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On stationary solutions to the non-vacuum Einstein field equations

We derive a local curvature estimate for four-dimensional stationary solutions to the inheriting Einstein-Maxwell-Klein-Gordon equations. In particular, it implies that any such stationary geodesically complete solution with vanishing Poynting vector and proper coupling constants (like dark energy) is flat. We also generalize the result to higher dimensions.


Introduction
The purpose of this paper is to generalize the results in [2] for vacuum Einstein field equations to non-vacuum Einstein field equations. In [2], we proved that any geodesically complete 4-d spacetime with a timelike Killing field satisfying the vacuum Einstein field equation with nonnegative cosmological constant is actually flat. This result generalized the previous results in [1] and [8] by removing the extra conditions (like chronological condition) imposed on their theorems.
Because the physical spacetime can never be flat, it must violate the assumptions in the theorem in [2], for instance, the geodesic completeness. Also, by the singularity theorem of Penrose and Hawking (see [5] Chapter 8), the geodesic completeness of the physical spacetime is also hardly valid. Moreover, we know that many stationary spacetimes contain black holes, see [3] [10]. In [2], in order to prove the above result, we actually derived a local curvature estimate (see Theorem 1.3 in [2]), which does not require the spacetime to be geodesically complete. So from this point of view, the local curvature estimate in [2] seems to be more important.
In this paper, we are mainly concerned with the Einstein field equations in the presence of two kinds of matters or fields, the electromagnetic fields and scalar fields. For the former case, the interaction of the gravity (a 4-d spacetime (M, g M )) and the electromagnetic field (a 2-form F ) is described by a system of differential equations called Einstein-Maxwell equation: Rg αβ + Λg αβ = κ(F αγ F βδ g γδ − 1 4 |F | 2 g αβ ) dF = d * F = 0, (1.1) where Λ, κ are constants, |F | 2 F ξγ F ηδ g ξη g γδ . The second equation in (1.1) satisfied by the 2-form F αβ (electromagnetic field) is called the Maxwell equation. A scalar field, in physics, associates a value (∈ R or C) to each point of the spacetime, like temperature, pressure etc. In this paper, we consider the Klein-Gordon scalar fields, which can be used to describe the π 0 , or π + , π − mesons. The Einstein equation coupled with Klein-Gordon equation is the following where Λ, κ ′ , m (mass), (reduced planck constant) are constants, φ : M → R or R 2 is a map (scalar field). The (uncharged) Einstein-Maxwell-Klein-Gordon equation is a combination of (1.1) and (1.2) (see [5] Chapter 3): (1. 3) It is easy to see that (1.3) reduces to (1.1) or (1.2) when φ = 0 or F = 0 in (1. 3). So instead of dealing with (1.1) and (1.2) separately, we can handle only one equation (1. 3).
Let X be a timelike Killing field on M , by which we say the spacetime (M, g M ) is stationary. Following [12] [13], we say the solution (g M , F, φ) in (1.3) is inheriting if F and φ satisfy L X (F ) = 0, dφ(X) = 0, (1. 4) which means that F and φ inherit the X-symmetry from the metric tensor g M . Denote the electric and magnetic fields (related to X) by respectively, where i X F is the tensor contraction of F with X. Let X * be the 1form on M obtained from X by lowering indices. We associate a Riemannian metric tensorĝ on M to g M (see [2]): (1.6) The first main result of the paper is the following local curvature estimate: (1.7) Then for any 0 < δ < 1, there is a constant C δ > 0 depending only on δ such that (1.8) The Hodge dual (on three-space) of the 2-form E ∧ B in (1.7) is a 1-form corresponding to the Poynting vector in physics. A corollary of Theorem 1.1 is the following Theorem 1.2 Let (M, g M ) be a geodesically complete 4-d spacetime with a timelike Killing field X such that g M satisfies the inheriting Einstein-Maxwell-Klein-Gordon We give a possible physical explanation why Theorem 1.1 or Theorem 1.2 hold for Λ ≥ 0, κ ≤ 0, κ ′ ≥ 0 (but not the contrary). It is well-known that positive cosmological constant Λ and negative energy (κ < 0, κ ′ > 0) will cause the acceleration of the universe. Intuitively, the acceleration will reduce the density of the matter or energy distribution of the universe, hence make the spacetime more "regular".
Remark 1.1 The spacetime being static implies that the Poynting vector vanishes, i.e. E ∧ B = 0.
We say a stationary spacetime is static if the orthogonal complement of the timelike Killing field is an integrable distribution. In general, when dim ≥ 5, the "curvature" measuring the difference between stationaryness and staticity is difficult to control. In order to generalize Theorems 1.1 and 1.2 to dim ≥ 5, we can assume for simplicity that the spacetime is static. In this case, we can obtain analogous results for general dimensions, see Theorems 3.1, 3.2.
The paper is organized as follows. In section 2, we prove Theorems 1.1 and 1.2. In section 3, we generalize the results to general dimensions.
Acknowledgement The author is grateful to Professors S. T. Yau, X. P. Zhu and Dr. J. B. Li for helpful discussions. The work was partially supported by grants NSFC11521101, 11025107.

4-d Einstein-Maxwell-Klein-Gordon equations 2.1 Preliminaries
In this section, we recall some useful formulas for stationary spacetimes, their computations could be found in [2], see also [1] [7]. Let {x α } be a local coordinate system on a 4-d stationary spacetime (M,ḡ, X) so that the metric tensorḡ = g αβ dx α dx β takes the following form:ḡ = −u 2 (dt + θ) 2 + g (2.1) where u = (− X, X ) 1 2 , θ = θ i dx i and g = g ij dx i dx j are t = x 0 -independent. Here we useḡ to denote the spacetime metic g M . We use the convention that for tensor indices, the Greek letters α, β, · · · are running from 0 to 3, the Latin letters i, j, · · · from 1 to 3. Let e 0 = ∂ ∂t , e i = ∂ ∂x i − θ i ∂ ∂t be a local tangent frame satisfying e i , e j = g ij and e 0 , e j = 0. Fix the local orientation of M (on this coordinate chart) given by the frame (e 1 , e 2 , e 3 , e 0 ). We use * 4 and * to denote the Hodge dual operators (with respect to the above orientation and meticsḡ and g) on spaces of dimensions 4 and 3 respectively.
We remark that the coordinate system {x α } and the local tangent frame {e α } will be used frequently through out of the paper. It is clear that their constructions are irrelevant to the dimensions.

(2.4)
For later use, theĝ-hessian of any time-independent smooth function f on the spacetime can be computed to be (see [2], (2.11)): In particular, we have△

Field equations
From L X F = 0, one can derive L X * 4 F = 0 since X is a Killing field. Denote the electric and magnetic fields related to X by which roughly say that E and B are spatial 1-forms on M . Let F = ω 1 (x)∧u(dt+θ)+ω 2 (x) be a local expression of the electromagnetic field F in the coordinate system {x α } in Section 2.1, where ω 1 (x) and ω 2 (x) are spatial forms of degrees 1 and 2. Since Maxwell's equation dF = d * 4 F = 0 is equivalent to a system of differential equations on the electric and magnetic fields E and B: Now we compute the components of the Ricci curvatureRic in terms of the frame {e α } in Section 2.1. From (2.7), we havē (2.10) Since i e j F = −E(e j )(dt + θ) + i e j ω 2 and

A Bochner formula
In this section, we assume that the condition always holds. When condition (2.20) holds, from the last 2 equations of (2.2), we know ω = dψ holds locally for some function ψ and From (2.14), the condition (2.20) is equivalent to i.e., the Poynting vector vanishes. Here we assume E = B = 0 if κ = 0.
Lemma 2.1 Let g H = y −2 1 dy 2 1 + c 2 y −l 2 1 dy 2 2 + · · · + c m y −lm 1 dy 2 m , c a > 0 be a Riemannian metric on m-dimensional upper half space H = {(y 1 , y 2 , · · · , y m ) : y i ∈ R, y 1 > 0}. In terms of the natural frame { ∂ ∂y a }, the only nonzero components of the Christoffel symbols Γ a bc of g H are: Up to the symmetries of curvature tensors, the only nonzero components of the curvature tensor of g H are where a, b = 1 and a = b.
Suppose ω, E = 0 at some point P , by (2.22), we have B = λE holds for some function λ near P . Using i) in Lemma 2.2 and the contracted 2nd Bianchi identity, Together with ii) in Lemma 2.2, we have This gives 2λ ω, E E = 0, which implies λ = 0. Hence B ≡ 0 near P , from the last equation in (2.19), we know ω, E ≡ 0 near P , which is a contradiction. If ω, B = 0, a contradiction can be derived by the same argument. ✷ Now we can apply the standard Bochner formula(for a map not necessarily harmonic, see [4]): By using (2.25), Lemma 2.3 and the following formula, one can get (2.33) Now we compute the term |∇ αβ Φ| 2 at a fixed point P = (x ′ , t ′ ). By definition, Let {x i } be a spatial normal coordinate system around the fixed point x ′ . Let Combining (2.28) and (2.5), we get (2.36) Similarly, one can compute (2.37) To compute the fourth term on the right hand side of (2.31), we need (2.40) To compute the third term on the right hand side of (2.31), we rewrite (see (2.11)) (2.41) Note that E ∧ B = 0 holds by condition (2.20). Combininĝ The most notable feature in formula (2.45) is that each term in the right hand side of (2.45) is nonnegative if Λ ≥ 0, κ ≤ 0 and κ ′ ≥ 0.

Proof of Theorems 1.1 and 1.2.
Proof. of Theorem 1.1. The idea of the proof is analogous to Theorem 5.3 in [2]. We assume ∂B(x 0 , a) = φ.
For any fixed 0 < δ < 1, we have two cases, So we may assume case 2) always holds, and we will show that f (x) ≤ C δ for some constant C δ depending only on δ. Note that (2.47) We will argue by contradiction. Suppose there are a sequence of 4-Lorentzian manifolds (M l ,ḡ l ) satisfying the equations (1.3) (1.4), and a sequence ofĝ l -balls (2.48) Scaling u l andḡ l by u l (x l ) −1 and h l (x l ) respectively, one can assume u l (x l ) = 1 and h l (x l ) = 1. We still use the same notations u l , ω l , E l , B l , h l ,ḡ l , etc. to denote the corresponding scaled quantities. For any fixed 0 < ǫ < 1, any x ∈B(x 0 , a l ) with If there is no ambiguity, we can also omit the subscript l from h l , u l , etc. It can be shown from (2.49): h(x) ≤ 4 onB(x, D).
By differentiating the last equation of (2.58), we get are uniformally bounded and m 2 2 ≤ δ −1 , κ ′ m 2 2 |φ| 2 ≤ 8, by L pestimate for elliptic equations, one can show the C 1,α -norms of √ κ ′ φ i are uniformally bounded. So the C 3,α -norms ofg l are uniformally bounded (using the harmonic coordinates {z ı } in (2.16)). By repeating the above arguments, one can show that for any k ∈ Z + , the C k,α -norms of log u l , ω l , . Note that the condition (1.7) imply that the limit must satisfy ( |κ|E) ∞ ∧( |κ|B) ∞ = 0. Hence equation (2.45) holds for the limit. The smooth convergence of h l and (2.49) imply that h ∞ (x) will achieve its maximum (= 1) at the origin z i = 0. Since each term in the right hand side of (2.45) is nonnegative, we can apply the strong maximum principle for the limit. This implies that h ∞ ≡ const. = 1 and each term on the right hand side of (2.45) vanishes. From |∇ log u ∞ | 4 = 0, we know The first equation in (2.58) for the limit will give a contradiction: In conclusion, we have proved (2.60) From the above proof (on the regularity of log u), supB (x 0 , 5a 8 ) |u −1 ∇ 2 u| + u −2 |∇ω| can also be bounded by Cδ m 2 2 + CC δ a −2 . Now combining (2.10) in [2] and formula (2.2), we know supB (x 0 , a 2 ) |Rm|ĝ ≤ Cδ m 2 2 + CC δ a −2 . The estimate (1.8) follows by redefining the constants.
If ∂B(x 0 , a) is empty, by using strong maximum principle on equation (2.45), we conclude that Rm ≡ 0, which fulfills the estimate (1.8).

Higher dimensional static fields
In this section, we generalize our estimate to general dimensions. We may assume the scalar fields can take their values on some manifold. Let (M, g M ) be a spacetime of dimension n+1, (W, g W ) a Riemannian manifold equipped with a Riemannian metric g W , V : W → R be a fixed function. We consider the (uncharged) Einstein-Maxwell-Klein-Gordon equation, which is the following system of differential equations on the Lorentzian metric g M , a 2-form F on M , and a map (scalar field) φ : M → W : where Λ, κ, κ ′ are constants, △ g M ,g W φ is the harmonic map Laplacian of φ.
ii) the magnetic field B = i X * F vanishes; iii) (W, g W ) has nonpositive sectional curvature; iv) the function V : W → R is nonnegative and convex, i.e., Hess(V ) ≥ 0.
LetB(x 0 , a) be aĝ−metric ball centered at x 0 of radius a > 0 with compact closure in M . Then there is a universal constant C > 0 such that

3)
where E = i X F is the electric field.
Note that the function f (x) is invariant under the scaling of the metric. Scaling u and the metricḡ by u(x) −1 and h(x), we may assume u(x) = 1 and h(x) = 1. It is not hard to prove (see (2.49) (2.50)): h(x) ≤ 4 onB(x, D),   since L(x) ≥ L(x) = 1. Hence D ≤ C √ n for some universal constant C. From (3.19), we get the desired estimate f (x) ≤ Cn.
If ∂B(x 0 , a) is empty, we can apply the strong maximum principle on equation (3.17) directly to conclude h ≡ 0. ✷ A corollary of Theorem 3.1 is the following theorem, which is a generalization of Theorem 1.2 in [2]. Theorem 3.2 Let (M, g M ) be a static spacetime of dimension n + 1 satisfying the assumptions in Theorem 3.1. If we assume (M, g M ) is geodesically complete, then the universal cover of (M, g M ) is isometric to a product R × N equipped with a product metric −dt 2 + g N , where g N is a complete Ricci flat Riemannian manifold.