Accelerating FJNW Metric

TL;DR

This work constructs a new exact Einstein-scalar spacetime representing an accelerating form of the Fisher-Janis-Newman-Winicour (FJNW) metric by combining perturbative methods with Buchdahl transformations. The solution depends on $m$, $\lambda$, and $a$, and is shown to be equivalent across Weyl and C-metric–like coordinate representations, with the scalar field $\phi = \sqrt{\frac{1-\lambda^2}{2}}\ log\left(\frac{f h}{\Omega^2}\right)$. Curvature invariants reveal genuine singularities at specified locations in both the $(x,y)$ and $(r,\theta)$ forms, and a detailed geodesic analysis via an effective potential uncovers the existence and stability conditions for circular orbits for both massive and massless test particles. The results offer new insights into acceleration in scalar-field spacetimes and set the stage for further studies of quasi-normal modes and related dynamical phenomena in this class of solutions.

Abstract

We derive exact form of accelerating Fisher-Janis-Newman-Winicour (FJNW) metric by a simple perturbative method. We also argue that by using Buchdahl transformations one can obtain the same accelerating FJNW metric. We investigate singularities of the accelerating FJNW metric and study their effects on global and local structures of this spacetime. We also study geodesics and stability of circular orbits.

Figures (11)

• Figure 1: Variations of the scalar field Eq.\ref{['q']} based on the parameters $\lambda$, $r$ and $\theta$, where $m=1$ and $a=\dfrac{1}{5}$. In diagram (a), the value of $\lambda$ is held constant, and the variations of the scalar field are plotted as a function of $r$ and $\theta$. In diagram (b), r is held constant while examining the behavior of the scalar field. Note that, based on the chosen coordinate system, only the region between the two green lines is physically meaningful.
• Figure 2: (a): Kretschmann scalar as a function of $y$ at $x=0$, plotted for three values of $\lambda$ and for $m=1$ and $a=\dfrac{1}{5}$. The curvature singularities occur at $y=-1$, $y=1$ and $y=\frac{1}{2ma}$. (b): Kretschmann scalar as a function of $r$ at $\theta=\dfrac{\pi}{2}$, plotted for three values of $\lambda$ and for $m=1$ and $a=\dfrac{1}{5}$. The Kretschmann scalar exhibits singular behavior at $r=2m$ and $r=\frac{1}{a}=5$, indicating curvature singularities in the space-time.
• Figure 3: Representation of the $xy$-coordinate plane based on the metric given in Eq. \ref{['xymetric']}. In this figure, the entire range of $x$ and $y$, i.e., $\left(-\infty,+\infty\right)$, is displayed. The coordinates $r$ and $\theta$, which are related to $x$ and $y$ via the transformations $y=-\dfrac{1}{a r}$ and $x=\cos\left(\theta\right)$, are also shown. The double red lines indicate singularities, the green lines mark the roots of the function $g\left(x\right)$, and the dashed blue lines correspond to $x=0$ and $y=0$.
• Figure 4: The set $I$ represents the admissible region. The subset $I_{+}$corresponds to the part of $I$ where the function $g\left(x\right)$ is positive, and in this region, $\tau$ plays the role of time. Similarly, $I_{-}$ denotes the admissible region where $g\left(x\right)$ is negative, representing the domain in which the variable $y$ plays the role of time.
• Figure 5: In part (a), the graph of the function $f\left(y\right)$ is plotted. The admissible regions, where $f\left(y\right)>0$, are shown in green, while the inadmissible regions, where $f\left(y\right)<0$, are shown in red. In part (b), the graph of the function $g\left(x\right)$ is displayed. The regions where $g\left(x\right)$ is positive are marked in blue, and the regions where $g\left(x\right)$ is negative are marked in yellow.