# Riemann Dual Tensor and Scalar Field Theory

I'm trying to find the component equation of motion for the action in a paper. The action for the system is, $$S=\frac{m_P^2}{8\pi}\int d^4x\sqrt{-g}\bigg(\frac{R}{2}-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi+\alpha\phi\mathcal{G}\bigg),$$ where $$\mathcal{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2$$ is the Gauss-Bonnet invariant. The modified Einstein equation of motion for the action is, $$G_{\mu\nu}=T_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi-\frac{1}{2}g_{\mu\nu}(\partial\phi)^2-\alpha(g_{\rho\mu}g_{\delta\nu}+g_{\rho\nu}g_{\delta\mu})\nabla_{\sigma}(\partial_{\gamma}\phi\epsilon^{\gamma\delta\alpha\beta}\epsilon^{\rho\sigma\lambda\eta}R_{\lambda\eta\alpha\beta}),$$ where the final term in the brackets is the Riemann Dual Tensor (it's divergenceless I think). The scalar field is a function of $$r$$ only and hence $$\gamma=r$$ for the third term to be non-zero. In the first part of the paper I have linked, the metric, $$ds^2=-e^{A(r)}dt^2+e^{B(r)}dr^2+r^2(d\theta^2+sin^2(\theta)d\varphi^2),$$ is used as an ansatz as a solution. This is then subbed into the Einstein equation and the $$tt, rr$$ and $$\theta\theta$$ equations are found.

I have tried doing the contractions in MAPLE for the $$tt$$ component, ensuring that the indices are correct (eg. $$\rho=t$$ since $$\mu=t$$ etc.). However, I keep getting terms of the form, $$-8\alpha e^A\frac{(1-e^B)(\phi''-\frac{B'}{2}\phi')}{e^{2B}r^2}+\frac {1}{2}e^{A}\phi'^2,$$ which is close to the answers they produce in the appendix except the $$B'\phi'$$ term in the appendix carries a ($$e^{B}-3$$) and I don't know where this 3 comes from. In finding my answer I use the divergenceless nature of the Riemann dual ($$*R*$$), in order to write the last term on the RHS of the Einstein equation as

$$\nabla_{\sigma}(\partial_{\gamma}\phi\epsilon^{\gamma\delta\alpha\beta}\epsilon^{\rho\sigma\lambda\eta}R_{\lambda\eta\alpha\beta})=\epsilon^{rt\varphi\theta}\epsilon^{tr\theta\varphi}R_{\theta\varphi\varphi\theta}\nabla_{r}\partial_r\phi=(*R*)_{\theta\varphi\varphi\theta}\bigg(\phi''-\frac{B'}{2}\phi'\bigg),$$ where in the last equality I have expanded out the covariant derivative and use the $$\Gamma^{r}_{rr}$$ Christoffel symbol.

Further problems develop when I look at the $$rr$$ term as I missing at least 4 terms from the appendix.

I'm unsure if there is a problem in my understanding, or if there is something I should know about the Riemann Dual that I don't have here, or whether my use of only the $$\Gamma^r_{rr}$$ symbol is correct. If anyone could give me a helping hand seeing where my calculations are going awry I would really appreciate it.

2. $$\gamma\neq r$$ all of the time due to the covariant derivative.