Conical spacetimes and global hyperbolicity

0.21 MB·English

by  Guenther Hörmann

CONICAL SPACETIMES AND GLOBAL HYPERBOLICITY
5 GÜNTHERHÖRMANN
1 FAKULTÄTFÜRMATHEMATIK,UNIVERSITÄTWIEN,AUSTRIA
0
2
Abstract. VickersandWilson([26])haveshownglobalhyperbolicityofthe
r conical spacetime in the sense of well-posedness of the initial value problem
p
for the wave equation in generalized functions. We add the aspect of metric
A
splittingandpreliminarythoughtsonCauchyhypersurfacesandcausalcurves.

1
2

] 1. Introduction and set-up of the cosmic string model
h
p
Ourbasicreferencesare[27]forgeneralrelativity,[17]forsmoothdifferentialand
-
h Lorentziangeometry,and[11]forgeneralizedfunctionsandnon-smoothdifferential
t geometry. A spacetime (M,g) shall consist of a connected smooth manifold M
a
m with a symmetric covariant 2-tensor field g that is measurable (with respect to
some[henceany]LebesguemeasureonM inthesenseof[6,Section16.22]),locally
[
bounded(withrespecttosome[henceany]RiemannianmetriconM),andisalmost

3 everywherenon-degenerateandofindex1(dimensionofamaximalsubspaceinthe
v
tangent space where the metric is negative definite). Furthermore, (M,g) shall be
2
time-oriented bytheexistenceandchoiceofanowherevanishingcontinuousvector
7
6 field that is timelike almost everywhere.
0 We consider a simple model of a cosmic string in terms of the so-called coni-
0 cal spacetime (cf. [10, Section 3.4] and [22, Section 7]), the metric of which with
.
1 parameter 0 < α < 1 can be specified in cylindrical coordinates (t,r,ϕ,z)
0 R ]0, [ ] π,π[ R=:W in the form ∈
5 × ∞ × − ×
l (t,r,ϕ,z):= dt2+dr2+α2r2dϕ2+dz2.
1 α
−
:
v Itarisesfromaconstructionwiththe universalcoveringspaceuponremovingfrom
i Minkowskispacethefixedpointsetofrotationbytheangle2παinthe(x,y)-plane.
X
In [24, Section 2] James Vickers discussed the more general geometric situation
r
a producing the class of cosmic string models based on methods for ‘elementary’
quasi-regularsingularities by Ellis and Schmidt in [7, Appendix A].
The above local form of the metric clearly possesses a continuous extension l
α
to r =0 (and to arbitrary values of ϕ), which is degenerate at r =0. The smooth
transformation Φ: (t,r,ϕ,z) (t,x,y,z) := (t,rcosϕ,rsinϕ,z) is a diffeomor-
phism of W onto V :=Φ(W)7→= (t,x,y,z) R4: (x,y)=(0,0) , hence we obtain
{ ∈ 6 }
the corresponding Cartesian representation of the Lorentz metric l on the open
α
dense region V R4 as push-forward Φ l with matrix
∗ α
⊂
−1 0 0 0
0 x2+α2y2 (1−α2) xy 0
G (t,x,y,z)= x2+y2 x2+y2 
α 0 (1−α2) xy α2x2+y2 0
 x2+y2 x2+y2 
 0 0 0 1
Date:April22,2015.
1
that has smooth and bounded components on V, which define unique classes in
L∞(R4),sinceR4 V hasLebesguemeasure0. HencethisL∞-extensionofG toR4
α
isthematrixofam\etricg . The“pull-back”Φ∗g ,whereΦisconsideredasmapR
α α
[0, [ [ π,π] R R4,givesl inthesensethatDΦ(t,r,ϕ,z)t G (Φ(t,r,ϕ,z)×)
α α
∞ × − × → · ·
DΦ(t,r,ϕ,z)=diag( 1,1,α2r2,1) holds almost everywhere, namely on W.
Thus, in the Carte−sian description we have M = R4 and the Lorentz metric g
α
is given in global coordinates (t,x,y,z) by
1+α2 1 α2
g (t,x,y,z):= dt2+ + − f (x,y) dx2+(1 α2)f (x,y)dxdy
α 1 2
− (cid:18) 2 2 (cid:19) −
1+α2 1 α2
+ − f (x,y) dy2+dz2
1
(cid:18) 2 − 2 (cid:19)
with f ,f L∞(R2) represented by following functions on R2 (0,0) :
1 2
∈ \{ }
x2 y2 (x+iy)2 2xy (x+iy)2
f (x,y):= − =Re and f (x,y):= =Im .
1 x2+y2 x2+y2 2 x2+y2 x2+y2
The eigenvalues of the matrix representing g at any point (t,x,y,z) with (x,y)=
α
6
(0,0) are 1, 1 (double), and α2, thus showing non-degeneracy in the tangent
−
spaces over these points. As time-orientation of the conical spacetime we declare
the (constant) timelike vector field (1,0,0,0)to be future-directed.
We may rewrite g by splitting off the non-smooth part in the form
α
1+α2 1+α2
g (t,x,y,z):= dt2+ dx2+ dy2+dz2
α
− 2 2
ηα(t,x,y,z)
| 1 α2 {z }
+ − f (x,y)dx2 f (x,y)dy2+2f (x,y)dxdy .
1 1 2
2 −
(cid:0) (cid:1)
h(x,y)
| {z }
Notethatlim g =lim η istheMinkowskimetric andhcanbe considered
α→1 α α→1 α
a discontinuous L∞ Lorentz metric on R2, since its corresponding matrix in the
2-dimensional tangent spaces at each (x,y)=(0,0) has eigenvalues 1 and 1.
6 −
James Vickers has studied the conical spacetime and also identified it as the
‘prototype’ of a more general class of spacetimes with 2-dimensional quasi-regular
singularities already in his doctoral thesis and later in a series of research articles
(cf. [24,25] and references therein). Geometric and distributional methods have
been employed to argue that these models satisfy the Einstein equations with cer-
tain components of the energy-momentum tensor being proportional to a delta
distribution in (x,y)-subspace, or, in fact, the Lorentz metric producing distribu-
tional curvature components of such type. Clarke, Vickers, and Wilson then in
[5] for the first time set-up the model of the conical spacetime within the frame-
workof Colombeau’s generalizedfunctions and confirmedVickers’earlierresult by
investigating the distributional shadow of the generalized curvature tensor.
In [22, Section 2] Steinbauer and Vickers discuss an analysis going back to Ge-
roch and Traschen which strives for ‘minimal sufficient conditions’ on the metric
components to guarantee a well-defined distributionalcurvature (employing multi-
plicationinspacesofmeasurablefunctionsasnonlinearoperations). Inthissetting,
the notion of a so-called gt-regular metric is introduced, requiring the components
2
of the metric g with respect to coordinates to be in L∞ H1 . As noted in [22]
loc loc
∩
the conical spacetime metric is not gt-regular,but it narrowly it fails to be so.
Lemma 1.1. Thefirst-orderdistributionalderivativesoff andf belongtoL1
1 2 loc
\
L2 , thus, f ,f (L∞ W1,1) H1 .
loc 1 2 loc loc loc
∈ ∩ \
Proof. Itsufficestoargueforf ,f asfunctions(ordistributions)onR2,sincethey
1 2
areconstantwithrespecttothevariablestandz. Weobservethatboth,f andf ,
1 2
arehomogeneousoforder0,hencethefirst-orderderivatives∂ f (1 j,k 2)are
j k
homogeneous distributions of order 1. Moreover,due to smoothnes≤s on R≤2 0 ,
− \{ }
the derivatives there are given by the classical pointwise formulae, which define
locally integrable functions (due to the homogeneity degree 1 in two dimensions,
−
or as can be seen from direct inspection). By unique extension of a homogeneous
distributiononRn 0 to Rn,ifthe degreeofhomogeneityis notaninteger n
\{ } ≤−
(cf. [12, Theorem 3.2.3]), we obtain that the distributions ∂ f (1 j,k 2)
j k
≤ ≤
are given by these formulae as functions in L1 . On the other hand, since each
loc
∂ f 2 =0andishomogeneousoforder 2onR2,∂ f cannotbesquare-integrable
j k j k
|over c|om6 pact neighborhoods of 0. − (cid:3)
As a consequence of g being not gt-regular it is natural to embed the metric
into Colombeau-generalized tensor fields and analyze the conical spacetime model
as one with a generalized Lorentzian metric in the sense of [11, Subsection 3.2.5]
employing the concept of the so-called special Colombeau algebras on manifolds.
By abuse of notation, we think of g as being represented by a net (gε) of
α α ε∈]0,1]
covariant two-tensor fields (with components obtained by mollification—in fact, it
suffices to regularize f and f in this way), such that on any relatively compact
1 2
subset gε is a smooth Lorentz metric when ε is sufficiently small. In [26] Vickers
α
andWilsonhaveshownthattheconicalspacetimemaybecalledglobally hyperbolic
inthesenseof[4],namelyintermsofwell-posednessforthewaveequation. Itisour
aim here to add a few observations and aspects of generalized global hyperbolicity
to their fundamental result.
2. (cid:3)-global hyperbolicity
Clarke advocated the view that a singularity of a non-smooth spacetime, i.e.,
with g below regularity C1,1, should be manifest as an obstruction to the Cauchy
development of the physical fields in that spacetime – contrary to the standard
notion as an “obstruction to extending geodesics” (cf. [4]). A spacetime (M,g)
is therefore called (cid:3)-globally hyperbolic, if there exists a C1 spacelike hypersurface
S M withunitfuturepointing(continuous)normalvectorfieldξ suchthatM S
⊂ \
is the disjoint union of two open (connected) subsets M+ and M−, ξ points from
M− to M+, and the Cauchy problem
(cid:3) u=f, u =v, ξu =w,
g S S
| |
iswell-posedinappropriateSobolevspaces. IncaseofamerelyL∞ Lorentzmetric,
initial data (v,w) H1(S) L2(S) and f = 0, say, the distributional solution u
∈ ×
could not be expected to be better than H1 (at least in “spatial directions”) and
hence (cid:3) u does not even make sense due to ill-defined distributional products
g
involving derivatives of metric components with derivatives of u.
FortheconicalspacetimemetricVickersandWilsonin[26]interpretedthemodel
intheframeworkofColombeau-generalizedfunctionsand,basedondelicateenergy
3
estimates, proved well-posedness of the generalized Cauchy problem with data on
S given by t = 0, thus showing that the cosmic string spacetime is generalized
(cid:3)-globally hyperbolic. Moreover,they succeed in showing that in case of embedded
Cauchydatafrom(v,w) H1(S) L2(S)theColombeausolutiontothegeneralized
∈ ×
Cauchyproblempossessesa distributionalshadowin D′(M)and they were able to
identify its analytic and geometric structure to quite some detail.
A further remarkable fact about [26] is that the well-posedness result of Vickers
and Wilson for the conical spacetime is still not covered by successive theoretical
developmenton the Cauchy problem for wave equations in non-smoothspacetimes
as in [9,15].
3. Globally hyperbolic metric splitting
Let us take here as a basis for the notion of a smooth globally hyperbolic space-
time the variant defined in [13, Section 6.6] and [18, Definition 5.24], namely, the
requirements of strong causality and compactness of all intersections of causal fu-
tures and causalpasts for arbitrarypairs of points in the manifold. Among several
equivalentwaystocharacterizeglobalhyperbolicityofsmoothspacetimesthathave
been established, Bernal and Sánchez in [2] gave the most detailed “normal form
description” in terms of the following global metric splitting statement.
Theorem 3.1. (M,g) is globally hyperbolic if and only if it is isometric with a
spacetime (R S,λ) such that:
×
(a) each t S is a (smooth spacelike) Cauchy hypersurface,
{ }×
(b) λ(t,x) = θ(t,x)dt2 +ρt(x) with a smoothly parametrized family (ρt)t∈R of
−
Riemann metrics on S, and
(c) θ C∞(R S) and positive.
∈ ×
We showthat the conicalspacetime satisfiesnon-smoothversionsofthis kind of
metric splitting in both senses, one directly with L∞ Riemann metrics ρ in (b) as
t
above, and also in the sense of Colombeau-generalizedspacetimes.
Metric splitting for the conical spacetime in the L∞-setting. The natural guess for
an L∞-type metric splitting in case of the conical spacetime M = R R3 is, of
course,toputS =R3,theisometrybeingtheidentitymap,andλ=g ×= dt2+ρ
α
−
with (t-independent) ρ defined by
1+α2 1 α2 1+α2 1 α2
ρ(x,y,z):= + − f (x,y) dx2+ − f (x,y) dy2
1 1
(cid:18) 2 2 (cid:19) (cid:18) 2 − 2 (cid:19)
+(1 α2)f (x,y)dxdy+dz2.
2
−
WewillcheckthatρisanL∞ RiemannmetriconR3 andthat t R3 isaCauchy
1 2 { }×
hypersurface for g in an appropriate sense :
α
1
Note that the condition ‘spacelike’ will in general not make sense, since a hypersurface is a
zero setforanyLebesgue measure onthe manifoldandthe L∞ Lorentz metric mightbe“blind”
tothissubset.
2
Namely, defined as usual by requiring that timelike inextendible curves meet the surface
exactly once, but with some caution about the notion of timelike: With general L∞ metrics
we cannot be sure that the composition of a Lebesgue measurable metric component with a
continuous curve tangent is measurable. However, gα is Borel measurable, since f1 and f2 are
Borel measurable; this simplifies the situation, because then s 7→ gα(γ(s))(γ˙(s),γ˙(s)) is Borel
measurableforanyC1 curveγ.
4
1. Boundedness of ρ is clear, hence it suffices to show that ρ is a smooth Riemann
metric on R3 (0,0,z): z R , which is clear from the following estimate for any
tangent vecto\r{(v ,v ,v ) ∈T } S =R3
1 2 3 ∈ (x,y,z) ∼
1+α2 1 α2
ρ(v,v)= (v2+v2)+ − (f v2+2f v v f v2)+v2
2 1 2 2 1 1 2 1 2− 1 2 3
1+α2 1 α2
=v2+ (v2+v2)+ − (xv +yv )2 (yv xv )2
3 2 1 2 2(x2+y2) 1 2 − 1− 2
(cid:0) (cid:1)
≥−(yv1−xv2)2≥−(x2+y2)(v12+v22)
| {z }
1+α2 1 α2
v2+ − (v2+v2)=α2(v2+v2)+v2.
≥ 3 2 − 2 1 2 1 2 3
(cid:0) (cid:1)
2. t R3isaCauchyhypersurface: Letγ =(γ ,γ )beaC1curvethatistimelike,
0 1
{ }×
meaning γ˙(s) = 0 for every s and g (γ(s))(γ˙(s),γ˙(s)) < 0 for almost every s, and
α
inextendible; l6et denote the Euclidean norm on R3, then we have by the above
kk
estimate 0 > g (γ˙,γ˙) = γ˙2 +ρ(γ˙ ,γ˙ ) γ˙2 +α2 γ˙ 2 (almost everywhere),
α − 0s 1 1 ≥ − 0 k 1k
hence γ (s) γ (s ) > α γ˙ (τ) dτ if s = s . If γ was bounded above or
| 0 − 0 0 | | s0k 1 k | 6 0 0
below then the Euclidean leRngth of γ were bounded, hence γ (by monotonicity)
1 0
as well as γ would have endpoints, i.e., γ could not be inextendible. If γ is
1 0
unboundedaboveandbelow,then the imageof γ isR, hence the curveγ musthit
0
t R3 at least once, and by strict monotonicity of γ at most once.
0
{ }×
Metric splitting in the Colombeau-setting. Considering now the conical spacetime
as a generalizedLorentzmanifold, we show that it also qualifies for anappropriate
variantof the metric splitting. The correspondinggeneralizationof this notionhas
been introduced in [15] and been shown to provide a sufficient condition for well-
posedness of the Cauchy problem for the wave equation; [14] discusses a slightly
extendeddefinitionofthemetricsplitting,inordertoallowformoregeneralisome-
tries that were needed to cover models of discontinuous wave-type spacetimes.
Definition 3.2. Let g be a generalized Lorentz metric on the smooth (n +1)-
dimensional manifold M. We say that (M,g) allows a globally hyperbolic metric
splitting if there exists a generalized diffeomorphism Φ: M R S, where S is
→ ×
an n-dimensional smooth manifold such that the following holds for the pushed
forwardgeneralized Lorentz metric λ:=Φ g on R S:
∗
×
(a) There is a representative (λ ) of λ such that every λ is a Lorentz metric
ε ε ε
and each slice t S with arbitrary t R is a (smooth, spacelike) Cauchy
0 0
{ } × ∈
hypersurface for every λ .
ε
(b) We have the metric splitting of λ in the form
λ= θdt2+H,
−
wθheGre(RH S∈)iΓsGg(lpobr∗2a(lTly20bSo)u)nidseda atn-ddelpoecnadlleynutngifeonremralylizpeodsitRivieem,ia.en.n,ifaonrsmometer(icheanncde
∈ ×
any) representative (θ ) of θ and for every relatively compact subset K S we
ε ε
⊂
can find a constant C >0 such that θ (t,x) C holds for small ε>0 and x K.
ε
≥ ∈
(c) ForeveryT >0thereexistsarepresentative(H ) ofH andasmoothcomplete
ε ε
Riemannian metric h on S which uniformly bounds H from below in the following
sense: for all t [ T,T],x S, v T S, and ε ]0,1]
x
∈ − ∈ ∈ ∈
(H ) (v,v) h(v,v).
ε t
≥
5
Remark 3.3. In fact, condition (a) follows from (b) and (c), though it might
be more instructive for the notion itself to have it still explicitly in the list of
basic properties. To show the redundancy of (a) we argue as follows to prove that
t S is a Cauchy hypersurface for every λ : Timelike curves γ = (γ ,γ ) with
0 ε 0 1
{ }×
respect to λ are characterized by 0 > (λ ) (γ˙(s),γ˙(s)) = θ (γ(s))γ˙ (s)2 +
ε ε (γ(s)) ε 0
−
(H ) (γ˙ (s),γ˙ (s)), which implies that s γ (s) is strictly monotonic, since
ε γ0(s) 1 1 7→ 0
γ˙2 >H (γ˙ ,γ˙ )/θ (γ)>0;if γ is supposedto be timelike inextendible, then either
0 ε 1 1 ε
γ is unbounded (above and below) or γ has no endpoints; if γ were bounded
0 1 0
aboveorbelow,then(bymonotonicity)γ˙ 0towardsanintervalboundaryofthe
0
→
curve, hence H (γ˙ ,γ˙ ) 0 and therefore γ˙ (s) = 0 for every s near the interval
ε 1 1 1
→
boundary by (c), which implies that γ has an endpoint; thus, γ is unbounded
1 0
aboveand below, i.e., there exists s in the curve interval suchthat γ (s )=t , so
0 0 0 0
that γ(s) t S. By strict monotonicity of γ , the curve hits t S at most
0 0 0
∈{ }× { }×
once.
Forthe generalizedfunction versionofthe conicalspacetimewe trywithasplit-
ting of the metric representative in the form gε = dt2+ρε with
α −
1+α2 1 α2 1+α2 1 α2
ρε(x,y,z):= + − fε(x,y) dx2+ − fε(x,y) dy2
(cid:18) 2 2 1 (cid:19) (cid:18) 2 − 2 1 (cid:19)
+(1 α2)fε(x,y)dxdy+dz2.
− 2
Clearly, (b) is satisfied with S =R3, Φ the identity map, and θ =1.
We will identify conditions on the regularization and/or on α for the following
to hold: β >0, (x,y,z) R3, v =(v ,v ,v ) T S =R3
∃ ∀ ∈ ∀ 1 2 3 ∈ (x,y,z) ∼
(1) ρε(x,y,z)(v,v) β(v2+v2)+v2 0<ε 1.
≥ 1 2 3 ≤
Then condition (c) will hold as well. Thus, the explicit globally hyperbolic metric
splitting of the conical spacetime will be established, once (1) is guaranteed.
We observe that ρε(x,y,z)(v,v)=
:=Fε(x,y)
1+α2 1 α2 fε(x,y) fε(x,y) v
v2+ (v2+v2)+ − (v ,v ) 1 2 1
3 2 1 2 2 1 2 ·z(cid:18)f2ε(x,y) }|−f1ε(x,y)(cid:19){·(cid:18)v2(cid:19)
1+α2 1 α2
v2+ (v2+v2)+ − µ (x,y)(v2+v2),
≥ 3 2 1 2 2 ε 1 2
where µ (x,y) is the smaller eigenvalue of F (x,y). Since det(F (x,y) µI ) =
ε ε ε 2
−
µ2 fε(x,y)2 fε(x,y)2,wehaveµ (x,y)= fε(x,y)+ifε(x,y). Hence,ourfinal
− 1 − 2 ε −| 1 2 |
taskistoprovideagoodupperboundfor fε+ifε . Notethatfε+ifε isobtained
| 1 2| 1 2
from f +if by convolution regularization with a δ-net in the form φ (x,y) :=
1 2 ε
φ(x/ε,y/ε)/ε2, where φ S(R2) is real-valued (to ensure real components of
∈
gε) with φ = 1; hence fε+ifε f +if φ = 1 φ and
α k 1 2kL∞ ≤ k 1 2kL∞k εkL1 · k kL1
µ (x,y) R φ , therefore ρε(x,y,z)(v,v) v2+ 1−kφkL1+α2(1+kφkL1)(v2+v2),
ε ≥−k kL1 ≥ 3 2 1 2
i.e., (1) holds, if 2β := 1 φ +α2(1+ φ ) is positive, which in turn is
−k kL1 k kL1
equivalent to requiring
φ 1
(2) α2 > k kL1 − =:c .
φ
φ +1
k kL1
Note that 0 c < 1, since φ 1. We have several regularization policies to
≤ φ k kL1 ≥
guarantee (2), and thereby also (1):
6
Variant A: Choosing φ nonnegative we obtain c =0, hence (1) holds with β =α2.
φ
But then we cannot require the higher moment conditions ∂l∂kφ(0)=0 (l,k 1),
1 2 ≥
thus we drop the strict equality of smooth functions with their (classes of) φ-
b
convolvedregularizations within the algebra of generalized functions.
Variant B: We choose φ in the standard way, i.e., including the higher moment
conditions,butobtaintheabovekindofmetricsplitting(withβ = α2−cφ )only
2(kφkL1+1)
for restricted values of α in the interval ]√cφ,1]. Or, in partial reverse, if α>0 is
given, then we may choose a mollifier φ (including the higher moment conditions)
with φ 1 so small that c < α2 holds (the possibility of this follows from
k kL1 − φ
more general results concerning constructions of one-dimensional mollifiers with
prescribedmomentsin[21]andtakingtensorproductstoobtainhigherdimensional
variants).
Variant C: Instead of using a model delta regularization based on scaling a single
mollifier φ we could follow the smoothing method introduced in [23, Lemma 4.3]
by employing a strict delta net (ψ ) with ψ 1 (as ε 0) and putting
ε k εkL1 → →
fε = f ψ ; then the inequality analogous to (2) (with ψ in place of φ) will be
j j ∗ ε ε
valid for any α, if ε is sufficiently small.
Remark 3.4. On first impression one might expect that a discussion of condition
(2) would be avoidable by employing the full Colombeau algebra instead of the
special version used here. Then (at least with the algebra on R4) essentially the
mollifier sets A ,A ,..., where those in A satisfy moment conditions up to order
0 1 p
p, take over the role of the parameter ε; however, as far as we could see, we would
still run into an estimation argument very similar to the above and end up with
the inequalityin(2),unless onlynonnegativemollifiersfromA willbe considered.
p
4. Cauchy hypersurface in a generalized sense
The conicalspacetime appears in [7] as an example showing a quasi-regular sin-
gularity alongthetwo-dimensionalsubmanifoldwhere(x,y)=(0,0),whichcannot
be detected by unboundedness of curvature components (measured in a parallely
propagated frame) near the singularity; the singularity was identified as one of a
topological nature, so that the spacetime cannot be extended into that region in
a globally consistent way. Thus Ellis and Schmidt concluded that “there can be
no global Cauchy surfaces in these space-times”. James Vickers later discussed the
extendabilityofthespacetimeinadistributiontheoreticsettingandinawidergeo-
metric context in [24,25],prior to moving on to a full analysis in the framework of
generalizedfunctionsstartingwith[5]andwithfundamentalresultsonthe Cauchy
problem for the wave equation in [26], which implicitly indicate that it should be
possibletoconsidertheinitialvaluehypersurface t=0 aCauchyhypersurfacein
{ }
anappropriatelygeneralizedsense. Infact,thiswasoneofClarke’smajorpointsof
motivation in establishing (cid:3)-global hyperbolicity in [4]: Several models with non-
smooth metrics are known in general relativity that violate the cosmic censorship
hypothesis–hencearenotgloballyhyperbolic–duetonecessarydeletionofpoints
of non-smoothness from the universe in order to save the usual geometric methods
(basedongeodesicsetc),despitethefactthatonecanachieveglobalwell-posedness
of the Cauchy problems for the physical fields in a weaker sense; still, the initial
value surface then cannot be a Cauchy hypersurface in the classical sense, since
7
Geroch proved in [8] that global hyperbolicity of a spacetime (M,g) is equivalent
to the existence of a Cauchy hypersurface S.
AstandarddefinitionofaCauchy hypersurface S M requiresthateveryinex-
⊂
tendible timelike curve intersects S exactly once ([17, Chapter 14, Definition 28]).
In the context of differential geometry with generalized objects, such as discontin-
uous vector and tensor fields, a difficulty arises as how to define the notion of an
inextendible generalized curve, because therein the required non-existence of limits
toward the interval boundaries seems to have no natural generalization. However,
theremightbeanalternativeapproachbyrememberingthatforaC1 curvedefined
as map on an open interval I into the manifold M one can resort to an arclength
parametrization with respect to a complete Riemannian metric, more precisely:
Choose a complete Riemannian metric η on M and let γ: I M be C1. Pick
t →
t I, then the arclength parameter s(t) := η(γ(τ))(γ˙(τ),γ˙(τ))dτ (giving
n0eg∈ative values, if t < t ) defines a continuousRnto0npdecreasing map s: I R such
0
→
that
(3) γ is inextendible if and only if s(I)=R.
Proof. If s(I) = R, then s is bounded above or below near one of the interval
6
boundaries of I and, by monotonicity, s(t ) has a limit as t I approaches
n n
that boundary; in particular, (s(t )) is a Cauchy sequence in R∈. Let d denote
n
the distance function on M induced by the complete Riemann metric η, then
d(γ(t ),γ(t )) s(t ) s(t ) (length of the corresponding curve segment),
n m n m
≤ | − |
showing that (γ(t )) is a Cauchy sequence in the metric space (M,d). Hence γ
n
is extendible. On the other hand, if γ is extendible, say the limit of γ(t) exists
as t approaches the right boundary point r, then s has finite length and s is
[t0,r[
bounded above, hence s(I)=R. (cid:3)
6
Thus,amongallregularC1curves,i.e.,C1mapsλ: I M suchthatλ˙(t)=0for
→ 6
all t I, precisely the inextendible curves possess a reparametrizationin the form
γ: R∈ M withη(γ(s))(γ˙(s),γ˙(s))=1foralls R. Wewillpickthisasastarting
→ ∈
pointforourfirstapproachtothenotionofinextendibilityinthegeneralizedsense,
but we refrain from turning this into an official definition, because its dependence
on the allowed class of generalized reparametrizations is troublesome.
As for the notion of a timelike generalized curve with respect to a generalized
Lorentzmetric,thenaturaldefinitionclearlywouldbebetorequirestrictnegativity
of g(γ˙,γ˙) as generalized function on an interval. However, for technical reasons in
the proof following below we strengthen this by a uniformity requirement. As
above, we do see this only as a preliminary concept (because it is parametrization
dependent) not justifying a definition of such notion yet.
More systematic and deeper attempts to investigate appropriate notions of a
generalized Cauchy hypersurface for generalized spacetimes will remain a future
task. Asapreliminaryobservationwemaypresentthefollowingstatement,inwhich
ecouldbereplacedbyanycompleteRiemannianmetric,theconditionofnormalized
tangentvectorstogetherwithrequiringthatthegeneralizedmapdefiningthecurve
hasdomainR mimics the inextendibility condition, andauniformstrictnegativity
property implements a strong variant of the property “timelike”.
Proposition4.1. Let(R4,g )bethegeneralizedconicalspacetime,S = 0 R3,
α
and e denote the Euclidean metric on R4. Suppose the generalized cu{rv}e×γ
∈
8
G[R,R4] is parametrized such that e(γ˙,γ˙) = 1 holds in G(R) and g (γ˙,γ˙) is uni-
α
formly strictlynegative,i.e.,foranyrepresentatives(γε) γ and(gε)
ε∈]0,1] ∈ α ε∈]0,1] ∈
g there exist q [0, [ and ε ]0,1] such that
α 0
∈ ∞ ∈
gε(γ (s))(γ˙ (s),γ˙ (s)) εq (s R,0<ε<ε ).
α ε ε ε ≤− ∈ 0
Then γ meets S in a unique (compactly supported) generalized point value, i.e.,
there is a unique s˜ (R˜)c such that γ(s˜) S˜c.
∈ ∈
Proof. Letγ berepresentedbyγ =(γε,γε): R R R3 andg berepresentedby
ε 0 1 → × α
gε. We will also employ the metric splitting established in Section 3, in particular
α
the estimate (1). The property of γ being uniformly timelike then implies that we
have with some q R and ε ]0,1]:
0
∈ ∈
εq gε(γ˙ ,γ˙ )= (γ˙ε)2+ρε(γ˙ε,γ˙ε) (γ˙ε)2+β γ˙ε 2 (0<ε<ε ),
− ≥ α ε ε − 0 1 1 ≥− 0 k 1k 0
where β here denotes the minimum of 1 and the constant occurring in (1); in
particular,
1 1
(⋆) γ˙ε(s) 2 (γ˙ε(s)2 εq) γ˙ε(s)2 s R,0<ε<ε .
k 1 k ≤ β 0 − ≤ β 0 ∀ ∈ 0
Theparametrization(“inextendibility”)conditionforγ gives(γ˙ε)2+ γ˙ε 2 =1+n
0 k 1k ε
with negligible (n ) , and combination with (⋆) implies
ε ε∈]0,1]
1 β+1
1+n (s) γ˙ε(s)2+ γ˙ε(s)2 = γ˙ε(s)2,
ε ≤ 0 β 0 β 0
hence γ˙ε(s)2 β (1+n (s)) β 1, if ε>0 is sufficiently small; in summary,
0 ≥ β+1 ε ≥ β+1· 2
there is some ε ]0,1] such that
1
∈
β
(⋆⋆) γ˙ε(s)2 s R,0<ε<ε .
0 ≥ 2(β+1) ∀ ∈ 1
This implies that γε: R R is bijective, if 0 < ε < ε . Let s denote the unique
point in R such tha0t γε(s→)=0 and put p :=γ (s )=1(0,γε(sε )) S.
0 ε ε ε ε 1 ε ∈
To see that (s ) is c-bounded we write
ε
0 0
d dτ
s =(γε)−1(0)= (γε)−1(τ)dτ =
ε 0 Z dτ 0 Z γ˙ε((γε)−1(τ))
0 0
γε(0) γε(0)
0 0
and note that (⋆⋆) implies 1/γ˙ε 2(β+1)/β, thus s γε(0) 2(β+1)/β
| 0| ≤ | ε| ≤ | 0 |
and we may call on the c-boundednepss of γ0. Thus (sε) defines a comppactly sup-
ported generalized point s˜ (R)c and p˜ = γ(s˜) in the sense of generalized point
∈
valueswith p˜representedby (p ). Since γ is c-boundedthe generalizedpoint value
ε
e
p˜=γ(s˜) belongs to Sc.
To prove uniqueness let us assume that q˜= γ(t˜) S with t˜ (R)c. We have
e ∈ ∈
γ (t˜)=0=γ (s˜),hence0=γ (t˜) γ (s˜)= t˜γ˙ (τ)dτ (the integralunderstoodε-
0 0 0 − 0 s˜ 0 e e
wiseonrepresentatives,includingtheupperaRndlowerlimit). By(⋆⋆)wehavethat
γ˙ isstrictly nonzero(evenboundedawayfromzerobyaclassicalconstant),there-
0
fore it is impossible to find any interval of length with non-negligible asymptotics
between s and t , hence t˜=s˜(and q˜=p˜). (cid:3)
ε ε
9
Example 4.2. (i) The generalized curve γ G[R,R4] with representative given
∈
by γ (s) = (s,sin(1),0,0) satisfies the hypothesis of the proposition. It meets the
ε ε
“Cauchy hypersurface” S in a generalized point with accumulation points forming
the subset 0 [ 1,1] (0,0) .
(ii)TheCol{om}b×ea−u-gene×ra{lizedm}apλ G(R,R4)representedbyλ (s)=(s,1,0,0)
∈ ε ε
is not c-bounded, hence not considered a generalized curve, though it satisfies the
other conditions in the above proposition. The “image” of λ meets S in a gener-
alized point which is not compactly supported and has no accumulation points of
e
corresponding representative nets.
5. Causal curves and approximations
Whatisageneralizedcausalcurve? Anotionofcausalcurveinthegeneralizedsense
seemsdelicate fortworeasons: First,aconditionlikeg(γ˙,γ˙) 0 inthe generalized
≤
functionsensewouldnotguaranteetohaveg -causalrepresentativesγ ofγ;second,
ε ε
the existence of zero divisors in the set of generalized numbers raises the question
whether a nonzero, but non free, tangent vector v with g(v,v) = 0 should be
consideredlightlike, spacelike,or neither. Even requiring regularizationsto consist
ofcausalcurvesbringsustothepracticalquestionofapproximation. Itcouldhavea
varietyofmeanings,atleasttwoofwhichonewouldhopetobeessentiallyequivalent
infairlygeneralclassicalsituations: Causalcurveshavingimagesetswithinagiven
openset,orcausalcurvesbeinguniformlycloseasmappingsfromintervalsintothe
manifold. Well-knownresultsonthesequestionsaddressthesequentialconvergence
in strongly causal Lorentz manifolds or abstract order theoretic constructions for
the globally hyperbolic case. We want to use the opportunity here to a provide
(or give a detailed review of) a precise statement, in what sense the versions using
classes of curves and specific parametrizations are topologically comparable.
Two topologies for continuous causal curves: A generalized Lorentz metric satis-
fying the globally hyperbolic metric splitting according to Definition 3.2 possesses
representatives consisting of smooth globally hyperbolic Lorentz metrics. Thus, in
these regularizations we may take advantage of very good causality properties, in
particular, including strong causality etc. As discussed, e.g., in the review article
[20] by Sánchez global hyperbolicity of a smooth spacetime (M,g) has been de-
scribed in [3] in terms of compactness with respect to the compact-open topology
of the set of causal curves connecting two points and parametrized proportionally
to arclengthonthe interval[0,1],whereas moreoften the so-calledC0-topologyon
the set of equivalence classes or images of causal curves is being employed instead
(e.g. in [16,18,27]). We investigate the relation between the corresponding topo-
logicalspacesofcausalcurvesorclassesofcausalcurves. Wemayreferto[19](and
the references therein) for more details on the analysis of comparability of several
related topologies for Lorentzian metrics that are merely continuous.
Let h be a complete Riemannian metric with induced metric d on M and
h
denotethe compact-opentopologyonC([0,1],M)(andsubsetsthereof)byτc.o.. It
is metrizable by the metric ρ of uniform convergenceρ(λ,γ)=sup d (λ(s),γ(s))
h
{ |
s [0,1] . Let p,q M satisfy p < q in the sense of the causal relation (as in
∈ } ∈
[17, Chapter 14]); p q means p<q or p=q.
≤
Recall that a continuous curve γ: [0,1] M is said to be causal, if for any
→
convex open subset U of M and s ,s [0,1], s s with γ([s ,s ]) U, we
1 2 1 2 1 2
∈ ≤ ⊂
have γ(s ) γ(s ) (relative U). Note that limits of continuous causal curves in
1 2
≤
10