PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(1117) (2018), 199–210 DOI: https://doi.org/10.2298/PIM1817199R

EINSTEIN–HILBERT TYPE ACTION ON SPACETIMES

Vladimir Rovenski

Abstract. The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler–Lagrange equations for the total mixed scalar curvature (as analog of Einstein–Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler–Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation with the new Ricci type curvature instead of Ricci curvature.

Introduction

Distributions on a manifold M (i.e., subbundles of T M) appear in various situations, e.g. as fields of tangent planes of foliations or kernels of differential forms, and many models in physics are foliated, e.g., twisted or warped products.

Let (M, g) be a pseudo-Riemannian manifold endowed with non-degenerate distribution D ⊂ T M. The mixed scalar curvature, S_{mix} (that is an averaged sectional curvature of planes that non-trivially intersect D and its orthogonal complement), see (1.1) and [11], is one of the simplest curvature invariants of a manifold endowed with a distribution. Note that from a mathematical point of view, a spacetime is described as a d-dimensional, time-orientable manifold, equipped with a Lorentzian metric, there also exists a timelike unit vector field N, whose orthogonal distribution is not necessarily integrable. Recall [3, 4] that stably causal and, in particular, globally hyperbolic spacetimes are naturally endowed with a codimension-one foliation (that is level hypersurfaces of a time-function). If D is spanned by N such that g(N, N) = ε_{N} ∈ {−1, 1}, then S_{mix} = ε_{N} Ric_{N,N} , where Ric_{N,N} is the Ricci curvature in the N-direction.

The mixed Einstein–Hilbert action has been introduced in [1] as analog of the Einstein–Hilbert action, with the scalar curvature replaced by S_{mix} :

2010 Mathematics Subject Classification: Primary 53C12; Secondary 53C44.

Key words and phrases: pseudo-Riemannian metric, distribution, variation, mixed scalar curvature, mixed Ricci tensor, mixed gravitational field equations.

The work is based on the talk given by the author on the XIX Geometrical Seminar, 2016.

(0.1)  J_{D,Ω} : g ↦ ∫_{Ω} \{ \frac{1}{2𝔞} (S_{mix} (g) − 2 Λ) + L(g)\} d vol_{g} .

Here Λ is the cosmological constant, L-Lagrangian describing the matter contents, and 𝔞-the coupling constant. The integral is taken over M if it converges; otherwise, one integrates over an arbitrarily large, relatively compact domain Ω containing supports of variations g_{t} with g_{0} = g. The physical meaning of (0.1) has been discussed in [1] for the case of a globally hyperbolic spacetime (M^{4}, g) when D = Span(N) and hence S_{mix} = g(N, N) Ric_{N,N} , and the following have been obtained: the Euler–Lagrange equations, see (3.12), called the mixed gravitational field equations; sufficient conditions for existence of solutions for empty space, the conservation law analogous to conservation law of stress-energy tensor in relativity; equations of motion of a particle on isoparametric foliations in the “mixed gravitational field"; the linearized mixed field equations about the Minkowski metric; the value of coupling constant 𝔞 in the weak field and low velocity limit.

In this paper, we explore (0.1) for any spacetime, the obtained mixed gravitational field equations generalize the result of [1]. In fact, we work in arbitrary number of dimensions of a pseudo-Riemannian manifold endowed with a distribution, and also generalize certain results of [2, 8], where the particular case of variations of metric (called adapted variations) has been examined. In this setting, we present the Euler–Lagrange equations for (0.1) in the classical form of the Einstein field equations:

(0.2)  Ric_{D} −(1/2) Scal_{D} ·g + Λ g = 𝔞 Θ,

where Θ is the stress-energy tensor, while the Ricci tensor and scalar curvature are replaced by a new Ricci type curvature and its trace. Consequently, (0.2) contains the new mixed Einstein tensor G_{D} := Ric_{D} −(1/2) Scal_{D} ·g, whose properties should be further investigated. The following equations with Ric_{D} seem to be interesting to solve w.r.t. g (this issue will be addressed in further work): (i) (0.2) in vacuum, i.e., for Λ = Θ = 0; (ii) Ric_{D}(g) = λ g, i.e., the “mixed Einstein metrics"; (iii) the mixed Ricci equation: Ric_{D} = r for a given (0, 2)-tensor r; (iv) the mixed Ricci flow: ∂_{t}g_{t} = −2 Ric_{D}(g_{t}); (v) the Yamabe-type problem: Scal_{D}(g) = const.

The paper has an introduction and three sections. Section 1 reviews necessary definitions of tensors. Section 2 starts with variation formulas for quantities on a manifold with a distribution. Using them, we deduce Euler–Lagrange equations of (0.1), which generalize results in [2, 8] and use them to build the mixed Ricci tensor Ric_{D} obeying (0.2). In Proposition 2.2 we observe that the replacement of S_{mix} by Scal_{D} in (0.1) leads to the same Euler–Lagrange equations (0.2). In Section 3 we explore Ric_{D} for spacetimes, see (3.6); for integrable D we compare the tensor, see (3.10), with its initial prototype defined in [1] for globally hyperbolic spacetimes.

1. Preliminaries

The section reviews necessary definitions of tensors, some of them were introduced in [8, 11]. We consider a connected manifold M^{n+p} with a pseudo-Riemannian metric g and a non-degenerate distribution D of rank dimD_{x} = n for every x ∈ M. A distribution (i.e., a subbundle) D ⊂ T M is non-degenerate, if D_{x} ⊂ T_{x}M is a non-degenerate subspace of g_{x} for every x ∈ M; in this case, its g-orthogonal complement D^{⊥} is also non-degenerate. A pair (D, D^{⊥} ) of complementary orthogonal distributions is called an almost product structure on (M, g), [5].

Let 𝔛_{M} (resp., 𝔛_{D}) be the module over C^{∞} (M) of all vector fields on M (resp. on D). The “musical" isomorphisms ♯ and ♭ are used for rank one tensors, e.g. if ω ∈ T_{0}^{1}(M) is a 1-form and X ∈ 𝔛_{M} then ω(X) = g(ω^{♯}, X) = X^{♭}(ω^{♯}). For (0, 2)- tensors A and B we have ⟨A, B⟩ = Tr_{g}(A^{♯}B^{♯}) = ⟨A^{♯}, B^{♯}⟩. We get X = X^{⊤} + X^{⊥} , where X^{⊤} is the D-component of X ∈ 𝔛_{M} w.r.t. g. Thus, g = g^{⊤} + g^{⊥} , where

g^{⊥} (X, Y) = g(X^{⊥} , Y^{⊥} ), g^{⊤} (X, Y) = g(X^{⊤} , Y^{⊤} ), (X, Y ∈ 𝔛_{M}).

The following convention is adopted for the range of indices:

a, b, c, … ∈ {1, …, n}, i, j, k, … ∈ {1, …, p}, n + p > 2.

It will be convenient to use orthogonal frames with certain nice properties. It is not hard to show that the local adapted orthonormal frame {E_{a}, E_{i}}, where {E_{a}} ⊂ D, and ε_{i} = g(E_{i}, E_{i}) ∈ {−1, 1}, ε_{a} = g(E_{a}, E_{a}) ∈ {−1, 1}, always exists on M.

We will define several tensors for one of distributions (say, D; similar tensors for D^{⊥} will be denoted using ^{⊥} notation). Let ∇ be the Levi-Civita connection of g. The integrability tensor and the second fundamental form of D, are given by

T (X, Y) = (1/2) [X^{⊤} , Y^{⊤} ]^{⊥} , h(X, Y) = (1/2) (∇_{X^{⊤}} (Y^{⊤} ) + ∇_{Y^{⊤}} X^{⊤} )^{⊥} ,

respectively. We use inner products of tensors, e.g.

⟨h, h⟩=∑ _{a,b} ε_{a}ε_{b}g(h(E_{a}, E_{b}), h(E_{a}, E_{b})),
⟨T, T ⟩=∑ _{a,b} ε_{a}ε_{b}g(T (E_{a}, E_{b}), T (E_{a}, E_{b})).

The mean curvature vector of D is H = Tr_{g} h = ∑ _{a} ε_{a}h(E_{a}, E_{a}). A distribution D is called totally umbilical, harmonic, or totally geodesic, if h = \frac{1}{n} Hg_{|D} , H = 0, or h = 0, resp. The Weingarten operator A_{Z} of D w.r.t. Z ∈ 𝔛_{D^{⊥}} , and the operator T_{Z}^{♯} are defined by

g(A_{Z} (X), Y) = g(h(X, Y), Z), g(T_{Z}^{♯}(X), Y) = g(T (X, Y), Z).

We will use the following convention for indices of various tensors: T_{i}^{♯} := T_{E_{i}}^{♯}, A_{i} := A_{E_{i}}, etc. Define a self-adjoint (1, 1)-tensor T = ∑ _{i} ε_{i}(T_{i}^{♯})^{2} and a self-adjoint (1, 1)- tensor with zero trace

K := ∑ _{i} ε_{i} [T_{i}^{♯}, A_{i}] = ∑ _{i} ε_{i} (T_{i}^{♯}A_{i} − A_{i} T_{i}^{♯}),

which vanishes when D is either integrable or totally umbilical. Define (1, 2)-tensors

α(X, Y) = (A_{X^{⊥}} (Y^{⊤} ) + A_{Y^{⊥}} (X^{⊤} ))/2,
θ(X, Y)= (T_{X^{⊥}}^{♯} (Y^{⊤} ) + T_{Y^{⊥}}^{♯} (X^{⊤} ))/2,
δ_{Z} (X, Y)= (g(∇_{X^{⊤}} Z, Y^{⊥} ) + g(∇_{Y^{⊤}} Z, X^{⊥} ))/2.

Note that α, θ and δ_{Z} are symmetric and vanish for (X, Y) ∈ (D^{⊥} × D^{⊥} ) ∪ (D × D).

For any (0, 2)-tensors P, Q and S on T M, define a tensor Υ_{P,Q} by

⟨Υ_{P,Q} , S⟩ = ∑ _{ν,μ} ε_{ν}ε_{μ}[S(P (e_{ν}, e_{μ}), Q(e_{ν}, e_{μ})) + S(Q(e_{ν}, e_{μ}), P (e_{ν}, e_{μ}))],

where {e_{ν}} is a local orthonormal basis of T M and ε_{ν} = g(e_{ν}, e_{ν}) ∈ {−1, 1}. Notice the properties Υ_{P,Q} = Υ_{Q,P} and Υ_{P,Q_{1} +Q_{2}} = Υ_{P,Q_{1}} + Υ_{P,Q_{2}} .

The curvature tensor is given by R_{X,Y} Z = ∇_{[X,Y]}Z−[∇_{X} , ∇_{Y}]Z. The function
(1.1)  S_{mix} = ∑ _{a,i} ε_{a}ε_{i} g(R_{a,i} E_{a}, E_{i})

is called the mixed scalar curvature of D (or D^{⊥} ) w.r.t. g. The following formula, see [11], has many applications and is used in the proof of Theorem 2.1 below:

S_{mix} = div(H + H^{⊥} ) + g(H, H) − ⟨h, h⟩ + ⟨T, T ⟩ (1.2)

+ g(H^{⊥} , H^{⊥} ) − ⟨h^{⊥} , h^{⊥} ⟩ + ⟨T^{⊥}, T^{⊥}⟩ .

The projection of the gradient of f ∈ C^{1}(M) onto D is ∇^{⊤} f := (∇f)^{⊤} . The D-Laplacian of f ∈ C^{2}(M) is Δ^{⊤} f = div^{⊤} (∇^{⊤} f). The D^{⊥} -divergence of X ∈ 𝔛_{M} is defined by div^{⊥} X = ∑ _{i} ε_{i} g(∇_{i}X, E_{i}). Thus, divX = div^{⊥} X + div^{⊤} X. For X ∈ 𝔛_{D^{⊥}} we have div^{⊥} X = divX + g(X, H). Similarly, for a (1, 2)-tensor P, define a (0, 2)-tensor div P = div^{⊤} P + div^{⊥} P, where

(div^{⊥} P)(X, Y) = ∑ _{i} ε_{i} g((∇_{i} P)(X, Y), E_{i}), X, Y ∈ 𝔛_{M} .

For a D^{⊥} -valued (0, 2)-tensor P, using ⟨P, H⟩(X, Y) := g(P (X, Y), H) we obtain

div^{⊥} P = divP + ⟨P, H⟩ . (1.3)

2. Euler–Lagrange equations for variations of metric

The section contains the Euler–Lagrange equations of the variational principle δJ_{D,Ω} (g) = 0 on a relatively compact domain Ω of a manifold M endowed with a distribution D. We consider variations {g_{t}}_{|t|<ε} of metric g_{0} = g on M such that the induced infinitesimal variations, presented by a symmetric (0, 2)-tensor B_{t} ≡ ∂g_{t}/∂t, are supported in Ω ⊂ M. We adopt the notations ∂_{t} ≡ ∂/∂t, B ≡ ∂_{t}g_{t} _{| t=0} . For a (0, 2)-tensor C we have ⟨C, B⟩ = ⟨Sym(C), B⟩.

Given D, its g_{t}-orthogonal complement D_{t}^{⊥} depends on t; hence, the g_{t}-projecti-ons onto D (denoted by ^{⊤} ) and onto D_{0}^{⊥} = D^{⊥} (denoted by ^{⊥} ) also depend on t. If (g_{t}) preserve orthogonality of D and D^{⊥} then projections ^{⊤} and ^{⊥} do not depend on t, and we obtain adapted variations, see [2, 8]. Among adapted variations, those preserving metric on D will be called g^{⊥} -variations.

Recall [10] that ∇^{t} (the Levi-Civita connection of g_{t}) is evolved as

(2.1)  2g_{t}(∂_{t}(∇_{X}^{t} Y), Z) = (∇_{X}^{t} B)(Y, Z) + (∇_{Y}^{t}B)(X, Z)
− (∇_{Z}^{t} B)(X, Y), X, Y, Z ∈ 𝔛_{M} .

Using (2.1), one may choose a nice evolution of a local orthonormal frame.

Lemma 2.1. Let a local (D, D^{⊥} )-adapted and orthonormal for t = 0 frame {E_{a}(t), E_{i}(t)} be evolved by {g_{t}}_{|t|<ε} according to

∂_{t}E_{a} = −(1/2) B_{t}^{♯}(E_{a})^{⊤} , ∂_{t}E_{i} = −(1/2) B_{t}^{♯}(E_{i})^{⊥} − B_{t}^{♯}(E_{i})^{⊤} .

Then {E_{a}(t), E_{i}(t)} is a g_{t}-orthonormal frame and {E_{a}(t)} ⊂ D, {E_{i}(t)} ⊂ D_{t}^{⊥} .

Using (2.1), Lemma 2.1 and decomposition of tensor fields w.r.t. g = g^{⊤} + g^{⊥} , we obtain variation formulas (for detailed proof the reader is referred to [9]).

Proposition 2.1. For any variation g_{t} of metric on M we have at t = 0

∂_{t}⟨h^{⊥} , h^{⊥} ⟩= ⟨divh^{⊥} − 4Υ_{α^{⊥} ,θ} + K^{♭} − \frac{1}{2}Υ_{h^{⊥} ,h^{⊥}} , B⟩− div(⟨h^{⊥} , B⟩),
∂_{t}g_{t}(H^{⊥}, H^{⊥} ) = ⟨(divH^{⊥} )g^{⊥} + 4⟨θ, H^{⊥} ⟩ − H^{⊥♭} ⊗ H^{⊥♭} , B⟩ − div((Tr_{D^{⊥}} B^{♯})H^{⊥} ),
∂_{t}⟨h, h⟩= ⟨divh + K^{♭}− 2 divα − 2Υ_{α,α^{⊥} +θ^{⊥}} − \frac{1}{2}Υ_{h,h} , B⟩+ div(⟨2α − h, B⟩),
∂_{t}g_{t}(H, H) = ⟨2⟨θ^{⊥} − α^{⊥} , H⟩+ 2 Sym(H^{♭}⊗ H^{⊥♭} )

− 2δ_{H} + (divH)g^{⊤}− H^{♭} ⊗ H^{♭}, B⟩+ div(2(B^{♯}H)^{⊤} − (Tr_{D} B^{♯})H),

∂_{t}⟨T^{⊥}, T^{⊥} ⟩= ⟨2T^{⊥♭} + 2Υ_{θ^{⊥} ,θ−α} − 2 divθ^{⊥} + \frac{1}{2}Υ_{T^{⊥} ,T^{⊥}} , B⟩+ div(2⟨θ^{⊥}, B⟩),
∂_{t}⟨T, T ⟩= ⟨\frac{1}{2}Υ_{T,T} + 2T^{♭}, B⟩,

Remark 2.1. Formulas of Proposition 2.1 can be presented in three-component form, in which D×D-and D^{⊥} ×D^{⊥} -components are dual, i.e., of the same form with interchanged D and D^{⊥} . For example, for g^{⊤} -variations we obtain the following:

∂_{t}⟨h, h⟩= ⟨divh + K^{♭}, B⟩ − div(⟨h, B⟩),
∂_{t}g_{t}(H, H) = ⟨ (divH) g^{⊤} , B⟩ − div((Tr_{D} B^{♯})H),
∂_{t}⟨h^{⊥} , h^{⊥} ⟩= −⟨Υ_{h^{⊥} ,h^{⊥}} , B⟩/2,
∂_{t}g_{t}(H^{⊥} , H^{⊥} ) = −⟨H^{⊥♭} ⊗ H^{⊥♭} , B⟩,
∂_{t}⟨T, T ⟩= 2 ⟨ T^{♭}, B ⟩,
∂_{t}⟨T^{⊥}, T^{⊥}⟩= ⟨Υ_{T^{⊥} ,T^{⊥}} , B⟩/2.

Assume that the Lagrangian can be written as a sum of two terms L_{gr} + L, where L_{gr} = \frac{1}{2𝔞} (S_{mix} −2 Λ) is the gravity Lagrangian, see (0.1), and the matter Lagrangian L depends only on g and not on its derivatives. Set J : g ↦ ∫_{Ω} L(g) d vol_{g} on M. Variation of the metric g with B = ∂_{t}g_{t |t=0} produces the stress-energy tensor Θ such that

\frac{d}{dt} J(g_{t})_{| t=0} = \frac{1}{2} ∫_{Ω} ⟨Θ, B⟩ d vol_{g},

and which governs the interaction of the “gravitational field" with the given field representing the matter. Note that, see [1],

Θ_{μν} = −2 \frac{∂L}{∂g^{μν}} + g_{μν} L, ( 2 \frac{∂L}{∂g_{μν}} + g^{μν} L) B_{μν} = Θ^{μν} B_{μν} = ⟨Θ, B⟩.

The product T M × T M is the sum of three subbundles, D^{⊥} × D^{⊥} , D × D and

V := (D × D^{⊥} ) + (D^{⊥} × D).

Definition 2.1. The symmetric (0, 2)-tensor Ric_{D} defined by

Ric_{D | D^{⊥} ×D^{⊥}} = divh^{⊥} + K^{⊥♭} − 2T^{⊥♭} + H^{♭} ⊗ H^{♭} − \frac{1}{2}Υ_{h,h} − \frac{1}{2}Υ_{T,T} + η g^{⊥} ,
Ric_{D | V} = 2 δ_{H} + 2 div(θ^{⊥} − α) − 4⟨θ, H^{⊥} ⟩ − 2⟨θ^{⊥} − α^{⊥} , H⟩

− 4Υ_{α^{⊥} ,θ} − 2Υ_{α,α^{⊥}} − 2Υ_{θ^{⊥} ,θ} − 2 Sym(H^{♭} ⊗ H^{⊥♭} ), (2.2)

Ric_{D| D× D} = divh + K^{♭} − 2T^{♭} + H^{⊥♭} ⊗ H^{⊥♭} − \frac{1}{2}Υ_{h^{⊥} ,h^{⊥}} − \frac{1}{2}Υ_{T^{⊥} ,T^{⊥}} + μ g^{⊤} ,

where

(2.3)  η = −\frac{n − 1}{p + n − 2} div(H^{⊥} − H), μ = \frac{p − 1}{p + n − 2} div(H^{⊥} − H),

is referred to as the mixed Ricci curvature. The trace of Ric_{D} is then

(2.4)  Scal_{D} = S_{mix} + \frac{p − n}{n + p − 2} div(H^{⊥} − H).

Proposition 2.2. Euler–Lagrange equations for (0.1) are the same as for the action

(2.5)  Ĵ_{D,Ω} : g ↦ ∫_{Ω} \{ \frac{1}{2𝔞} (Scal_{D}(g) − 2 Λ) + L(g)\} d vol_{g} .

Proof. For any X_{t} supported on Ω and any g_{t} we have

∫_{Ω} divX_{t} d vol_{g_{t}} = ∫_{∂Ω} g(X_{t}, ν) dA_{g_{t}} ,

ν being outward-pointing normal unit vector field to ∂Ω. If ∂_{t}g and ∂_{t}X are supported in Ω then the rhs integral does not depend on t: \frac{d}{dt} ∫_{Ω} divX_{t} d vol_{g_{t}} = 0. Thus, for X_{t} = \frac{p−n}{n+p−2} (H^{⊥} − H) and by (2.4), if we replace S_{mix} by Scal_{D} in (0.1) then get the same Euler–Lagrange equations. □

Theorem 2.1 (Euler–Lagrange equations). A metric g on M with nondegenerate D is critical for (0.1), or equivalently, for (2.5), if and only if g is a solution to (0.2), where Ric_{D} is given by (2.2).

Proof. We derive for the gravitational part J_{mix ,D,Ω} : g ↦ ∫_{Ω} S_{mix} (g) d vol_{g} of (0.1). By (1.2),

\frac{d}{dt} J_{mix ,D,Ω} (g_{t}) = \frac{d}{dt} ∫_{Ω} Q(g_{t}) d vol_{g_{t}} ,

where Q(g) := S_{mix} (g) − div(H + H^{⊥} ) is given using (1.2) as

(2.6) Q(g) = g(H, H) − ⟨h, h⟩_{g} + g(H^{⊥} , H^{⊥} ) − ⟨h^{⊥} , h^{⊥} ⟩_{g} + ⟨T, T ⟩_{g} + ⟨T^{⊥}, T^{⊥}⟩_{g} .

Recall, see [10], that the volume form is evolved as

(2.7)  ∂_{t} (d vol_{g_{t}} )_{| t=0} = \frac{1}{2} (Tr_{g} B) d vol_{g},

and Tr_{g} B = ⟨g, B⟩, where B = {∂_{t}g_{t}}_{| t=0} . Applying Proposition 2.1 to (2.6), using (1.3), (2.7) and removing integrals of divergences of vector fields compactly supported in Ω, we obtain

\frac{d}{dt} J_{mix ,D,Ω} (g_{t})_{| t=0} =

= ∫_{Ω} (∂_{t}Q(g_{t})_{| t=0} + \frac{1}{2} (Tr_{g} B)(S_{mix} (g) − div(H + H^{⊥} ))) d vol_{g}

= ∫_{Ω} ⟨4 ⟨θ, H^{⊥} ⟩ − H^{⊥♭} ⊗ H^{⊥♭} − divh^{⊥} + 4Υ_{α^{⊥} ,θ} − K^{♭} + \frac{1}{2} Υ_{h^{⊥} ,h^{⊥}}

− H^{♭} ⊗ H^{♭} + 2 ⟨θ^{⊥} − α^{⊥} , H⟩ + 2 Sym(H^{♭} ⊗ H^{⊥♭} ) − 2 δ_{H} + 2 div(α − θ^{⊥} )

+ 2 Υ_{α,α^{⊥} +θ^{⊥}} + \frac{1}{2}Υ_{h,h} − divh − K^{♭} + 2 T^{⊥♭} + 2 Υ_{θ^{⊥} ,θ−α} + \frac{1}{2} Υ_{T^{⊥} ,T^{⊥}}

(2.8) + \frac{1}{2} Υ_{T,T} + T^{♭} + \frac{1}{2}S_{mix} · g + \frac{1}{2} div(H^{⊥} − H)( g^{⊥} − g^{⊤} ), B⟩d vol_{g} .

If g is critical for J_{mix ,D,Ω} w.r.t. all variations (g_{t}) then the integral in (2.8) is zero for arbitrary symmetric (0, 2)-tensor B. We can further decompose the resulting Euler–Lagrange equations into three independent parts: its V -, D×D- and D^{⊥}×D^{⊥} - components. In this way, for the action (0.1) we obtain (2.2). The D^{⊥} × D^{⊥} - component (2.2)_{3} is dual to the D × D-component (2.2)_{1}. Substituting (2.2) with arbitrary (μ, η) into (0.2), and comparing the result with (2.2), we find (μ, η) in (2.3) as solution of a linear system

(p − 2) η + n μ = div(H^{⊥} − H), (n − 2) μ + p η = − div(H^{⊥} − H).  □

Remark 2.2. Components (2.2)_{1} and (2.2)_{3} can be obtained using adapted variations of metric, see also [8]. Component (2.2)_{2} is not symmetric under the change D^{⊥} ↔ D, because S_{mix} (g_{t}) (when D is fixed) is different from the mixed scalar curvature for g_{t} when D^{⊥} is fixed while D_{t} varies. In order to organize better the div-terms in (2.2)_{2}, we introduce the symmetric (0, 2)-tensor:

Ric^{⊤} (X, Y) = ∑ _{a} ε_{a}(g(R_{E_{a} ,X^{⊤}} E_{a}, Y^{⊥} ) + g(R_{E_{a} ,Y^{⊤}} E_{a}, X^{⊥} )).

Using the Codazzi equation of [5, Theorem 2.4], V -component (2.2)_{2} can be written as

(2.9) Ric_{D| V} = Ric^{⊤} + 2 div(θ^{⊥} + θ) + 2⟨α − θ, H^{⊥} ⟩
+ 2⟨α^{⊥} − θ^{⊥}, H⟩ − 2 Sym(H^{♭} ⊗ H^{⊥♭} ) − 2Υ_{α,α^{⊥}} + 2Υ_{θ,θ^{⊥}} .

As an immediate consequence of (2.9), we get: if D and D^{⊥} are integrable then

Ric_{D | V} = Ric^{⊤} + 2⟨α, H^{⊥} ⟩ + 2⟨α^{⊥} , H⟩ − 2 Sym(H^{♭} ⊗ H^{⊥♭} ) − 2Υ_{α,α^{⊥}} .

Corollary 2.1. A metric g on M with non-degenerate D is critical for the action (0.1) w.r.t. all adapted variations of g if and only if g solves (0.2), where Ric_{D} is built from (2.2)_{1} and (2.2)_{3} (that is Ric_{D | V} = 0).

Next, we will examine (0.2) for generalized product metrics, having many applications. Let M = M_{1} ×M_{2} be a product of pseudo-Riemannian manifolds (M_{i} , g_{i}) (i = 1, 2), and let π_{i} : M → M_{i} and dπ_{i} : T M → T M_{i} be the canonical projections. Given positive twisting functions ϕ, ψ ∈ C^{∞} (M), a double-twisted product M_{1} ×_{(ϕ,ψ)} M_{2} is M_{1} × M_{2} with the metric ḡ = ϕ g^{⊤} + ψ g^{⊥} , where g^{⊤} = π_{1}^{∗}g_{1} and g^{⊥} = π_{2}^{∗}g_{2}. If either ϕ or ψ is constant, then we have a twisted product. The leaves M_{1} ×{y} (tangent to D) and the fibers {x}×M_{2} (tangent to D^{⊥} ) are totally umbilical in (M, ḡ) and this property characterizes double-twisted products, see [6]. Recall that Hess_{ψ} (X, Y) = ḡ( ¯∇_{X} ( ¯∇ ψ), Y) for all X, Y ∈ 𝔛_{M}.

The next proposition describes Ric_{D} for double-twisted products, and shows that the stress-tensor in (0.2) should be of a particular structure. The result can be also used for finding special solutions of the mixed Ricci equation and mixed Einstein metrics mentioned in the introduction.

Proposition 2.3. Let g = g^{⊤} +g^{⊥} be a direct product metric on M = M_{1}×M_{2}.

Then a double-twisted product metric ḡ = ψ g^{⊤} + ϕ g^{⊥} is critical for (0.1) if and only if ḡ solves (0.2) with

Ric_{D | D^{⊥} ×D^{⊥}} = (F_{1} + Λ) g^{⊥} + F_{2}(∇^{⊥} ϕ)^{♭} ⊗ (∇^{⊥} ϕ)^{♭},
(2.10)  Ric_{D | V} = F_{3}(∇^{⊤}ϕ)^{♭} ⊗ (∇^{⊥} ψ)^{♭} + F_{4} (Hess_{ψ} )_{|V} ,
Ric_{D | D× D} = (F_{5} + Λ) g^{⊤} + F_{6}(∇^{⊤}ψ)^{♭} ⊗ (∇^{⊤} ψ)^{♭},

where (F_{5}, F_{6}) are dual under the change (D^{⊥}, ψ) ↔ (D, ϕ) to (F_{1}, F_{2}), resp., and

F_{1} = \frac{n(n − 1)}{4} ψ^{−2} g(∇^{⊥} ψ, ∇^{⊥}ψ) + ( 1 − \frac{p}{4}) (p − 1)ψ^{−1} ϕ^{−1} g(∇^{⊤} ϕ, ∇^{⊤} ϕ)

− (p − 1)ψ^{−1} Δ^{⊤} ϕ + ( 1 − \frac{n}{2}) (p − 1)ψ^{−2} g(∇^{⊤} ψ, ∇^{⊤} ϕ),

F_{2} = −\frac{n(n − 1)}{2} ϕ^{−2} ψ^{−2} ,
F_{3} = −(n − 1)ψ^{−1} ( \frac{p − 2}{4} ϕ^{−1} − \frac{1}{2} ψ^{−1} ) , F_{4} = \frac{1 − n}{2} ψ^{−1} .

Note that constant ϕ, ψ are the only solutions of (2.10) with Λ = 0 = Θ.

3. The mixed Ricci tensor on spacetimes

Let n = 1. In this case, D is spanned by a nonsingular vector field N, i.e., the distribution tangent to one-dimensional foliation of (M^{p+1} , g) by N-curves. An example of such foliations is provided by a circle action S^{1} ×M → M without fixed points. Now, let g(N, N) = ε_{N} ∈ {−1, 1}, then g = g^{⊤} + g^{⊥} , where

(3.1)  g^{⊤} = ε_{N} N^{♭} ⊗ N^{♭}.

For physically relevant applications, one should take p = 3 and ε_{N} = −1, while g_{| D^{⊥}} > 0. Note that S_{mix} = ε_{N} Ric_{N,N} , and the action (0.1) reduces itself to

(3.2)  J_{N,Ω} : g ↦ ∫_{Ω} \{ \frac{1}{2𝔞} (ε_{N} Ric_{N,N} (g) − 2 Λ) + L(g)\} d vol_{g} .

For a foliation by N-curves we have T = 0, while ⟨T^{⊥}, T^{⊥}⟩ = ε_{N} ⟨T_{N}^{⊥♯} , T_{N}^{⊥♯} ⟩. It follows easily that h = H g^{⊤} and ⟨h, h⟩ = g(H, H), where H = ε_{N} ∇_{N} N is the curvature of N-lines, and

(3.3)  H^{⊥} = ε_{N} τ_{1}N, h^{⊥} = h_{sc}^{⊥}N, g(H^{⊥} , H^{⊥} ) = τ_{1}^{2}, ⟨h^{⊥} , h^{⊥} ⟩ = τ_{2},

where h_{sc}^{⊥} = ε_{N} ⟨h^{⊥} , N⟩ is the scalar second fundamental form of D^{⊥} and τ_{i} is the trace of the ith power of A_{N}^{⊥}. By (1.3), we obtain

(3.4)  divh^{⊥} = ∇_{N} h_{sc}^{⊥} − τ_{1}h_{sc}^{⊥}, divh = (divH) g^{⊤} .

Note that divN = −τ_{1} and div(τ_{1}N) = N(τ_{1}) − τ_{1}^{2}. Denote by R_{N} : X → R_{N,X} N the Jacobi operator of N. Recall the identities, see [7, 8],

(3.5) ϵ_{N} (R_{N} + (A_{N}^{⊥})^{2} + (T_{N}^{⊥♯} )^{2})^{♭} = N(h_{sc}^{⊥}) − H^{♭} ⊗ H^{♭} + Def_{D} H,

ϵ_{N} Ric_{N,N} = divH + ϵ_{N} (N(τ_{1}) − τ_{2}) + ⟨T^{⊥}, T^{⊥}⟩.

The D-deformation tensor of Z ∈ 𝔛_{M} is the symmetric part of ∇Z restricted to D:

2 (Def_{D} Z)(X, Y) = g(∇_{X} Z, Y) + g(∇_{Y}Z, X), (X, Y ∈ 𝔛_{D}).

Proposition 3.1. For D spanned by N, the symmetric tensor Ric_{D} is given by

Ric_{D | D^{⊥} ×D^{⊥}} = ∇_{N} h_{sc}^{⊥} − τ_{1}h_{sc}^{⊥} − ε_{N} (2 (T_{N}^{⊥♯} )^{2} + [T_{N}^{⊥♯} , A_{N}^{⊥}])^{♭},
(3.6)  Ric_{D}(· , N)_{| D^{⊥}} = div^{⊥} (T_{N}^{⊥♯} ) + 2 (T_{N}^{⊥♯} H)^{♭}
Ric_{D}(N, N) = ε_{N} (N(τ_{1}) − τ_{2}) − ⟨T^{⊥}, T^{⊥}⟩,

and its trace is

Scal_{D} = ε_{N} Ric_{N,N} + div(ε_{N} τ_{1}N − H).

Due to (3.5), an equivalent form of Ric_{D} just involving the curvature tensor is (3.7)

Ric_{D | D^{⊥} ×D^{⊥}} = ε_{N} (R_{N} + (A_{N}^{⊥})^{2} − (T_{N}^{⊥♯} )^{2} + [T_{N}^{⊥♯} , A_{N}^{⊥}])^{♭} + H^{♭} ⊗ H^{♭}

− τ_{1}h_{sc}^{⊥} − Def_{D} H − \frac{1}{2} (ε_{N} Ric_{N,N} + div(ε_{N} τ_{1}N − H)) g^{⊥} ,

Ric_{D}(· , N)_{| D^{⊥}} = div^{⊥} (T_{N}^{⊥♯} ) + 2 (T_{N}^{⊥♯} H)^{♭},
Ric_{D}(N, N) = ε_{N} Ric_{N,N} −4 ⟨T^{⊥}, T^{⊥}⟩ − div(ε_{N} τ_{1}N + H).

Proof. By (2.3), we have η = 0 and μ = ε_{N} (N(τ_{1})−τ_{1}^{2})−divH. Substituting values (3.3), (3.4) and

Υ_{h,h} = 2 H^{♭} ⊗ H^{♭}, T = ε_{N} (T_{N}^{⊥♯} )^{2}, Υ_{T,T} = 0

into (2.2)_{1} yields (3.6)_{1}. Substituting values (3.3), (3.4) and

h = H g^{⊤} , H^{⊥♭} ⊗ H^{⊥♭} = ε_{N} τ_{1}^{2} g^{⊤} , K = 0 = T ,
Υ_{h^{⊥} ,h^{⊥}} = 2 ε_{N} τ_{2} g^{⊤} , Υ_{T^{⊥} ,T^{⊥}} = 2 ⟨T^{⊥}, T^{⊥}⟩ g^{⊤}

into (2.2)_{3} yields (3.6)_{3}. Note also that θ = 0; hence,

2 (divα)(X, N) = g(∇_{N} H − τ_{1}H, X),
2 ⟨θ^{⊥} − α^{⊥} , H⟩(X, N) = −g(T_{N}^{⊥♯} (H) + A_{N}^{⊥} (H), X),
2 Sym(H^{♭} ⊗ H^{⊥♭} )(X, N) = g(τ_{1}H, X),
2 δ_{H} (X, N) = g(∇_{N} H, X),
2Υ_{α,α^{⊥}} (X, N) = g(A_{N}^{⊥} (H), X)

for all X orthogonal to N, and the V -component (2.2)_{2} is

(3.8)  Ric_{D | V} = divθ^{⊥} − ⟨θ^{⊥} , H⟩.

The lhs of (3.8) vanishes on all pairs of vectors except those from V. We have

2 divθ^{⊥} (X, N) = ∑ _{i} ε_{i} g((∇_{i} T_{N}^{⊥♯} )(X), E_{i}) + ε_{N} g(∇_{N} (T_{N}^{⊥♯} (X)), N)
= div^{⊥} (T_{N}^{⊥♯} )(X) + g(T_{N}^{⊥♯} (H), X)

for X ∈ 𝔛_{D^{⊥}} Hence, (3.8) is written as (3.6)_{2}. Using (3.5) in (3.6) yields (3.7). The above and (3.1) yield (0.2). □

Theorem 2.1 and Proposition 3.1 yield the following.

Corollary 3.1 (The mixed gravitational field equations). A metric g on M with D spanned by a unit vector field N is critical for (3.2) if and only if g is a solution of (0.2) with Ric_{D} given in (3.6). In this case, (0.2) splits into the system

(3.9)

∇_{N} h_{sc}^{⊥} − τ_{1}h_{sc}^{⊥} − ε_{N} (2(T_{N}^{⊥♯} )^{2} + [A_{N}^{⊥} , T_{N}^{⊥♯} ])^{♭}
+ ⟨T^{⊥}, T^{⊥}⟩ − \frac{1}{2} (ε_{N} (2 N(τ_{1}) − τ_{1}^{2} − τ_{2}) − 2Λ) g^{⊥} = 𝔞 Θ_{| D^{⊥} ×D^{⊥}} ,
div^{⊥} (T_{N}^{⊥♯} ) + 2 (T_{N}^{⊥♯} H)^{♭} = 𝔞 Θ_{▪ ,N | D^{⊥}} ,
\frac{1}{2} ε_{N} (τ_{1}^{2} − τ_{2}) − \frac{3}{2} ⟨T^{⊥}, T^{⊥}⟩ + Λ = 𝔞 Θ_{N,N} .

Example 3.1. Let a unit vector field N generates a geodesic Riemannian flow (i.e., h^{⊥} = 0 = h) on a pseudo-Riemannian manifold (M^{p+1} , g). In this case,

Ric_{D}(X, N) = − Ric_{X,N} .

As an example, consider the Hopf fibration S^{2m+1} → CP^{m}, for which N is tangent to the fibers {S^{1}}. We have T^{⊥}(X, Y) = g(JX, Y)N for the standard almost complex structure J on ℝ^{2m+2} and metric g > 0, and hence T_{N}^{⊥♯} (X) = J(X) and ⟨T^{⊥} , T^{⊥} ⟩ = 2m. It follows that (∇_{Z} T_{N}^{♯})(X) = (∇_{Z} J)(X) = 0, and (3.9)_{2} is valid with Θ_{| D^{⊥} ×D} = 0. Moreover, (3.9)_{1} and (3.9)_{3} are satisfied with

𝔞 Θ_{| D^{⊥} ×D^{⊥}} = (2 − m + Λ) g^{⊥} , 𝔞 Θ_{N,N} = Λ − 3.

The reader can find more examples in [8, 9], and for spacetimes, in [1].

Corollary 3.2. A metric g on M with non-degenerate D = Span(N) and integrable D^{⊥} is critical for (3.2) if and only if g solves (0.2), where the nonzero components of Ric_{D} are given by

(3.10)  Ric_{D |D^{⊥} ×D^{⊥}} = ∇_{N} h_{sc}^{⊥} − τ_{1}h_{sc}^{⊥}, Ric_{D}(N, N) = ε_{N} (N(τ_{1}) − τ_{2}).

In this case, (0.2) splits into the system

(3.11) \begin{aligned}∇_{N} h_{sc}^{⊥} − τ_{1}h_{sc}^{⊥} − \frac{1}{2} (ε_{N} (2 N(τ_{1}) − τ_{1}^{2} − τ_{2}) − 2Λ) g^{⊥} & = 𝔞 Θ_{| D^{⊥} ×D^{⊥}} , \\ \frac{1}{2} ε_{N} (τ_{1}^{2} − τ_{2}) + Λ & = 𝔞 Θ_{N,N} . \end{aligned}

Proof. Since (3.9)_{2} is valid for integrable D^{⊥} , in this case, variations g_{t} constant on D yield the same Euler–Lagrange equations as adapted g^{⊥} -variations. □

Action (3.2) has been studied in [1] for a globally hyperbolic spacetime (M^{4}, g), which is naturally foliated, see [3]. There, D = Span(N) with ε_{N} = −1, and D^{⊥} is integrable (i.e., T^{⊥} = 0). Although Euler–Lagrange equations for (3.2) coincide with (3.11), they are written in [1] in terms of a non-symmetric tensor Ric_{D}^{1}, which is different from (3.10):

Ric_{D}^{1}_{| D^{⊥} ×D^{⊥}} = ∇_{N} h_{sc}^{⊥} − τ_{1}h_{sc}^{⊥},
Ric_{D}^{1}(X, N) = − Ric_{D}^{1}(N, X) = div(A_{N}^{⊥} (X)), X ∈ 𝔛_{D^{⊥}} ,
Ric_{D}^{1}(N, N) = divH,

with the trace Scal_{D}^{1} = ε_{N} Ric_{N,N} −ε_{N} (τ_{1}^{2} − τ_{2}). Consequently, the mixed gravitational field equations given in [1] are slightly different from (0.2):

(3.12)  Ric_{D}^{1} −\frac{1}{2} Scal_{D}^{1} ·g + ε_{N} Ric_{N,N} ( \frac{1}{2} g − ε_{N} N^{♭} ⊗ N^{♭}) + Λ g = 𝔞 Θ.

Since we actually use the symmetric part of Ric_{D}^{1} in (3.12), its left hand side vanishes when evaluated on (X, N). Thus, comparing Ric_{D} with Ric_{D}^{1}, we find

Sym(Ric_{D}^{1}) − Ric_{D} = (divH − ε_{N} (N(τ_{1}) − τ_{2})) g^{⊤} ,
Scal_{D}^{1} − Scal_{D} = divH − ε_{N} (N(τ_{1}) − τ_{2}).

Certainly, (3.12) reduces to (3.11) when evaluated on D^{⊥} ×D^{⊥} and (N, N), respectively. This corresponds to the following values of (η, μ) in (2.2) when n = 1:

η = − div(H^{⊥} − H) + ε_{N} Ric_{N,N} , μ = div(H^{⊥} − H) − ε_{N} Ric_{N,N} .

Comparing (3.12) with (0.2) we conclude that (2.3) is the best choice of (η, μ).

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Department of Mathematics University of Haifa Haifa Israel

 vrovenski@univ.haifa.ac.il