General-relativistic rotation laws in rotating fluid bodies

0.13 MB·English

by  Patryk Mach

General-relativistic rotation laws in rotating fluid bodies
Patryk Mach and Edward Malec
Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagiellon´ski, L ojasiewicza 11, 30-348 Krak´ow, Poland
We formulate new general-relativistic extensions of Newtonian rotation laws for self-gravitating
stationary fluids. They have been used to re-derive, in the first post-Newtonian approximation,
the well known geometric dragging of frames. We derive two other general-relativistic weak-field
effects within rotating tori: the recently discovered dynamic anti-dragging and a new effect that
measuresthedeviation from theKeplerian motion and/or thecontribution of thefluidsselfgravity.
Onecan use the rotation laws to study the uniqueness and the convergence of the post-Newtonian
approximations, and theexistence of the post-Newtonian limits.
PACSnumbers: 04.20.-q,04.25.Nx,04.40.Nr,95.30.Sf
I. INTRODUCTION whereρisthebaryonicrest-massdensity,histhespecific
enthalpy, and p is the pressure. The 4-velocity uα is
Stationary Newtonian hydrodynamic configurations normalized, gαβuαuβ = 1. The coordinate (angular)
−
5 are characterized by a variety of rotation curves. The velocity reads~v =Ω∂φ, where Ω=uφ/ut.
1 angular momentum per unit mass j can be any func- Weassumeabarotropicequationofstatep=p(ρ). To
0
tion of r, where r is the distance from the rotation axis. bemoreconcrete,onecantakethepolytropicequationof
2
Other restrictions arise from stability considerations [1]. state p(ρ,S) = K(S)ργ, where S is the specific entropy
un In contrastto that, for a long time the only known rota- of fluid. Then one has h(ρ,S) = K(S)γ−γ1ργ−1. The
tion law in general-relativistic hydrodynamics had been entropy is assumed to be constant.
J
that with j as a linear function of the angular velocity. Define the square of the linear velocity
2
RecentlyGaleazzi,YoshidaandEriguchi[2]havefounda
ω 2
] nonlinearangularvelocityprofile,thatmayapproximate V2 =r2 Ω e2(β−ν)/c2. (4)
c the Newtonian monomial rotation curves Ω = w/rλ (cid:16) − c2(cid:17)
q 0
in the nonrelativistic limit. In this paper we define Thepotentialsα, β, ν,andω satisfyequationsthathave
-
r general-relativistic rotation curves j = j(Ω) that in the beenfoundbyKomatsu,EriguchiandHachisu[3]. They
g
[ nonrelativistic limit exactly coincide with Ω0 = w/rλ constituteanoverdetermined,butconsistent,setofequa-
(0 λ 2, λ = 1). We are able to obtain the general- tions. The general-relativistic Euler equations are solv-
≤ ≤ 6
2 relativisticKeplerianrotationlawthatpossessesthefirst able, assuming anintegrability condition— that the an-
v post-Newtonianlimit(1PN)andexactlyencompassesthe gular momentum per unit mass,
9
solutioncorrespondingtothemasslessdiskofdustinthe
3
Schwarzschildspacetime. V2
5 j =u ut = , (5)
4 φ Ω ω 1 V2
− c2 − c2
0 (cid:0) (cid:1)(cid:0) (cid:1)
. II. HYDRODYNAMICAL EQUATIONS depends only on the angular velocity Ω; j j(Ω). In
1 ≡
such a case the Euler equations reduce to a general-
0
5 Werecapitulate,following[3],theequationsofgeneral- relativisticintegro-algebraicBernoulliequation,thatem-
1 relativistic hydrodynamics. Einstein equations, with the bodiesthehydrodynamicinformationcarriedbythecon-
: signature of the metric ( ,+,+,+), read tinuityequations Tµν =0andthebaryonicmasscon-
v − servation (ρuµ∇)µ=0. It is given by the expression
i µ
X R G ∇
R g =8π T , (1)
r µν − µν 2 c4 µν h ν 1 V2 1
a ln 1+ + + ln 1 + dΩj(Ω)=C.
where T is the stress-momentum tensor. The station- (cid:18) c2(cid:19) c2 2 (cid:18) − c2 (cid:19) c2 Z
µν
(6)
ary metric reads
ds2 = ec22ν(dx0)2+r2e2cβ2 dφ ω (r,z)dx0 2 III. ROTATION LAWS
− (cid:16) − c3 (cid:17)
+e2cα2 dr2+dz2 . (2) The general-relativistic rotation law employed in the
(cid:0) (cid:1) literature [3–7] has the form
Here x0 =ct is the rescaled time coordinate, and r, z, φ
are cylindrical coordinates. We assume axial symmetry j(Ω)=A2(Ωc Ω), (7)
−
and employ the stress-momentum tensor
where A and Ω are parameters. In the Newtonian limit
c
Tαβ =ρ(c2+h)uαuβ +pgαβ, (3) and large A one arrives at the rigid rotation, Ω = Ω ,
c
2
while for small A one gets the constant angular momen- expansions, that they are non-unique. Damour, Jara-
tum per unit mass. A three-parameter expression for j nowskiandScha¨fer[8] demandthata testbody rotating
is proposed in [2]. circularly in a Schwarzschild space-time satisfies exactly
Below we define a new family of rotation laws, the Keplerian rotation law with Ω2 = GM/R3, where
R is the areal radius. Inspired by this we impose a Fix-
w1−δΩδ ingCondition(F-Conditionthereafter)—thatarotat-
j(Ω) , (8)
≡ 1 κw1−δΩ1+δ+ Ψ ing infinitely thin disk made of dust in a Schwarzschild
− c2 c2
space-time satisfies exactly the Bernoulli equation and
wherew, δ,κandΨareparameters. The rotationcurves the Keplerian rotation law.
Ω(r,z) ought to be recoveredfrom the equation Considera rotating,infinitely thin andweightlessdisk
ofdustintheSchwarzschildgeometry. Thisisatextbook
w1−δΩδ V2 knowledgethatthereexistsastationarysolution—each
= , (9)
1 κw1−δΩ1+δ+ Ψ Ω ω 1 V2 particle of dust can move along a circular trajectory of
− c2 c2 (cid:0) − c2(cid:1)(cid:0) − c2 (cid:1) a radius R with the angular velocity Ω = GM/R3.
For δ = 1, the general-relativistic Bernoulli equation We shall present this solution in conformal copordinates,
6 −
(6) acquires a simple algebraic form using our formalism. The conformal Schwarzschild met-
ricreadsds2 = Φ2/f2 dx0 2+f4 dr2+dz2+r2dφ2 ,
−
where Φ = 1 GM/(cid:0)(2c2(cid:1)√r2+(cid:0)z2) and f = 1 +(cid:1)
h V2
1+ eν/c2 1 GM/(2c2√r2+z−2). The angular velocity is equal to
(cid:18) c2(cid:19) r − c2 ×
−1 the Keplerian velocity Ω2 = GM/(√r2+z23f6) and
1 κw1−δΩ1+δ+ Ψ (1+δ)κ =C. (10) R=√r2+z2f2. The totalenergy per unit mass Ψ van-
(cid:18) − c2 c2(cid:19) ishes for a test dust. Let the disk lie on the z =0 plane
and assume the rotation law with δ = 1/3 and κ=3:
We shallexplainnow the meaning andstatus of the four −
constants w, δ,κ and Ψ. Assume that there exists the w4/3Ω−1/3 V2
Newtonianlimit(thezerothorderofthepost-Newtonian 1 3w4/3Ω2/3 = Ω 1 V2 . (12)
expansion — 0PN) of the rotation law. This yields − c2 − c2
(cid:0) (cid:1)
Here V2 = Ω2r2f6/Φ2. Notice that h = 0; the enthalpy
w perunitmassvanishesfordust. Thisisasimpleexercise
Ω = . (11)
0 2 toshowthatw =√GM andΩ2 =GM/(r3f6)solveboth
r1−δ
Eq.(12)andtheBernoulliequation(10);the constantin
Thusw andδ canbeobtainedfromtheNewtonianlimit. (10) equals to unity.
Moreover, the constant w is any real number, while δ
is nonpositive — due to the stability requirement [1] —
andsatisfiesthebounds δ 0andδ = 1. These IV. 1PN CORRECTIONS TO ANGULAR
twoconstantscanbegive−n∞ap≤riori≤withinthe6 g−ivenrange VELOCITY
of values. Let us remark at this point that the rotation
law (8), and consequently the Newtonian rotation (11), Taking into account the above, we shall prove
applies primarily to single rotating toroids and toroids that if κ = (1 3δ)/(1 + δ) + (c−2) and Ψ =
rotatingaroundblackholes. Inthe caseofrotatingstars 4c + (c−2), wh−ere c is the NOewtonian hydro-
0 0
O
onewouldhavetoconstructaspecialdifferentiallyrotat- dynamic energy per unit mass, then the exact
inglaw,withthe aimto avoidsingularityatthe rotation solution satisfies the first post-Newtonian (1PN)
axis. equations. We shall use the formalism of [3] and the
The two limiting cases δ =0 and δ = correspond rotationlaw(9),andrecovermostoftheresultsobtained
−∞
to the constantangularmomentum per unit mass (Ω0 = in the 1PN approach employed in [11]. Notice that if
w/r2) and the rigid rotation (Ω = w), respectively. The δ = 1/3, then κ = 3 — one recovers the coefficient
Keplerian rotation is related to the choice of δ = 1/3 in fro−nt of w4/3Ω2/3 in (12) that is required by the F-
and w2 = GM, where M is a mass [9]. The case−with Condition.
δ = 1 should be considered separately, but we expect The 1PN approximation corresponds to the choice of
−
that the reasoning will be similar. metric exponents α=β = ν = U with U c2 [10].
ThevaluesofκandΨareproblematical. Onepossibil- Define ω r−2A . The spa−tial p−art of the|m|et≪ric
φ
≡
ity to get them is to apply the PN expansion. The rota-
tionlawinthePNexpansionschemeshouldnotbegiven ds2 = 1+ 2U + 2U2 (dx0)2 2c−3A dx0dφ
apriori,but is expected to build up — in the subsequent −(cid:18) c2 c4 (cid:19) − φ
orders of c−2 — from the Newtonian rotation law. The 2U
Newtonian rotation curves are specified arbitrarily, but + 1 dr2+dz2+r2dφ2 . (13)
(cid:18) − c2 (cid:19)
thenextPNcorrectionsshouldbedefineduniquely. This (cid:0) (cid:1)
is,however,awellknownpropertyofthepost-Newtonian is conformally flat.
3
We split different quantities (ρ, p, h, U, and vi) into in the case of test fluids, at the symmetry plane z = 0.
their Newtonian (denoted by subscript ‘0’ and 1PN (de- For the dust, in the Schwarzschildgeometry, we get
noted by subscript ‘1’) parts. E.g., for ρ, Ω, Ψ, and U
this splitting reads vφ w
Ω=Ω + 1 = ; (21)
ρ=ρ +c−2ρ , (14a) 0 c2 r˜3/2
0 1
Ω=Ω +c−2vφ, (14b) the 1PN correction to Ω0 is equal to 32Uc20Ω0. Thus the
0 1 FC condition is satisfied in the 1PN order.
Ψ=Ψ + (c−2), (14c) After these consideration we are able to interpret the
0
O meaning of various contributions to the 1PN angularve-
U =U +c−2U . (14d) locityΩ. ThefirsttermissimplytheNewtonianrotation
0 1
law rewritten as a function of the geometric distance, as
Notice that, up to the 1PN order, given at the 1PN level of approximation, from the rota-
tion axis. The second term in (19) vanishes at the plane
1
∂ p=∂ h +c−2∂ h + (c−4), (15) ofsymmetry, z =0, for circularKeplerianmotionoftest
i i 0 i 1
ρ O
fluidsinthemonopolepotential GM/R. Thusitissen-
−
wherethe 1PNcorrectionh to the specific enthalpycan sitive both to the contribution of the disk self-gravity at
1
be writtenas h = dh0ρ . For the polytropicequationof the plane z =0 and the deviation from the strictly Kep-
1 dρ0 1 lerian motion. The third term is responsible for the geo-
state this gives h =(γ 1)h ρ /ρ .
1 0 1 0
− metric frame dragging. The last term represents the re-
Makinguseoftheintroducedabovesplittingofquanti-
cently discovered dynamic anti-dragging effect; it agrees
ties into Newtonian 0PN and 1PN parts one can extract
(for the monomial angular velocities Ω = r−2/(1−δ)w)
fromEq.(6)the0PN-and1PN-levelBernoulliequations. 0
— with the result obtained earlier in [11].
The 0PN equation reads
A comment on the term 2 Ω3r2, that has been
−c2(1−δ) 0
h +U δ−1 Ω2r2 =c , (16) missingin [11]. Thereasonforthisomissionisfollowing.
0 0 0
− 2(1+δ) There is a gauge freedom in choosing an integrability
condition for the 1PN hydrodynamic equation; due to
where c is a constantthat can be interpreted as the en-
0 that the Bernoulli equationof the 1PN order is specified
ergy per unit mass. At the Newtonian level this is sup-
uptoafunctionF(r). Weassumedin[11],inordertoget
plemented by the Poisson equation for the gravitational
the1PNBernoulliequationasin[10],thatF(r)=0;but
potential
that is not consistent with the F-Condition. It appears
thattherightvalueisF(r)= Ω4r4/(1+δ),whichleads
∆U0 =4πGρ0, (17) − 0
to the emergence of the term in question.
where ∆ denotes the flat Laplacian. The first correction The vectorial component Aφ satisfies the following
vφ tothe angularvelocityΩ isobtainedfromthe pertur- equation
1
bation expansion of the rotation law (9) up to terms of
the order c−2. Assuming that Ψ0 =4c0, one arrives at ∆Aφ 2∂rAφ = 16πGr2ρ0Ω0. (22)
− r −
2 A 4Ω h
vφ = Ω3r2+ φ 0 0, (18)
1 −1 δ 0 r2(1 δ) − 1 δ The 1PN Bernoulli equation does not influence the
− − − 1PN correction to the angular velocity. It has the form
where we applied Eqs. (11) and (16).
3
Remember that in the Newtonian gauge imposed in c = h U Ω A +2r2(Ω )2h h2
the line element (13) the geometric distance to the ro- 1 − 1− 1− 0 φ 0 0− 2 0
tation axis is given by r˜ = r(1 U0/c2)+ (c−4). It 4h U 2U2 δ−1 r4Ω4+F(r), (23)
is enlightening to write down the−full expressOion for the − 0 0− 0 − 4(1+δ) 0
angular velocity, up to the terms (c−4):
O where c is a constant. In order to derive (23) we again
1
Ω=Ω + v1φ = w 2 Ω U +Ω2r2 used Ψ0 =4c0. This result agrees with the 1PN calcula-
0 c2 r˜2/(1−δ) − c2(1 δ) 0 0 0 tion of [11] up to the term F(r).
− (cid:0) (cid:1) The1PNpotentialcorrectionU canbeobtainedfrom
A 4 1
φ
+ Ω h . (19)
r2c2(1 δ) − c2(1 δ) 0 0 ∆U =4πG ρ +2p +ρ (h 2U +2r2(Ω )2) .
− − 1 1 0 0 0 0 0
−
This expression reduces to (cid:0) (cid:1)(24)
Equations (22) and (24) have been derived in [11] in
vφ the framework of 1PN approximation. They can be also
Ω=Ω + 1 =
0 c2 obtaineddirectlyfromtheEinsteinequationswrittenfor
w 4 the metric (2), as derived e.g. in [3]. Here we recall a
Ω h , (20)
r˜2/(1−δ) − rc2(1 δ) 0 0 versionsimilartothatusedin[6];itturnsouttobemore
−
4
conveninent than the original form of [3]. The relevant the parameterκ is agivenfunctionofδ. The fullsystem
equations read ofEinstein-Bernoulliequationscanbesolvednumerically
within this class of data and the resulting solutions are
G 1+V2/c2
∆ν =4π e2α/c2 ρ(c2+h) +2p expected to possess 0PN and 1PN limits.
c2 (cid:20) 1 V2/c2 (cid:21) V. CONCLUDING REMARKS
−
1 1
+ r2e2(β−ν)/c2 ω ω (β+ν) ν
2c4 ∇ ·∇ − c2∇ ·∇ We write down the general-relativistic rotation laws,
recover the well known geometric dragging of frames
and
and derive a full form of the two other weak-fields ef-
∆+ 2∂ ω = 16πGe2α/c2ρ(c2+h) Ω−ω/c2 fects, including the dynamic anti-dragging effect of [11].
(cid:18) r r(cid:19) − c2 1 V2/c2 The latter can be robust according to the numerics of
−
1 [11],butthe ultimate conclusionrequiresafully general-
+ (ν 3β) ω,
c2∇ − ·∇ relativistic treatment, that is the use of the new rota-
tion laws. The frame dragging occurs — through the
where denotes the “flat” gradient operator. The re-
Bardeen-Pettersoneffect[16]—insomeAGN’s[17]. The
∇
maining Einstein equations yield corrections of higher
two other effects canlead to its observable modifications
orders.
in black hole systems with heavy disks.
In summary, we have shown that — for <
−∞ In the weak field approximation of general relativity
w < and δ 0, δ = 1 — the choice
κ = (∞3 δ)/(1−+∞δ)≤+ ≤(c−2) an6 d−Ψ = 4c + (c−2) the angular velocity of toroids depends primarily on the
− O 0 O distance from the rotation axis — as in the Newtonian
in the rotation law (9) guarantees that if there
hydrodynamics—buttheweakfieldscontributionsmake
exists an exact solution analytic in powers of c−2,
the rotation curve dependent on the height above the
then it satisfies the 0PN and 1PN approximating
symmetry plane of a toroid.
equations.
The new rotation laws would allow the investigation
One easily finds out that the rotation law (8) satisfies
of self-gravitating fluid bodies in the regime of strong
the generalized Rayleigh criterion [12] for stability dj <
dΩ gravity for general-relativistic versions of Newtonian ro-
0 up to 1PN order, assuming that δ is strictly negative.
tation curves. In particular, they can be used in order
Comments on the 1PN corrections to the angular ve-
to describe stationary heavy disks in tight accretionsys-
locity. In the following considerations we assume w >0,
tems with central black holes. These highly relativistic
whichmeansΩ >0,butthe reasoningis symmetricun-
0 systems can be created in the merger of compact bina-
der the parity operationw w. The specific enthalpy
ries consisting of pairs of black holes and neutron stars
h ≥ 0 is nonnegative, then→ce−−4Ω1−0hδ0 is nonpositive — [14, 15], but they might exist in some active galactic nu-
the discovered in [11] instantaneous 1PN dynamic reac- clei.
tion slows the motion: it “anti-draggs” a system. In The new general-relativistic rotation laws can be ap-
contrasttothat,the wellknowngeometrictermwithAφ plied to the study of various open problems in the post-
ispositive[11],andthecontribution Aφ totheangu- Newtonian perturbation scheme of general-relativistic
r2(1−δ)
lar velocity is positive — it pushes a rotating fluid body hydrodynamics. We demonstrate in this paper that an
forward. Thus the two terms in (18) counteract. adaptationoftheconditionusedin[8]ensuresuniqueness
Dust is special — the specific enthalpy h vanishes, up the 1PN order. Further applications include the in-
0
hence dusttestbodies areexposedonlyto the geometric vestigation of convergence of the post-Newtonian pertu-
effect — the frame dragging. Even more special is the bationscheme,as wellas the existence of the Newtonian
rigid (uniform) rotation — the correction terms vφ are and post-Newtonian limits of solutions.
1
proportional to 1/(1 δ) and they vanish, because now
−
δ = . Uniformly rotating disks are already known to
−∞
minimize the total mass-energy for a given baryon num- ACKNOWLEDGMENTS
ber and total angular momentum [13]. The vanishing
of the 1PN correction vφ is their another distinguishing PM acknowledges the support of the Polish Ministry
1
feature. of Science and Higher Education grant IP2012 000172
It follows from our discussion that assuming the F- (IuventusPlus). EMthanksPiotrJaranowskifordiscus-
Condition, one has three free parameters: w,δ and Ψ; sions on the PN approximations.
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