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Non static local string in Brans-Dicke theory

A recent investigation shows that a local gauge string with a phenomenological energy momentum tensor, as prescribed by Vilenkin, is inconsistent in Brans-Dicke theory. In this work, it has been shown that such a string is indeed consistent if one introduces time dependences in the metric. A set of solutions of full nonlinear Einstein's equations for the interior region of such a string are presented.

The importance of nonminimally coupled scalar tensor theories such as Brans-Dicke theory, in the context of inflationary scenario has generated a considerable interest in the gravitational field of certain topological defects, particularly cosmic strings in such theories [1], [2], [4], [5], [6]. Very recently it has been shown by the present authors that a local static gauge string, given by the energy momentum tensor components T t t = T z z = 0 and all other T µ ν = 0 [7] is inconsistent in B-D theory of gravity [8], although it is quite consistent in a more general scalar tensor theory [9], where the parameter ω is a function of the B-D scalar field.
A very recently communicated investigation by Dando and Gregory [10] has shown that one can have a nonsingular spacetime for a global string in dilaton gravity if one includes time dependences in the line element. In this work, we have shown in a very similar manner that one can obtain a consistent set of nonstatic solution for local gauge string with T µ ν mentioned above in B-D theory. The gravitational field equations in B-D theory are given by where ω is a dimensionless constant parameter of the theory and φ = φ(r) is the B-D scalar field.
The wave equation for the scalar field φ is In these equations, T µν represents the energy momentum tensor components for all the fields except the scalar field φ and T is the trace of T µν .
The local gauge string is characterised by an energy density and a stress along the symmetry axis, given by [7] The conservation of matter is represented by the equation It should be noted that (5) and (2) are not independent equations, as in view of (1) and the Bianchi identity, one yields the other.
So D and A are constants and in what follows we shall take D = 1 and e 2A = 1 which leads only to a rescaling of coordinates and no loss generality.
In view of (6a) and (6b), the field equations (1) can be written as, The wave equation (2) can be written as From (7b) and (7c) one can writë where b 0 is a separation constant.
Hence the equation to be solved are Here we have four unknowns C, φ, σ, B and we have five equations. However one can show that (10a) is not independent equation but can be obtained from rest of the equations.
In what follows we shall try to solve the system of equations for b 0 = 0 and for b 0 = 0.
In this case, from (10e), one can solve B which becomes where the constants of integration have been absorbed by rescaling the time coordinate without any loss of generality.
A coordinate transformation of the form [12] ξ = tSinhz and τ = tCoshz reduces this solution to the static form. That is the case b 0 = 0 essentially represents a static case.
From (10d), however, one can see that either σ = 0 for finite value of ω, or ω → ∞ for finite value of σ. But we know that for large value of ω, the B-D theory is indistinguishable from GR when T = T µ µ = 0 [13]. Hence we cannot have a cosmic string with T µ ν given by (4a) and (4b) in B-D theory for b 0 = 0 which is a static case. This result is completely on agreement with the previous work [8] that a static local gauge string of Vilenkin type does not exist in B-D theory of gravity.
In this case, one should note from eqn (10d) that if φ > 0, which ensures the positivity of G, the gravitational constant, and (2ω + 3) is also positive, then to have positive energy density σ, b 0 should be negative and hence we take where b 1 is a real constant.
It deserves mention at this point that Gregory [14] found a nonsingular solution for a global string in GR for nonstatic case. For that, one has to take a positive value of the constant b 0 , and even for the global string, the energy density corresponding to the string (T 0 0 , to be precise) takes a negative value for a certain value of the radial distance from the axis of the string.
where M and N are constants of integration.
Again from (10b) and (10c) one gets, Now if one assumes φ ′ φ = 0, then one can integrate the above equation to yield where K is a constant of integration.
The string energy density σ can be found from (10d) which becomes Finally the line element becomes One should note that for r → 0 i.e. near the axis of the string, the line element becomes, As a conclusion, one can say that a consistent set of interior solutions of the Einstein's equations for a Vilenkin type local gauge string can be obtained in B-D theory if it is nonstatic, although a static string of this type is inconsistent in this theory.
The nonstatic metric (20) shows that the proper volume becomes zero periodically in time and hence there is disk like singularity in spacetime periodically with time.
But this fact is not reflected in curvature scalar given by which is evidently time independent. This result is similar to the case for nonstatic global string both in GR and in dilaton gravity earlier obtained by Gregory et.al [14][10]. There also it can be shown that for b 0 < 0, the proper volume periodically vanishes with time, whereas the curvature scalar does not. It can also be seen from (22) that the space time becomes singular periodically at finite distances from the axis of the string.