Black hole mirages: electron lensing and Berry curvature effects in inhomogeneously tilted Weyl semimetals

We thank the Referee for his/her report, and like to add the following remarks on point 3.1 and 3.2. In Schwarzschild coordinates, the matrix representation of the metric $g_{\mu\nu}$ reads
\begin{align}
g =
\begin{pmatrix}
1-\frac{u(r)^2}{c^2} & 0 & 0 & 0 \\
0 & -\frac{1}{1-\frac{u(r)^2}{c^2}} & 0 & 0 \\
0 & 0 & -r^2 & 0 \\
0 & 0 & 0 & -r^2 \sin^2(\theta )
\end{pmatrix}.
\end{align}
This leads to the Hamiltonian
\begin{align}
\mathcal H =
\frac{1}{2} \left(\frac{k_1^2 u(r)^2}{c^2}+\frac{E^2}{c^2-u(r)^2}-k_1^2-\frac{k_2^2}{r^2}-\frac{k_3^2 \csc^2(\theta )}{r^2}\right),
\end{align}
with a dispersion relation on the $(x,y)$ plane, assuming $r>r_t$
\begin{align}
E_\pm = \pm c\sqrt{1-\frac{u(r)^2}{c^2}}\sqrt{k_1^2+\frac{L_z^2}{r^2}}
\end{align}
and therefore a different dynamics parametrized by the temporal coordinate. We note that the analog setup could be naively realized by a spatially modulated Weyl cone of the form $E_\pm=\pm v(r) k$ (using $k_z=0$, like in our manuscript, such that contributions to the dynamics by anomalous terms vanish). The associated Hamiltonian is given by $\mathcal H = \chi\,v(r)\,{\vec k}\cdot {\vec\sigma}$, where the tilt is translated into an isotropic velocity field $v(r)=\sqrt{c^2-u(r)^2}$. Note that, contrary to Gullstrand-Painlevé coordinates, Schwarzschild coordinates only make sense for $r>r_t$ because the metric is divergent at $r=r_t$, and the dispersion becomes complex at $r<r_t$. Despite these differences, we find by straightforward evaluation that the effective potential identified from $\dot r(\tau)$ for $r>r_t$ is equal in both coordinate systems.

We are indebted to Mr. van Wezel for being persistent on his last doubts. Before we submit a revised version, we would like to eliminate the last point of confusion.

We believe the main reason for his doubt is rooted in the fact that radial trajectories are accelerated during their outwards movement, such that we are focussing on this discussion in more detail here. Consider initial coordinates $x>\rho_t$, $y=z=0$ and $k_x>0$, $k_y=k_z=0$, such that the radial trajectory is along the $x$-axis. The equations of motion can then be solved for the momentum coordinate
$$
k_x(t)=k_x(0)^2\exp\left(-\int_0^t u'(x(\tau))d\tau\right).
$$
From the solution of $k_x$, we recognize $\dot x(t) = u(x) + v_F>0$. One can also show that the acceleration is $\ddot x(t) = u'(x)(u(x) + v_F)>0$. Also, since $L_z=0$, the effective potential vanishes, and $\dot x^2(t) = E_+^2/k^2=E_+^2\exp(2\int_0^t u'(x(\tau))d\tau)/k_x(0)^2$.

We want to stress that the radial velocity and the effective potential in our manuscript are consistent with Eqs. (12), (13) [for $\sigma=0$] and Fig. 1 of Phys. Rev. D 92, 084042. The quantitative differences in the amplitude vs. distance of Fig. 1 compared to our Fig. 2 are caused by choosing dimensionless units $GM=c=1$, which means $r_t/2=v_F=1$ in our setting. The crucial difference to the dynamics we present in the main text is that we parametrize the differential equations using the temporal coordinate, which differs with respect to Eq. (12) by an additional factor $({dt}/{d\tau})^{-1}$ (for details, we refer to the revised App. A). Note that this factor is equal to $\pm c/k$ when $L=L_z$, which gives rise to a decreasing effective mass for radially escaping trajectories,
$$
m_{\rm eff}(t)=2k^2=2k(0)^2\exp\left(-2\int_0^tu'(\rho(\tau))d\tau\right),
$$
and the particle is accelerated away from the origin.

The geodesic equation is parametrized by an affine parameter instead, resulting in $\dot \rho(\tau)^2=\pm E_\pm^2$ (for radial trajectories with $L_z=0$), such that the velocity of particles escaping radially is constant in $\tau$. In conclusion, the "repulsion" of radially escaping particles is caused by the effective mass as a result from the temporal coordinate parametrization of the trajectories, not by the effective potential.

Compared to arXiv:1911.04305v1 mentioned by the Referee, note that the authors use a different notation, i.e. $h^2=L^2/(m\mu)$ and $E=kmc^2$ (see also the Wikipedia article on Schwarzschild geodesics, and our reply to the second report). Therefore, their effective potential in Eq. (3.21) is written in different units, which, upon multiplying the right hand side of Eq. (3.21) by $m$, followed by $m\rightarrow0,\mu\rightarrow1$ leads to the equations in our main text.

We admit that we could have given more details in App. A relating the geodesic equation parametrized by an affine parameter to the semiclassical equations of motion parametrized by the temporal coordinate, which we will incorporate if Mr. van Wezel agrees to the conclusion of this discussion.