Nonlocal extension of relativistic non-equilibrium thermostatics

The description of the neutron star structure involves into consideration the formalism of equations of state. We believe that the nonlocal rheological-type extension of the relativistic non-equilibrium irreversible thermodynamics and thermostatics is the most interesting trend in this direction, and in two recent works [1,2] we have prepared the corresponding mathematical basis. In the work [3] we have made the trial step towards the description of rheological-type equation of state for the neutron stars at zero temperature. In this note we suggest the new nonlocal model adapted for static spherically symmetric rheological systems. In this context we consider the heat-flux four-vector q^k and the traceless

“The work was supported by Russian Foundation for Basic Research (Grant No. 20-52-05009).

part of the non-equilibrium pressure tensor Птп to be vanishing. We search for the scalar part of the pressure tensor П as a function of the radial variable r.

• 1.    The formalism

  • 1.1.    Geometrical aspects of the theory

We use the canonic line element for the model of static spherically symmetric spacetime ds2 = B(r)dt2 — A(r)dr2 — r2 (d92 + sin2 Qdtp2^ , B(to) = 1 = A(^).(1-1)

The symmetry of this spacetime is described by the following Killing vectors:

• ^(o) = 5o ,   ^(i) =sin^9 + c°t 6c°s^5£ ,   ^(2) = cosip53e — cot 6sinv5y ,   £,^v) = 5^ .(1.2)

One obtains from (1.2) and (1.1) that the squares of the Killing vectors give the following relationships:

• ^(0) = 9kj^(0)^(0) = B,   ^(y) = 9k(£,(v)£,3(v) = —r sin 0 ,   ^(1) + ^(2) + ^(y) = — 2r .(1-3)

• 1.2.    The velocity four-vector and the director

The velocity four-vector and its covariant derivative are, respectively, of the form

′′

Uг = 50 -= ,  VkU . = —505[ —= ^ DUг = UkVkU . = —5[ — .(1.4)

V B              2 у B

The prime denotes the derivative with respect to the radial variable. For these quantities we deal with the vanishing expansion scalar (О = 0), shear tensor ^ст_гк = 0), and vorticity tensor Uw_kk = 0); the only acceleration four-vector DU_г is non-vanishing. With (1.4) the Eckart’s version of thermodynamics predicts that, when the heat-flux four-vector q^k vanishes [4], the temperature T (r) is predetermined to be of the following form:

⊥

■■ = 0 ^ _TV k T = DU k ^ T (r) = - BL .                  (1-5)

The director ^' can be defined as the unit spacelike four-vector orthogonal to the velocity four-vector, which inherits the symmetry of the spacetime, i.e.,

TO =0 , ^X = —1 , ^_н At = ^k ^ - ^ Э к &) =0 ,         (1.6)

where £$_(а) is the Lie derivatives along all the Killing vectors (1.2). All these requirements are satisfied for the four-vector Al = 5^^д; clearly, it possesses the following properties:

h = ---_r----⁰------у ^Vk Qi^ ) ^(0) [^21) + ^22) + ^2y)]

1 d Г h₀ r²-AB dr [ 2

(1.8)

• 1.3.    Extension of the entropy flux four-vector

The decomposition of the entropy flux four-vector S^k can be organized, e.g., as follows:

S^k = S₍^kis) + Mk^k /р—W ( T )] + 2 [Р^-1П]2 +| т П²! ,                (1.9)

where the contribution S^k_IS ) appeared in the Israel-Stewart theory [5]. Here т is a constant and /(T) is some function of the temperature. The inverse operator P^-1 is defined as

РР-1П = П ^ Р-1П =

/r∞

с/гП(г)^Л(г).

(1.10)

For the static spherically symmetric model the entropy production scalar takes the form

П2

a = VkS^k = К П [/ ( T ) + Р^-1П + т РП] = — > 0 , ⁹s

(1.П)

where S is some function of the temperature. Thus, the non-equilibrium pressure П has to satisfy the integro-differential equation

• т РП - ^П+ / ( T )+ Р^-1П = 0 . 9 (h

  (1.12)

  
  

The pure differential version of this equation is

^{T Р2П —} 9^^{РП + П}

¹ -Pf о¹?)] + TFrPT = 0. 9^^      dT

(1.13)

We deal with the linear differential equation of the second order known as the Burgers equation [6,7]. The inhomogeneity of the temperature with the law PT = — ^т(y^ys ^r ~ predetermines the features of the pressure distribution П(г); wlien /(T) = const the equation (1.13) admits the vanishing pressure

П = 0.

Conclusion

The equation (1.13) gives us the new equation describing the profiles of the non-equilibrium pressure H(r) in the model of static spherically symmetric object. The next step we have to do is to formulate the extended equation of hydrostatic equilibrium, which is the key element of the analysis of the star structure. We hope to analyze this problem in the nearest future.