Holographic relationships in Lovelock type brane gravity

TL;DR

This paper demonstrates that Lovelock type brane gravity inherently exhibits holographic properties, establishing a direct correspondence between bulk and surface terms in the action, and relating conserved tensors and geometric degrees of freedom.

Contribution

It establishes a natural holographic relationship in Lovelock brane gravity, linking bulk and surface terms and conserved tensors, which was not previously demonstrated.

Findings

Bulk and surface Lagrangian terms are fully related.

Conserved tensors in the theory are interconnected.

Geometric degrees of freedom correspondence is discussed.

Abstract

We show that the Lovelock type brane gravity is naturally holographic by providing a correspondence between bulk and surface terms that appear in the Lovelock-type brane gravity action functional. We prove the existence of relationships between the $\mathcal{L}_{\mbox{\tiny bulk}}$ and $\mathcal{L}_{\mbox{\tiny surf}}$ allowing $\mathcal{L}_{\mbox{\tiny surf}}$ to be determined completely by $\mathcal{L}_{\mbox{\tiny bulk}}$. In the same spirit, we provide relationships among the various conserved tensors that this theory possesses. We further comment briefly on the correspondence between geometric degrees of freedom in both bulk and surface space.

Equations

$$S[X^{\mu }]={\int }_m{d}^{p+1}x\sqrt{-g}\sum _{n=0}^{p+1}{\alpha }_n{L}_n(g_{ab},K_{ab}),$$

$$L_n(g_{ab},K_{ab})={\delta }_{b_1{b}_2{b}_3\cdots b_n}^{a_1{a}_2{a}_3\cdots a_n}{K}^{b_1}{}_{a_1}{K}^{b_2}{}_{a_2}{K}^{b_3}{}_{a_3}\cdots K^{b_n}{}_{a_n},$$

$$\delta^{a_{1}a_{2}\ldots a_{n-1}a_{n}}_{b_{1}b_{2}\ldots b_{n-1}b_{n}}=\left|\matrix{\delta^{a_{1}}_{b_{1}}&\delta^{a_{1}}_{b_{2}}&\ldots&\delta^{a_{1}}_{b_{n-1}}&\delta^{a_{1}}_{b_{n}}\cr\delta^{a_{2}}_{b_{1}}&\delta^{a_{2}}_{b_{2}}&\ldots&\delta^{a_{2}}_{b_{n-1}}&\delta^{a_{2}}_{b_{n}}\cr\vdots&\vdots&\ddots&\vdots&\vdots\cr\delta^{a_{n-1}}_{b_{1}}&\delta^{a_{n-1}}_{b_{2}}&\ldots&\delta^{a_{n-1}}_{b_{n-1}}&\delta^{a_{n-1}}_{b_{n}}\cr\delta^{a_{n}}_{b_{1}}&\delta^{a_{n}}_{b_{2}}&\ldots&\delta^{a_{n}}_{b_{n-1}}&\delta^{a_{n}}_{b_{n}}}\right|.$$

$$L_{(2n)}$$

$$L_{(2n+1)}$$

$$\delta S={\int }_m{d}^{p+1}x\left[\delta (\sqrt{-g})\sum _{n=0}^{p+1}{\alpha }_n{L}_n+\sqrt{-g}\sum _{n=0}^{p+1}{\alpha }_n\delta L_n\right].$$

$$\delta L_n$$

$$J_{(n)b}^a:={\delta }_{b{b}_1{b}_2{b}_3\dots b_n}^{a{a}_1{a}_2{a}_3\dots a_n}{K}^{b_1}{}_{a_1}{K}^{b_2}{}_{a_2}{K}^{b_3}{}_{a_3}\dots K^{b_n}{}_{a_n},$$

$$\delta S={\int }_m{d}^{p+1}x\sqrt{-g}\sum _{n=0}^{p+1}{\alpha }_n\left[\left(J_{(n)}^{ab}-\frac{1}{2}{g}^{ab}{L}_n\right)\delta g_{ab}+n{J}_{(n-1)}^{ab}\delta K_{ab}\right].$$

$$\delta g_{ab}$$

$$\delta K_{ab}$$

$$\delta S={\int }_m{d}^{p+1}x\sqrt{-g}\left[\mathcal{E}(L)n_{\mu }\delta X^{\mu }+{\int }_m{d}^{p+1}x\sqrt{-g}{\nabla }_a{Q}^a\right],$$

$$\mathcal{E}=\sum _{n=0}^{p+1}{\alpha }_n{J}_{(n)}^{ab}{K}_{ab}=0,$$

$$Q^a:=\sum _{n=0}^{p+1}{\alpha }_n\left(J_{(n)}^{ab}{e}_{\mu b}\delta X^{\mu }-n{J}_{(n-1)}^{ab}{n}_{\mu }\delta {\nabla }_b{X}^{\mu }\right).$$

$$J_{(2n)b}^a$$

$$J_{(2n+1)b}^a$$

$$\sqrt{-g}L=\sqrt{-g}{Q}_a{}^{bcd}{\mathcal{R}}^a{}_{bcd},$$

$${\mathcal{L}}_n=\sqrt{-g}{}_{(n)}{\mathcal{Q}}_a{}^{bcd}{\mathcal{R}}^a{}_{bcd},$$

$$L_{(2n)}={}_{(2n)}{Q}_{b_{2n-1}}{}^{a{a}_{2n-1}{a}_{2n}}{\mathcal{R}}^{b_{2n-1}}{}_{a{a}_{2n-1}{a}_{2n}},$$

$${}_{(2n)}{Q}_{b_{2n-1}}{}^{a{a}_{2n-1}{a}_{2n}}:=\frac{1}{2^n}{\delta }_{b_1{b}_2\cdots b_{2n-1}{b}_{2n}}^{a_1{a}_2\cdots a_{2n-1}{a}_{2n}}{g}^{a{b}_{2n}}{\mathcal{R}}^{b_1{b}_2}{}_{a_1{a}_2}\cdots {\mathcal{R}}^{b_{2n-3}{b}_{2n-2}}{}_{a_{2n-3}{a}_{2n-2}}.$$

$${}_{(2n)}{Q}^{abcd}={-}_{(2n)}{Q}^{abdc}={-}_{(2n)}{Q}^{bacd}={}_{(2n)}{Q}^{cdab},$$

$${}_{(2)}{Q}^{abcd}$$

$${}_{(4)}{Q}^{abcd}$$

$${\mathcal{L}}_{(2n)}=\sqrt{-g}{}_{(2n)}{Q}_a{}^{bcd}{\mathcal{R}}^a{}_{bcd}.$$

$${\mathcal{L}}_{(2n)}$$

$${\mathcal{L}}_{(2n)}={\mathcal{L}}_{(2n)\text{mbox}bulk}+{\mathcal{L}}_{(2n)\text{mbox}sur},$$

$${\mathcal{L}}_{(2n)\text{mbox}bulk}$$

$${\mathcal{L}}_{(2n)\text{mbox}sur}$$

$${\delta }_t^r\frac{\partial {\mathcal{L}}_{2n\text{mbox}bulk}}{\partial {\Gamma }_{st}^r}=2\sqrt{-g}\left[{}_{(2n)}{Q}_r{}^{bcs}{\Gamma }_{bc}^r+{}_{(2n)}{Q}_a{}^{srd}{\Gamma }_{dr}^a\right],$$

$${\mathcal{L}}_{(2n)\text{mbox}sur}=-{\partial }_a\left({\delta }_b^c\frac{\partial {\mathcal{L}}_{(2n)\text{mbox}bulk}}{\partial {\Gamma }_{ab}^c}\right).$$

$${\mathcal{L}}_{(2n)}={\mathcal{L}}_{(2n)\text{mbox}bulk}-{\partial }_a\left({\delta }_b^c\frac{\partial {\mathcal{L}}_{(2n)\text{mbox}bulk}}{\partial {\Gamma }_{ab}^c}\right).$$

$$e^{\mu }{}_r\frac{\partial {\mathcal{L}}_{(2n)\text{mbox}bulk}}{\partial ({\partial }_c{e}^{\mu }{}_r)}=-2\sqrt{-g}{Q}_a{}^{bcd}{\Gamma }_{bd}^a.$$

Full text

1 Introduction

Holography is a concept that has received widespread attention in a large number of physical theories. Its application covers a wide extent of scenarios ranging from optics, condensed matter, string theory and gravitation, mainly. In theoretical physics the most common conception we have is that of a correspondence between the dynamics of a system occurring in a $(D-1)$-dimensional space (termed surface, $\partial\mathcal{M}$) and the dynamics in a $D$-dimensional space (termed bulk, $\mathcal{M}$) [1, 2, 3]. In particular, for string theory the concept of holography is used to relate two different theories: one of them describing the surface while the other the bulk. From another point of view, the holography term can be used with a slightly different nuance. It is known in gravitation that the Einstein-Hilbert action, and its geometric generalization viz. the Lanczos-Lovelock gravity action [4], can be splitted into bulk and surface terms closely related each other. This fact leads to a correspondence giving account of the same theory both in the surface and in the bulk which serves to relate the degrees of freedom of the same theory either on the bulk or in the surface. Following the observation of Padmanabhan and collaborators [5, 6, 7], this latter equivalence can be considered as a holograpy of the action functional, avoiding confusion with the notion coming from string theory. Certainly, this holographic property of gravitation and its deep connection with the thermodynamics of black holes has been well exploited [8, 9, 10].

Within the framework of relativistic extended objects, often referred to as branes, floating in a background spacetime usually treated as fixed, the dynamics is obtained from an action constructed by the formation of suitable higher-order scalars, using the induced metric $g_{ab}$, and the extrinsic curvature tensor $K_{ab}$, which represent the geometry of the worldvolumes [11, 12]. To have congruence with the mathematical structure, it is required that both the worldvolume reparametrization invariance and the diffeomorphisms invariance of the background spacetime must be satisfied, which reduces the number of possible geometrical scalars. However, a not so pleasant feature is that this type of actions yield Euler-Lagrange equations involving higher-order derivatives of the field variables. In this regard, the Lovelock type brane gravity is a second-order theory that involves a set of geometric invariants defined on a $(p+1)$-dimensional hypersurface, the trajectory swept out by a $p$-dimensional brane, embedded whitin a flat manifold with an extra dimension with the particularity that the associated equations of motion remain of second-order in the derivatives of the fields [13, 14]. This particular aspect makes a given theory free from many of the pathologies that plague higher-order derivative theories [15]. This fact is important because it assures no propagation of extra degrees of freedom. The underlying common structure which gives rise to this theory is that these invariants are polynomials of degree $n\leq p+1$, but now in the extrinsic curvature of the hypersurface. The interest in this effective field theory is not only for its rich geometric structure but also for its possible implications in the description of cosmological acceleration behaviors in the brane-like universe scenarios [16, 17, 18, 19]. These Lovelock brane invariants are similar in form either to the original Lanczos-Lovelock invariants in pure gravity or to their necessary counterterms in order to have a well posed variational problem [4, 20, 21, 22]. This analogy needs a word of caution. While for even values of $n$ the Lovelock brane invariants look like the Gauss-Bonnet invariants, for odd values of $n$ the corresponding Lovelock brane invariants acquire the form of the Gibbons-Hawking-York-Myers boundary terms [20, 23] which are seen as counterterms if we have the presence of bulk Lovelock invariants. Unlike what happens with the counter-terms either in the Einstein theory or more generally in the Lanczos-Lovelock gravity, for the odd terms in this type of gravity we have the presence of time derivatives of the field variables, which is a sign that we have dynamics on this type of hypersurfaces. This important fact leads to a handful of attractive features.

In this spirit, the holographic scheme of our interest considers that from a second-order Lagrangian density $\mathcal{L}$ we can establish a splitting of it into two parts; the first being a first-order derivative Lagrangian density, $\mathcal{L}_{\mbox{\tiny bulk}}$, and the second a divergence absorbing the second-order derivative content, $\mathcal{L}_{\mbox{\tiny surf}}$, with the attribute of being able to establish a specific relationship between these such that it allows to determine $\mathcal{L}_{\mbox{\tiny surf}}$ completely in terms of $\mathcal{L}_{\mbox{\tiny bulk}}$. This type of correspondences were termed also as holographic relationships [5]. With this understanding, the bulk and the surface terms in Lovelock type brane gravity must encode the same amount of dynamical content. Within the quantum framework of brane-like universes [24, 25, 26, 27, 28, 29] and the Dirac’s extensible model for the electron [30, 31, 32, 33, 34] the usage of this holographic concept has been used in order to extract information from one term of the splitted Lagrangian, based in the other.

This paper is devoted to show that the Lovelock type brane gravity also possesses a holographic nature by providing the existence of holographic relationships. To some extent this fact is reasonable since this type of gravity belongs to the set of theories known as affine in acceleration, i.e., linear in second order derivatives of the fields [35], allowing the identification of a divergence term. In our approach, we have a larger number of holographic relationships since we have at most the double of the geometric invariants in comparison with the original Lanczos-Lovelock gravity. Additionally, we highlight the role played by the different conserved tensors that this theory possesses in our development as well as the relationships among them. Moreover, it will be interesting to explore whether these holographic relationships may also connect black hole solutions with their termodinamical properties in the braneworld scenario once suitable geometries be adapted, as occur in the case of the original Lovelock gravity theory. For some of the black holes solutions in this direction see [36, 37, 38].

The remainder of the paper is organized as follows. In section 2 we provide an overview of the Lovelock theory for extended objects geodesically floating in a flat Minkowski spacetime. We derive holographic relationships for the Lovelock type brane gravity in section 3. In section 4 we provide some relationships among the different conserved tensors of this theory. In section 5 we provide a discussion about the splitting into two parts of the first Lovelock brane invariant, named $K$ brane action. Conclusions and comments are presented in section 6. Throughout our analysis the convention for the worldvolume Riemann tensor we follow is $\mathcal{R}_{abc}{}^{d}=-2\partial_{[a}\Gamma^{d}_{b]c}+2\Gamma^{e}_{c[a}\Gamma^{d}_{b]e}$ where $T^{[ab]}$ indicates anti-symmetrization under the convention $T^{[ab]}=(T^{ab}-T^{ba})/2$. Similarly, $T^{(ab)}$ indicates symmetrization according to $T^{[ab]}=(T^{ab}+T^{ba})/2$. Somewhat larger computations were put in an Appendix.

2 Lovelock type brane theory

The system of interest is a $p$-dimensional spacelike extended object, $\Sigma$, geodesically floating in a $N=(p+2)$-dimensional fixed Minkowski spacetime ${\cal M}$ with metric $\eta_{\mu\nu}$ $(\mu,\nu=0,1,\ldots,p+1)$. To specify the $\Sigma$ trajectory, known as worldvolume and denoted by $m$, we set $y^{\mu}=X^{\mu}(x^{a})$ where $y^{\mu}$ are local coordinates of ${\cal M}$ and $x^{a}$ are local coordinates of $m$, being $X^{\mu}$ the embedding functions $(a,b=0,1,\ldots,p)$. Within the geometric framework of extended objects the essential derivatives of $X^{\mu}$ enter the game through the induced metric tensor $g_{ab}=\eta_{\mu\nu}e^{\mu}{}_{a}e^{\nu}{}_{b}$ and the extrinsic curvature $K_{ab}=-\eta_{\mu\nu}n^{\mu}\nabla_{a}e^{\nu}{}_{b}=K_{ba}$ where $e^{\mu}{}_{a}=\partial_{a}X^{\mu}$ are the tangent vectors to $m$, $n^{\mu}$ is the spacelike unit normal vector to $m$, and $\nabla_{a}$ is the worldvolume covariant derivative, $\nabla_{a}g_{bc}=0$.

We are interested in the Lovelock type brane gravity theory [13]. For a $(p+1)$-dimensional worldvolume described parametrically by the field variables $X^{\mu}$, the action functional

[TABLE]

where

[TABLE]

ensures that the Euler-Lagrange equations for the field variables are of second order. Here, $\alpha_{n}$ are constants with appropriate dimensions, $g:=\det(g_{ab})$ and $\delta^{a_{1}a_{2}a_{3}\ldots a_{n}}_{b_{1}b_{2}\,b_{3}\ldots\,b_{n}}$ denotes the alternating tensor known as the generalized Kronecker delta (gKd). Certainly, taking as a guideline the original Lovelock theory of gravity, the action (1) has been formed but now considering the antisymmetric products of the extrinsic curvature $K^{a}{}_{b}$. Written out in full, the gKd is given by the determinant made of Kronecker delta functions

[TABLE]

Regarding the definition (2), we set $L_{0}=1$. We must to stress that the action functional (1) is invariant under reparametrizations of the worldvolume. In view of the fact that the Lagrangian (2) is a polynomial of degree $n\leq p+1$ in the extrinsic curvature, the action (1) is a second-order derivative theory. The geometrical invariants (2) are known as Lovelock brane invariants (LBI). By construction, these terms vanish for $n>p+1$ whereas the term with $n=p+1$ corresponds to a topological invariant not contributing to the field equations. Since the independent variables to describe the worldvolume are the embedding functions instead of the metric, we then have one greater number of Lovelock type brane terms contrary to the pure gravity case. For even $n$ we recognize the form of the Gauss-Bonnet (GB) terms but expressed now in terms of the worldvolume geometry; for $n=0$ we have the DNG Lagrangian, for $n=2$ we have the Regge-Teitelboim (RT) model [39, 40, 41, 42, 24, 26, 27, 28], for $n=4$ we have the standard GB Lagrangian which for $p>3$ produces non-vanishing equations of motion with ghost-free contribution [19, 43, 44]. On the other side, for odd $n$ the corresponding LBI are seen as the Gibbons-Hawking-York-Myers boundary terms which may exist if we have the presence of bulk Lovelock invariants (see [4, 13] for further details). In short, for a $p$-brane there are at most $p+1$ possible terms leading to second-order equations of motion [13, 14].

Given that the background spacetime is flat Minkowski, in order to have a surface that propagates here, both intrinsic and extrinsic geometries of the surface can not be chosen arbitrarily but must satisfy the Gauss-Codazzi and Codazzi-Mainardi integrability conditions given by ${\cal R}_{abcd}=K_{ac}K_{bd}-K_{ad}K_{bc}$ and $\nabla_{a}K_{bc}-\nabla_{b}K_{ac}=0$, respectively. Here, ${\cal R}_{abcd}$ denotes the worldvolume Riemann tensor. In this sense, the repeated application of the Gauss-Codazzi condition in (2), gives rise to express the LBI for the even and odd cases in terms of ${\cal R}^{a}{}_{bcd}$. Indeed, for $n=0,1,2,3,\ldots$ we have

[TABLE]

These relations will prove useful to carry out the computations leading to holographic relationships in this type of gravity. Indeed, when the LBI are expressed in this fashion, it may be suggested to consider alternative sets of independent variables instead of $X^{\mu}$.

In view of the importance of some conserved tensors inherent in this theory, as well as to get a better understanding of our development, we glance at the manifestly covariant variation approach of the action in order to obtain the extrema conditions of the model (for more details see [13, 14]). We perform the variation of the action (1) in steps. We start with

[TABLE]

Now, we must recall the well known expression for the first variation, $\delta(\sqrt{-g})=(\sqrt{-g}/2)g^{ab}\delta g_{ab}$. Afterwards, from (2), the second variation yields

[TABLE]

where have introduced the worldvolume tensor [13]

[TABLE]

for the $n$th order Lovelock type brane invariant. Additionally, $\delta K^{a}{}_{b}=\delta(g^{ac}K_{cb})=-g^{ad}K^{c}{}_{b}\delta g_{dc}+g^{ac}\delta K_{cb}$. Thence, the variation (7) becomes $\delta L_{n}=\left(J^{ab}_{(n)}-g^{ab}L_{n}\right)\delta g_{ab}+n\,J^{ab}_{(n-1)}\delta K_{ab}$, in which, we have considered that the tensor (8) obeys the useful recurrence relation $J^{a}_{(n)b}=\delta^{a}_{b}\,L_{n}-nK^{a}{}_{c}J^{c}_{(n-1)b}$, [13, 14]. Thus, the variation (6) can be written in the form

[TABLE]

Some important remarks, before continuing, are in order. The tensors (8) are symmetric since they inherit the symmetries from the extrinsic curvature. These are conserved, $\nabla_{a}J^{ab}_{(n)}=0$, which may readily be proved by using the Codazzi-Mainardi integrability condition for extended objects, $\nabla_{a}K_{bc}=\nabla_{b}K_{ac}$. The term Lovelock brane tensors, for expression (8), has been coined to a large extent because they play a similar role to those conserved tensors appearing in the pure Lanczos-Lovelock gravity. For a $(p+1)$-dimensional worldvolume there are at most an equal number of conserved tensors $J^{ab}_{(n)}$. Finally, it is fairly direct to verify that $J^{ab}_{(n)}=(1/n)(\partial L_{n}/\partial K_{ab})$, for $n\neq 0$.

Let us return to the variation (9). Since the field variables are the functions $X^{\mu}$, the response of the action (1) to small changes in the worldvolume, $X^{\mu}(x^{a})\rightarrow X^{\mu}(x^{a})+\delta X^{\mu}(x^{a})$, is done through the variations $\delta g_{ab}$ and $\delta K_{ab}$. The deformation $\delta X^{\mu}$ can be decomposed into its parts, normal and tangential to the worldvolume, $\delta X^{\mu}=\Phi\,n^{\mu}+\Phi^{a}\,e^{\mu}{}_{a}$. The motions transverse to the worldvolume are the physically relevant ones so the common strategy is consider only that $\delta_{\perp}X^{\mu}=\Phi n^{\mu}$ where $\Phi=\eta_{\mu\nu}n^{\mu}\delta X^{\nu}$ it is assumed small in addition to be an arbitrary function of $x^{a}$. The tangential deformation can be identified with a reparametrization of the worldvolume, so that $\delta_{\parallel}S$ contributes only a total derivative, $\delta_{\parallel}S=\nabla_{a}(\Phi^{a}\,S)$. Taking into account both types of deformations, $g_{ab}$ and $K_{ab}$ respond as [45]

[TABLE]

After substituting these into (9), a lengthy but straightforward computation leads to $\delta S$ to take the form

[TABLE]

where

[TABLE]

is the Euler-Lagrange derivative which in turn can be expressed in terms of the LBI. Further, $Q^{a}$ denotes the Noether current given by the worldvolume vector

[TABLE]

It should be pointed out that we have only one equation of motion which is second-order in the field variables. This is related to the fact that we have only one relevant degree of freedom for this type of branes which is associated with the geometric configuration itself of the worldvolume. Certainly, as already mentioned, the physically observable measure of the deformation of the worldvolume, is the breathing mode provided by the scalar field $\Phi$, [14].

Following a similar strategy to the one made to obtain (4) and (5), we have a close relationship between the $J^{ab}_{(n)}$ and the worldvolume Riemann tensor. A repeated application of the Gauss-Codazzi integrability condition in the definition (8) yields the handy identities

[TABLE]

for $n=0,1,2,3,\ldots$.

The framework outlined above serves to argue that, except for the $K$ term, all the LBI satisfy the structure

[TABLE]

where $Q_{a}{}^{bcd}$ is a tensor that has all the symmetries of the worldvolume Riemann tensor, made from the induced metric, the Riemann tensor and the extrinsic curvature itself, besides a zero divergence, $\nabla_{c}Q_{a}{}^{bcd}=0$. This will be proved shortly.

In passing we remark that, due to the fact that the worldvolume is an oriented timelike manifold embedded in $\mathcal{M}$, the odd Lovelock type brane invariants, unlike what happens with the common counter-terms in the pure Lovelock gravity, they have the presence of time derivatives, which indicates that we have dynamical content on this type of hypersurfaces [28, 29, 30, 31, 32, 33, 34]. It should also be noted that there are formal descriptions in which some hypersurfaces are considered to play the role of boundaries for gravitational spaces, and which are described by actions that represent counter-terms of the original action. Such hypersurfaces can have a timelike, spacelike or null-like causal structure [46, 47].

3 Holographic relationships

The way in which the geometric invariants (2) were splitted into (4) and (5) becomes important because, except for the $K$ brane term, any $n$th Lovelock type brane density acquires the form

[TABLE]

where ${}_{(n)}\mathcal{Q}_{a}{}^{bcd}$ is a tensor with specific properties. We turn now to prove that any $n$th order Lovelock type brane model allows a decomposition in terms of both a bulk and surface components of the action functional in which both terms are directly related. To do this we will proceed in two parts.

3.1 Even case

Clearly, (4) furnishes us with the structure given in (18) for $n=1,2,3,\ldots,$

[TABLE]

with

[TABLE]

With the indices conveniently placed, it is evident that this tensor inherits the symmetries from the Riemann tensor

[TABLE]

and it is constructed from $g^{ab}$ and $\mathcal{R}^{a}{}_{bcd}$ or, from a brane point of view, from $g^{ab}$ and $K_{ab}$ when the worldvolume Riemann tensor is expressed in terms of the extrinsic curvature via the Gauss-Codazzi equations for the worldvolume. The first values of this $Q$ tensor are

[TABLE]

with $G^{ab}=\mathcal{R}^{ab}-(\mathcal{R}/2)g^{ab}$ being the worldvolume Einstein tensor, and $J^{ab}_{(0)},J^{ab}_{(1)}$ and $J^{ab}_{(2)}$ are the first Lovelock brane tensors given by (8), [13].

Consequently, relationship (19) allows us to express the Lagrangian densities for the even case as

[TABLE]

With the aid of the Riemann tensor, $\mathcal{R}_{abc}{}^{d}=-2\partial_{[a}\Gamma^{d}_{b]c}+2\Gamma^{e}_{c[a}\Gamma^{d}_{b]e}$, the former equation can be written in the form

[TABLE]

Now, by virtue of the identity $\partial_{a}\sqrt{-g}=\sqrt{-g}\,\Gamma^{b}_{ba}$, we have

[TABLE]

It follows therefore that, as long as the condition $\nabla_{c}\left[\,{}_{(2n)}Q_{a}{}^{bcd}\right]=0$ is satisfied, the previous equation takes the form

[TABLE]

where

[TABLE]

Some aspects of this framework are in order. It should be notice that for the RT model (26) is quadratic in $\Gamma^{a}_{bc}$; this evident by using (22). In this approach the form of $\mathcal{R}_{abc}{}^{d}$, expressed entirely in terms of the Christoffel symbols and its derivatives without requiring $g^{ab}$ and $K_{ab}$, has been an important ingredient in arriving to (25). This fact, allows the decomposition of the Lagrangian densities in terms of a surface and bulk terms. Relation (26) can be considered as a variant of the so-called Dirac-Schrödinger Lagrangian, or $\Gamma\Gamma$ Lagrangian for short, for Einstein theory [48]. According to the symmetries (21) the divergence of the tensor $Q$ on any of the indices vanishes. This is proved in A.1.

It is necessary to stress that $Q_{a}{}^{bcd}=Q_{a}{}^{bcd}(g^{ab},\mathcal{R}^{a}{}_{bcd})$ so the differentiation of (26) with respect to the Christoffel symbols infers that

[TABLE]

where $\partial\Gamma^{a}_{bc}/\partial\Gamma^{r}_{st}=\delta^{a}_{r}\delta^{s}_{b}\delta^{t}_{c}$ has been considered. Due to the symmetry properties of the tensor (20), it is necessary that the second term of this expression should vanish. Clearly, comparison of the former expression with relation (27) furnishes us with a holographic relation

[TABLE]

We find then that (25), with the aid of the identity (28), can be expressed as

[TABLE]

This way of approaching the problem is similar in the spirit to the one carried out in [5] for the case of pure Lovelock gravity. It remains to verify that $\nabla_{c}\left[\,{}_{(2n)}Q_{a}{}^{bcd}\right]=0$. In fact, on geometrical grounds, the the invariance under reparametrizations of the worldvolumes in this type of gravity framework requires this condition.

Alternatively, there is another form of express the holographic relation (28). By using the well known relation for the Christoffel symbols in terms of the geometry of the worldvolume, $\Gamma^{c}_{ab}=g^{cd}e_{\mu\,d}\partial_{a}e^{\nu}{}_{b}$, when a fixed metric exists in the bulk; this is the case, for example, when the Minkowski metric is expressed in spherical coordinates. As before, the differentiation of (26) with respect to the gradients of the tangent vectors leads to

[TABLE]

In arriving to the last equality we have used (21) and the symmetries of the Christoffel symbols. In this sense, comparison with the relation (27) furnishes us with another holographic relationship

[TABLE]

Such transformation leads to express (25) as

[TABLE]

This particular expression serves to make contact with the Hamiltonian framework behind this effective field theory. With support with the Ostrogradsky Hamiltonian formalism [49, 51, 52], the canonical momentum conjugate to $e^{\mu}{}_{a}=\partial_{a}X^{\mu}$ is

[TABLE]

so that the relation (32) becomes

[TABLE]

This is precisely the form of the so-called $``d(qp)$” structure introduced by Padmanabhan in [5, 6, 10], which assumes that it is valid for many gravitational theories.

Regarding the last alternative, for the RT model when the definition of the Christoffel symbols, $\Gamma^{a}_{bc}=e_{\mu}{}^{a}\partial_{b}e^{\mu}{}_{c}$, is introduced into (26) and after a arrangement of the various terms floating around, the corresponding bulk Lagrangian density (26) becomes

[TABLE]

where

[TABLE]

Notice that (35) is now quadratic in $e^{\mu}{}_{a}$.

3.2 Odd case

In analogy with the analysis performed for the even case, from the expression (5) it follows that the structure (18) holds in the odd case. For $n=1,2,3,\ldots$

[TABLE]

where

[TABLE]

This tensor also inherits the symmetries from the Riemann tensor

[TABLE]

It turns out that (38) is constructed from $g^{ab},\mathcal{R}^{a}{}_{bcd}$ and $K_{ab}$ unlike what happens with the even case. A few values of the $\mathsf{Q}$ tensor are

[TABLE]

where $J^{ab}_{(3)},J^{ab}_{(2)}$ and $J^{ab}_{(1)}$ are conserved Lovelock brane tensors. It is therefore possible to write a Lagrangian density for the odd case, analogous to (24), by writing

[TABLE]

In building the splitting of the Lagrangian density for this case into a bulk term and a surface term, is clear that the treatment performed for the even case, is applied directly. We thus find

[TABLE]

where

[TABLE]

as long as the condition $\nabla_{c}\left[\,{}_{(2n+1)}\mathsf{Q}_{a}{}^{bcd}\right]=0$ is satisfied.

It must be pointed out that from the functional dependence of the variables of the tensor (38), $\mathsf{Q}=\mathsf{Q}(g^{ab},\mathcal{R}^{a}{}_{bcd},K_{ab})$, it immediately follows that the structure of (3.1), (28) and (29), holds for the odd case. Certainly, the structures given in (28) and (29) are still maintained for this case

[TABLE]

We find then that (40), with the aid of the identity (43), can be expressed as

[TABLE]

thus establishing holographic relationship for the odd case.

In analogy with the even case, the expression (30) remains valid in this case as long as we take into account that the tensor (38) does not explicitly depend on the gradient of the tangent vectors. This serves to establish another holographic relationship for the odd case. Additionally, the relationship (34) is still valid for this case so we can also ensure a “$d(qp)$” structure inherent in this context. The proof that the tensor (38) is conserved is provided in A.2.

4 Relationships among conserved Lovelock type brane tensors

We do not attempt to attribute any sophisticated interpretation, either physical or geometrical, to the $Q$ tensors. In turn, we believe that tensors $J$ have a clearer geometric interpretation since they come from the invariance under reparametrizations of the worlvolume. In fact, it turns out that these tensors are anchored in some manner.

It follows from the contraction of definition (8) with the extrinsic curvature, an expression for the $n$th order Lovelock type brane invariant

[TABLE]

with $p=0,1,2,\ldots$. With this identity we shall relate tensors $Q$ in favor of tensors $J$. Certainly, when the Gauss-Codazzi integrability condition, $\mathcal{R}_{abcd}=K_{ac}K_{bd}-K_{ad}K_{bc}$, is inserted into the expression (19) and by invoking the symmetries of the $Q$ tensor, we get

[TABLE]

Now, when the expression (45) enters the game, this allows to identify a relationship among the conserved tensors of the Lovelock brane gravity theory

[TABLE]

It is important to remark that the conservation of the tensors (20) and (38) is due to the invariance under reparametrizations of the worldvolume. Knowing that $\nabla_{b}J^{ab}_{(n)}=0$ must be fulfilled, it is necessary that the $Q$ tensors must be conserved, as it happens.

Clearly, a similar procedure with the relations (37) and (45), helps to identify another relationship among the conserved tensors of the theory

[TABLE]

It is worthwhile to remark that (47) and (48) are valid for $n=1,2,3,\ldots$.

In [13, 14] was proved that for a $p$th order Lovelock type brane model, the associated field equation is

[TABLE]

In this sense, from the relations (47) and (48), it is fairly easy to express the equations of motion in terms of the conserved $Q$ tensors.

For an even Lovelock type brane model, $L_{(2n)}$, the equation of motion results $L_{(2n+1)}=J^{ab}_{(2n)}K_{ab}=0$. That is, $2\,_{(2n+1)}\mathsf{Q}^{acbd}K_{cd}K_{ab}=0$, where the expression (48) has been invoked. Thus, by using back the Gauss-Codazzi condition, we have that the associated equation of motion is

[TABLE]

Similarly, for an odd Lovelock type brane model, $L_{(2n+1)}$, the equation of motion is $L_{(2n+2)}=J^{ab}_{(2n+1)}K_{ab}=0$. In such a case, by considering (47), this equation of motion also takes a compact form

[TABLE]

Doubtless, (50) and (51) make up a rendering of the expression (49) in terms of the Riemann tensor.

Furthermore, in passing we would like to mention the relation among the conserved stress tensor $f_{(n)}^{a\,\mu}$ for the $n$th order LBI with the $Q$ tensors in dependence of the nature of $n$, [13]. This is quite straightforward by considering identities (47) and (48). For the even case we have

[TABLE]

whereas for the odd case, we get

[TABLE]

What remains to be done is an analysis of both the physical utility as well as a deep geometric interpretation of this type of holographic relationships.

5 On the $K$ brane action

Within the framework behind (25) and (40), the dependence on the curvature tensor has been essential in our development when we perform the covariant separation of the Lovelock type brane invariants. A glance at (5) shows that for $n=1$, the scheme adopted here rules out the case of the $K$ brane model because this can not be written in terms of the Riemann tensor. This is quite evident since this model is linear in the mean extrinsic curvature. Thence, this particular case needs a careful handling, or different, since the simple expression, $\sqrt{-g}\,K$, does not help to reach the purpose. Indeed, the Lagrangian density for this model in presence of a fixed background spacetime is

[TABLE]

where $G_{\mu\nu}$ is the bulk metric, $\Gamma^{\mu}_{\alpha\beta}$ are the background Christoffel symbols and $\mathcal{H}^{\mu\nu}:=g^{ab}e^{\mu}{}_{a}e^{\nu}{}_{b}$ is the projection operator in $\mathcal{M}$ onto $m$. Although the strategy of identifying a surface term seems feasible, here is not directly of much use since by considering an integration by parts, and making use of the definition of the extrinsic curvature this fails due to one faces the orthonormality of the worldvolume basis, $e_{a}\cdot n=0$.

An alternative way of handling this issue is by breaking the worldvolume covariance and thus perform a Hamiltonian analysis supported by a geometric ADM framework. Then, inspired by the geometric approach made in [50, 51] we can express (54) as

[TABLE]

where $\dot{X}^{\mu}=\partial_{t}X^{\mu}$, $t$ is a coordinate that labels the leafs of the foliation of the worldvolume by $\Sigma_{t}$ and $h:=\det(h_{AB})$ with $h_{AB}$ being the spacelike metric defined on $\Sigma_{t}$ such that $\sqrt{-g}=N\sqrt{h}$. Further, $N$ and $N^{A}$ are the lapse function and the shift vector, respectively. Here is more explicit the linear dependence of this Lagrangian density on the accelerations of the brane so that this model also is affine in acceleration [35]. A possibility to continue with the computation is to take advantage from the fact that for a codimension one worldvolume, the normal vector can be written in terms of the velocity of the brane, $\dot{X}^{\mu}$, the Levi-Civita symbols and the geometry of the brane $\Sigma_{t}$ in order to reduce the expression. However, a further develop gives rise to a splitting rather difficult at this time so that we are unable to immediately assess the accuracy of this alternative. Therefore, this issue needs to be further analyzed before to provide a total conclusion. We will report this development elsewhere.

6 Concluding remarks

We have shown that Lovelock type brane gravity has holographic relationships similar in spirit to those of pure Lanczos-Lovelock gravity. These relationships allow us to extract information about a part of the splitted Lagrangian density, in terms of the other one. A distinctive feature of this framework is the strong dependence on the conservation condition on the $Q$ tensors, (20) and (38). As shown, this is a consequence of the invariance under reparametrizations of the action functional in this type of gravity. In each surface governed by this action, there is only one degree of freedom, corresponding to the breathing mode of the worldvolume so its nature is only geometric [14]. In this sense, it is expected that the Lagrangians, either $\mathcal{L}_{\mbox{\tiny bulk}}$ or $\mathcal{L}_{\mbox{\tiny surf}}$ reflect this fact, thereby encoding the same amount of dynamical content by describing the same degree of freedom. The next step in this development is to explore the physical consequences coming from the holographic relationships. We expect to show eventually how these relationships get reflected at the boundary surface of the worldvolumes governed by this gravity. Furthemore, the holographic relationships that were found here also preserve the so-called “$d(qp)$” structure [5, 6], which can be thought of as a part of the transition from a coordinate representation to a momentum representation in dependence of the chosen variables. Additionally, in order to continue the quantum analysis for these brane models, the Hamiltonian formalism can benefit from this approach by considering only the Lagrangian $\mathcal{L}_{\mbox{\tiny bulk}}$ since the second-order nature of the full Lagrangian density is encoded in the Lagrangian $\mathcal{L}_{\mbox{\tiny surf}}$ thus avoiding a cumbersome Ostrogradski Hamiltonian approach.

Acknowledgments

I am grateful to R. Capovilla, R. Cordero, M. Cruz and C. Campuzano for the stimulating discussions and valuable comments. The author also thanks ProdeP-México, CA-UV-320: Algebra, Geometría y Gravitación. This work was partially supported by SNI (México).

Appendix A Proofs of the divergenless of the $Q$ terms

In this Appendix we provide the proofs of the divergenless property of the tensors (20 and (38).

A.1 Proof of $\nabla_{c}\left[{}_{(2n)}Q_{a}{}^{bcd}\right]=0$.

From Eq. (20) we have

[TABLE]

In view of the antisymmetry of the gKd it follows the relation

[TABLE]

From the usual Bianchy identity, $\nabla_{[a}\mathcal{R}_{bc]de}=0$, we infer that

[TABLE]

so that, the assertion is proved.

A.2 Proof of $\nabla_{c}\left[\,{}_{(2n+1)}\mathsf{Q}_{a}{}^{bcd}\right]=0$.

From Eq. (38) we have

[TABLE]

As before, in view of the antisymmetry of the gKd as well the Codazzi-Mainardi integrability condition for extended objects of arbitrary dimensions in Minkowski spacetime, $\nabla_{a}K_{bc}=\nabla_{b}K_{ac}$, we have

[TABLE]

By invoking the Bianchy identity, $\nabla_{[a}\mathcal{R}_{bc]de}=0$, as well the antisymmetry of the gKd and the symmetry of the extrinsic curvature, it follows that

[TABLE]

This proves our assertion.

References