a r X i v :g r -q c /0102013v 1 5 F e
b 2001Gravitational-wave dynamics and black-hole dynamics:
cond quasi-spherical approximation
Sean A.Hayward
Asia Pacific Center for Theoretical Physics,
The Korea Foundation for Advanced Study Building 7th Floor,Yoksam-dong 678-39,Kangnam-gu,Seoul 135-081,Korea
and
Department of Physics,Konkuk University,93-1Mojin-dong,Kwangjin-gu,Seoul 143-701,Korea
hayward@mail.apctp
(2nd February 2001)
Gravitational radiation with roughly spherical wavefronts,produced by roughly spherical black
holes or other astrophysical objects,is described by an approximation scheme.The first quasi-
spherical approximation,describing radiation propagation on a background,is generalized to in-
clude additional non-linear effects,due to the radiation itlf.The gravitational radiation is locally
defined and admits an energy tensor,satisfying all standard local energy conditions and entering
the truncated Einstein equations as an effective energy tensor.This cond quasi-spherical approx-
imation thereby includes gravitational radiation reaction,such as the back-reaction on the black
hole.With respect to a canonical flow of time,the combined energy-momentum of the matter and
gravitational radiation is covariantly conrved.The corresponding Noether charge is a local gravi-
tational mass-energy.Energy conrvation is formulated as a local first law relating the gradient of
the gravitational mass to work and energy-supply terms,including the energy flux of the gravita-丢书大作战
tional radiation.Zeroth,first and cond laws of black-hole dynamics are given,involving a dynamic
surface gravity.Local gravitational-wave dynamics is described by a non-linear wave equation.In
terms of a complex gravitational-radiation potential,the energy tensor has a scalar-field form and
the wave equation is an Ernst equation,holding independently at each spherical angle.The strain
to be measured by a distant detector is simply defined.
04.30.-w,04.25.-g,04.70.Bw,04.20.Ha
I.INTRODUCTION Gravitational waves and black holes are among the most popular and intensively investigated topics in physics at the turn of the millenium.Both are predictions of Einstein’s General Relativity [1]which involve esntially relativistic gravitational effects.Gravitational waves were originally introduced by Einstein himlf [2]and the first black-hole solution was also almost immediately discovered [3],though the term black hole was coined much later [4].In the remainder of the twentieth century,astrophysical evidence eventually accumulated to the point where it is nowadays believed that black holes are not only common in the univer,but astrophysically dominant energy sources and appreciable mass concentrations,being the final remnant of any sufficiently massive star,with supermassive black
holes powering active galactic nuclei and lurking at the heart of most other [5].Gravitational waves are expected to make a similar transition from theory to obrvation with the operation of veral new gravitational-wave [6].Obrvation of gravitational waves from black holes would provide the first direct evidence for the existence of the latter,rather than the impressive but indirect evidence of their effect on other astrophysical bodies or surrounding matter.The combination of the two intertwined topics,gravitational waves and black holes,pos an exceptional challenge for theorists.
For instance,an expected source of detectable gravitational waves is the inspiral and coalescence of a binary black-hole system.Although the earliest and latest stages are understood in terms of post-Newtonian [7]and clo-limit approximations [8]respectively,the coalescence is generally thought to be tractable only by full numerical simulations
[9,10].Textbook theory simply does not suffice to understand the process in physical terms.Stationary black holes are well understood [11–13],but the black holes of interest are highly dynamical.Asymptotic or weak gravitational waves are well understood,respectively by Bondi-Penro theory [14–20]and Wheeler-Isaacson high-frequency (or merely linearized)theory [21–26],but strong gravitational waves produced by a distorted,rapidly evolving black hole are not unders
tood at all.Indeed,it is sometimes argued that there is nothing to physically understand,merely complicated equations to numerically integrate,as gravitational waves cannot be localized [25].The main purpo of this article is to address this lack of relevant theory by providing an astrophysically realistic approximation scheme in which both gravitational radiation and black holes are locally defined,along with their physical attributes,with each dynamically influencing the other.The approximation can be simply stated:it holds where the gravitational wavefronts and black
五行图hole(or other astrophysical object)are roughly spherical.This does not contradict the classically quadrupolar nature of gravitational waves,as the wave amplitude at different angles may be arbitrarily aspherical.The wavefronts,not
爱鸟护鸟的宣传语
the waves themlves,should be roughly spherical.
With such a general premi of rough sphericity,this quasi-spherical approximation scheme intuitively promis a
wide range of validity,in particular including a just-coalesced black hole.It may also prove applicable to neutron stars or supernovas,or indeed any astrophysical situation which has rough spherical symmetry.Further,for any other
山药肉丸
process which can be enclod by roughly spherical surfaces,it provides a mid-zone and,assuming isolation,far-zone approximation.Mathematically this was originally achieved[27]by linearizing certainfields which would vanish in
exact spherical symmetry,having made a decomposition of space-time adapted to the roughly spherical surfaces, henceforth called transver surfaces.This will henceforth be called thefirst quasi-spherical approximation.Since
there is no assumption of cloness to stationarity,the approximation holds for arbitrarily fast dynamical process. The approximation has also been tested against the main expected source of asphericity,angular momentum,by
applying it to Kerr black holes[28]:the error in the strain waveform is much lower than expected signals from binary black-hole coalescence.
The approximation allows a local definition of gravitational radiation esntially becau the decompodfields naturally divide into quasi-spherical variables and wave variables,the latter satisfying a wave equation and yielding
韩国党派
the Bondi news,or equivalently,the obrvable strain.Specifically,the gravitational radiation is encoded in the shear tensors of the outgoing and ingoing wavefronts.Then by retaining non-linear terms in the(previously linearized)
fields,one would expect to obtain an approximation which is more accurate for the gravitational-radiation ctor. This cond quasi-spherical approximation is also prented here.Bothfirst and cond approximations share the remarkable feature that,to compute the obrvable waveforms,no transver derivatives need be considered.The
truncated equations form an effectively two-dimensional system,already written in characteristic form by virtue of the first-order dual-null formulation,to be integrated independently at each angle of the sphere.Numerically this is much
easier to implement and computationally inexpensive when compared to the full Einstein system.Numerical codes exist for bothfirst and cond approximations[28].Moreover,the wave equation can be written as an Ernst equation in
terms of a gravitational-radiation potential.Thus the gravitational-wave dynamics is amenable to analytical methods. Thefirst approximation describes gravitational-wave propagation on a backgroun
d,though the background is neither
necessarily spherical norfixed in advance;this merely means that the quasi-spherical equations decouple from the gravitational-wave equation.There is no such decoupling in the cond approximation,and therefore no background
which is independent of the waves.Back-reaction of the waves on the geometry is thereby included.Specifically,one may define a gravitational-radiation energy tensor[29]which acts just like a matter energy tensor in the truncated
Einstein equations.Thus there is a fully relativistic inclusion of gravitational radiation reaction for dynamic black holes.For instance,if a black hole emits gravitational radiation,it backscatters to produce ingoing radiation which is
absorbed by the black hole,thereby increasing its mass and area.This last property follows from a local cond law [30,31],part of a general theory of black-hole dynamics[32],where a black hole is defined by a type of trapping horizon.
This local theory of dynamic black holes,not to be confud with textbook black-hole statics and asy
mptotics,is extended here by deriving quasi-spherical generalizations of the spherically symmetricfirst[33]and zeroth[32]laws
of black-hole dynamics,involving local definitions of mass and surface gravity.
The article is organized as follows.§II reviews the formalism of dual-null dynamics,which provides a geometrical
description of the wavefronts and the corresponding decomposition of Einstein gravity.§III describes bothfirst and cond quasi-spherical approximations and the resulting truncatedfield equations.§IV locally defines the gravitational radiation and its energy tensorΘ.A truncated Einstein tensor illustrates the role ofΘas an effective matter energy
tensor in the cond approximation.§V locally defines dynamic black holes,along with mass m and surface gravityκ.§VI derives an energy conrvation law for the combined energy-momentum of the matter and gravitational radiation. This is also formulated as afirst law relating the gradient of m to work and energy-supply terms,including the energyflux of the gravitational radiation.§VII gives zeroth,first and cond laws of black-hole dynamics and various
inequalities.§VIII introduces the complex gravitational-radiation potential,in terms of whichΘtakes a scalar-field form and the gravitational-wave equation takes an Ernst form.§IX defines conformally rescaledfields more suited to asymptotics,including a localized Bondiflux and a conformal strain tensor.§X concludes.
II.DUAL-NULL DYNAMICS
The wavefronts of outgoing and ingoing gravitational radiation form two families of null hypersurfaces,intercting in the two-parameter family of transver spatial surfaces.This geometry is described by the formalism of dual-null dynamics[34,35],summarized in this ction.Denoting the space-time metric by g and labelling the null hypersurfaces by x±,the normal1-forms n±=−dx±therefore satisfy
g −1(n ±,n ±)=0.
(1)The relative normalization of the null normals may be encoded in a function f defined by
e f =−g −1(n +,n −).
(2)Then the induced metric on the transver surfaces,the spatial surfaces of interction,is found to
be
h =g +2e −f n +⊗n −
(3)where ⊗denotes the symmetric tensor product.The dynamics is generated by two commuting evolution vectors u ±:[u +,u −]=0(4)
where the brackets denote the Lie bracket or commutator.Thus there is an integrable evolution space spanned by (u +,u −).The evolution derivatives,to be discretized in a numerical code,are the projected Lie derivatives
∆±=⊥L u ±
(5)
where ⊥indicates projection by h and L denotes the Lie derivative.There are two shift vectors
s ±=⊥u ±.(6)In a coordinate basis (u +,u −,e 1,e 2)such that u ±=∂/∂x ±,where e a =∂/∂x a is a basis for the transver surfaces,the metric takes the form
g =h ab (dx a +s a +dx ++s a −dx −)⊗(dx b +s b +dx ++s b −dx −)−2e −f dx +⊗dx −.(7)
Then (h,f,s ±)are configuration fields and the independent momentum fields are found to be linear combinations of
θ±
=¯∗L ±¯∗1(8)σ±
=⊥L ±h −θ±h (9)ν±
=L ±f
(10)
ω=12θ2±−1
2R −|
12Df ±ω)♯ +8πT +−(14)L ±ν∓=1
2θ+θ−−e −f 14|Df |2±ω♯·Df +8π T +−+12(D ♯·σ±−Dθ±+Dν±−θ±Df )∓8πT ±(16)
政治学习计划⊥L ±σ∓=σ+·h ♯·σ−±12Df ±ω)⊗(12Df ±ω) −e −f |12Df ±ω)♯ h +8πe −f ⊥T −1
where D is the covariant derivative and R the Ricci scalar of h,a dot denotes symmetric contraction,a colon denotes double symmetric contraction,a sharp(♯)denotes the contravariant dual with respect to h−1=h♯(index raising), aflat(♭)will denote the covariant dual with respect to h(index lowering),|ω|2=ω·ω♯and|σ|2=σ:σ♯.Units are such that Newton’s gravitational constant is unity.This is the Einstein system infirst-order dual-null form;the equations will simply be called thefield equations.The cond-order version obtained by eliminating the momentum fields and symmetrizing in L(+L−)are the Einstein equations themlves.
III.QUASI-SPHERICAL APPROXIMATIONS
男生有好感的初期表现
The dual-nullfields and operators fall into two class,tho which vanish in spherical symmetry,(σ±,ω,s±,D), and tho which generally do not,(θ±,ν±,h,f,∆±)[33,36].Thefirst quasi-spherical approximation[27]therefore consisted of linearizing in(σ±,ω,s±,D).It was then noticed that the gravitational radiation is encoded in the shear tensorsσ±[29],as will be explained in more detail in the following.This suggests that retaining non-linear terms in σ±would give a more accurate approximation for the gravitational-radiation ctor of the theory.Thus the cond quasi-spherical approximation consists of linearizing in(ω,s±,D)only.For a given matter model,one would also have to decide which decompod matterfields to linearize.In this article,no specific matter model will befix
ed,but the matter energy tensor T will be retained for generality and assumed to be consistently truncated.
More formally,one may introduce two expansion parameters into the fullfield equations,ǫ1precedingσ±andǫ0 preceding(ω,s±,D).Then both approximations ignore terms o(ǫ0),whereas thefirst approximation ignores terms o(ǫ1).Thenǫ1measures the strength of the gravitational radiation,whereasǫ0measures other asphericities,due to angular momentum or other transver effects.The terminology“first and cond order”has been carefully avoided becau the true cond-order quasi-spherical approximation would be full Einstein gravity,since the only non-linear terms in the fullfield equations are cond-order in the dynamicalfields and operators.However,one may say that the cond approximation is cond-order in the gravitational radiation.
It is uful to decompo the transver metric h into a conformal factor r and a transver conformal metric k by
h=r2k(21)
such that
∆±ˆ∗1=0(22)
whereˆ∗is the Hodge operator of k,satisfying¯∗1=ˆ∗r2.The Ricci scalar of h is found to be
R=2r−2(1−D2ln r)(23)
by using the coordinate freedom on a given surface tofix k as the metric of a unit sphere.One may take quasi-spherical coordinates x a=(ϑ,ϕ)on the transver surfaces to obtain the standard area form of a unit sphere:
ˆ∗1=sinϑdϑ∧dϕ(24)
where∧denotes the exterior product of forms.Then r is the quasi-spherical radius.Approximations for rough cylindrical or plane symmetry could similarly be produced.A uful truncation identity,holding in bothfirst and cond approximations,is
∆±=⊥L±.(25)
This will be ud throughout the article without further reference,with∆±rather than L±appearing explicitly.
In thefirst approximation,the truncatedfield equations decouple into a three-level hierarchy.Thefirst-level equations
∆±r=1
θ2±−8πT±±(28)
2
∆±θ∓=−θ+θ−−e−f r−2+8πT+−(29)
e−f h♯:T)(30)
∆±ν∓=−1
2
have the same form as in spherical symmetry.The equations,the quasi-spherical equations,therefore determine a quasi-spherical background.The cond-level equations
∆±k=r−2σ±(31)
(h♯:T)h)(32)∆±σ∓=1
2
constitute a wave equation for k,describing the gravitational-wave propagation,as explained in detail in the following. The quadratic shear term in the shear propagation equation(32)was omitted in the original reference[27].The third-level equations for(ω,s±)need not be solved for the radiation problem.This is becau,fixing u+to be the outgoing direction,the Bondi news at null infinityℑ±is esntiallyσ∓/r[37].This determines the strain to be measured by a gravitational-wave detector,as explained in the penultimate ction.
In the cond approximation,the truncatedfield equations also decouple,this time into only two levels,with the last level for(ω,s±)again being irrelevant to the radiation problem.The remaining equations are
∆±r=1
θ2±−1
2
θ+θ−−e−f r−2+12e−f h♯:T)(38)
2
(h♯:T)h).(39)∆±σ∓=1
2
The dual-null initial-data formulation is bad on a spatial surface S and the null hypersurfacesΣ±locally generated from S in the u±directions.The structure of the truncatedfield equations shows that one may specify(θ±,r,f,k)on S,(σ+,ν+)onΣ+and(σ−,ν−)onΣ−.In particular,the initial data is freely specifiable.In summary,the vacuum system consists of ninefirst-order differential equations and their duals.
Mathematically,the difference betweenfirst and cond approximations is that in thefirst approximation,the equations for the quasi-spherical variables(θ±,ν±,r,f)decouple from the equations for the wave variables(σ±,k). Physically this describes gravitational-wave propagation on a quasi-spherical background.The background is not fixed in advance and need not be spherically symmetric,so even thefirst approximation is widely applicable.There is no such decoupling in the c
ond approximation:the gravitational-wave terms(σ±,k)now enter the equations for the quasi-spherical part of the geometry,so there is no longer a background which is independent of the waves.Physically this corresponds to including radiation reaction,as clarified in the next ction.
Nevertheless,bothfirst and cond approximations share the remarkable feature that,to compute the obrv-able waveforms,no transver D derivatives need be considered.The truncated equations form an effectively two-dimensional system,to be integrated independently at each angle of the sphere.Physically this means that the obrved gravitational-wave signal depends only on the line of sight to the source,surely a plausible result.Moreover, by virtue of the dual-null formulation,the equations are already written in characteristic form,the mathematically standard form for analysis of hyperbolic equations.Numerical implementation is conquently straightforward and computationally inexpensive.Numerical codes exist for bothfirst and cond approximations[28].
IV.GRA VITATIONAL RADIATION:LOCAL ENERGY TENSOR
The truncated equations in the cond quasi-spherical approximation,(33)–(39),differ from tho of thefirst,(26)–(32),by terms quadratic in the shear tensorsσ±,which appear additively with terms involving the energy tensor of the matter.Specifically,the cond approximation may be obtained from thefirst by replacing the matter energy tensor T with T+Θ,where
Θ±±=σ±:σ♯±/32π(40)
Θ+−=0(41)
⊥Θ=e f(σ+:σ♯−)h/32π.(42) ThusΘplayes the role of an effective energy tensor for the gravitational radiation.Written covariantly in terms of the transver conformal metric k,this defines the energy tensor of the gravitational radiation:
Θαβ= ∆αk,∆βk −1
32π
(43)
where α,β =k ab k cdαacβbd is the transver conformal inner product and∆=dx+∆++dx−∆−is the1-form evolution derivative.
More formally,in terms of the expansion parametersǫ0andǫ1,one may introduce a truncated Einstein tensor C defined in terms of the full Einstein tensor G by
C±±=lim
ǫ1→0lim
ǫ0→0
G±±=−∆±θ±−ν±θ±−1
2
(h♯·C)h and the C±components are irrelevant to the radiation problem.Then
C=8πT(48) are the truncated Einstein equations of thefirst approximation,obtained from(26)–(32),whereas
C=8π(T+Θ)(49) are the truncated Einstein equations of the cond approximation,obtained from(33)–(39).This demonstrates thatΘis an energy tensor for the gravitational radiation,in the n that it is included as an effective matter energy tensor in the cond approximation.Gravitational radiation reaction,the back-reaction of the radiation on the space-time, has thereby been included.
It should be stresd that(i)mathematically,Θis a genuine tensor,but depends on the dual-null foliation,not just on the space-time;(ii)the physical interpretation ofΘas energy requires the quasi-sph
erical approximation to be valid,meaning that the transver surfaces must indeed be roughly spherical.This will not be made preci here, as the range of validity of the approximation is not clear in advance and best explored in applications.The intuitive meaning of roughly spherical should be clear by any standards.
In summary,the quasi-spherical approximation allows a local definition of the energy-momentum-stress of gravita-tional radiation,and therefore of the radiation itlf:gravitational radiation is prent at a given point if and only if Θis non-zero there.With the orientation such that u+is the outgoing null direction,there is outgoing radiation if and only if∆−k(equivalentlyσ−)is non-zero,and ingoing radiation if and only if∆+k(equivalentlyσ+)is non-zero. The terminology gravitational radiation rather than wave is generally preferable,since∆±k need not be oscillatory. Instead,frequency spectra for ingoing and outgoing radiation may be defined by Fourier transformations to frequency f+and f−respectively:
k±(f±)= γe−2πif±x±k(x±)dx±(50)
whereγis a curve of constant(x∓,ϑ,ϕ).If the Fourier transform is peaked in frequency space,one may say that there is a gravitational wave.In contrast,gravitational radiation is defined even when there is no typical frequency. This clearly indicates that the approximation has a different physical basis to th
at of the Isaacson high-frequency approximation[24,25],which is usually quoted to make n of linearized gravitational waves.三国刘封
The non-zero components ofΘmay be written as
Θ±±=||∆±k||2/32π(51)
⊥Θ=e f ∆+k,∆−k h/32π(52) where||α||2= α,α is the transver conformal norm.Then theΘ±±components have a similar form to tho of the gravitational-wave energy tensor of the high-frequency approximation[24,25],with k replacing the transver traceless metric perturbation.However,the high-frequency approximation requires averaging over veral wavelengths and has no term proportional to g.In this connection,it ems that the quasi-spherical situation allows a natural choice of transver traceless gauge and eliminates the need for averaging.Earlier attempts to construct pudotensors for gravitational waves by Einstein and others might be converted to genuine tensors by similar gauge-fixing adapted to the transver surfaces,but currently such pudotensors are generally not accepted.
