1 Introduction

In Ref. [1] Junpei Harada proposed an extension named “Cotton gravity” of the Einstein equations, where the geometric term (the Einstein tensor) is replaced by the Cotton tensor, and the source (the energy-momentum tensor) is replaced by gradients of the energy-momentum. In a space-time of dimension n:

$$\begin{aligned} \textsf{C}_{jkl} = \nabla _j T_{kl}-\nabla _k T_{jl}- \frac{g_{kl}\nabla _j T -g_{jl}\nabla _k T}{n-1} \end{aligned}$$
(1)

T is the trace \(T^k{}_k\) and Newton’s constant is absorbed in \(T_{jk}\). The Cotton tensor is defined as

$$\begin{aligned} \textsf{C}_{jkl} = \nabla _j R_{kl}-\nabla _k R_{jl}- \frac{g_{kl}\nabla _j R -g_{jl}\nabla _k R}{2(n-1)} \end{aligned}$$
(2)

It is related to the Weyl tensor, \(\textsf{C}_{jkl} = -\frac{n-2}{n-3}\nabla _m C_{jkl}{}^m\), and contains third-order derivatives of the metric tensor.

Harada showed that his gravity Eq. (1) descend from a variational principle with action, in \(n=4\), \( S= \int d^4 x \sqrt{-g} (C_{jklm} C^{jklm} - R_{jklm} T^{jklm}) \). The variation is taken in the connection. \(T^{jklm}\) is a curvature tensor built with the energy-momentum and the metric tensors (Eq. 14 in [1]). Conformal gravity has the same action, but the variation in the metric tensor gives the equations \(-4(2\nabla ^j\nabla ^m C_{jklm} +R^{jm} C_{jklm})= T_{kl}\), that are fourth-order in the derivatives of the metric tensor.

While solving (1) for a vacuum (\(T_{kl}=0\)) static spherically symmetric space-time, Harada obtained a generalization of the Schwarzschild solution:

$$\begin{aligned} ds^2 = -b^2(r) dt^2 + \frac{1}{b^{2}(r)} dr^2 + r^2d\Omega ^2_2 \end{aligned}$$
(3)

with \(b^2(r)= 1-2M/r+ \gamma r -\frac{1}{3}\Lambda r^2\). He remarked the similarity with a vacuum solution of conformal gravity, where \(b^2(r)\) is replaced by \({\hat{b}}^2(r) = [ 1-3\beta \gamma \) \(- \beta (2-3\beta \gamma )/r +\gamma r -K r^2 ] \) (see [2, 3]).

In ref. [4], Harada applied his theory to describe the rotation curves of several galaxies, where the effect of the possible dark-matter halo is supplanted by the modified gravitational potential.

After its appearance, Harada’s paper was criticized by objecting that it adds nothing to standard General Relativity [5, 6]. This paper shows that it is not the case: there is a rich structure in the theory, that includes the standard one.

The difference of Eqs. (1) and (2) shows that

$$\begin{aligned} {\mathscr {C}}_{kl} =R_{kl}-T_{kl}-g_{kl} \frac{R-2T}{2(n-1)} \end{aligned}$$
(4)

is a Codazzi tensor:

$$\begin{aligned} \nabla _j {\mathscr {C}}_{ik} = \nabla _i {\mathscr {C}}_{jk} \end{aligned}$$
(5)

Equations (4) and (5) are equivalent to the Harada equation (1) for Cotton gravity. In fact, with \(R_{kl}={\mathscr {C}}_{kl}+T_{kl}+g_{kl}\frac{R-2T}{2(n-1)}\) the Cotton tensor (2) is constructed, and the Codazzi condition ensures that (1) is obtained.

The third order character of (1) is reduced to second order in (4), with the appearance of a supplemental term, the Codazzi tensor. The latter may be thought of as a modification of the Ricci tensor, or the energy-momentum tensor, or both.

The case \({\mathscr {C}}_{kl}=0\) in (4) restores the Einstein equations, and Eq. (1) is identically true. The “trivial” case \({\mathscr {C}}_{jk}=B g_{jk}\), adds a cosmological constant.

If \({\mathscr {C}}_{kl}\ne 0\), Eq. (4) can still be interpreted as the Einstein equation, with a modified energy-momentum tensor:

$$\begin{aligned} R_{kl} - \tfrac{1}{2} R g_{kl} = T_{kl} + {\mathscr {C}}_{kl} - g_{kl} {\mathscr {C}}^r{}_r \end{aligned}$$
(6)

Let us mention that Codazzi tensors appear in the geometry of hypersurfaces [7]. A Lorentzian hypersurface in a Minkowski space-time has Riemann tensor \(R_{jklm} = \Omega _{jl}\Omega _{km} -\Omega _{jm}\Omega _{kl}\) where \(\Omega _{jk}\) is a Codazzi tensor. The trivial case \(\Omega _{jk} = \frac{R}{n(n-1)} g_{jk}\) corresponds to a constant-curvature hypersurface, and the tensor has a single, constant eigenvalue.

A non-trivial Codazzi tensor poses important limitations on the geometry of the hosting space-time.

Among the possible tensors, we choose to investigate two simple and physically relevant ones, that often appear in the expressions of the Ricci or of the energy-momentum tensors. They involve the basic kinematic quantities \(u_i\) and \(\dot{u}_i\).

We begin with the “perfect fluid” tensor \({\mathscr {C}}_{jk}=Au_j u_k +Bg_{jk}\) with the Codazzi property. Andrzej Derdziński [8] proved that if \({\mathscr {C}}^k{}_k\) is a constant, then the space-time is warped (GRW, generalized Robertson-Walker space-time), i.e. there are coordinates such that

$$\begin{aligned} ds^2 = -dt^2 + a^2(t) g^\star _{\mu \nu } (\textbf{x}) dx^\mu dx^\nu \end{aligned}$$
(7)

with Riemannian metric \(g^\star _{\mu \nu }\). The hypothesis was weakened by Gabe Merton [9], who showed that a necessary and sufficient condition for the GRW space-time is \(v^j\nabla _j {{\mathscr {C}}}^k{}_k=0\) for all vectors \(v^ju_j=0\) (the result was proven in Riemannian signature, but it also holds in Lorentzian).

In Theorem 2.1 we prove that a perfect fluid tensor is Codazzi if and only if the space-time is “doubly twisted”, i.e. there are coordinates such that

$$\begin{aligned} ds^2 = - b^2(t,\textbf{x}) dt^2 + a^2(t,\textbf{x}) g_{\mu \nu }^\star (\textbf{x}) dx^\mu dx^\nu \end{aligned}$$
(8)

with the special condition that \((\partial _t\log a)/b\) only depends on time t. Remarkably, this metric with the constraint happens to be a generalization of the well known Stephani Universes.

We discuss special cases, including Merton’s result, and obtain the general form of the Ricci tensor.

Next we study the “current flow” tensor \({\mathscr {C}}_{jk}=\lambda (u_j \dot{u}_k +\dot{u}_j u_k)\) with the Codazzi condition and closed vector field \(\dot{u}_j\). The field \(u_j\) turns out to be vorticity-free but not shear-free. This makes the metric more general than doubly-twisted, Eq. (28). If it is constrained to be static, a useful form of the Ricci tensor is obtained. We list some of the several examples that can be found in the literature.

Finally, we consider Yang Pure space-times. They are characterised by a Ricci tensor that is a Codazzi tensor. Among the examples, we show that a Friedmann-Robertson-Walker metric is Yang Pure if and only if \(\nabla _j R=0\).

This concludes Sect. 2 of the paper.

In Sect. 3 we show that these results are interesting for the Cotton gravity by Harada. If nontrivial, the Codazzi tensor introduces geometric or unconventional matter content in the Einstein equation, depending on the point of the view, in a way different than other extended theories of gravity.

This suggests a solution to the Harada equations which goes as follows: given the form of a Codazzi tensor, this determines a class of space-times that host the tensor. The space-time in turn determines the Ricci tensor. Finally, the Codazzi and the Ricci tensors in Eq. (4) determine the energy-momentum tensor of the Harada equation.

The two Codazzi tensors that are here studied, modify the energy-momentum in its perfect-fluid component or in the current component.

We end with a discussion of constant curvature space-times, for which Ferus [10] identified the general form of Codazzi tensors.

We employ the Lorentzian signature \((-+...+)\), latin letters for space-time components and greek letters for space components. A dot on a quantity X is the operator \(\dot{X} =u^k\nabla _k X\). The symbols \(\eta \), \(\epsilon \) are the scalar functions \(\eta = \dot{u}^k \dot{u}_k\) and \(\epsilon = \dot{u}^k\nabla _k\eta \).

2 Codazzi tensors and their space-times

In refs. [11, 12] we showed that a Codazzi tensor always satisfies an algebraic identity with the Riemann tensor (it is “Riemann compatible”):

$$\begin{aligned} {\mathscr {C}}_{im} R_{jkl}{}^m +{\mathscr {C}}_{jm} R_{kil}{}^m +{\mathscr {C}}_{km} R_{ijl}{}^m =0. \end{aligned}$$
(9)

This property implies that a Codazzi tensor is also Weyl compatible, with the Weyl tensor \(C_{jklm}\) replacing \(R_{jklm}\). The contraction with the metric tensor \(g^{il}\) gives \({\mathscr {C}}_{jm}R_k{}^m = {\mathscr {C}}_{km}R_j{}^m\), i.e. a Codazzi tensor commutes with the Ricci tensor.

As anticipated, we investigate two forms of Codazzi tensor. We name them in analogy with terms in an energy-momentum tensor: \({\mathscr {C}}_{jk}= Au_j u_k + Bg_{jk}\) (perfect fluid) and \({\mathscr {C}}_{jk}= \lambda (u_j \dot{u}_k + \dot{u}_j u_k)\) (current flow). \(A\ne 0\), B, \(\lambda \) are scalar fields. The vector field \(u_j\) is time-like unit, \(u^j u_j =-1\), and is named velocity. The vector field \(\dot{u}_j =u^k\nabla _k u_j\) is spacelike, orthogonal to the velocity, and is named acceleration.

We show that the Codazzi property of such tensors strongly restricts the space-times they live in.

2.1 Perfect fluid Codazzi tensors and Stephani universes.

Theorem 2.1

The perfect fluid tensor \({\mathscr {C}}_{jk}= Au_j u_k + Bg_{jk}\) with \(u^ju_j=-1\) is Codazzi if and only if

$$\begin{aligned} \nabla _i u_j = \varphi (g_{ij}+u_i u_j) - u_i \dot{u}_j \end{aligned}$$
(10)

where \((n-1)\varphi \) is the usual expansion parameter,

$$\begin{aligned} \nabla _i A&= -u_i \dot{A} - \dot{u}_i A \end{aligned}$$
(11)
$$\begin{aligned} \nabla _i B&= -u_i \dot{B} \end{aligned}$$
(12)
$$\begin{aligned} \varphi&= -\dot{B}/A \end{aligned}$$
(13)

This relation is useful:

$$\begin{aligned} \nabla _i\varphi = - u_i {\dot{\varphi }} \end{aligned}$$
(14)

Proof

See Appendix 1. \(\square \)

Proposition 2.2

If \({\mathscr {C}}_{jk}\) is a perfect-fluid Codazzi tensor, the velocity \(u_i\) is Riemann compatible, \( u_i R_{jklm}u^m + u_j R_{kilm}u^m + u_k R_{ijlm}u^m =0 \), and it is an eigenvector of the Ricci tensor, \(R_{jk}u^k =\gamma u_j\), with eigenvalue

$$\begin{aligned} \gamma = (n-1) ({\dot{\varphi }} +\varphi ^2) - \nabla _k \dot{u}^k \end{aligned}$$
(15)

The following identity for the acceleration holds:

$$\begin{aligned} (\varphi \dot{u}_k +\ddot{u}_k) u_l - u_k(\varphi \dot{u}_l + \ddot{u}_l) = \nabla _k\dot{u}_l - \nabla _l\dot{u}_k. \end{aligned}$$
(16)

Proof

The first statement is an obvious consequence of (9) and of the first Bianchi identity. For the eigenvalue we evaluate:

$$\begin{aligned} R_{jklm}u^m =&\nabla _j \nabla _k u_l - \nabla _k \nabla _j u_l \\ =&\nabla _j [\varphi (g_{kl}+u_k u_l)-u_k\dot{u}_l] - \nabla _k [\varphi (g_{jl}+u_j u_l)-u_j\dot{u}_l] \nonumber \\ =&- (g_{kl} u_j -g_{jl} u_k){\dot{\varphi }} +(\nabla _j u_k-\nabla _k u_j) (\varphi u_l -\dot{u}_l) \\&+ u_k ( \varphi \nabla _j u_l -\nabla _j \dot{u}_l) - u_j ( \varphi \nabla _k u_l -\nabla _k \dot{u}_l) \nonumber \\ =&- (g_{kl} u_j -g_{jl} u_k)({\dot{\varphi }} +\varphi ^2)- (u_j \dot{u}_k- u_k \dot{u}_j) (\varphi u_l -\dot{u}_l) - u_k \nabla _j \dot{u}_l + u_j \nabla _k \dot{u}_l \end{aligned}$$

The contraction with \(g^{jl}\) gives: \(R_{km}u^m = (n-1)({\dot{\varphi }} +\varphi ^2) u_k + \varphi \dot{u}_k - u_k \eta - u_k \nabla _j \dot{u}^j + u^j \nabla _k \dot{u}_j \). Since \(\dot{u}^j u_j=0\), the last term is: \(-\dot{u}^j \nabla _k u_j = -\varphi \dot{u}_k +u_k \eta \) by Eq. (10), and cancels three terms. The eigenvalue \(\gamma \) is read.

The contraction with \(u^j\) gives the symmetric tensor

$$\begin{aligned} u^jR_{jklm}u^m = (g_{kl} +u_l u_k)({\dot{\varphi }} +\varphi ^2) + \dot{u}_k (\varphi u_l -\dot{u}_l) - u_k \ddot{u}_l - \nabla _k \dot{u}_l \end{aligned}$$
(17)

Subtraction with indices k, l exchanged gives the identity for the acceleration. \(\square \)

With the aid of the Weyl tensor, we obtain the expression of the Ricci tensor on a space-time with a perfect fluid Codazzi tensor.

Proposition 2.3

(The Ricci tensor)

$$\begin{aligned} R_{kl} =&\frac{R-n\gamma }{n-1} u_k u_l + \frac{R-\gamma }{n-1}g_{kl} +\Pi _{kl} \nonumber \\ \Pi _{kl}=&\tfrac{1}{2} (n-2) [u_k(\varphi \dot{u}_l - \ddot{u}_l) + u_l (\varphi \dot{u}_k - \ddot{u}_k) - (\nabla _k \dot{u}_l + \nabla _l \dot{u}_k)] \nonumber \\&- (n-2) [\dot{u}_k \dot{u}_l +E_{kl} ] + \frac{n-2}{n-1} (g_{kl}+u_k u_l) \nabla _p \dot{u}^p \end{aligned}$$
(18)

where \(\gamma \) is the eigenvalue (15), \(\Pi _{kl}\) is symmetric traceless with \(\Pi _{kl} u^l=0\), and \(E_{kl}=u^j u^m C_{jklm}\) is the electric tensor. It is symmetric, traceless, with \(E_{jk}u^k=0\).

Proof

The general expression of the Weyl tensor is:

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

The contraction with \(u^j u^m\) and (17) give:

$$\begin{aligned} E_{kl} =&(g_{kl} +u_l u_k)({\dot{\varphi }} +\varphi ^2) + \dot{u}_k (\varphi u_l -\dot{u}_l) - u_k \ddot{u}_l - \nabla _k \dot{u}_l \\&- \frac{ R_{kl} +2 \gamma u_k u_j +\gamma g_{kl} }{n-2} + R \frac{g_{kl}+u_ku_l}{(n-1)(n-2)}. \end{aligned}$$

The Ricci tensor is obtained:

$$\begin{aligned} R_{kl} =&\left[ \frac{R-n\gamma +(n-2)\nabla _p\dot{u}^p}{n-1} \right] u_k u_l + \left[ \frac{R-\gamma +(n-2)\nabla _p\dot{u}^p}{n-1} \right] g_{kl}\\&- (n-2)[ \dot{u}_k \dot{u}_l -\varphi \dot{u}_k u_l + u_k\ddot{u}_l +\nabla _k \dot{u}_l +E_{kl}]. \end{aligned}$$

The expression is symmetrized with the identity (16) and the correction to the perfect fluid part is made traceless by subtraction. \(\square \)

We discuss the geometric restrictions posed by a perfect-fluid Codazzi tensor. The presence of a shear-free and vorticity-free velocity field, Eq. (10), classifies the space-time as doubly-twisted [13], i.e. there is a coordinate frame such that the metric has the form (8).

In this frame, with the Christoffel symbols

$$\begin{aligned} \Gamma _{00}^0 =\frac{\partial _t b}{b}, \quad \Gamma _{\mu 0}^0 =\frac{\partial _\mu b}{b}, \quad \Gamma _{\mu \nu }^0 = \frac{\partial _t a}{ab^2} g^\star _{\mu \nu }, \quad \Gamma _{0\mu }^\nu = \frac{\partial _t a}{a}\delta _\mu ^\nu \end{aligned}$$

Eq. (10) for \(u_j\) and \(\dot{u}_j=u^k\nabla _k u_j\) give: \(u_0=-b(t,\textbf{x})\), \(u_\mu =0\), and

$$\begin{aligned} \dot{u}_0=0, \; \dot{u}_\mu = \frac{\partial _\mu b(t,\textbf{x})}{b(t,\textbf{x})};\quad \varphi = \frac{1}{b(t,\textbf{x})} \frac{\partial _t a(t,\textbf{x})}{a(t,\textbf{x})} . \end{aligned}$$
(19)

By Eq. (14), the doubly twisted metric has the constraint that \(\varphi \) only depends on time. With \(a=1/V(\textbf{x},t)\), the metric (8) with the constraint becomes:

$$\begin{aligned} ds^2 = -\left[ \frac{1}{\varphi (t)} \frac{\partial _t V}{V}\right] ^2 dt^2 + \frac{g_{\mu \nu }^\star (\textbf{x})dx^\mu dx^\nu }{V^2(\textbf{x},t)} \end{aligned}$$
(20)

This metric generalizes the well known Stephani metrics, presented in the following example.

Example 2.4

Remarkably, Eqs. (10)–(13) in \(n=4\) coincide with eqs. 37.32–37.34 in the book by Stephani et al. [15]. They were derived for a Riemann tensor of the form \(R_{jklm}={\mathscr {C}}_{jl}{\mathscr {C}}_{km}-{\mathscr {C}}_{jm}{\mathscr {C}}_{kl}\), with \({\mathscr {C}}_{jk}=Au_ku_l+ Bg_{jk}\) (note that if \({\mathscr {C}}_{jk}\) is invertible then the Bianchi identity implies that it is a Codazzi tensor [14]). Such space-times are conformally flat and are named Stephani universes [15, 16]. They are solutions of the Einstein equations with a perfect fluid source \(T_{jk}\).

The Stephani metric is

$$\begin{aligned} ds^2 = -\left[ \frac{1}{\varphi (t)} \frac{\partial _t V}{V}\right] ^2 dt^2 + \frac{dx^2+dy^2+dz^2}{V^2(\textbf{x},t)} \end{aligned}$$

with \(V (\textbf{x},t)= V_0(t) +\frac{B^2(t) -\varphi ^2(t)}{4V_0(t)} \Vert \textbf{x} - \textbf{x}_0(t)\Vert ^2\), where \(V_0\), \(\varphi \) and \(\textbf{x}_0\) are arbitrary functions of time.

We now consider some special conditions of the perfect fluid Codazzi tensor.

Lemma 2.5

If the acceleration is closed, \(\nabla _j \dot{u}_k =\nabla _k \dot{u}_j\), then \(b(t,\textbf{x}) = {\hat{b}}(t) b(\textbf{x})\), and \(\ddot{u}_k = \eta u_k - \varphi \dot{u}_k \).

Proof

The condition that matters is \(\nabla _0 \dot{u}_\mu = \nabla _\mu \dot{u}_0\) i.e. \(\partial _t \dot{u}_\mu - \Gamma _{0\mu }^\nu \dot{u}_\nu = -\Gamma _{\mu 0}^\nu \dot{u}_\nu \). By the symmetry of the Christoffel symbols, we remain with \(0=\partial _t \dot{u}_\mu \) i.e. \(\dot{u}_\mu =\partial _\mu \log b\) is independent of t. Then \(b(t,\textbf{x}) = b_1(t)b_2(\textbf{x})\).

Equation(16) now is: \( (\varphi \dot{u}_k +\ddot{u}_k) u_l - u_k(\varphi \dot{u}_l + \ddot{u}_l) =0\). Contraction with \(u^l\) is: \(\ddot{u}_k = - \varphi \dot{u}_k - u_k (u^l\ddot{u}_l)\). The identity \(u^l\dot{u}_l=0\) gives \(u^l\ddot{u}_l = - \dot{u}^l \dot{u}_l \equiv -\eta \). \(\square \)

\(\bullet \) If \(\nabla _k A=-u_k \dot{A} \) i.e. \(\dot{u}_k=0\), then \(b(t,\textbf{x})\) is only a function of time. It is \(b=1\) after a rescaling of time. The equations \(\partial _\mu \varphi =0\) show that a only depends on time. Therefore, the space-time is a generalised Robertson Walker (GRW) space-time, Eq. (7) [17, 18].

This agrees with Theorem 1.2 in [9], stating that (in a Riemannian setting) a perfect fluid Codazzi tensor such that \(h^{jk}\nabla _k {\mathscr {C}}^i{}_i =0\) implies a warped metric.

With \(\xi \equiv (n-1)({\dot{\varphi }} +\varphi ^2)\), the Ricci tensor now is:

$$\begin{aligned} R_{jk} = \frac{R-n\xi }{n-1} u_j u_k + \frac{R-\xi }{n-1} g_{jk} -(n-2) E_{jk}. \end{aligned}$$
(21)

\(\bullet \) If \(B=0\), i.e. \({\mathscr {C}}_{jk}=A u_j u_k\), then \(\nabla _i u_j = -u_i\dot{u}_j\) and A solves (11). The equation \(\varphi =0\) gives that \(a(t,\textbf{x})\) is independent of time, and can be absorbed in the space metric to give

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

Its conformally flat and spherically symmetric version generalises the Schwarzschild interior solution, Eq. 37.39 in [15]. If moreover \(\dot{u}_i\) is closed, then the metric is static ( [15], page 283):

$$\begin{aligned} ds^2 = - b^2(\textbf{x}) dt^2 + g_{\mu \nu }^\star (\textbf{x}) dx^\mu dx^\nu . \end{aligned}$$
(22)

\(\bullet \) In General Relativity the vanishing of the Cotton tensor \(\textsf{C}_{jkl}=0\) means that \(R_{kl}-g_{kl} \frac{R}{2(n-1)}\) is a Codazzi tensor. The Einstein equations then imply that also \({\mathscr {C}}_{kl}= T_{kl}-\frac{T}{n-1}g_{kl}\) is a Codazzi tensor.

2.2 Current-flow Codazzi tensors

We investigate Codazzi tensors with the form of a current-flow tensor \({\mathscr {C}}_{jk} =\lambda (u_j \dot{u}_k + \dot{u}_j u_k),\) with closed \(\dot{u}_i\).

The eigenvalues are 0 and \(\pm i\lambda \sqrt{\eta }\), the latter being non-degenerate with complex eigenvectors \(V^{\pm }_k =\pm \sqrt{\eta }u_k + i\dot{u}_k\), \(g^{jk}V^+_j V^-_k =0\). Since the Codazzi tensor commutes with the Ricci tensor, \(V^{\pm }_k\) are also eigenvectors of the Ricci tensor. From \(0=V^+_j R^{jk}V^-_k\) one obtains

$$\begin{aligned} \dot{u}^j R_{jk}\dot{u}^k = -\eta u^j R_{jk} u^k \end{aligned}$$
(23)

Theorem 2.6

The tensor \({\mathscr {C}}_{jk} =\lambda (u_j \dot{u}_k + \dot{u}_j u_k)\) with closed acceleration is Codazzi if and only if:

$$\begin{aligned} \nabla _j u_k&= -\frac{{\dot{\lambda }}}{\lambda }\frac{ \dot{u}_j \dot{u}_k }{\eta }- u_j \dot{u}_k \end{aligned}$$
(24)
$$\begin{aligned} \nabla _j \lambda&= -u_j {\dot{\lambda }} -\lambda \dot{u}_j \left( 2+\frac{\dot{u}^p\nabla _p \eta }{2\eta ^2}\right) \end{aligned}$$
(25)
$$\begin{aligned} \nabla _j \dot{u}_k&= - \eta u_j u_k -\frac{{\dot{\lambda }}}{\lambda } (\dot{u}_j u_k + u_j \dot{u}_k) +\dot{u}_j \dot{u}_k \frac{\dot{u}^p\nabla _p\eta }{2\eta ^2} \end{aligned}$$
(26)

Proof

See Appendix 2. \(\square \)

A useful relation found in the proof is

$$\begin{aligned} \nabla _k \eta = -2 \frac{{\dot{\lambda }}}{\lambda }\eta u_k + \dot{u}_k \frac{\dot{u}^p \nabla _p \eta }{\eta }. \end{aligned}$$
(27)

We discuss the geometric restrictions posed by a current-flow Codazzi tensor with closed acceleration.

Since the velocity has non-zero shear tensor

$$\begin{aligned} \sigma _{jk} = \frac{{\dot{\lambda }}}{\lambda }\left[ \frac{g_{jk}+u_j u_k}{n-1}- \frac{\dot{u}_j \dot{u}_k}{\eta }\right] \end{aligned}$$

there are coordinates such that the metric has the structure [19]:

$$\begin{aligned} ds^2 = - b^2(t,\textbf{x})dt^2 + G^\star _{\mu \nu } (t,\textbf{x})dx^\mu dx^\nu \end{aligned}$$
(28)

with Christoffel symbols \(\Gamma _{00}^0 =\frac{\partial _t b}{b}\), \(\Gamma _{\mu 0}^0 =\frac{\partial _\mu b}{b}\), \(\Gamma ^\mu _{00} = G^{\star \mu \nu } b\partial _\nu b\), \( \Gamma _{\mu \nu }^0 = \frac{\partial _t G^{\star }_{\mu \nu }}{2b^2}\), \(\Gamma _{0\nu }^\mu = \tfrac{1}{2}G^{\star \mu \rho } \partial _t G^\star _{\nu \rho }\) and \(\Gamma _{\rho \sigma }^\mu = \Gamma _{\rho \sigma }^{\star \mu }\). The equations for u, \(\dot{u}\) give:

$$\begin{aligned} u_0=-b(t,\textbf{x}),\; u_\mu =0, \qquad \dot{u}_0=0, \; \dot{u}_\mu = \frac{\partial _\mu b(t,\textbf{x})}{b(t,\textbf{x})} \end{aligned}$$

In this frame, the equations \(\nabla _\mu u_\nu = -\frac{{\dot{\lambda }}}{\lambda } \frac{\dot{u}_\mu \dot{u}_\nu }{\eta } \) and \(\nabla _0\dot{u}_\mu = -\frac{{\dot{\lambda }}}{\lambda }u_0\dot{u}_\mu \) are:

$$\begin{aligned} - \tfrac{1}{2} b \,\frac{\partial G^{\star }_{\mu \nu }}{\partial t} =\frac{{\dot{\lambda }}}{\lambda } \frac{\partial _\mu b \,\partial _\nu b}{\eta } , \quad \frac{\partial \dot{u}_\mu }{\partial t} -\tfrac{1}{2} \dot{u}_\nu G^{\star \nu \rho }\frac{\partial G^\star _{\mu \rho } }{\partial t} = \frac{{\dot{\lambda }}}{\lambda }b\, \dot{u}_\mu \end{aligned}$$
(29)

We now specialize to static space-times.

2.2.1 Static space-times

If \({\dot{\lambda }} =0\), Eq. (29) shows that \(G^\star _{\mu \nu } \) is independent of time t, as well as \(\dot{u}_\mu \). Then \(b(t,\textbf{x}) =\beta (t) b(\textbf{x})\). The product \(\beta ^2(t)dt^2\) in \(ds^2\) redefines the time, and the metric is static, Eq. (22).

Theorem 2.6 becomes: the current-flow tensor with \({\dot{\lambda }} =0\) and closed acceleration is Codazzi if and only if:

$$\begin{aligned} \nabla _j u_k&= - u_j \dot{u}_k \end{aligned}$$
(30)
$$\begin{aligned} \nabla _j \lambda&= -\lambda \dot{u}_j \left( 2+\frac{\dot{u}^p\nabla _p \eta }{2\eta ^2}\right) \end{aligned}$$
(31)
$$\begin{aligned} \nabla _j \dot{u}_k&= - \eta u_j u_k +\dot{u}_j \dot{u}_k \frac{\dot{u}^p\nabla _p\eta }{2\eta ^2} \end{aligned}$$
(32)

Eq. (30) and closedness of \(\dot{u}_i\) covariantly confirm the space-time as static.

Proposition 2.7

In a static space-time with Eqs.(30)–(32) with closed \(\dot{u}_i\), the vectors \(u_i\) and \(\dot{u}_i\) are eigenvectors of the Ricci tensor with the same eigenvalue.

Proof

(1) For brevity, put \(\epsilon =\dot{u}^p\nabla _p \eta \).

Equation(27) with \({\dot{\lambda }} =0\) is: \(\nabla _j \eta ^2 = 2 \epsilon \dot{u}_j\) Now it is \(\nabla _k \nabla _j \eta ^2 = 2\dot{u}_j \nabla _k \epsilon + 2 \epsilon \nabla _k \dot{u}_j\). Antisimmetrization, with the property that \(\nabla _k\dot{u}_j = \nabla _j \dot{u}_k\) gives: \(\dot{u}_j \nabla _k \epsilon = \dot{u}_k \nabla _j \epsilon \) i.e.

$$\begin{aligned} \nabla _j \epsilon = \dot{u}_j \frac{\dot{u}^p\nabla _p \epsilon }{\eta } \end{aligned}$$

(2) \(R_{jklm}u^m = \nabla _j \nabla _k u_l - \nabla _k \nabla _j u_l = -\nabla _j (u_k\dot{u}_l)+\nabla _k (u_j\dot{u}_l) = (u_j \dot{u}_k - u_k\dot{u}_j)\dot{u}_l - u_k \nabla _j\dot{u}_l + u_j \nabla _k \dot{u}_l = (u_j \dot{u}_k - u_k\dot{u}_j)\dot{u}_l (1 + \epsilon /2\eta ^2)\). Contraction with \(g^{jl}\):

$$\begin{aligned} R_{km} u^m = - (\eta + \frac{\epsilon }{2\eta } ) u_k \end{aligned}$$
(33)

(3) \(R_{jklm}\dot{u}^m = \nabla _j \nabla _k \dot{u}_l - \nabla _k \nabla _j \dot{u}_l = \nabla _j (-\eta u_k u_l + \dot{u}_k \dot{u}_l \frac{\epsilon }{2\eta ^2}) - \nabla _k (-\eta u_j u_l + \dot{u}_j \dot{u}_l \frac{\epsilon }{2\eta ^2})= -(u_k \nabla _j\eta - u_j\nabla _k\eta ) u_l +\eta (u_j \dot{u}_k - \dot{u}_j u_k)u_l + (\dot{u}_k \nabla _j \dot{u}_l - \dot{u}_j \nabla _k \dot{u}_l)\frac{\epsilon }{2\eta ^2} +(\dot{u}_k\nabla _j \frac{\epsilon }{2\eta ^2} - \dot{u}_j \nabla _k \frac{\epsilon }{2\eta ^2})\dot{u}_l \). The last parenthesis is zero because \(\nabla _j \frac{\epsilon }{2\eta ^2}\) is proportional to \(\dot{u}_j\). Then: \(R_{jklm}\dot{u}^m = (u_j \dot{u}_k - \dot{u}_j u_k)u_l (\eta + \frac{\epsilon }{\eta }) + (\dot{u}_k \nabla _j \dot{u}_l - \dot{u}_j \nabla _k \dot{u}_l)\frac{\epsilon }{2\eta ^2}\). Contraction with \(g^{jl}\):

$$\begin{aligned} R_{km} \dot{u}^m = - \left( \eta +\frac{\epsilon }{2\eta } \right) \dot{u}_k \end{aligned}$$
(34)

\(\square \)

The Ricci tensor is now obtained.

In Prop.2.7 we evaluated \(R_{jklm}u^m = (u_j \dot{u}_k - u_k\dot{u}_j)\dot{u}_l (1 + \epsilon /2\eta ^2)\). Contraction with \(u^j\) is \(u^j R_{jklm}u^m = - \dot{u}_k \dot{u}_l (1 + \epsilon /2\eta ^2)\). The contraction of the Weyl tensor and (33) give

$$\begin{aligned} E_{kl} = -\dot{u}_k \dot{u}_l \left( 1 + \frac{\epsilon }{2\eta ^2}\right) - \frac{ R_{kl} - (g_{kl}+2u_k u_l) (\eta + \epsilon /2\eta )}{n-2} + R \frac{g_{kl}+u_k u_l}{(n-1)(n-2)} \end{aligned}$$

We then find:

$$\begin{aligned} R_{kl}&= \left[ \frac{R}{n-1}+ 2\eta +\frac{\epsilon }{\eta }\right] u_k u_l +\left[ \frac{R}{n-1}+ \eta +\frac{\epsilon }{2\eta }\right] g_{kl } \nonumber \\&\quad - (n-2) \left[ E_{kl} +\dot{u}_k \dot{u}_l \left( 1 + \frac{\epsilon }{2\eta ^2}\right) \right] . \end{aligned}$$
(35)

In particular, by Eq. (34), one has the eigenvalue equation

$$\begin{aligned} (n-2)E_{kl} \dot{u}^l = \left[ \frac{R}{n-1}-(n-4)\left( \eta +\frac{\epsilon }{2\eta }\right) \right] \dot{u}_k \end{aligned}$$

Eq. (32) with \(\epsilon =0\) (then \(\eta \) is a constant) was obtained by Rao and Rao [20] in a static metric to characterize the relativistic generalisation of the uniform Newton force at a spatial hypersurface.

2.2.2. We restrict the static space-time to be spherically symmetric, and give some examples in the end:

$$\begin{aligned} ds^2 = -b^2(r) dt^2 + f_1^2(r) dr^2 + f_2^2(r) d\Omega _{n-2}^2 \end{aligned}$$
(36)

In spherical symmetry \(\dot{u}\) is radial and \(\dot{u}_r = b'(r)/b(r)\) (a prime is a derivative in r). The definition \(\eta = \dot{u}^k \dot{u}_k\) gives:

$$\begin{aligned} \eta (r)= \frac{1}{f_1^2(r)} \frac{b'^2(r)}{b^2(r)} \end{aligned}$$
(37)

In such coordinates the solution of Eq. (31) is

$$\begin{aligned} \lambda (r) = \kappa \frac{f_1(r)}{b(r)b'(r)} \end{aligned}$$
(38)

with a constant \(\kappa \).

Since \(\dot{u}\) is a radial vector, the angular components of Eq. (32) are \(\nabla _a \dot{u}_{a'} =0\) (where \(a, a' = 1,...,n-2\) enumerate the angles). It implies \(\Gamma _{a,a'}^r \dot{u}_r= 0\) i.e. \(\Gamma _{a,a'}^r = 0\). With the expression in [21] Appendix 9.6, one gets the condition on the metric:

$$\begin{aligned} \frac{df_2}{dr}=0. \end{aligned}$$

In conclusion, a static spherically symmetric space-time with Codazzi tensor \({\mathscr {C}}_{jk}=\lambda (u_j\dot{u}_k + \dot{u}_j u_k)\) with closed acceleration has the form:

$$\begin{aligned} ds^2 = - b^2(r) dt^2 + f_1^2(r) dr^2 + L^2 d\Omega _{n-2}^2 \end{aligned}$$
(39)

where L is a positive constant.

The electric tensor and the scalar curvature of the space manifold are obtained from Eq. (33) in [21] with \(a=1\), \(f_2=L\), and the relations (37) and (38):

$$\begin{aligned} E_{jk}(r)&= \frac{n-3}{n-2}\frac{1}{f_1^2} \left( \frac{f_1^2}{L^2} +\frac{b'}{b} \frac{f_1'}{f_1} - \frac{b''}{b} \right) \ \left[ \frac{\dot{u}_j \dot{u}_k}{\eta } - \frac{h_{jk}}{n-1}\right] \end{aligned}$$
(40)
$$\begin{aligned} R^\star&= \frac{1}{L^2}(n-2)(n-3) \end{aligned}$$
(41)

where \(h_{jk} = g_{jk}+ u_k u_l\).

The Ricci tensor (35) in spherical coordinates is sum of three tensors, proportional to \(u_j u_k\), \(g_{jk}\) and \(\dot{u}_j \dot{u}_k\).

We list some examples of the metric (39). They share the same form of Ricci tensor (35), with electric tensor (40). Moreover, they are endowed with a current-flow Codazzi tensor with non-zero components \({\mathscr {C}}_{0r}={\mathscr {C}}_{r0} =\kappa f_1(r)/b(r)\) in the coordinates of each below-listed metric.

Example 2.8

Nariai space-times solve Einstein’s equations in vacuo [22,23,24,25]:

$$\begin{aligned} ds^2= \frac{1}{\Lambda } \left[ -a\cos \log \left( \frac{r}{r_0} \right) dt^2 + \frac{dr^2}{r^2} + d\Omega _2^2\right] . \end{aligned}$$

where \(\Lambda \) is the cosmological constant. A coordinate change (see [22]) brings the metric to the direct product of de Sitter \((dS)_2\) with the sphere \(S_2\) of radius \(\sqrt{3/\Lambda }\): \( ds^2 = -b\, d\tau ^2 +b^{-1} d\rho ^2 +(3/\Lambda ) d\Omega _2^2\), with \(b(\rho ) =1-\frac{1}{3}\Lambda \rho ^2\).

Example 2.9

Bertotti-Robinson space-times are conformally flat solutions of the source-free Einstein-Maxwell equations with non-null e.m. field [26,27,28]:

$$\begin{aligned} ds^2 = \frac{r_0^2}{r^2}\left[ -dt^2 + dr^2 +r^2 d\Omega _{n-2}^2 \right] . \end{aligned}$$
(42)

The Ricci tensor is (35) with \(R=0\), \(\epsilon =0\), \(\eta =\frac{1}{r_0^2}\) \(\lambda = -\kappa \frac{r^2}{r_0^2}\):

$$\begin{aligned} R_{kl}=\frac{1}{r_0^2}(2 u_k u_l +g_{kl})-2 \dot{u}_k \dot{u}_l. \end{aligned}$$
(43)

Eqs. (30) and (32), that reads \(\nabla _j \dot{u}_k =\frac{1}{r_0^2} u_j u_k\), imply \(\nabla _i R_{jk}=\nabla _j R_{ik}\). Therefore, Bertotti-Robinson space-times have two Codazzi tensors: the Ricci tensor and

$$\begin{aligned} {\mathscr {C}}_{jk}=-\kappa \frac{r^2}{r_0^2}(u_j \dot{u}_k + \dot{u}_j u_k) \end{aligned}$$

A coordinate change [29] brings the metric to: \(ds^2 = -b\, d\tau ^2 + b^{-1}d\rho ^2 + r_0^2 d\Omega _{n-2}^2\) with \(b(\rho ) =1+\rho ^2/r^2_0\). It is the direct product of (AdS)\(_2\) with the sphere \(S_{n-2}(r_0)\).

Example 2.10

In [30] black holes are studied in string-corrected Einstein-Maxwell theory coupled to a dilaton field. The solution displayed in Eq. 35 is

$$\begin{aligned} ds^2 = -(ar^2+br +c) dt^2 + \frac{dr^2}{ar^2+br +c} + L^2 d\Omega _2^2. \end{aligned}$$

Depending on the constants, it may reduce to (AdS)\(_2\times S_2(L)\) or (dS)\(_2\times S_2(L)\).

Example 2.11

In [31] the Bertotti-Robinson-type black hole solutions of string theory are obtained, by CFT methods. This one (Eq. 38) is an example:

$$\begin{aligned} ds^2 = -\left[ \tfrac{r^2}{\ell ^2} + \tfrac{J^2}{r^2}-M\right] dt^2 + \frac{dr^2}{\left[ \tfrac{r^2}{\ell ^2} + \tfrac{J^2}{r^2}-M\right] } + L^2 d\Omega ^2_2 \end{aligned}$$

where M is the mass, J is the angular momentum, \(\ell ^2\) is proportional to the cosmological constant.

Example 2.12

In [32] spherical black hole solutions of the Einstein-Maxwell-scalar equations are found, where the scalar field is non-minimally coupled to the Maxwell invariant. Among others the following metric is given (Eq. 4.11), where a is a constant:

$$\begin{aligned} ds^2 = a \left( -r^2 dt^2 +\frac{1}{r^2}dr^2 \right) + L^2 \,d\Omega _2^2. \end{aligned}$$

The examples describe “near horizon geometries of extremal black-holes” [33]. They are direct products of a Lorentzian 2D spacetime with the 2-sphere, whose general properties were investigated by Ficken [34].

2.3 Yang Pure space-times

A Yang Pure space-time is defined by a Ricci tensor that is a Codazzi tensor:

$$\begin{aligned} \nabla _j R_{kl} = \nabla _k R_{jl} \end{aligned}$$
(44)

equivalent to \(\nabla ^m R_{jklm}=0\). Contraction with \(g^{jl}\) gives \(\nabla _k R=0\). They were introduced by Chen Ning Yang in 1974 in the geometry of Yang-Mills theories [35, 36]. These are examples of solutions of Yang’s equation (44):

  • Vacuum solutions of Einstein’s equations: \(R_{kl}=\Lambda g_{kl}\).

  • Wei-Tou Ni obtained the conformally-flat non-static solution [37]

    $$\begin{aligned} ds^2 = \left[ C+\frac{f(r-t)}{r}+\frac{g(r+t)}{r}\right] (-dt^2 +dr^2 +r^2 d\Omega _2^2) \end{aligned}$$

    where C is a constant, f and g are arbitrary functions, and also the solution:

    $$\begin{aligned} ds^2 = -dt^2 + \left[ 1 + \frac{a}{r} + b r^2 \right] ^{-1} dr^2 + r^2 d\Omega _2^2 \end{aligned}$$

  • In 1975 A. H. Thompson [38] found geometrically degenerate solutions of Yang’s gravitational equations. In particular, he showed that the Bertotti-Robinson metric Eq. (42) is Yang Pure.

  • Friedmann-Robertson-Walker (FRW) space-times

    $$\begin{aligned} ds^2 = -dt^2 + a^2(t)\left[ \frac{dr^2}{1- k r^2} + r^2 d\Omega _2^2 \right] \end{aligned}$$
    (45)

    may be also characterized by a “perfect fluid” Ricci tensor

    $$\begin{aligned} R_{jk} = \tfrac{1}{3}(R-4\xi ) u_j u_k + \tfrac{1}{3} (R-\xi ) g_{jk} \end{aligned}$$
    (46)

    and zero Weyl tensor. Here (see [39]): \(u^k u_k=-1\), \(\nabla _i u_j = H (u_iu_j + g_{ij})\) where \(H=\dot{a}/a\) is Hubble’s parameter and \(\xi = 3(H^2+\dot{H}) = 3\ddot{a}/a\). The Cotton tensor being zero, a FRW space-time is Yang Pure if and only if \(\nabla _j R=0\).

The flat case \(k=0\) was solved by the authors [39]. While the two geometric constraints fix the Ricci tensor, the Einstein equations provide a source which is a perfect fluid with equation of state \(p=w(t)\mu \) that evolves from \(w=1/3\) (pure radiation) to \(w=-1\) (accelerated expansion, without a cosmological constant \(\Lambda \)).

3 Harada-Cotton gravity

The results of the previous section are interesting for Harada’s Cotton gravity. The symmetries of the Weyl tensor imply two important facts [1]:

  1. (1)

    \(g^{kl}\textsf{C}_{jkl}=0\), then (1) mantains the law \(0=\nabla _k T^{jk}\).

  2. (2)

    \(\nabla ^l \textsf{C}_{jkl}=0\) implies that \(R_{jk}\) and \(T_{jk}\) commute:

    \(0 = \nabla _l (\nabla _j T_k{}^l- \nabla _k T_j{}^l) = [\nabla _l,\nabla _j ]T_k{}^l -[\nabla _l,\nabla _k]T_j{}^l + \nabla _j (\nabla _l T_k{}^l) - \nabla _k (\nabla _l T_j{}^l) = R_{ljkm}T^{ml}+R_{jm} T_k{}^m - R_{lkjm}T^{ml} - R_{km} T_j{}^m\).

    The first term cancels the third one by the first Bianchi identity: \(0=(R_{ljkm}+R_{jklm}+R_{kljm})T^{ml}\) (the second term vanishes).

As stated in the introduction, Eq. (1) naturally provides the Codazzi tensor in Eq. (4). Depending on its form, there are different levels of Cotton gravity, that are extensions of the Einstein gravity.

The choice of the Codazzi tensor restricts the space-time which, in turn, provides the structure of the Ricci tensor. Together, the Ricci and the Codazzi tensors determine the energy-momentum tensor:

$$\begin{aligned} T_{kl} = R_{kl} - \frac{1}{2} R g_{kl} -{\mathscr {C}}_{kl} +g_{kl}{\mathscr {C}}^j{}_j \end{aligned}$$
(47)

By construction, the metric of the space-time solves the Cotton-gravity equation (1) with the energy-momentum tensor (47). This approach reverses the standard one, where the matter tensor is the input. Of course it is simpler, but not alternative, than solving high order differential equations for the metric. Here a form of the metric is given a priori through the Codazzi tensor.

Equation (47) is Einstein’s equation corrected by a Codazzi tensor, Eq. (6), in analogy with other theories of extended gravity (the H-term of Eq. 26 in [40]).

3.1 Yang Pure spaces.

Since the Ricci tensor is Codazzi, the definition (2) of Cotton tensor shows that Yang Pure spaces are solutions of the vacuum Harada equations \(\textsf{C}_{jkl}=0\). Harada’s vacuum solution (3) is not a Yang Pure space.

Now we present the simplest Codazzi tensors, with examples that only aim at illustrating the procedure.

3.2 .

The trivial Codazzi tensors \({\mathscr {C}}_{jk}=0\) and \({\mathscr {C}}_{jk} = B g_{jk}\) (with B constant by the Codazzi condition) give the Einstein equations without or with a cosmological constant.

3.3 Case \({\mathscr {C}}_{jk} {= A u_j u_k+ Bg_{jk}}\), \({u^ku_k=-1}\)

The generalized Stephani Universes are solutions of the Harada equation with energy-momentum tensor (47) built with the Ricci tensor (18) and the Codazzi tensor. Such inhomogeneous cosmological models may provide an explanation of the observed accelerated expansion of the universe and bypass the dark energy problem (see for example [41, 42] and references therein).

We here give the Ricci tensor for the simpler Stephani Universe in \(n=4\):

$$\begin{aligned} R_{kl} = 2AB \, u_k u_l +g_{kl} (3B^2 -AB) \end{aligned}$$
(48)

Its perfect fluid form implies a perfect fluid source in the Einstein equations, as well as in the Harada equations (with different density and pressure).

3.4 Case \({\mathscr {C}}_{jk} = {A u_j u_k}\), \({u_k u^k=-1}\), \({\dot{u}}\) closed

The Codazzi condition is equivalent to \(\nabla _i u_j = -u_i\dot{u}_j\) and (11). The space-time is static and the velocity is eigenvector of the Ricci tensor: \(R_{jk}u^k=-u_j (\nabla _k \dot{u}^k)\).

This example in \(n=4\) is static and spherically symmetric:

$$\begin{aligned} ds^2 = -b^2(r) dt^2 + f(r)^2 dr^2 + r^2 d\Omega _2^2 \end{aligned}$$
(49)

The function A(r) solves (11), where the time component is an identity and \(A' = -A b'/b \) (a prime is a derivative in r). The equation is solved by

$$\begin{aligned} A(r) = \frac{k}{b(r)} \end{aligned}$$

where k is a constant. The covariant form of the Ricci tensor on static isotropic space-times was obtained in [21] (Eq. 49 with \(\varphi =0\)):

$$\begin{aligned} R_{jk}&= u_j u_k \frac{R+4\nabla _p \dot{u}^p}{3}+ g_{jk}\frac{R + \nabla _p \dot{u}^p}{3} + \Pi _{jk} \nonumber \\ \Pi _{jk}&=\left[ \frac{\dot{u}_j \dot{u}_k}{\eta } -\frac{h_{jk}}{3}\right] \left[ \nabla _p \dot{u}^p - 3 \left( \eta + \frac{\dot{u}^i\nabla _i\eta }{2\eta } \right) - 2 E(r) \right] \end{aligned}$$
(50)

where \(\eta = \dot{u}^j \dot{u}_j = b'^2/(b^2 f^2)\). E(r) is the amplitude of the electric tensor

$$\begin{aligned} E(r)&= \frac{1}{2f^2}\left[ \frac{f^2}{r^2} - \frac{1}{r^2} - \frac{f'}{f r} +\frac{b'}{b}\frac{d}{dr} \log (fr) - \frac{b''}{b} \right] \nonumber \\ R&=\frac{2}{r^2}\left( 1-\frac{1}{f^2}\right) + \frac{4}{r} \frac{f'}{f^3} -\frac{2}{bf^2} \left( b'' - b' \frac{f'}{f} + 2\frac{b'}{r}\right) \nonumber \\ \nabla _p \dot{u}^p&= \frac{1}{bf^2} \left( b'' - b' \frac{f'}{f} + 2\frac{b'}{r} \right) \end{aligned}$$
(51)

The traceless tensor \(\Pi _{jk}\) modifies the perfect fluid term. It is \(\Pi _{jk}u^k=0\) and \(\Pi _{jk}\dot{u}^k \propto \dot{u}_j\).

The Ricci tensor has three eigenvalues and builds a Cotton tensor \(\textsf{C}_{jkl}\) that, by construction, solves Harada’s equation (1) for the following energy-momentum tensor:

$$\begin{aligned} T_{jk} =&u_j u_k \frac{R+4\nabla _p \dot{u}^p}{3}+ g_{jk}\frac{R + \nabla _p \dot{u}^p}{3} +g_{kl} \frac{T}{3} - g_{kl} \frac{R}{6}\\&-{\mathscr {C}}_{kl} +\left[ \frac{\dot{u}_j \dot{u}_k}{\eta } -\frac{h_{jk}}{3}\right] \left[ \nabla _p \dot{u}^p - 3 \left( \eta + \frac{\dot{u}^i\nabla _i\eta }{2\eta } \right) - 2 E(r) \right] . \end{aligned}$$

A simplification is done with the expression of the trace T, and with the following identity (Lemma 3.4 in [21]):

$$\begin{aligned} \nabla _p \dot{u}^p - 3\left( \eta + \frac{\dot{u}^i\nabla _i \eta }{2\eta } \right) = - \frac{2}{f^2}\,\left[ \frac{b''}{b} - \frac{b'}{b} \frac{d}{dr} \log (r f) \right] . \end{aligned}$$

The result is:

$$\begin{aligned} T_{jk} =&u_j u_k \left[ \frac{R}{3}+\frac{4}{3}\nabla _p \dot{u}^p - \frac{k}{b}\right] + g_{jk} \left[ -\frac{R}{6} + \frac{1}{3}\nabla _p \dot{u}^p -\frac{k}{b} \right] \\&+\left[ \frac{\dot{u}_j \dot{u}_k}{\eta } -\frac{h_{jk}}{3}\right] \frac{1}{f^2} \left[ - \frac{b''}{b} + \frac{b'}{b} \left( \frac{1}{r}+\frac{f'}{f}\right) - \frac{f^2-1}{r^2} + \frac{f'}{f r} \right] . \end{aligned}$$

The tensor specifies the parameters of an anisotropic fluid

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

with \(P=\frac{1}{3}p_r + \frac{2}{3}p_\perp \) (effective pressure), density \(\mu \), radial pressure \(p_r\), transverse pressure \(p_\perp \), constructed with the free parameters b(r), f(r), k. Note the pressure anisotropy despite the spherical symmetry of the metric.

3.5 Case \({\mathscr {C}}_{jk}{=\lambda (u_j\dot{u}_k + \dot{u}_j u_k)}\) with closed \({\dot{u}_j}\)

The metrics in examples 2.8–2.12 are static spherically symmetric solutions of equations of various gravity theories, Einstein, Einstein-Maxwell, low energy string, with their own matter or radiation content. However, since they all contain a current-flow Codazzi tensor, they all solve the Harada equation (1) with a proper energy-momentum tensor that is obtained below, characterized by a current-flow term.

The metrics determine the Ricci tensor (35) with radial symmetry. The energy-momentum tensor is (47) with \(R=R^\star -2\eta - \epsilon /\eta \) and \(R^\star =2/L^2\):

$$\begin{aligned} T_{kl}=&u_k u_l \frac{R^\star }{2}+ h_{kl}\left[ -\frac{R^\star }{6}+\frac{2\eta }{3}+\frac{\epsilon }{3\eta }\right] \\&-{\mathscr {C}}_{kl} - 2\left[ \frac{\dot{u}_k \dot{u}_l}{\eta }- \frac{h_{kl}}{3}\right] \left[ \eta +\frac{\epsilon }{2\eta } +E(r)\right] \\ E(r)=&\frac{1}{2}\frac{1}{f_1^2} \left( \frac{f_1^2}{L^2} +\frac{b'}{b} \frac{f_1'}{f_1} - \frac{b''}{b} \right) . \end{aligned}$$

It is the energy-momentum tensor of a fluid with velocity \(u_j\), acceleration \(\dot{u}_j\), energy density \(\mu =\frac{1}{2}R^\star \), pressure anisotropy \(p_r -p_\perp = -2\eta -\epsilon /\eta -2E(r)\) and effective pressure \(3P=p_r +2p_\perp = -\frac{R^\star }{2}+2\eta +\frac{\epsilon }{\eta }\).

Example 3.1

Consider the Bertotti-Robinson metric in 2.9: \(b(r)=f_1(r)= r_0/r\), \(L=r_0\). It is \(\eta =1/r_0^2\), \(\epsilon =0\) and \(\lambda =-\kappa r^2/r_0\).

The Ricci tensor for this metric is: \( R_{kl} = \frac{1}{r_0^2}(2u_k u_l + g_{kl} ) - 2\dot{u}_k \dot{u}_l \) and \(R=0\). The metric (42) solves the Harada equation with the traceless energy-momentum tensor

$$\begin{aligned} T_{kl}= \frac{1}{r_0^2}(2u_k u_l +g_{kl}) -2 \dot{u}_k \dot{u}_l +\frac{\kappa }{r_0} r^2 (u_k\dot{u}_l + \dot{u}_k u_l). \end{aligned}$$

3.6 Cotton gravity in constant curvature space-times

Harada made the remark that a de Sitter metric is a vacuum solution (\(T_{jk}=0\)) of the Cotton gravity equation (1). We extend his remark.

Constant curvature space-times are defined by the Riemann tensor

$$\begin{aligned} R_{jklm} = \frac{R}{n(n-1)} (g_{jl}g_{km} - g_{jm} g_{kl}) \end{aligned}$$

They are conformally flat \((C_{jklm}=0)\) and Einstein (\(R_{jk}=g_{jk} R/n\)). In the Lorentzian signature there are exactly three cases: Minkowski (\(R=0)\), de Sitter (\(R>0\)) and anti-de Sitter (\(R<0\)) (see [43], pages 124, 131).

Ferus [10] proved that the only non-trivial Codazzi tensor in a constant curvature space-time has the form

$$\begin{aligned} {\mathscr {C}}_{jk} =\nabla _j \nabla _k \phi +\frac{R\phi }{n(n-1)}g_{jk} \end{aligned}$$
(52)

where \(\phi \) is a smooth scalar field (see also [44] p.436). Then

$$\begin{aligned} T_{kl} =&\frac{R}{n}g_{kl} - \frac{R}{2} g_{kl} -{\mathscr {C}}_{kl} + g_{kl}{\mathscr {C}}^j{}_j \nonumber \\ =&g_{kl}\left[ \nabla _j\nabla ^j \phi + \frac{1}{n}R\phi -\frac{n-2}{2n}R\right] -\nabla _k \nabla _l \phi \end{aligned}$$
(53)

is the most general energy-momentum tensor for Cotton gravity in constant curvature space-times. Therefore, Minkowski, de Sitter, anti-de Sitter space-times solve the Cotton gravity equation. The Codazzi tensor introduces a coupling of gravity with a scalar field.

4 Conclusion

Codazzi tensors have an intrinsic geometric importance. They also naturally enter in the recently proposed Cotton gravity by Harada. The specific form of a Codazzi tensor restricts the space-time it lives in.

We presented a strategy to find solutions of Cotton gravity. In essence, we showed that the equation for Cotton gravity is the Einstein equation modified by the presence of a Codazzi tensor. We investigated two specific forms of Codazzi tensors: the perfect fluid and the current flow.

In the first case the hosting metric turns out to be a generalization of Stephani Universes. Stephani Universes are conformally flat cosmological solutions of the Einstein equations with perfect fluid source.

In the second case, a static current flow Codazzi tensor generates metrics that embrace Nariai and Bertotti-Robinson space-times, and extensions. In the literature they are solutions of various gravity theories, such as Einstein, low energy string, Einstein-Maxwell and so on. By construction, all these metrics solve the Harada-Cotton gravity in geometries selected by the Codazzi tensor, with stress-energy tensors different from the original theory.

An interesting question is whether other forms of Codazzi tensors may give rise to new solutions of Cotton gravity of physical interest, using the same strategy.