Spacetimes with continuous linear isotropies II: boosts

Conditions are found which ensure that local boost invariance (LBI), invariance under a linear boost isotropy, implies local boost symmetry (LBS), i.e. the existence of a local group of motions such that for every point $P$ in a neighbourhood there is a boost leaving $P$ fixed. It is shown that for Petrov type D spacetimes this requires LBI of the Riemann tensor and its first derivative. That is also true for most conformally flat spacetimes, but those with Ricci tensors of Segre type [1(11,1)] may require LBI of the first three derivatives of curvature to ensure LBS.


Introduction
In the first paper of this series (MacCallum, 2021) spacetimes with a local spatial rotational invariance were (re)-investigated. Here the corresponding issues for local boost invariance are studied. The issues arising and the methods to be used are set out in Sections 1 and 2 of the previous paper. Only those key points required to make this paper reasonably self-contained will be repeated here.
In studying local rotational symmetry of spacetime (LRS), Ellis (1967) introduced three definitions which he showed to be equivalent for dust spacetimes. The results were later extended to spacetimes with a perfect fluid and electromagnetic field (Stewart and Ellis, 1968). Here Ellis's definition (Am) will be studied. It reads: (Am) At each point P in an open neighbourhood U of a point Po, there exists a nondiscrete subgroup g of the Lorentz group in the tangent space T P which leaves invariant the curvature tensor and all its covariant derivatives to the m-th order. Implicit in this definition are conditions on the smoothness of the manifold and the correspondence between the g at separated points. This paper considers the case where the group g of linear isotropies contains boosts. This is called local boost invariance (LBI), 'local' meaning that the same g applies throughout U . 'Spacetime' here just means a four-dimensional Lorentzian manifold. The field equations of general relativity will not be used, but in cases where the Ricci tensor takes the form that would be implied by specific matter content in general relativity that interpretation is referred to.
The starting conjecture (based on a claim by Siklos (1976) now known to be false in general) is that (A 1 ) is sufficent to imply Ellis's definition (C), i.e.
There exists a local group of motions Gr in an open neighbourhood W of a point Po which is multiply transitive on some q [-dimensional] surface through each point P of W .
With LBI this would imply that the spacetime had local boost symmetry (LBS), i.e. that the group Gr contained, for every P ∈ W , a subgroup of boosts leaving P fixed. A theorem of Hall (1989) implies, under appropriate topological and smoothness conditions, that W is a region of a manifold in which the same Gr acts globally.
Only Petrov type D and conformally flat spacetimes have a Weyl tensor that can satisfy (A 0 ) (or (Am) for larger m) with a group g containing boosts. There are rather more Ricci tensor types that can satisfy (A 0 ) for boosts. For (A 0 ) to apply to the whole Riemann tensor, the Weyl and Ricci tensors must of course be appropriately aligned, and there may be a nonzero Ricci scalar. The tracefree part of the Ricci tensor can be characterized by its Segre type. The possible invariance groups of the Ricci tensor were listed by Segre type in Table 5.2 of Stephani et al. (2003). Table 1 here lists those with nontrivial invariance groupsÎ 0 which include a boost. Cahen and Defrise (1968) showed that for Petrov type D spacetimes with boost (or spatial rotation) invariance and any compatible Ricci tensor, (A 2 ) was a sufficient criterion for the spacetime to be LRS or LBS. Subsequently Goode and Wainwright (1986) gave criteria for the LRS Petrov type D case in terms of the spin coefficients and curvature expressed in a Newman-Penrose (NP) null tetrad. These criteria were shown in MacCallum (2021) to be equivalent to (A 1 ). The discrepancy with Cahen and Defrise's use of (A 2 ) is shown in the Appendix of MacCallum (2021) to be due to a less suitable choice of frame in the calculations.
Here it is shown in Section 2, by arguments parallel to those of Goode and Wainwright (1986), that (A 1 ) is also sufficient for LBS in Petrov type D spacetimes. In Section 3, the corresponding question for conformally flat spacetimes is studied. In both these sections the detailed arguments are closely related to those of MacCallum (2021) by the asterisk operation of the GHP formalism (Geroch et al., 1973). A recent preprint addresses local boost invariance of higher dimensional manifolds (McNutt et al., 2019).
As in the previous paper, the Cartan-Karlhede procedure for characterizing spacetimes and testing their equivalence, as outlined in Section 2 of MacCallum (2021), is used. It relies on the computation of "Cartan invariants", the components of the Riemann tensor and its covariant derivatives in canonically chosen frames. The implementation used here employs the Newman-Penrose formalism as set out in Chapter 7 of Stephani et al. (2003). The "Newman-Penrose equations" (the Ricci equations) and Bianchi identities [(7.21a)-(7.21r) and (7.32a)-(7.32k) in Stephani et al. (2003)] will be referred to below as (NPa)-(NPr) and (Ba)-(Bk).
A minimal set of Cartan invariants sufficient for the above procedure, was defined by MacCallum andÅman (1986). It consists of totally symmetrized spinor derivatives of the Newman-Penrose curvature quantities. Here the shorthand notation for such spinors, as defined in MacCallum (2021), will be used. If Q ABC...
is a relevant curvature quantity then the notation Q AB ′ denotes the component of Q (ABC...) (E ′ F ′ ...) in which A of the m unprimed indices and B of the n primed indices are contracted with the basis spinors ι andῑ respectively (and the others with the basis spinors o andō). χ is said to have valence (m, n). The set defined in MacCallum andÅman (1986) consists of the totally symmetrized derivatives of Ψ , Φ and Λ, together with, at order 1, Ξ DEF W ′ = ∇ C W ′ Ψ CDEF and at order q + 2, the d'Alembertians of quantities at order q. For a totally-symmetrized spinor of valence (m, n), only components with 2(A + B) = m + n are LBI.

Petrov type D spacetimes with local boost invariance
In the calculations, the boost invariance is assumed to act in the (k, l) plane of a Newman-Penrose tetrad adapted to the Petrov type D Weyl tensor and to leave the Riemann tensor and its first derivative unchanged 1 . Λ and Ψ 2 ( = 0) are boost invariant, so from (A 1 ) DΛ = ∆Λ = DΨ 2 = ∆Ψ 2 = 0. In Φ AB ′ only the components Φ 11 ′ and Φ 02 ′ can be nonzero. Boost invariance of the Cartan invariants ∇Ψ AB ′ requires that These spacetimes are members of Kundt's class. Therefore, in Petrov type D, invariance of the Riemann tensor and its first derivatives under a boost implies that there is a Newman-Penrose tetrad (a canonical one for Petrov type D, fixed up to a spatial rotation and boost) in which the following criteria hold.
To complete the check of the equivalence of the conditions (C1 † )-(C3 † ) with the assumption that the Riemann tensor and its first derivatives are boost invariant, one has to show that once the remaining frame freedom, a spatial rotation, has been fixed, so that Φ 02 ′ is an invariant, DΦ 02 ′ = ∆Φ 02 ′ = 0, which follows if ε and γ are real. (The boost invariance of Ξ AB ′ is readily checked.) These and other restrictions on the spin coefficients analogous to those in Section 3 of MacCallum (2021) are now sought, following analogous steps in Goode and Wainwright (1986). They will enable the LBI of higher derivatives of the curvature to be checked. From (Bh) and (Bj) one finds Applying the [δ, D] commutator to (−Ψ 2 − Φ 11 ′ + Λ) and using (NPc) [which tells us that Dτ = (ε −ε)τ ], (Bb) and Ψ 2 = 0 one obtains Dπ = −(ε −ε)π.
One can show, as in the LRS case, that in Petrov type D one cannot have s = 1 and t 2 = 1, t 1 = t 0 = 0 which would require (A 4 ) to be checked. To eliminate the possibility, the calculations follow a similar logic to those in MacCallum (2021). Necessarily Ψ 2 = 0, and Φ 11 ′ and Φ 02 ′ would be constant. That ∇Ψ AB ′ is constant implies τ and π are constant. If at least one of them is nonzero, then (NPg) and/or (NPp) imply thatᾱ − β is constant (possibly zero). Direct calculation (using CLASSI) then shows t 2 = 0 i.e. all terms in the second derivatives are also constant, so the Cartan-Karlhede procedure terminates. If both π and τ are zero, (NPg) implies Φ 02 ′ = 0, and then inspection (using CLASSI) shows that all first derivatives of the Riemann tensor, and hence all higher derivatives, are zero, and the Cartan-Karlhede procedure terminates at step 1.
Thus for LBS Petrov type D spacetimes it is sufficient to check (A 3 ) and the Cartan-Karlhede procedure must terminate at the third step or earlier. This proves the following.
Theorem 2 If a spacetime of Petrov type D is such that the Riemann tensor and its first derivative are invariant under a local boost invariance, then the spacetime is locally boost symmetric and admits a local isometry group Gr (r ≥ 3).
The converse of Theorem 2 is obvious, and by the equivalence shown above this proves Theorem 1. Note that as in Section 3 of MacCallum (2021) the invariance of the derivatives of the Ricci tensor has not been used to derive the results, only checked, and Ψ 2 = 0 was used only in deriving (C1 † ) and (2.5). The LBS conclusion depends only on (2.1), (2.10) and (2.8).
In the following section it is shown that Theorem 2 is still true with 'Petrov type D' replaced by 'conformally flat', unless the Ricci tensor is of Segre type [1(11,1)] when some cases require LBI of the curvature and its first three derivatives to ensure LBS.

Conformally flat spacetimes with local boost invariance
The conformally flat cases to be considered are those Ricci tensor types appearing in Table 1. (Am) is assumed to hold with a group g which contains a boost. By the same argument as in MacCallum (2021), Ricci tensors of Segre type [(111,1)] are easily disposed of: the spacetimes are of constant curvature, the subgroup g in (Am) is the trivial one comprising the whole Lorentz group, s = 6, tp = 0 = t 0 , and there is a group G 10 transitive on the whole spacetime. The Cartan-Karlhede procedure terminates at the first step and (A 0 ) suffices because it will imply (A 1 ).
In the rest of this section the actual (Am) required for local LBS in conformally flat spacetimes with the various Ricci tensors which admit a boost invariance, but are of less symmetry than Segre type [(111,1)], are studied.

The first derivatives and Bianchi identities
The first step is to impose LBI on the first derivatives of Φ and Λ. Then one can try to derive (C1 † ), which were obtained in Petrov type D cases from invariance of ∇Ψ , and look for an appropriate variant of the rest of the arguments in Section 2.
(2.6) holds (being just (NPc) and (NPi)) and (2.5) then follows simply from it, usingπ + τ = 0. One can then obtain (2.7) and (2.8), and complete the proof that (A 1 ) is sufficient to imply LBS in this case as in Section 2. One may note that since DΦ 02 ′ = ∆Φ 02 ′ = 0, c is constant in the timelike two-planes determined by the boost.
If s = 2 one must have t 1 = 0 and so the Cartan-Karlhede procedure terminates at step 1 since neither s nor t has changed and as in Section 3 of MacCallum (2021) this gives the Bertotti-Robinson type solutions with a G 6 transitive on the whole spacetime, and (A 1 ) suffices.

Ricci tensors of Segre type [1(11,1)]
Here the invariance group is SO(2,1), generated by null rotations about k and about l and a boost in the (k, l) plane. The SO(2,1) group acts in a hyperplane and leaves invariant one direction in the (m, m) plane. The Ricci tensor represents a tachyonic fluid, and a canonical form for it which is manifestly null rotation invariant (as in MacCallum (2020)) about each of the null directions has only Φ 11 ′ and Φ 02 ′ non-zero with 2|Φ 11 ′ | = |Φ 02 ′ |. (This is a specialization of the form for Segre type [11(1,1)], treated above.) Using the remaining freedom of spatial rotation in the (m, m) plane one can set Φ 02 ′ = 2Φ 11 ′ : the parameters of both null rotations are then pure imaginary and the vector orthogonal to the hyperplane in which the SO(2,1) acts is in the direction m + m.
The boost rescales the parameters of the null rotations. One might therefore have invariance under a two-dimensional subgroup of SO(2,1) generated by the boost and one of the null rotations (see entry R6 in Table 6.1 of Hall (2004)). Among the quantities defined by MacCallum andÅman (1986), only ∇ k Φ AB ′ and ∇ k Λ AB ′ , both of which are Hermitian, and d'Alembertians thereof, have to be considered. Boost invariance implies that of the components of ∇ k Φ AB ′ , only those with A + B = 2 + k can be nonzero. If then ∇ k Φ AB ′ is invariant under one of the null rotations, then from the Hermitian symmetry it will also be invariant under the other. Thus ∇ k Φ AB ′ will be SO(2,1) invariant. The same applies to ∇ k Λ AB ′ and the d'Alembertians of spinors of lower derivative order.
So s = 2 is impossible and either s = 3 or s = 1. If s = 3 there can be at most one independent function of position (as there is just one spacelike direction fixed under g). The spacetime admits a G 6 (or in special cases a G 7 , cf. Rebouças and Teixeira (1992)) acting on timelike hypersurfaces of constant curvature. The metrics include analogues of the FLRW metrics for perfect fluids.
In case (a), we find that ∇ k Φ and ∇ k Λ are invariant under a null rotation about k for k = 1 . . . 3 and therefore s = 3 by the earlier argument that s i cannot be 2. We thus again have at least a G 6 if ℑ αβ = 0. Note that (A 2 ) was checked but does not give extra conditions in this case.

Conclusion
The work in Sections 2 and 3 gives the following analogue of Theorem 3 of Mac-Callum (2021).
Theorem 3 In spacetimes with a Ricci tensor of Segre type [1(11, 1)] whose distinguished spacelike eigenvector is not geodesic, local boost invariance of the curvature and its derivatives up to the third holds if and only if the spacetime is locally boost symmetric. In all other cases, local boost invariance of the curvature and its first derivatives holds if and only if the spacetime is locally boost symmetric.