Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and electromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from the metric are unified and extended to arbitrary dimensions, to include a cosmological constant, and to include any self-interaction potential. Necessary and sufficient conditions on a four-dimensional spacetime metric for it to be an electrovacuum and formulas for constructing the electromagnetic field from the metric are generalized to include a cosmological constant. Both null and non-null electromagnetic fields are treated. A number of examples and applications of these results are presented.
In the general theory of relativity it is often possible to eliminate the matter fields from the Einstein-matter field equations and express the equations as local geometric conditions on the spacetime metric alone. This possibility was discovered by Rainich,1 who showed how to eliminate the Maxwell field from the Einstein-Maxwell equations, arriving at the “Rainich conditions,” which give necessary and sufficient conditions on a spacetime metric for it to be a non-null electrovacuum. Rainich’s work was made prominent by Misner and Wheeler,2 who advanced the “geometrization” program in which all matter was to be modeled as a manifestation of spacetime geometry. Over the subsequent years, a variety of additional geometrization results have been found pertaining to electromagnetic fields, scalar fields, spinor fields, fluids, and so forth. See, e.g., Refs. 3–13. Results such as these provide, at least in principle, a new way to analyze field equations and their solutions from a purely geometric point of view, just involving the metric.
The geometrization conditions which have been obtained over the years, while conceptually elegant, are generally more complicated than the original Einstein-matter field equations. For example, the Einstein-Maxwell equations are a system of variational second-order partial differential equations, while the Rainich conditions involve a system of non-variational fourth-order equations. For this reason, one can understand why geometrization results have seen relatively little practical use in relativity and field theory. The current abilities of symbolic computational systems have, however, made the use of geometrization conditions viable for various applications. Indeed, the bulk of the non-null electrovacuum solutions presented in the treatise of Ref. 14 were verified using a symbolic computational implementation of the classical Rainich conditions. The geometrization examples for various null electrovacua found in Ref. 13 were obtained using the DifferentialGeometry package in Maple.15 The purpose of this paper is to compile a set of geometrization results for the Einstein field equations which involve the most commonly used matter fields and which are as comprehensive and as general as possible while at the same time in a form suitable for symbolic computational applications.
This last point requires some elaboration as it significantly constrains the type of geometric conditions which we shall deem suitable for our purposes. A suitable geometrization condition for our purposes will define an algorithm which takes as input a given spacetime metric and which determines, solely through algebraic and differentiation operations on the metric, whether the metric defines a solution of the Einstein equations with a given matter content. When the metric does define a solution, the matter fields shall be constructed directly from the metric via algebraic operations, differentiation, and integration.
We summarize our treatment for each of three types of matter fields and compare to existing results as follows.
A. Perfect fluids
We give necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations. Formulas for constructing the fluid from the metric are obtained. These results apply to spacetimes of any dimension greater than two and allow for a cosmological constant. No energy conditions or equations of state are imposed. Existing geometrization conditions for fluids can be found in Ref. 12, which generalize those found in four spacetime dimensions in Ref. 11. The conditions given in Refs. 12 and 11, while elegant, involve the existence of certain unspecified functions and so do not satisfy the computational criteria listed above. The results of Ref. 11 also include conditions which enforce equations of state, while the results of Ref. 12 enforce the dominant energy condition. Our conditions enforce neither of these since we are interested in geometrization conditions which characterize any type of fluid solutions. The geometrization conditions we obtain are built algebraically from the Einstein tensor and so involve up to two derivatives of the metric.
B. Scalar fields
Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from the metric have been obtained by Kuchař for free fields in four spacetime dimensions without a cosmological constant.4 These results apply to massless and massive fields. The results of Ref. 4 subsume related results in Refs. 3 and 6. More recently, conditions for a symmetric tensor to be algebraically that of a free massless scalar field in any dimension, without cosmological constant, have been given in Ref. 12. These conditions are necessary algebraically but are not sufficient for geometrization since additional differential conditions are required. Here we give necessary and sufficient conditions for a metric to define a solution of the Einstein-scalar field equations which generalize all these results. In particular, the results we obtain here hold in arbitrary dimensions, they allow for a cosmological constant, they allow for a mass, and they allow for a freely specifiable self-interaction potential. Null and non-null fields are treated. The geometrization conditions we have found for a scalar field necessarily involve both algebraic and differential conditions on the Einstein tensor; they involve up to three derivatives of the spacetime metric.
C. Electromagnetic fields
Necessary and sufficient conditions on a four-dimensional spacetime metric for it to be a non-null electrovacuum were given by Rainich1 and Misner and Wheeler.2 The null case has been investigated by Misner and Wheeler,2 Geroch,7 Bartrum,8 and Ludwig.9 Building upon Ludwig’s results, a set of geometrization conditions for null electrovacua has been given by one of us in Ref. 13 via the Newman-Penrose formalism. Here, we generalize both the non-null results of Rainich, Misner, and Wheeler and the null results of Ref. 13 to include a cosmological constant. While this represents a very modest generalization of the classical Rainich conditions in the non-null case, it represents a less obvious generalization in the null case. In any case, we hope there is some value in assembling all these results in one place and in a unified form which is amenable to computational algorithms. All these results hold in four dimensions. The geometrization conditions for non-null electromagnetic fields involve up to four derivatives of the metric, while the geometrization of null electromagnetic fields involves as many as five derivatives of the metric.
Besides proving various geometrization theorems, we provide a number of modest illustrations of the theorems which hopefully serve to clarify their structure and usage. All these illustrations were accomplished using the DifferentialGeometry package in Maple, amply demonstrating the amenability of our results to symbolic computation. Software implementation of our geometrization results, computational details of some of the applications presented here, along with additional applications, can be found at http://digitalcommons.usu.edu/dg/.
II. PERFECT FLUIDS
Let (M, g) be an n-dimensional spacetime, n > 2, with signature (− + + ⋯ + ). Let μ:M → R and p:M → R be functions on M. Let u be a unit timelike vector field on M, that is, gabuaub = − 1. The Einstein equations for a perfect fluid are
Here Rab is the Ricci tensor of gab, R = gabRab is the Ricci scalar, Λ is the cosmological constant, and with being Newton’s constant. We note that the cosmological and Newton constants can be absorbed into the definition of the fluid. With
the Einstein equations take the form
If there exist functions and and a timelike unit vector field u on M such that (2.3) holds, we say that (M, g) is a perfect fluid spacetime. Note that if in some open set then the spacetime is actually an Einstein space on . In what follows, when we speak of perfect fluid spacetimes we assume that at each point of the spacetime. Note also that we have not imposed any energy conditions, equations of state, or thermodynamic properties. These additional considerations are examined, for example, in Ref. 11.
In the following we will use the trace-free Ricci (or trace-free Einstein) tensor Sab,
We will also need the following elementary result.
The following theorem gives a simple set of Rainich-type conditions for a perfect fluid spacetime.
From the proof of this theorem we obtain a prescription for construction of the fluid variables from a metric satisfying conditions (1)–(3).
A. Example: A static, spherically symmetric perfect fluid
Here we use Theorem 1 to find fluid solutions. Consider the following simple ansatz for a class of static, spherically symmetric spacetimes:
Here, f is a function to be determined. Computation of the tensor field K and imposition of the quadratic condition Ka[bKc]d = 0 leads to a system of non-linear ordinary differential equations for f(r) which can be reduced to
This has the 1-parameter family of solutions
where λ and r are restricted by 1 + λr2 > 0 to give the metric Lorentz signature. With f(r) so determined, the scalar α and the tensor K are computed to be
from which it immediately follows that all three conditions of Theorem 1 are satisfied.
From Corollary 1 the energy density, , pressure , and 4-velocity u are given by
If desired, one can interpret this solution as admitting a cosmological constant and a stiff equation of state, μ = p = 1/(2qr2).
B. Example: A class of 5-dimensional cosmological fluid solutions
In this example, we use Theorem 1 to construct a class of cosmological perfect fluid solutions on a 5-dimensional spacetime (M, g) where M = R × Σ4 with Σ4 = R3 × R being homogeneous and anisotropic. We start from a 4-parameter family of metrics of the form
where r0, R0, b, and β are parameters to be determined. This metric defines a family of spatially flat 3+1 dimensional homogeneous and isotropic universes with (x, y, z) coordinates, each with an extra dimension w described with its own scale factor. Using Theorem 1 we select metrics from this set which solve the perfect fluid Einstein equations.
Condition (2) of Theorem 1 leads to a system of algebraic equations for b and β from which we have found three solutions,
Case (i) can be eliminated from consideration since
which violates condition (3) of Theorem 1. Cases (ii) and (iii) satisfy all the conditions of Theorem 1. Using Corollary 1 we obtain solutions of the Einstein equations as follows:
Case (ii) is anisotropic (except when ) and allows for any combination of expansion and contraction for the (x, y, z) and w dimensions. Case (iii) is isotropic in all four spatial dimensions. Both cases are singular as t → 0.
III. SCALAR FIELDS
Kuchař has given Rainich-type geometrization conditions for a minimally coupled, free scalar field in four dimensions without a cosmological constant.4 Here, we generalize his treatment to a real scalar field with any self-interaction, in any dimension, and including the possibility of a cosmological constant.
The Einstein-scalar field equations for a spacetime (M, g) with a minimally coupled real scalar field ψ, with self-interaction potential V(ψ), and with cosmological constant Λ are given by
where and we use a semicolon to denote the usual torsion-free, metric-compatible covariant derivative.
We distinguish two classes of solutions to the Einstein-scalar field equations. If a solution has ψ;aψ;a ≠ 0 everywhere we say that the solution is non-null. If the solution has ψ;aψ;a = 0 everywhere we say that the solution is null.
The Rainich-type conditions we shall obtain require the following extension of Proposition 1 (cf. Ref. 4).
We shall use the following notation:
The geometrization theorems we shall obtain will depend upon whether the self-interaction potential V(ψ) is present. We begin with a free, massless, non-null scalar field.
From the proof just given, it is clear the scalar field is determined from the metric by solving a system of quadratic equations followed by a simple integration.
We turn to the special case of free, massless, null solutions, that is, solutions in which gabψ;aψ;b = 0.
We now turn to geometrization conditions which describe the case with V(ψ) ≠ 0. We define . We always assume that V(ψ) has been specified such that there exists an inverse function W:R → R with , and .
Finally we consider the null case with a given self-interaction potential, invertible as before.
A. Example: A non-inheriting scalar field solution
so that, according to Corollary 2, the scalar field is given by
and the cosmological constant is given by . This solution (with λ = 0) was exhibited in Ref. 16. We remark that while the spacetime is static, the scalar field is clearly not static and so represents an example of a “non-inheriting” solution to the Einstein-scalar field equations. Non-inheriting solutions of the Einstein-Maxwell equations are well-known.14 Geometrization conditions, which depend solely upon the metric, treat inheriting and non-inheriting matter fields on the same footing.
We have been able to find an analogous family of non-inheriting solutions in 2+1 dimensions from an analysis of the geometrization conditions in Theorem 2. In coordinates (t, r, θ) the spacetime metric takes the form
where b is a constant. This metric yields A = Λ and
so that from Corollary 2 the scalar field is given by
B. Example: No-go results for spherically symmetric null scalar field solutions
We use Theorem 3 to show that there are no null solutions to the free, massless Einstein-scalar field equations if the spacetime is static and spherically symmetric, provided the spherical symmetry orbits are not null. We also show that there are no spherically symmetric null solutions with null spherical symmetry orbits. Both results hold with or without a cosmological constant. Since these results follow directly from the geometrization conditions, they apply whether or not the scalar field inherits the spacetime symmetries.
We first consider a static, spherically symmetric spacetime in which the spherical symmetry orbits are not null. We use coordinates chosen such that the metric takes the following form:
These conditions force the trace-free Ricci tensor to vanish, whence the scalar field vanishes and we have an Einstein space. Consequently, there are no non-trivial null solutions to the Einstein-scalar field equations in which the spacetime is static and spherically symmetric with non-null spherical symmetry orbits.
Next, we consider a spherically symmetric spacetime in which the spherical symmetry orbits are null. In this case, there exist coordinates (v, r, θ, ϕ) such that the metric takes the form
for some functions w ≠ 0 and u. Calculation of conditions (3.28) and (3.29) for metrics (3.62) reveals they are incompatible. Consequently there are no null solutions to the Einstein-scalar field equations in this case. Since (3.62) is not actually static, but merely spherically symmetric, this proves that there are no Einstein-free-scalar field null solutions for spacetimes which are spherically symmetric with null symmetry orbits.
C. Self-interacting scalar fields
Fonarev17 has found a 1-parameter family of non-null spherically symmetric solutions to the Einstein-scalar field equations with a potential energy function which is an exponential function of the scalar field. Here we verify these solutions directly from the metric using Theorem 4.
In coordinates (t, r, θ, ϕ) and with q = 1, the metric in Ref. 17 takes the form
where m > 0 is a free parameter,
and α and β parametrize the scalar field potential and cosmological constant via
The inverse of the potential function is given by
Calculating the tensor H in (3.40) yields
and it follows that (3.38) is satisfied. Therefore, from Theorem 4, the metric (3.63) does indeed define a scalar field solution with the potential (3.65). Using Corollary 4, the scalar field is calculated to be
in agreement with Fonarev.17
IV. ELECTROMAGNETIC FIELDS
Necessary and sufficient conditions for a metric to be a non-null electrovacuum were first given by Rainich1 and enhanced by Misner and Wheeler.2 Necessary and sufficient conditions for a metric to be a null electrovacuum have been obtained in Ref. 13. In each case, procedures for constructing the electromagnetic field from the metric have also been obtained. These results apply to the Einstein-Maxwell equations in four spacetime dimensions with no electromagnetic sources and with no cosmological constant. Here, we summarize all these results while generalizing them to include a cosmological constant.
The Einstein-Maxwell equations for a spacetime (M, g) with electromagnetic field Fab = F[ab] and cosmological constant Λ are given by
Here , ∇ is the torsion-free derivative determined by the metric, Rab and R are the Ricci tensor and Ricci scalar of the metric. We shall refer to (4.1) alone as the Einstein equations and we shall refer to (4.2) alone as the Maxwell equations.
If the two scalar invariants of the electromagnetic field vanish in some region,
we say that the electromagnetic field is null in that region. Otherwise, the electromagnetic field is non-null in that region. If ζab is a 2-form, the Hodge duality operation is given by
Conditions (4.5) generalize the classical algebraic Rainich conditions; they apply equally well for non-null and null electromagnetic fields. In the null case, they take a simpler form.
With the substitution of Sab for Rab, the proof is identical to that found in Ref. 2. □
From Proposition 3 and the general form of the electromagnetic 2-form in (4.6), the classical results of Rainich, Misner, and Wheeler generalize to the Einstein-Maxwell equations with a cosmological constant as follows.
with all other scalar products vanishing. We will use the Newman-Penrose formalism for this tetrad as defined, e.g., in Ref. 18. In particular, we recall the definitions of the following Newman-Penrose spin coefficients. The twist of the null congruence with tangent field ka is given by
The shear of the null congruence is defined by
The acceleration of the congruence is defined by
The quantities ω, , and are intrinsic properties of the null congruence determined by ka and are independent of the choice of adapted tetrad. The congruence is surface forming if and only if ω = 0. The congruence consists of geodesics precisely when κ = 0.
Finally, we denote by the basis of 1-forms dual to , α = 1, 2, 3, 4. The dual basis satisfies
Let (M, g) be a 4-dimensional spacetime. The following three conditions on g are necessary and sufficient for the existence of a null 2-form Fab such that (g, F) satisfy the Einstein-Maxwell equations (4.1) and (4.2).
We remark that condition (4.23) is one real condition depending upon as many as four derivatives of the metric. Condition (4.24) is complex; it represents two real conditions and depends upon as many as five derivatives of the metric. In either case, the conditions for an electrovacuum described in Theorem 7 do not depend on the choice of tetrad adapted to k.13
From the proof of Theorem 7, the null electromagnetic field is constructed from the metric as follows.
We remark that, just as in the case with vanishing cosmological constant, F is determined up to a duality rotation in the twisting case, and F involves an arbitrary function of one variable in the twist-free case.
A. Example: LBR solution
There is a class of solutions to the Einstein-Maxwell equations with a spacetime that is the product of two-dimensional constant curvature spaces, which is due to Levi-Civita,20 Bertotti,21 and Robinson;22 we call it the LBR solution. As an illustration of Theorem 6, we give a novel derivation of the LBR solution in its most general form using symmetry methods.
Consider the set of spherically symmetric spacetimes. As is well known (see, e.g., Ref. 14), the set of such spacetimes can be partitioned into three classes according to whether the “warp factor” has a spacelike, null, or vanishing gradient. We consider here the latter case in which the spacetime (M, g) is necessarily the product of two-dimensional geometries. We denote this product by M = L × S, where S has constant positive curvature and Riemannian signature (these are the spherical symmetry group orbits), and L has Lorentzian signature. Using standard spherical polar coordinates on S and null coordinates on L, the spacetime metric takes the following form:
Here r0 > 0 is a constant. We now consider the conditions of Theorem 6. We begin by imposing the condition that the scalar curvature is constant, R = 4Λ. This condition takes the form
The first term on the left-hand side represents the scalar curvature of L, so this condition forces L to have constant curvature, as might have been expected from the product form of the spacetime geometry. Thus, there are three types of solutions to this geometrization condition according to whether the constant curvature of L is positive, negative, or zero. Spacetimes satisfying (4.32) are symmetric spaces; the curvature tensor is covariantly constant.
Here is the electromagnetic field F−, constructed from the metric g− in (4.34) via Corollary 8,
where α ∈ [0, 2π). The fields (g−, F−) satisfy the Einstein-Maxwell equations with cosmological constant .
B. Example: A stationary null electrovacuum with cosmological constant
We use Theorem 7 to construct a new hypersurface homogeneous solution to the Einstein-Maxwell equations with a cosmological constant and with a null electromagnetic field. We begin by defining a class of metrics, parametrized by an arbitrary function f, which generalizes a homogeneous null electrovacuum with cosmological constant due to Siklos.23 In coordinates (u, v, y, z), our generalization is
where Λ < 0. This metric satisfies R = 4Λ. The trace-free Ricci tensor takes the pure radiation form
with a twist-free, shear-free null geodesic generator
From Theorem 7, Eq. (4.23), the metric defines a null electrovacuum if and only if f(y) satisfies
The Siklos solution has f(y) = y4. A new, non-trivial solution to this equation which leads to a metric satisfying all of the hypotheses of Theorem 7 in the twist-free case is given by
From Corollary 9, the null electromagnetic 2-form F which serves as the source for this spacetime is given by
where α(v) is an arbitrary function.
This null electrovacuum has Petrov type N and admits four Killing vector fields, (∂u, ∂v, ∂z, 2z∂u − v∂z), generating an isometry group with three-dimensional timelike orbits and null rotation isotropy.25 The electromagnetic field does not inherit all the spacetime symmetry; for example, F is not translationally invariant in the z coordinate.
The authors gratefully acknowledge fruitful discussions with Ian Anderson and Läg Avulin. This work was supported in part by Grant No. OCI-1148331 from the National Science Foundation.
Conditions (4.22) correspond to the Mariot-Robinson theorem,19 which states that the repeated principal null direction of a null electromagnetic field is necessarily tangent to a shear-free geodesic null congruence, and is also a repeated principal null direction of the algebraically special spacetime upon which the null electromagnetic field resides. This condition is necessary and sufficient (at least in the analytic setting) for the existence of a null solution to the Maxwell equations on a given spacetime, but is only a necessary condition for a solution to the Einstein-Maxwell equations with null electromagnetic field.
The Siklos solution, f(y) = y4, yields a spacetime which has five Killing vector fields generating a transitive isometry group.