ALGEBRAIC AND CAUSAL STRUCTURES | STATIONARY STRUCTURES - part 1

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For any point x∈M let Nǫx be the image under ex pxof the ball of radiusǫin K⊥x, for ǫ >0 sufficiently small ; letB(x,ǫ) ={t·z|z∈Nǫxand|t|< ǫ}. For some ǫ0(x)>0,ǫ < ǫ0(x) implies thatfor allz∈Nǫx,γz(the integral curve ofKwithγz(0) =z) intersectsNǫxpreciselyinz: For otherwise, there are sequences of points{zn}and{ ̄zn}inNǫx(anyǫsmallenough) approachingxwithγzn(tn) = ̄znfor sometn6= 0; we can assume alltn>0.Forǫsufficiently small,Nǫxis achronal withinB(x,ǫ) (viewed as a spacetime in itsown right), soγznmust exitB(x,ǫ) before returning to it and encountering ̄zn;hence,tn>2ǫ. Fornsufficiently large, this means (sinceγznis not an arbitrarytimelike curve but is constrained to be one of theK-curves)γznenters the futureofxbefore exitingB(x,ǫ), soγzn(sn)≫xfor somesnwith 0< sn< ǫ. Similarly,fornsufficiently large, asγznentersB(x,ǫ) on its way to ̄zn, it must enter the pastofxwithinB(x,ǫ), soγzn(s′n)≪xfor somes′nwithtn−ǫ < s′n< tn. As we alsohaveǫ < tn−ǫ, this gives usx≪γzn(sn)≪γzn(s′n)≪x, violating chronology.Letǫx= (1/2)ǫ0(x), and letUx=B(x,ǫx). The entire tubeTx=R·Nǫxxiswell behaved, with (t,z)7→t·zproviding a diffeomorphismR×Nǫxx→Tx; indeed,this is true even usingǫ0(x) forǫxinstead of half that value. Now consider anyy∈M. Ify∈Tx, then for some uniquez∈Nǫxxand some uniquet0,y=t0·z.LetWx,y=B(t0·x,ǫx); then fortsufficiently large (t·Ux)∩Wx,yis empty, astpushesUxwell above or below thet0level. Ify∈∂(Tx), we do essentially the samething:Wx,y=B(t0·x,2ǫx), as the tube works equally well at that radius. Finally,ify∈M−closure(Tx), letWx,y=M−closure(Tx), since theR-action keepsTxwithin itself.The following definitions and constructions are taken from [GHr].Any line bundle such asπ:M→Qhas a cross-sectionz:Q→M, i.e.,π(z(q)) =q; this amounts to giving a global choice of starting-time forthe clockscarried by the stationary observers. Any such choice allowsthe definition of a mapτz:M→Rbyτz(x)·z(π(x)) =x, i.e.,τzgives the elapsed time of an eventencountered by a stationary observer from that observer’sz-starting-time. (In theabsence of any confusion, this function will just be calledτ.) Note that for anycross-sectionz, (dτz)K= 1.Conversely, define aKilling time-functionas any functionτ:M→Rsatisfying(dτ)K= 1. Then sinceτnecessarily takes on all values on any oneK-orbit, wecan define a cross-sectionzτ:Q→Mby the requirement thatτ◦zτ= 0. Clearlyτ=τzτ. Also, for any cross-sectionz,zτz=z.For a Killing time-functionτ, note thatdτandαhave the same effect onK,and both are invariant under theK-action; therefore, there is onQa unique 1-formω(orωτfor specificity) such thatα−dτ=π∗ω; call this theKilling drift-formassociated with the Killing time-functionτ(or, alternatively, associated with thecorresponding cross-sectionz=zτ). Thus, we can write the spacetime metric as(1.1)g=−(Ω◦π)(dτ+π∗ω)2+π∗h,andMis static iffdω= 0 (i.e.,ωis a closed 1-form). We have a diffeomorphism(τ,π) :M→R×Q, but this is not a metric product nor even conformal to one,so long asωis non-zero.It should be noted that this formulation, encapsulated in (1.1), is slightly moregeneral than what is sometimes called astandard stationaryspacetime (see, for
example [JS]): That is a spacetimeMwith mapsτ:M→Randπ:M→Q, withQbearing a 1-formω, a Riemannian metricgQ, and a positive function Ω :Q→R,with the spacetime metric given by(1.1a)g= (Ω◦π)(−dτ2−dτ⊗π∗ω−π∗ω⊗dτ+π∗gQ).This matches up with (1.1) forh= Ω(ω2+gQ). But the difference is that inthe standard stationary formulation, it is not sufficient that the metrichinducedon the stationary observer spaceQbe Riemannian; rather, it is required that(1/Ω)h−ω2(that is,gQ) be Riemannian, i.e., that||ω||<1 for norm calculatedusing ̄h= (1/Ω)h—theconformal metriconQ, mentioned earlier. (Note thatthis restriction on the conformal norm ofωis equivalent to the image ofz—thatis, theτ= 0 hypersurface—being locally spacelike. Thus, a presentation of thespacetime as standard stationary is the same as having a spacelike cross-section;this observation is the same as Lemma 3.3 in [JS].) As will be seen in section 2,this additional requirement for the standard stationary formulation (i.e., conformalnorm ofωless than 1) is closely related to M being causal, so the extra generality allowed in the present formulation is perhaps not of great use; but this generalitymakes it possible to treat cases that might nota prioribe known to be causal.In essence,ω(or, rather,−ω) measures, from one stationary observer to theones infinitesimally close to it, the difference in starting-times for theirz-clocks, asmeasured by the universal clockK. More precisely: IfXis a vector inQand ̄Xis a lift ofXtoM, perpendicular toK, then−ω(X) = (dτ) ̄X; thus,−ωmeasuresinfinitesimal change inτ(i.e., in starting-time) in directions perpendicular to theKilling field.A good physical interpretation of the drift-formωcan be gleaned from Proposi-tion 1.3 below.At any one point, photons travel at a speed of oneτ-unit of time per one confor-mal unit of length, so a naive expectation is that a closed path of conformal lengthLinQ, traced about by a photon, would lead to the photon coming back withan elapsedτ-time ofL. It is the integral ofωover such a loop that specifies theextent to which this naive expectation is incorrect. This iswhat inspires the term“drift-form”, as one can think of what is being measured by∫cωas being an inher-ent drift or wind felt in the observer-spaceQ, affecting the transmission of a signalalong the pathcof observers; however,ωdepends on the choice of cross-section, sowe will develop another object, the “fundamental cocycle”,that does not dependon cross-section.There is a gauge freedom in the choice of the cross-sectionz: For any mapη:Q→R, we can changeztozη, defined byzη(q) =η(q)·z(q) (and this en compasses all possible cross-sections). Thenτη=τ−η◦π(i.e.,τzη=τz−η◦π)andωη=ω+dη. Thus,ωis defined up to an exact 1-form. In the static case, we canchoose the cross-section so thatωis zero on any simply-connected neighborhood (sothe static observers have the same starting-times, as measured byK), but whetherthis can be globalized depends on the global topology of M and whether the closed form ω interacts with that global topology to prevent its representation as an exactform. In the general stationary case, things are even more complicated.The static case has a simple explication of the 1-form ω: As it is closed anddefined up to an exact 1-form, it precisely defines an elementβof the first deRham cohomology ofM,H1(M;R); call it thefundamental cohomology classof
the static spacetimeM. This is most easily interpreted as being a real-valued groupmapρωon elements of the fundamental group ofQ,G=π1(Q): Choose a basepointq0inQ(better yet, choose a base pointx0∈M, and letq0=π(x0)). Thenfor any elementa∈G,ais represented by a base-pointed loopcinQ(i.e.,cstartsand ends atq0); we writea= [c]. We defineρω(a) =∫cω; thenρω:G→Risa group morphism, asρω(aa′) =∫c·c′ω=∫cω+∫c′ω, wherea= [c],a′= [c′],andc·c′indicates concatenation of curves (firstc′, thenc). As noted above,∫cωhas the physical significance of being the difference betweenconformal length ofthe loopcand change in observer-time between emission and receptionof a photonalong that path. It can be expressed in an explicitly gauge-independent manner asβ([c]) =∫ ̄cα, where ̄cis any loop inMwhich is a lift ofc(the independence amongrepresentative loopscis due to Stokes’ Theorem, as two homotopic loops form theboundary of a 2-surface—the homotopy between them).For the general stationary case, things aren’t as neat. But we can still employa form of algebraic structure by considering the Abelian groupZ(Q) (thecyclesofQ) generated by all base-pointed loopscinQ(parametrized with ̇cnever 0),subject to these relations:(1) same-direction reparametrization is irrelevant: ifc′is a reparametrizationofcin the same direction, thencandc′represent the same element;(2) reverse-parametrization is inverse: ifc′iscwith the reverse parametrization,thencandc′represent inverse elements; and(3) concatenation is sum: the concatenation ofcandc′represents the sameelement as their group sum.(This exposition will work for any manifoldQ, not just the observer space of astationary spacetime.) We write that the loopcrepresents the element〈c〉inZ(Q). Call a loopsimpleif it has no base-pointed sub-loops that are reverseparametrizations of one another. Any cycleζis then represented by an essentiallyunique base-pointed simple loopcinQ; more precisely, for anyζ∈Z(Q), thereis a unique collection of oriented (but unparametrized) base-pointed loops{ci}(with none of thecithe reversal of anothercj) such thatζis represented by theconcatenation (in any order) of the{ci}.DefineZ∗(Q) = Hom(Z(Q),R) (thecocyclesofQ). Any 1-formθonQdefinesa cocycle{θ}via integration: Ifζ=〈c〉is a cycle, then{θ}(ζ) =∫cθ, which isindependent of the representation forζ. Note that if we add any exact form toθ,it doesn’t change the cocycle: For any η:Q→R, for any loopc,∫cdη= 0, so{θ+dη}={θ}.(The definition of cycles provides perhaps the minimum identifications needed so that cycles are understood to be platforms for integration of 1-forms. But doesthis definition perhaps allow other cocycles than those coming from 1-forms? Itmight be provident to sharpen the notion of cycles so as to capture only 1-forms ascocycles. For instance, we could employ additional identifications:(4) Segment interchange: If loopscandc′are composed of concatenated seg-mentsc=σ2·σ1andc′=σ′2·σ′1, with the join-point of the segments incthesame as that inc′, then withc′′=σ′2·σ1andc′′′=σ2·σ′1,c+c′∼c′′+c′′′(i.e.,σ2·σ1·σ′2·σ′1∼σ′2·σ1·σ2·σ′1).(5) Segment reversal: If a loopcis composed of concatenated segmentsσ4·σ3·σ2·σ1withσ2andσ3the same except for being reverse-oriented, then withc′=σ4·σ1,c′∼c(i.e.,σ4·(σ2)−1·σ2·σ1∼σ4·σ1).

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FALL 2017 STATIONARY STRUCUTURES AND STATIC SPACETIMES PRESENTED AND ORGANISED BY WICEA - SPEAKER'S CONFERENCE - KARAM OUHAROU - 

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