Covariant higher order perturbations of branes in curved spacetime

Brane mechanics refers to the study of the dynamics of branes embedded in a higher dimensional space, usually called ambient space, background space, or target space. This is a generalization of the notion of a relativistic particle, a 0-brane. Then we can have strings or 1-branes. The term branes is leaving the dimension arbitrary. Branes are highly relevant in the description of different physical scenarios where the relevant physical degrees of freedom are confined to an appropriate sub-region of interest. Branes are described by a local action that is a functional of the geometry of the worldvolume spanned by the brane in its evolution. The guiding principles in the construction of the action are reparametrization invariance and background diffeomorphism invariance. This turns the action into a geometric model. The term ‘brane mechanics’ was introduced by Carter [1, 2], where it is emphasized the range of applications of the subject. Since then, the range has been widely extended in important ways.

At the classical level, even the simplest geometric models produce equations of motion that are highly non-linear. An important special case are strings propagating in a flat background, where the equations of motion turn out to be linear, in an appropriate gauge [3]. Besides this special case, in general one confronts a serious challenge in the analysis of the possible solutions of the equations of motion. In addition, the non-linearity can be made even worse if one considers a non trivial background spacetime, with non vanishing curvature, see e.g. [4] for the case of a string, where the background curvature plays the role of an external force. In face of these difficulties, one has to resort to perturbation theory. The first step is to encounter a relevant exact solution to perturb about. To attain this, symmetries are invoqued and exploited, to make the problem tractable. The next obvious step is a direct linearization of the equations of motion that produce a set of equations for field deviations of fields that (on-shell) reduce to coupled quadratic oscillators, and so on. This ‘direct approach’ is the one more commonly used in the literature [5]. There is also a ‘covariant direct approach’ that exploits a perturbation taken as a directional covariant derivative along a deviation vector, or deformation, and that is especially advantageous in a curved background, see e.g. [6].

The main subject of this paper is to offer a complete covariant variational approach that does not use any gauge-fixing at any stage, where by gauge-fixing it is meant a split of brane perturbations in their normal and tangential modes with respect to the worldvolume. This approach sets brane mechanics as a covariant classical field theory, trying to avoid any idiosyncratic notation, and using standard familiar language, that hopefully will be useful to the reader. The covariance is with respect to both background diffeomorphisms and worldvolume reparametrization invariance. The main tool that we exploit is a covariant variational derivative, inspired by the pioneering work of Ba$\dot{\mbox{z}}$ański [7], and its higher order extensions. From this point of view, conservation laws become easily accessible via Noether’s theorem, as well as how they behave under perturbations [8]. In general, one has at its disposal the whole weaponry of the calculus of variations, properly generalized to a field theory. The emphasis on covariance is not only desirable, but needed because gauge invariance is a subtle point in perturbation theory. As one goes to higher orders in perturbation theory, it becomes more and more difficult to disentangle physical modes from gauge unphysical modes. As emphasized especially by Carter, explicit covariance allows one to sidestep many artificial complications. In any case, once covariant expressions are obtained, with reasonable minimal pain, one has always the option to gauge-fix to one heart’s desire. In the classical case this is especially important to get to a seed solution of physical interest. Perturbations then would need to respect the gauge fixing conditions. When considering possible quantum corrections, it is mandatory in order to arrive to a proper definition of physical states.

The action for the brane is defined as an integral over the worldvolume of a Lagrangian density of weight one, to ensure reparametrization invariance. Its first order variation produces the Euler-Lagrange derivative and a Noether current. The latter is instrumental in the derivation of the canonical linear momentum, that will play a central role in our approach. The vanishing of the first variation, under suitable boundary conditions, gives the equations of motion for the system. The second order variation of the action has been considered by many authors in a ‘direct approach’. This corresponds to a linearization of the equations of motion [9, 10, 4]. An approach related to ours has been offered in the important work by Battye and Carter [11]. The motivation to go to second order is manifold, see e.g. [12] for hypersurfaces in arbitrary spacetimes. In brane mechanics, the second order variation is used to study stability of solutions, see e.g. [13] for a covariant study of perturbations in Lovelock type brane gravity. In addition, like the geodesic deviation equation for a relativistic particle, the vanishing of the second variation, on-shell, produces Jacobi equations that describe the relative motion of neighboring branes connected by a deviation vector. In a curved background, they take the form of a kinetic term that involves the Hessian for the model, with the background Riemann curvature playing the role of an external force. The second variation also gives the index of the geometric model. The index is of paramount importance in various geometrical contexts, including geometry in the large, see e.g. [14] for minimal surfaces. In addition, in the study of quantum or thermal fluctuations the second variation appears as the ‘shape operator’ that in a path integral determines both one-loop contributions and the semi-classical regime.

In many instances, second order variations is not enough, as degeneracies may emerge, and a third variation of the action is needed. In any case, it is useful to establish a pattern in perturbation theory that potentially can be extended to higher orders. This should be of interest especially in relativistic astrophysical applications, where observational data are becoming more and more precise, and a covariant perturbation theory is a needed theoretical tool. In this context, the works [15, 16] are a recent attempt to match upcoming cosmological data with perturbative theoretical models beyond General Relativity. The third variation we derive exhibits an interesting structure, with a kinetic term that depends on the Jacobi operator, inherited from the second variation, acting on the second order perturbations, and a source term that depends on the known first order perturbations. This structure was known previously in classical field theory, see [7]. One previous study on second order perturbation of Dirac-Nambu-Goto [DNG] branes in a curved background by Kiosses and Nicolaidis needs to be mentioned in this context [17] (see also [18]). As far as we know, it was the first attempt to go to a description of second order perturbations in full generality. Their strategy is a ‘direct approach’ splitting the perturbations in parts tangential and normal to the worldvolume. Although extremely convenient at first order, where tangential deformations can be associated to a mere reparametrization, and ignored safely, at higher order things become quite complicated, and the risk is to miss the forest for the trees. A covariant variational approach allows to understand the geometric structure of perturbation theory. In addition, we consider any codimension for the brane, although codimension one, for a hypersurface and codimension two are simpler to deal with. The presence of source terms that depend not only on the background Riemann curvature, but also on its covariant derivative is identified.

In recent years, extremal surfaces and their perturbations have emerged as actors in the context of AdS/CFT or Gauge/Gravity duality. In [19] it was conjectured that the entanglement entropy of a spatial subregion in a CFT can be calculated perturbatively using the area of a codimension-2 spacelike minimal surface in the dual gravity theory. A covariant version of this holographic entanglement entropy for arbitrary regions and ‘extremal’ surfaces was given in [20]. Based on this, the authors of [21, 22] argued that the Jacobi operator for 2-dimensional surfaces in a particular coordinate system is uniquely fixed using only boundary data. Classically in the gravity side, holographic entanglement entropy is proportional to a surface area. However, probably this is not the whole story. One may include quantum fields adding to the entanglement, that in turn transform the extremal surfaces into ‘quantum extremal surfaces’. They are defined as stationary points of Bekenstein’s generalized entropy [23]. Branes have also been considered for the physics of the large. One imagines the observable universe as being a brane embedded in a higher dimensional place, with all of its fields and particles constrained to the brane, only gravity is free to ‘leak’ into the bulk at high energies. These brane-world models offer possible phenomenological corrections to General Relativity [24, 25].

There are other contexts, unrelated, where a brane geometrical description is useful and has been proved very successful both theoretically and in its experimental confirmation in the description of equilibrium configurations of lipid membranes in soft matter physics, see e.g. [26] and references therein. There is a common geometrical language, but with respect to more speculative relativistic settings, there is also the crucial advantage of experimental verification, to test theoretical endeavour. A geometric approach to fluid lipid membranes in space, using a covariant variational approach is proposed in [27, 28]. Hydrodynamics on curved surfaces can be studied with a geometric description of submanifolds [29]. In this context, the theoretical work by Armas and collaborators is worth mentioning,e.g. in [30], a thorough study of brane and edge dynamics is given.

This paper is organized as follows. Section II gives a brief overview of the embedding geometry and sets our notation. Section III establishes a general covariant variational approach for relativistic brane models that depend at most on first order derivatives on the embedding functions. We construct systematically the first, second and third order variations of the action, by considering covariant variations with respect to a one parameter family of embeddings. For the sake of clarity, the third order variation is performed first in a flat background, then we add the complication of a curved background. Section applies the general formalism to the case of the DNG action. Finally, in Section V we offer a brief discussion.

In this section, we describe briefly the geometry of the embedding of a time-like submanifold, or worldvolume, that depicts the evolution of a relativistic object, or brane, in a fixed background spacetime.

The semi-Riemannian background spacetime is $\{M,g_{\mu\nu}\}$ of dimension $N+1$ with local coordinates $x^{\mu}$ ($\mu,\nu,\dots=0,1,\dots,N$), and endowed with a metric $g_{\mu\nu}$ of signature with mostly plus signs. Spacetime indices are lowered and raised with the spacetime metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. The background spacetime covariant derivative $\nabla_{\mu}$ is assumed to be torsionless and compatible with the background metric $g_{\mu\nu}$, $\nabla_{\rho}g_{\mu\nu}=0$, with Riemann curvature tensor defined as $[\nabla_{\mu},\nabla_{\nu}]V^{\rho}=-R_{\mu\nu\sigma}{}^{\rho}V^{\sigma}$, where we follow the conventions of [31].

The brane of dimension $d$ sweeps in its evolution a worldvolume $w$ of dimension $d+1$. The worldvolume $w$ can be described in a parametric form by the (time-like) local embedding functions $x^{\mu}=X^{\mu}(\xi^{a})$, in $\{M,g_{\mu\nu}\}$, where $\xi^{a}$ are local coordinates for $w$ ($a,b,\dots=0,1,\dots,d$). The $d+1$ tangent vectors to the worldvolume $w$ are $X^{\mu}_{a}=\partial_{a}X^{\mu}=\partial X^{\mu}/\partial\xi^{a}$, and the induced metric, or first fundamental form, on $w$ is

with determinant $\gamma$. The time-like character of the wordvolume is assured by requiring $\gamma<0$. Tangential indices are raised with the inverse induced metric $\gamma^{ab}$ and lowered with $\gamma_{ab}$. We denote with $\overline{\nabla}_{a}$ the worldvolume torsion-less covariant derivative compatible with the induced metric $\gamma_{ab}$, $\overline{\nabla}_{a}\gamma_{bc}=0$, with Riemann curvature ${\mathcal{R}}_{abc}{}^{d}$.

The extrinsic geometry of the worldvolume can be introduced via the spacelike normal vectors $n^{\mu}{}_{i}$ to the worldvolume $(i,j\dots=1,2,\dots,N-d)$ that are defined up to a sign and a rotation by the orthogonality condition $g_{\mu\nu}X^{\mu}_{a}n^{\nu}{}_{i}=0$ and normalized with $g_{\mu\nu}n^{\mu}{}_{i}n^{\nu}{}_{j}=\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. We leave the co-dimension arbitrary. Of course, there are noticeable simplifications for a hypersurface, $N-d=1$. We also note the completeness relationship

that defines the tangential and normal projectors $h^{\mu\nu}$ and $n^{\mu\nu}$, respectively.

The generalization of the classical Gauss-Weingarten equations for submanifolds to a curved background is given by

Here $\gamma_{ab}{}^{c}$ is the Christoffel symbol associated with the worldvolume covariant derivative $\overline{\nabla}_{a}$, and $\nabla_{a}=X^{\nu}_{a}\nabla_{\nu}$ is the projection of the background spacetime covariant derivative onto the worldvolume. The extrinsic curvature tensor, or second fundamental form, is

and $K^{i}=\gamma^{ab}K_{ab}{}^{i}$ denotes the mean extrinsic curvature. The twiddle on the covariant derivative of the normal vector in (4) denotes the $O(N-d)$ rotation covariant derivative $\widetilde{\nabla}_{a}n^{\mu}{}_{i}=\nabla_{a}n^{\mu}{}_{i}-\omega_{a}{}_{i}{}^{j}n^{\mu}{}_{j}$, where the connection $\omega_{a}{}^{ij}=-\,\omega_{a}{}^{ji}$ is the twist potential, see e.g. [6]. In this brief introduction, we prefer a ‘kinematical’ description, without getting into the more essential aspects of the embedding geometry, that involve a proper treatment of the relationship of the intrinsic and extrinsic geometry of the worldvolume, via the Gauss-Codazzi-Mainardi equations.

We consider a geometric model that describes the dynamics of a relativistic brane that satisfies the requirements of reparametrization invariance and that is a scalar under background diffeomorphism transformations. In addition, we focus our attention on models that depend at most on first derivatives of the embedding functions

where the Lagrangian density of weight one ${\mathcal{L}}(X^{\mu}_{a})$ depends only on the intrinsic geometry of the worldvolume $w$. We have absorbed the differential $d^{d+1}\xi$ in the integral sign, henceforth. For the sake of simplicity we assume that the brane has no boundary, and with appropriate boundary conditions $\partial w=0$. The simplest example that will be the focus of our attention in the next section is the DNG model, proportional to the volume of $w$, with $\mu$ a constant tension,

In the following, we consider the first, second, and third variations of the general intrinsic geometric model (6). This general approach allows to establish a covariant perturbation theory that can be extended to more complex physical systems, once a proper framework for geometric models is introduced.

The DNG action is a first approximation in this approach to constructing geometric models to describe physical systems, where we put to use the general covariant variational formulation we have described above. Originally, it was proposed by Dirac in a geometric relativistic model for an ‘extensible electron’ [40]. Later, Nambu and independently Goto applied the same idea to a relativistic string to model gluons in QCD [41, 42]. The rest is a long history.
From the viewpoint of brane mechanics, the DNG action is an obvious starting point. This action is proportional to the worldvolume of the brane and is invariant under diffeomorphisms. The DNG action is given by

where $\mu$ is the membrane tension. Using the framework previously developed, for the first variation we only need the canonical momentum. The canonical momentum is

Note that it is tangential to the worldvolume, as is the case for any intrinsic geometric model and isotropic. The equations of motion that are given by the vanishing of the first variation are

Since the action is invariant under reparametrizations, the tangential variations do not influence the dynamics because these can be identified as a divergence. Thus when one projects the equations of motion onto the basis of the background spacetime, only the normal part survives, the latter is given by

i.e. the vanishing of the worldvolume mean extrinsic curvature $K_{i}=\gamma^{ab}K_{ab\,i}$. This is the relativistic version of the equilibrium condition for a minimal surface with arbitrary codimension.

For the second variation, we need only to calculate the Hessian,

where we have used the completeness relationship (2) that defines the normal projector $n^{\mu\nu}$. The tangential bivector that appears in (49) is

with indices raised with $\gamma^{ab}$ and lowered with $g_{\mu\nu}$. Substituting in the general expression for the second variation (26) we have

where we identified the tangential projector defined in the completeness relation (2), given by

For this case the Jacobi equations (28) read

Notice that normal and tangential projections of the deviation vector arise naturally in the expression, showing the mixture of both contributions. These equations provide a generalization to branes of the well known geodesic deviation equation for particles.

For the third variation we need to obtain the tensor $\cal T$, defined in (36). When the algebraic dust settles down, we obtain

where there is an abundance of raised and lowered indices, and now we have a tangential antisymmetric trivector

Putting it all together, we use the expressions (46),(49), and (54) to insert them into (43), taking into account (44). Then, we can write the full third variation for the DNG action

Of course, the vanishing of the last integral gives us the second order perturbation to the equations of motion

here $\mathcal{J}_{\mu}(\delta^{2}X)$ is given by the left side of (53), with $\delta X\rightarrow\delta^{2}X$. The right side of (57) is the source term of Jacobi’s nonhomogeneous equation. Notice that the source term is influenced by the Riemann tensor of the background spacetime and its covariant derivative. One might think that the covariant derivative on the Riemann tensor should vanish since the background spacetime is fixed. However, when making variations, local changes in the embeddings must be taken into account, which justifies the appearance of this term.

In this paper we have developed a systematic covariant variational approach to perturbation theory for intrinsic brane models. Perturbations are realized as covariant derivatives along the flow of a deformation vector, with respect to a one parameter family of the worldvolume corresponding to the brane’s evolution. These vectors came from coordinate variations of a two parameters family of embeddings $X^{\mu}(\xi^{a};s)$. This is the standard approach used in the calculus of variations, but it has to be modified appropriately for systems with an infinite number of degrees of freedom.

As a first step, we consider models that depend only on the intrinsic geometry, i.e. that contain only first derivatives of the embedding functions. This set-up allows for a generalization to higher derivatives, relevant in many contexts, with the obvious complications involved.

We introduce a general covariant variational approach that to first order reproduces well known equations of motion. At second order, we obtain the Jacobi equations that describe the evolution of deviation vectors that depict the behavior of a one-parameter family of branes under the external force provided by the background projected Riemann curvature. In the kinetic term that appears in the Jacobi equations it is emphasized the central role of the Hessian matrix of the geometrical model. In the special case of a DNG geometric model, the kinetic term takes the form of a ‘mass matrix’ that separates neatly in a normal and tangential contribution. It should be mentioned at this point that a fully covariant approach is needed especially when considering boundary contributions. This consideration is relevant when issues of boundary vs. bulk contributions are deemed to be relevant. The Jacobi equation itself can be obtained from an accessory variational principle, as its Euler-Lagrange equation.
In our search for a suitable geometric structure for higher order variations, we extend the approach to a third variation of the action, making contact with previous investigations. What emerges is that the structure of the third variation is qualitatively different. The kinetic term is related to the Jacobi operator that appears in the second variation, but there is also a source term. This indicates the existence of a generic structure in perturbation theory that needs to be explored in more detail.

The equations obtained are very general, and can be applied readily to any specific background and any geometric model of interest. Unlike previous related works, we have not separated the perturbation into tangential and normal modes. Although this means we have non-physical degrees of freedom in our results, one can always substitute normal deformations to eliminate this, recovering the results of previous works. The variational approach allowed us to notice that all perturbations may be expressed as conservation laws plus curvature terms.

The Dirac-Nambu-Goto action was taken as an example. We found that beyond first order, the tangential and normal projections of the deformation vectors are naturally present, even though the vector itself remained covariant. Even in this simpler case, the second order perturbation equations can be very complicated and the contribution of the spacetime curvature is highly relevant. The framework developed here can be applied to any intrinsic model that depends at most on first order derivatives of the embedding functions, the case for higher order dependance may be developed in a similar way.

GC and EYL thank CONACyT for their graduate fellowships. RC thanks Sistema Nacional de Investigadores for partial support.