Invariant Construction of Cosmological Models Erik Jörgenfelt

Home , Dust solution, Fluid solution, Ricci decomposition

Spring 2016 Thesis, 15hp Bachelor of physics, 180hp Department of physics

Abstract Solutions to the Einstein field equations can be constructed from invariant objects. If a solution is found it is locally equivalent to all solutions constructed from the same set. Here the method is applied to spatially-homogeneous solutions with pre-determined algebras, under the assumption that the invariant frame field of standard vectors is orthogonal on the orbits. Solutions with orthogonal fluid flow of Bianchi type III are investigated and two classes of orthogonal dust (zero pressure) solutions are found, as well as one vacuum energy solution. Tilted dust solutions with non-vanishing vorticity of Bianchi types I–III are also investigated, and it is shown that no such solution exist given the assumptions.

Sammanfattning Lösningar till Einsteins fältekvationer kan konstrueras från invarianta objekt. Om en lösning hittas är den lokalt ekvivalent till alla andra lösningar konstruerade från samma mängd. Här appliceras metoden på spatialt homogena lösningar med förbestämda algebror, under anta- gandet att den invarianta ramen med standard-vektorer är ortogonal på banorna. Lösningar med ortogonalt ﬂuidﬂöde av Bianchi typ III undersöks och två klasser av dammlösningar (noll tryck) hittas, liksom en vakum-energilösning. Dammlösningar med lut och noll-skild vorticitet av Bianchi typ I–III undersöks också, och det visas att inga sådana lösningar existerar givet antagandena.

Contents

1 Introduction 1

2 Preliminaries 3 2.1 Smooth Manifolds...... 3 2.2 The Tangent and Co-tangent Bundles...... 4 2.3 Tensor Fields and Diﬀerential Forms...... 6 2.4 Lie Groups...... 13

3 Spacetime 15 3.1 Metric...... 15 3.2 Connection and Curvature...... 16 3.3 The Cartan Equations...... 20 3.4 Stress-Energy Tensor and Fluid Dynamics...... 22 3.5 Spatially-Homogeneous Solutions...... 26

4 Invariant Construction of New Solutions 31

5 Orthogonal Bianchi Type III 35 5.1 Dust Solutions...... 36 5.2 Vacuum Energy Solutions...... 39

6 Dust Solutions with non-Vanishing Vorticity 41 6.1 Bianchi II...... 42 6.2 Bianchi III...... 43

7 Conclusions 47

8 Appendix 49

1 Introduction

Study of the equivalence problem in general relativity has lead to the development of a method to construct solutions to the Einstein field equations in terms of invariant objects[14]. Integrability conditions were found in [5], and in [6] the method was extended to determining the isometry group and full line element, along with application in a fixed frame. In this essay that method is applied to find orthogonal Bianchi type III solutions, whence we briefly show, in subsection 3.5, how the method can be applied to a pre-determined spatially- homogeneous isometry group algebra. This is done through choosing a preferred co-moving Lorentz frame, dependent on the tilt. The Bianchi types classify all possible simply invariant three-dimensional isometry groups into nine types, Bianchi I–IX, based on their algebra. The preferred frame determines all Ricci rotation coefficients in terms of the tilt and the normalization factors of invariant orbit frame fields, and the derivatives of both, along with the group algebra. With the method we find two classes of dust (zero pressure) solutions, and one vacuum energy solution that is found to be a special foliation of de Sitter and anti-de Sitter space. Observation of the redshift from distant objects indicates an accelerating expansion of the universe and the large scale isotropy of cosmic microwave background radiation is indicative of large scale homogeneity[10], although the latter result has been called into question[20]. There have also been reports of results indicative of large-scale anisotropy in the current state of the universe[21]. Such observations, among others produce observational evidence to compare the physical properties of any cosmological model against, and indicate deviations from the current standard model of cosmology[10]. The computer algebra programs CLASSI by Jan Åman[1] and EXTERIOR by Michael Bradley[3] are used in the application of the theory. Solution of the differential equations that arise is aided by the program REDUCE. This essay is structured as follows. In section2 we cover preliminary results in differential geometry, and introduce some necessary concepts from Lie group theory. In section3 we cover the mathematical structure of spacetime, some fluid dynamics, and some fundamental results concerning spatially-homogeneous models. In section4 we review the method of construction of solutions from invariant objects in a fixed frame, and in particular give the necessary and sufficient conditions for a set to produce a geometry. In section5 we apply the method to orthogonal Bianchi type III solutions, and in section6 we apply it to attempt to find dust solutions with non-vanishing vorticity. If the standard invariant frame fields can be assumed to be orthogonal on the orbits, it is shown that no such solution exists in Bianchi types I – III, and the method could be further applied to Bianchi types IV – IX, although the complexity of the equations is expected to rise for higher Bianchi types. These results also hold for constant, non-zero pressure. The mathematical framework laid out here is based on [11], and more stringent formalism can be found there, although we prove proposition 2.5, which is left without proof in [11]. Some inspiration is also taken from [15]. We will use units such that the speed of light c = 1, and such that 8πG = 1, where G is the gravitational constant. We shall follow the Landau-Lifshitz timelike convention, and let indices i, j, k . . . denote frame fields on the tangent and co-tangent bundle unless otherwise noted. For coordinate frames we will use µ, ν, σ, . . ..

2 Preliminaries

2.1 Smooth Manifolds Smooth manifolds are the fundamental mathematical object behind the general relativistic description of gravity. Although smooth manifolds are special topological manifolds we will make no further mention of this fact. Instead we will deﬁne smooth manifolds directly via the existence of an atlas of smooth charts. In this approach focus shall be placed on achieving a workable model of spacetime and much of the mathematics behind the formalism will be glanced over. Deﬁnition 2.1. Let M be a set. Then a chart of M is a bijection of an open subset U ⊂ M n n m onto an open subset of some R . Two charts x : U1 → V1 ⊂ R and y : U2 → V2 ⊂ R are said −1 to be compatible if either x ◦ y is a smooth bijection or if U1 ∩ U2 = ∅. In this manner a chart x : U → V provides a corresponding set of coordinates for the subset U, namely xi := ri ◦ x, where ri denotes the i:th canonical coordinate function on Rn. The chart may be used to “chart” its domain mathematically, but obviously the coordinates themselves carry no deeper meaning. This is an important realization in general relativity, and one that must not be forgotten.

Definition 2.2. An atlas of a set M together with a collection of charts (Uα, xα)α∈A of M such that S (i) i Ui = M. (ii) All charts are compatible. Here A can be any set. Thus an atlas of M provides each region of M with at least one set of coordinates, and in any overlap there exists a smooth coordinate transformation between the coordinates. Now note that chart compatibility is an equivalence relation. A chart is said to be compatible with an atlas if it is compatible with all charts in the atlas, or equivalently with any chart in the atlas. In this manner we can extend the compatibility relation to hold between atlases: Definition 2.3. Two atlases are said to be compatible if their union is also an atlas. An equivalence class of atlases of some set M is called a smooth structure on M. In other words, if any chart from the first atlas is compatible with any chart from the second atlas, they belong to the same differentiable structure. Each atlas thus gives rise to a unique differentiable structure. A smooth manifold will consist of a set M together with a smooth structure on M, but since the union of all atlases in such a structure is also an atlas, and is uniquely determined by the original atlas, we may use this maximal atlas interchangeably with the differentiable structure. Definition 2.4. A smooth manifold is a set M and a smooth structure on M such that the topology induced by the structure is Hausdorff1 and second countable2. If the charts are in Rn then we say that the dimension of the manifold, dim(M) = n, or simply that M is a smooth n-manifold. We will not explicitly refer to the differentiable structure, but be satisfied with that there is such a structure chosen. Instead we may refer to a chart as admissible if it belongs to some atlas in the differentiable structure.

1 See below for the deﬁnition. 2 See below for the deﬁnition.

3 Our definition of smooth manifolds also includes some topological considerations, seen above. Although strictly not necessary, and often omitted, the following result does provide an easy way to ensure that a desired model fulfills the necessary axioms. Indeed, with this result in hand it is trivial to show the necessary topological properties in nearly any actual model of spacetime. It also provides any reader unfamiliar with point-set topology with an accessible description of the topological requirements. The open sets of the induced topology are precisely those sets that are the domain of some admissible chart. Formally, the induced topology is Hausdorff if for any two points x and y there are open sets U and V such that x ∈ U, y ∈ V , and U ∩ V = ∅. The topology is second countable if there is a countable collection of open sets {Ui} such that any open subset can i∈N be written as a union of some sub-collection {Ui}i∈A where A ⊂ N. Proposition 2.5. Let M be a set with a smooth structure given by an atlas A. Then

(i) If for every two distinct points p, q ∈ M, we have that either p and q are respectively in disjoint chart domains Uα and Uβ from the atlas, or they are both in a common chart domain, then the induced topology is Hausdorﬀ.

(ii) If A is countable, or there is an atlas contained in A that is, then the topology induced is second countable.

Proof. We start by proving (i). If p, q ∈ M are distinct points such that they are contained in respectively disjoint chart domains Uα and Uβ then these very domains are the necessary neighbourhoods. Suppose instead that they are contained in the domain of a common chart (U, x). Then x(p) and x(q) are distinct points in some Rn, whence there are open sets X(p) ∈ O, −1 −1 x(q) ∈ B such that O ∩ B = ∅. Take V = x (O) and W = x (B). Clearly (V, x |V ) and (W, x |W ) are compatible with (U, x) and thus all charts in A, whence they are charts in some atlas in the diﬀerentiable structure. Then V and W are the necessary neighbourhoods. To prove (ii) we let A = (Ui, xi) be the countable atlas, and suppose O ⊂ M is open. i∈N Obviously O is covered by {Ui} . If every Ui with O ∩ Ui 6= is contained in O we are done, i∈N ∅ so suppose there is some Uj such that O ∩ Uj 6= ∅ and Uj 6⊂ O. Then O ∩ Uj is open, whence n n xj(O ∩ Uj) is open in R . Since R is second countable it follows that xj(O ∩ Uj) is the union of a countable family of open balls B (i ∈ ). If we denote V = x−1(B ), then (V , x | ) i N j(i) j i j(i) j Vj(i) is a compatible chart for all i, and by adjoining these charts to A we get a new countable atlas that provides a countable cover for O where Uj has been replaced by sets Vj(i), which are all contained in O. This process can be repeated until O is the union of chart domains of some countable atlas.

In general relativistic cosmology one almost always works within the domain of a single chart. This is because one is primarily concerned with local effects, which can be taken to mean exactly that. A less mathematical description would be effects hat affect a sufficiently small region of spacetime. Of course, the extent of a sufficiently small region depends on the situation.

2.2 The Tangent and Co-tangent Bundles The formalism of the tangent bundle of a manifold requires tedious mathematics, but the principle is easy to understand: each point in spacetime can be endowed with a vector. The vector is allowed to point in any direction along spacetime. Thus if a table surface is spacetime the vector is not allowed to point into the surface or out of the surface, but along the surface in any direction. Furthermore, it can be given any (real) magnitude. This is exactly how we work with vectors in Newtonian mechanics.

4 Here, the formalism of tangent vectors as differential operators on functions is defined, and a qualitative motivation is given. Then the nature of vector fields is described, and co-vector fields (or differential 1-forms) is given a similar treatment. For more stringent formalism regarding these matters see e.g. [11].

Definition 2.6. The tangent space of a point p ∈ M is the vector space TpM spanned by the set ∂ ∂ ∂ , ,..., , ∂x1 p ∂x2 p ∂xn p where vector addition is taken to be addition of differential operators and scalar multiplication is taken to be scalar multiplication of differential operators. A tangent vector is an element of TpM. It is not difficult to realize that the tangent space at a point contains all directional derivatives at the point. In fact, all derivatives (defined by requiring that they obey the Leibniz law) are directional derivatives, and so contained in the tangent space. If we consider a tangent vector intuitively as the velocity of some curve passing through the point, the derivative that represents the vector mathematically is simply the directional derivative along that curve. That tangent vectors are derivatives should therefore not be difficult to accept. On the odd occasions that we refer explicitly to the vectors above, we will use ∂µ to denote ∂ . ∂xµ A vector field assigns to each point in its domain a single tangent vector, and it does so in a sufficiently smooth manner. Mathematically, the tangent spaces of all points combine to form a smooth manifold referred to as the tangent bundle, TM. A vector field may then be considered a smooth map from the base manifold to the tangent bundle, where smoothness is defined via composition with charts. Since there will be no reason to attempt to consider non-smooth vector fields, the precise mathematics is omitted. The space of vector fields is denoted X(M). In Rn vectors and co-vectors are rarely treated separately. Fortunately co-vectors are not difficult to grasp, but requires the concept of the dual of a vector space: Definition 2.7. The dual space of a vector space V is the set of all linear maps ω : V → R. ∗ We denote it by V , and call its elements dual vectors. If (e1, . . . , en) is a basis of V , and 1 n ∗ i i 1 n ω , . . . , ω are elements of V such that ω (ej) = δj, then ω , . . . , ω is called the dual basis of (e1, . . . , en). The co-tangent space at a point is simply the dual of the tangent space, and the co-tangent spaces of all points combine to form the co-tangent bundle, T ∗M. A co-vector field, defined analogously to vector fields, is often called a differential 1-form (or simply a 1-form). The reason for this shall become clear in Subsection 2.3. In Rn we are not used to distinguishing tangent vectors from co-vectors, since both are intrin- sically identified with elements of the vector space Rn. However, while tangent vectors appear naturally as velocity vectors, co-vectors appear naturally as normal vectors to hypersurfaces. To see this consider a hypersurface Σ ⊂ M defined locally by f(x1, . . . , xn) = 0. Then df (Definition 2.8) is a 1-form such that any vector v tangent to Σ at p fulfills df(v) = 0. Definition 2.8. Let M be a smooth manifold and let p ∈ M. For any smooth function f on M we define the differential of f to be the 1-form df : TM → R given by df(v) = v(f), for all v ∈ TM.

5 From definition 2.8 it becomes immediately clear that the set n o dx1,..., dxn provides each point in its domain with a dual basis to the canonical (coordinate) basis of the tangent space, see definition 2.6. We now extend the notion of basis to fields. Note that an ex- tension of definition 2.9 applies in general to what is called vector bundles, which not surprisingly includes the co-vector bundle, but also the tensor bundles (subsection 2.3). Definition 2.9. Let M be a smooth n-manifold. Then a local frame field of a subset U ⊂ M is a set of vector fields {e1, . . . , en} such that any vector field X can be written as a linear combination of the ei:s in U: i X |U = X ei, for smooth functions Xi, which are referred to as the components of X (relative to the local frame field). From here on out we will often use Xi to denote the vector field X when the local frame field i is implied. Similarly if ω is the dual basis of ei at each point, then we will often use ηi to refer i i to the 1-form η = ηiω , and may alternatively refer to ω as a local frame field or the dual frame field of ei. The local frame fields provided by the coordinate functions are referred to as a coordinate frames or holonomic frame fields. However, there are potentially many local frame fields that cannot be given as a coordinate frame (even accounting for the degree of freedom in choosing coordinates). Such a local frame field is referred to as non-holonomic. Any frame field is referred to as co-moving if it may be considered attached to a preferred observer.

2.3 Tensor Fields and Differential Forms Tensor fields are highly prevalent in general relativity, and in semi-Riemannian geometry in general. There is no easily accessible concrete geometric picture of what a tensor field represents, because tensor fields are so generalized, but we shall discuss a specific subclass of tensor fields called differential forms that do have a concrete geometric representation, which is very enlightening, and carry fundamental information about the underlying manifold. Definition 2.10. A tensor on a real valued vector space V is a multilinear map

T : V1 × ... × Vr × Vr+1 × ... × Vr+s → R, ∗ ∗ where all Vi are either copies of V or V , and such that there are r copies of V and s copies of V . We say that t is contravariant of dimension r and covariant of dimension s, and write r T ∈ Ts (V ). r Obviously, to each point p ∈ M, we may assign a tensor t ∈ Ts (TpM). A sufficiently smooth r assignment of tensors this way produces a tensor field, and for each (s) we have several corresponding tensor bundles. Several because technically each tensor field need the same ordering of copies of V and V ∗. Now, crucially, any linear map V → R can be represented by a covector, while any linear map V ∗ can be represented by a vector. Therefore a local frame field of tensors can be given from any local frame field of the tangent bundle, and its dual. We may write r r Ts (M) for the space of (s) tensor fields on M. A concrete example will be the most enlightening: suppose T : TM × TM ∗ × TM → C∞(M)

6 is a tensor ﬁeld on M. Then

j i k T = Ti kω ⊗ ej ⊗ ω .

j for some smooth functions Ti k, which we again refer to as the components of the tensor field. We will, perhaps somewhat confusingly, often use the components of tensor fields in general to refer to the tensor field, but will continue to use bold font when we refer to the tensor field without using the component, in order to reduce initial confusion. The reader should realize that the tensor product,⊗, simply “stitches” two tensors together. In terms of components, if j i T = Ti and Q = Q j we write

j k T ⊗ Q = Ti Q `, and it becomes immediately obvious that the tensor product is associative. The contraction of a tensor field is performed by summing over all frame field elements in one contravariant argument and one covariant argument. In component notation we simply write i Ti and sum as is required by the Einstein summation convention. We can also contract a tensor j k product and write e.g. Ti Q j. Finally then we note that passing an argument to a tensor field is equivalent to contracting the tensor product:

j i T (X, η) ≡ Ti X ηj.

A tensor, say Qijk`, is symmetric in the arguments, say i, j, and k, if

Qijk` = Qσiσj σk` for all permutations σ = σi, σj, σk of (i, j, k). It is completely symmetric if it is symmetric in all of its arguments. Conversely, it is anti-symmetric or skew-symmetric in the arguments if

Qijk` = sgn(σ)Qσiσj σk` for all permutations, and completely anti-symmetric if it is anti-symmetric in all of its arguments. Here sgn σ = 1 if σ is an even permutation, and sgn σ = −1 if σ is an odd permutation. Note that a tensor obviously can be symmetric respectively anti-symmetric in both covariant and contravariant arguments separately, but never mixed. We introduce the following shorthand notation: 1 X Q ≡ Q , (ijk)` 3! σiσj σk` σ∈S3 1 X Q ≡ sgn(σ)Q , [ijk]` 3! σiσj σk` σ∈S3 and respectively for more or fewer arguments, and where S3 is the symmetric group of 3 letters. Thus Q[ijk]` is anti-symmetric in i,j, and k, and Q[ijk]` = Qijk` if Q is already anti-symmetric in the arguments. We call completely anti-symmetric k-dimensional covariant tensor fields differential k-forms, and denote the space of k-forms by Ωk(M), or Ω(M) for the space of all differential forms. We take 0-forms to be smooth functions on M, and note that 1-forms are co-vector fields. Differential forms in general are, however, more difficult to understand intuitively than co- vector fields are. With the results of the rest of the subsection in hand it might be difficult to convince ourselves that differential k-forms can be thought of as some sort of measurement of an object determined by an ordered set of k vectors. Such an object corresponds to an oriented

7 k-plane element in the same way that a vector corresponds to a line element via the velocity vector. Further understanding is best gained from the Grassman algebra, which we do not exhibit here for brevity. The interested reader is instead referred to e.g. [11]. However, to connect with the way we explained that 1-forms naturally appear (as normal vectors to hypersurfaces) we will present an incomplete picture. We will also use the exterior product (Definition 2.11), but present it here because its use in understanding differential forms. Suppose that a k-form can be written as the exterior product of k 1-forms (such a k-form is said to be decomposable). Each 1- form determines a hypersurface locally at each point, but this says nothing of its magnitude. Let each 1-form determine a family of (identical) hypersurfaces, and let the “distance” between two hypersurfaces in the same family be inversely proportional to the magnitude of the 1-form, such that the 1-form “measures” how many hypersurfaces a vector, or alternatively a line element, pierces (where a vector tangential to the hypersurface is thus intuitively taken to pierce none). A decomposable 2-form similarly determines a family of “tubes”, and measures how many tubes an oriented 2-surface element cover. A decomposable k-form can be taken to determine a family of ‘cells” (limited in k dimensions), and measure how many such cells an oriented k-plane element fill. The orientation allows for positive or negative values, depending on how it matches the order of multiplication of the 1-forms, or correspondingly the orientation of the cells. Each k-form can be written as a sum of decomposable k-forms. The exterior product between differential forms is an anti-symmetric tensor product. In- vestigation of its properties is revealing about the structure of forms and we shall prove some revealing results. The proofs in this section follows [11] closely, and the next immediate goal is proposition 2.15.

Deﬁnition 2.11. Let M be a smooth manifold. Given ω ∈ Ωk(M) and η ∈ Ω`(M) we deﬁne the exterior product ω ∧ η ∈ Ωk+`(M) by

(k + `)! ω ∧ η := ω η . k! `! [i1...ik j1...j`]

Proposition 2.12. Let α ∈ Ωk(M), β ∈ Ω`(M), and γ ∈ Ωm(M). Then

(i) ∧ :Ωk(M) × Ω`(M) → Ωk+`(M) is R-bilinear; (ii) α ∧ β = (−1)k`β ∧ α;

(iii) α ∧ (β ∧ γ) = (α ∧ β) ∧ γ.

Proof. We start by proving i). For c, c0 ∈ R and β0 ∈ Ω`(M) we have, by the deﬁnition

0 0 k + `! 0 0 α ∧ cβ + c β = α[i ...i cβ + c β k! `! 1 k j1...j`] 1 X = sgn (σ) α cβ + c0β0 σi1 ...σik σ ...σ k! `! j1 j` σ∈Sk+` c X = sgn (σ) ασ ...σ βσ ...σ + k! `! i1 ik j1 j` σ∈Sk+` 0 c X 0 + sgn (σ) ασi ...σi βσ ...σ k! `! 1 k j1 j` σ∈Sk+` = c (α ∧ β) + c0 α ∧ β0

8 Clearly the symmetric argument goes through unedited for the ﬁrst term in the product, which gives us i). To prove ii) we let τ ∈ Sk+` be such that τ(1, . . . , k + `) = (k + 1, . . . , k + `, 1, . . . , l). Suppose ` ≥ k; the converse argument for k ≥ ` is nearly identical. Then

(k + `)! α ∧ β = α β k! `! [i1...ik j1...j`] 1 X = sgn (σ) ασ ...σ βσ ...σ k! `! i1 ik j1 j` σ∈Sk+` 1 X = sgn (σ) αστ ...στ βστ ...στ ...στ k! `! j`−k+1 j` i1 ik j`−k σ∈Sk+` 1 X = sgn (τ) sgn (στ) βστ ...στ ...στ αστ ...στ k! `! i1 ik j`−k j`−k+1 j` σ∈Sk+`

To see this recall that sgn(στ) = sgn(σ) sgn(τ) and that sgn(τ) sgn(τ) = 1 for all σ, τ ∈ Sk. We thus have

α ∧ β = sgn(τ)β ∧ α, since if σ varies over all of Sk+` then so does στ. Now, since τ obviously contains k` inversions, sgn(τ) = (−1)k`. This gives us ii). Finally, for iii) we simply compute

(k + ` + m)! (` + m)! α ∧ (β ∧ γ) = α β γ k!(` + m)! [i1...ik `! m! [j1...jk n1...nm]] (k + ` + m)! = α β γ . k! `! m! [i1...ik j1...j` n1...nm] Symmetric computation gives us

(k + ` + m)! (α ∧ β) ∧ γ = α β γ , k! `! m! [i1...ik j1...j` n1...nm] and by the associativity of the tensor product, we have iii).

We note here that combining properties i) and iii) of the proposition yields k-linearity in R of a wedge product of k terms. The following easily proved result is both suggestive and useful in providing us with coordinate local frame ﬁelds for diﬀerential forms.

1 k 1 Lemma 2.13. Let α ,..., α be elements of Ω (M), and let v1,..., vk be elements of TM. Then

1 k α ∧ · · · ∧ α (v1,..., vk) = det A,

h i i i i where A = aj is the k × k matrix whose ij:th entry is aj = α (vj).

Proof. From proposition 2.12, we have

1 k 1 k α ∧ · · · ∧ α = k! α [i1 ⊗ · · · ⊗ α ik],

9 or,

1 k X 1 k α ∧ · · · ∧ α (v1, . . . , vk) = sgn(σ)α (vσ1 ) ··· α (vσk )

σ∈Sk = det A, where A is as above.

For Corollary 2.14, which we will use to find coordinate local frame fields for differential k-forms from those for 1-forms, it is convenient to first introduce the Levi-Civita symbol:  1 if j , . . . , j is an even permutation of i , . . . , i ,  1 k 1 k i1,...,ik = j1,...,jk −1 if j1, . . . , jk is an odd permutation of i1, . . . , ik,  0 otherwise.

1 n Corollary 2.14. Let (e1, . . . , en) be a local frame ﬁeld for TM, and let (ω , . . . , ω ) be its dual frame for T ∗M. Then

ωi1 ∧ · · · ∧ ωik (e ,..., e ) = i1...ik . j1 jk j1...jk

Proof. Suppose there is some j` ∈/ {i1, . . . , ik}. Then the matrix A from lemma 2.13 will contain a zero column and det A = 0. Suppose instead that j1, . . . , jk is an even permutation of i1, . . . , ik. Then A will be the matrix resulting from an even number of row inversions of the identity matrix. Since each inversion adds a factor of −1 to the determinant and the identity matrix has a determinant of 1, clearly det A = 1. Similarly, if j1, . . . , jk is an odd permutation of i1, . . . , ik, then A is the result of an odd number of row inversions. Thus det A = −1.

Let us now introduce multi-index notation

I = (i1, i2, . . . , ik)

I i1 ik and write eI for (ei1 ,..., eik ) and ω for ω ∧ · · · ∧ ω , where e1,..., en as usual is a local frame ﬁeld of TM, and ω1,..., ωn is its dual frame. Thus we could, with some abuse of notation, I I write ω (eJ ) = J , where I and J are taken to be multi-indices of the same length, or similarly I i α (vJ ) = det α (vj) . Additionally we let an arrow over the multi-index indiciate a strictly increasing sequence. We let Ik denote the set of multi-indices of length k, in case the length of the multi-index beneﬁts from being explicitly stated.

Proposition 2.15. If (ω1,..., ωn) is a local frame ﬁeld for T ∗M = Ω1(M), then the set

n I~ ko ω : I~ ∈ I ,

k k n k is a local frame ﬁeld for Ω (M). Thus dim Ω (M) = k , and in particular dim Ω (M) = 0 for k > n.

Proof. To prove linear independence suppose that

I~ 0 = cI~ω .

10 Applying both sides to eJ~, we get, by Corollary 2.14

I~ 0 = cI~ω (eJ~) = c I~ I~ J~

= cJ~. where the last equality follows from the fact that the only permutation of a strictly increasing sequence that is also a strictly increasing sequence is the identity permutation (an even permu- n ~ o tation). Since this holds for any J~, it holds for all J~, and ωI : I~ ∈ Ik is linearly independent. n ~ o To prove that ωI : I~ ∈ Ik spans Ωk(M), we ﬁrst note that if two alternating k-linear maps agree on eI~, then by k-linearity and the alternating property (both from proposition 2.12), they k I~ are equal. Now let α ∈ Ω (TM) and β = α(eI~)ω . Then

I~ β(eJ~) = α(eI~)ω (eI~) = α(e )I~ I~ J~

= α(eJ~).

k Thus α and β agree on all eJ~, so α = β, and since α was arbitrary, all elements of Ω (M) can be written as a linear combination of ωI~.

Before moving on to the topic of spacetime we will extend the diﬀerential (deﬁnition 2.8) uniquely to a map, called the exterior derivative, d: Ω(M) → Ω(M) such that

d: Ωk(M) → Ωk+1(M) and such that

(i) d is R-linear; (ii) d(α ∧ β) = dα ∧ β + (−1)k α ∧ dβ for all α ∈ Ωk(M) and all β ∈ Ω(M);

(iii) d commutes with restrictions;

(iv) d◦ d≡ 0.

With proposition 2.12 in hand, the first three demands should seem natural for something we would like to call a derivative. The last demand on the list, however, may seem arbitrary. It is not as arbitrary as it may seem, though. Start with the pictorial view of differential forms presented earlier, where a k-form on an n-manifold represents a grid of k-cells extending in (n − k) dimensions. Say we want to know if any of these k-cells end somewhere in any of the (n − k) dimensions, and furthermore would like to place a new hypersurface there, turning them into (k + 1)-cells, thus going from a k-form to a (k + 1)-form. Now, there is a topological fact that states that the (oriented) boundary of an (oriented) boundary is zero; it is not difficult to convince yourself of this fact in lower dimensions by considering simple objects (see Figure1), and for manifolds it is trivially true (see e.g. [11]). So we would expect that applying the same reasoning again should result in zero. This view also matches the differential, if we allow the “density of boundaries” of a scalar field (0-form), or equivalently a smooth function, to be equated with the density of its levels

11 , 1 M -cells (rep- 1) − n ( ) we have locally 6= 0 k ( ) M

( k Ω . . ~ I ∈ ω x ~ I d α to represent the boundary (ignoring for the x ∂M ∧ The boundary of the cube is made up by six d Z ~ I ~ I will not pass through the boundary, since each α α = 12 ∂M = ω M d := d α M α Z d , and we define ~ I , then we should expect that this number equals the number of α M must pass straight through the manifold, thus being counted twice with M can be viewed as a statement of a clear connection between the exterior The boundary of a boundary is zero. d = 0 ) that end inside ◦ ω d -cells that pass through the boundary, i.e. the right hand side. Because any cell that 1) For a proof that the demands we made defines a unique map see e.g. [11]. We will instead Thus − Figure 1: squares, and the boundaryedge of is the counted boundary twice, is in made opposite up directions, of so the summing edges gives of zero. each square. However, each Namely those that are traditionally called Gauss’ theorem, Stokes’ theorem, and Green’s theorem n Then we can intuitively view the left hand side as counting the number of The map thus defined hasready the to desired move properties, on. as the interested reader may verify, and we are resenting ( does not end inside opposite signs anda summed to cell zero. entirely contained Via in orientation, the this interior view of also accounts for the fact that end is counted,smooth and manifolds, again which extends with and opposite subsumes several signs. familiar theorems The from mathematics vector analysis, behind Stokes’ theorem on for smooth functions (0-forms) 1 and write is slightly more complicated, but we have justfocus found on the the essence application of ofidentical the it. to exterior the derivative. differential For (definition 0-forms the2.8), exterior and derivative for is of course surfaces, i.e. itsthe rate exterior of derivative. change. Having formed this picture, we can glimpsemoment the that importance our of current definition of smooth manifolds does not allow for a boundary) of derivative and boundaries. Indeed, if we use 2.4 Lie Groups Lie groups are abstract groups that are also smooth manifolds. There is extensive theory on the subject, and Lie groups are important in many areas of physics. We shall be brief, and provide only a cursory introduction. Our primary interest lies in the use of Lie groups as isometry groups. Isometry groups represent actions under which the metric is invariant. This may include rotations, translations, and boosts. Most of these considerations will have to be postponed until after the metric has been introduced. We shall cover these topics at the end of section3, but introduce some necessary concepts here. We shall also introduce the Lie algebra of the tangent bundle. Definition 2.16. An absract group G is a Lie group if it is also a smooth manifold, and if the multiplication map and inverse map are both smooth. We can let a Lie group act on the base manifold by defining a map ` : G × M → M such that for all g1, g2 ∈ G and p ∈ M we have (i) ` g2, ` (g1, p) = ` (g2g1, p), (ii) ` (e, p) = p, where e ∈ G is the identity element. Since the map, which we refer to as an action, is often to be implicitly understood we may use g · p to denote ` (g, p). The set G · p = {g · p : g ∈ G} is called the orbit of the point p, and we note that the orbits are equivalence classes and so partition M. We say that G acts transitively on M if for any p, q ∈ M there is a g ∈ G such that g ·p = q, that is to say all points in M belong to the same orbit. Furthermore we say that the group is simply transitive if the dimension of the orbits is equal to the dimension of the group. If N ⊂ M is a manifold such that G · N ⊂ N then we say that N is a homogeneous space. For any point p ∈ M we can define the isotropy group Gp = {g ∈ G : g · p = p}. The smoothness of a Lie group gives rise to a Lie algebra, defined in general as any vector space a equipped with a bilinear map a × a → a, which we call the corresponding Lie bracket and denote by (u, v) 7→ [u, v], such that

[u, v] = − [v, u] and such that it obeys the Jacobi identity:

u, [v, w] + w, [u, v] + v, [w, u] = 0.

Although the way it does so is non-trivial to exhibit, and shall be omitted for brevity, it is important to realise that elements of the (Lie) algebra of a Lie group can be considered inﬁnitesimal group elements. Another important, but non-trivial, result is that if the Lie group is connected then representations of the algebra are equivalent to representations of the group. Intuitively we can consider a map that takes us “along” the algebra elements to the rest of the group. There is in fact such a map, called the exponential map, but we shall not dwell more on the subject. Now, for any Lie algebra with a basis e1, . . . , en we can additionally deﬁne the associated k structure constants c ij by

k ei, ej = c ijek,

k k which obviously statisfy c ij = −c ji. To simplify understanding of the subject note that for all linear algebras, which include all groups of interest to us, the bracket is deﬁned as the commutator: ei, ej := eiej − ejei.

13 The above reasoning should have made clear that the group action of any connected Lie group can be equivalently described by the action of its algebra, providing a smooth application of the group action, through the exponential map. This allows us to remain in the neighbourhood of the point p which is acted upon, and an arbitrary point p is carried into a point q such that

µ µ µ xq = xp + εvp ,

µ where v = v ∂µ is a vector representing a line element, and ε is taken to be some very small num- ν ber; in the inﬁnitesimal case we let ε → 0. The Jacobian, J µ, of this coordinate transformation µ and its inverse, K ν , are given by

ν ν ν µ µ µ 2 J µ = δ µ + εv ,µ,K ν = δ ν − εv ,ν + O(ε ),

µ µ where we used comma in index notation to denote partial diﬀerentiation, so v ,ν ≡ ∂ν v . Since we are interested ultimately in the inﬁnitesimal case we take the terms of order ε2 or higher to be identically zero. Now, tensor components are transformed by the Jacobian and its inverse:

µ µ µ σ τ µ T ν |q = Te ν |p + εv ,σTe ν |p − εv ,ν Te τ |p , where Te is to be taken to be T before the group action. If we specialize this transformation to a vector u the last term vanishes and we ﬁnd

µ µ µ σ u − u = u | ν ν − u | ν − εv ,σu , e xp +εvp xp p so that

u − ue µ ν µ ν £vu := lim = u ,ν v − v ,ν u ε→0 ε = v ◦ u − u ◦ v.

We call it the Lie derivative, and it effectively allows us to describe how vectors change infinites- imally. However, the commutator expression that we end up with can also readily be verified to be a Lie bracket on the tangent bundle. Indeed, the tangent bundle is a Lie algebra with the bracket thus defined if we take scalars to be smooth functions on M, and we may alternatively denote £vu by [v, u]. We will refer to the structure constants associated with some local frame i field as structure coefficients. Similarly, we can of course take, for T = T j

i k k i i k £vT := T j,kv − T jv ,k + T kv ,j, but note that there is no natural way to define a bracket in this case. If £vT = 0 the change in T can be seen as a pure coordinate transformation. The Lie derivative of a function is defined as the ordinary vector derivative introduced earlier, and in this manner straightforward calculation verifies that for a tensor T and vectors or covectors u, and v, we have £ξ T (u, v) = £ξT (u, v) + T £ξu, v + T u, £ξv , i.e. the Lie derivative is natural with respect to contraction. This can be viewed as the Lie derivative obeying the Leibniz rule if we view contraction as a multiplication map.

14 3 Spacetime

General relativity models spacetime as a smooth manifold, but endows it with additional structure. One of these is the metric tensor, which provides a way to measure distance and angles; another is the connection, which allows us to measure infinitesimal changes in directions; a third is the stress-energy tensor, which fills spacetime with content. We shall dwell briefly on these matters before moving on to some of the implications. We shall also consider some purely physical demands on our smooth manifold. These limit the stress-energy tensor to describe content that is physically plausible. Along these line, physical interpretation of certain mathematical quantities will also be presented. Finally, we present the Einstein field equations — the mathematical laws that govern general relativity — and cover some fundamental results concerning spatially-homogeneous solutions.

3.1 Metric A metric tensor is a purely symmetric two-dimensional covariant tensor (field), g, that is non- i j degenerate. Meaning that for some point p ∈ M if giju v = 0 for all v ∈ TpM then necessarily u = 0 at p. From now on we may refer to subspaces of TM, which should be taken to mean subspaces of the TpM in the domain. We may also often drop the “field” term, for brevity. If g is a non-degenerate metric on M, then upon choosing an orthonormal frame, e1,..., en, of TM we must have either gii = 1 or gii = −1, which gives us a list of +1’s and −1’s. Permuting the vector fields so that all +1’s come first, we get a list (1,..., 1, −1,..., −1) that we refer to as the signature of tangent bundle. Lorentzian manifolds are semi-Riemannian manifolds of a specific signature:

Definition 3.1. Let M be a smooth manifold and let g ∈ T2(M). If g is symmetric, nondegen- erate, and of constant signature on M, we call g a semi-Riemannian metric on M, and we say that (M, g) is a semi-Riemannian manifold. If (the dimension of M is ≥ 2 and) the signature of the metric is (1, −1, −1, −1,...) we also say that (M, g) is a Lorentzian manifold. The proper time between two events along an observer world line, L(t), with velocity vector ui is defined by Z q i j ∆τ = giju u dt. L Spacetime in general relativity is modeled as a four-dimensional Lorentzian manifold. Vectors, i j i j u such that giju u > 0 are called timelike, while vectors, v, such that gijv v < 0 are called i j spacelike. Finally vectors, n, such that gijn n = 0 are called null or light-like. Observers with mass have timelike velocity vectors representing that the proper time increases along the observer’s world line . Mass-less observers, such as light, have null velocity vectors because the proper time remains constant along the world line. A spacelike velocity vector would correspond to an observer moving faster than the speed of light, and the proper time along its world line would be imaginary. Obviously this is generally regarded as unphysical. We will always take timelike and null vector fields to be future-pointing (causal) for obvious reasons. Type changing, i.e. index raising and lowering, is mathematically defined through so called musical isomorphisms between the tangent bundle and the co-tangent bundle. However, it can be introduced simply as contraction with the metric tensor or its inverse. Thus, given a manifold ij ik i endowed with a metric gij define the tensor g by requiring g gkj = δ j. Note the use of the same kernel letter for the two tensors. Indeed we will consider both tensors as different manifestations of the metric tensor.

15 j Now note that gijv is itself a co-vector, which we may denote by vi, following the same ij i reasoning as above. Similarly g αi is a vector, which we may denote by α . Since by the ik j i j i deﬁnition we also have g gkjv = δ jv = v we observe that there is no risk of confusion i i ij between whether v stands for the vector v or the type changed co-vector v = g vj. In other words, the maps thus deﬁned are the previously mentioned musical isomorphisms. Finally for a general tensor, T we have

k1...kj−1kj kj+1...kr k1...kj−1 kj+1...kr gikj T `1...`s = T i `1...`s .

Before moving on to the connection, note that we may refer to a frame on TM as rigid if the components of the metric are constants. Such frames are in general non-holonomic, because if there is a rigid coordinate frame then there is a coordinate frame in which the metric is orthonormal. We shall see later that this corresponds to ﬂat space. A rigid frame may be classiﬁed as e.g. Lorentz if

1 0 0 0  0 −1 0 0  gij =   . 0 0 −1 0  0 0 0 −1

The reader should also be aware of the fundamental nature played by the metric in general relativity. Indeed, one often speciﬁes the model used by writing out the line element in coordinate frame, i.e.

2 i j ds = gij dx dx , from which the connection and the curvature may be found. This is because the connection in general relativity is deﬁned by the metric, as we shall see. Note also that the fact that a coordinate frame gives us all this information should not be taken to be indicative of any fundamental nature of the coordinates. Rather it is an indication of the opposite.

3.2 Connection and Curvature The connection on a smooth manifold is a mathematically rich topic, but general relativity limits us to a uniquely defined connection, called the Levi-Civita connection. We shall therefore limit ourselves to covering the absolute basics and the application of the covariant derivative, which is equivalent to the connection. We shall also describe how this gives a description of the curvature of a manifold, and briefly mention the conditions behind the Levi-Civita connection. To begin with a connection is something that exists on a vector bundle (mentioned earlier), and provides a structure to lift curves from the base manifold onto the bundle. This view is not important for us, although it is to some degree in general relativity, being the principle behind parallel transport. Rather, we consider specifically the connection on the tangent bundle TM, but note that it gives rise to a connection on the co-tangent bundle T ∗M by demanding that it acts naturally with respect to contraction. In this manner, the connection considered can be extended to tensor bundles in general. Secondly, we are interested in the connection primarily because it is equivalent to the covariant derivative (indeed, these terms are often used interchangeably) which allows us to measure infinitesimal changes in direction, as well as magnitude.

Deﬁnition 3.2. The covariant derivative (on the tangent bundle) is a map ∇ : X(M)×X(M) → X(M), or more generally ∇ : X(M) × T (M) → T (M), where ∇(v, T ) is written ∇vT or in i c component form T j;cv , such that

16 (i) ∇fv+guT = f∇vT + g∇uT ;

(ii) ∇v (T + Q) = ∇vT + ∇vQ;

(iii) ∇v (fT ) = (vf) T + f∇vT .

∞ r r For scalar fields f, g ∈ C (M). Clearly this also defines a map ∇ : Ts (M) → Ts+1(M), which we may denote by ∇T (without a subscript for the vector argument). See further below. To differ between the two maps one often refers to the former one as the covariant derivative along a vector v, and the latter as alternatively simply the covariant derivative or the gradient. One important property that is obfuscated by definition 3.2 is the fact that the covariant derivative of a tensor along a vector, v, at a point, p, depends on the value of v at p only. See e.g. [11] for a complete proof. Because of the linearity in the first argument, we can define the so called connection forms. So suppose as usual that e1,..., en is a frame field, and let v be an arbitrary vector field. Then

i ∇vej = γ j(v)ei,

i for some 1-form γ j. This matrix of 1-forms are referred to as connection forms and the compo- i nents γ jk are referred to as Ricci rotation coefficents in rigid frames, because they in fact detail the rotation of any vector field as you move through the manifold. Here rotation is to be taken to be relative to parallel transported vectors (a topic that we will not delve into, for brevity). In a coordinate frame the Ricci rotation coefficients are equivalent to the Christoffel symbols, but since we will be working mainly in non-holonomic frames the coordinate formalism is of little interest to us. To see how the covariant derivative of vector fields can be extended to 1-forms by demanding that it acts naturally with respect to contraction, recall that it should act on functions as vector derivation (see subsection 2.2) and consider

i c ∇u α (v) = αiv u |c i c = αiv u ;c i c i c = αi;cv u + αiv ;cu i c i c = αi|cv u + αiv |cu , where we used the | in index notation to denote ordinary vector differentiation of the component functions, and the final expression is simply the first line expanded by the product rule. Imposing this rule as a demand, it is clear that we can define the covariant derivative of a 1-form using only vector differentiation and the covariant derivative of vector fields. We are now ready to write down the covariant derivative of vector fields and 1-forms, and thus tensors in general:

i c i c i j c v ;cu = v |cu + γ jcv u , c c j c αi;cu = αi|cu − γ icαju , where the ﬁrst expression follows from the deﬁnitions above, and the second from the contraction demand. To clarify how this applies to tensors in general we provide one example:

i c i c i k c k i c T j;cu = T j|cu + γ kcT ju − γ jcT ku .

17 (s, t) (0, t)

y v y0

u (s, 0) (0, 0)

0 Figure 2: Parallel transport around a small loop: the vector y returns as the vector y . Θuvy = 0 lims,t→0 y .

Let us now deﬁne the curvature operator Θuv : T (M) → T (M), which is deﬁned for pairs u, v ∈ X(M) as

Θuv := [∇u, ∇v] − ∇[u,v].

If a vector y is parallel transported around a small loop, as in figure2, then Θuvy is the limit of the change as the loop shrinks to zero. We omit the formal proof, but see e.g. [11]. Intuitively, the first bracket measures the difference between “the change in the change in y along u along v” and “the change in the change in y along v along u”. The last term accounts for the fact that u may change along v and vice versa. The Levi-Civita connection, which is always used in general relativity is uniquely defined by requiring

(i) gij;c ≡ 0; i c i c (ii) v ;cu − u ;cv = [u, v]. The proof is actually straight-forward, but we will omit it for brevity, and focus instead on the i i i i implications. The ﬁrst demand can easily be seen to imply that u vi |c ≡ u vi ;c = u ;cvi+u vi;c and is intuitively understood as requiring that the covariant derivative treat the metric as a product, thus obeying the Leibniz rule of standard derivatives. The second demand may be a bit more tricky, but [15] has gone so far as to say that any geometric theory of gravity obeying the equivalence principle must have a connection obeying this demand. It is referred to as having zero torsion, and can be seen to equate the Lie bracket with commuting covariant derivatives of vectors. This means that the Lie derivative can be taken with respect to the connection, for

i i i £u v αi = £uv αi + v (£uαi) i c i c i = v ;cu αi − u ;cv αi + v (£uαi) i = ∇u v αi i c i c = v ;cu αi + v αi;cu . Combining the two expressions we ﬁnd

j j £uαi = αi;ju + αjv ;i,

18 which is exactly the formula for the Lie derivative, but with the covariant derivative taking the place of vector differentiation of the components. This expression in turn gives the same result for vectors, which gives the result for tensors in general. Qualitatively this can be understood to mean that the connection does not “twist” around paths in the manifold. First observe that the first demand gives gij|k = 2γ(ij)k and second demand may be equiva- i i i lently stated as 2γ [jk] = C kj, where C kj are defined by

k ei, ej = C ijek, for a given frame. This is in fact a Lie algebra when we consider smooth functions to be scalars. We will use capital letter C to denote what are formally the structure constants of this specific algebra, but here take the shape of smooth functions, and refer to them instead as structure coefficients. Now note that the two demands together allow us to write down the rotation coefficents as 1 γ = g + g − g + C + C − C , ijk 2 ij|k ik|j jk|i kij jik ijk where we let the metric lower the indices on both structure coefficients and rotation coefficients. In particular, in a rigid frame 1 γ = C + C − C , (1) ijk 2 kij jik ijk in which case we observe γijk = γ[ij]k. Now, for vectors, the curvature operator accepts three vectors and outputs a fourth. Direct calculation can verify that applying the metric to the output vector and a fifth vectors turns it into a well-defined tensor:

R(y, w, u, v) := g(Θuvw, y),

The tensor R = Rijkl is called the Riemann curvature tensor. It is difficult to overstate the importance of this tensor; the Riemann tensor and a finite number of its covariant derivatives gives a complete local classification of the manifold, if expressed in a rigid frame. It obeys the symmetries Rijkl = R([ij][kl]), which should be taken to mean anti-symmetry in each pair but symmetry on interchanging the pairs. The anti-symmetry of the first pair is obvious from the definition of the curvature operator, but proving the others is an unnecessary exercise in mathematics. Applying the (Levi-Civita) covariant derivative twice to a vector u we calculate

i c d i c d i c d u ;cv w = u ;cdv w + u ;cv ;dw , ;d and find 1 ui wdvc = [∇ , ∇ ] − ∇ u. [dc] 2 v w [v,w] So, from the definition of the Riemann tensor we obtain the Ricci identity 1 u = R ub. (2) a;[dc] 2 abcd k From the Riemann tensor, the Ricci curvature tensor Rij = R ikj can be defined. The Einstein field equations, which we shall encounter later, links the Ricci tensor to the stress- energy tensor, and makes clear that the Ricci tensor describes the curvature of the manifold as

19 an eﬀect of the immediate content. From the Ricci tensor one can deﬁne the curvature scalar i 1 R = R i and the traceless part Sij = Rij − 4 Rgij. With this in hand one can decompose the Riemann tensor according to the irreducible Ricci decomposition:

Rijk` = Cijk` + Eijk` + Gijk`, (3) where 1 E ≡ g S + g S − g S − g S , ijk` 2 ik j` j` ik i` jk jk i` R G ≡ g g − g g , ijk` 12 ik j` i` jk and the tensor Cijk` is referred to as the Weyl tensor. The Weyl tensor obviously obeys the same k symmetries as the Riemann tensor, but additionally is trace free: C ikj ≡ 0. Thus the Weyl tensor complements the Ricci tensor and describes the curvature of spacetime that does not arise from immediate content. Relative to an observer with velocity vector uk the Weyl tensor can split into an electric part,

k ` Eij ≡ Cikjù u (4) and a magnetic part, 1 H ≡ C`m ukun (5) ij 2 ik`m jn see e.g. [10], that carry direct physical significance, see e.g. [16],[18] and [17]. Briefly, one can say that the electric part represents a tidal field, which drives the separation of a cloud of test particles, while the magnetic part represents a frame dragging field. Both are present in gravitational waves, and the analogy to electromagnetic fields is strong.

3.3 The Cartan Equations The Cartan equations give an efficient tool for calculating both the connection and the Riemann tensor from the metric in rigid frames. Furthermore they are revealing about the structure of spacetime, and they are used in the theory of invariant construction of new solutions. In this subsection we follow [15]. The first Cartan equation gives us a new description of how the connection forms give the change in the local frame field. Importantly, it does so in the language of differential forms and shows that we can use the exterior derivative rather than the covariant derivative in this case. Therefore it can be used to determine the connection forms. The second Cartan equation provides a new description of how curvature and connection are linked. Again, with the use of differential forms it becomes an application in exterior calculus which is straightforward to carry out in rigid frames. Before moving on to the equations it is important to extend the exterior derivative to tensor- valued forms. A tensor-valued form is a differential form that produces a tensor field rather than a real number. Thus a tensor-valued form can be seen as an element of T (M) ⊗ Ω(M), and in particular all tensor fields are at least tensor valued 0-forms. For a tensor valued form we extend the exterior derivative with the help of the covariant derivative. For η = Q ⊗ ω we take

dη = ∇Q ∧ ω + Q ⊗ dω,

20 where the gradient of the tensor Q is taken to be a tensor-valued 1-form, and

(Q ⊗ ω) ∧ α ≡ Q ⊗ (ω ∧ α) .

Thus we demand that the exterior derivative follows the same rules as the covariant derivative, but lose the property d2 ≡ 0 unless applied to a ordinary differential form (or flat space, as we shall see). Note also that if we choose to consider e.g. a 1-form to be a co-vector-valued 0-form we will get a different result than if we consider it a 1-form. To be rigorous it may therefore be important to specify to what bundle the exterior derivative is extended. Alternatively, one can demand that any such bundle is completely contravariant for clarity, thus demanding T ∈ T r(M) in the above example. We shall take this approach. i i Now consider eiω ≡ n, by the definition of dual frames. Thus d eiω = 0, but expanding the product we find

i i 0 = ∇ei ∧ ω + ei dω i j i = ei γ j ∧ ω + dω , by relabeling of indices. Thus we can conclude

i j i dω = ω ∧ γ j. (6)

i 1 i j k This is the ﬁrst Cartan equation, and may alternatively be written dω = − 2 C jkω ∧ ω = i j k γ jkω ∧ ω . i i i j Now consider an arbitrary vector v = v ei. Then we can similarly ﬁnd dv = ei dv + γ jv , and thus

2 i i j 2 i i j i j d v = ∇ei ∧ dv + γ jv + ei d v + dγ jv − γ j ∧ dv

i j i k = ei γ j ∧ γ k + dγ k v .

Clearly the two-form in parenthesis carries some signiﬁcance. First note that for an arbitrary i j1...j tensor-valued 1-form T ⊗ α = Tiω , where Ti is taken to mean T k αi, we have

j i j i d(T ⊗ α) = Ti;jω ∧ ω + Tiω ∧ γ j, whence

i j i j k d(T ⊗ α)(u, w) = 2T[i;j]w u + 2Tiγ [jk]u w i j i j k = 2T[i;j]w u − TiCjku w .

Now, recall that dv(u) = ∇uv by the deﬁnition, so that

2 d v (u, w) = ∇u∇wv − ∇w∇uv − ∇[u,w]v

= Θuwv. and we have arrived at the second Cartan equation: 1 dγi = γk ∧ γi + Ri ωk ∧ ω`. (7) j j k 2 jk` Now since d2 ωi ≡ 0 we have 1 d ωj ∧ γi = ωk ∧ γj ∧ γi + ωj ∧ γi ∧ γk − Ri ωi ∧ ωk ∧ ω` j k j k j 2 jk`

21 1 = − Ri ωi ∧ ωk ∧ ω` = 0, 2 jk` by application of equation (6) and equation (7). This yields the ﬁrst Bianchi identiy or sometimes the cyclic identity:

i R [jk`] ≡ 0. (8)

2 i Similarly, as we did for equation (8) we note that since d γ j ≡ 0 we must have

1 d −γi ∧ γk + Ri ωk ∧ ω` = γi ∧ γ` ∧ γk − γi ∧ γk ∧ γ` + k j 2 jk` ` k j k ` j 1 1 − Ri ω` ∧ ωm ∧ γk + Rk γi ∧ ω` ∧ ωm+ 2 k`m j 2 j`m k 1 + Ri ωm ∧ ωk ∧ ω`+ 2 jk`,m 1 1 + Ri ωm ∧ γk ∧ ω` + Ri ωk ∧ ωm ∧ γ` 2 jk` m 2 jk` m = 0, or 1 Ri − Ri γn + Rn γi − Ri γn − Ri γn ωm ∧ ωk ∧ ω` = 2 jk`,m nk` jm jk` nm jn` km jkn `m 1 = Ri ωm ∧ ωk ∧ ω` = 0 2 jk`;m which gives us the second Bianchi identity:

i R j[k`;m] ≡ 0. (9)

Let us pause for a moment and consider equations (6) and (7) in the light of our pictorial view of differential forms. The local frame field fills the region of spacetime with (imaginary) hypersurfaces. By equation (6) it becomes clear that to turn these into tubes where the densities change it is sufficient to make use of the connection forms. Equation (7) then tells us how these 1-forms change throughout the manifold. In particular, the first term tells us that some of this structure can be described by the same family of connection forms, and importantly the second term tells us that the remainder describes the curvature of the manifold. This should hint at the importance of the Cartan equations, and it is important to realize that the equations are fundamental in that they must hold for all general relativistic geometries. Conversely, if the Cartan equations can be shown to hold for a set of corresponding values then a general relativistic geometry can be constructed. This is used in the invariant construction of solutions to the Einstein field equations that we apply in this essay, see section4

3.4 Stress-Energy Tensor and Fluid Dynamics The stress-energy tensor describes the matter content of our model, and depending on the type of matter content it can take on different shapes. We will only be working with perfect fluids, defined by isotropic pressure and no dissipative effects, for which the stress-energy tensor takes the form (see e.g. [15])

T ij = (µ + p) uiuj − pgij, (10)

22 where p is the internal pressure of the fluid, µ is its rest mass density, and ui is its normalized i velocity vector: u ui = 1. When plugged into the Einstein field equations it describes a fluid that does not self-interact through other means than through gravitation and pressure, which is a reasonable model for the current state of the universe. See e.g. [10] for more information. First a brief description of the physical interpretation of the components of the stress-energy tensor is presented. Here we also follow [15]. Then the so called kinematic quantities shall be introduced, see e.g. [23], before the Einstein field equations are finally presented. The stress-energy tensor is two-dimensional and always completely symmetric, but consider it briefly, in a somewhat unorthodox manner, to be a vector-valued 1-form. The two dimensions j i have very real physical interpretation. Namely, Ti v describes the flux of the v component of j i energy-momentum. Observe then that if v is the velocity vector of some observer, then Ti v describes the flux of energy as observed by that observer. We shall also need to introduce the projection tensor relative to some observer with velocity vector v. When contracted with any vector the projection tensor produces a co-vector that is perpendicular to the velocity vector v. We define it as hij := vivj − gij, and observe that any vector y can be split into a component perpendicular to v and one parallel to v. Thus

i i i hijy = hijy⊥ + hijyk

= −yj⊥ + yjk − yjk

= −yj⊥, and we see why it is named a projection tensor. With this tool in hand the covariant derivative of the ﬂuid ﬂow can be written in terms of the kinematic quantities 1 u = Ω + σ − h θ + a u , (11) i;j ij ij 3 ij i j where

k ` Ωij := hi hj u[k;`] is the vorticity of the ﬂuid;

1 σ := h kh ` u + θh ij i j (k;`) 3 kl is the shear of the ﬂuid;

i θ := u ;i is the expansion of the ﬂuid; and

j ai := ui;ju is the acceleration of the fluid. Firstly, note that any object in free fall follows a geodesic through j spacetime, defined precisely by ui;ju ≡ 0, so the acceleration clearly gives the deviation from geodesic motion, or in other words the deviation from free fall. Next, the expansion is given as i the divergence u ;i of the fluid velocity. For suppose all other kinematic quantities are zero, then all components of the fluid velocity grow only along themselves and they do it with equal speed. To understand the shear and vorticity consider the space orthogonal to the fluid velocity. If only the shear is non-zero, then we can choose a frame such that the shear is diagonal, and it becomes

23 clear that all the components grow along themselves, but they do so in a way that produces no overall expansion. Finally then, if only the vorticity is non-zero then the components are rotated around the vector Ω = (Ω23, Ω13, Ω12). In cosmological perfect fluid models all observers are considered to be moving with the fluid. In a co-moving Lorentz frame the velocity vector of the fluid is thus u = (1, 0, 0, 0), and we also have

k 0 ui;j = −γ ijuk = −γ ij = −γ0ij, or 1 −γ = Ω + σ − h θ + a u . (12) 0ij ij ij 3 ij i j We will somewhat informally adopt the term non-kinematic for Ricci rotation coeﬃcients that are not given by equation (12). Contracting equation (2) with ud we obtain

b d d d d Rabcdu u = ua;du − ua;c u − ua;du ;c, ;c ;d and further contraction on the index pair a, c yields

b d c d d;c Rbdu u = a ;c − θ|du − uc;du 1 = ac − θ ud − θ2 − σ σcd + Ω Ωcd. ;c |d 3 cd cd On rearranging the terms this gives us the Raychaudhuri-Ehlers equation for the evolution of θ along the ﬂuid ﬂow 1 θ uc = ai − θ2 − σ σij + Ω Ωij − R uiuj. (13) |c ;i 3 ij ij ij For a more complete treatment of the time evolution of the kinematic quantities see e.g. [10]. The stress-energy tensor must also obey some purely physical conditions, together referred to as energy conditions, see e.g. [19]. For a discussion on the current standings of these conditions in cosmology see e.g. [22]. Firstly, we may demand that any observer observes matter density to be non-negative. This is the null energy condition (for lightlike observers) or alternatively the weak energy condition (for timelike observers), and gives, for any timelike or null vector vi

i j Tijv v ≥ 0 (null/weak energy condition). Note that by continuity the weak energy condition implies the null energy condition. We shall however continue to refer to them separately. Secondly, we may also demand that no observer can observe energy to ﬂow faster than the speed of light. This is the dominant energy condition and gives mathematically, for any two timelike or null vectors vi and uj

j j Tijv u ≥ 0 (dominant energy condition). Clearly this in turn implies both the weak and null energy conditions. Thirdly, we may further demand that gravity is locally attractive to any observer. This is the strong energy condition. Mathematically it gives, for all timelike vectors vi

i j Rijv v ≥ 0 (strong energy condition).

24 The strong energy condition can easily be seen to imply the Table 1: Implications of energy con- null energy condition but not the weak energy condition. ditions for a perfect fluid. Note that all These have very concrete implications for a perfect fluid, other energy conditions also imply the see table1. Note that we give implications in such a way null energy condition, and further that that demands made by weaker energy conditions appear the dominant energy condition implies only once, though clearly, as outlined above, e.g. the weak the weak energy condition. energy condition also implies the null energy condition. Note that the conservation of energy-momentum in any Energy Condition Implication inertial frame (normal coordinates, see e.g. [11] or [15]) Null µ + p ≥ 0 implies that the stress-energy tensor has a vanishing di- Weak µ ≥ 0 ij vergence. Thus T ;j ≡ 0 is given by the equivalence prin- Dominant µ2 ≥ p2 ciple. Proving this mathematically without the use of the Strong µ + 3p ≥ 0 Einstein field equations is not particularly difficult, but requires substantial and otherwise irrelevant theory. For a perfect fluid in a co-moving frame this gives us the following conservation equations:

µ|0 = −θ (µ + p) , (14)

p|i = ai (µ + p) , (15) where the index i is taken to be spatial (i ∈ {1, 2, 3}) only. Note that the null energy condition thus also guarantees that the fluid density drops as it expands, and that the fluid accelerates towards lower pressure regions from higher pressure ones, as one would expect. Finally, then, the Einstein field equations give the Ricci curvature tensor from the stress- energy tensor. So not only does the curvature of spacetime determine free fall motion of its content, but the content determines the Ricci curvature of spacetime. Although not obvious in its standard form, which we present here, the Einstein field equations is a (complicated) coupled system of non-linear differential equations (see e.g. [15]). The field equations (without cosmological constant) are most commonly written in the way we present them here: 1 R − Rg = T , ij 2 ij ij but may equivalently be written as 1 R = T − T g , (16) ij ij 2 ij

i where T = T i. This allows us to easily give the Ricci curvature of a perfect ﬂuid solution in its co-moving frame: 1 1 R = (µ + 3p) ,R = R = R = (µ − p) . 00 2 11 22 33 2 This in turn gives us the Riemann curvature tensor: 1 1 R = E − (µ + 3p) ,R = E − (µ + 3p) , 0101 11 6 0202 22 6 1 1 R = − (E + E ) − (µ + 3p) ,R = E + E − µ, 0303 11 22 6 1212 11 22 3 1 1 R = −E − µ, R = −E − µ, 1313 22 3 2323 11 3 R0102 = R1323 = E12,R0103 = −R1213 = E13, (17)

25 R0203 = R1213 = E23,R0112 = R0323 = B13,

R0113 = −R0223 = −B12,R0313 = −R0212 = −B23,

R0123 = B11,R0213 = −B22,

R0312 = − (B11 + B22) , where we used the fact that electric and magnetic parts of the Weyl tensor are symmetric and i i trace free: E i = B i = 0.

3.5 Spatially-Homogeneous Solutions We now move firmly into the topic of cosmological models, following [23]. We will consider in particular spatially-homogeneous models, which are defined by the existence of a group (action) which leaves the metric invariant, and is such that the orbits are (not necessarily spacelike) hypersurfaces. Such an action is called an isometry, and the vector, ξ, describing it is called a Killing vector. Thus £ξg = 0. But then c c c £ξg = gij;cξ + gcjξ ;i + gicξ ;j (18) = 2ξ(i;j) = 0. We call this the Killing equation, and it is useful for finding and testing isometries. Furthermore, for vectors u, v we have £ξ g (u, v) = £ξg (u, v) + g £ξu, v + g u, £ξv = g [ξ, u] , v + g u, [ξ, v] .

So that for a frame field such that [ξ, ei] = 0 we have ξgij ≡ 0 for the component functions gij. In other words the frame is rigid when restricted to an arbitrary orbit. Such a frame field is referred to as invariant, and the group associated with the algebra spanned by the killing vectors of a solution is referred to as the symmetry group. We will consider only cases where the symmetry group is simply transitive, so that it is by necessity three-dimensional, but see e.g. [23]. Continuing we will make the implicit assumption that the symmetry group is three-dimensional and acts simply transitively on the manifold. An invariant frame for an arbitrary orbit, H, can easily be constructed by picking a basis for the tangent space TpH = TpM |H for some p ∈ H. This basis can be chosen such that the inner product on TpH induced by the metric takes any symmetric and non-degenerate shape, since if ξ is a killing vector and a is a constant, then aξ is also a killing vector. Parallel transport of this basis with help of the Lie derivative with respect to the Killing vectors yields an invariant frame field of the orbit. Since a one-parameter family of orbits fills the manifold we can refer to the orbits as H(t) where t ∈ R is the parameter. If the evolution is to occur along timelike curves then we can conclude that the orbits are spacelike, but we are free to select any timelike curve as our time axis. Note that the invariant frame field for the orbit H(t) can vary smoothly with t. Since we wish to proceed with a co-moving frame we fix the time axis to be the world line(s) i of the fluid, and the fluid velocity is given by u = ∂t = e0. If the fluid flow is orthogonal to the hypersurfaces we can use the smoothly evolving frame fields of H(t) for our spacelike frame vector i i i 1 2 3 fields, thus taking ej = nj(t)Aj∂i, where nj is the normalizing factor and Aj = Aj(x , x , x ) for i = 1, 2, 3. Consider therefore an invariant local frame field of an arbitrary orbit, H, spanned by Killing vectors ξi say j j j ui = ui ξj, ui |p = δi ,

26 k k k for some point p ∈ H. Then the invariance gives ξiuj = −c ij at p, where c ij are the structure k constants of the symmetry group: ξi, ξj = c ijξk. But then

h k ` i ui, uj = ui ξk, ujξ`

k ` ` k k ` = ui ξkuj ξ` − uj ξùi ξk + ui uj [ξk, ξ`] ` k ` k ` m k = −ui c `jξk + ujc ìξk + ui uj c `mξk k k k = −c ijξk = −c ijuk = C ijuk, but since the structure coefficients are constants in H this equation holds in all of H. We proceed under the assumption that the standard invariant basis is orthogonal on the orbits, which may potentially limit our solution space. Because of the smooth evolution with t we can therefore conclude that in in the orthogonal fluid flow case we can choose our co-moving Lorentz frame such that the structure coefficients are given by

0 i ni,t C 0i = 0,C 0i = ni j 0 C 0i = 0,C ij = 0, (19) k ninj k C ij = − c ij, nk where we have taken i, j, k 6= 0 and furthermore i 6= j 6= k, i 6= k, and no summation is to be performed. This completely determines the Ricci rotation coeﬃcients in terms of the quantities ni, their derivatives, and the group algebra. Observe in particular that

γ0j0 = γij0 = γ0ji = 0 (i 6= j).

Thus the shear is always diagonalized, and aj and Ωji always vanish. Note that by equation (13) the strong energy condition thus demands θ,t < 0 in orthogonal solutions. We ﬁnd the following explicit expressions for the remaining kinematic quantities:

1 ni,t σii − θ = . (20) 3 ni If the fluid flow is not orthogonal we refer to the solution as tilted. Suppose therefore that we have a local frame ei such that ge0i 6= 0 for at least one spatial index i. The frame can be orthonormalized by standard procedures, but applying e.g. the Gram-Schmidt process yields an unwieldy tangent bundle algebra. Let the indices a, b, . . . temporarily denote frame vectors ea such that ge0a = 0. Then let i temporarily denote one specific frame vector ei such that ge0i 6= 0, let j denote a second specific frame vector ej such that ge0j 6= 0, and let k denote the third specifc frame vectors ek such that ge0k 6= 0. Note that if we e.g. have tilt along two basis vectors then instead of ek we have ea. Then we can construct a co-moving Lorentz frame ei by taking

−g t = e0i , |i p 2 1 + ge0i −|ge0i| ge0j t|j = r , 2 2 2 ge0i 1 + ge0i 1 + ge0i + ge0j

− ge0j ge0k t|k = r . 2 2 2 2 2 ge0j 1 + ge0i + ge0j 1 + ge0i + ge0j + ge0k

Equivalently, we may write

−t|i g0i = , e q 2 1 − t|i

−t|j ge0j = r , 2 2 2 (21) 1 − t|i 1 − t|i − t|j

−t|k ge0k = r , 2 2 2 2 2 1 − t|i − t|j 1 − t|i − t|j − t|k since we can always choose ge0i > 0 and ge0j > 0 (and ge0k > 0) by inverting the basis vectors in the origin. This process could obviously be extended to higher dimensions if necessary. Note 2 2 2 that we must always have 1 − t|i − t|j − t|k > 0 since t is timelike. The structure coefficients 0 a b i j C 0a, C 0a, C 0a, C 0a, and C 0a are given by equations (19). For for the remaining structure coefficients see the Appendix. Finally then, the three-dimensional groups that may serve as the isometry group of a four- dimensional Lorentzian manifold have been classified by their algebra into nine classes, referred to as Bianchi types I through IX (see e.g. [23], and e.g. [24] for more information on the particulars). Note that these groups may well appear as subgroups of the complete isometry group of any solution. The group algebras, see table2 is with respect to standard basis, corresponding to standard Killing vectors, which gives rise to a standard invariant frame field. This essay works with orthogonal solutions of Bianchi type III, and from table2 it is clear that in our preferred co-moving Lorentz frame, must have

1 1 1 1 1 2 2 2 γ 21 ≡ γ 22 ≡ γ 23 ≡ γ 32 ≡ γ 33 ≡ γ 31 ≡ γ 32 ≡ γ 33 ≡ 0, 1 3 (22) γ 31 = −γ 11 6= 0.

In section6 we consider dust solutions with non-vanishing vorticity of Bianchi type I–III. This restriction allows an eﬀective statement of the Ricci rotation coeﬃcients dependent on the tilt.

28 Table 2: The group algebras of Bianchi types I – IX with respect to a standard basis. This corresponds concretely to a standard invariant frame ﬁeld of the orbits, and can be used to determine the structure coeﬃcients in terms of the tilt and normalization factors.

Bianchi Type Non-zero structure constants I All structure constants are zero. 1 II c 23 = 1. 1 III c 13 = 1. 1 1 2 IV c 13 = c 23 = c 23 = 1. 1 2 V c 13 = c 23 = 1. 1 2 VI c 13 = 1, c 23 = h where h 6= 0, 1. 2 1 2 2 VII c 13 = −c 23 = 1, c 23 = h where h < 4. 3 1 2 VIII c 12 = −c 23 = −c 13 = 1. 1 2 3 IX c 23 = −c 13 = c 12 = 1.

4 Invariant Construction of New Solutions

Any neighbourhood of a manifold may be described by several different coordinate functions, and understandably the metric takes on different shapes for different sets of coordinates. This gives rise to the so called equivalence problem: how can one tell whether a metric describes the same local geometry as another metric? This is obviously the case if and only if there is a coordinate transformation that relates the two metrics, but the problem is aggravated by the fact that general relativity allows for arbitrary smooth coordinate transformations. In 1980 Anders Karlhede developed an algorithm to classify metrics invariantly[13], building on the work of Élie Cartan[9]. The theory has since been developed further. In particular the method uses the following theorem, which we state without proof, but see e.g. [9].

Theorem A. Two charts (U, x) and (U,e xe) of two n-dimensional semi-Riemannian manifolds are equivalent if and only if, in rigid frames, the set of equations

Rijk`(x, ξ) = Reijk`(x,e ξe) R (x, ξ) = R (x, ξ) ijk`;c1 eijk`;c1 e e . (23) . R (x, ξ) = R (x, ξ) ijk`;c1...cp+1 eijk`;c1...cp+1 e e A 1 is compatible. Here ξ , A ∈ 1,..., 2 n(n − 1) , are rotation parameters, i.e. the ﬁbre coordinates of the bundle of Lorentz frames over the manifold(s), and Rijk` is the Riemann curvature tensor. The integer (p + 1) is the smallest integer such that the set

p+1 R = Rijk`,Rijk`;c1 ,...,Rijk`;c1...cp+1 is functionally dependent on the set Rp. Certain discrete transformations may also need to be checked for. For a review of the equivalence problem and the Cartan-Karlhede algorithm see e.g. [13] again. Based on the Cartan-Karlhede algorithm for the invariant classification of solutions, which we shall not go further into detail on, a method for constructing solutions to the Einstein field equations has been developed, see e.g. [14], [2], [4], [5], [6], [7], and [8]. In its most basic form one chooses a set Rp+1 and attempts to construct a solution from it. The necessary and sufficient conditions for the existence of a solution are given in [5]. In [6] the method was extended to solve for the line element and determine the isometry group of the solution. In addition the formalism in a fixed frame, which we shall be using, was developed. A comparison to other tetrad methods is also given in [6]. Importantly, due to theoremA, any solution for a given set is locally equivalent to all other solutions. 1 The original theory is formulated on the the bundle of Lorentz frames, F (M), itself a 2 n(n+1)- manifold, and in fact a fiber bundle over M, see e.g. [11]. It can be constructed from the tangent bundle and the generalized orthogonal group. We shall however proceed directly with the fixed frame formalism, and refer the interested reader to e.g. [6] for more rigorous formalism. We let i i A i i i τ j = τ jA dξ denote the connection forms on the fibres and let ω j = γ j + τ j denote the total connection forms. In [13] it was shown that the Cartan equations (6) and (7) still hold for i i the connection forms ω j on F (M). Additionally τ j generate the orthogonal group and we have e.g. (see [12])

m m m m dξ Rijk` = Rmjk`τ i + Rimk`τ j + Rijm`τ k + Rijkmτ `,

31 where dξ denotes the exterior derivative on each ﬁber. Note that we here consider the exterior derivative of the component function Rijk`. This gives us

m m m m DRijk` = Rmjk`ω i + Rimk`ω j + Rijm`ω k + Rijkmω ` + Rijk`;m, where D= d+ dξ. i i i i I 1 Because ω j are linearly independent of ω the set ω , ω j = ω (I = 1,..., 2 n(n + 1)) can be taken as a frame ﬁeld of F (M). Therefore, if {Iα} denotes a maximal set of functionally p n α o n µ i o independent elements of R then the set I |K is equivalent to the set x |k, γ jk . We will refer to Iα as essential coordinates. Furthermore, in analogy with the ﬁrst Cartan equation we may write 1 dωI = − CI ωJ ∧ ωK , (24) 2 JK

I I for smooth functions C JK , and by the second Cartan equation conclude that C JK contain all the information of Rijk`. In essential coordinates

I I L I α L dC JK = C JK|Lω = C JK,αI |Lω ,

p+1 n I α o and similarly for higher derivatives. So the set R is given by the set S = C JK ,I |K , and we may use

α α K dI = I |K ω

n i µ i o for the derivatives. Equivalently, as we have seen, we may take S = R jk`, x |k, γ jk, gij . If p+1 n i i o R depends only on the coordinates t, as it does in our case then S is reduced to R jk`, t|k, γ jk, gij . Invariance under rotation in one or more planes may further reduce the set by reducing the number of rotation coeﬃcients that appear in the covariant derivatives of Rijk`. A set S will produce a geometry if it satisﬁes the Cartan equations ([5]) on the frame bundle. From the reduced set S we can solve for

0 −1 p ω = t|0 dt − t|pω , where p = 1, 2, 3. The remaining 1-forms, ωp, will satisfy the Cartan equations if d2 ωp ≡ 0, but we have already seen that this is equivalent to equation (8), the ﬁrst Bianchi identity. Similarly p 2 p any missing connection forms γ q will satisfy the Cartan equations if d γ q ≡ 0, which we have seen to be equivalent to equation (9), the second Bianchi identity. Given thus that ωp satisfy the Cartan equations, expanding 2 0 2 −1 p d ω ≡ d t|0 dt − t|pω

2 0 2 i similarly we ﬁnd that d ω ≡ 0 if and only if d t ≡ d t|iω ≡ 0. Expanding gives us the ﬁrst integrability condition

k t|[ij] − t|kγ [ij] = 0, (25)

i which corresponds to equation (6). The ﬁxed frame formalism restricts us to τ j = 0, and the i i i k condition dτ j = d ω j − γ jkω ≡ 0 must be imposed so that the Cartan equations are also

32 guaranteed to hold on the frame bundle F (M). Expanding gives us the second integrability condition

i i im i m R jk` = 2 γ j[`|k] + γjm[kγ `] + γ jmγ [k`] , (26)

a p which corresponds to equation (7). For the non-essential coordinate x and the planes q such that the metric is invariant under rotation in them we must, as we have seen, impose the Bianchi identities:

a R [ijk] = 0, p (27) R q[ij;k] = 0.

These are the necessary and sufficient conditions for a reduced set S to give rise to a general relativistic geometry, although further physical conditions, such as the energy conditions, may need to be applied on top. Since we will be proceeding with a preferred co-moving Lorentz frame i and take R jk` to be given by equations (17) the first Bianchi identity is automatically satisfied. The integrability conditions give us a set of differential equations to be solved for the reduced set S. Once all elements of S has been determined, equation (6) gives us differential equations to solve for the frame field 1-forms, which in turn gives us the line element.

5 Orthogonal Bianchi Type III

In the orthogonal case equation (25) is automatically fullﬁlled once we put the acceleration and vorticity to zero. Proceeding with equations (26) and applying equations (17) and (22), we ﬁrst note that σ22 = −2σ11, E22 = −2E11, Bij ≡ 0 for all i, j, and Eij = 0 if i 6= j. Observe that since all physical quantities are invariant under frame change 1 7→ 3 and 3 7→ 1, all solutions 1 given will be locally rotational symmetric in the 3 plane, meaning they are invariant under 1 rotations in the 3 plane. We take γ131 ≡ `(t). Then the remaining integrability conditions are given below: 1 ` = ` σ − θ , ,t 11 3 1 σ = −θσ − `2, 11,t 11 3 3 θ = −`2 − (µ + p) − 9σ2 ,t 2 11 3 = 2`2 − θ2 + (µ − p) (28) 2 1 = `2 − θ2 − 3p − 9σ2 2 11 1 µ = θ2 − `2 − 3σ2 , 3 11 1 1 E = `2 + θσ − σ2 , 11 3 3 11 11

Note that the expressions for θ,t are mutually equivalent, and also equivalent to equation (13). Similarly the expression for `,t is equivalent to equation (20). The weak energy condition demands that 1 θ2 ≥ `2 + 3σ2 , (29) 3 11 which implies `,t 6= 0, by equations (28). The null energy condition can immediately be seen to imply θ,t < 0, as is to be expected from all orthogonal solutions

2 2 θ,t ≤ −` − 9σ11.

2 2 Furthermore the dominant energy condition demands θ,t > 2` − θ , and equations (28) directly imply that σ11 6= 0. Not surprisingly σ11 can also not be constant. Because inputting σ11,t = 0 gives us 1 θ = − `2, 3σ11 so that 1 2 `,t = ` σ11 + ` , 9σ11 or

3σ eσ11t ` = ±√ 11 , 1 − e2σ11t

35 where we translated the time coordinate to remove the integration constant. Note the (possibly coordinate) singularity at t = 0. If we take σ11 < 0 the solution will be deﬁned for all t > 0. However, this solution breaks the dominant energy condition for

1 8 t > ln ; 2σ11 9 the weak energy condition for √ ! 1 1 + 13 t > ln ; 2σ11 6 the null energy condition for

1 3 t > ln ; 2σ11 5 and the strong energy condition for

1 1 t < ln . 2σ11 3 Thus it can clearly be discarded as unphysical.

5.1 Dust Solutions (p = 0) Dust solutions are characterized by p = 0. This gives us, via equations (28)

` 5 ` 2 1 ,tt = ,t − `2, (30) ` 2 ` 2 or   √  √ C` + 1 − 1  C` + 1 C  t = ±  + ln  √  ,  ` 2  C` + 1 + 1  for some integration constant C, and where we translated the time coordinate to remove the second integration constant. An explicit solution is not possible unless we take C = 0. This special case may be treated separately and for C 6= 0, we ﬁnd d` dt = ∓ √ , `2 C` + 1 or d` √ = ∓`2 C` + 1, dt which allows us to conclude that 1 √ σ − θ = ∓` C` + 1. 11 3

36 √ Introducing the coordinate u = C` + 1, we have ! u 1 1 + u t = ±C + ln , 1 − u2 2 1 − u 2 du dt = ± , C`2 and because we want du to be causal we select 2 du dt = , C`2 if C > 0 and correspondingly for the coordinate transformation. The above equations show that we can take C > 0 without restriction. Note that u = 0 corresponds to t = 0 and u = 1 corresponds to t = ∞. In our new coordinate we ﬁnd ! σ −2 σ 2 σ 11 = 9 11 + 1 − 6 11 u . ` ,u 3C` ` ` This Riccati equation may be solved to get 1+u 1 2 u ln 1−u + Du − 2 3u − u − u σ11 = , ` 1+u 6 u ln 1−u + Du − 2 for some new integration constant D. The system of equations is now solved, and we have found

2 2 u2 − 1 µ = , 2 1+u C u ln 1−u + Du − 2

Observe that there will be a singularity where 2 1 + u = ln + D, u 1 − u or alternatively −1 CD = t + . u` 2 There is one such value for u (and for positive t), and we may denote it by S. For non-negative D

lim µ = ∞, u→S+ and µ > 0 for all u > S. Additionally θ behaves similarly since we have taken C to be positive, so that ` ∈ −1/C, 0: 1+u 3 1 2 2 u ln 1−u + Du − 2 −3u + 2u + u − u u − 1 θ = . 1+u 2C u ln 1−u + Du − 2

37 As u approaches one we conversely ﬁnd that both θ and µ approach zero. Introducing the function 1 + u k := u ln + Du − 2, 1 − u the non-zero rotation coeﬃcients are given below, up to symmetries:

u 1 − u2 γ1 = 01 C 3 = γ 03, 2 k 1 − u2 γ2 = ,u , 02 2Ck u2 − 1 γ1 = . 31 C By equation (6) this gives us a set of ordinary diﬀerential equations to solve for the frame ﬁeld ωi , and we can solve for the line element:

4C2 e2z/C 1 ds2 = du2 − dx2 − k2 dy2 − dz2, (31) (1 − u2)4 (1 − u2)2 (1 − u2)2 defined for S < u < 1. The singularity at u = S, defined by k = 0, is gravitational because the Kretschmann scalar becomes infinite when k approaches zero. The case C = 0 immediately gives us 1 ` ` = ± , ` = − , t ,t t and we are able to conclude 1 1 σ − θ = − , 11 3 t which in turn easily gives us 2C ln t + C + 2 θ = , t (C ln t + 1) −C ln t + C − 1 σ = , 11 3t (C ln t + 1) for some new constant C. We have found 2C µ = . t2 (C ln t + 1)

Note the singularity at t = e−1/C and that µ > 0 for all t > e−1/C . This is well defined for all C 6= 0, but C = 0 corresponds to vacuum, and is therefore not included in this analysis, since there would be no fluid flow to reference. Although there is an additional singularity at t = 0 we will consider t ∈ e−1/C , ∞ . The non-zero rotation coefficients are given below, up to symmetries 1 γ1 = 01 t

38 3 = γ 03, C γ2 = , 02 t (C ln t + 1) 1 γ1 = ± . 31 t The line element is

ds2 = dt2 − t2e±2z dx2 − (C ln t + 1)2 dy2 − t2 dz2. (32)

The singularity at t = e−1/C is once again gravitational because the Kretschmann scalar approaches inﬁnity.

5.2 Vacuum Energy Solutions (µ + p = 0) Vacuum energy solutions are characterized by µ + p = 0. This yields, via equations (28)

2 3 4 `,ttt` − 6`,tt`,t` + 5 `,t + `,t` = 0. (33)

Although this equation does not lend itself to an immediate general solution, a particular solution can easily be found: 1 2 ` = ± , θ = , t + D t + D 1 σ = − . 11 3 (t + D) This corresponds to the second dust solution, equation (32), with C = 0, and gives a Bianchi III foliation of Minkowski space. Another particular solution is given by C ` = ± , sin C (t + D) θ = C 2 cot C (t + D) − tan C (t + D) , −2C σ = . 11 3 sin 2C (t + D) but this gives µ = −3C2, and so violates the weak energy condition for real C. Of course C ` = ± , cos C (t + D) is equivalent by translation of the time coordinate. For complex C the solution is complex unless C = iB, for some real B, in which case we ﬁnd B ` = ± , sinh B (t + D) θ = B 2 coth B (t + D) + tanh B (t + D) . −2B σ = , 11 3 sinh 2B (t + D)

39 but note that translation cannot give a hyperbolic cosine solution. This time, as expected, µ = 3B2. As usual, translating the time coordinate allows us to set D = 0, and solve for the line element:

ds2 = dt2 − e2z sinh2 (Bt) dx2 − cosh2 (Bt) dy2 − sinh2 (Bt) dz2. (34)

The solution is invariant under sign change on B. Since µ = −p = 3B2 the model can be taken to describe an empty universe with negative (attractive) cosmological constant. Similarly, the analogous solution with non-hyperbolic functions can be taken to describe an empty universe with positive (repulsive) cosmological constant. We ﬁnd that E11 ≡ 0 so that it is a conformally 2 2 ﬂat solution. Furthermore Sij ≡ 0 and R = 12C or R = −12B . Thus the singularity at t = 0 is a coordinate singularity and furthermore we can conclude that these are special foliations of the de Sitter (and anti-de Sitter) space(s). Note also that

±C ±1 lim = . C→0 sin C (t + D) t + D

Thus the Minkowski solution, as would be expected, appears as a special case of the de Sitter solution by setting C = 0, or equivalently µ = 0. Having found two integration constants seems to indicate that there should be a more general solution, and indeed `,t = 0 is one formal solution (although of no physical interest) that cannot appear as a special case of the de Sitter solution. However, no successful ansatz was found to generalize the de Sitter solution.

40 6 Dust Solutions with non-Vanishing Vorticity

In this section we will allow for a cosmological constant, Λ, which modiﬁes equation (16) to

1 R = T − T g − Λg . ij ij 2 ij ij

The cosmological constant can thus be seen to correspond to vacuum energy, for if µm and pm denote the matter density and pressure, we can take µe = µm + Λ and pe = pm − Λ to get an eff effective stress energy tensor Tij = (µe + pe) uiuj − pegij obeying the equation 1 R = T eff − T effg , ij ij 2 ij i.e. the standard Einstein field equations. Most of the implications of the energy conditions remain unchanged, as can be seen from their definitions, but the strong energy condition now implies

µ + 3p − 2Λ ≥ 0.

Dust models (p = 0) are generally considered a good model for the (standard) matter content at the current state of the universe. By equation (15) such models must have vanishing acceleration unless also µ = 0. Equation (13) then reduces to

1 1 θ = − θ2 − σ σij + Ω Ωij − µ + Λ, (35) ,t 3 ij ij 2 in a comoving Lorentz frame. In particular, we note that θ,t > 0 can only be achieved if the strong energy condition is broken or if there is a non-vanishing vorticity. Dust models are a special, physically significant, case of the more general constant-pressure models, for which the acceleration always vanishes by equation (15) unless the model can be considered purely vacuum energy. Assuming vanishing acceleration we find, from the structure coefficients in the Appendix, that ni ∝ ge0i, nj ∝ ge0j, and nk ∝ ge0k. Furthermore, the following equations can be extracted with the help of equations (21):

1 1 t = t σ − θ , t = t σ − θ + t σ , |i,t |i ii 3 |j,t |j jj 3 |i ij 1 1 t|jt|i σii − 3 θ t|k,t = t|k σkk − θ + t|jσjk + t|iσik, σij = − 2 , (36) 3 1 − t|i 1 1 t|kt|j σjj − 3 θ + t|kt|iσij t|kt|i σii − 3 θ σjk = − 2 2 , σik = − 2 , 1 − t|i − t|j 1 − t|i 1 n = n σ − θ . a,t a aa 3

The remaining oﬀ-diagonal shear components are zero (i.e. if there are less than three non-zero tilt components some shear components vanish). The vorticity components are directly linked to the functions (here indices i, j, k, . . . are temporarily free again)

k ninj k n ij ≡ − c ij, nk

41 which are determined by the normalization functions from Subsection 3.5 and the group algebra. k k k k In particular note that n ij is zero if and only if c ij is and n ij = n [ij]. We postpone the explicit statements of the vorticity components because the expressions are significantly simplified once a specific algebra is chosen, and once the indices i, j, k are chosen. Note however that it is immediately obvious that there can be no Bianchi type I dust solutions with non-vanishing vorticity. 0

6.1 Bianchi II

For Bianchi type II solutions we ﬁrst note that the vorticity vanishes unless t|1 6= 0, so we are free to choose i = 1. We consider ﬁrst cases where only t|1 6= 0. Then we add to equations (36) the following equation:

Ω23 = Cn2n3, for some non-zero constant C and where the other vorticity components vanish. The remaining (non-kinematic) Ricci rotation coeﬃcients are given below:

γ120 = 0, γ130 = 0,

γ230 = Ω23, γ121 = 0, 1 Ω23 γ122 = −t|1 σ22 − θ , γ123 = − , 3 t|1

Ω23 γ131 = 0, γ132 = , t|1 1 Ω23 γ133 = −t|1 σ33 − θ , γ231 = , 3 t|1

γ232 = 0, γ233 = 0. With these substitutions equations (25) are automatically fulﬁlled, while equations (26) demand µ + p = 0. Note that although such a solution does not demand vanishing acceleration, and additionally does violate the strong energy condition, there is obviously nothing further stopping us from considering a vacuum energy solution with vanishing acceleration. However, the resulting equations are diﬃcult to solve, and we shall proceed instead with the quest for dust solutions. Proceeding with the case of t|1 6= 0 and t|2 6= 0, we add to equations (36) the following non-zero vorticity component

Ct|2 Ω23 = 2 n3, 1 − t|1 for some constant C. The non-kinematic Ricci rotation coeﬃcients are given below:

γ120 = −σ12, γ130 = 0, 1 γ = −Ω , γ = t σ − θ − t σ , 230 23 121 |2 11 3 |1 12 1 Ω23 γ122 = −t|1 σ22 − θ , γ123 = − , 3 t|1

Ω23 γ131 = 0, γ132 = , t|1

42 1 Ω23 γ133 = −t|1 σ33 − θ , γ231 = , 3 t|1 1 γ = 0, γ = −t σ − θ . 232 233 |2 33 3

Equations (25) demand σ12 ≡ 0 or equivalently σ11 ≡ θ/3, upon which equations (26) demand θ ≡ 0. Furthermore equations (26) then demands

2 2 p − Λ = σ22 1 − t|1 , but making the substitutions we ﬁnd that equations (26) have no solutions. For Bianchi II solutions the case t|1 6= 0 and t|3 6= 0 is equivalent to the case t|1 6= 0 and t|2 6= 0, as is apparent from the group structure constants (table2) and the factor C in Ω23. We therefore immediately move on to consider the case with no zero tilt components, letting j = 2 and k = 3. We add to equations (36) the only non-zero vorticity component

Ct|2t|3 Ω23 = . 2 2 2 1 − t|1 1 − t|1 − t|2

The non-kinematic Ricci rotation coeﬃcients are:

γ120 = −σ12, γ130 = −σ13, 1 γ = Ω − σ , γ = t σ − θ − t σ , 230 23 23 121 |2 11 3 |1 12 1 Ω23 γ122 = −t|1 σ22 − θ , γ123 = −t|1σ23 + t|2σ13 − t|3σ12 − , 3 t|1 1 Ω23 γ131 = t|3 σ11 − θ − t|1σ13, γ132 = −t|1σ23 − t|2σ13 + t|3σ12 + , 3 t|1 1 Ω23 γ133 = −t|1 σ33 − θ , γ231 = −t|1σ23 − t|2σ13 + t|3σ12 + , 3 t|1 1 1 γ = t σ − θ − t σ , γ = −t σ − θ . 232 |3 22 3 |2 23 233 |2 33 3

Equations (25) immediately demand σ11 = σ22 = θ/3, and then equations (26) demand θ ≡ 0. Upon making the substitutions we ﬁnd that equations (26) have no solutions.

6.2 Bianchi III As for Bianchi type II note that all vorticity components vanish in Bianchi type III solutions unless t|1 6= 0. We are therefore once again free to choose i = 1, and for the case with only t|1 6= 0 we ﬁnd 1 Ω = t n , 13 2 |1 3 to be the only non-zero vorticity component. The non-kinematic Ricci rotation coeﬃcients are given below:

γ120 = 0, γ130 = Ω13,

43 γ230 = 0, γ121 = 0, 1 γ = −t σ − θ , γ = 0, 122 |1 22 3 123

γ131 = n3, γ132 = 0, 1 γ = −t σ − θ , γ = 0, 133 |1 33 3 231

γ232 = 0, γ233 = 0.

Equations (25) are automatically fulﬁlled, and equations (26) demand σ11 = −θ/6. Making the 2 2 substitutions we ﬁnd from (26) that µ + p + n3 1 − t|1 = 0, whence any solution must violate the weak energy condition. The case with t|1 6= 0 and t|2 6= 0 yields

2 1 −t|1t|2n3 Ω13 = t|1n3, Ω23 = , 2 2 2 1 − t|1 as the only non-zero vorticity components. The non-kinematic Ricci rotation coeﬃcients are given below

γ120 = −σ12, γ130 = Ω13, 1 γ = Ω , γ = t σ − θ − t σ , 230 23 121 |2 11 3 |1 12 1 Ω23 γ122 = −t|1 σ22 − θ , γ123 = − , 3 t|1

Ω23 γ131 = n3, γ132 = , t|1 1 Ω23 γ133 = −t|1 σ33 − θ , γ231 = , 3 t|1 1 γ = 0, γ = −t σ − θ . 232 233 |2 33 3

Equations (25) immediately yield σ11 = θ/3, but then equations (26) have no solutions. We proceed instead with the case t|1 6= 0 and t|3 6= 0. The only non-zero vorticity component is

Ct|1t|3 Ω13 = 2 , 1 − t|1 and the non-kinematic Ricci rotation coeﬃcients are given below.

γ120 = 0, γ130 = Ω13 − σ13,

γ230 = 0, γ121 = 0, 1 γ = −t σ − θ , γ = 0, 122 |1 22 3 123 1 Ω13 γ131 = t|3 σ11 − θ − t|1σ13 + , γ132 = 0, 3 t|1

44 1 γ = −t σ − θ , γ = 0, 133 |1 33 3 231 1 γ = t σ − θ , γ = 0. 232 |3 22 3 233

However, equations (25) have no solutions. We consider ﬁnally the case with no zero tilt components, and select j = 2 and k = 3. The non-zero vorticity components are given by

2 Ct|1t|3 Ct|1t|2t|3 Ω13 = 2 2 , Ω23 = , 1 − t|1 − t|2 2 2 2 1 − t|1 1 − t|1 − t|2 and the non-kinematic Ricci rotation coeﬃcients are given below.

γ120 = −σ12, γ130 = Ω13 − σ13, 1 γ = Ω − σ , γ = t σ − θ − t σ , 230 23 23 121 |2 11 3 |1 12 1 Ω23 γ122 = −t|1 σ22 − θ , γ123 = −t|1σ23 + t|2σ13 − t|3σ12 + , 3 t|1 1 Ω13 Ω23 γ131 = t|3 σ11 − θ − t|1σ13 + , γ132 = −t|1σ23 − t|2σ13 + t|3σ12 − , 3 t|1 t|1 1 Ω23 γ133 = −t|1 σ33 − θ , γ231 = −t|1σ23 − t|2σ13 + t|3σ12 − , 3 t|1 1 1 γ = t σ − θ − t σ , γ = −t σ − θ . 232 |3 22 3 |2 23 233 |2 33 3

Again, equations (25) have no solutions.

7 Conclusions

With reservation for the fact that restricting ourselves to cases where the standard invariant frame field is orthogonal on the orbits may limit the solution space, we will now consider the results. Thus any conclusion made in the general context of some symmetry group should be taken to be true under this restriction. In section5 we studied orthogonal Bianchi type III solutions, and in general found that all such solutions exhibit local rotational symmetry around the 2-axis, because all physical quantities are invariant under the frame change 1 7→ 3 and 3 7→ 1. It is well known, see equation (13), that orthogonal solutions cannot exhibit θ,t > 0 unless the strong energy condition is violated. We found two dust solutions — (31) and (32) — and one vacuum energy solution — (34). The vacuum energy solution (34) turned out to be a special foliation of de Sitter space. By equation (30) we can conclude that (31) and (32) are all orthogonal dust solutions of Bianchi type III. Equation (33) seems to be indicative of that there is a more general orthogonal Bianchi type III vacuum energy solution, although it was not found. The first dust solution, (31) displays typical behaviour associated with its gravitational singularity at u = S. The rest mass density approaches infinity, as does the expansion, although it is unlikely to be physically plausible that the pressure would remain at zero close to the singularity. As u approaches one, the density and expansion approach zero. Thus (31) describes an anisotropic evolutionary “flat” universe. Note also that

1 u u2 − 1 σ − θ = , 11 3 C

1 3 2 2 2 which corresponds to u ;1 = u ;3 = u 1 − u /C, and u ;2 = θ − 2u 1 − u /C. In particular, the model is expanding in all directions. The second dust solution, (32) also displays typical behaviour associated with its gravitational singularity at t = e−1/C , as well as evolutionary flat behaviour. Again the model is expanding in all directions. Within the region considered, it thus differs from (31) only in quantitative behaviour. Allowing for a non-zero cosmological constant in section5 would only change the equation for µ in equations (28), and correspondingly the two latter equations for θ,t. This would alter the weak energy condition, equation (29), and might e.g. allow for `,t = 0. In section6 we studied tilted dust solutions of Bianchi types I – III in an attempt to find one with non-vanishing vorticity, thus allowing for θ,t > 0. Instead we found that any dust solution must have vanishing vorticity, although Bianchi types IV – IX were left unexplored. It is also dubious at best to draw far-gone conclusions based on these results since the restriction to standard invariant frame fields that are orthogonal on the orbits may be severe. To solve this one may first orthogonalize the invariant frame, but this will complicate the form of the k functions n ij in terms of the group algebra, see e.g. [23] and [24]. Obviously, this can also be done in orthogonal cases. Orthogonalizing the frame in the special case of Bianchi type I does k not, however, change n ij, and the result for Bianchi type I stands in general. According to [23] it is a known result that Bianchi type I solutions must have vanishing vorticity, a result that is not immediately obvious from the structure coefficients in the Appendix once we allow for non-zero acceleration.

8 Appendix

Here we give the structure coeﬃcients of tilted solutions in the frame given in Section3. The functions

k ninj k n ij ≡ − c ij, nk are determined by the normalization functions from Subsection 3.5 and the group algebra. In k k k k particular note that n ij is zero if and only if c ij is and n ij = n [ij].

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