[Paper Review] Cylindrically Symmetric-Static Brans-Dicke-Maxwell Solutions

Ahmet Baykaland, Özgür Delice|ArXiv.org|Dec 23, 2005

Fractional Differential Equations Solutions5 references3 citations

TL;DR

This paper derives exact static cylindrically symmetric electrovac solutions in Brans-Dicke theory with a scalar field coupled to the Maxwell field. It shows that these solutions reduce to Einstein-Maxwell solutions when the Brans-Dicke parameter $k=1$, to Brans-Dicke vacuum solutions when the field strength vanishes, and to the Levi-Civita solution in the general relativistic limit, but unlike Einstein-Maxwell solutions, they are singular in the Brans-Dicke case unless $k=1$. The scalar field is independent of the Brans-Dicke parameter $\omega$, but the electromagnetic field strengths depend on $\omega$, indicating that the $\omega \to \infty$ limit does not recover general relativity; instead, setting $k=1$ achieves this.

ABSTRACT

We present static cylindrically symmetric electrovac solutions in the framework of the Brans-Dicke theory and show that our solution yields some of the well-known solutions for special values of the parameters of the resulting metric functions.

Motivation & Objective

• To derive exact electrovac solutions in the Brans-Dicke theory with cylindrical symmetry.
• To examine how these solutions reduce to known Einstein-Maxwell and vacuum Brans-Dicke solutions under specific parameter limits.
• To investigate the role of the Brans-Dicke parameter $\omega$ and metric parameter $k$ in determining the structure of the solutions and their physical limits.
• To compare the singularities and regularity of Brans-Dicke-Maxwell solutions with those of Einstein-Maxwell solutions, particularly in the case of axial and radial electromagnetic fields.

Proposed method

• A static cylindrically symmetric metric ansatz is used, with the line element expressed in terms of radial, axial, and angular coordinates.
• The field equations are derived from a Lagrangian density involving the Brans-Dicke scalar field $\phi$, the metric, and the electromagnetic 2-form $F=dA$, using exterior calculus and Hodge duality.
• The solutions are obtained by solving the field equations for three configurations: axial magnetic field, radial electric field, and combined fields.
• The metric, scalar field $\phi$, and electromagnetic field strength $F_{\mu\nu}$ are expressed as functions of the radial coordinate $r$, with parameters $\sigma$, $k$, $c$, $W_0$, and $\omega$.
• The solutions are checked for consistency with Maxwell’s equations and the scalar field equation $ (2\omega+3)d*d\phi = 0 $, which vanishes due to the traceless nature of the Maxwell energy-momentum tensor.
• The behavior of the solutions is analyzed in the limits $k=1$ (general relativity), $c=0$ (vacuum), and $\sigma=0$ (Minkowski limit), with attention to singularities and regularity.

Experimental results

Research questions

• RQ1 How do static cylindrically symmetric electrovac solutions in Brans-Dicke theory differ from their Einstein-Maxwell counterparts in terms of structure and regularity?
• RQ2 What is the role of the Brans-Dicke parameter $\omega$ in determining the electromagnetic field strengths, and does it affect the scalar field $\phi$?
• RQ3 Can the general relativistic limit of the Brans-Dicke-Maxwell theory be recovered by setting $k=1$ rather than taking the $\omega \to \infty$ limit?
• RQ4 Why are Brans-Dicke-Maxwell solutions singular at the axis even when the Einstein-Maxwell solution (e.g., Bonnor-Melvin) is regular, and how does this change for $k=1$?
• RQ5 What is the physical significance of the parameter $k$ in the metric and scalar field, and how does it control the transition to known solutions like Levi-Civita or Minkowski?

Key findings

• The solution with an axial magnetic field is singular at the axis unless $k=1$, at which point it reduces to the Einstein-Maxwell Bonnor-Melvin solution.
• The radial electric field solution is singular at $r = c^{-2/(4\sigma+1-k)}$, which can be interpreted as the boundary of the source, and reduces to the Levi-Civita solution when $c=0$ and $k=1$.
• When $k=1$, the Brans-Dicke-Maxwell solution reduces to the Einstein-Maxwell solution with a radial electric field, and the scalar field becomes $\phi = W_0^{-1} r^{1-k}$, which is constant when $k=1$.
• For $\sigma=0$, the Levi-Civita solution reduces to Minkowski spacetime, but in the Brans-Dicke case with $\sigma=0$, the electric field does not vanish, indicating a non-vacuum solution even in the absence of curvature.
• The scalar field $\phi$ is independent of the Brans-Dicke parameter $\omega$, but the electromagnetic field strengths $F_{\mu\nu}$ explicitly depend on $\omega$, showing that $\omega$ affects the dynamics of the electromagnetic field.
• The general relativistic limit is recovered not by $\omega \to \infty$, but by setting $k=1$, which implies $1/\phi = W_0$, indicating that $k$ plays a more direct role in the physical limit than $\omega$.