Non-linear electrodynamics non-minimally coupled to gravity:
symmetric-hyperbolicity and causal structure.

Well-posedness of the initial value problem stands out as a basic requirement to be satisfied by any relativistic field theory. Broadly speaking, it implies that solutions for a given problem exist, are unique, and depend continuously on the initial data. Hence, well-posedness is at the roots of physics, for it amounts to the predictability power of a given theory. While it is difficult to decide whether a given non-linear theory has a well-posed initial value problem, a necessary and sufficient condition for well-posedness around a given solution is that all the linearised problems obtained by linearising near such a solution are well-posed, see for instance Kreiss and Lorenz (1989); Papallo and Reall (2017).

In the linearized regime, well-possedness means that the equation of motion is hyperbolic. At least three notions of hyperbolicity can be distinguished Kreiss and Lorenz (1989); Beig (2006): (a) weak hyperbolicity, in which all the roots of the characteristic equation are real, (b) strong hyperbolicity implies that there is an energy estimate that sets a bound for the energy of a solution at a given time in terms of the initial energy ¹¹1In fact, strong hyperbolicity is a necessary and sufficient condition for the initial-value problem to be well-posed., and (c) symmetric hyperbolicity, which is a sufficient condition for well-posedness Sarbach and Tiglio (2012). It follows that symmetric hyperbolicity implies strong hyperbolicity, which in turn implies weak hyperbolicity.

There are many examples in which the requirement of some kind of hyperbolicity imposes severe restrictions on the Lagrangian of a given theory. For instance, as shown in Papallo and Reall (2017), the equations of motion of Lovelock’ss theory are always weakly hyperbolic for weak fields but not strongly hyperbolic in a generic weak-field background. The well-posedness of Horneski theory has been analyzed in Papallo and Reall (2017); Papallo (2017). As shown in the latter reference, the most general Horndeski theory which is strongly hyperbolic for weak fields in a generalized harmonic gauge is simply k-essence coupled to Einstein’s gravity (see also Bernard et al. (2019)). The well-posedness of scalar-tensor effective field theory was studied in Kovács and Reall (2020), where it was shown that the equations of motion are strongly hyperbolic at weak coupling.

We would like to explore here the constraints imposed by the requirement of well-posedness in the case of nonlinear electromagnetic theories coupled to gravity. Nonlinear electromagnetism has been widely studied in several contexts. A non-exhaustive list of applications and references includes black holes Moreno and Sarbach (2003); Bretón and Perez Bergliaffa (2015); Toshmatov et al. (2018); Magos and Bretón (2020); Falciano et al. (2021), astrophysics Heyl and Hernquist (2004); Harding and Lai (2006); Turolla et al. (2015) and cosmology Novello et al. (2004, 2007); Campanelli et al. (2008); Vollick (2008); Kruglov (2015). There are also several articles devoted to different aspects of the propagation of perturbations in nonlinear electromagnetic theories, such as Gutierrez et al. (1981); Deser et al. (1999); Goulart de Oliveira Costa and Esteban Perez Bergliaffa (2009); Perlick et al. (2018). The matter of (symmetric) hyperbolicity for nonlinear electromagnetism minimally coupled to gravity was analyzed in a flat spacetime background in Abalos et al. (2015), while the hyperbolicity of Maxwell’s equations with a local (and possibly nonlinear) constitutive law in flat spacetime was considered in Perlick (2011). Our aim here is to determine the restrictions that follow from the requirement of symmetric hyperbolicity on nonlinear electromagnetism non-minimally coupled to gravity, several aspects of which have been studied in detail in Balakin and Lemos (2005); Balakin et al. (2008, 2010); Dereli and Sert (2011). We shall restrict to couplings linear in the curvature, since higher-order couplings produce higher-than-second order equations for the gravitational field Balakin and Lemos (2005).

As mentioned above, a sufficient condition for well-posedness to hold is that the system under study admits a symmetric-hyperbolic representation. The theory of first-order symmetric hyperbolic systems, originally due to Friedrichs Friedrichs (1954), has been extensively developed (see for instance Beig (2006) and references therein). To study the evolution of a system we shall adopt here the modern geometric approach to the subject outlined by Geroch in Hall et al. (1996), in which covariance is kept during most of the calculation, instead of using a 3+1 decomposition of spacetime (as for instance in Perlick (2011)) ²²2For a list of other approaches to the analysis of the evolution of minimally coupled electromagnetism, see Abalos et al. (2015).. We shall see that such approach leads to a general form for the symmetrizer, and to conditions for symmetric hyperbolicity that are easier to evaluate than those for other types of hyperbolicity.

The structure of the paper is as follows. In Section II the basic notation used in the equations of motion is presented. Symmetric hyperbolicity, the related concept of symmetrizer, and the role of the constraints are analyzed in Section III. The equations defining the characteristic cones will be deduced in Section IV. Our main result, namely the explicit form of the matrices that are needed to investigate the symmetric hyperbolicity of any nonlinear electromagnetic theory non-minimally coupled to gravity can be found in Section V. In Section VI, some examples of the restrictions imposed by symmetric hyperbolicity are presented for different theories. Our closing remarks are presented in Section VII.

To begin with, let $\mathcal{M}$ denote a smooth four-dimensional spacetime with a Lorentzian metric $\boldsymbol{g}$ of signature $(+,-,-,-)$. For the sake of concreteness we assume $\mathcal{M}$ to be also oriented and globally hyperbolic i.e. $M\cong\mathbb{R}\times\Sigma$, with $\Sigma$ a codimension-$1$ hypersurface. Sticking to the conventions $R_{ab}\equiv R^{c}_{\phantom{a}acb}$ and $[ab]\equiv ab-ba$, the canonical decomposition of the Riemman tensor into its irreducible parts reads

where $W_{abcd}$ is the Weyl conformal tensor, $S_{ab}$ is the traceless part of the Ricci tensor, $R$ is the scalar curvature and $g_{abcd}=g_{a[c}g_{d]b}$ is the Kulkarni-Nomizu product of the metric with itself. Clearly, each factor in the decomposition has the same algebraic symmetries as the full Riemann tensor, that is

An arbitrary rank four covariant tensor satisfying Eqs. (2) has 21 independent components and is often referred to as a double symmetric $(2,2)$-form (the skew pairs can be interchanged). If, in addition, the tensor satisfies Eq. (3), it is called an algebraic curvature tensor (see for instance Besse (1987)).

If we are given a double symmetric $(2,2)$-form on $\mathcal{M}$, say $\chi$, we may obtain its left-dual $\star\chi$ and right-dual $\chi\star$ by

where $\varepsilon_{abcd}$ is the Levi-Civita tensor with $\varepsilon_{0123}=\sqrt{-g}$. Notice that the place of the $\star$ indicates the pair of skew indices which are Hodge-dualized. A direct consequence is that

We aim at constructing Lagrangians describing the coupling of gravity with the electromagnetic field using invariants that are at most linear in the curvature ³³3As discussed in Balakin and Lemos (2005), higher-order couplings lead to equations of motion with derivatives of the metric higher than two., and respecting the $U(1)$ gauge invariance of electromagnetism. Recalling that every algebraic curvature tensor satisfy the Ruse-Lanczos identity, we may construct the following independent rank four tensors (see the Appendix for details)

Notice that ${\phantom{a}}^{(\Gamma)}\chi_{abcd}$ $(\Gamma=1,2,...,7)$ are double symmetric $(2,2)$-forms by construction and it can be checked that they exhaust the interesting possibilities: the first pair, Eqs. (6), involves only algebraic terms in the metric, while the remaining Eqs. (7)-(9) contain at most linear terms in the curvature.

Let $F_{ab}=\partial_{[a}A_{b]}$ denote a test electromagnetic field propagating on $\mathcal{M}$. We shall focus on the propagation properties of such a field on a fixed gravitational background. The EM field will be described by the gauge-invariant action functional of the form

where the factor $1/4$ is introduced for future convenience and the Lagrangian density is taken as an arbitrary smooth function of the following scalars

In analogy with electrodynamics in material media, we call ${\phantom{a}}^{(\Gamma)}H^{ab}$ the $\Gamma$-th induction tensor and ${\phantom{a}}^{(\Gamma)}\chi^{ab}_{\phantom{a}\phantom{a}cd}$ the $\Gamma$-th constitutive tensor. Particular instances described by Eq. (10) include Maxwell’s theory, minimally-coupled nonlinear electrodynamics, and the three-parameter non-minimal Einstein–Maxwell model originating from QED vacuum polarization in a background gravitational field (see for instance Drummond and Hathrell (1980)), among others. With the conventions presented above, the variation of the action with respect to the 4-potential yields a coupled system of first-order quasi-linear PDEs for the fields, given by

Here, $\nabla_{a}$ is the covariant derivative compatible with the metric and the full induction tensor is defined by

with $\mathcal{L}_{\Gamma}\equiv\partial\mathcal{L}/\partial I^{\Gamma}$, for conciseness. Notice that the system defined by Eqs. (12) is composed of eight equations for only six unknowns. In relevant physical situations, the equations must include six dynamical equations (defined with respect to some time coordinate, to be identified later) and two constraints, which are to be imposed on initial data.

In order to study whether the equations of motion admit a symmetric-hyperbolic representation, it is convenient to recast the system of equations Eqs. (12) in a unified manner as

where $x\in{\mathcal{M}}$, $K_{A\phantom{a}\beta}^{\phantom{a}m}$ is the principal part of the PDE and $J_{A}(x,\Phi)$ stands for semi-linear contributions (whose explicit form is unnecessary for our discussion). Here capital Latin indices $(A=1,...,8)$ stand for the space of multi-tensorial equations, lowercase Latin indices $(m=0,1,2,3)$ stand for space-time indices, and greek indices $(\beta=1,..,6)$ for tensorial unknowns. To start with, we introduce an ordering of the antisymmetric indices to obtain the six possible collective quantities

Making the identification $\Phi^{\beta}\rightarrow F^{bc}$ and performing simple manipulations in Eqs. (12), the principal part is then written as

with

$X_{abcd}$ consists of a main term involving only first partial derivatives of the Lagrangian density and a nonlinear term including the Hessian matrix of the latter. Clearly, if the Lagrangian is a linear combination of the invariants $I^{\Gamma}$, the last term vanishes and linear equations of motion follow. More importantly, $X_{abcd}$ is always a symmetric double (2,2)-form independently on the specific form of the Lagrangian. This is a direct consequence of the symmetries of the constitutive tensors together with the symmetry of the Hessian matrix, i.e. $\mathcal{L}_{\Gamma\Lambda}=\mathcal{L}_{\Lambda\Gamma}$.

In what follows we shall use the covariant approach for first-order symmetric-hyperbolic systems outlined in Hall et al. (1996): a symmetric hyperbolization of Eqs. (14) means that there exist a smooth symmetrizer $h^{A}_{\phantom{a}\alpha}$ and a covector field $n_{m}$, such that:

1. 1.

   $\hat{K}_{\alpha\phantom{a}\beta}^{\phantom{a}m}\equiv h^{A}_{\phantom{a}\alpha}K_{A\phantom{a}\beta}^{\phantom{a}m}$ is symmetric in the indices $\alpha,\ \beta$.
   

2. 2.

   The matrix $\hat{K}_{\alpha\beta}(n)\equiv\hat{K}_{\alpha\phantom{a}\beta}^{\phantom{a}m}n_{m}$ is positive-definite.
   

Roughly speaking, the first statement means that it should be possible to construct from Eqs. (14) a new subsystem of first-order quasi-linear PDEs given by

with $\hat{J}_{\alpha}\equiv h^{A}_{\phantom{a}\alpha}J_{A}$. Clearly, the new system contains only evolution equations and its dimension is equivalent to the number of unknown fields. The second statement means that the new system can be solved uniquely for any given set of initial conditions on a hypersurface $\Sigma$ with normal covector $n_{m}$. What are the remaining equations the symmetrizer does not capture? For the system to be consistent they should not be of the evolution type. In other words, they must be satisfied automatically once they are satisfied initially i.e., they should be what we normally call the constraints⁴⁴4See references Hall et al. (1996); Beig (2006); Kato (1975) for additional details..

The geometrical meaning of the covectors introduced in the previous paragraph is as follows. At a spacetime point $p\in\mathcal{M}$, the collection of all covectors satisfying condition (2) is denoted by $S_{p}$. This set defines a nonempty open, convex cone at $p$ and the tangent vectors $p^{a}\in T_{p}\mathcal{M}$ such that $p^{a}n_{a}>0$ for all $n_{a}\in S_{p}$ determine the cone of influence of the physical field, i.e., the maximal speed of propagation in any given direction. It turns out that the latter is also a nonempty open, convex cone at $p$.

In the next subsection, we start by showing that a family of symmetrizers parametrized by a vector field always exists for the system of first order PDEs given by Eqs. (12). It is important to point out that our result is general in the sense that it does not depend on the specific form of the Lagrangian density. We then obtain the conditions that a theory must satisfy for positive-definiteness (see condition (2) above) to hold. This is first obtained for a particular covector and henceforth generalized by inspecting the characteristic varieties (dispersion relations), which are necessarily well-behaved for symmetric hyperbolic systems.

Suppose that we manage to find a symmetric hyperbolization with constraints, as described in the previous sections. We have seen that this choice, however, is far from unique. Indeed, by the continuity of Eq.(40), any small deformation of $t_{a}$ will be such that condition (2) of symmetric hyperbolization is satisfied. It turns out that the set of all admissible covectors $S_{p}$ in $T_{p}^{*}\mathcal{M}$ is determined by the unique connected, open, convex, positive cone containing the initial covector Beig (2006) . Its existence is related to the hyperbolicity of the characteristic polynomial, defined by

which is a homogeneous multivariate polynomial of degree $6$ in our case. Recall that, at a spacetime point, such a polynomial is called hyperbolic in a direction $t_{a}$ if $p(t_{a})>0$ and the univariate polynomial $p(u_{a}+\lambda t_{a})$ only has real roots for all covectors $u_{a}\neq t_{a}$. Now, the vanishing set of the characteristic polynomial will define an algebraic variety: the cone of characteristic conormals (or characteristic cone, for brevity). In general, it will consist of different codimension 1 sheets which may be nested, intersect along lines, or even coincide. Geometrically, hyperbolicity in the direction of $t_{a}$ is the requirement that every line parallel to $t_{a}$ intersects this algebraic variety at exactly $``6"$ points (counting multiplicities). Clearly, this condition severely constrains the topology of the characteristic cones, thus guaranteeing well-behaved propagation for small wavy excitations. In particular, it was shown by Garding Gårding (1959) that the closure of the connected component of $t_{a}$ in the set $\{n_{a}|p(n_{a})\neq 0\}$ is necessarily convex: the hyperbolicity cone of the polynomial. That symmetric hyperbolicity implies the hyperbolicity of the characteristic polynomial is direct. To see this one simply observes that the equation

characterizes the eigenvalues of the quadratic form $\hat{K}_{\alpha\beta}(a_{m})$ relative to the metric $\hat{K}_{\alpha\beta}(t_{m})$ - and these eigenvalues have to be real (see Beig (2006) for further details).

In order to compute the characteristic polynomial explicitly, we substitute Eq. (40) into Eq. (46) and use Schur’s determinant identity to obtain the product of determinants

where

Notice that $q(t)$ is necessarily positive definite, since $\mathbb{P}>0$ and $\mathbb{S}<0$ for this particular covector. Interestingly, the determinant in Eq. (49) has been calculated several times in literature, see e.g. Perlick (2011),Rivera and Schuller (2011). In particular, it can be shown that it factorizes as

where

In other words, the sixth order polynomial always reduces to a product of a quadratic polynomial and a quartic polynomial. Since the vanishing set of the quadratic polynomial gives a non-compact variety (plane) inconsistent with the constraint equations, the characteristic cone is given by the covectors $k_{a}$ that satisfy the fourth-order Fresnel equation

The fourth rank tensor defined by the terms between parentheses is called the Kummer tensor, whereas its totally symmetrized version is usually called the Tamm-Rubilar tensor. If symmetric hyperbolicity holds, the above polynomial is necessarily hyperbolic in the direction of $t_{a}$, its vanishing set determining the causal structure of the theory up to a conformal factor. In specific situations, where $X_{abcd}$ is sufficiently simple, Eq. (52) will reduce to the more familiar product of quadratic polynomials ⁶⁶6A general Ansatz leading to bi-metricity in nonlinear electromagnetism was presented in Visser et al. (2003).

In these cases, symmetric hyperbolicity guarantees that the rank-2 contravariant tensors $g^{ab}_{(1)}$ and $g^{ab}_{(2)}$ are nondegenerate, necessarily of Lorentzian type and with the same signature. Furthermore, if these tensors coincide ⁷⁷7Covariant conditions on the Fresnel surface for birefringence to be absent were derived in Itin (2005). , then only one of them must be considered in the dispersion relation (reduced polynomial) , i.e.,

It is important to point out that the abovementioned result stating that symmetric hyperbolicity leads to a cone strongly constraints the possible shape of the Fresnel surfaces, defined in a convenient 3-space: they must be topologically equivalent to those obtained from the 4-d hyperbolicity cone. Hence, open surfaces are not allowed by symmetric-hyperbolic propagation ⁸⁸8For examples of Fresnel surfaces in Maxwell´s theory non-minimally coupled to gravity, see Balakin and Zayats (2018). .

We have seen that symmetric hyperbolicity imposes restrictions on admissible physical theories. Importantly, they are always expressed in terms of matrix inequalities which are much easier to guarantee than to check the hyperbolicity of the corresponding characteristic polynomial. In order to obtain explicit expressions for the matrix inequalities, it is convenient to introduce the projection tensor $h_{ab}=g_{ab}-t_{a}t_{b}$, which projects arbitrary tensors onto the “rest spaces” orthogonal to $t^{q}$, i.e,

Similarly, we decompose the electromagnetic 2-form, the Weyl tensor and the traceless part of the Ricci tensor as

with the following definitions

According to the latter, the tensor fields $\{E_{a},B_{a},Q_{a},\mathcal{E}_{ab},\mathcal{B}_{ab},N_{ab}\}$ are automatically orthogonal to the auxiliary vector field.

In order to compute $\mathbb{P}_{ab}$ and $\mathbb{S}_{ab}$ from Eq. (17), the following relations involving the constitutive tensors are useful

Similarly, for the induction tensors, Eq. (11), one obtains

Using the above relations and the Ruse-Lanczos identities (see the Appendix A), it can be checked that the tensors $\mathbb{P}_{ab}$ and $\mathbb{S}_{ab}$ are given by:

where the dots stand for possible nonlinear terms in the irreducible parts of the curvature tensor, which we shall not discuss in this work. The tensors $\mathbb{P}_{ab}$ and $\mathbb{S}_{ab}$ given above are the most general ones when dealing with symmetric hyperbolicity in models of nonlinear electromagnetism coupled linearly to curvature. The inequalities given in Eqs. (41), defined in terms of such tensors, lead to drastic restrictions on admissible theories: there must be a strong compromise between the electromagnetic quantities $\{E^{a},B^{a}\}$, the spacetime curvature expressed via $\{\mathcal{E}_{ab},\mathcal{B}_{ab},Q_{a},N_{ab}\}$ and the partial derivatives of the Lagrangian density.

Let us present next several examples that illustrate the restrictions obtained so far. For the sake of completeness, we display, in each case, the tensor $X_{abcd}$, the relevant inequalities and the corresponding characteristic polynomial. A generic feature of the latter is non-factorization: in general, the characteristic varieties are described by vanishing sets of fourth-order polynomials which do not split into products of second-order polynomials. Needless to say, this fact makes it considerably difficult to guarantee hyperbolicity straight from the characteristic polynomial. In constrast with this fact, symmetric-hyperbolicity automatically implies that the characteristic varieties are necessarily well-behaved.