematical framework for considering duality in several physical scenarios, including p-branes and M-theory. Therefore, our work may be useful in the analysis of some aspects of duality in p-branes and M-theory. Furthermore, we argue that our approach can also be relevant in the context of 2t physics and cosmology.

We start the analysis with a metric $\gamma_{AB}$ in a $d-$dimensional manifold $M$, which will be splitted into a two-blocks metric corresponding to $M=M_{p}\times M_{q}$. The first block metric, of dimension $p$, will be denoted by $g_{\mu\nu}$, and the second of dimension $q=d-p$, will be denoted by $g_{ij}$, where Greek indices ($\alpha$, $\beta$, …) run from $1$ to $p$, lowercase Latin indices ($i$, $j$, …) from $p+1$ to $d$, and capital Latin indices ($A$, $B$, …) from $1$ to $d$. With this prescription, we have [12][13]:

[TABLE]

where for consistency, the upper zero [math] corresponds to a $p\times q$-matrix and the lower zero [math] corresponds to a $q\times p$-matrix. Here, $x$ refers to coordinates in $M_{p}$, while $y$ refers to coordinates in $M_{q}$.

As usual, the Riemann tensor is defined in terms of the Christoffel symbols as

[TABLE]

where in turn, the Christoffel symbols are

[TABLE]

From (3), using the distinction of the indices $\mu,\nu,...etc.$ and $i,j,...etc.$, we find that the Christoffel symbols can be splitted in six classes:

[TABLE]

Here, $\left\{{}_{\alpha\beta}^{\mu}\right\}$ and $\left\{{}_{jk}^{i}\right\}$ refer to Christoffel symbols in terms of the metric $g_{\alpha\beta}$ and $g_{ij}$, respectively.

Now, the components of the Riemann tensor can also be splitted in similar classes. For example:

[TABLE]

where

[TABLE]

Using (4), the relation (5) becomes

[TABLE]

Similarly, we get the component of the Riemann tensor with Latin indices:

[TABLE]

We also have

[TABLE]

Defining the covariant derivatives $\mathcal{D}_{\nu}\Gamma_{ij}^{\mu}=\partial_{\nu}\Gamma_{ij}^{\mu}+\Gamma_{\nu\lambda}^{\mu}\Gamma_{ji}^{\lambda}$ and $\mathcal{D}_{j}\Gamma_{\nu i}^{\mu}=\partial_{j}\Gamma_{\nu i}^{\mu}-\Gamma_{\nu k}^{\mu}\Gamma_{ji}^{k}$, the expression (9) is reduced to

[TABLE]

Thus, by using (4), we find

[TABLE]

With a similar calculation, we obtain

[TABLE]

From the Ricci tensor

[TABLE]

we can construct the curvature scalar $\mathcal{R}$:

[TABLE]

We can explicitly rewrite (14) as follows:

[TABLE]

Using the symmetric property $g^{ij}\mathcal{R}_{\ i\alpha j}^{\alpha}=g^{\mu\nu}\mathcal{R}_{\ \mu k\nu}^{k}$, (15) can be simplified to

[TABLE]

Considering (7), (8), (11) and (12), and writing as ${}^{1}R=g^{\mu\nu}R_{\ \mu\alpha\nu}^{\alpha}$ and ${}^{2}R=g^{ij}R_{\ ikj}^{k}$ the Ricci scalar in $p$ and $q$ dimensions, respectively, (16) leads to

[TABLE]

This expression can be simplified to

[TABLE]

Therefore, we obtain the action

[TABLE]

It is reasonable to assume that at large distances $\int_{M}\sqrt{{}^{1}g}\mathcal{D}_{\mu}(\sqrt{{}^{2}g}g^{ij}g_{ij}^{\quad,\mu})=0$ and $\int_{M}\sqrt{{}^{2}g}\mathcal{D}_{i}(\sqrt{{}^{1}g}g^{\mu\nu}g_{\mu\nu}^{\quad,i})=0$. Therefore (19) can be simplified as follows [17]

[TABLE]

Thus, we have achieved our first goal of splitting the action in terms of the $M^{p}$ and $M^{q}$ spaces. However, observe that the last two terms in (20) are interacting terms between the two metric fields $g_{ij}$ and $g_{\alpha\beta}$.

As an application of (20), we shall now discuss several examples. First, one may assume that $g_{\mu\nu}=g_{\mu\nu}(x)$ and $g_{ij}=g_{ij}(y)$. In this case $\mathcal{S}$ is reduced to

[TABLE]

which is a well known result.

More interesting cases may arise if we assume that $g_{\mu\nu}=\psi^{2}(x,y)\tilde{g}_{\mu\nu}(x)$ and $g_{ij}=\varphi^{2}(x,y)\tilde{g}_{ij}(y)$. Let us first substitute this assumption in the last two terms of (20):

[TABLE]

Since the Ricci scalars ${}^{1}R$ and ${}^{2}R$ become

[TABLE]

and

[TABLE]

respectively, after some rearrangements, we obtain

[TABLE]

Integrating by parts, (25) yields

[TABLE]

Here, we are interested in exploring a possible duality symmetry in (26). For this purpose, let us consider the special case $\psi=\varphi^{-1}$. We have

[TABLE]

which can also be rewritten as

[TABLE]

Since

[TABLE]

and

[TABLE]

we can further simplify (28) to

[TABLE]

where

[TABLE]

and

[TABLE]

We are interested in the possible invariance of the action (31) under the duality transformation $\varphi\rightarrow\varphi^{-1}$. Applying this transformation to (31), we obtain

[TABLE]

while

[TABLE]

Therefore we observe that if $p\rightarrow q+2$ and $q\rightarrow p-2$, then $\mathcal{S}_{p}$ remains invariant, but

[TABLE]

that is, $\mathcal{S}_{q}$ is not invariant. Conversely, if $p\rightarrow q-2$ and $q\rightarrow p+2$, then $\mathcal{S}_{q}$ is invariant, while $\mathcal{S}_{p}$ becomes

[TABLE]

which means that $\mathcal{S}_{p}$ is not invariant. Therefore, we have the peculiar situation that under the duality transformation $\varphi\rightarrow\varphi^{-1}$, $\mathcal{S}_{p}$ or $\mathcal{S}_{q}$ remains invariant, but not both, depending on having $p\rightarrow q+2$ and $q\rightarrow p-2$, (that is $p\rightleftarrows q+2$), or $p\rightarrow q-2$ and $q\rightarrow p+2$ ( $p\rightleftarrows q-2$) , respectively.

This duality property for the action (31) resembles the Farkas property in oriented matroid theory. In order to fully appreciate this comment, let us explain briefly what the Farkas property means. First, let us consider a total tangent bundle $T=(H,V)$ associated with $M$, where $H$ and $V$ denote the horizontal and vertical parts of $T$. Assume that $L\subseteq M$ corresponds to $H$ and  that the orthogonal complement $L_{\perp}$ corresponds to $V$. Now, just as $(H,V)$ determine the structure of $T$, the dual pair $(L,L_{\perp})$ determines the structure of the total space $M$. It turns out that one can introduce the concept of an oriented matroid in terms of the structure $(L,L_{\perp})$, rather than only in terms of the subspace $L$. (For details in oriented matroids, see Ref. [6].) One can prove that a transition of the form

[TABLE]

ensures a duality symmetry, which is one of the main subjects of oriented matroid theory. In fact, there exist a formal definition of an oriented matroid in terms of the analogue of $(L,L_{\perp})$. Such a definition uses the concept of Farkas property, which we shall now proceed to discuss briefly (see Ref. [5] for details).

Let us first describe the sign vector concept. Let $E\neq\varnothing$ be a finite set. An element $X\in\{+,-,0\}^{E}$ is called a sign vector. The positive, negative and zero parts of $X$ are denoted by $X^{+}$, $X^{-}$ and $X^{0}$ respectively. Further, we define $suppX\equiv X^{+}\cup X^{-}$. Consider two sets $S$ and $S^{\prime}$ of signed vectors. The pair $(S,S^{\prime})$ is said to have the Farkas property, if $\forall e\in E$ either

(Fa)$\exists X\in S$, $e\in suppX$ and $X\geq 0$

or

(Fb) $\exists Y\in S^{\prime}$, $e\in suppY$ and $Y\geq 0$,

but not both. Here, $X\geq 0$ means that $X$ has a positive (+) or a zero (0) entry in every coordinate. Observe that $(S,S^{\prime})$ has the Farkas property if and only if $(S^{\prime},S)$ has it. Let $S$ be a set of signed vectors, and let $I$ and $J$ denote disjoint subsets of $E$. Then

[TABLE]

is called a minor of $S$ (obtained by deleting $I$ and contracting $J$). Here, the symbol ”$\ast$” denotes and arbitrary value. If $S$ and $S^{\prime}$ are sets of sign vectors on $E$, then $(S\backslash I/J,S\prime\backslash J/I)$ is called minor of $(S,S^{\prime})$. Similarly,

[TABLE]

is called the reorientation of $S$ on $I$. Further, $(_{I}S,_{I}S^{\prime})$ is the reorientation of $(S,S^{\prime})$ on $I$. Moreover, $S$ is symmetric if $S=-S$, where $-S$ is the set of signed vectors which are opposite to the signed vectors of $S$.

We can now give a definition of oriented matroids in terms of the Farkas property. Let $E\neq\varnothing$ be a finite set and let $S$ and $S^{\prime}$ two sets of sign vectors. The pair $(S,S^{\prime})$ is called an oriented matroid on $E$, if

(O1) $S$ and $S^{\prime}$ are symmetric, and

(O2) every reorientation of every minor of $(S,S^{\prime})$ has the Farkas property.

From this definition it follows that $(S^{\prime},S)$ is also an oriented matroid as it is every reorientation and every minor of $(S,S^{\prime})$.

Two sign vectors $X$ and $Y$ are orthogonal if

[TABLE]

Accordingly, we denote the orthogonal complement of $S$ by $S_{\perp}$, and it is defined by

[TABLE]

If $S^{\prime}\subseteq S_{\perp}$, then $S$ and $S^{\prime}$ can be considered as orthogonal.

Coming back to our problem at hand we have that, due to the form of the metric (1), $\mathcal{S}_{p}$ and $\mathcal{S}_{q}$ can be associated with the two orthogonal subspaces $M_{p}$ and $M_{q}$ of $M$, respectively. This suggests to introduce the analogue of the Farkas property for the action (31):

For every transformation $\varphi\rightarrow\varphi^{-1}$,

(Ha) $\exists p$ for $M_{p}$, such that $\mathcal{S}_{p}$ is invariant

or

(Hb) $\exists q$ for $M_{q}$, such that $\mathcal{S}_{q}$ is invariant,

but not both.

It is interesting to write (31) in the alternative form

[TABLE]

Here, we considered the fact that ${}^{1}\tilde{R}=\tilde{g}^{\mu\nu}\tilde{R}_{\mu\nu}$, ${}^{2}\tilde{R}=\tilde{g}^{ij}\tilde{R}_{ij}$, $\gamma^{\mu\nu}=\varphi^{2}(x,y)\tilde{g}^{\mu\nu}(x)$ and $\gamma^{ij}=\varphi^{-2}(x,y)\tilde{g}^{ij}(x)$. This expression can be written in the more compact form

[TABLE]

Let us make some final comments. Usually, one is interested in the invariance of an action $\mathcal{S}$ under certain infinitesimal transformations. Here we have shown that, if an action can be divided in two complementary ”orthogonal” actions $\mathcal{S}_{p}$ and $\mathcal{S}_{q}$, then the invariance of the total action $\mathcal{S=S}_{p}+\mathcal{S}_{q}$ under duality transformations is not what really matters, but rather whether $\mathcal{S}_{p}$ or $\mathcal{S}_{q}$ are invariant, but not both, as the analogue of the Farkas property should require. Since the results obtained above are valid for arbitrary dimensionality -albeit the conditions imposed for the metric-, they can be of interest for cosmology with extra dimensions, in particular when those extra dimensions are non-compact [14][15]. Also, since a priori it has been not chosen any signature for the metric $\gamma_{AB}$, our analysis may give some insight in 2t physics [16].

Acknowledgments

This work was partially supported by PROFAPI 2008 and 2009.