J. Astrophys. Astr. (1984) 5, 317–322
Cosmological Solution with an Energy Flux Bijan Modak
Department of Physics, Presidency College, Calcutta 700073
Received 1984 February 25; accepted 1984 June 4
Abstract. The paper presents some spherically symmetric cosmological Solutions in which the velocity field is shear-free but there is a flux of energy. The solutions are believed to be new and the previous known solutions of this class due to Bergmann and Maiti may be obtained as special cases of our metrics.
Key words: heat flow—cosmology
1. Introduction Recently, Bergmann (1981) and Maiti (1982) have given spherically symmetric metrics corresponding to distributions of fluid with a heat flux. Bergmann imposed the condition that the fluid velocity vector is both geodetic and shear-free and then integrating the condition of isotropy of fluid pressure he obtained the metric (1) with (2) where R and k are undetermined functions of t alone. Obviously the essential difference between the Bergmann metric and the Friedmann metric of the isotropic universe is that in the latter the parameter k is a constant and hence, with a suitable choice of the radial variable, can be reduced to any of the values 0, +1, – 1. Maiti, on the other hand, started with the condition that the space-time is conformally flat. Plugging in the requirements of spherical symmetry and absence of shear, Maiti was led to the solution (3) Here k, as in isotropic cosmology, is a constant (reducible to 0, +1, — 1) and A, a and R are three arbitrary functions of time. It may be noted that (as Maiti also noticed) one of the functions, A, may be removed by a transformation of the time variable:
There are thus essentially two arbitrary functions of time as in the Bergmann solution. This paper presents a number of such solutions some of which contain Bergmann’s and maiti’s solutions as special cases but we claim them to be more general and new.
Besides assuming spherical symmetry and shear-free velocity field, some additional ad-hoc mathematical relations are introduced to have the solution in simple closed form. The procedure is similar to that adopted by Tolman (1939) in his classical paper on static spherically symmetric fluid spheres.
2. The field equation and their integration Consider the spherically symmetric metric (1) where ν and µ are functions of radial coordinate r and time t. With the velocity vector νµ = e –v/2 δµ this ensures that the 0 velocity field is shear-free. Allowing for the possibility of a heat flux the energymomentum tensor may be written as (4) Here, the coordinates t, r, θ, φ are numbered 0,1,2,3 respectively, and qµ is the heat flux vector. With the metric (1), the non-trivial field equations are (5) (6) (7) (8) where primes and dots indicate partial differentiation with respect to r and t respectively. Introducing the variable ξ = [eµr2]1/4 and x = ln r we may reduce Equation (5) to the form (cf. Raychaudhuri 1953) (9) We obtain a family of solutions by assuming (10) in an ad-hoc manner so that Equation (9) may be integrated in closed form. We then get (11) where Rn = [(n + 6)k/h2]4/( n+4), α = (n + 4)/2, h and k are arbitrary functions of t alone and n 0, Rn, a function of t through k, h and n respectively. Eliminating p from Equations (6) and (7) we get (12)
Cosmological solution with an energy flux
where Ζ == (v’/2r)[l + krα/4]4/α. An integral of (12) in closed form with arbitrary n, k has not been found. Therefore we discuss the solution (12) with particular values of n and functional dependence of k. When n = 0, the solution becomes (13) where a is an arbitrary function of t alone. The solution (13) differs from Maiti’s solution only in that k is an arbitrary function of time rather than a constant. Bergmann’s solution may also be obtained from Equation (13) by putting a = 0, which makes the t-lines geodetic. The solution (13) is again conformally flat, as is the case in both Bergmann’s and Maiti’s metric. When both k = 0 and a = 0, the solution (13) goes over to the standard homogeneous and isotropic model. Considered in generality, Equation (13) is thus a new solution. We write down the expressions for ρ, p, q1, and the expansion θ for our solution (13) (14a)
(14b) (14c) (14d) and the nonzero component of acceleration vector (15a) and the 3-space curvature is (15b) The parameter k is thus associated with the scalar curvature R* and the parameter a measures the non-geodesicity of the space-time. Another new class of simple solutions may be obtained by letting k = 0 in Equation (11). Equation (12) then leads to (16) where R and b are arbitrary functions of t alone. It is interesting to note that solution (16) does not belong to Maiti’s form. Maiti’s solution goes over to the Friedmann metric with flat space-section if one puts k = 0 and this is obtained from (16) only if
b = 0. We give below the expressions for ρ, p, q1 and θ with the metric (16). (17a, b) (17c, d) and the nonzero component of acceleration vector
The metric (16) apparently has a singularity as r → ∞, because ev blows up, however ρ, p, q1 and θ all vanish as r → ∞. It can be shown that the scalar curvature Rµvρσ Rµvρσ vanishes as r → ∞. So the space-time seems asymptotically flat.
3. Solution by the method of Glass and Bergmann In the present section we shall follow the procedure of Bergmann to obtain a solution more general than that given by him. Glass (1979) shows that for a spherically symmetric fluid in shear-free motion, the condition of isotropy of pressure gives (18) where A2 = ev F2 = eµ and x = r2. The above equation holds irrespective of whether qµ vanishes or not. The integration is easiest if one puts ev = A2 = 1 (in which case the motion becomes geodetic) and this was indeed done by Bergmann. However one may integrate the equations easily by plugging in other functional forms of A (or F). We exhibit only one case. Assuming
we readily find (19) where R, k, b are arbitrary functions of t alone. The role of R as the curvature parameter no longer holds now, although it does have that significance in the neighbourhood of the centre of symmetry. We go over to the Bergmann solution by putting b= 0, while k =0 leads to our solution (16). The general solution gives non-geodetic metric as (ev)' ≠ 0. The solutions (13) and (16) are conformally flat but with nonzero value of k and b, whereas (19) is not. It is interesting that in this non-conformally flat solution, the physical variables like energy density etc. diverge as r → ∞.
4. Concluding remarks Maiti started with the condition of conformal flatness. One may therefore wonder why he did not obtain our general solutions (13) and (16). The reason is that on feeling that
Cosmological solution with an energy flux
the three space metrics (t = constant) are spaces of constant curvature he wrote down the metric as
and inadvertently introduced the assumption of k being independent of time which is however not demanded by the situation. We have already noted that the essential generality obtained in our solution is that we can go over to both Maiti’s metric and Bergmann’s metric as special cases of Equation (13). What, if any, is the significance of the functions a and b appearing in our solution (13) and (16)? Obviously they have resulted in non-geodesicity of the velocity field. Naturally they are also linked with the heat flux and spatial dependence of the expansion scalar. Thermodynamically, one associates a temperature non-uniformity with heat flux; the heat flux in the rest frame (i.e. the comoving frame as used here) may occur either by radiation or by conduction. A radiation flux would complicate matters as then the Maxwell equations have to be considered. Here we therefore take the simple case of conduction and assume the phenomenological relation (20) where K is the heat conductivity and T, the temperature. In the present case this yields (21) A non-vanishing heat flux thus means a temperature gradient as may be expected to occur following the formation of a gravitational condensation. However, in the present paper, we do not make any attempt to develop a physically realistic picture of such a situation, but we estimate the temperature in a simple case. With ev = 1 as in Bergmann’s solution, this would directly link up the heat flux with the temperature gradient. In our case the calculation of Τ in general leads to quite complicated expressions. A comparatively simple formula is obtained with the metric (16). From Equation (21) we get (22) where we have assumed the conductivity K to be constant and f is an arbitrary function of t alone.
Acknowledgement The author thanks Professor A. K. Raychaudhuri for valuable guidance, the referee for helpful suggestions which have lead to an improved presentation and CSIR for the award of a fellowship.
Bijan Modak References
Bergmann, Ο. 1981, Phys. Lett., 82A, 384. Glass, E. N. 1979, J. Math. Phys., 20, 1908. Maiti, S. R. 1982, Phys. Rev., D25, 2518. Tolman, R. C. 1939, Phys. Rev., 55, 364. Raychaudhuri, A. K. 1953, Phys. Rev., 89, 417.