Codazzi tensors and their space-times and Cotton gravity

We study the geometric properties of certain Codazzi tensors for their own sake, and for their appearance in the recent theory of Cotton gravity. We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdziński. We also give necessary and sufficient conditions for a space-time to host a current-flow Codazzi tensor. In particular, we study the static and spherically symmetric cases, which include the Nariai and Bertotti-Robinson metrics. The latter are a special case of Yang Pure space-times, together with spatially flat FRW space-times with constant curvature scalar. We apply these results to the recent Cotton gravity by Harada. We show that the equation of Cotton gravity is Einstein’s equation modified by the presence of a Codazzi tensor, which can be chosen freely and constrains the space-time where the theory is staged. In doing so, the tensor (chosen in forms appropriate for physics) implies the form of the Ricci tensor. The two tensors specify the energy-momentum tensor, which is the source in the equation of Cotton gravity for the metric implied by the Codazzi tensor. For example, we show that the Stephani, Nariai and Bertotti-Robinson space-times are characterized by a “current flow” Codazzi tensor. Because of it, they solve Cotton gravity with physically sensible energy-momentum tensors. Finally, we discuss Cotton gravity in constant curvature space-times.


Introduction
In Ref. [1] Junpei Harada proposed an extension named "Cotton gravity" of the Einstein equations [2,3], 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: T is the trace T k k , and the Newton constant is absorbed in T jk .The Cotton tensor is related to the Weyl tensor, C jkl = − n−2 n−3 ∇ m C jkl m , and contains third derivatives of the metric tensor.While solving (1) for a vacuum (T kl = 0) static spherically symmetric solution, Harada obtained a generalization of the Schwarzschild solution: with b 2 (r) = 1 − 2M/r + ar + br 2 .Next, in Ref. [4], he applied the 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.The difference of equations ( 1) and (2) shows that is a Codazzi tensor: Equations ( 3) and ( 4) are equivalent to the Harada equation ( 1) for Cotton gravity.In fact, with R kl = C kl + T kl + g kl R−2T 2(n−1) the Cotton tensor tensor (2) is constructed, and the Codazzi condition ensures that (1) is obtained.It turns out that the third order character of (1) manifests in the supplemental terms provided by the Codazzi tensor either to the Ricci tensor or to the energymomentum tensor, or both in (3).The case C kl = 0 in (3) restores the Einstein equations, and eq.( 1) is identically true.The "trivial" case C jk = Bg jk , adds a cosmological constant.If C kl = 0, eq.( 3) can still be interpreted as the Einstein equation, but with a modified energymomentum tensor: Let us mention that Codazzi tensors appear in the geometry of hypersurfaces [5].A Lorentzian hypersurface in a Minkowski space-time has Riemann tensor R jklm = Ω jl Ω km − Ω jm Ω kl where Ω jk is a Codazzi tensor.The trivial case Ω jk = 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 energymomentum tensors.They involve the basic kinematic quantities u i and ui .
We begin with the "perfect fluid" tensor C jk = Au j u k + Bg jk with the Codazzi property.Andrzej Derdziński [6] proved that if C k k is a constant, then the spacetime is warped (GRW, generalized Robertson-Walker space-time), i.e. there are coordinates such that with Riemannian metric g ⋆ µν .The hypothesis was weakened by Gabe Merton [7], who showed that a necessary and sufficient condition for the GRW space-time is v j ∇ j C k k = 0 for all vectors v j u 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 with the special condition that (∂ 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 C jk = λ(u j uk + uj u k ) with the Codazzi condition and closed vector field uj .The field u j turns out to be vorticity-free but not shear-free.This makes the metric more general than doubly-twisted, eq.( 27).However, 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 examples, we show that a Friedmann-Robertson-Walker metric is Yang Pure if and only if ∇ j R = 0.This concludes Section 2 of the paper.
In Section 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 tensor in eq.(3) 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 De Sitter space-times, for which Ferus [8] 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 Ẋ = u k ∇ k X.The symbols η, ǫ are the scalar functions η = uk uk and ǫ = uk ∇ k η.

Codazzi tensors and their space-times
In refs.[10,9] we showed that a Codazzi tensor always satisfies an algebraic identity with the Riemann tensor (it is "Riemann compatible"): 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 C jm R k m = 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 of an energy-momentum tensor: C jk = Au j u k + Bg jk (perfect fluid) and C jk = λ(u j uk + uj u k ) (current flow).A = 0, B, λ are scalar fields.The vector field u j is time-like unit, u j u j = −1, and is named velocity.The vector field uj = u k ∇ 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.
0, and it is an eigenvector of the Ricci tensor, R jk u k = γu j , with eigenvalue The following identity for the acceleration holds: Proof.The first statement is an obvious consequence of (8) and of the first Bianchi identity.For the eigenvalue we evaluate: The contraction with g jl gives: R km u m = (n−1)( φ+ϕ 2 )u k +ϕ uk −u k η −u k ∇ j uj + u j ∇ k uj .Since uj u j = 0, the last term is: − uj ∇ k u j = −ϕ uk + u k η by eq.( 9), and cancels three terms.The eigenvalue γ is read.The contraction with u j gives the symmetric tensor Subtraction with indices k, l exchanged gives the identity for the acceleration.
We the aid of the Weyl tensor, we obtain the expression of the Ricci tensor on a space-time with a perfect fluid Codazzi tensor.
Proof.The general expression of the Weyl tensor is: A double contraction and (16) give: .
The Ricci tensor is obtained: The expression is symmetrized with the identity (15) and the correction to the perfect fluid part is made traceless by subtraction.
We discuss the geometric restrictions posed by a perfect-fluid Codazzi tensor.The presence of a shear-free and vorticity-free velocity field, eq.( 9), classifies the space-time as doubly-twisted [11], i.e. there is a coordinate frame such that the metric has the form (7). In this frame, with the Christoffel symbols eq.( 9) for u j and uj = u k ∇ k u j give: u 0 = −b(t, x), u µ = 0, and By eq.( 10), the doubly twisted metric has the constraint that ϕ only depends on time.With a = 1/V (x, t), the metric (7) with the constraint becomes: This metric generalizes the well known Stephani metrics, presented in the following example.
Example 2.4.Remarkably, equations (9)-( 13) coincide with eqs.37. 32-37.34 in the book by Stephani et al. [12].They were derived for a Riemann tensor of the form is invertible then, the Bianchi identity implies that it is a Codazzi tensor [13]).Such space-times are conformally flat and are named Stephani universes [12] [14].They are solutions of the Einstein equation with a perfect fluid source T jk .The Stephani metric in n = 4 is x − x 0 (t) 2 , where V 0 , ϕ and x 0 are arbitrary functions of time.
We now consider some special conditions of the perfect fluid Codazzi tensor.
Eq.( 15) now is: after a rescaling of time.The equations ∂ µ ϕ = 0 show that a only depends on time.Therefore, the space-time is a generalised Robertson Walker (GRW) space-time, eq.( 6) [15,16].This agrees with Theorem 1.2 in [7], stating that (in a Riemannian setting) a perfect fluid Codazzi tensor such that h jk ∇ k C i i = 0 implies a warped metric.With ξ ≡ (n − 1)( φ + ϕ 2 ), the Ricci tensor now is: • If B = 0, i.e.C jk = Au j u k , then ∇ i u j = −u i uj and A solves (11).The equation ϕ = 0 gives that a(t, x) is independent of time, and can be absorbed in the space metric to give Its conformally flat and spherically symmetric version generalises the Schwarzschild interior solution, eq.37.39 in [12].If moreover ui is closed, then the metric is static ( [12], page 283): • In General Relativity the vanishing of the Cotton tensor is a Codazzi tensor.The Einstein equations then imply that also

Current-flow Codazzi tensors
We investigate Codazzi tensors with the form of a current-flow tensor C jk = λ(u j uk + uj u k ), with closed ui .The eigenvalues are 0 and ±iλ √ η, the latter being non-degenerate with complex eigenvectors Since the Codazzi tensor commutes with the Ricci tensor, V ± k are also eigenvectors of the Ricci tensor.From Theorem 2.6.The tensor C jk = λ(u j uk + uj u k ) with closed acceleration is Codazzi if and only if: A useful relation found in the proof is We discuss the geometric restrictions posed by a current-flow Codazzi tensor with closed acceleration.Since the velocity has non-zero shear tensor there are coordinates such that the metric has the structure [17]: The equations for u, u give: In this frame, the equations ∇ µ u ν = − λ λ uµ uν η and ∇ 0 uµ = − λ λ u 0 uµ are: We now specialize to static space-times.

2.2.1.
Static space-times.If λ = 0, eq.( 28) shows that G ⋆ µν is independent of time t, as well as uµ .Then b(t, x) = β(t)b(x).The product β 2 (t)dt 2 in ds 2 redefines the time, and the metric is static, eq.( 21).Theorem 2.6 becomes: the current-flow tensor with λ = 0 and closed acceleration is Codazzi if and only if: Eq.( 29) and closedness of ui covariantly confirm the space-time as static.
Proposition 2.7.In a static space-time eqs.( 29)- (31) with closed ui , the vectors u i and ui are eigenvectors of the Ricci tensor with the same eigenvalue.
The Ricci tensor is now obtained.In Prop.2.7 we evaluated R jklm u m = (u j uk − u k uj ) ul (1 + ǫ/2η 2 ).Contraction with u j is u j R jklm u m = − uk ul (1 + ǫ/2η 2 ).The contraction of the Weyl tensor and (32) give We then find: In particular, by eq.( 33), one has the eigenvalue equation Eq.( 31) with ǫ = 0 (then η is a constant) was obtained by Rao and Rao [18] 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: In spherical symmetry u is radial and ur = b ′ (r)/b(r) (a prime is a derivative in r).
The definition η = uk uk gives: In such coordinates the solution of eq.( 30) is with a constant κ.Since u is a radial vector, the angular components of eq.( 31) are ∇ a ua ′ = 0 (where a, a ′ = 1, ..., n − 2 enumerate the angles).It implies Γ r a,a ′ ur = 0 i.e.Γ r a,a ′ = 0.With the expression in [19] Appendix 9.6, one gets the condition on the metric: In conclusion, a static spherically symmetric space-time with Codazzi tensor C jk = λ(u j uk + uj u k ) with closed acceleration has the form: where L is a positive constant.
The electric tensor and the scalar curvature of the space manifold are obtained from eq.( 33) in [19] with a = 1, f 2 = L, and the relations ( 36) and ( 37): where h jk = g jk + u k u l .The Ricci tensor (34) in spherical coordinates is sum of three tensors, proportional to u j u k , g jk and uj uk .
We list some examples of the metric (38).They share the same form of Ricci tensor (34), with electric tensor (39).Moreover, they are endowed with a currentflow Codazzi tensor with non-zero components C 0r = C r0 = κf 1 (r)/b(r) in the coordinates of each below-listed metric.
Example 2.8.Nariai space-times solve the Einstein equations in vacuum [20,21,22,23]: Example 2.9.Bertotti -Robinson space-times are conformally flat solutions of the source-free Einstein-Maxwell equations with non-null e.m. field [24,25]: The Ricci tensor is (34) Eqs.( 29) and (31), that reads ∇ j uk = 1 r 2 0 u j u k , imply that ∇ i R jk = ∇ j R ik .Therefore, Bertotti-Robinson space-times have two Codazzi tensors: the Ricci tensor and Example 2.10.In [26] black holes are studied in string-corrected Einstein-Maxwell theory coupled to a dilaton field.The solution displayed in eq.35 is Example 2.11.In [27] the Bertotti-Robinson-type black hole solutions of string theory are obtained, by CFT methods.This one (eq.38) is an example: where M is the mass, J is the angular momentum, ℓ 2 is proportional to the cosmological constant.
Example 2.12.In [28] spherical black hole solutions of the Einstein-Maxwellscalar 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:

Yang Pure space-times
A Yang Pure space-time is defined by a Ricci tensor that is a Codazzi tensor: equivalent to ∇ m R jklm = 0. Contraction with g jl gives ∇ k R = 0.They were introduced by Chen Ning Yang in 1974 in the geometry of Yang-Mills theories [29].
These are examples of solutions of Yang's equation (43).
• Wei-Tou Ni obtained the conformally-flat non-static solution [31] where C is a constant, f and g are arbitrary functions, and also the solution: • In 1975 A. H. Thompson [32] found geometrically degenerate solutions of Yang's gravitational equations.In particular, he showed that the Bertotti-Robinson metric eq.( 41) is Yang Pure.
• Friedmann-Robertson-Walker (FRW) space-times may be also characterized by a "perfect fluid" Ricci tensor and zero Weyl tensor.Here: u k u k = −1, ∇ i u j = H(u i u j + g ij ) where H = ȧ/a is Hubble's constant and ξ = 3(H 2 + Ḣ) = 3ä/a.The Cotton tensor being zero, a FRW space-time is Yang Pure if and only if ∇ j R = 0.The flat case k = 0 was solved by the authors [33].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)ρ that evolves from w = 1/3 (pure radiation) to w = −1 (accelerated expansion, without a cosmological constant Λ).

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) g kl C jkl = 0, then (1) mantains the law 0 = ∇ k T jk .
(2) ∇ l C jkl = 0 implies that R jk and T jk commute: The first term cancels the third one.As stated in the introduction, eq.( 1) naturally provides the Codazzi tensor in eq.( 3).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: By construction, the metric of the space-time solves the Cotton-gravity equation ( 1) with the energy-momentum tensor (44).This approach reverses the standard one, with the matter tensor as input.

Case
The generalized Stephani Universes are solutions of the Harada equation with energy-momentum tensor (44) built with the Ricci tensor (17) 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 [35,36] and references therein).We here give the Ricci tensor for the simpler Stephani Universe in n = 4: 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).

Case
The Codazzi condition is equivalent to ∇ i u j = −u i uj and (11).The space-time is static and the velocity is eigenvector of the Ricci tensor: R jk u k = −u j (∇ k uk ).This example in n = 4 is static and spherically symmetric: The function A(r) solves (11), where the time component is an identity and A ′ = −Ab ′ /b (a prime is a derivative in r).The equation is solved by where k is a constant.The covariant form of the Ricci tensor on static isotropic space-times was obtained in [19] (eq.49 with ϕ = 0): where η = uj uj = b ′2 /(b 2 f 2 ).E(r) is the amplitude of the electric tensor The traceless tensor Π jk modifies the perfect fluid term.It is Π jk u k = 0 and Π jk uk ∝ uj .The Ricci tensor has three eigenvalues and builds a Cotton tensor C jkl that, by construction, solves Harada's equation (1) for the following energy-momentum tensor: A simplification is done with the expression of the trace T , and with the following identity (Lemma 3.4 in [19]): The result is: The tensor specifies the parameters of a static fluid with P = 1 3 p r + 2 3 p ⊥ (effective pressure), density µ, radial pressure p r , transverse pressure p ⊥ , constructed with the free parameters b(r), f (r), k.Note the pressure anisotropy despite the spherical symmetry of the metric.

3.5.
Case C jk = λ(u j uk + uj u k ) with closed uj .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 currentflow Codazzi tensor, they all solve the Harada equation ( 1) with a proper energymomentum tensor that is obtained below, characterized by a current-flow term.The metrics determine the Ricci tensor (34) with radial symmetry.The energymomentum tensor is (44 It is the energy-momentum tensor of a fluid with velocity u j , acceleration uj , energy density µ = 3.6.Cotton gravity in De Sitter space-times.Constant curvature space-times are defined by the Riemann tensor and include De Sitter, anti De Sitter, Milne, Lanczos space-times.They are conformally flat (C jklm = 0), and Einstein (R jk = g jk R/n).
Harada made the remark that a De Sitter metric is a vacuum solution (T jk = 0) of the Cotton gravity equation (1).Ferus proved that the following is the only non-trivial Codazzi tensor in a De Sitter space-time [8] and [37] p.436: where φ is a smooth scalar field.Then is the most general energy-momentum tensor for Cotton gravity in a De Sitter space-time.
We now discuss an extension.Consider C jk = ∇ j ∇ k φ + Kφg jk with K constant and time-like ∇ k φ.The vector u k = ∇ k φ/ −∇ j φ∇ j φ is time-like unit.By the Ricci identity, the condition that C jk is a Codazzi tensor is Contraction with g jl shows that u m is eigenvector of the Ricci tensor: R k m u m = (n − 1)Ku k .In analogy with Prop.2.3 one evaluates the Ricci tensor: where E jk is the electric tensor.With the Ricci and Codazzi tensors, the tensor is the energy-momentum tensor for the Cotton gravity equation (1) with any metric such the given C jk is Codazzi.
In the cases described above, the Codazzi tensor introduces a coupling of gravity with a scalar field.

Conclusion
Codazzi tensors have an intrinsic geometric importance, and naturally enter in the recently proposed Cotton gravity by Harada.The specific form of a Codazzi tensor restricts the space-time it lives in.These facts allow for a strategy to find solutions of the Cotton gravity.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.In the literature, 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.and the contraction with uk : 1  2 λ∇ j η = − ληu j − uj (2λη + up ∇ p λ).In particular, 2λ + λ uj ∇ j η 2η 2 + up ∇ p λ η = 0 (59) This relation in (56) and in (58) respectively gives equations ( 24) and (25).7) Contraction of (57) with uk is: 0 = ( ui u j − u i uj )( up ∇ p λ) + λη(∇ i u j − ∇ j u i ) + 1  2 λ(u j ∇ i η − u i ∇ j η) − λ( uj üi − ui üj ) Now specify ük and ∇ k η: 0 = (u i uj − u j ui ) + (∇ i u j − ∇ j u i ).This statement means that the velocity is vorticity-free.8) Contraction of (57) with ui : 0 =u j ( ληu k + uk up ∇ p λ) + λ ui (∇ i u j − ∇ j u i ) uk + λ(u j ui ∇ i uk + uj ui ∇ i u k − η∇ j u k ) =u j ( ληu k + uk up ∇ p λ) + ληu j uk + λ(u j ui ∇ i uk + uj ui ∇ i u k − η∇ j u k ) Note that ui ∇ i uk = ui ∇ k ui = The addends cancel and ∇ i C jk − ∇ j C ik = 0.

Theorem 2 . 1 .
The perfect fluid tensor C jk = Au j u k + Bg jk with u j u j = −1 is Codazzi if and only if

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 C jkl = 0. Now we present the simplest Codazzi tensors, with examples that only aim at illustrating the procedure.3.2.The trivial Codazzi tensors C jk = 0 and C jk = Bg jk (with B constant by the Codazzi condition) give the Einstein equations without or with a cosmological constant.