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Spinning String Fluid Dynamics in General Relativity

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Spinning String Fluid Dynamics in General Relativity Class. Quantum Grav. 14 (1997) 3501–3512. Printed in the UK PII: S0264-9381(97)84861-7 Spinning string fluid dynamics in general relativity Larry L Smalley and Jean P Krisch † ‡ Department of Physics, University of Alabama in Huntsville, Huntsville, AL 35899, USA † Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA ‡ Received 9 June 1997 Abstract. The general, energy–momentum tensor for a dynamical, spinning string fluid in general relativity is presented using the Ray–Hilbert variational principle. The calculations are given for both the standard and the extended thermodynamics versions in which the latter includes both the spin and string as thermodynamic variables. Both versions yield the same Fermi–Walker transport of the spin with a correction term due to the string. As an unexpected general feature, it is shown that the string and spin are dual 2-forms. Examples of solutions to the field equations are given for the extension of static black holes for non-spinning, string fluids to stationary, slowly rotating black holes for spinning string fluids. These solutions are then compared with solutions obtained from the postulated energy–momentum tensor of Letelier. The general feature of these solutions for positive density is that the spin causes the event horizon for the stationary black hole to contract whether or not the spin is considered as a thermodynamic variable. PACS numbers: 0440N, 0570, 9880H 1. Introduction to spinning string fluids in general relativity The discussion of the cosmological implications of strings goes back nearly 20 years to the work of Zeldovich [1] and Vilenkin [2]. A string dust model was introduced by Letelier [3] from the description of the string function as a spacetime, surface-forming bi-vector ∂xµ ∂xν 6µν AB (1) = ∂λA ∂λB where 1ifA0,B 1 == AB 1ifA1,B 0 (2) = − = = 0ifAB. = In a previous work on the dynamics of a string fluid [4], it was noted that the form of equation (1) follows closely the form of the spin bi-vector developed by Halbwachs [5], and therefore the description of a string fluid energy density will use the parallel description of the velocity matrix used in discussions of fluid dynamics in continuum mechanics (see, for example, the discussions of fluids with spin and twist in metric affine geometry for a more complete discussion [6]). The basic discussion of string fluid dynamics plus applications to static black hole models and cosmological solutions in general relativity (GR) are described by [7] (referred to as paper I). In this work, a much more extensive discussion extends the concept of the fluid dynamics of strings to the case of spinning string fluids. This means that 0264-9381/97/123501+12$19.50 c 1997 IOP Publishing Ltd 3501 3502 L L Smalley and J P Krisch in a fluid context, a Ray–Hilbert variational principle [8] can be developed for a spinning string fluid by introducing a complete set of tetrads for the fluid . 6ij ρλ(x) a4i a3j a4j a3i ρσij (3) = − = along with the usual spin density . Sij ρk(x) a1i a2j a1j a2i ρsij (4) = − = µi where ρ is the string density, a are the tetrads, where the Latin indices, i 0, 1, 2, 3, are the holonomic coordinates, the Greek indices, µ 1, 2, 3, 4 are the non-holonomic= coordinates, and where k(x) and λ(x) are the spin and string= (module) functions, respectively (considered as a parametric function), which will not be varied directly. The holonomic metric has signature, gij ( 1, 1, 1, 1), and the non-holonomic metric, ηµν (1, 1, 1, 1). We also identify the tetrad= a−4i ui . Using equations (3) and (4), the equivalent= ‘angular’− velocity matrix takes the form ≡ ij αi j ω a aα (5) =˙ so that the string energy density is given by 1 ij 4 3i TST 6 ωij ρλa ia (6) = 2 = ˙ in comparison with the spin energy density 1 ij 1 2i TS S ωij ρλa ia . (7) = 2 = ˙ Some additional justification for this string formulation is given by Nieto [9]. It is interesting to note the similarity of the spin/string energy density in the fluid framework to that which occurs in general spin/twist formulations [6]. In a previous work [4], the generality of the spacetime manifold was not restricted to the Riemannian geometry of GR. However, this work will restrict itself to the Riemannian manifold since the examples discussed in the penultimate section are compared to the behaviour of strings in general relativity (GR). In the next section, the treatments of spinning string fluids using both the improved and unimproved thermodynamic versions are described. Examples of spinning string solutions are given in section 3, and the conclusions and discussions are given in the final section. 2. Spinning string fluid Lagrangian The Ray–Hilbert Lagrangian for a spinning string fluid in GR spacetime becomes (8) LG = LM + LG where is the matter Lagrangian LM µi νj µν i e ρ[1 (ρ,s)] λµν a a gij η λ2 i ρu LM = − + + − + ∇ i i 1i 2 4 3i λ3u X,i λ4u s,i kρa a ρλa ia (9) + + − ˙ i − ˙ and is the gravitational field Lagrangian LG eR (10) LG = 2κ where e √ g, κ 8πG, R is the Riemann scalar in GR spacetime, λµν are the Lagrange = − = multipliers and we write λ44 λ1 in the variation. Note that all components of the tetrads occur in the Lagrangian (9), and= are therefore active in the variational equations. In the first part of this section, the spin and string variables are not included in the thermodynamics Spinning string fluid dynamics in general relativity 3503 as in the non-improved energy–momentum formulations of spinning fluids [11]. In the following abbreviated subsection, the spin and string variables are included in the extended thermodynamics as in the improved energy–momentum formulations of spinning fluids [12]. 3. Development of the energy–momentum tensor 3.1. Standard thermodynamics The standard thermodynamics for a perfect fluid for a Ray–Hilbert [8] variational principle is given by p d T ds dρ. (11) = + ρ2 αi The variational variables are then gij ,ρ,s,X,a and the various Lagrange multipliers. 3.1.1. Variational equations. (i) Tetrad and density variational equations. The variational equations become: Density variation δρ: p i 3 ik 2 1 λ2 λu a i ka a k 0. (12) − + + ρ − ˙ − ˙ − ˙ = 4-velocity δu: 1 2 3 ρ iλ2 λ3X,i λ4s,i 2λ1ui 2λ41a i 2λ42a i 2λ43a i − ∇ + + 3 + +3k 1+k 2 + ρλa i ρλuk ia ρka ia k 0. (13) − ˙ − ∇ − ∇ = Tetrads 1i 1 2 3 2 δa :2λ11a i 2λ12a i 2λ13a i 2λ41ui ρka i 0 (14) 2i 2 + 1 + 3 + − k˙ = 1 δa :2λ22a i 2λ12a i 2λ23a i 2λ42ui ρu k ka i 0 (15) + + + + ∇ = 3i 3 1 2 k δa :2λ33a i 2λ13a i 2λ23a i 2λ43ui ρu k(λui ) 0. (16) + + + + ∇ = The tetrad and density variations reveal the following relevant results for the Lagrange multipliers which will be useful in the analysis of the metric field equations and the transport properties of the spinning string. 2λ1 [ρ(1 ) p] TST (17) = + + − ρλ2 [ρ(1 ) p] TST TS (18) − ˙ = + + + + λ12 0 (19) = k 0 (20) ˙ = 2k 2λ41 ρkuka (21) = ˙ 1k 2λ42 ρkuka (22) =− ˙ 2λ ρλ (23) 43 ˙ 1=−i 1j 2i 2j 3i 3j (1i 3j) (2i 3j) (i 1j) (i 2j) λ11a a λ22a a λ33a a λ13a a λ23a a λ41u a λ42u a + (i+3j) 1 +(2i 1j) 1 +(1i 2j) 1 +k (i 3+j) λ43u a ρka a ρka a ρu k λu a 0 (24) + − 2 ˙ + 2 ˙ + 2 ∇ = (i 1j) (i 2j) 1 (i j)k λ41u a λ42u a u S uk (25) + = 2 ˙ (i 3j) 1 (i 3j) λ43u a ρλu a . (26) =−2 ˙ 3504 L L Smalley and J P Krisch Adding equations (24)–(26) gives the useful result necessary to resolve the metric field equation 1i 1j 2i 2j 3i 3j (1i 3j) (2i 3j) (i 1j) λ11a a λ22a a λ33a a 2λ13a a 2λ23a a 2λ41u a + +(i 2j) +(i 3j) + + 2λ42u a 2λ43u a +1 (2i 1j) +1 (1i 2j) 1 k (i 3j) 1 (i j) 1 (i 3j) ρka a ρka a ρu k λu a u S uk ρλu a . = 2 ˙ + 2 ˙ − 2 ∇ + 2 ˙ − 2 ˙ (27) The final set of identities in this section comes from the antisymmetric combination of the tetrad variational equations similar to equations (24) and (25), [i 1j] [i 2j] [1i 2j] [2i 1j] [i 3j] 2λ41u a 2λ42u a ρka a ρka a ρλu a 0 (28) [i 1j] + [i 2j] + ˙[i j]k − ˙ + ˙ = 2λ41u a 2λ42u a ukuS (29) + =˙ and since k 0 ˙ = ρsij 2ρk a[1i a2j] a[1i a2j] . (30) ˙ = ˙ + ˙ In addition, we note the obvious identity [i 3j] [i j]k ρλu a uku 6 . (31) ˙ = ˙ Finally, combining equations (28)–(31) gives ij [i j ]k [i j ]k ρs 2u S uk 2u 6 uk 0 (32) ˙ + ˙ + ˙ = which represents the generalization of Fermi–Walker transport for the spinning string. The variations have been made assuming that the spin vector and string vector are both along the same tetrad. With this assumption, the spin module function k(x) is constant along the flow lines equation (20). If a different choice is made, for example, the string vector is written with the e1i tetrad, then the dot derivative of the spin module function is proportional to the λ12 multiplier and is not necessarily zero. The transport equation, as written in equation (32), is the same in both cases and therefore independent of the calculation’s choice of tetrads.
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