Electromagnetic effect on scalar field collapse in higher curvature gravity

Abstract

We consider a “Scalar-Maxwell–Einstein–Gauss–Bonnet” theory in four dimension, where the scalar field couples non-minimally with the Gauss–Bonnet (GB) term. This coupling with the scalar field ensures the non topological character of the GB term. In such higher curvature scenario, we explore the effect of electromagnetic field on scalar field collapse. Our results reveal that the presence of a time dependent electromagnetic field requires an anisotropy in the background spacetime geometry and such anisotropic spacetime allows a collapsing solution for the scalar field. The singularity formed as a result of the collapse is found to be a curvature singularity which may be point like or line like depending on the strength of the anisotropy. We also show that the singularity is always hidden from exterior by an apparent horizon.

Appendix I: Situation of isotropic spacetime

The non static isotropic metric ansatz is taken as,

$$\begin{aligned} ds^2 = -dt^2 + a^2(t) \bigg [dr^2 + r^2d\theta ^2 + dz^2\bigg ] \end{aligned}$$

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with a(t) is the scale factor of the spacetime characterized by the coordinates t ($=x^0$), r ($=x^1$), $\theta$ ($=x^2$) and z ($=x^3$) where t is the timelike one. Moreover the scalar field and the electromagnetic field are considered to be dependent only on t. Therefore $F_{\mu \nu }$ has three non zero independent components: $F_{01}$, $F_{02}$ and $F_{03}$. With these non zero components of $F_{\mu \nu }$, we obtain various components of $T_{\mu \nu }(A)$ from Eq. (3) and are given by,

$$\begin{aligned} T_{00}= & {} \frac{1}{2}\bigg [F_{01}F^{01} + F_{02}F^{02} + F_{03}F^{03}\bigg ]\nonumber \\ T_{11}= & {} -\frac{1}{2}a^2\bigg [F_{01}F^{01} - F_{02}F^{02} - F_{03}F^{03}\bigg ]\nonumber \\ T_{22}= & {} -\frac{1}{2}a^2\bigg [-F_{01}F^{01} + F_{02}F^{02} - F_{03}F^{03}\bigg ]\nonumber \\ T_{33}= & {} -\frac{1}{2}a^2\bigg [-F_{01}F^{01} - F_{02}F^{02} + F_{03}F^{03}\bigg ]\nonumber \\ T_{10}= & {} T_{20} = T_{30} = 0\nonumber \\ T_{12}= & {} -a^2 F_{01}F^{02}, T_{13} = -a^2 F_{01}F^{03}, T_{23} = -a^2 F_{02}F^{03} \end{aligned}$$

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Using the above expressions of $T_{\mu \nu }(A)$, the non diagonal components of gravitational equation are simplified to the following form:

$$\begin{aligned} F_{01}F^{02} = F_{01}F^{03} = F_{02}F^{03} = 0 \end{aligned}$$

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which has the solution as $F_{01} = F_{02} = F_{03} = 0$. Thus a spatially flat isotropic spacetime cannot support the time dependent electromagnetic field. However a Bianchi-I spacetime, although it is spatially flat, can sustain the gauge field by virtue of its anisotropy.