Aspects of $(d+D)$-dimensional anisotropic conformal gravity

Abstract

We discuss various aspects of anisotropic gravity in $(d+D)$-dimensional spacetime, where D dimensions are treated as extra dimensions. It is based on the foliation preserving diffeomorphism invariance and anisotropic conformal invariance. The anisotropy is embodied by introducing a factor z which discriminates the scaling degree of the extra D dimensions against the d-dimensional base spacetime and Weyl scalar field which mediates the anisotropic scaling symmetry. There is no intrinsic scale but a physical scale $M_*$ emerges as a consequence of spontaneous conformal symmetry breaking. We discuss interesting lower dimensional gravity theories obtained from our model. In a $(d,D)=(2,2)$ case, we suggest a UV-complete unitary quantum gravity which might become Einstein gravity in IR. In a certain (2,1) case, we obtain CGHS model. Some vacuum solutions are also obtained and we discuss an issue of ‘size separation’ between the base spacetime and the extra dimensions. The size separation means large distinction between the scales (e.g. cosmological constant) appearing in the base spacetime and the extra dimensions respectively.

Appendices

Appendices

Appendix A: Construction of FPD invariant (d, D) gravity

In this section, we illustrate the detailed steps for construction of FPD Invariant (d, D) Gravity theory which is given in Sect. 2.1. The $(d+D)$-dimensional metric is expressed by employing ADM decomposition as

$$\begin{aligned} \mathrm{d}s^2= & {} G_{AB}\mathrm{d}X^A \mathrm{d}X^B~~ \nonumber \\= & {} g_{\mu \nu }(\mathrm{d}x^\mu +N^\mu _m \mathrm{d}y^m)(\mathrm{d}x^\nu +N^\nu _n \mathrm{d}y^n)+\gamma _{mn}\mathrm{d}y^m \mathrm{d}y^n, \end{aligned}$$

(38)

where again the capital Roman indices as A, B... run over $(d+D)$-dimensional spacetime, i.e. $A, B, ...=1,..., d+D$. We use $X^A$ for the coordinate variable of the $(d+D)$-dimensional spacetime. The $(d+D)$-dimensional coordinate is a direct sum of d-dimensional spacetime and D-dimensional space coordinates as $X^A= (x^\mu , y^m)$, where the sub-indices $\mu$, $\nu$ run from 1 to d and m, n run from $d+1$ to $d+D$. $G_{AB}$, $g_{\mu \nu }$ and $\gamma _{mn}$ are the metrics of $(d+D)$-, d- and D- dimensions respectively. $N_m^{\mu }$ is the shift vector along the d-dimensional base spacetime corresponding to the mth direction [28].

The inverse metric is obtained as

$$\begin{aligned} G^{AB}=\begin{pmatrix}g^{\mu \nu }+g^{mn}N_m^\mu N_n^\nu &{} -N^{\mu n} \\ -N^{m\nu } &{} \gamma ^{mn} \\ \end{pmatrix}, \end{aligned}$$

(39)

where $g^{\mu \nu }$ and $\gamma ^{mn}$ are inverse metrics of $g_{\mu \nu }$ and $\gamma _{mn}$ respectively. $N^{\mu m}=\gamma ^{mn}N_n^\mu .$

Let us consider FPD as we do in Sect. 2.1 and its coordinate transform being given by

$$\begin{aligned} x^{\mu }\rightarrow x^{\prime \mu }\equiv x^{\prime \mu }(x,y),~~~y^n\rightarrow y^{\prime n}\equiv y^{\prime n}(y), \end{aligned}$$

(40)

whose infinitesimal transformations can be written as

$$\begin{aligned} x^{\prime \mu }=x^\mu +\xi ^\mu (x,y),~~~y^{\prime m}=y^{m}+\eta ^m(y);~~~\partial _{\mu } \eta ^m=0, \end{aligned}$$

(41)

where $\eta ^m(y)$ is a function of $y^m$ only. The finite coordinate transformations of each component of the metric under FPD of (40) are given as follows [9]:

$$\begin{aligned} g^{'}_{\mu \nu }(x',y')= & {} \frac{\partial x^\rho }{\partial x^{'\mu }}\frac{\partial x^\sigma }{\partial x^{'\nu }}g_{\rho \sigma }(x,y), \end{aligned}$$

(42)

$$\begin{aligned} {N'}_{m}^{\mu }(x',y')= & {} \Big (\frac{\partial y^n}{\partial y^{'m}}\Big )\Big [\frac{\partial x^{'\mu }}{\partial x^{\nu }}N_{n}^{\nu }(x,y)-\frac{\partial x'^{\mu }}{\partial y^{n}}\Big ], \end{aligned}$$

(43)

$$\begin{aligned} \gamma '_{mn}(x',y')= & {} \frac{\partial y^p}{\partial y^{'m}}\frac{\partial y^q}{\partial y^{'n}}\gamma _{pq}(x,y). \end{aligned}$$

(44)

One can easily check that the above transformations (42)–(44) leave the line element (39) invariant.

For further computation, it is more convenient to introduce orthogonal basis in which the computation of the curvatures scalar R is simplified to a great extent. We employ the orthogonal basis ${\hat{X}}^A = ( {\hat{x}}^\mu , {\hat{y}}^m)$ [29] and rewrite line element (39) as

$$\begin{aligned} ds^2= {{\hat{G}}_{AB}d{\hat{X}}^A d {\hat{X}}^B} \equiv {\hat{g}}_{\mu \nu }d{\hat{x}}^\mu d{\hat{x}}^\nu +\hat{\gamma }_{mn}d{\hat{y}}^md{\hat{y}}^n, \end{aligned}$$

(45)

where the transformations between the coordinate and the orthogonal bases are given by

$$\begin{aligned} d{\hat{x}}^\mu =dx^\mu +N_m^\mu dy^m, ~~ d{\hat{y}}^m=dy^n. \end{aligned}$$

(46)

The relations between the metrics are

$$\begin{aligned} {\hat{g}}_{\mu \nu }=g_{\mu \nu }, ~~ \hat{\gamma }_{mn}=\gamma _{mn}. \end{aligned}$$

(47)

We note that these transformations are non-integrable.

Next, we develop an algebra between translation operators(partial derivative operators) in orthogonal basis. An observation of (46) suggests that

$$\begin{aligned} \frac{\partial {\hat{x}}^\nu }{\partial x^\mu }=\delta ^\nu _\mu , ~ \frac{\partial {\hat{y}}^m}{\partial x^\mu }=0, ~\frac{\partial {\hat{x}}^\nu }{\partial y^m}=N^\nu _m, ~\frac{\partial {\hat{y}}^m}{\partial y^n}=\delta ^m_n,~ \end{aligned}$$

(48)

and their inverse relations,

$$\begin{aligned} \frac{\partial x^\nu }{\partial {\hat{x}}^\mu }=\delta ^\nu _\mu , ~ \frac{\partial y^m}{\partial {\hat{x}}^\mu }=0, ~\frac{\partial x^\nu }{\partial {\hat{y}}^m}=-N^\nu _m, ~\frac{\partial y^m}{\partial {\hat{y}}^n}=\delta ^m_n. \end{aligned}$$

(49)

Using these facts, the derivative operators in orthogonal basis is given by

$$\begin{aligned} \frac{\partial }{\partial {\hat{x}}^\mu }=\frac{\partial }{\partial x^\mu }, ~~\frac{\partial }{\partial {\hat{y}}^m}=\frac{\partial }{\partial y^m}- N^\mu _m\frac{\partial }{\partial x^\mu }, \end{aligned}$$

(50)

in terms of the old variables. We also have formulas for their second derivatives as

$$\begin{aligned} \frac{\partial }{\partial {\hat{x}}^\mu }\frac{\partial }{\partial {\hat{x}}^\nu } =\frac{\partial }{\partial {\hat{x}}^\nu }\frac{\partial }{\partial {\hat{x}}^\mu },~~\frac{\partial }{\partial {\hat{x}}^\mu } \frac{\partial }{\partial {\hat{y}}^m}-\frac{\partial }{\partial {\hat{y}}^m} \frac{\partial }{\partial {\hat{x}}^\mu }= -\frac{\partial N^\nu _m}{\partial {\hat{x}}^\mu } \frac{\partial }{\partial {\hat{x}}^\nu } \end{aligned}$$

(51)

and

$$\begin{aligned} \frac{\partial }{\partial {\hat{y}}^m}\frac{\partial }{\partial {\hat{y}}^n}-\frac{\partial }{\partial {\hat{y}}^n}\frac{\partial }{\partial {\hat{y}}^m}=-F^\mu _{mn}\frac{\partial }{\partial {\hat{x}}^\mu }, \end{aligned}$$

(52)

where

$$\begin{aligned} F^{\mu }_{mn}= & {} \partial _{m}N_{n}^{\mu }-\partial _{n}N_{m}^{\mu }-N_{m}^{\nu }\partial _{\nu }N_{n}^{\mu } +N_{n}^{\nu }\partial _{\nu }N_{m}^{\mu }. \end{aligned}$$

(53)

We package these relations as a compact form of larger commutation relations as

$$\begin{aligned}{}[\hat{\partial }_A, \hat{\partial }_B]=f^C_{~AB}\hat{\partial }_C, \end{aligned}$$

(54)

where the structure constant $f^C_{~AB}$ is given by

$$\begin{aligned} f^m_{~\mu \nu }=f^\sigma _{~\mu \nu }=0, ~f^\nu _{~\mu m}=-f^\nu _{~m\mu }= -\frac{\partial N^\nu _m}{\partial {\hat{x}}^\mu },~f_{~mn}^\mu =-F_{mn}^\mu . \end{aligned}$$

(55)

Geometrically, the field strength $F^{\mu }_{mn}$ is a Diff(d) vector-valued curvature associated with the holonomy of the shift vector $N_n^\mu$ along the extra dimensions.

Now we are ready to compute curvature tensor in $d+D$ dimension. First, we define the connections. To do this, we introduce vectors ${ \vec{{r}} =(X^0, X^1, \dots , X^{d+D})}$ and $\vec{{V}}$ in the basis ${ \vec{\hat{{{e}}}}_A\equiv \frac{\partial \vec{ {r}}}{\partial {\hat{X}}^A}}$ such that $\vec{{V}}={\hat{V}}^A \vec{\hat{{{e}}}}_A$. These basis vectors are directly related with the metric coefficients of (45) by

$$\begin{aligned} \vec{\hat{{{e}}}}_A\cdot \vec{\hat{{{e}}}}_B={\hat{G}}_{AB}. \end{aligned}$$

(56)

Equation for parallel transport is defined as

$$\begin{aligned} \frac{{\mathrm{d}}\vec{{V}}}{\mathrm{d}u}=\frac{{\mathrm{d}}{\hat{X}}^A}{{\mathrm{d}}u}\left[ \frac{\partial {\hat{V}}^B}{\partial {\hat{X}}^A} \vec{\hat{{e}}}_B+ {\hat{V}}^B\frac{\partial \vec{\hat{{{e}}}}_B}{\partial {\hat{X}}^A}\right] =0 \end{aligned}$$

(57)

along a curve parametrized by an affine parameter u. The next step is to introduce connections in this orthogonal coordinate by

$$\begin{aligned} \frac{\partial \vec{\hat{{{e}}}}_B}{\partial {\hat{X}}^A}=\hat{\Gamma }^C_{~AB}\vec{\hat{{{e}}}}_C, \end{aligned}$$

(58)

and using (56), we obtain

$$\begin{aligned}&\hat{\Gamma }^C_{~AB}=\hat{\Gamma }^{C}_{(C)AB}+ \frac{1}{2}{\hat{G}}^{CD}\nonumber \Big (f_{DAB}-f_{ABD}-f_{BAD}\Big ), ~~(f_{ABC}={\hat{G}}_{AD}f^D_{~BC}), \end{aligned}$$

(59)

where $\hat{\Gamma }^{C}_{(C)AB}$ is the usual form of Chrisoffel connection being given by

$$\begin{aligned}&\hat{\Gamma }^{C}_{(C)AB}=\frac{1}{2}{\hat{G}}^{CD}\nonumber \left( \frac{\partial {\hat{G}}_{DA}}{\partial {\hat{X}}^B}+ \frac{\partial {\hat{G}}_{DB}}{\partial {\hat{X}}^A}- \frac{\partial {\hat{G}}_{AB}}{\partial {\hat{X}}^D}\right) . \end{aligned}$$

(60)

The covariant derivative in this orthogonal coordinate is defined with the connection $\hat{\Gamma }^C_{~AB}$ by

$$\begin{aligned} \hat{\nabla }_A {\hat{V}}^B\equiv \frac{\partial {\hat{V}}^B}{\partial {\hat{X}}^A}+\hat{\Gamma }^B_{~AC}{\hat{V}}^C, \end{aligned}$$

(61)

and one can prove explicitly the metric compatibility,

$$\begin{aligned} \hat{\nabla }_A{\hat{G}}_{BC}=0. \end{aligned}$$

(62)

The Riemann tensor is defined as

$$\begin{aligned}{}[\hat{\nabla }_A, \hat{\nabla }_B]{\hat{V}}^C={\hat{R}}^C_{~DAB}{\hat{V}}^D, \end{aligned}$$

(63)

where

$$\begin{aligned}&{\hat{R}}^A_{~BCD}=\hat{\partial }_C\hat{\Gamma }^A_{~DB} -\hat{\partial }_D\hat{\Gamma }^A_{~CB}\nonumber \\ & + \hat{\Gamma }^A_{~CE}\hat{\Gamma }^E_{~DB} -\hat{\Gamma }^A_{~DE}\hat{\Gamma }^E_{~CB}-f^E_{~CD}\hat{\Gamma }^A_{~EB}. \end{aligned}$$

(64)

A straightforward computation yields the scalar curvature (7). Each term in (7) is separately invariant under FPD. To show this, one may use the following facts:

$$\begin{aligned} \hat{\partial }^{'}_m=\left( \frac{\partial y^n}{\partial y^{'m}}\right) \hat{\partial }_n. \end{aligned}$$

(65)

${{\mathcal {D}}}_m g_{\mu \nu }$ and $F_{mn}^{\mu }$ change covariantly as

$$\begin{aligned} ({{\mathcal {D}}}_m g_{\mu \nu })^{'}= & {} \frac{\partial y^{n}}{\partial y^{'m}}\frac{\partial x^{\rho }}{\partial x^{'\mu }}\frac{\partial x^{\sigma }}{\partial x^{'\nu }}{{\mathcal {D}}}_n g_{\rho \sigma }, \end{aligned}$$

(66)

$$\begin{aligned} (F^{\mu }_{mn})^{'}= & {} \frac{\partial y^{p}}{\partial y^{'m}}\frac{\partial y^{q}}{\partial y^{'n}}\frac{\partial x^{'\mu }}{\partial x^{\nu }}F^{\nu }_{pq}, \end{aligned}$$

(67)

and the transformation of $\gamma _{mn}$ is also covariant, i.e.

$$\begin{aligned} (\partial _\mu \gamma _{mn})^{'}= & {} \frac{\partial x^{\nu }}{\partial x^{'\mu }}\frac{\partial y^{p}}{\partial y^{'m}}\frac{\partial y^{q}}{\partial y^{'n}}(\partial _\nu \gamma _{pq}) \end{aligned}$$

(68)

under FPD transformation, where to derive that, one may use a fact, ${\partial }^{'}_{\mu }=\left( \frac{\partial x^{\nu }}{\partial x^{'\mu }}\right) \partial _{\nu }.$