3+1 formulation of non-ideal hydrodynamics

a r X i v :g r -q c /9710107v 1 22 O c t 1997Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 7February 2008(MN L A T E X style file v1.4)3+1formulation of non-ideal hydrodynamicsJochen Peitz 1⋆and Stefan Appl 2,3†1LandessternwarteK¨o nigstuhl,69117Heidelberg,Germany 2ObservatoireAstronomique,Universit´e Louis Pasteur,11,rue de l’Universit´e ,67000Strasbourg,France3Institut f¨u r Angewandte Mathematik,Universit¨a t Heidelberg,Im Neuenheimer Feld 294,69120Heidelberg,Germany Accepted for publication in the MNRAS ABSTRACT The equations governing dissipative relativistic hydrodynamics are formu-lated within the 3+1approach for arbitrary spacetimes.Dissipation is ac-counted for by applying the theory of extended causal thermodynamics (Israel-Stewart theory).This description eliminates the causality violating infinite signal speeds present in the conventional Navier-Stokes equation.As an ex-ample we treat the astrophysically relevant case of stationary and axisymmet-ric spacetimes,including the Kerr metric.The equations take a simpler form whenever the inertia due to the dissipative contribution can be neglected.Key words:relativity –hydrodynamics –black hole physics –accretion,accretion discs –galaxies:active –stars:neutron1INTRODUCTIONThe motion of dissipative fluids in strong gravitational fields is of considerable interest in various fields of astrophysics and cosmology.Examples include accretion discs around compact objects,rotating relativistic fluid configurations such as supermassive stars,neutron stars or strange (boson)stars,the collapse of stellar objects and the merging of compact objects.Examples in cosmology cover inflationary cosmological scenarios and the evolution of density fluctuations.The non-stationary modeling of relativistic matter is most conveniently performed within ⋆e-mail:jpeitz@lsw.uni-heidelberg.de†e-mail:appl@gaia.iwr.uni-heidelberg.dec0000RAS2J.Peitz and S.Applthe3+1formalism,where the equations of motion for gravitational-and matterfields are decomposed w.r.t.a congruence offiducial observers(FIDOs),allowing to express time derivatives on a per-unit-universal-time basis.The3+1representation of the equations for ideal,non-dissipative(relativistic)hydrodynamics as well as Maxwell’s equations for the electric and magneticfields and Einstein’s equations for the gravitationalfields have been discussed by many authors,both in a cosmological context(e.g.Durrer&Straumann1988) and in the case of black hole spacetimes(e.g.Thorne&Macdonald1982).Modeling dissipative processes requires non-equilibrium or irreversible thermodynamics. Standard(or classical),irreversible thermodynamics(in the following referred to as standard thermodynamics)wasfirst extended from Newtonian to relativisticfluids by Eckart(1940). However,the Eckart theory,and a variation thereof by Landau&Lifshitz(1959)shares with its Newtonian counterpart serious problems.Notably that dissipativefluctuations propagate at an infinite speed.In addition,generic short wavelength secular instabilities driven by dissipative processes exist(Lindblom&Hiscock1983)andfinally,no well-posed initial value problem exists for rotatingfluid configurations.At the origin of these problems in standard thermodynamics is the description of non-equilibrium via the local equilibrium states alone,i.e.it is assumed that local thermodynamic equilibrium is established on an infinitely short time-scale(see e.g.Jou,Casas-V´a zquez& Lebon1997for an introduction).In extended theories of irreversible thermodynamics the set of thermodynamic variables is extended to include the dissipative variables.This re-stores causality and stability under a wide range of conditions(Hiscock&Lindblom1983).A non-relativistic extended theory was proposed by M¨u ller(1967),and was then general-ized to the relativistic case by Israel(1976)and Stewart(1977).The extended theory is commonly referred to as causal thermodynamics,second-order thermodynamics or transient thermodynamics.The problem of non-causality has recently received attention in the context of transonic accretion discs.Two different approaches have been proposed to overcome this difficulty. For steadyflow Narayan(1992)has established causality by calculating the coefficient of kinematic viscosity within an extended version offlux limited diffusion theory(Levermore& Pomraning1981),assuming a particular steady state phase-space distribution function for the turbulentfluid elements in the disc.The influence of this modified viscosity coefficient was studied in stationary accretion discs by Popham&Narayan(1994)and Syer&Narayan (1993).A relativitic generalization of the modified viscosity has been proposed,and used inc 0000RAS,MNRAS000,000–0003+1formulation of non-ideal hydrodynamics3 models of stationary relativistic accretion discs(Peitz&Appl(1997)).A different approach by Papaloizou&Szuskiewicz(1994),which is related to a causal description for the thermo-dynamics,was used by Kley&Papaloizou(1997)in time-dependent models for accretion disc boundary layers.Gammie&Popham(1997)have recently considered a similar extension to stationary relativistic accretion discs.In cosmology the theory of causal thermodynamics is currently attracting growing interest predominantly in the contexts of re-heating processes after inflation(Zimdahl,Pav´o n&Maartens1997)and in linear perturbation theory for the evolution of densityfluctuations(Maartens&Triginer1997).This paper provides a complete set of equations for dissipativefluid mechanics in their 3+1representation,using a causal description of thermodynamics.In Section2the ba-sic elements of both standard and causal dissipative hydrodynamics are reviewed in their spacetime description,and their3+1representation is derived in Section3.As a particular application of astrophysical interest we specify the system of equations given in Section3 to the case of a stationary,axisymmetric back ground in Section4.2DISSIPATIVE RELATIVISTIC HYDRODYNAMICSThe equations of ideal relativisticfluid mechanics and the equations for dissipative relativis-ticfluid mechanics in both the standard irreversible and the extended causal thermodynam-ics description are reviewed.For a detailed discussion see Israel&Stewart(1979),Hiscock &Lindblom(1983)or a recent treatment by Maartens(1997).2.1NotationWe use geometrized units such that c=1=G.Tensorfields defined on spacetime(M,g) with metric g of signature(−,+,+,+)appear in roman(e.g.u,T),while scalar functions are in italic.The velocity u of thefluid is normalized to u·u=−1.The tensor h=g+u⊗u projects into the3-space orthogonal to u,the local rest frame of thefluid(LRF).Total projections(i.e.projection in any free index)parallel to and orthogonal to u are denoted by()u and()h.If A,B and C are a scalar,vector and rank-2tensorfield on(M,g), respectively,thenA u=A h=A,B u=B u=u·B,B h=h·B,C u=C u=u·C·u,C h=h·C·h.(1) c 0000RAS,MNRAS000,000–0004J.Peitz and S.ApplTwo covariant differential operators˙D and D are defined as projections of the affine con-nection∇on(M,g)into directions parallel and orthogonal to u,˙D≡∇=u·∇,(2)uD≡∇h=h·∇.(3) For2-tensorfields C we further introduce(anti-)symmetrization operators a and s by C= C a if C is anti-symmetric and C= C s if C is symmetric and trace-free.Finally,the irreducible decomposition of∇u yields∇u=σ+ω+1/3Θh−a⊗u,(4) with shearσ≡ ∇u s h,vorticityω≡ ∇u a h,acceleration a≡˙D u and expansionΘ≡D·u=∇·u of thefluid trajectories.2.2PerfectfluidsA perfectfluid is described by the velocity u,baryon number density n,mass-energy density ρ,isotropic pressure p and specific entropy s,which are subject to the conservation laws0=∇·n,(5) 0=∇·T.(6) The particle current vector n and the symmetric stress-energy tensor T are given byn=n u,(7) T=ρu⊗u+p h.(8) The LRF conservation laws for energy and momentum result from projecting(6)parallel and orthogonal to u.With(8)one can write0=(∇·T)u and0=(∇·T)h as0=˙Dρ+(ρ+p)Θ,(9) 0=D p+(ρ+p)a.(10) The metric g is coupled to stress-energy T by Einstein’s equationsG=8πT,(11) where G is the Einstein tensor.Thermodynamic scalar functions are defined in the LRF.The entropyfluxs=s n(12)c 0000RAS,MNRAS000,000–0003+1formulation of non-ideal hydrodynamics5 is conserved alongflow lines(adiabaticflow),0=∇·s.(13) The temperature T is defined via the Gibbs equationT d s=d(ρ/n)+p d(1/n),(14) where d is the exterior derivative on(M,g).In general,two thermodynamic scalars are needed as independent variables,which we choose n andρ.A scalar equation of state,e.g. p=p(n,ρ),closes the system of equations(5),(9)-(11)for the dynamical variables{n,ρ,u, g}.2.3DissipativefluidsChoosing the particle current for dissipativefluids by(7)corresponds to selecting an average velocity u such that the particleflux in the associated rest frame vanishes.This so-called particle frame or Eckart frame(Eckart1940)is the natural frame in systems where particle number is conserved(see Israel&Stewart1979for the alternative energy frame description). The state of thefluid is assumed close to afictitious thermodynamic equilibrium state, characterized by the local thermodynamic equilibrium scalars n0,ρ0,p0,s0,T0and the local equilibrium velocity u0,which in the Eckart frame can be chosen such that only the pressure p deviates from the local equilibrium pressure p0by the bulk viscous pressureΠ=p−p0, whereas n=n0andρ=ρ0.Dropping subscripts0allows then to write the general stress-energy tensor for dissipativefluids asT=ρu⊗u+(p+Π)h+q⊗u+u⊗q+π,(15) where the heatflux q relative to the particle frame and the anisotropic stress tensorπare orthogonal to u,q=q h,π= π s h.(16) Conservation laws again hold for n and T.However,in irreversible thermodynamics the entropy is no longer conserved.According to the second law,the rate of entropy generation must therefore be positive definite,∇·s≥0,(17) implying that s has a dissipative vector contribution R in excess to(12),s=s n+R/T.(18) c 0000RAS,MNRAS000,000–0006J.Peitz and S.ApplFollowing the phenomenological Israel-Stewart approach,s=s0remains related to T=T0 by the Gibbs equation(14).The dissipative part R is assumed to be an algebraic function of n and T only,which vanishes in equilibrium(R0=0).The theories of standard irreversible thermodynamics and of extended causal thermodynamics differ in the forms of R,as given below.The equations of energy and momentum conservation for(15)can be written as0=˙Dρ+(ρ+p+Π)Θ+σ:π+(2a+D)·q,(19) 0=(ρ+p+Π)a+D(p+Π)+D·π+π·a+(˙D q)h+(σ+ω+4/3Θh)·q(20) where B:C≡tr(B·C)for2-tensorfields B,C.The above treatment applies for a single-componentfluid and allows a natural extension to multi-componentfluids.2.4Standard thermodynamicsStandard thermodynamics assumes a linear dependence of R on the thermodynamicfluxes {Π,q,π},which is possible only if(18)takes the forms=s n+q/T.(21) Using(5),(19)and(20)yields the entropy generation rateT∇·s=−ΘΠ−(D ln T+a)·q−σ:π.(22) The simplest relation to be imposed between the thermodynamicfluxes{Π,q,π}and the thermodynamic forces{3Θ,D ln T+a,σ}in agreement with(17)is linear,Π=−ζΘ,(23) q=−λT(D ln T+a),(24)π=−2ησ,(25) with non-negative coefficients of bulk viscosityζ(ρ,n),thermal conductivity,λ(ρ,n)and shear viscosityη(ρ,n).This brings(22)toT∇·s=Π2/ζ+q·q/(λT)+π:π/(2η),(26) which on using(5),(14),(19)and(20)yields an evolution equation for the entropy density s,T n˙Ds=−ΘΠ−∇·q−a·q−π:σ.(27)c 0000RAS,MNRAS000,000–0003+1formulation of non-ideal hydrodynamics7 As a consequence of(23)-(25),the thermodynamicfluxes react instantaneously to the cor-responding thermodynamic forces,implying propagation of signals at(causality violating) infinite speed.In standard thermodynamics thedynamics of thefluid is governed by the(com-pressible)Navier-Stokes equation,which results from substitution of(23)-(25)into(20).2.5Causal thermodynamicsKinetic theory can motivate that R is second-order in the dissipative terms(Israel&Stewart 1979).Truncation atfirst order removes terms necessary for causality and stability.The most general algebraic form for R of at most second-order in the dissipativefluxes leads tos=s n+q/T−1/2 β0Π2+β1q·q+β2π:π u/T+α0Πq/T+α1π·q/T.(28) The entropy density measured in LRF then becomes−s·u=sn−1/2 β0Π2+β1q·q+β2π:π .(29) The negative sign of the non-equilibrium contributions reflects the fact that the entropy den-sity is maximum in equilibrium.The thermodynamic coefficientsβj(ρ,n)≥0in(28)model deviations of the physical entropy density from sn due to scalar/vector/tensor dissipative contributions to R.Theαi(ρ,n)model contributions due to viscous/heat coupling,which do not influence the physical entropy density(29).The entropy generation rate associated with(28)follows from(5),(14),(19)and(20)to T∇·s=−ΠX−q·Y−π: Z s(30) with the scalar,vector and rank-2tensorfieldsX=Θ+β0˙DΠ−α0D·q−κ0T q·∇(α0/T)+1/2ΠT∇·(β0u/T),(31) Y=∇ln T+a+β1˙D q−α0∇Π−α1∇·π−(1−κ0)ΠT∇(α0/T)−(1−κ1)Tπ·∇(α1/T)+1/2T q∇·(β1u/T),(32) Z=∇u+β2˙Dπ−α1∇q−κ1T q⊗∇(α1/T)+1/2Tπ∇·(β2u/T).(33) The simplest evolution equations for the causal thermodynamicfluxes{Π,q,π}in agreement with the second law(17)are again linear relationships,Π=−ζX,(34) q=−λT Y h,(35)π=−2η Z s.(36)hc 0000RAS,MNRAS000,000–0008J.Peitz and S.ApplTwo additional thermodynamic coefficientsκk(ρ,n)had to be introduced in(31)-(33)as a consequence of the ambiguity involved in factoring terms which involve productsΠq and π·q in(28).Furthermore,(34)-(36)contain terms involving gradients of theαi andβj. Since theκk are unknown a priori,these terms could be important even if the gradients themselves are small(Hiscock&Lindblom1983).Finally,in(31)-(33)we neglected further contributions due to additional coupling terms between{Π,q,π}and a,ω,which can be shown to exist in kinetic theory(Israel&Stewart1979).The complexity of the full evolution equations(34)-(36)makes applications tractable only if certain simplifications are made.A particularly simple set of evolution equations results from the assumptions(Maartens1997)0=κ0=κ1,(37) 0=α0=α1,(38) 0≃∇·(βj u/T),(39) where(37)reflects essentially the lack of knowledge on theκk while(38)neglects the cou-pling between heatflux and viscosity.Implications of(39)are multifold and need to be justified after a particular solution was found using a parametrization forβj(ρ,n).The evo-lution equations resulting from(34)-(36)under the assumptions(37)-(39)are of covariant relativistic Maxwell-Cattaneo form,τ0(˙DΠ)h+Π=˜Π,(40)τ1(˙D q)h+q=˜q,(41)τ2(˙Dπ)h+π=˜π,(42) with the relaxation timesτj(ρ,n)given byτ0=ζβ0,τ1=λTβ1,τ2=2ηβ2,,(43) and{˜Π,˜q,˜π}the(re-named)standard thermodynamicfluxes as in(23)-(25).In contrast to the algebraic constraint equations(23)-(25),the evolution equations(40)-(42)arefirst order partial differential equations,which assure that in the LRF the viscous bulk/shear stresses and the heatflux relax towards their standard limits{˜Π,˜q,˜π}on time-scalesτj. The relaxation timesτj follow in principle from kinetic theory,but can be estimated as mean collision times,1/τ∼nΣv,withΣthe collision cross section and v the mean particle speed.c 0000RAS,MNRAS000,000–0003+1formulation of non-ideal hydrodynamics 9For later use we re-write (40)-(42)as˙DΠ=1τ1˜q −q +(q ·a)u ,(45)˙D π=1=−ζΘ=−λT D ln T ,(48)˜π,(49)with the thermodynamic forces calculated from the kinematic propertiesΘ≡a |a=0=0,σh .(50)The causal evolution equations (44)-(46)simplify to˙DΠτ0˜Π ,(51)˙D q τ1˜q ,(52)c 0000RAS,MNRAS 000,000–00010J.Peitz and S.Appl˙Dπτ2 ˜π .(53)3DISSIPATIVE HYDRODYNAMICS IN3+1FORMULATION Particularly useful for time-dependent calculations in general relativity is the3+1formu-lation,where time derivatives are always with respect to globally defined universal time. Applications of3+1hydrodynamics and3+1magnetohydrodynamics have been mostly re-stricted to idealfluids(see Bonazzola et al.1993for an exception).A collection of numerous general relativistic equations in the3+1representation can be found in Durrer&Straumann (1988).We give a3+1representation of relativistic dissipative hydrodynamics for both,stan-dard and causal thermodynamics.For a detailed derivation of the equations presented in this section we refer to Peitz(Peitz1998).3.1Generalities on the3+1formalismAssuming that spacetime(M,g)admits a slicing by slicesΣt,i.e.there is a diffeomorphism Φ:M→Σ×I,I⊂I R,such that the manifoldsΣt=Φ−1(Σ×{t})are spacelike and the curvesΦ−1(m,t)are timelike.These curves define a vectorfield∂t which can be decomposed into normal and parallel components relative to the slicing,∂t=αˆn+β.(54) Hereˆn is the timelike unit normalfield(congruence offiducial observers=FIDOs)andβis tangent to the slicesΣt.αis the lapse function andβis the shift vectorfield.A coordinate system{x i}onΣinduces natural coordinates on M,i.e.Φ−1(m,t)has coordinates(t,x i)if m∈Σhas coordinates x i.The timelike curves∂t have constant spatial coordinates (preferred timelike curves).Now setβ=βi∂i(where∂i≡∂/∂x i).From g(ˆn,∂i)=0one finds g(∂t,∂t)=−(α2−βiβi)and g(∂t,∂i)=βi.In coordinates co-moving with the FIDOs the metric thus readsg=−(α2−βiβi)d t⊗d t+βi d t⊗d x i+βi d x i⊗d t+γij d x i⊗d x j(55) =−α2d t⊗d t+γij(d x i+βi d t)⊗(d x j+βj d t).(56) The forms d t and d x i+βi d t are thus orthogonal.γis the metric induced onΣt,and the affine connection on(Σt,γ)is denoted by∇.The tangent and cotangent spaces of M have two natural decompositions,which give rise to two types of bases of vectorfields and1-forms.These are the dual pair{∂µ}andc 0000RAS,MNRAS000,000–000{d xµ}for comoving coordinates{xµ}and,on the other hand,the dual pair{ˆn,∂i}and {αd t,d x i+βi d t}.Instead of the coordinate basis{∂i}one may also use an orthonormal horizontal basis{e i}with g(e i,e j)=δij,together with the dual basis{ϑj}for{d x i}.Then one has the two dual pairs{∂t,e i},{d t,ϑi}and{e0=ˆn,e i},{θµ}with the orthonormal tetrad{θµ}given byθ0=αd t andθi=ϑi+βi d t,withβi defined byβ=βi e i.From(54) follows the relation1e0=ˆn=on(Σt,γ),and the horizontal parts of only two spatial tensorfields,namelyˆa=∇lnα,(61) K=TK.(62) Here K is the extrinsic curvature tensor(second fundamental form),defined on(M,g)by K≡−∇ˆn=−13ˆΘγ+1α ∂t−Lβ E A+˙D E A,(65)E ˙D B =γα ∂t−LβS B+˙D S B−E B F−γK·S B,(67)E ˙D C =γα ∂t−LβS C+˙D S C−γK·T C− E Cγ+T C ·F,(69)T ˙D C =γirreducible decomposition∇u=σ+ω+1α ∂t−Lβ(γn)+1α ∂t−Lβ E T−tr(K)E T+2(S T·∇)lnα+∇·S T−tr(K·T T),(80)0=1α∇·(αTT).(81)For completeness we give the3+1representation of the Einstein equations(e.g.Durrer c 0000RAS,MNRAS000,000–000&Straumann1988).They consist of evolution equations forγand K,obtained from the definition of the extrinsic curvature(63)and from the(space,space)components of(11). These dynamical equations arefirst order differential equations,1α ∂t−Lβ K=Ri(γ)−2K2+tr(K)K−8πT T+4π tr(K)−E T γ−He(lnα),(83)where K2≡K·K,Ri()is the Ricci tensor and He()is the Hessian on(Σt,γ),respectively. The(time,time)and(time,space)components of(11)yield the Hamiltonian and momentum constraints,R−tr(K)2−tr(K2)=16πE T,(84)∇·K−∇tr(K)=8πST,(85) where R is the curvature scalar on(Σt,γ).3.3The standard constraint equationsThe3+1representation of the constraint equations(23)-(25)for the thermodynamicfluxes {˜Π,˜q,˜π}depends on the3+1representation of the kinematic propertiesΘ,a,σof thefluid, which can be calculated toEΘ=W+ϑ−γtr(K),(86)Ea=γW+˙Dγ−K u,(87)Sa=γW+a,(88)E σ=12(γ2−1)W+γ2 Dγ+γa − 12(γ2−1)W+13 W+ϑ−γtr(K) u− 12 W⊗u+u⊗W−1α ∂t−Lβγ+(u·∇)lnα,(92)W≡1(23)-(25),E ˜Π=−ζEΘ,(94)E˜q=−λT γ2−1 1α ∂t−Lβ ln T+D ln T+S a ,(96)E ˜π=−2ηEσ,(97)S ˜π=−2ηSσ,(98)T ˜π=−2ηTσ.(99)3.4The causal evolution equationsFor the sake of simplicity the following discussion will be restricted to the causal evolu-tion equations(44)-(46).Generalization to the full evolution equations(34)-(36)is straight-forward.The required3+1representations of˙DΠ,˙D q,˙Dπfollow readily from(65)-(70). Remains to decompose products q·a in(45)andπ·a in(46),E q·a =EqEa+Sq·S a,(100)E π·a =EπEa+Sπ·S a,(101)S π·a =EaSπ+Tπ·S a.(102)The3+1evolution equations for the causal thermodynamicfluxes{EΠ,E q,S q,Eπ,Sπ,Tπ} then follow from(44)-(46)toγτ0 E˜Π−EΠ ,(103)γτ1 E˜q−E q +γE q·a+F·S q,(104)γτ1 S˜q−S q +E q·a u+E q F+γK·S q,(105)γτ2 E˜π−Eπ +2γEπ·a+2F·Sπ,(106)γτ2 S˜π−Sπ +γSπ·a+Eπ·a u+γK·Sπ+ Eπγ+Tπ ·F,(107)γτ2 T˜π−Tπ +u⊗Sπ·a+Sπ·a⊗u+2γK·Tπ+F⊗Sπ+Sπ⊗F,(108)with F according to(71).c 0000RAS,MNRAS000,000–0003.5The weakly dissipative limitThe limit of weak dissipation discussed in section 2.6implies three simplifications of the 3+1equations of causal hydrodynamics.The first simplfication concerns the 3+1conservation laws (80)and (81)for energy and momentum,where dissipative contributions {Π,q,π}to the matter sources {E T ,S T ,T T }in (75)-(78)can be dropped wherever they enter alge-braically,and only temporal and spatial gradients remain.The second simplification affects the 3+1representation of the kinematic properties of the fluid,(86)-(91).Using the geodesic conditions E a =S a =0,(87)and (88)yield simplified expressions for W and W ,W =−˙D ln γ+K u /γ,(109)W =−a /γ,(110)which are no longer time-dependent.Therefore,any kinematic properties (86)-(91)simplify to time-independent expressionsΘE =0,(112)aE =13γ2−1ϑ−γtr(K )=23ϑ−γtr(K ) +1S =13˙D ln γ−K u /γ−ϑ+γtr(K ) γu ,(115)σ−γK +1= ∇u s −1/2ϑh .Finally,the weak dissipation limit affects the 3+1evolution equa-tions (103)-(108)for the thermodynamic fluxes {E Π,E q ,S q ,E π,S π,T π},where productsE q ·a ,E π·a ,S π·a as in (100)-(102)are dropped.This yields the weak dissipative evolution equations for the thermodynamic fluxes {E Π,S q ,S π},γ+˙D E Πτ0 E ˜Π ,(117)γ+˙D E q τ1 E ˜q +F ·S q α ∂t −L β S q =1−S q F +γK ·S q α ∂t −L β E π=1−E π,(120)c0000RAS,MNRAS 000,000–000γ+˙D Sπτ2 S˜π +γK·Sπγ+Tπα ∂t−Lβ Tπ=1−Tπ+F⊗Sπ⊗F.(122)4STATIONARY AND AXISYMMETRIC BACKGROUND SPACETIMES The3+1equations of dissipative hydrodynamics are specified to the class of stationary,ax-isymmetric background spacetimes.This situation is realized if thefluid under consideration has negligible influence on the gravitationalfield of a central object,and in addition this field is known to be stationary and axisymmetric.These assumptions include most appli-cations related to accretion/ejectionflows in the vicinity of compact objects.For rotating black holes the vacuum metric is Kerr,and we give the equations also for this special case.4.1Implications of symmetriesThe general form of a stationary,axisymmetric vacuum spacetime can be put in a form which is symmetric under a simultaneous change of sign of t andφ,the Killing coordinates associated with the commuting time and axial Killing vectorfields k and m.Choosing the remaining two meridional coordinates as spherical coordinates allows to write g asg=−α2d t⊗d t+˜ω2(dφ−ωd t)⊗(dφ−ωd t)+e2µd r⊗d r+e2νdθ⊗dθ,(123) with the invariant metric coefficients˜ω2=m2,ω=−k·m/m2,α2=−k2+k·m/m2.(124) For the physical interpretation of˜ω,ωandαsee e.g.Bardeen(1970).The generic choice of thefiducial congruence is(see e.g.Thorne&Macdonald(1982)for criteria that uniquelyfix this choice)ˆn=e0=1and the metric induced on (Σt ,γ)becomes γ=˜ω2d φ⊗d φ+e 2µd r ⊗d r +e 2νd θ⊗d θ.(127)The 3+1representation of Killing’s equation for k and m,together with their commutivity,allows to establish the following relations (Thorne &Macdonald 1982)∂t µ=0,∂t ν=0,∂t ω=0,∂t m =0,∂t γ=0,(128)m ·∇µ=0,m ·∇ν=0,m ·∇ω=0, ∇m s =0,(129)ˆΘ=0,ˆσ=12αm ⊗∇ω+∇ω⊗m .(131)Therefore K measures the shear of hypersurfaces Σt ,which vanishes for ω=0.4.2Gravitomagnetic and gravitoelectric tensor fieldsA characteristic phenomenon in axisymmetric spacetimes is the dragging of inertial frames.Physical implications due to this effect (see Thorne et al.1986for the case of the Kerr metric)are often described in terms of the gravitomagnetic tensor field H ,defined on (Σt ,γ)by H ≡1α∇(ωm )=−12(H ·T +T ·H ).(133)The antisymmetric part of H can be expressed as an axial vector field on (Σt ,γ),namely the gravitomagnetic vector field J ≡12J ∧S .(135)The Lie derivatives L βin the 3+1equations derived in section 3can then be expressed as L βE =(β·∇)E ,(136)L βS =(β·∇)S −(S ·∇)β=(β·∇)S −αS ·H ,(137)c0000RAS,MNRAS 000,000–000LβT=(β·∇)T−2 (T·∇)β s=(β·∇)T−2α T·H s=(β·∇)T−2T·K.(138) The term S·H in(137)may be expressed by either H as in(132)or in terms of J and K via(135).In addition to the gravitomagnetic tensorfields H and J,we introduce the gravitoelectric vectorfield G,defined on(Σt,γ)byG≡−ˆa=−∇lnα.(139) Thisfield measures the gravitational acceleration measured by thefiducial observers.4.3Conservation lawsThe3+1conservation laws for particle number(79),energy(80)and momentum(81)in a stationary,axisymmetric background read0=1∂t−β·∇(γn)+1α ∂α ∂α∇(αT T)=1∂t−β·∇S T+∇·T T−H·S T− E Tγ+T T ·G.(142)4.4Evolution equationsThe3+1representation(86)-(91)of the kinematic properties of thefluid in stationary, axisymmetric background readEΘ=W+ϑ,(143)Ea=γW+˙Dγ+ˆσu,(144)Sa=γW+a,(145)E σ=32 γ2−1 W+γ2Dγ+γa − ˆσuγ+ˆσ ·u,(147)T σ=γ3W h+σ+γˆσ(148)with time derivatives contained inW=1∂t−β·∇γ−G u,(149)W=1∂t−β·∇u−H·u−γG.(150)c 0000RAS,MNRAS000,000–000The 3+1representation of the standard constraint equations (94)-(99)can be written as E ˜Π=−ζE Θ,(151)E ˜q =−κTγ2−11∂t −β·∇ ln T +γ˙D ln T +E a ,(152)S ˜q =−κT γu1∂t −β·∇ln T +D ln T +S a ,(153)E ˜π=−2ηE σ,(154)S ˜π=−2ηS σ,(155)T ˜π=−2ηT σ.(156)The 3+1representation ofthe causal evolution equations (103)-(108)reads γ∂t −β·∇E Π+˙DE Π=1α∂τ1E ˜q −Eq+γE q ·a+F ·S q,(158)γ∂t −β·∇S q +˙DS q=12γJ ∧S q,(159)γ∂t −β·∇E π+˙DE π=1α ∂τ2 S ˜π−S π+γS π·a +E π·au +1α ∂τ2T ˜π−T π+u ⊗S π·a+S π·a⊗u +γF ⊗S π+γS π⊗F ,(162)withF =γG −ˆσ·u .(163)4.5Weakly dissipative limitThe 3+1representation of the kinematic properties of the fluid in the weak dissipation limit can be written asΘE =0,(165)aE =13γ2−1ϑ=23ϑ+1σ2 a /γ+∇γ +ˆσ·u +1T =σ3˙D ln γ−ˆσu /γ h .(169)The 3+1weak dissipation evolution equations (103)-(108)for {E Π,S q ,S π}can be written asγ∂t−β·∇ E Π=1−E Πα ∂+˙D E q τ1 E ˜q +F ·S q α ∂+˙D S q τ1S ˜q +E q 2γJ ∧S q α ∂+˙D E πτ2E ˜π +2F ·S πα ∂+˙D S πτ2S ˜π +1+ E π ·F ,(174)γ∂t −β·∇ T π=1−T π+S π∆Σ2,˜ω=Σ∆,e 2ν=̺2.(177)A consistent treatment of viscous hydrodynamics in the vicinity of a rotating black hole would require a careful analysis of the boundary conditions at the horizon.Such an analysis is best performed within the concept of a streched horizon ,defined by a small value of lapse α.We refer to the membrane paradigma of Thorne et al.(1986),where this approach is applied to ideal magnetohydrodynamics and Maxwell’s equations.This work also contains a coordinate representation of G and H .。