The Geometry of the Frame Bundle over Spacetime

Fredrik Ståhl

A b s tr a c t . One of the known mathematical descriptions of singularities in Gen­ eral Relativity is the b-boundary, which is a way of attaching endpoints to inex­ tendible endless curves in a spacetime. The b-boundary of a manifoldM with connection T is constructed by forming the Cauchy completion of the frame bundle LM equipped with a certain Riemannian metric, the b-metricG. We study the geometry of {LM, G) as a Riemannian manifold in the case whenT is the Levi-Cività connection of a Lorentzian metricg on M. In particular, we give expressions for the curvature and discuss the isometries and the geodesics of {LM, G) in relation to the geometry of {M,g).

1. Introduction

In general relativity, the concept of singularities is unavoidable. Known solutions to Einsteins field equations display a variety of non-trivial singularities, giving rise to such diverse phenomena as black holes, the big bang and topological anomalies. The situation is very different from most other field theories, where singular solu­ tions may be explained as artefacts of idealised modelling (like point charges) or differentiability restrictions (like shock waves), for example. In order to study sin­ gularities, it is important to have a mathematical machinery that allows us to treat questions like convergence or divergence of physical quantities when approaching the singularity. In other words, one would like to incorporate the singularities to­ gether with the regular spacetime points in some abstract set, equipped with a suit­ able topology that allows one to define statements such as close to the singularity’ in a mathematically precise sense. One definition of this kind is the b-boundary [15]. Given a manifoldM with connection T, the b-boundary Mof is formed by constructing a suitable metric, the b-metric G, on the bundle of framesLM . The pair {LM , G) can then be viewed as a Riemannian manifold, and in particular, as a metric space. The b-boundary is formed by taking the Cauchy completion of LM( , G), and a projection then gives an extensionM of the original manifoldM . We leave the details to the next section. The b-boundary construction has several drawbacks however, the most import­ ant being that the topology on the extended Mset is non-Hausdorff in general (see,

1 e.g., [3, 5, 17]). There has been some attempts to remedy the situation (see [6, 7]), although they have not been entirely successful. In order to obtain a more complete understanding of what goes wrong with the b-boundary, we need to understand the geometry LMof (, G) better. This is the subject at hand. Since the object of interest is spacetime, we restrict attention to the case when T is the Levi-Cività connection of a Lorentzian metricg on M. The outline of the paper is as follows: in §2, we go through the essential steps of the b-boundary definition. In §3 and §4, we calculate the connection and curvature of {LM , G) and discuss some of the implications. Finally, §5 and §6 are devoted to a discussion of the isometries and the geodesics ofLM ( , G) in relation to the corresponding structures on (Af,g).

2. The b-boundary

We will attach an abstract boundary set to a Lorentzian manifold (Af,g)i where M is a smooth n -dimensional connected orientable Hausdorff manifold with a smooth metric g of signature n — 2. The case of interest in relativity theory is of course when n = 4. The construction of the b-boundary may be carried out in different bundles over M (see [15], [5] or [11] for some background). It is often convenient to work with the bundle of pseudo-orthonormal frames OAT, consisting of all pseudo­ orthonormal frames at all points ofM. In our case this leads to complications because of the amount of algebra involved, so we choose instead to construct the b-boundary via the bundle of general linear framesLM. LM is a principal fibre bundle with the general linear groupGL{n) on R” as its structure group. We write the right action of an elementA G GL{n) as Ra : E >—> EA for E G LM. Given an orientation of Af, the frame bundleLM has two connected components which we denote byL^M andL~M , corresponding to frames with positive and negative orientation, respectively. We will use the notationL'M for any one of these two components. Clearly, anyA G GL(n) which changes the orientation sets up a 1-1 correspondence betweenL^M andL~M, and we may regard the component of the identityGL\n) C GL{n) as the structure group of L'M. The fibre bundle structure ofLM gives a canonical 1-form0 : T (LM ) —» R”, and the Levi-Cività connection corresponds to a connection form T{LM)u; : —> flï(»), where Qi(n) is the Lie algebra of GL{n) [13]. Let (•, •)]&« and (*, be Euclidian inner products with respect to fixed bases in R” and £jl(«), respectively. We define a Riemannian metricG on LM , the b-metric or Schmidt metric , by G{X,Y) := (e(X),0(Y))w + (u,(X)my))*k*) (2*!)

2 (c.f. [15, 5]). Then G can be shown to be uniformly equivalent under a change of bases in R” andgi(n) [15, 5]. If 7 is a curve in LM , the b-length of 7 is the length of 7with respect to the b-metric G, and is denoted by7 /(). Thus

f(7 ):= / ( |0 ( 7 ) |2 + lk(7)l|2) 1/2^ , (2.2) where |*| and ||-|| are the fixed Euclidian norms in R” andgi(n), respectively, and7 is the tangent of7. If 7is horizontal, 0^(7) =0 and7 may be written as a pair (À, E) where À = 71-07 is a curve in M andE is the parallel frame along À given by 7. By definition, 0(7 ) is the vector of the components of the tangent A in the frameE. So the b-length of a horizontal curve7 is equivalent to the length measured by a comoving observer along 71-07 (which is called generalised affine parameter length’ in [11]). This motivates further study of the b-metric, since the presence of an endless curve with finite generalised affine parameter length is often taken as the criterion for a spacetime to be singular. Following Schmidt [15], we now use the b-metricG to construct a topological boundary of the base manifoldM , providing endpoints for all endless curves with finite length. Since(L'M, G) is a Riemannian manifold, it is a metric space with topological metric d (the b-metric distance function), and so the Cauchy comple­ tion L’M of L'M is well defined.L'M is a complete metric space, and we define the boundary ofL'M as dL'M := L'M \ L'M. The topological metric d then has a unique extensiond to L'M. It can be shown that the action ofGL'(n) is uniformly continuous on(L'M, G), viewed as a metric space [15, 5]. It follows that there is a unique uniformly continuous extension of the right action ofGL'(n) to (L'M, G). Justified by the above, we may now construct the topological space

M — ÜM/GLXn), (2.3) the set of orbits of GL'(n) in L'M. We can also define a continuous projection 7f : L'M —> M taking a point in L'M to the corresponding orbit ofGL'(n). By definition,n coincides with7 r on L'M. Thus it (L'M) = M may be regarded as a subset of M, and we define theb-boundary of M as dM := M \ M. It is important to emphasise that the topological structure of L'M may be quite complicated. In particular, in many relevant casesL'M is non-Hausdorff [1, 12, 3, 17]. This is related to the possibility of ‘fibre degeneracy’, as the fibre bundle structure usually cannot be extended toL'M. A boundary orbit may even be a single point [2, 17].

3 For the rest of this paper we will, somewhat sloppily, write LM instead ofL!M andGL{n) instead ofG L \n ).

3. Cartans equations and the connection

It is clear from the definition (2.1) ofG that El := 0l together with := (3.1) is a global orthonormal coframe field onLM ( , G). If we let uppercase latin indices take values in the extended index set {/ 0... = n} U { { / , /'}; /, = 0... «}, E1 is a basis for the cotangent spaceT*{LM). Note that the brackets denote ordered pairs and not unordered sets. If we denote the connection form{LM of, G) by uj, applying Cartans first equation with vanishing torsion gives

dE 1 + ܻ1; A EJ = 0. (3.2) We can also apply Cartans equations to6 andu) on M, which gives

àel + a ei = o (3.3) and tl) = dcj’j + ui‘k A u kj, (3.4) where fiy = \ R'jkl O^O1 is the curvature 2-form andR ljki are the frame components of the Riemann tensor of{M ,g). Combining (3.2) with (3.3) and (3.4) we get the system

Cblj A E^ = ujj A (3.5) ù {y ‘ ] A EJ = u>) R'i’jk & 0 k- We also have the metric condition CJy — —ÙjJ/. (3.6) Solving (3.5-3.6) for G: we get ù,' = îk -^)-ïE âV ‘. k,l + W - \Y^R]r,kek, (3.7) k

= i Slj(u)lj> — i>) -f- à1 j’{

4 4. The curvature

Our objective is now to compute the curvature of (LM , G) expressed in the basis E 1. If we denote the curvature form onLM by Û, Cartans second equation on LM gives

j = d c +c A U^j. (4.1)

In order to calculate d6j we first need an expression forRljkh d which is given by the following lemma. Lemma4 .1. The exterior derivatives o f the frame components o f the Riemann tensor, viewed as functions on LM , are

àR*jkl = — Rmjkl + R'mkl U™j + U)™ + R)km U™I + R'jkhm Q™ • (4-2) Here Rljkl-,m we the frame components o f the covariant derivative o f the Riemann tensor of{M ,g).

Proof Let Ei be the standard horizontal vector fields dual 6l,to fixed by the choice of basis for R”. Similarly, let -£{;/} be the fundamental vertical vector fields dual to corresponding to the choice of basis for gl(w). ClearlyEj = {£;, -£[/,*'}} is a basis for T(LM). Moreover, it is the unique dual ofE 1. From the definition of the exterior derivative,

dR)kl = Y . E^ R)kl) 0W + E E{m,m'}(R,jkl) « V (4.3) m m,rri To proceed further we introduce coordinates onLM as follows. Let p £ M and let at1 be coordinates forM on a neighbourhoodU of p. Given a frame F = (Fi) at a point q ÇlU, we may express each Fi as

* - ( * ? ) « • <4-4) The determinant of the matrixX is nonzero, so we may use(xa,X^i) as coordinates for LM on ir~ 1 (U). The coordinate expressions for the horizontal vector fieldsEi and the vertical vector fields <£{*•/} are then

<4-5)

- («!?)«■ (46) where Tbac are the Christoffel symbols of (M ,g ) in the coordinatesx a [13]. Now the Riemann tensor frame componentsRljki are related to the coordinate compon­ ents R%cd by

R)ki = (4.7) where X -Ms the inverse of X . Applying (4.5—4.6) to (4.7) and inserting into (4.3) then gives the desired result. □

With the help of (3.7), (3.3-3.4) and Lemma 4.1, it is possible to solve (4.1) for the curvature form Û, and then calculate the Riemann tensor, the Ricci tensor and the curvature scalar. It is a trivial but long and tedious exercise, so we will not describe it here. Some of the results are given in Appendix A, here we just give the expression for the curvature scalar R:

R = - \ n 2{n + 3) - \ £ (R)kl)2 + <4-8)

* where Rljki andRij are the frame components of the Riemann and Ricci tensor of (M,g), respectively. What is the relevance of (4.8) in relation to singularities? A b-incomplete endless curve À in M has an endpointp on the b-boundarydM. The horizontal lift 7of A has finite b-length and endsdLM. at In many cases, some frame component of the Riemann tensor will diverge along7 (c.f. [11, 3, 4, 8]). Using the terminology of [8], we say that p is a curvature singularity. We treat the case when the divergence is unbounded. Proposition 4.2. Let 7be a horizontal curve ending at dLM, and suppose that some frame component o f the Riemann tensor o f (M,g) tends to ±00 along 7. Then the curvature scalar R o(LM, f G) tends to — 00 along 7.

Proof The last term in (4.8) may be written as

(4-9) which is clearly dominated by the second term if someRljki —* ±00. □

We now turn to the case when the frame components of the Riemann tensor are bounded on a curve ending at the boundary point. In this case the obstruction to extendingM as a spacetime could be either some oscillatory divergence of the curvature, or purely topological. One might ask if it is possible that (LM, G) can be extended as a Riemannian manifold even(M, if g) is inextendible. We answer this question in the case when the boundary fibre is totally degenerate, in a sense which we now specify. The b-boundary construction outlined §2in can also be carried out using the bundle of pseudo-orthonormal framesOM. OM is a fibre bundle with the Lorentz group C as its structure group, and there is a natural inclusionOM C LM such that OM is a reduced subbundle ofLM [13]. The b-metric is defined onOM by (2.1), i.e., exactly in the same way as for LM. Since the connection inOM is simply the reduction of the connection inLM, (OM, G) is a Riemannian submanifold of (LM,G). We write OM anddOM for the Cauchy completion and boundary of OM, respectively. Then an alternative definition (see [5, 9]) of the b-boundary is dM := M \M with

M := ÖM/C. (4.10)

In [17], it was shown that for many exact solutions in general relativity, the ‘fibre’ (or, more correctly, the orbit of the extended group action) over a pointp G dM in OM is totally degenerate, i.e., a single point. SincedO M C dLM, the fibre in LM is degenerate as well, though possibly not completely.

Proposition 4.3. Suppose that (L M , G) is (locally) extendible through it~l(p), where p G dM, and that the corresponding boundary fibre in OM is completely degenerate. Then (M,g) is asymptotically a conformally flat Einstein space, i.e. the curvature R o f (M,g) is given by Rljki = ^Rg^gijj in the limit at p.

Proof. Let À: [0, 1] —> OM C LM be a horizontal curve ending at7f-1(/>) such that the restriction to [0,1) is contained inOM. Since (LM,G) is extendible through 7r_1(/>), the curvature scalar R must have a well defined limit along A(*) ast —> 1. The restriction of7 f-1 (p) to OM is a single point, so the limit ofR must be invariant under the action of any Lorentz transformationL G C, changing the curve according to XL. Now the curvature scalar R is given by (4.8), which may be rewritten as

R = - \n2{n + 3 ) - I i + R - 2 J2 ( < (hŸ + 2*oo, (4.11) Oi,ß, 7 where 0 is the timelike index, greek indices go from 1 ton — 1, R is the curvature scalar andI is the scalar invariantR*Jkl Rijki of (M , g ). Since / andR are scalar invariants, we only need to consider the last two terms in (4.11). Given the frame E at A(*), we apply Lorentz transformations in then — 1 timelike planes spanned byE$ andEa, a = 1 , 2 , — 1. After some algebra we

7 find that (4.11) is invariant if and only if

R%y = Rßo^n (4.12) * % i9 = R \- tß, (4.13) R^aOß= 0 ifa^/3, (4.14) /?V = ° if a ^ {/3,7 ,e}. (4.15) From (4.13) and (4.14), the Ricci tensor is given by

Rij = \R g ij , (4.16) which is the condition for an Einstein space [16]. Applying the firstBianchi iden­ tity to (4.12) and permuting the indices shows that i?°a/57 = 0. Thus (4.12-4.15) implies

R)kl = \Rg\k9D r (4.17) which means that the Weyl tensor vanishes. □

5. Isometries

5.1. Horizontal isometries. We seek isometries of (LM , G) which have horizontal orbits. Any transformationp of M induces an automorphismp of LM taking a frame E = (Eq, E2,..., ü»_i) at p E M to the frame p — (p*Eo, p*E \,..., ptEn-\) at p(p) G M. A transformation of(M,g) is said to be affine if it preserves the connection. Obviously, the group of affine transformations includes the isometry group. From [13] we have the following.

Proposition 5.1. (1) For any p, p leaves the canonical 1-form 0 invariant. Moreover; any fibre- preserving automorphism o f LM that leaves 6 invariant is induced by a transformation o f M. (2) The fibre-preserving automorphisms o f L M that leaves both the canonical 1-form 6 and the connection form lj invariant are exactly those that are induced by the affine transformations o f (M , g ). It is worth noting thatp maps OM into itself if and only ifp is an isometry of (M , g) [13]. From Proposition 5.1 we draw the following conclusion which is apparent from the definition2 .(1) of the b-metric. Corollary 5.2. Any affine transformation p of (M,g) induces an isometry p o f (LM, G).

8 There might of course be other isometries of (LM, G) induced by non-affine transformations of(M,g). They are characterised by the property that they pre­ serve the inner product(u:,^)^ny This includes transformations that change the connection form according tou; i—> AwB where A andB belong to the rotation group 0(n).

5.2. Vertical isometries. Our objective is now to find all vertical isometries of (LM, G), i.e., the isometries where each orbit is contained in a single fibre LM.of Assume that V = V1 Ej is a vertical Killing vector field. Then the corresponding covector Vj is given by Vi = 0 and , (5.1) for some function a from LM to the Lie algebra ßt(w), and satisfies the Killing equation V{I.j) = 0. (5.2) Here the semicolon denotes a covariant derivative with respect toG and the paren­ theses denote symmetrisation. From the expressions (3.7) for the connection formCj we may identify the con­ nection coefficients in the frameE on LM. Then (5.2) becomes *(i/) = (5.3) Ei(ajk) = 0, (5.4) E{i,j}(akl) + E{k,l}(aij) - 4(ik)àji + a{ji)&ik = (5-5) From (5.3), a is skew, and from (5.4), a is a function of the fibre coordinates only. Loosely speaking, a is the same on all fibres (locally). Thus (5.5) gives E{i,j}(aki) + E{k,l}(aij) = 0. (5.6) This equation is obviously fulfilled ifa is a constant element ino(n), the Lie algebra of the rotation group 0(n). Then V is the Killing vector field of a global isometry generated by the right actionR a of A = exptf E 0(n). (This can of course be seen directly from the definition2 .1 () of the b-metric, using the transformation properties of 0 and u; under the actionRa .) Instead of searching for non-constant solutions to (5.6), we reformulate the prob­ lem as finding local isometries of a single fibre which is warranted by (5.4). Since T is isomorphic to GL(n), we may view a^j as the Killing vector field of a local isometry of GL(n) with the metric Gp induced byG. By the definition of the connection form u>,Gjr is invariant under theleft action ofGL(n ), but not necessarily under the right action. To study this in more detail, we introduce co­ ordinates onT as in §4. Letxl be coordinates in a neighbourhood of the base point 7r(^r) G M. Then any frameF = (F,) G T can be expressed byFi = for some X G GL(n), and the mapip : T —> GL(n) given by Z7 i—>X is an isomorphism. The restriction of the connection formu to the fibre is

(5.7) andGjr is given by

G?{X,Y) = (u>{X)MY))Mny (5.8) Apparently, we may view Gjr as a metric on GL(n). The invariance ofGjr under left translationsX —> AX is apparent from (5.7). But we also see that G? is not invariant under the right action ofGL(n). Underp, the orthonormal basis E{ij} of T(T) is mapped to the orthonormal basis eij of gl(n) used when defining the inner product 0(•,i(w). -) From the fact that a is skew and (5.6 ) we get

(eij + eji)(ak/) = 0, (5.9) which means that a is completely specified by its value on 0(n) C GL(n). Also, from (5.7 ) and 5(.8), the restriction ofG? to 0[n) is invariant under both left and right translations of0(n). Since 0(n) is a compact Lie group, there is only one choice of a bi-invariant metric on0(n), up to a constant factor [10]. We have thus arrived at a complete characterisation of the vertical isometries of (ZAf,G).

Proposition 5.3. The group of (local) vertical isometries o f (LM , G) is isomorphic to the group o f (local) isometries of0(n) equipped with the 'canonical bi-invariant metric.

6. Geodesics

Suppose that 7 is a geodesic of(LM, G)y with tangent vectorvl Ei 4- VlJ E^jy and affine parameter t. Reading off the connection coefficients from (3.7) and inserting into the geodesic equationVyV = 0 gives

** = E vjipj + E Rkujvk‘vi > (6.1) j V”' = -v ‘v1' + Y^{VkiVki' - VikVik), (6.2) k where the dot denotes an ordinary derivative with respectt. to Note that the right hand side of6 (.2) is symmetric, which means that the skew part of Vn must be constant.

10 6.1. Vertical geodesics. As we saw in §5.2, any fibre T with the metric Gjr induced by G is isometric to GL(n) with the metric generated by the inner product (•, -)öi(w)- Thus the vertical geodesics are simply the geodesics ofGL(n ) with respect to this metric, so the vertical geodesics are not coupled to the geometry of(M,g) at all. Still, we clarify a few points. If v1 = 0, (6.1) is automatically satisfied and (6.2) becomes

V“ = J2 (vk>vkl' - VikVik). (6.3) k

We see that if V is constant and skew, it satisfies (6.3). In other words, orbits ofR a are geodesics if and only Aif E 0(n). This is not surprising since0(n) is a compact Lie group, andGjr corresponds to the canonical bi-invariant metric onO(w). Thus (0(n), Gjr) is a normal homogeneous space, and it is a property of such spaces that the geodesics are given by orbits of right actions 10 [ ]. Note that this means that (0(n), Gjr) is a totally geodesic submanifold ofGL(n ( ), Gjr). Constant symmetricV are also solutions of (6.3). Although it looks nonlinear, (6.3) actually has a simple structure. Let W andC be the symmetric and skew part of V, respectively. As noted above,C must be constant, so (6.3) becomes

W“' = - 2 Y, («Cikw ki' + Ci>kW ki), (6.4) k i.e., a system of linear ordinary differential equations W.in

6.2. Horizontal geodesics. If 7 is horizontal, Vlt =0 along 7 . It follows from (6 .2) that v1 = 0, so there are no horizontal geodesics ofLM ( , G). In particular, no geodesic of(LM, G) is a horizontal lift of a geodesic of M,( g ). In [18], properties of curves 7 in M with extremal length was studied, with the length measured in a parallel pseudo-orthonormal frame along7 . It was found that for 7 to be a geodesic of(M, g), it is necessary that

5tjW !RJklmW kW l = 0, (6.5) where W l is the tangent vector of7 andRJklm are the components of the Riemann tensor, both expressed in the parallel frame. This is a severe restriction on(M,g). Locally, geodesics may be considered as extremal curves of the length functional. We can reformulate the result in [18] as the following: even if we restrict attention to horizontal curves inOM, an extremal of the b-length functional cannot be expected to be a lift of a geodesic in(M, g).

11 7. Discussion

There are many open questions as to the structure of(LM, G) for physically inter­ esting spacetimes (M,g ). In fact, even for the 2-dimensional Schwarzschild space­ time, it is not known if the b-boundary is of dimension0 or 1. Hopefully, geometric methods in(LM, G) may help to shed some light on the situation. But it should be remembered that actual calculations with the b-boundary is difficult even in the two-dimensional case. Proposition 4.2 also raises some questions. Loosely speaking, thatR —» — oo may be interpreted as that the geometry becomes ‘infinitely hyperbolic’ at the boundary point. However, not all sectional curvatures have to diverge, which means that the circumference’ of the boundary point may be finite in some planes while it is infinite in other planes. Also, the geodesics of (LM, G) may be very complicated in general, with no obvious connection to the geodesics of(M,g). The geodesic equations 6 ( .1- 6 .2) suggest the possibility of a kind of oscillatory behaviour of the horizontal compon­ ents vl of the tangent vector.

Appendix A. Expressions for the curvature

As described in §3, the curvature formÛ of (LM, G) can be found by applying (3.3-3.4) and Lemma 4.1 to (3.7). Because of the different metrics involved, some care is needed to keep track of which metric is used for contractions and raising and lowering of indices. Note that covariant and contravariant components in the frame Ej — (6l, u>V) may be identified, since the frame components Gof is given by

Gij := Sij, Gi{jj'} — Gjjj’^i = 0 and = (A. 1)

So we may use 5, the Kronecker delta, for index operations. The frame components canonical form0 , the connection form u; and the curvature tensorR of (M, g ) refer to the components0*, u*j andRljki in the frame on(M,g) specified by the loca­ tion inLM . If we raise and lower indices withö, an ambiguity arises when applying symmetries and contractions to the Riemann tensor. For example,Rijki will not be equal to Rjikl since the first index is lowered withS instead ofg . Therefore we keep the index positions fixed and adopt the convention that all repeated indices repres­ ent contractions, regardless of their variance. We contend ourselves with showing the results, since the calculations involve a substantial amount of algebra.

12 The components of the curvature form ft in the frame Ej = (0*, u>V) are n*. = \ 0 ‘A0f + \ (R ‘lkj - R’lkt + Rklij - R mm'ik Rmm'jl - R mm'ijRmm'kl) & A 01 — \ R ll'ij-,k 6k A <*>'/'— \ { d k + d ì ) A + w *)

+ 5 ÿ A U>”/ + AU3mI + d m A u /m)

+ \ Rklim'd*I A (u>”y + U^m) + I RkIjm d I A (w” + l*/OT)

- \ Rkk’im RlI'jm d k ' A U>//, (A.2)

+ I ( V 'ad i -O’ A d j> - S ij 0 k A ( d ' k + dj>) - öif 0k A d j )

- \ (Rlrif O’ + Rllnj O’) A J v - \ R’m 0k A ( d t + d t)

— \ Rlj'ik 6k A {d[ + u>lj) + I R’nk 0k A { d i + d j )

+ \ R’llk OkA d t - \ Rljik 0k A d j' - \ Rli'imR’j'mk 0k A d r , (A.3)

;,/} = ~\{&i'j' 0‘ A O’ + 6i>j O' A O’’ + 6i f O' A O’ + 5ij 0'' A O’) + \ O’ + R \ji O’ ' - R’j'i'i 0 i - R’j'ii 0 ‘) A 0l

+ \ (ôi’j>(Rjkt - Rjikl) + àij{R‘j'kl - R-'i'kl) + öi'jRfkl - ôj'iR’i'kl) 9k a 0l - \ R'i'km R)'lm 0k A 0 l + \ {dj A dj' - di A d'i) — \ fy {(dk + di') A (dj' + d k) + d i ' A d j )

— \ fy' {{d + d i) A (d j + d k) + d k A d k ) - \ fy {d k A dk+ di' A d j) - \ ôi>j {dk A d k + d i A d j). (A.4)

From (A.2-A.4) we can obtain the components of the Riemann tensor, and a con­ traction then gives the Ricci tensor.

Rij — — &ij — 5 Rkk'uRkk’ji + R'j- (A.5) RiVJ'i = R{jj'}> = \ R)'ik;k j (A.6 ) R{i,i'}{j,j'} = - ^ 2 ^ d ' d ~ ”2" fySpj' + \ ôu'ôjj' + \ R'i'kk'R-’j'kk'- (A.7)

Contracting again gives the curvature scalarR as given in §4, equation (4.8).

13 Acknowledgements

The calculations of the connection and curvature forms were checked with a modi­ fied version of theRicci package [14] for Mathematica by Wolfram Research, Inc.

References

[1] B. Bosshard, On the b-boundary of the closed Friedman-model, Comm. Math. Phys. 46 (1976), 263-268. [2] C. J. S. Clarke,The singular holonomy group, Comm. Math. Phys. 58 (1978), 291-297. [3] ______, The analysis of space-time singularities, Cambridge University Press, Cambridge, 1993. [4] C. J. S. Clarke and B. G. Schmidt,Singularities: The state o f the art, Gen. Rei. Grav. 8 (1977), no. 2, 129-137. [5] C. T. J. Dodson, Space-time edge geometry, Int. J. Theor. Phys. 17 (1978), no. 6, 389-504. [6] ______, A new bundle completion for parallelizable space-times, Gen. Rei. Grav. 10 (1979), no. 12, 969-976. [7] C. T. J. Dodson and L. J. Sulley,On bundle completion for parallelizable manifolds, Math. Proc. Cambridge Philos. Soc. 87 (1980), 523-525. [8] G. R R. Ellis and B. G. Schmidt,Singular space-times, Gen. Rei. Grav. 8 (1977), no. 11, 915— 953. [9] H. Friedrich, Construction and properties o f space-time b-boundaries, Gen. Rei. Grav. 5 (1974), 681-697. [10] S. Gallot, D. Hulin, andj. Lafontaine,Riemannian geometry, Springer, Berlin, 1987. [11] S. W. Hawking and G. F. R. Ellis,The large scale structure of space-time, Cambridge Univ. Press, Cambridge, 1973. [12] R. Johnson, The bundle boundary in some special cases, J. Math. Phys. 18 (1977), 898-902. [13] S. Kobayashi and K. Nomizu,Foundations of differential geometry, vol. I, John Wiley & Sons, New York, 1963. [14] J. M. Lee, Ricci — a mathematica package for doing tensor calculations in differential geometry, Available on the internet athttp : //www.math. washi ngton. edu/~l ee/Ri cci /. [15] B. G. Schmidt, A new definition of singular points in general relativity, Gen. Rei. Grav. 1 (1971), no. 3, 269-280. [16] M. Spivak,A comprehensive introduction to differential geometry, vol. IV, Publish or Perish, Bo­ ston, Mass., 1975. [17] F. Ståhl,Degeneracy of the b-boundary in general relativity, Comm. Math. Phys. 208 (1999), 331-353. [18] ______, On imprisoned curves and b-length in general relativity, Preprint, University of Umeå, 2000.

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