1 Introduction

The field equations of gravitational theories are covariant, and equate a geometric tensor (e.g. Einstein, Cotton, Bach tensor) to a tensor describing matter. Solutions are usually found in coordinates that exploit the symmetries. However, there are advantages in keeping the coordinate-free tensor description as far as possible. Besides the formal elegance, it naturally addresses scalar identities. We do so in this study of static space-times, beginning with local geometry and then discussing gravity.

A covariant characterization of a static space-time involves an equation for a time-like velocity \(u_k\) with a closed space-like acceleration \(\dot{u}_k=u^j\nabla _j u_k\). The two defining vectors are the natural start for the expansion of the relevant tensors. In absence of symmetries they are complemented by two others. This is basically the spirit of the 1+1+2 formalism introduced by Clarkson and Barrett [15, 16]. Here the second vector is fixed by the context. Carloni used the formalism to specify the stress tensor, and the Ricci tensor ensuing from the Einstein equation, in spherically symmetric metrics [14]. In this work the two unspecified orthogonal vectors are not explicitly required, as the Ricci tensor is constructed via the integrability conditions and linked to the electric part of the Weyl tensor.

With the vectors \(u_j\), \(\dot{u}_j\) and an orthogonal space-like pair, we consider the antisymmetric tensor (\(\eta =\dot{u}^k \dot{u}_k\))

$$\begin{aligned} F_{jk} ={\mathbb {E}} \tfrac{1}{\sqrt{\eta }} (u_j \dot{u}_k - \dot{u}_j u_k ) + {\mathbb {B}} (y_jz_k-y_kz_j) \end{aligned}$$

and obtain the conditions on \({\mathbb {E}}\) and \({\mathbb {B}} \) to yield a Faraday tensor. The terms correspond to the electric and magnetic fields. In the Einstein theory, the equations of linear and non-linear electrodynamics constrain the Ricci tensor to a simple structure, with coefficients R and \(R^\star \). The weak energy condition imposes a non-negative spatial curvature scalar, \(R^\star \ge 0\), while the scalar curvature R is zero if and only if the electrodynamics is linear.

After the general setting, we turn to the much studied spherically symmetric static space-times, with line element

$$\begin{aligned} ds^2 = - B(r) dt^2 +\frac{dr^2}{B(r)} +r^2(d\theta ^2 + \sin ^2\theta \, d\phi ^2). \end{aligned}$$

The early solutions in General Relativity, named after Schwarzschild, de Sitter, Reissner and Nordstrøm, belong to this class. In the years many others were found as black hole (BH) or compact star solutions, that fit in the present covariant description.

Beginning with geometry, we obtain the relevant static spherical tensors (Ricci, Weyl, Cotton, Bach) as combinations of \(u_i u_j\), \(g_{ij}\) and \(\dot{u}_i \dot{u}_j\). Remarkably, also the energy–momentum tensor of linear and non-linear electrodynamics is a combination of the same elementary tensors, with the magnetic term \({\mathbb {B}}\) being forced to be the field of a magnetic monopole. By constraining the tensor coefficients to simple forms, we obtain scalar equations that recover the early metrics and others.

By the equality inherent gravity theories mentioned in the beginning, the geometric Ricci, Cotton and Bach tensors fix the form of the energy–momentum tensor respectively in Einstein, Cotton and Conformal gravity. A similar construction is made in f(R) theory, where the Ricci tensor and the Hessian provide the form of the matter tensor. In these four theories, it has the form of energy–momentum tensor of an anisotropic fluid or of linear or nonlinear electrodynamics.

The identification of the geometric and physical coefficients of the tensors in the left and right sides of the field equations, provides scalar equations. The functions B(r) found on geometric grounds specify as solutions of field equations.

In a century of gravity theories, many static spherical solutions were found. This is a very short and partial recount.

In 1968 James Bardeen (son of John B. of BCS theory) obtained the first singularity-free BH solution of the Einstein equation [2, 4]:

$$\begin{aligned} B(r) = 1- \frac{2M r^2}{(r^2 + g^2)^{3/2}} \end{aligned}$$

Ayon-Beato and Garcia reinterpreted it as a magnetic monopole solution in Einstein-non-linear electrodynamics [1].

In 2003 Kiselev published a new exact solution of the Einstein equation for quintessential matter surrounding a BH [32]

$$\begin{aligned} B(r) = 1 -\frac{2M}{r} -\frac{K}{r^{1+3w}} \end{aligned}$$

It raised a debate, until Visser showed in 2020 that the Kiselev BH is neither a perfect fluid nor a quintessence [57]. Generalisations of Kiselev space-times were recently used in the framework of gravitational lensing [49].

In 1996 Hayward [27] discovered a line element describing the local formation of a BH out of vacuum, its Bardeen-like static quiescence and final evaporation.

Bronnikov [9] showed that Einstein gravity coupled to nonlinear electrodynamics has nontrivial spherical solutions with global regular metric if and only if the electric charge is zero and the Lagrangian \({\mathscr {L}}(F)\) has a finite limit as \(F\rightarrow \infty \). In the same context, Dymnikowa [20] studied the existence of regular spherically symmetric electrically charged solutions. The effects of torsion were considered by Cotton [18].

Gravastars (gravitational vacuum stars) were introduced in 2001 [46] as an alternative to BH that avoid the problems associated with horizons and singularities. Models in nonlinear electrodynamics were constructed by Lobo and Arellano [37].

Among a great variety of spherical metrics in Einstein gravity, we quote the Yukawa BH [45], the Van der Waals BH [50], the global monopole [5], the Rindler–Grumiller metric [23], the logotropic BH [13], quantum corrections to Reissner–Nordstrøm BH [58]. The study of non-linear electrodynamics coupled to f(R) gravity was started by Hollenstein and Lobo [29, 52].

In 1989 Mannheim and Kazanas [39] obtained an exact vacuum solution of Conformal gravity, and applied it to describe the rotation curve of galaxies without dark matter

$$\begin{aligned} B(r) = -\frac{1}{r}\beta (2-3\beta \gamma ) + (1-3\beta \gamma ) + \gamma r - \kappa r^2 \end{aligned}$$

Topological black holes in conformal gravity were studied by Klemm [31].

Based on the conformal action, but with variation in the connection, a theory named Cotton gravity was recently proposed by Harada [25]. We showed that the field equation can be recast as Einstein equations, with the freedom of a Codazzi tensor. As such, they are second order in the derivatives of the metric tensor [44].

This is the ouline of the paper. In Sect. 2 we discuss static space-times in general, the Ricci and the Weyl tensors. It is partly based on our study of the larger family of doubly-warped space-times [43]. Useful equations are collected in “Appendix 1”.

In Sect. 3 we introduce the Faraday tensor and prove necessary and sufficient conditions on the scalars \({\mathbb {E}}\) and \({\mathbb {B}}\) (the proof is in “Appendix 2”). The general discussion of Einstein gravity coupled to linear (LE) and non-linear electrodynamics (NLE) is in Sect. 4. An interesting form of the Ricci tensor is obtained, with conclusions about the curvature space-time and space scalars R and \(R^\star \). Section 5 discusses the anisotropic fluid source, concluding that the energy density is proportional to \(R^\star \ge 0\).

In Sect. 6 we discuss spherical symmetry, where the full form of the Ricci, Weyl, Cotton and Bach tensors are obtained, as combinations of the basic tensors \(g_{jk}\), \(u_ju_k\) and \(\dot{u}_j \dot{u}_k\) with coefficients that are linear or at most quadratic in B(r), \(B'(r)\) and \(B''(r)\). Simple conditions yield the early static metrics. In Sect. 7 we consider the Einstein gravity, where the metrics are solutions of field equations. Pure dust or perfect fluid solutions are not possible, unless \(p=-\mu \). We then discuss LE and NLE, with some identities that allow for reconstructing the Lagrangian \({\mathscr {L}}(F)\) from B(r) (actually from a coefficient of the Ricci tensor).

NLE coupled to f(R) gravity is presented in Sect. 8, with immediate recognition of the known property that \(f_R(r)\) is constrained to be linear in r. This allows the integration of one field equation in presence of point charges. Since only the solution appears in previous papers, we offer its deduction in “Appendix 3”.

In Cotton gravity (Sect. 9), after a brief presentation of our interpretation as an Einstein equation, we show that the vacuum solution by Harada is also a solution of the Einstein theory with an anisotropic energy–momentum. We then present two solutions: with perfect fluid and LE. Finally, in Sect. 10 we turn to Conformal gravity. We recall the vacuum solution by Mannheim and Kazanas, and present a solution in LE. We end with the conclusions.

In this paper the static four-dimensional Lorentzian spacetimes have signature \((-,+++)\). A dot denotes the action of \(u^k\nabla _k\).

2 Static space-times

The velocity is eigenvector of the Ricci tensor, the Electric tensor

evaluates the Weyl tensor, the form of the Ricci tensor is obtained.

There are various characterisations of static spacetimes:

\(\bullet \) The existence of a time-like vector field \(u_k\) (named velocity) that is normalized, \(u^ku_k=-1\), with gradient

$$\begin{aligned} \nabla _j u_k = -u_j \dot{u}_k \end{aligned}$$
(1)

such that the ‘acceleration’ \(\dot{u}_k = u^j\nabla _j u_k\) is a closed vector field [54]:

$$\begin{aligned} \nabla _j \dot{u}_k = \nabla _k \dot{u}_j \end{aligned}$$
(2)

The acceleration is spacelike (\(u^k\dot{u}_k=0\)) with normalization \(\eta =\dot{u}^p\dot{u}_p>0\). Contraction of (2) with \(u^j\) shows that \({\ddot{u}}_k = u^j\nabla _k \dot{u}_j = - \dot{u}^j \nabla _k u_j = \dot{u}^j u_k \dot{u}_j = \eta u_k\).

\(\bullet \) The existence of coordinates \((t,\textbf{x})\) where the metric tensor has the static form

$$\begin{aligned} ds^2 = -B(\textbf{x})dt^2 + g^\star _{\mu \nu } (\textbf{x}) dx^\mu dx^\nu \end{aligned}$$

In this frame: \(u_k=(-\sqrt{B}, \textbf{0})\), \(\dot{u}_k =(0, \partial _\mu \log \sqrt{B})\).

\(\bullet \) The existence of a time-like hypersurface orthogonal Killing vector: \(\nabla _i\xi _j+\nabla _j\xi _i=0\). The vector is \(\xi _j=u_j\sqrt{B}\), and \(\dot{u}_j =\nabla _j \log \sqrt{B}\).

Proposition 2.1

The velocity is an eigenvector of the Ricci tensor,

$$\begin{aligned} R_{jk} u^k = - (\nabla _p \dot{u}^p) u_j \end{aligned}$$
(3)

and it is Riemann compatible:

$$\begin{aligned} (u_i R_{jklm}+u_j R_{kilm} + u_k R_{ijlm}) u^m =0. \end{aligned}$$
(4)

Proof

\(R_{jklm}u^m = (\nabla _j\nabla _k -\nabla _k\nabla _j)u_l = (u_j\dot{u}_k -\dot{u}_j u_k)\dot{u}_l + u_j\nabla _k \dot{u}_l - u_k\nabla _j \dot{u}_l\). Contraction with \(g^{jl}\) gives property (3). Multiplication by \(u_i\) and cyclic sum gives compatibility. \(\square \)

As expected, the “time derivative” of geometric invariants is zero:

Proposition 2.2

Let \(\eta = \dot{u}^k\dot{u}_k\) and \(R=g^{jk} R_{jk}\),

$$\begin{aligned} {\dot{\eta }} =0,\quad \dot{R}=0, \quad u^k\nabla _k (\nabla _p\dot{u}^p)=0 \end{aligned}$$
(5)

Proof

1) \({\dot{\eta }} = u^k\nabla _k (\dot{u}^p \dot{u}_p) = 2{\ddot{u}}^p u_p = 2\eta u^p\dot{u}_p =0\). 2) \(u^k\nabla _k \nabla _p\dot{u}^p = u^k R_{kp}{}^p{}_m \dot{u}^m + u^k\nabla _p\nabla _k \dot{u}^p = u^k R_{kp}\dot{u}^p + \nabla _p({\ddot{u}}^p) - (\nabla _p u^k)(\nabla _k\dot{u}^p) = \nabla _p (\eta u^p) + u^p\dot{u}^k \nabla _p \dot{u}_k = {\dot{\eta }} + \frac{1}{2}{\dot{\eta }} = 0\). 3) Eq. (3) and \(\nabla _j R^j{}_k = \frac{1}{2}\nabla _k R\) give \(\dot{R} = -2 u^j\nabla _j (\nabla _p\dot{u}^p)=0\). \(\square \)

Being Riemann compatible, the velocity is also “Weyl compatible" (Theorem 2.1 in [42]): \((u_i C_{jklm}+u_j C_{kilm} + u_k C_{ijlm}) u^m =0\). The Weyl tensor is

$$\begin{aligned} C_{jklm} {=} R_{jklm} {+} \frac{1}{2}(g_{jm} R_{kl} {-} g_{km} R_{jl} {+} g_{kl}R_{jm} - g_{jl} R_{km}) {-} \frac{R}{6} \, ( g_{jm}g_{kl}{-} g_{km}g_{jl} ). \end{aligned}$$

The contraction \(E_{kl} = u^j C_{jklm} u^m\) is the Electric tensor; it is symmetric, traceless and \(E_{jk}u^k=0\). Weyl compatibility is equivalent to the relation

$$\begin{aligned} C_{jklm}u^m = u_k E_{jl} - u_j E_{kl} \end{aligned}$$

The explicit evaluation of the Electric tensor gives an identity with the Ricci tensor:

$$\begin{aligned} E_{kl} = - \frac{1}{2} R_{kl} +\frac{1}{2}(\nabla _p\dot{u}^p) (g_{kl} + 2u_l u_k) + \frac{1}{6} R (g_{kl}+ u_k u_l ) + u^jR_{jklm} u^m \end{aligned}$$
(6)

where \(u^j R_{jklm}u^m = -\dot{u}_k \dot{u}_l - \nabla _k \dot{u}_l - \eta u_k u_l\) (see Prop. 2.1).

Proposition 2.3

(Weyl tensor) In a four-dimensional static spacetime the Weyl tensor is solely determined by the electric tensor:

$$\begin{aligned} C_{jklm}&= (g_{kl}+2 u_k u_l )E_{jm} -(g_{jl}+2u_ju_l )E_{km} \nonumber \\&\quad + (g_{jm} +2u_j u_m) E_{kl} - (g_{km}+2u_k u_m) E_{jl} \end{aligned}$$
(7)

and \(C_{jklm}C^{jklm} = 8 E^{kl}E_{kl}\).

Proof

In \(n=4\) the following identity by Lovelock holds [38]: \(0= g_{ir} C_{jklm} + g_{jr} C_{kilm}+ g_{kr} C_{ijlm} +g_{im} C_{jkrl} + g_{jm} C_{kirl}+ g_{km} C_{ijrl} +g_{il} C_{jkmr} + g_{jl} C_{kimr}+ g_{kl} C_{ijmr} \). The contraction with \(u^i u^r\) and Weyl compatibility give the Weyl tensor. \(\square \)

At each point we choose a basis of vectors formed by \(u_i\) and three orthonormal space-like vectors \(\frac{1}{\sqrt{\eta }}\dot{u}_i\), \(y_i\), \(z_i\):

$$\begin{aligned} g_{ij} = -u_i u_j + \frac{\dot{u}_i \dot{u}_j}{\eta } +y_i y_j + z_i z_j \end{aligned}$$
(8)

Lemma 2.4

In a \(n=4\) static space-time:

$$\begin{aligned}&\nabla _k y_j = - Y_k \dot{u}_j - \Omega _k z_j \end{aligned}$$
(9)
$$\begin{aligned}&\nabla _k z_j = -Z_k\dot{u}_j +\Omega _k y_j \end{aligned}$$
(10)
$$\begin{aligned}&\nabla _k \dot{u}_j = -\eta u_k u_j +\tfrac{1}{2\eta }\dot{u}_j\nabla _k\eta +\eta (y_j Y_k + z_j Z_k) \end{aligned}$$
(11)

where \(Y_k =\frac{1}{\eta } y^p\nabla _k \dot{u}_p\), \(Z_k =\frac{1}{\eta } z^m\nabla _k \dot{u}_m\) and \(\Omega _k= y^p\nabla _k z_p\). It is also:

$$\begin{aligned} u^k Y_k=0,\quad u^kZ_k=0, \quad y^k Z_k = z^k Y_k \nonumber \\ \eta ( y^k Y_k + z^k Z_k ) = - \eta + \nabla ^p\dot{u}_p - \frac{\dot{u}^p \nabla _p\eta }{2\eta } \end{aligned}$$
(12)

Proof

The gradient of (8) is: \( 0 = u_k (u_i \dot{u}_j + \dot{u}_i u_j) +\frac{1}{\eta } (\dot{u}_i \nabla _k \dot{u}_j + \dot{u}_j \nabla _k \dot{u}_i)- \dot{u}_i \dot{u}_j \frac{\nabla _k \eta }{\eta ^2}+y_i \nabla _k y_j +y_j \nabla _k y_i + z_i \nabla _k z_j+ z_j \nabla _k z_i \). The contractions with \(y^i\) or \(z^i\) give the first two relations. While contracting with \(\dot{u}^i\) note that \(\dot{u}^i\nabla _k y_i = -y^i\nabla _k \dot{u}_i =-\eta Y_k\), and \(\dot{u}^i\nabla _k z_i =-\eta Z_k\).

Similarly: \(u^kY_k= \frac{1}{\eta } y^j {\ddot{u}}_j = y^ju_j=0\) and \(u^j Z_j=0\), and \(y^kZ_k=z^kY_k\). Finally, (12) results from the contraction \(g^{jk}\) of (11). \(\square \)

With this choice of basis vectors, the Ricci tensor (6) is:

$$\begin{aligned} R_{kl}&=u_k u_l \left[ \frac{R}{3} +2\nabla _p \dot{u}^p\right] + g_{kl}\left[ \frac{R}{3} +\nabla _p \dot{u}^p\right] -2\dot{u}_k \dot{u}_l -2E_{kl} \nonumber \\&\quad -\frac{1}{\eta } \dot{u}_l \nabla _k \eta -2\eta (Y_k y_l + Z_k z_l). \end{aligned}$$
(13)

In static space-times the curvature scalar R and the space curvature scalar \(R^\star \) are related by the identity (see [43] Eq. 34):

$$\begin{aligned} R=R^\star -2\nabla _p \dot{u}^p. \end{aligned}$$
(14)

3 The Faraday tensor in static space-times

The conditions for a Faraday tensor and the conserved current are given.

The antisymmetric tensors \(u_i\dot{u}_j - u_j \dot{u}_i\) and \(y_i z_j - y_jz_i\) are “time independent”: \( u^k\nabla _k (u_i\dot{u}_j - u_j \dot{u}_i)=0\) and \(u^k\nabla _k (y_i z_j - y_jz_i) =0 \). The other antisymmetric combinations of the basis vectors do not share this property.

Therefore, we consider the following antisymmetric tensor

$$\begin{aligned} F_{jk} = \frac{{\mathbb {E}}}{\sqrt{\eta }} (u_j \dot{u}_k - \dot{u}_j u_k) + {{\mathbb {B}}} (y_j z_k - y_k z_j) \end{aligned}$$
(15)

where \({\mathbb {E}}\) and \({\mathbb {B}}\) are scalar fields with \(\dot{{\mathbb {E}}}=\dot{{\mathbb {B}}}=0\). Since \({\dot{\eta }}=0\), it is \(\dot{F}_{jk}=0\).

\(F_{jk}\) is a Faraday tensor if:

$$\begin{aligned} \nabla _j F_{kl} + \nabla _k F_{lj} + \nabla _l F_{jk} = 0. \end{aligned}$$
(16)

Theorem 3.1

(The Faraday tensor) In a static space-time, the tensor (15) with \(\dot{{\mathbb {E}}}=\dot{{\mathbb {B}}}=0\) is Faraday if and only if

$$\begin{aligned}&\nabla _k \frac{{\mathbb {E}}}{\sqrt{\eta }} = \varkappa \dot{u}_k, \end{aligned}$$
(17)
$$\begin{aligned}&\dot{u}^k \nabla _k {\mathbb {B}} = {\mathbb {B}} \left[ \eta - \nabla ^k\dot{u}_k + \frac{\dot{u}^k \nabla _k\eta }{2\eta } \right] . \end{aligned}$$
(18)

Then it is \({\dot{\varkappa }} =0\).

Proof

See “Appendix 2”. \(\square \)

The vector field \(J_k =\nabla _j F^j{}_k \) is a conserved current \(\nabla _k J^k=0\).

Proposition 3.2

(The current)

$$\begin{aligned}&J^k = J u^k + \big (z^k y^m - y^kz^m\big ) \left( \nabla _m {\mathbb {B}} - {\mathbb {B}} \frac{\nabla _m \eta }{2\eta }\right) \end{aligned}$$
(19)
$$\begin{aligned}&J= -\eta \varkappa +{\mathbb {E}}\sqrt{\eta }-\frac{{\mathbb {E}}}{\sqrt{\eta }}\nabla _j \dot{u}^j \end{aligned}$$
(20)

Proof

Equation (92) gives \( J_k = -\eta \varkappa u_k +\frac{{\mathbb {E}}}{\sqrt{\eta }}({\ddot{u}}_k - u_k \nabla _i \dot{u}^i) + (y^j\nabla _j {\mathbb {B}} - {\mathbb {B}}\dot{u}^j Y_j) z_k - (z^j\nabla _j {\mathbb {B}} - {\mathbb {B}} \dot{u}^j Z_j)y_k + {\mathbb {B}}(Y_j z^j - Z_j y^j) \dot{u}_k \). The last term is zero and \({\ddot{u}}_k =\eta u_k\). It is also \( \dot{u}^j Y_j = \frac{1}{\eta } y^m \dot{u}^j \nabla _j \dot{u}_m =\frac{1}{\eta } y^m \dot{u}^j \nabla _m \dot{u}_j =\frac{1}{2\eta } y^m \nabla _m \eta \), and \(\dot{u}^j Z_k = \frac{1}{2\eta } z^m \nabla _m \eta \). The current is obtained, and is orthogonal to \(\dot{u}_k\). \(\square \)

4 Linear/non-linear electrodynamics in static Einstein gravity

The equations of the Einstein—LE and NLE theory are discussed.

The Einstein equations of gravity coupled to an electromagnetic field descend from the action (see [1])

$$\begin{aligned} S=\frac{1}{2}\int d^4x \sqrt{-g} \left[ R - 4 {\mathscr {L}} ( F) \right] \end{aligned}$$

where \({\mathscr {L}}\) is a scalar function of the squared Faraday tensor \(F=\tfrac{1}{4} F_{jk}F^{jk} \). In linear electrodynamics \({\mathscr {L}}(F)=F\).

The vanishing of the variations of the action in the metric tensor and in the vector potential (\(F_{jk}=\nabla _j A_k - \nabla _k A_j\)) respectively give:

$$\begin{aligned}&R_{jk} -\tfrac{1}{2} g_{jk} R= 2 {\mathscr {L}}_F(F) F_{jm} F_k{}^m - 2 g_{jk} {\mathscr {L}} (F) \end{aligned}$$
(21)
$$\begin{aligned}&\nabla _j ({\mathscr {L}}_F(F) F^{jk}) =0 \end{aligned}$$
(22)

where \({\mathscr {L}}_F = d{\mathscr {L}}/dF\). The right-hand-side of Eq. (21) is the energy–momentum tensor \(T_{jk}^{nlin} \) of non-linear electrodynamics.

In the static setting with \(F_{jk}\) given by (15), it is \(F = \tfrac{1}{2}({\mathbb {B}}^2-{\mathbb {E}}^2)\) and

$$\begin{aligned} T_{jk}^{nlin} = 2 ({\mathbb {E}}^2+{\mathbb {B}}^2) \left[ u_j u_k -\frac{\dot{u}_j \dot{u}_k}{\eta } \right] {\mathscr {L}}_F (F) + 2 g_{jk} [{\mathbb {B}}^2 {\mathscr {L}}_F(F) - {\mathscr {L}}(F)]. \end{aligned}$$
(23)

Note the disappearance of the space-like vectors \(y_j \) and \(z_j\).

In the linear case the tensor is traceless:

$$\begin{aligned} T_{jk}^{lin} = 2 ({\mathbb {E}}^2+{\mathbb {B}}^2) \left[ u_j u_k +\frac{1}{2}g_{jk} -\frac{\dot{u}_j \dot{u}_k}{\eta } \right] . \end{aligned}$$
(24)

The contractions of the Einstein equation (21) with \(u^j\) and \(g^{jk}\) give two interesting relations between the geometry and the scalars of electrodynamics:

$$\begin{aligned}&{\mathbb {E}}^2 {\mathscr {L}}_F (F) + {\mathscr {L}} (F)=\tfrac{1}{2}\nabla _p\dot{u}^p + \tfrac{1}{4}R \end{aligned}$$
(25)
$$\begin{aligned}&F {\mathscr {L}}_F (F) - {\mathscr {L}}(F) = - \tfrac{1}{8} R \end{aligned}$$
(26)

These are immediate consequences:

Proposition 4.1

(1) \(R=0\) if and only if \({\mathscr {L}}(F)=cF\) (we take \(c=1\)).

(2) In linear electrodynamics: \({\mathbb {B}}^2+{\mathbb {E}}^2 = \nabla _p\dot{u}^p \).

The sum of (25) and (26) is: \(({\mathbb {B}}^2+{\mathbb {E}}^2){\mathscr {L}}_F=\nabla _p\dot{u}^p +\tfrac{1}{4}R\). The Einstein equation of non-linear electrodynamics becomes a geometric prescription for the Ricci tensor, which acquires a form much simpler than the general one in static space-times (13):

$$\begin{aligned} R_{jk} = \left( R^\star -\frac{1}{2}R\right) \left[ u_j u_k +\frac{1}{2} g_{jk}-\frac{\dot{u}_j \dot{u}_k}{\eta } \right] + \frac{1}{4} g_{jk} R \end{aligned}$$
(27)

\(u^k\) and \(\dot{u}^k\) are eigenvectors of the Ricci tensor with eigenvalue \(- (\nabla _p \dot{u}^p) \), while \(y^k\) and \(z^k\) are eigenvectors with eigenvalue \(\frac{R}{2}+\nabla _p\dot{u}^p\) (in the linear case: \(R=0\)).

The second field equation (22) describes the current. It is equivalent to the following three equations

$$\begin{aligned}&\nabla _j \left[ \tfrac{\dot{u}^j}{\sqrt{\eta }} {\mathbb {E}} {\mathscr {L}}_F\right] = \sqrt{\eta }\left[ {\mathbb {E}}{\mathscr {L}}_F\right] \end{aligned}$$
(28)
$$\begin{aligned}&\nabla _j [y^j {\mathbb {B}} {\mathscr {L}}_F] = - (z^j\Omega _j) [ {\mathbb {B}} {\mathscr {L}}_F] \end{aligned}$$
(29)
$$\begin{aligned}&\nabla _j [z^j {\mathbb {B}} {\mathscr {L}}_F ] = (y^j\Omega _j) [{\mathbb {B}} {\mathscr {L}}_F ] \end{aligned}$$
(30)

Proof

The expression (15) of the Faraday tensor is placed in (22):

$$\begin{aligned} 0&=\nabla _j \left[ \tfrac{{\mathbb {E}}}{\sqrt{\eta }} {\mathscr {L}}_F \left( u^j \dot{u}_k - u_k \dot{u}^j\right) \right] + \nabla _j \left[ {\mathbb {B}} {\mathscr {L}}_F (y^j z_k - y_k z^j)\right] \\&=\dot{u}_k u^j\nabla _j \left[ \tfrac{{\mathbb {E}}}{\sqrt{\eta }} {\mathscr {L}}_F \right] - u_k \dot{u}^j \nabla _j \left[ \tfrac{{\mathbb {E}}}{\sqrt{\eta }} {\mathscr {L}}_F \right] + \left[ \tfrac{{\mathbb {E}}}{\sqrt{\eta }} {\mathscr {L}}_F \right] \left( {\ddot{u}}_k - u_k \nabla _j \dot{u}^j\right) \\&\quad + z_k y^j\nabla _j \left[ {\mathbb {B}} {\mathscr {L}}_F \right] - y_k z^j \nabla _j\left[ {\mathbb {B}} {\mathscr {L}}_F \right] + \left[ {\mathbb {B}} {\mathscr {L}}_F \right] \left( z_k\nabla _j y^j + y^j\nabla _j z_k \right. \\&\quad \left. - y_k \nabla _j z^j - z^j \nabla _j y_k\right) = - u_k \dot{u}^j \nabla _j \left[ \tfrac{{\mathbb {E}}}{\sqrt{\eta }} {\mathscr {L}}_F \right] + u_k \left[ \tfrac{{\mathbb {E}}}{\sqrt{\eta }} {\mathscr {L}}_F \right] \left( \eta - \nabla _j \dot{u}^j\right) \\&\quad + z_k y^j\nabla _j \left[ {\mathbb {B}} {\mathscr {L}}_F \right] - y_k z^j \nabla _j \left[ {\mathbb {B}} {\mathscr {L}}_F \right] \\&\quad +\left[ {\mathbb {B}} {\mathscr {L}}_F \right] \left( z_k\nabla _j y^j - y^jZ_j\dot{u}_k +y^j\Omega _j y_k - y_k \nabla _j z^j + z^j Y_j \dot{u}_k +z^j\Omega _j z_k\right) \end{aligned}$$

The coefficient of \(\dot{u}_k\) is proportional to \((z^j Y_j -y^j Z_j)=0\). The vector equation gives three conditions for the coefficients of the components along \(u_k\), \(z_k\) and \(y_k\). \(\square \)

An extension with \({\mathscr {L}}(F,{}^*F)\), where the invariant scalar \({}^*F\) is built with the dual Faraday tensor, is studied by Bokulić et al. [7].

5 Anisotropic perfect fluid in static Einstein gravity

Absence of convective term. Positive energy means positive space-curvature scalar.

The Einstein equation for a static anisotropic fluid with velocity \(u_i\) is:

$$\begin{aligned} R_{jk} - \frac{1}{2} g_{jk} R = (p+\mu ) u_j u_k + p g_{jk} + \Pi _{jk} \end{aligned}$$

where \(\Pi _{jk}\) is the stress-tensor (traceless and \(\Pi _{jk} u^k=0\)), p is the effective pressure and \(\mu \) is the energy density. A convective term \((u_j q_k + u_k q_j)\) is forbidden in static space-times as it would violate Eq. (3).

By the general property \(R_{jk}u^k = (-\nabla _p \dot{u}^p) u_j\) the contraction of the Einstein equation with \(u^k\) gives \(\nabla _p\dot{u}^p + \frac{R}{2}= \mu \). Now use (14) and obtain the simple relation:

$$\begin{aligned} \mu = \frac{1}{2} R^\star \end{aligned}$$
(31)

The trace and the previous equation give the pressure:

$$\begin{aligned} 3p= \frac{1}{2} R^\star -R \end{aligned}$$
(32)

Remark 5.1

In general, in a static space-time the Einstein equations relate the positive energy constraint to the space curvature scalar:

$$\begin{aligned} T_{ij} u^i u^j = R_{jk}u^ju^k +\tfrac{1}{2}R = \nabla _p \dot{u}^p +\tfrac{1}{2} R =\tfrac{1}{2}R^\star \ge 0 \end{aligned}$$

In spherical symmetry (see “Appendix 1”): \(R^\star = 2 (\frac{1-B}{r^2} +\frac{B'}{r})\). The condition becomes

$$\begin{aligned} \frac{d}{dr} \frac{B(r)-1}{r} \ge 0 \end{aligned}$$
(33)

6 Spherical symmetry

Expressions of the Ricci, Weyl, Cotton and Bach tensors in terms of

\(u_ju_k\), \(g_{jk}\) and \(\dot{u}_j \dot{u}_k\).

Natural constraints give notorious metrics.

The magnetic part of the Faraday tensor is a monopole.

The majority of static spherical metrics discussed in the literature depend on a single scale function \(B(r)>0\):

$$\begin{aligned} ds^2 = - B(r) dt^2 + \frac{dr^2}{B(r)} + r^2 ( d\theta ^2 + \sin ^2 \theta \, d\phi ^2 ). \end{aligned}$$
(34)

In coordinates \((t,r,\theta ,\phi )\), the acceleration is the radial vector \(\dot{u}_k=(0,\frac{B'}{2B},0,0)\), where a prime is a derivative in r.

If X(r) is a scalar function, its gradient is parallel to \(\dot{u}_j\):

$$\begin{aligned} \nabla _j X = \frac{\dot{u}_j\dot{u}^k}{\eta }\nabla _k X= \dot{u}_j \frac{2B}{B'} X' \end{aligned}$$
(35)

The following scalars are obtained from expressions valid for the broader class of spherical doubly-warped space-times (see “Appendix 1” or equations 51, 40 and 53 in [43]):

$$\begin{aligned} \eta = \frac{1}{4} \frac{B'^2}{B} , \qquad \frac{\dot{u}^p\nabla _p \eta }{\eta } = B'' - \frac{B'^2}{2B}, \qquad \nabla _p\dot{u}^p = \frac{B''}{2} + \frac{B'}{r}. \end{aligned}$$
(36)

A key quantity is the following:

Proposition 6.1

$$\begin{aligned} \nabla _j \dot{u}_k = \left[ -\eta + \frac{B'}{2r}\right] u_j u_k + \frac{B'}{2r} g_{jk} + \left[ \frac{B''}{2} - \frac{B'}{2r} -\eta \right] \frac{\dot{u}_j\dot{u}_k}{\eta }. \end{aligned}$$
(37)

Proof

In spherical coordinates, with \(y^j=(0,0,r,0)\) and \(z^j=(0,0,0,r\sin \theta )\) one evaluates

$$\begin{aligned} Y_j = \frac{2B}{rB'}y_j, \qquad Z_j = \frac{2B}{rB'}z_j \end{aligned}$$

Then \(Y_j y_k + Z_j z_k = \frac{2B}{rB'} (g_{jk} + u_j u_k -\frac{ \dot{u}_j\dot{u}_k}{\eta } )\). The static expression (11) becomes

$$\begin{aligned} \nabla _j \dot{u}_k&= -\eta u_j u_k +\frac{\dot{u}_j \dot{u}_k}{2\eta ^2}\dot{u}^p\nabla _p\eta +\eta \frac{2B}{rB'} \left( g_{jk} + u_j u_k -\frac{ \dot{u}_j\dot{u}_k}{\eta } \right) \\&=u_j u_k \left[ -\eta + \frac{B'}{2r}\right] + g_{jk} \frac{B'}{2r} + \frac{\dot{u}_j\dot{u}_k}{\eta } \left[ \frac{1}{2\eta }\dot{u}^p\nabla _p \eta - \frac{B'}{2r} \right] \end{aligned}$$

With insertions of the scalars (36), the result is obtained. \(\square \)

We now obtain the covariant expressions of the Ricci, the Weyl, the Cotton and the Bach tensors. They will appear as combinations of the tensors \(u_j u_k\), \(g_{jk}\) and \(\dot{u}_j\dot{u}_k\), with coefficients that are scalar functions of r.

Proposition 6.2

(The Ricci tensor)

$$\begin{aligned}&R_{jk} = g_{jk} \frac{R(r)}{4} + \left[ u_j u_k +\frac{1}{2}g_{jk} -\frac{\dot{u}_j \dot{u}_k}{\eta } \right] A(r) \nonumber \\&A(r) = \frac{1-B}{r^2} +\frac{B''}{2} \end{aligned}$$
(38)

Proof

Insert the expression for \(Y_ky_l +Z_kz_l\) evaluated in Prop. 6.1 in Eq. (13):

$$\begin{aligned} R_{kl}&=u_k u_l \left[ \tfrac{R}{3} +2\nabla _p \dot{u}^p\right] + g_{kl}\left[ \tfrac{R}{3} +\nabla _p \dot{u}^p\right] -2\dot{u}_k \dot{u}_l -2E_{kl}\\&\quad -\tfrac{\dot{u}_k \dot{u}_l}{\eta } \tfrac{\dot{u}^p \nabla _p\eta }{\eta } -2\tfrac{B'^2}{4B} \tfrac{2B}{rB'} (g_{kl} + u_k u_l -\tfrac{ \dot{u}_k\dot{u}_l}{\eta } )\\&= u_k u_l \left[ \tfrac{R}{3} +2\nabla _p \dot{u}^p -\tfrac{B'}{r}\right] + g_{kl}\left[ \tfrac{R}{3} +\nabla _p \dot{u}^p -\tfrac{B'}{r}\right] \\&\quad -\tfrac{\dot{u}_k \dot{u}_l}{\eta } \left[ 2\eta + \tfrac{\dot{u}^l \nabla _l\eta }{\eta } -\tfrac{B'}{r} \right] -2E_{kl}\nonumber \end{aligned}$$

The spherical scalars \(\eta \), \(\dot{u}^p\nabla _p\eta \) and \(\nabla _p\dot{u}^p\) are given in Eq. (36). The electric tensor and the curvature scalar are obtained from equations (86) and (90) in “Appendix 1”:

$$\begin{aligned}&E_{kl} = E(r) \left[ \frac{\dot{u}_k \dot{u}_l}{\eta } - \frac{u_k u_l+g_{kl}}{3}\right] \end{aligned}$$
(39)
$$\begin{aligned}&E(r) = \frac{1}{2}\left[ \frac{1-B}{r^2} + \frac{B'}{r} - \frac{B''}{2}\right] \nonumber \\&R(r)= 2\frac{1-B}{r^2} - 4\frac{B'}{r} - B'' \end{aligned}$$
(40)

The Ricci tensor is then written as a sum with a trace-less term. \(\square \)

Proposition 6.3

(The Weyl tensor) With the expression of the electric tensor, the static Weyl tensor (7) becomes:

$$\begin{aligned} C_{jklm}&= E(r) \Big [ (g_{kl}+2 u_k u_l )\frac{\dot{u}_j \dot{u}_m}{\eta } -(g_{jl}+2u_ju_l )\frac{\dot{u}_k \dot{u}_m}{\eta } + (g_{jm} +2u_j u_m) \frac{\dot{u}_k \dot{u}_l}{\eta } \nonumber \\&\quad - (g_{km}+2u_k u_m) \frac{\dot{u}_j \dot{u}_l}{\eta } - u_k u_l g_{jm} +u_ju_l g_{km} - u_j u_m g_{kl} + u_k u_m g_{jl} \nonumber \\&\quad -\frac{2}{3} (g_{kl} g_{jm} - g_{jl}g_{km}) \Big ]. \end{aligned}$$
(41)

The Riemann tensor can then be obtained.

Proposition 6.4

The Cotton tensor (Cotton [17]) The Cotton tensor \(C_{jkl} = \nabla _j R_{kl} - \nabla _k R_{jl} - \frac{1}{6}(g_{kl} \nabla _j R - g_{jl} \nabla _k R) \) is proportional to \(\nabla _m C_{jkl}{}^m \). Here it is

$$\begin{aligned} C_{jkl} = \frac{B'}{2\eta } \left( A' +\frac{A}{r}\right) \left[ \dot{u}_j (u_k u_l + \frac{1}{3} g_{kl} ) - (u_j u_l +\frac{1}{3} g_{jl} )\dot{u}_k \right] . \end{aligned}$$
(42)

Proof

The evaluation is rather long. Let us specify some building steps. With (37) we obtain:

$$\begin{aligned} \nabla _j \frac{\dot{u}_k \dot{u}_l}{\eta } = - u_j (u_k \dot{u}_l +u_l \dot{u}_k) + \frac{B'}{2r\eta }\Big (h_{jk}\dot{u}_l +h_{jl}\dot{u}_k - \frac{2}{\eta } \dot{u}_j \dot{u}_k \dot{u}_l \Big ) \end{aligned}$$
(43)

Next, with the spherical static Ricci tensor (38):

$$\begin{aligned} \nabla _j (R_{kl} -\frac{R}{6}g_{kl} )&= -A(r)\frac{B'}{2r\eta }(h_{jk}\dot{u}_l +h_{jl}\dot{u}_k - \frac{2}{\eta }\dot{u}_j \dot{u}_k \dot{u}_l ) \\&\quad + \frac{B'}{2\eta } \dot{u}_j \left[ \left( u_k u_l +\frac{1}{2}g_{kl} -\frac{\dot{u}_k \dot{u}_l}{\eta } \right) A' + g_{kl} \frac{R'(r)}{12} \right] . \end{aligned}$$

The subtraction with j, k exchanged gives

$$\begin{aligned} C_{jkl} =\frac{B'}{2\eta }\left[ ( \dot{u}_j u_k u_l - \dot{u}_k u_j u_l)\left( A'+ \frac{A}{r}\right) + (\dot{u}_j g_{kl} - \dot{u}_k g_{jl})\left( \frac{A'}{2}+\frac{R'}{12} + \frac{A}{r}\right) \right] . \end{aligned}$$

With the identity \( R' = -8\frac{A}{r} -2A' \) the Cotton tensor gains the useful form (42). \(\square \)

Proposition 6.5

The Bach tensor (Bach [3]) With the Weyl tensor \(C_{jkl}{}^m\), the Bach tensor is the only algebraically independent one that is invariant for a conformal transformation \(g'_{ij}(x) = e^{2\phi (x)}g_{ij}(x)\) in \(n=4\) [56]:

$$\begin{aligned} {\mathscr {B}}_{kl} = 2\nabla ^j\nabla ^m C_{jklm} + R^{jm} C_{jklm} = -\nabla ^j C_{jkl} + R^{jm} C_{jklm} \end{aligned}$$

where \(C_{jkl}\) is the Cotton tensor. It is symmetric, traceless and divergence-free. In the metric (34) we find:

$$\begin{aligned}&{\mathscr {B}}_{kl} = B_1 u_k u_l + \frac{1}{4} (B_1-B_2) g_{kl} + B_2 \frac{\dot{u}_k \dot{u}_l}{\eta } \end{aligned}$$
(44)
$$\begin{aligned}&B_1+B_2 = - \frac{2B}{r}\left( A' +\frac{A}{r}\right) - \frac{2B}{3}\left( A' +\frac{A}{r}\right) ^\prime \end{aligned}$$
(45)
$$\begin{aligned}&B_1 - B_2= -\frac{8}{3} AE - \frac{4}{3}\left( A' +\frac{A}{r}\right) \left( B' +\frac{B}{r}\right) -\frac{4B}{3}\left( A' +\frac{A}{r}\right) ^\prime \end{aligned}$$
(46)

Proof

With Eq. (7) and using \(E_{kl}u^l=0\), it is:

$$\begin{aligned} R^{jm} C_{jklm} =A(r) \left[ - (g_{kl}+2 u_k u_l )E_{jm} \frac{\dot{u}^j \dot{u}^m}{\eta } + \frac{\dot{u}_l E_{km}\dot{u}^m}{\eta } + \frac{\dot{u}_k E_{jl} \dot{u}^j}{\eta }\right] \end{aligned}$$

Now use \(E_{jk}\dot{u}^k = \frac{2}{3}E(r) \dot{u}_l\). Then:

$$\begin{aligned} R^{jm} C_{jklm} =- \frac{4}{3}A(r) E(r) \left[ u_k u_l + \frac{1}{2} g_{kl} - \frac{\dot{u}_k \dot{u}_l}{\eta } \right] \end{aligned}$$
(47)

The divergence of the Cotton tensor is

$$\begin{aligned} \nabla ^j C_{jkl}&= \frac{B'}{2\eta }\frac{d}{dr} \left[ \frac{B'}{2\eta } \left( A' +\frac{A}{r}\right) \right] \dot{u}^j \left[ \dot{u}_j \left( u_k u_l + \frac{1}{3} g_{kl} \right) - \left( u_j u_l +\frac{1}{3} g_{jl} \right) \dot{u}_k \right] \nonumber \\&\quad + \frac{B'}{2\eta } \left( A' +\frac{A}{r}\right) \nabla ^j \left[ \dot{u}_j \left( u_k u_l + \frac{1}{3} g_{kl} \right) - \left( u_j u_l +\frac{1}{3} g_{jl} \right) \dot{u}_k \right] \nonumber \\&= \frac{1}{3}\left( A' +\frac{A}{r}\right) \left[ 2B' \left( u_k u_l +\frac{1}{2} g_{kl} -\frac{\dot{u}_k \dot{u}_l}{\eta }\right) +\frac{B}{r} \left( 5u_k u_l + g_{kl} +\frac{\dot{u}_k \dot{u}_l}{\eta }\right) \right] \nonumber \\&\quad + B\left( A' +\frac{A}{r}\right) ^\prime \left[ u_k u_l + \frac{1}{3}g_{kl} - \frac{1}{3} \frac{\dot{u}_k \dot{u}_l}{\eta }\right] . \end{aligned}$$
(48)

It turns out that \({\mathscr {B}}_{kl} \) is a traceless linear combination of \(u_k u_l\), \(g_{kl}\) and \(\dot{u}_k \dot{u}_l\). The expressions \(B_1\pm B_2\) result from the scalars \({\mathscr {B}}_{jk}u^j u^k\) and \({\mathscr {B}}_{jk}\dot{u}^j \dot{u}^k\) evaluated with (47) and (48). \(\square \)

Now we pin down some space-times that solve special geometric constraints.

Proposition 6.6

A spherically symmetric static space-time

(a) has \(\underline{\mathsf{zero~scalar~curvature}}\) \((R=0)\) if \(2\frac{1-B}{r^2} - 4\frac{B'}{r} - B'' =0\) i.e.

$$\begin{aligned} B(r) = \frac{b_{-2}}{r^2}+ \frac{b_{-1}}{r} +1 \end{aligned}$$
(49)

The Ricci tensor (38) is traceless, with \(A(r) = 2 b_{-2}/r^4 \). (b) is \(\underline{\mathsf{conformally~flat}}\) \((C_{jklm}=0)\) if \(E(r)=0\) i.e.

$$\begin{aligned} B(r) =1+b_1r + b_2 r^2 \end{aligned}$$
(50)

(c) is \(\underline{\textsf{harmonic}}\) \((\nabla ^m C_{jklm}=0)\) if

$$\begin{aligned} B(r) = \frac{b_{-1}}{r} +1 + b_1 r +b_2 r^2 \end{aligned}$$
(51)

Proof: Equation (42) gives \(A'+A/r=0\) i.e. \(A=-b_1/r\). Then: \( B'' + 2\frac{1-B}{r^2} + \frac{2b_1}{r} =0 \), with the above solution. \(\square \)

(d) is \(\underline{\mathsf{bi-harmonic}}\) \((\nabla ^j C_{jkl}=0)\) if it is harmonic (51), or if

$$\begin{aligned} B(r) = \kappa r^2 \end{aligned}$$
(52)

with arbitrary constant (note that it is not a special case of harmonic).

Proof: The expression (48) for \(\nabla ^j C_{jkl}\) is zero if the components \(g_{kl}\) and \(\dot{u}_k \dot{u}_l\) vanish (the component \(u_k u_l\) vanishes because of the trace condition). The difference gives:

$$\begin{aligned} \left( A'+\frac{A}{r} \right) \left[ -\frac{B'}{3} + \frac{2}{3} \frac{B}{r}\right] =0 \end{aligned}$$

The first factor vanishes for harmonic space-times, the other gives \(B=\kappa r^2\), that also solves the other constraint and sets \(A(r)=1/r^2\), \(E(r)=1/(2r^2)\). \(\square \)

(e) is \(\underline{\textsf{Einstein}}\) if \(A(r)=0\) i.e.

$$\begin{aligned} B(r) = \frac{b_{-1}}{r} +1 + b_2 r^2 \end{aligned}$$
(53)

(f) is \(\underline{\mathsf{Constant~Curvature}}\) if \(A(r)=0\) and \(C_{jklm}=0\).

$$\begin{aligned} B(r) = 1+ b_2 r^2 \end{aligned}$$
(54)

The Riemann tensor has the form \(R_{jklm} = \frac{R}{12}(g_{jl} g_{km} - g_{jm} g_{kl})\) with \(R= -12 b_2\). (g) has \(\underline{\mathsf{zero~Bach~tensor}}\) \(({\mathscr {B}}_{kl}=0)\) if

$$\begin{aligned} B(r) = -\frac{\beta (2-3b_1 \beta )}{r} + (1-3b_1\beta ) + b_1 r + b_2 r^2 \end{aligned}$$
(55)

Proof: the Bach tensor (44) is zero if \(B_1\pm B_2=0\). With \(B\ne 0\) the sum gives \( 3\left( A' +\frac{A}{r}\right) + r \left( A' +\frac{A}{r}\right) ^\prime =0 \) i.e \([r^2 (Ar)^\prime ]^\prime =0\), \(A(r) = \frac{a_{-2}}{r^2} + \frac{a_{-1}}{r}\). Then \( B(r) = \frac{b_{-1}}{r} + b_0 + b_1 r + b_2 r^2 \) with \(1-b_0=a_{-2}\) and \(b_1 = -a_{-1} \).

This expression in Eq. (46) gives: \(a_{-2}(1+b_0) = 3 a_{-1}b_{-1} \) and \(a_{-2}b_1 = - a_{-1}(1-b_0)\). The first one is the constraint \(1-b_0^2=-3b_1 b_{-1}\), the other equation is trivial.

A possible parameterization of B(r) is (55). \(\square \)

Proposition 6.7

(The Faraday tensor) In a static spherical-symmetric space-time the magnetic coefficient \({\mathbb {B}} (r)\) of the Faraday tensor (15) is

$$\begin{aligned} {\mathbb {B}}(r) = \frac{q_m}{r^2} \end{aligned}$$
(56)

where \(q_m\) is a magnetic charge. The current is time-like and independent of \({\mathbb {B}}\):

$$\begin{aligned} J^k = \left( -\eta \varkappa +{\mathbb {E}}\sqrt{\eta }-\frac{{\mathbb {E}}}{\sqrt{\eta }}\nabla _j \dot{u}^j\right) u^k \end{aligned}$$
(57)

Proof

The Eq. (17) for \({\mathbb {E}}/\sqrt{\eta }\) is satisfied by any function of r. The Eq. (18) for \({\mathbb {B}} \) becomes

$$\begin{aligned} \frac{1}{2} {\mathbb {B}}' = - {\mathbb {B}} \frac{1}{r}. \end{aligned}$$

with solution (56). The expression of the current (19) simplifies as directional derivatives other than \(\dot{u}^k\nabla _k\) are zero for scalars that only depend on r. \(\square \)

The geometric cases presented in Prop. 6.6 correspond to well known static spherically symmetric solutions of gravitational theories.

We consider the Einstein, the Cotton, the f(R) and the Conformal Gravity theories.

7 Static solutions in Einstein gravity

The imperfect fluid cannot be perfect. Early solutions. Properties of LE and NLE

The Einstein tensor \(G_{jk} = R_{jk}- \frac{R}{2} g_{jk} \) for the static spherical metric (34) is

$$\begin{aligned} G_{jk} = \frac{2}{3} A(r) u_j u_k + \left[ \frac{B''}{3} + \frac{B'}{r} -\frac{1-B}{3r^2}\right] g_{jk} - A(r) \left[ \frac{\dot{u}_j\dot{u}_k}{\eta } -\frac{u_ju_k+g_{jk}}{3}\right] . \end{aligned}$$
(58)

Its tensor structure and the Einstein equation \(G_{jk}=T_{jk}\) dictate that of the energy–momentum density \(T_{jk}\). In the picture of a fluid it is:

$$\begin{aligned} T_{jk} = (p+\mu ) u_j u_k + p g_{jk} + (p_r-p_\perp ) \left[ \frac{\dot{u}_j\dot{u}_k}{\eta } -\frac{u_ju_k+g_{jk}}{3}\right] \end{aligned}$$
(59)

The structure of the stress tensor is fully specified. The energy density \(\mu \), the effective pressure \(p=\frac{1}{3}(p_r+2p_\perp )\), the radial and transverse pressures \(p_r\), \(p_\perp \) are functions r:

$$\begin{aligned} \mu = -p_r = \frac{1-B}{r^2} - \frac{B'}{r}, \quad p_\perp = \frac{B''}{2} + \frac{B'}{r} \end{aligned}$$
(60)

Pressure isotropy (\(p_r=p_\perp \)) imposes \(A(r)=\frac{B''}{2} + \frac{1-B}{r^2}=0\), i.e. the space-time is Einstein with fluid equation of state \(\mu = - p\), and

$$\begin{aligned} B(r) = 1+\frac{b_{-1}}{r} + b_2 r^2 \end{aligned}$$
(61)

The field equation with a dust source, \(G_{jk} = \mu u_j u_k\) does not admit a static solution (34) (the source term must contain a pressure anisotropy to compensate the equality).

Remark 7.1

There has been a discussion whether a static spacetime with spherical metric (34) may host a perfect fluid. Faraoni et al. [21] and Visser [57] showed the inconsistency of the Kiselev metric with a perfect fluid source. A definite negative answer has been given by Lake and Bisson [6, 35]. Here, again, we have shown that it does not occur unless \(p=-\mu \).

For the static anisotropic fluid tensor (59), the equation \(\nabla _j T^j{}_k =0\) is:

$$\begin{aligned} 0&= (p+\mu ) \dot{u}_k +\nabla _k p {+} (p_r-p_\perp ) \left[ \nabla _j \frac{\dot{u}_j\dot{u}_k}{\eta } -\frac{\dot{u}_k}{3}\right] {+} \left[ \frac{\dot{u}_j\dot{u}_k}{\eta } -\frac{g_{jk}}{3}\right] \nabla _j (p_r-p_\perp )\\&=(\mu +p_\perp ) \dot{u}_k +\nabla _k p_\perp + (p_r-p_\perp )\nabla _j \frac{\dot{u}^j\dot{u}_k}{\eta } + \frac{\dot{u}_k}{\eta } \dot{u}^j \nabla _j (p_r-p_\perp ) \end{aligned}$$

A gradient is evaluated in (43): \( \nabla _j \frac{\dot{u}^j \dot{u}_k}{\eta } = \dot{u}_k + \frac{B'}{r\eta } \dot{u}_k \). In spherical symmetry: \(\nabla _k p_\perp = \frac{1}{\eta }\dot{u}_k \dot{u}^j \nabla _j p_\perp \). The derivative of the radial pressure is obtained:

$$\begin{aligned} 0=p'_r+ \frac{B'}{2B}(\mu +p_r ) + \frac{2}{r} (p_r-p_\perp ) \end{aligned}$$
(62)

7.1 Some static solutions of the Einstein equations

Simple conditions provide the classical static solutions

\(\bullet \) Schwarzschild space-time.

\(R_{jk}=0\) gives the famous vacuum spherical solution (61), with \(b_2=0\):

$$\begin{aligned} B(r) = 1 - \frac{2M}{r} \end{aligned}$$

\(\bullet \) Schwarzschild - de Sitter (SdS) space-time. The equation \(G_{jk} +\Lambda g_{jk}=0\) is solved by (61) with free parameter \(b_{-1}\). The coefficient of \(g_{jk}\) fixes \(b_2=-\frac{1}{3}\Lambda \).

\(\bullet \) Reissner-Nordstrøm space-time [19].

If \(R=0\) the Einstein equation \(R_{kl}=T_{kl}\) implies the energy–momentum tensor \(T_{jk} = \frac{b_{-2}}{r^4}\left[ u_j u_k -\frac{1}{2}g_{jk} -\frac{\dot{u}_j \dot{u}_k}{\eta }\right] \). In comoving coordinates the non-zero Faraday component

$$\begin{aligned} F_{tr} = \frac{{\mathbb {E}}}{\sqrt{\eta }}u_0 \dot{u}_r = {\mathbb {E}} \big (-\sqrt{B}\big ) \frac{1}{2} \frac{B'}{B} = -\frac{\sqrt{b_{-2}}}{r^2} \end{aligned}$$

corresponds to the radial electric field of a point charge \(q_e=\sqrt{b_{-2}}\).

The metric function is (Reissner 1916, Nordström 1913):

$$\begin{aligned} B(r) = 1 - \frac{2M}{r} + \frac{q_e^2}{r^2} \end{aligned}$$
(63)

\(\bullet \) Reissner-Nordstrøm-(anti) de Sitter space-time [36].

It is a variant of the previous metric where a cosmological term \(-\Lambda g_{jk}\) is added to the traceless \(T_{jk}^{em}\). The scale function is:

$$\begin{aligned} B(r) = 1 - \frac{2M}{r} + \frac{q_e^2}{r^2} - \frac{1}{3} \Lambda r^2 \end{aligned}$$

7.2 Linear and non-linear electrodynamics in Einstein gravity

While \({\mathbb {B}}(r)\) is fixed and equal to \(q_m/r^2\) by the Faraday conditon in spherical symmetry, \({\mathbb {E}}(r)\) is model dependent. In linear \(({\mathscr {L}}_F=1)\) or non-linear electrodynamics:

Proposition 7.2

\({\mathbb {E}}(r)\) solves the implicit equation (see [10, 24])

$$\begin{aligned} {\mathbb {E}} (r) = \frac{q_e}{r^2 {\mathscr {L}}_F (F)} \end{aligned}$$
(64)

Proof

In spherical symmetry the Eq. (28) for \({\mathbb {E}}\) is

$$\begin{aligned} \frac{B'}{2} \frac{d}{dr}\log \left[ \tfrac{{\mathbb {E}}}{\sqrt{\eta }}{\mathscr {L}}_F \right] =\frac{B'^2}{4B} - \frac{B''}{2} - \frac{B'}{r} \end{aligned}$$

Then \( \frac{d}{dr}\log [\frac{{\mathbb {E}}}{\sqrt{\eta }}{\mathscr {L}}_F] = \frac{d}{dr}\log \frac{\sqrt{B}}{B' r^2}\). The integration yields a constant \(q_e\). \(\square \)

In linear electrodynamics Eq. (64) is solved by a Coulomb field

$$\begin{aligned} {\mathbb {E}}^{lin}(r) = \frac{q_e}{r^2} \end{aligned}$$
(65)

and the electromagnetic energy–momentum density tensor is

$$\begin{aligned} T_{jk}^{lin} = 2\, \frac{q_e^2+q_m^2}{r^4} \left[ u_j u_k + \frac{1}{2} g_{jk} -\frac{\dot{u}_j \dot{u}_k}{\eta } \right] . \end{aligned}$$
(66)

This result recovers a generalization of Birkhoff’s theorem, stating that a spherical symmetric solution of the Einstein–Maxwell equations is necessarily a piece of the Reissner–Nordstrøm geometry with monopole charges (see [47], p. 844).

The expression \(T_{jk}^{nlin}\) in the Einstein equation (21) with the Einstein tensor (58) gives Eq. (26) and

$$\begin{aligned} 2 ({\mathbb {B}}^2+{\mathbb {E}}^2) {{\mathscr {L}}}_F (F) = A(r) \end{aligned}$$
(67)

The case \(R=0\) i.e. \({{\mathscr {L}}}_F=1 \) (linear electrodynamics) is the Reissner-Nördstrom solution with “dyonic charge”, i.e. \(b_{-2} = q_m^2 + q_e^2\).

Equation (67) has been exploited to infer the Lagrangian \({\mathscr {L}}\) from the metric function B(r) (through A(r)), or the opposite, in two situations: \({\mathbb {E}}=0\) or \({\mathbb {B}}=0\). The feasibility of the correspondence has been investigated by Bronnikov [11].

\(\bullet \) Purely magnetic (\({\mathbb {E}}=0\)). Then \(4F {\mathscr {L}}_F (F)=A(r)\) with \(F=q_m^2/(2r^4)\).

Ayón-Beato and Garcia [1] started with the metric of the Bardeen black-hole, and deduced the Lagrangian:

$$\begin{aligned} B(r) =1- \frac{2mr^2}{(r^2 + q_m^2)^{3/2}}\quad \rightarrow \quad {\mathscr {L}}(F) = \frac{3m}{q_m^3} \left[ \frac{\sqrt{2q^2_mF}}{1+ \sqrt{2q^2_mF}} \right] ^{5/2} \end{aligned}$$

Kruglov [34] obtained the Lagrangian of the Hayward black-hole [27]:

$$\begin{aligned} B(r) =1- \frac{2mr^2}{r^3 + q_m^3}\quad \rightarrow \quad {\mathscr {L}}(F) = \frac{3}{2^{3/4}} \frac{(2q_m^2F)^{3/2}}{(1+ (2q^2_mF)^{3/4})^2 } \end{aligned}$$

\(\bullet \) Purely electric (\(q_m=0\)). Then \(A(r) = - 4F {\mathscr {L}}_F (F)\) with \(F=-{\mathbb {E}}^2/2\).

With the aid of Eq. (64) Halilsoy et al. [24] obtained the metric from the Lagrangian:

$$\begin{aligned} {\mathscr {L}} (F) = \frac{a}{b\sqrt{2} - \sqrt{-4F}}\quad \rightarrow \quad A(r) = \frac{2bq_e}{r^2} - \sqrt{\frac{|a|q_e}{\sqrt{2}}} \frac{2q_e}{r} \end{aligned}$$

The function A(r) has the same form as the Mannheim–Kazanas solution (55) Prop. 6.6. The same metric function B(r) is also a vacuum solution of Conformal Gravity.

An interesting application has been the forecast of the shadow of the black hole in M87 [33].

8 Linear and non-linear electrodynamics in f(R) gravity

The equations and the charged solution by Hollenstein and Lobo

f(R) gravity is an extension of Einstein gravity, where a function f(R) replaces R in the Einstein-Hilbert action. The equations in spherical symmetry are studied by Capozziello et al. in [12]. With coupling to non-linear electrodynamics, the equations of motion, with \(f_R=df/dR\) are [29]:

$$\begin{aligned}&R_{jk} f_R(R) -\frac{1}{2}g_{jk}f(R) + [g_{jk} \square -\nabla _j\nabla _k] f_R(R) = T^{nlin}_{jk}\end{aligned}$$
(68)
$$\begin{aligned}&\nabla _j (F^{jk} {\mathscr {L}}_F(F)) =0 \end{aligned}$$
(69)

The second one is the same as in the Einstein theory. The equations are studied in the static metric (34). For any function g(r): \(\nabla _k g= \dot{u}_k \frac{2B}{B'} g' \) (Eq. (35)), and

$$\begin{aligned} \nabla _j \nabla _k g&= (\nabla _j \dot{u}_k) \Big (\frac{2B}{B'} g'\Big ) + \dot{u}_j \dot{u}_k \frac{2B}{B'}\frac{d}{dr}\Big (\frac{2B}{B'} g'\Big )\\&= u_j u_k \left[ \frac{B}{r} -\frac{B'}{2}\right] g' + g_{jk} \frac{B}{r}g' + \frac{\dot{u}_j \dot{u}_k}{\eta } \left[ Bg'' +\frac{1}{2}B' g' - \frac{Bg'}{r}\right] \end{aligned}$$

where \(\nabla _j \dot{u}_k\) is (37). In particular: \(\square g = 2\frac{B}{r}g' +B'g'+ Bg'' \).

Given the expressions of the Ricci tensor (38) and of \(T_{jk}^{nlin}\) (23), the first field equation corresponds to three scalar equations:

$$\begin{aligned}&-\frac{1}{2}f + f_R \left[ \frac{R}{4}+\frac{1}{2}A(r)\right] +\square f_R - \frac{B}{r} f_{R}' = 2[ {\mathbb {B}}^2 {\mathscr {L}}_F -{\mathscr {L}}] \end{aligned}$$
(70)
$$\begin{aligned}&f_R A(r) +\left[ \frac{B'}{2} -\frac{B}{r}\right] f_{R}' = 2({\mathbb {E}}^2+{\mathbb {B}}^2) {\mathscr {L}}_F \end{aligned}$$
(71)
$$\begin{aligned}&f_R A(r) + \left[ \frac{B'}{2} - \frac{B}{r}\right] f_{R}' + Bf_R'' = 2({\mathbb {E}}^2+{\mathbb {B}}^2) {\mathscr {L}}_F \end{aligned}$$
(72)

The difference of equations (71) and (72) is \(f_R'' =0\). Thus, we reobtain a simple general result by Hollenstein and Lobo [29]:

Proposition 8.1

In f(R)-nonlinear electrodynamics with static metric (34), it is \(f_R(R(r)) = cr + d \), where c and d are constants.

The case \(c=0\), \(d=1\) is Einstein’s gravity (\(f=R\)).

The result greatly simplifies Eqs. (70) and (71). With \(\frac{1}{4}R+\frac{1}{2}A= \frac{1-B}{r^2} -\frac{B'}{r}\) and \(\square f_R = 2\frac{B}{r}c +B'c\), they become:

$$\begin{aligned}&\frac{1}{2}f(R) = (cr+d) \left[ \frac{1-B}{r^2} -\frac{B'}{r}\right] + c\left[ \frac{B}{r} + B'\right] - 2[ {\mathbb {B}}^2 {\mathscr {L}}_F -{\mathscr {L}}] \end{aligned}$$
(73)
$$\begin{aligned}&(cr+d) \left[ \frac{1-B}{r^2} + \frac{B''}{2}\right] + c \left[ \frac{B'}{2}- \frac{B}{r}\right] = 2({\mathbb {E}}^2+{\mathbb {B}}^2) {\mathscr {L}}_F \end{aligned}$$
(74)

The spherical symmetry always forces \({\mathbb {B}}=q_m/r^2\). To go further, we consider linear electrodynamics \({\mathscr {L}}_F=1\), \({\mathbb {E}}=q_e/r^2\). Equation (74) can now be solved and is Eq. 34 in [29] (since it is without explanation, we offer a derivation in “Appendix 3”).

$$\begin{aligned} B(r) = 1&+\frac{cK}{2d^2} -\frac{K}{3dr}- \left[ 1+\frac{cK}{d^2} + 4 \frac{q_e^2+q_m^2}{d} (\frac{c}{d})^2\right] \left[ \frac{c}{d} r - (\frac{c}{d})^2 r^2 \log \frac{cr+d}{r} \right] \nonumber \\&+ \frac{q_e^2+q_m^2}{d}\left[ \frac{1}{r^2} - (\frac{c}{d}) \frac{4}{3r} + 2(\frac{c}{d})^2 \right] + K_0r^2. \end{aligned}$$
(75)

The solution B(r) has to produce in (73) a function f(R), and be compatible with \(f_R(R)=cr+d\).

The thermodynamics of a \(f(R)=R-2\alpha \sqrt{R} \) black hole with metric (34) are studied in [48], in power law electrodynamics.

9 Static solutions in Cotton gravity

Cotton gravity is Einstein gravity with a free Codazzi tensor.

Two new solutions: perfect fluid and LE.

Cotton gravity was introduced by Harada [25], as an extension of Einstein’s gravity. In the Harada equation, the Einstein tensor is replaced by the Cotton tensor, and the energy–momentum tensor is replaced by gradients of it:

$$\begin{aligned} C_{jkl} = \nabla _j T_{kl} - \nabla _k T_{jl} - \frac{1}{3}(g_{kl}\nabla _j T - g_{jl}\nabla _k T) \end{aligned}$$
(76)

where \(T=T^k{}_k\). As we showed in [44] the Harada equation is equivalent to the Einstein equation with an energy momentum modified by an arbitrary Codazzi tensor

$$\begin{aligned}&R_{jk} - \tfrac{1}{2}g_{jk} R = T_{jk} +{\mathscr {C}}_{jk} - g_{jk} {\mathscr {C}}^k{}_k\nonumber \\&\nabla _i {\mathscr {C}}_{jk} = \nabla _j {\mathscr {C}}_{ik} \end{aligned}$$
(77)

9.1 The Harada solution

Harada found a static spherical solution of his equation with \(C_{jkl}=0\). It is (51) in Prop. 6.6:

$$\begin{aligned} B(r) = 1 - \frac{2M}{r} - \frac{\Lambda }{3}r^2 +\gamma r \end{aligned}$$
(78)

It generalizes the Schwarzschild solution by a cosmological term, and corresponds to solving the Einstein equation with the energy momentum

$$\begin{aligned} T_{jk} =-\Lambda g_{jk}+ \frac{\gamma }{r} \left[ - \frac{2}{3} u_j u_k + \frac{4}{3} g_{jk} + \left( \frac{\dot{u}_j\dot{u}_k}{\eta } -\frac{u_ju_k+g_{jk}}{3}\right) \right] \end{aligned}$$

Therefore, the Harada vacuum solution is a solution of the Einstein equation for an exotic anisotropic fluid, with velocity \(u_k\), energy density \(\mu =-p_r= -\frac{2\gamma }{r} +\Lambda \) and transverse pressure \(p_\perp = \frac{\gamma }{r} - \Lambda \).

The same function B(r) appears as solution of a model for gravity at large distances studied by Grumiller [23], with an analogous energy–momentum tensor.

Harada numerically solved the equations for Cotton-gravity to describe the rotation curves of galaxies [26], where a linear term \(\gamma r\) provides the observed gravitational potential without the need of dark matter.

9.2 Perfect fluid solution

While in Einstein gravity there are no perfect fluid solutions with the static spherical metric (34), this is no longer true in Cotton gravity because of the freedom of choosing the Codazzi tensor.

The following one, with constants K and \(\kappa \),

$$\begin{aligned} {\mathscr {C}}_{jk} = \tfrac{K}{\sqrt{B(r)}} u_j u_k +\kappa g_{jk} \end{aligned}$$
(79)

is a Codazzi tensor in the metric (34) (see [44]). By choosing \(B(r)= \frac{b_{-1}}{r} +1 + b_2 r^2 \) (as (53), i.e. \(A(r)=0\)), the Ricci tensor is Einstein, \(R_{jk}= - 3 b_2 g_{jk}\). The energy–momentum tensor

$$\begin{aligned} T_{jk}&= R_{jk} - \tfrac{1}{2}R g_{jk} - {\mathscr {C}}_{jk} + g_{jk} {\mathscr {C}}^k{}_k \\&=-\tfrac{K}{\sqrt{B(r)}} (u_j u_k +g_{jk}) - \left[ \tfrac{R}{4}+ 4\kappa \right] g_{jk} \end{aligned}$$

is perfect fluid, and the Harada equation (76) is solved by the metric (34) with the function B(r). The perfect fluid has constant energy density \(\mu =4\kappa +R/4\), while \(p+\mu =-\frac{K}{\sqrt{B}(r)}\) is a function of r because of the Codazzi term.

  • If \({\mathscr {C}}_{jk}=0\) we recover GR with the cosmological law \(p=-\mu \).

  • If \(K =0\), \(\kappa = -\frac{R}{16}\) we get the empty solution \(p=\mu =0\). Thus in Cotton gravity the same metric (SdS or SadS) is compatible with different energy–momentum tensors.

  • The electric function is \(E(r)=-\frac{3}{2}\frac{b_{-1}}{r^3}\). If \(b_{-1}=0\) then \(C_{jklm}=0\). \(B(r)= 1+b_2 r^2\) gives the metric of a constant curvature space-time \(R_{jklm}=\frac{1}{12}R(g_{jl}g_{km}- g_{jm}g_{kl})\) in presence of a perfect fluid in Cotton gravity.

Remark 9.1

Apparently, the statement that (79) is a Codazzi tensor in a constant curvature space-time comes at odds with the theorem by Ferus [22] stating that the only Codazzi tensors is such spacetimes are \(\nabla _j\nabla _k \varphi + \frac{1}{12}R \varphi g_{jk}\), where \(\varphi \) is an arbitrary scalar field. Actually, it can be shown that the tensor is in this class with \(\varphi (r) = K \sqrt{B(r)}\). The term \(\kappa g_{jk}\) is the trivial Codazzi tensor.

9.3 Linear electrodynamics

We obtain a new solution for Cotton gravity in presence of the linear tensor of electrodynamics.

Proposition 9.2

The metric function B(r) solving the Cotton gravity equation in linear electrodynamics is:

$$\begin{aligned} B(r) = \left[ 1- \frac{2M}{r} -\frac{\Lambda }{3}r^2 +\gamma r \right] + \frac{q_e^2 + q_m^2}{r^2} \end{aligned}$$
(80)

It is the sum of the solution of \(C_{jkl}=0\) (in square brackets) and a dyonic charge term.

Proof

The traceless energy–momentum tensor \(T_{jk}^{lin}\) in Eq. (24), is entered in the Cotton gravity equation: \(C_{jkl}= \nabla _j T_{kl}^{lin} - \nabla _k T_{jl}^{lin}\):

$$\begin{aligned} C_{jkl} = \left[ K'+\frac{K}{r}\right] (u_k \dot{u}_j - u_j \dot{u}_k) u_l + \left[ \frac{K'}{2} + \frac{K}{r}\right] (\dot{u}_j g_{kl}- \dot{u}_k g_{jl}) \end{aligned}$$
(81)

where for brevity \(K=2[\frac{q_m^2}{r^4} +{\mathbb {E}}^2(r)] \). The static spherical Cotton tensor is (42).

The contraction with \(g^{kl}\) gives \(0=\frac{K'}{2}+\frac{2K}{r}\), with solution

$$\begin{aligned} {\mathbb {E}}(r)=\frac{q_e}{r^2} \end{aligned}$$

The contraction with \(u^k u^l\) is \( A' + \frac{A}{r} = \frac{3}{4} K'\) with solution \(A(r) = -\frac{\gamma }{r} + 2\frac{q_e^2+q_m^2}{r^4}\), with a constant \(\gamma \). The corresponding metric function is obtained. \(\square \)

10 Static solutions in conformal gravity

The field equations, the Mannheim-Kazanas and LE solutions.

The action of conformal gravity is \(S = -\alpha _G \int d^4x \sqrt{(-g)} C_{jklm} C^{jklm} + S_{matter}\) In \(n=4\) the Weyl term, that accounts for geometry, is invariant for the conformal transformationFootnote 1\(g'_{jk}(x) = e^{2\phi (x)} g_{jk}(x)\).

The variation in the metric tensor, neglecting boundary terms, is:

$$\begin{aligned} \delta S= 2\alpha _G \int d^4x \sqrt{(-g)} {\mathscr {B}}_{kl}\, \delta g^{kl} - \frac{1}{2}\int d^4x \sqrt{(-g)} \, T_{kl}\, \delta g^{kl} \end{aligned}$$

where \({\mathscr {B}}_{kl} = 2\nabla ^j\nabla ^m C_{jklm} + R^{jm} C_{jklm} = -\nabla ^j C_{jkl} + R^{jm} C_{jklm} \) is the Bach tensor and \(T_{kl}\) is the energy–momentum density tensor. The field equation of Conformal gravity is:

$$\begin{aligned} 4\alpha _G \, {\mathscr {B}}_{kl} = T_{kl} \end{aligned}$$
(82)

The property \(\nabla _j T^j{}_k=0\) is mantained by the identity \(\nabla _j {\mathscr {B}}^j{}_k=0\).

Equation (44) for the static spherical Bach tensor fixes the form of the energy–momentum tensor as an anisotropic fluid (59):

$$\begin{aligned}{} & {} 4\alpha _G \left[ B_1 u_j u_k +\frac{B_1-B_2}{4} g_{jk} +B_2 \frac{\dot{u}_j\dot{u}_k}{\eta } \right] \\{} & {} \quad =(\mu +p_\perp ) u_j u_k + p_\perp g_{jk} + (p_r-p_\perp ) \frac{\dot{u}_j\dot{u}_k}{\eta } \end{aligned}$$

with \(\mu = \alpha _G (3B_1 + B_2)\), \(p_r = \alpha _G(B_1+3B_2)\) and \(p_\perp = \alpha _G (B_1-B_2)\).

Since the Bach tensor is traceless, it is \(T^k{}_k=0\), i.e. in static conformal gravity the fluid always satisfies

$$\begin{aligned} p_r+ 2p_\perp =\mu \end{aligned}$$
(83)

The continuity equation for the energy momentum is Eq. (62).

Let us view some special cases:

10.1 Vacuum solution

Mannheim and Kazanas [39] obtained the vacuum spherical static solution for conformal gravity, \({\mathscr {B}}_{jk}=0\). It is the metric function B(r) in Eq. (55) Prop. 6.6. The solution arose much interest for the description of the rotation curves of galaxies, where the linear term \(\gamma r\) accounts for the plateau without need of dark matter [28, 30, 39, 41]. Constraints on the value of the constant \(\gamma \) were obtained by Sultana et al. [55], using data for perihelion shift.

Bach showed that every static spherically symmetric space-time that is conformally related to the Schwarzschild-de Sitter (SdS) metric solves \({\mathscr {B}}_{jk}=0\) [40]. The converse was later proved by Buchdahl (see [28]). In some papers it is actually proven that (55) is conformally equivalent to the SdS metric.

10.2 Perfect fluid

The anisotropic term is zero if \(B_2=0\), and \(p_r=p_\perp =\mu /3\).

The condition on \(B_2\) is a fourth order non-linear differential equation for B(r).

10.3 The anisotropic EoS \(\varvec{\mu = -p_r}\)

This occurs for \(B_1+B_2=0\). The trace-less energy momentum tensor takes the same form of \(T_{jk}^{lin}\) of linear electrodynamics, Eq. (24):

$$\begin{aligned} T_{kl} = \mu \left[ u_k u_l + \frac{1}{2} g_{kl} - \frac{\dot{u}_k \dot{u}_l}{\eta } \right] \end{aligned}$$

For this reason, the case is by far the most studied in the literature. Remarkably, Riegert [51] proved that Birkhoff’s theorem holds in conformal gravity and implies that a spherical symmetric solution of the Bach-Maxwell equations is necessarily static, with B(r) given below.

Equation (45) gives: \(0= \frac{1}{r}\left( A' +\frac{A}{r}\right) + \frac{1}{3}\left( A' +\frac{A}{r}\right) ^\prime \), with solution \(A(r) =\frac{1}{r^2}(1-b_0) - \frac{1}{r} b_1\). The equation has solution

$$\begin{aligned} B(r) = \frac{b_{-1}}{r} + b_0 + b_1 r + b_2 r^2 \end{aligned}$$

It follows that \(E(r)=\frac{1}{2r^2}(1-b_0^2)-\frac{3}{2} \frac{b_{-1}}{r^3}\). This, in Eq. (46) gives \(2B_1 = -\frac{4}{3r^4}[(1-b_0^2) +3b_1b_{-1}]\). The energy density is \(\mu =p_\perp =-p_r= 2\alpha _G B_1\):

$$\begin{aligned} \mu (r) = -\alpha _G \frac{8}{3r^4}[(1-b_0^2) +3b_1b_{-1}] \end{aligned}$$

The dependence \(r^{-4}\) agrees with the monopole field in linear electrodynamics. In this picture:

$$\begin{aligned} 2(q_e^2 + q_m^2) = -\alpha _G \frac{8}{3}[(1-b_0^2) +3b_1b_{-1}] \end{aligned}$$

The parameter \(b_2\) is free while \(b_{\pm 1}\) and \(b_0\) are constrained. The metric function B can be cast as follows [40]

$$\begin{aligned} B(r) = -\frac{1}{r}\left[ \beta (2-3\beta \gamma ) +\frac{q_e^2+q_m^2}{4\gamma \alpha _G }\right] + (1-3\beta \gamma ) + \gamma r -\kappa r^2 \end{aligned}$$

Let’s look at two subcases:

10.3.1 The harmonic solution (\(\nabla _mC_{jkl}{}^m=0\))

The function B(r) by Harada Eq. (78) solves \(C_{jkl}=0\). It implies \(A(r) =-\gamma /r\) and \(E(r) = 3\,M/r^3\), i.e. \(b_0=1\), \(b_1=\gamma \) and \(b_{-1}=-2\,M\). Therefore: the Harada metric (78) solves the field equation of conformal gravity in presence of electric and magnetic monopoles, with charge \(q_e^2+q_m^2 = 8 M \gamma \; \alpha _G\).

10.3.2 The bi-harmonic solution (\(\nabla _j\nabla _mC^j{}_{kl}{}^m=0\))

Besides the harmonic solution, the function \(B(r)=\kappa r^2\), see (52), cancels the term \(\nabla ^j C_{jkl}\). With \(b_0=0\), \(b_1=b_{-1}=0\), \(B(r)=\kappa r^2\) solves the field equation of conformal gravity coupled to monopole charges \(q_e^2+q_m^2 =-\frac{4}{3} \alpha _G\), independent of \(\kappa \), with \(\alpha _G<0\).

Some equations of state \(p_r=p_r(\mu )\) have been numerically studied by Brihaye and Verbin [8].

11 Conclusions

The initial effort of writing tensors with the vectors \(u_j\), \(\dot{u}_j\) that define static space-times, and two other orthogonal vectors, is rewarded by the simplicity of the study of the field equations in gravitation theories. In spherical symmetry the first two vectors suffice, the others being projected away with entrance of the metric tensor. In the field equations, a geometric tensor equals a matter tensor; the tensor form of the first determines that of the latter, and the equality of the coefficients are scalar field equations.

With this plan we obtain a list of solutions in Einstein, Cotton, f(R) and conformal gravity, with results on the Faraday tensor and (non)linear—electrodynamics, and new solutions in Cotton gravity.

New and old results are here obtained in the natural and simple covariant formalism.

This strategy may be applied to other extended theories, as Gauss–Bonnet gravity.