Some classes of gravitational shock waves from higher order theories of gravity

Abstract

We study the gravitational shock wave generated by a massless high energy particle in the context of higher order gravities of the form $F(R,R_{\mu\nu}R^{\mu\nu},R_{\mu\nu\alpha\beta}R^{\mu\nu \alpha\beta})$. In the case of $F(R)$ gravity, we investigate the gravitational shock wave solutions corresponding to various cosmologically viable gravities, and as we demonstrate the solutions are rescaled versions of the Einstein-Hilbert gravity solution. Interestingly enough, other higher order gravities result to the general relativistic solution, except for some specific gravities of the form $F(R_{\mu\nu}R^{\mu\nu})$ and $F(R,R_{\mu\nu}R^{\mu\nu})$, which we study in detail. In addition, when realistic Gauss-Bonnet gravities of the form $R+F(\mathcal{G})$ are considered, the gravitational shock wave solutions are identical to the general relativistic solution. Finally, the singularity structure of the gravitational shock waves solutions is studied, and it is shown that the effect of higher order gravities makes the singularities milder in comparison to the general relativistic solutions, and in some particular cases the singularities seem to be absent.

Appendix: Christoffel symbols and curvature tensors of the gravitational shock wave metric

Here we present in detail the Christoffel symbols and the components of the Riemann and Ricci tensors corresponding to the gravitational shock wave metric,

$$ \mathrm{d}s^{2}=-\mathrm{d}u\mathrm{d}v+H(u,x,y) \mathrm{d}u^{2}+ \mathrm{d}x^{2}+\mathrm{d}y^{2}. $$

(47)

The Christoffel symbols are given below,

$$\begin{aligned} \begin{aligned} & \varGamma^{2}_{\,1\,1}=-\partial_{u}\,H(u,x,y),\qquad \varGamma^{2}_{\,3\,1}=-\partial_{x}\,H(u,x,y), \\ &\varGamma^{2}_{\,4\,1}=-\partial_{y} \,H(u,x,y),\qquad \varGamma^{3}_{\,1\,1}=-\frac{1}{2}\partial_{x}\,H(u,x,y), \\ &\varGamma^{4}_{\,1\,1}=-\frac{1}{2}\partial_{y}\,H(u,x,y). \end{aligned} \end{aligned}$$

(48)

The only non-zero component of the Ricci tensor $R_{\mu\nu}$ is,

$$ R_{uu}=-\frac{1}{2} \biggl( \frac{\partial^{2}}{\partial x^{2}}+ \frac{ \partial^{2}}{\partial y^{2}} \biggr) H(u,x,y). $$

(49)

The Ricci scalar $R$, the Ricci tensor squared $R_{\mu\nu}R^{\mu \nu}$, the Riemann tensor squared $R_{\mu\nu k \lambda}R^{\mu \nu k \lambda}$ and the Gauss-Bonnet scalar, calculated for the metric (47), are equal to zero, that is,

$$ \begin{aligned} &R=0,\qquad R_{\mu\nu}R^{\mu\nu}=0, \qquad R_{\mu\nu k \lambda}R^{\mu\nu k \lambda}=0, \\ &\mathcal{G}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\lambda k}R^{\mu\nu\lambda k}=0. \end{aligned} $$

(50)

Finally, the non-zero components of the Riemann tensor are the following,

$$\begin{aligned} \begin{aligned} & R^{2}_{3\,1\,3}=\partial_{x}^{2}H(u,x,y), \qquad R^{2}_{3\,1\,4}=\partial_{(x,y)}H(u,x,y), \\ &R^{2}_{3\,3\,1}=-\partial_{x}^{2}H(u,x,y), \qquad R^{2}_{3\,4\,1}=-\partial_{(x,y)}H(u,x,y), \\ &R^{2}_{4\,1\,3}=\partial_{y}H(u,x,y),\qquad R^{2}_{4\,1\,4}=\partial_{y}^{2}H(u,x,y), \\ &R^{2}_{4\,3\,1}=-\partial_{(x,y)}H(u,x,y), \qquad R^{2}_{4\,4\,1}=-\partial_{y}^{2}H(u,x,y), \\ &R^{3}_{1\,1\,3}=\frac{1}{2}\partial_{x}^{2}H(u,x,y), \qquad R^{3}_{1\,1\,4}=\frac{1}{2}\partial_{(x,y)}H(u,x,y), \\ &R^{3}_{1\,3\,1}=-\frac{1}{2}\partial_{x}^{2}H(u,x,y),\quad\ \ \ R^{3}_{1\,4\,1}=-\frac{1}{2}\partial_{(x,y)}H(u,x,y), \\ & R^{4}_{1\,1\,2}=\frac{1}{2}\partial_{(x,y)}H(u,x,y),\qquad R^{4}_{1\,1\,4}= \frac{1}{2}\partial_{y}^{2}H(u,x,y), \\ &R^{4}_{1\,3\,1}=-\frac{1}{2}\partial_{(x,y)}H(u,x,y),\qquad R^{4}_{1\,4\,1}=-\frac{1}{2}\partial_{y}^{2}H(u,x,y) \end{aligned} \end{aligned}$$

(51)