Ideal Magnetohydrodynamics with Radiative Terms: Energy Conditions
Oscar M. Pimentel, F. D. Lora-Clavijo, Guillermo A. González
aa r X i v : . [ g r- q c ] A p r Ideal Magnetohydrodynamics with RadiativeTerms: Energy Conditions
Oscar M. Pimentel , ∗ , F. D. Lora-Clavijo , † Guillermo A.Gonz´alez , ‡ , Grupo de Investigaci´on en Relatividad y Gravitaci´on, Escuela de F´ısica,Universidad Industrial de Santander, A. A. 678, Bucaramanga 680002, Colombia. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]
PACS numbers: 04.20.-q; 04.40.-b; 47.10.-g
Abstract.
Nowadays, the magnetic and radiation fields are very important tounderstand the matter accretion into compact objects, the dynamics of binary systems,the equilibrium configurations of neutron stars, the photon diffusion, etc. The energyand the momentum associated to these fields, along with the matter one, need tosatisfy some conditions that guarantee an appropriate physical behavior of the sourceand its gravitational field. Based on this fact, we present the energy conditions fora perfect fluid with magnetic and radiation field, in which the radiation part of theenergy-momentum tensor is assumed to be approximately isotropic, in accordance withthe optically thick regime. In order to find these conditions, the stress tensor of thesystem is written in an orthonormal basis in which it becomes diagonal, and the energyconditions are computed through contractions of the energy-momentum tensor with thefour velocity vector of an arbitrary observer. Finally, the conditions for a magnetizedfluid are presented as a particular case in which the radiation contribution is zero.
1. Introduction
It is usually expected that any solution to the Einstein field equations, describing realisticsources of gravitational field, has to satisfy the energy conditions [1]. These conditionsfor a perfect fluid and for a viscous fluid have been widely used in some astrophysicaland theoretical contexts [2, 3, 4]. Nevertheless, recent researches are focused on themagnetic and radiation fields contributions to the energy-momentum tensor, becausethey allow describing a wider variety of astrophysical scenarios; for instance, the effectof a toroidal magnetic field on the equilibrium configurations of rotating neutron starswas treated in [5]. The structure of neutron stars when strong magnetic fields arepresented is studied in [6]. Additionally, recent calculations have demonstrated that thebinary neutron stars mergers, using magnetohydrodynamic simulations in full generalrelativity, produce relativistic jets and strong magnetic fields, which serve as a central deal Magnetohydrodynamics with Radiative Terms: Energy Conditions ρ and the velocity v i to ρ = ρ atm and v i = 0, once ρ ≤ ρ atm [14], or set to zero the initial value of magnetic field, instead of the velocity,in the low density regions [15], and additionally, impose constraints on the conservativevariables to reduce posible numerical errors that can drive to unphysical results [16].The correct values of ρ atm , in principle, only depends on the specific problem underconsideration; the only requirement is that the dynamics of the system should not beaffected by the presence of such atmosphere [17], which is usually treated as a low-densityfluid governed by the same equation of state of the bulk matter [18, 19]. Now, whenthe fluid has magnetic and radiation fields, the energy conditions may represent usefulconstraints on the physical variables, that motivate the atmospheres and the possibleequations of state for that kind of source. For all the reasons mentioned, it becomesimportant to find the energy conditions for a perfect fluid with magnetic and radiationfields.The energy conditions for a viscous fluid were calculated in [20]. Nevertheless,when the heat flux vector of the fluid is different from zero, the procedure to computethe energy conditions becomes more difficult. In 1988 Kolassis, Santos, and Tsoubelisfound the conditions for the energy-momentum tensor of a viscous fluid with heat flux,written in the Eckart approximation of small velocity gradients [21]. To do it, the authorsdiagonalized the energy-momentum tensor and wrote its energy conditions in terms ofthe corresponding eigenvalues [1]. With this method, it was necessary to deal witha fourth order polynomial equation, so additional assumptions were required in orderto find the eigenvalues. Recently, the energy conditions for the same dissipative fluidwere found in [22], without considering any approximation on the velocity gradients.The used method leave the energy conditions free from restrictions and consists indiagonalize the spatial part of the energy-momentum tensor, compute the physicalquantities as measured by an arbitrary observer, and finally, to write the conditionsso that they are independent of the observer. Now, in this work we are going to findthe energy conditions for a perfect fluid with magnetic and radiation field by using thesame algebraic procedure developed in [22], that leaves the conditions independent from deal Magnetohydrodynamics with Radiative Terms: Energy Conditions − , + , + , +), and with units in which the gravitational constant G and thespeed of light c are equal to one.
2. The Energy-Momentum tensor for Relativistic Magneto-Fluids withRadiation Field
In this section we are going to present the energy-momentum tensor for a perfect fluidwith magnetic and radiation fields. This tensor can be splitted into three terms, T αβ = T αβf + T αβm + T αβr , (1)where T αβf , T αβm and T αβr are the energy-momentum tensors of the fluid, the magneticfield, and the radiation field, respectively. Now, we will introduce each tensor byseparately, presenting the approximations and assumptions for each contribution. a. The fluid contribution We are going to compute the energy conditions for a non-dissipative matter field inwhich the energy fluxes are zero and there are not anisotropic stresses. In this case, theenergy-momentum tensor of the fluid can be written as T αβf = ( ρ + p ) u α u β + pg αβ , (2)where u α is the four-velocity vector of the fluid, ρ is the energy density in the comovingframe, and p is the thermal pressure of the fluid, also measured by the comoving observer.A fluid described by (2) is commonly know as perfect fluid , and is widely used to modelthe average properties of isolated rotating relativistic stars [23, 24], and to study thefluid dynamics around compact objects [25, 26]. b. The magnetic field contribution The energy-momentum tensor of the electromagnetic field is given by the followingexpression, T αβem = 14 π (cid:18) F αγ F βγ − F γδ F γδ g αβ (cid:19) , (3) deal Magnetohydrodynamics with Radiative Terms: Energy Conditions F αβ is the Faraday tensor. This tensor can be decomposed in terms of the electricfield E α , and the magnetic field, B α , measured by the comoving observer, as F αβ = E α u β − E β u α + 12 ǫ αβµν ( u µ B ν − u ν B µ ) (4)where ǫ αβγδ is the Levi-Civita tensor.On the other hand, according to the Ohm’s law, j α = σE α , where the conductioncurrent, j α , is related to the electric field through the constitutive relation, j α = σE α ,where σ is the conductivity. Now, if we suppose that the material is a perfect conductor,then σ → ∞ , and the only way to have a finite conduction current is that E α = 0. Thisapproximation is the basis for the ideal magnetohydrodynamics, which is very usefulto describe highly conducting astrophysical fluids where the effect of the magnetic fieldcannot be neglected, such as neutron stars, accretion fluids, magnetized winds, etc [27].As a consequence of this approximation, F αβ u α = 0; so it is possible to write theenergy-momentum tensor (3) as [27, 28] T αβm = | b | u α u β + 12 | b | g αβ − b α b β , (5)where | b | = b α b α and b α = B α / √ π . The 4-vector b α is spacelike and satisfies theproperty b α u α = 0. The tensor T αβm describes the energy and the momentum of themagnetic field in a well-conductor fluid. c. The radiation field contribution The last term in the right hand side of (1), which describes the energy and themomentum of the radiation field, is given by [29] T αβr = Z I ν N α N β dνd Ω , (6)where I ν = I ( x α ; N α , ν ) is the specific intensity of the radiation at position x α withfrequency ν and moving in the direction N α ≡ p α /h P ν , p α is the photon four-momentum, h P is the Planck constant, and d Ω is the differential of solid angle; ν , I ν and d Ωare measured in a comoving frame. Moreover, since the photon propagation directionbecomes N ˆ α = (1 , N ˆ i ), ˆ i = 1 , ,
3, in this comoving frame [9], we can define the radiationmoments [29], i.e. , the radiation energy density T ˆ0ˆ0 r = E r , the radiation flux T ˆ0ˆ ir = F ˆ ir ,and the radiation stress tensor T ˆ i ˆ jr = P ˆ i ˆ jr , as E r = Z I ν dνd Ω , F ˆ ir = Z I ν N ˆ i dνd Ω , P ˆ i ˆ jr = Z I ν N ˆ i N ˆ j dνd Ω . (7)Now, many current numerical codes are devoted to describe systems such asrelativistic stars or high density fluids in which the photons free paths are small, andtherefore they interact with the material and diffuse through it. As a consequence, theradiation is trapped in the interior of the compact object and I ν becomes isotropic,so that I ν = I ( x α ; ν ) [29]. Under this limit of isotropy, widely known as the opticallythick regime, the radiation stress tensor becomes P ˆ i ˆ jr = P r δ ˆ i ˆ j , with P r = E r /
3, andthe radiation flux vector vanishes [9]. Nevertheless, allowing F ˆ ir to have small non-zero deal Magnetohydrodynamics with Radiative Terms: Energy Conditions T αβr = ( E r + P r ) u α u β + F αr u β + F βr u α + P r g αβ , (8)where F αr is the radiation flux four-vector which is defined as F αr = h αβ Z I ν N β dνd Ω , (9)and h αβ = g αβ + u α u β is the induced metric of the normal space to the 4-velocity u α ,so it is easy to show that u α F αr = 0 .Finally, replacing (2), (5), and (8) in (1) we obtain [9] T αβ = ( ρ + p + | b | + E r + P r ) u α u β + (cid:18) p + P r + 12 | b | (cid:19) g αβ + F αr u β + F βr u α − b α b β , (10)which is the total energy-momentum tensor for a perfect magneto-fluid with a radiationfield.
3. The Energy Conditions
Now, we are interested in finding the energy conditions for a fluid described by (10).These conditions are restrictions on the energy-momentum tensor that ensure thesolutions to the Einstein equations to describe realistic gravitational systems. We canbriefly formulate the energy conditions as [20, 30] (a) The weak energy condition:
The energy density, ǫ , measured by an arbitraryobserver, defined by its four-velocity vector W α , must be positive, i.e. , ǫ = T αβ W α W β ≥ . (11) (b) The strong energy condition: Any timelike or null congruence of geodesics mustbe convergent. In other words, the gravitational field must be attractive. Thiscondition is satisfied if the following inequality holds for any arbitrary observer [1], µ = R αβ W α W β ≥ . (12)Now, by using the Einstein equations, the last expression becomes T αβ W α W β ≥ − T, (13)so we can say that the stress of the fluid can not be much larger than its energydensity. deal Magnetohydrodynamics with Radiative Terms: Energy Conditions (c) The dominant energy condition: The energy flux density measured by an arbitraryobserver, S α = − T αβ W β , (14)must be a future oriented timelike or null vector. This means that S > , (15) S α S α ≥ . (16)This condition can be physically interpreted as saying that the matter and energy cannot flow faster than light. We can see that the dominant energy condition contains theweak one, but all the three conditions are mathematically independent [31]. Therefore,the aim of this section is to write this conditions in terms of the energy density andpressure of the fluid, the energy density and pressure of the radiation, the radiation fluxvector, and the magnetic field.There are two ways for addressing the problem. The first one is to rewrite theenergy conditions in terms of the eigenvalues of T αβ [1, 21]. Nevertheless, computingthe eigenvalues of (10) is a difficult task because we need to deal with a fourth-orderpolynomial, so it becomes necessary to impose additional restrictions on the energy-momentum tensor in order to find solutions. The other way for finding the energyconditions is to define an orthonormal basis in which the spatial part of T αβ (the oneassociated with the normal and tangential stresses) becomes diagonal. In this basis,the four-velocity W α has arbitrary components, and therefore, the energy conditionsare written in terms of these components (cf. 11 - 16). Then, with the aim of ensuringthe invariance of the energy conditions, it is necessary to apply algebraic procedures todecouple them from the components of W α . This last method was applied in [22] inorder to find the energy conditions for a viscous fluid with heat flux.In this paper, we are going to use the same method used in [22], but now with theenergy-momentum tensor of a perfect magneto-fluid with radiation field (10). To doit, we start by computing the total energy of the system, E , measured by a comovingobserver, E = T αβ u α u β = ρ + 12 | b | + E r . (17)We also compute the isotropic pressure [32], which is written via the spatial trace of theenergy-momentum tensor asˆ P = 13 h αβ T αβ = p + P r + 16 | b | . (18)Now, in terms of E and ˆ P the energy-momentum tensor (10) takes the form T αβ = ( E + ˆ P ) u α u β + ˆ P g αβ + F αr u β + F βr u α + Π αβ , (19)where we have defined the deviatoric tensorΠ αβ = 13 | b | h αβ − b α b β . (20) deal Magnetohydrodynamics with Radiative Terms: Energy Conditions αβ satisfies the same properties of the deviatoric stresstensor from relativistic fluid dynamics [32]: it is traceless and its projection along u α iszero, so g αβ Π αβ = Π αα = 0 , Π αβ u α = Π αβ u β = 0 , (21)respectively. Finally, if we introduce the stress tensor S αβ = ˆ P h αβ + Π αβ , (22)then (19) reduces to T αβ = E u α u β + F αr u β + F βr u α + S αβ , (23)which has the same form as the one of the energy-momentum tensor for a viscous fluidwith heat flux. This make sense since the term b α b β may introduce tangential stressesdue to the magnetic field, in a similar manner as the deviatoric stress tensor does for aviscous fluid. Additionally, the radiation flux vector, F αr , plays an analogous roll to theone of the heat flux vector.The tensor S αβ is the spatial part of T αβ ; therefore, it describes the normal andtangential stresses due to the fluid, the magnetic field, and the radiation field. We canfind an orthonormal basis of eigenvectors { X α , Y α , Z α } in which S αβ takes the diagonalform, S αβ = P X α X β + P Y α Y β + P Z α Z β , (24)where, P i , i = 1 , , , are the eigenvalues. In this basis of eigenvectors, the radiationflux and the magnetic field have arbitrary components, so, F αr = F r X α + F r Y α + F r Z α , (25) b α = b X α + b Y α + b Z α , (26)in such a way that ( F r ) = ( F r ) + ( F r ) + ( F r ) and | b | = b α b α = ( b ) + ( b ) + ( b ) .Additionally, in the orthonormal tetrad { u α , X α , Y α , Z α } , the four-velocity vector of thearbitrary observer takes the form W α = γu α + A X α + A Y α + A Z α , (27)where γ and A i , i = 1 , , , satisfy the relation γ = √ A ≥
1, with A = A + A + A .Finally, the eigenvalues, P i can be computed as follows: P = S αβ X α X β = p + P r + 12 | b | − b , (28) P = S αβ Y α Y β = p + P r + 12 | b | − b , (29) P = S αβ Z α Z β = p + P r + 12 | b | − b , (30)where the different magnetic components are the responsible for the anisotropy of S αβ ,because in general, P = P = P .The following step is to use (23) and (27) to write the inequalities (11 - 16) and try todevelop an algebraic procedure to decouple the energy conditions from the components deal Magnetohydrodynamics with Radiative Terms: Energy Conditions W α . Nevertheless, this algebraic procedure will be the same as the one used in [22]to find the energy conditions for a viscous fluid with heat flux. Hence, we can statethat the energy conditions for the energy-momentum tensor presented in (23) are (seeAppendix A) E ≥ F r , (31) E + P i ≥ F r , i = 1 , , , (32) E + P + P + P ≥ F r . (33) E ≥ P i + F r + 2( E + P + P + P ) F r , i = 1 , , . (34)Therefore, using (17), (18), and (28-30), the energy conditions for a perfect magneto-fluid with radiation field are ρ + 12 | b | + E r ≥ F r , (35) ρ + | b | + E r + p + P r − b i ≥ F r , i = 1 , , , (36) ρ + | b | + E r + 3 p + 3 P r ≥ F r . (37)( ρ + 12 | b | + E r ) ≥ (cid:18) p + P r + 12 | b | − b i (cid:19) + F r + 2( ρ + | b | + E r + 3 p + 3 P r ) F r , i = 1 , , . (38)With the inequalities (35), (36), and (37) the weak energy condition and the strongenergy condition are both satisfied; while the dominant energy condition is equivalentto (38).As a particular case, when E r = 0, P r = 0 and F αr = 0, the radiation field vanishes,and (35-38) reduce to ρ + 12 | b | ≥ , (39) ρ + p + | b | − b i ≥ , i = 1 , , , (40) ρ + 3 p + | b | ≥ , (41) ρ ≥ | p − b i | , i = 1 , , . (42)These energy conditions are also of astrophysical and numerical interest because theymay be applied to the energy-momentum tensor of a perfect fluid with an arbitrarymagnetic field.
4. Conclusion
We have obtained the energy conditions for a perfect fluid with magnetic and radiationfields, whose physical description is the main objective of the general relativisticradiation magneto-hydrodynamics . We first presented the energy-momentum tensor ofthe total system (10), consisting of the fluid, the magnetic field, and the radiation field.With this tensor, we computed the energy density of the system, by projecting (10) deal Magnetohydrodynamics with Radiative Terms: Energy Conditions αβ , to the stress tensor of the fluid (22), in the same way as the deviatoric stress tensordoes for a viscous fluid.By writing the four-velocity vector of the arbitrary observer W α in the comovingtetrad, where the stress tensor (22) is diagonal, we can compute the energy density ǫ ,the energy flux density S α , and the scalar µ , as measured by this observer. Then, byapplying the same algebraic procedure presented in [22], we showed that the weak, andstrong energy conditions are satisfied if (35-37) are simultaneously satisfied. We alsoshowed that the dominant energy condition, for the system of interest, is equivalentto the inequality (38). As a particular case, when E r = 0, P r = 0, and F αr = 0, theradiation field vanishes, and the energy conditions reduce to those presented in (39-42), and correspond to a perfect fluid with magnetic field. These conditions are usefulsince many numerical simulations are carried out using this fluid as a test fluid in acurved spacetime background, or as a source of gravitational field to model the averageproperties of neutron stars and white dwarfs.Finally, it is worth mentioning that the procedure to compute the energy conditionsfrom the energy-momentum tensor (10) is quite general, and can be applied to any T αβ . In particular, it is possible to consider fluids in which the specific intensity ofthe radiation, I ν , is not isotropic, and therefore the radiation stress tensor can be non-diagonal. Nevertheless, we have worked with this approximation, first of all with theaim of simplifying the calculations and the results, and secondly because the opticallythick limit is used in most of the codes designed to study the radiation process associatedwith the dynamics of accretion disks around compact objects in the frame of the generalrelativity. Acknowledgments
O. M. P. wants to thanks the financial support from COLCIENCIAS and UniversidadIndustrial de Santander. F.D.L-C gratefully acknowledges the financial support fromUniversidad Industrial de Santander under grant number 1822. G. A. G. was supportedin part by VIE-UIS, under Grants No. 1347 and No. 1838, and by COLCIENCIAS,Colombia, under Grant No. 8840.
Appendix A. Derivation of the energy conditions
In this appendix we are going to present a brief summary of the algebraic procedure tocompute the energy conditions (31 - 34) for a perfect fluid with magnetic and radiationfields. Nevertheless, this procedure is explained with more detail in [22].To start, we use the energy-momentum tensor (23) and the four-velocity vector ofthe arbitrary observer (27), to write the energy density, ǫ , and the term µ = R αβ W α W β , deal Magnetohydrodynamics with Radiative Terms: Energy Conditions ǫ = E γ + X i =1 P i A i − γ ( F r · A ) , (A.1) µ = 12 E + X i =1 P i ! + X i =1 ( E + P i ) A i − γ ( F r · A ) , (A.2)respectively. Now, using the relations, γ = 1 − A and γ ≥ A , and taking into accountthat − F r A ≤ F r · A ≤ F r A , we can conclude, after some calculations, that ǫ ≥ ( E − F r ) + X i =1 ( E + P i − F r ) A i , (A.3) µ ≥ E + 3 ˆ P − F r X i =1 ( E + P i − F r ) A i . (A.4)Therefore, the necessary and sufficient conditions to have ǫ ≥ i.e. to satisfy the weakenergy condition for all values of A i are E ≥ F r , (A.5) E + P i ≥ F r , (A.6)with i = 1 , ,
3. Equivalently, the strong energy condition ( µ ≥
0) is satisfied for allvalues of A i if E + 3 ˆ P ≥ F r , (A.7) E + P i ≥ F r , (A.8)where i = 1 , , S α in a comovingtetrad through the transformation S ( µ ) = e α ( µ ) S α , where e α ( µ ) are orthonormal vectors. Inthis way, the metric tensor reduces to that of Minkowski and the calculations are easier.The condition (15) can be written as S (0) = E γ − F r · A > ( E − F r ) γ , and therefore, S (0) > E > F r .Finally, the term S α S α in the condition (16) becomes S α S α = S ( µ ) S ( µ ) = − N ( S )with N ( S ) = ( S (0) ) − ( S (1) ) − ( S (2) ) − ( S (3) ) and S ( i ) = F r i γ − P i A i . In this way, wecan show that N ( S ) = ( E γ − F r · A ) − X i =1 ( F r i γ − P i A i ) . (A.9)Then, expanding this expression, using the condition E > F r , and the fact that F r > N ( S ) ≥ [ E − F r − E + 3 ˆ P ) F r ]+ X i =1 [ E − F r − E + 3 ˆ P ) F r − P i ] A i . (A.10) deal Magnetohydrodynamics with Radiative Terms: Energy Conditions N ( S ) ≥
0) for all values of A i , then E ≥ F r + 2( E + 3 ˆ P ) F r , (A.11) E ≥ F r i + F r + 2( E + 3 ˆ P ) F r , (A.12)where i = 1 , ,
3. We can see that the condition (A.12) contains (A.11) and therefore, allthe energy conditions are satisfied if (A.5), (A.6), (A.7) and (A.12) are simultaneouslysatisfied.
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