We consider the reconstruction of a Lifshitz spacetime from three perspectives:
differential entropy (or ‘hole-ography’), causal wedges and entanglement wedges. We
find that not all time-varying bulk curves in vacuum Lifshitz can be reconstructed via
the differential entropy approach, adding a caveat to the general analysis of

Article funded by SCOAP3

0$. The work in~\cite{Headrick-ml-2014eia} also extends to a variety of other backgrounds that possess planar symmetry, including Lifshitz spacetime.\footnote{One can also extend the result to include spacetimes in higher dimensions with a generalised notion of `planar symmetry'. This effectively means that the additional dimensions can be `factored out' and intervals are replaced with strips. We consider three bulk dimensions for simplicity.} The relationship between the lengths of the boundary-anchored extremal curves $\gamma^m_\pm(\lambda)$ and the length of the bulk curve $\gamma_B$ is clearly preserved in Lifshitz spacetime. However, in order to actually relate the differential entropy to the gravitational entropy, we require that the entanglement entropy in the boundary theory is still computed holographically by~\cite{Hubeny-ml-2007xt}. It is not clear that this is the case for Lifshitz spacetime; for instance, the interpretation for the length of a bulk spacelike geodesic with endpoints at different times is unknown. The purpose of this section is to demonstrate the extent to which the geometric construction still goes through, regardless of the entropic interpretations, whilst also highlighting the differences between a Lifshitz and an asymptotically AdS spacetime. Now we turn to Lifshitz spacetime~\eqref{eq:Lifshitzmetric} with $z>1$. Consider a bulk curve that sits at a fixed radius $u=u_{\star}$ but varies in time and respects the periodicity in $x$. We parametrize this curve by \begin{align} \gamma_B(\lambda) &= \{ T(\lambda), \xi\, \lambda, u_{\star} \} \quad \text{with} \quad \lambda\in[0,1] \,, \label{eq:gammaB}\\ \gamma_B(0) &= \gamma_B(1) \,. \label{eq:periodicBCgammaB} \end{align} Since we want the bulk curve to be spacelike, its tangent vector must be everywhere spacelike: \begin{equation}\label{eq:spacelikecondition} T'(\lambda)^2 < \xi^2\, u_{\star}^{2(z-1)} . \end{equation} We can immediately write down length of this curve: \begin{equation}\label{eq:gravitationalentropy2} S_G = \frac{1}{4 G_N} \int_0^1 d\lambda\, \sqrt{-\frac{T'(\lambda)^{2}}{u_{\star}^{2z}} +\frac{\xi^2}{u_{\star}^2}} \,. \end{equation} We need a continuous family of bulk spacelike geodesics that begin and end at the boundary $u=0$ in order to construct the differential entropy. The equations we must solve are~\eqref{eq:gengeodesics},~\eqref{eq:Edef} and~\eqref{eq:Pdef}. It is convenient to parametrize the spacelike geodesics via the radius $u$. Consequently, we find pairs of functions $t_\pm(u)$ and $x_\pm(u)$ that describe the two halves of a geodesic, one on each side of its radial turning point. These two halves must match smoothly at the bulk curve $\gamma_B(\lambda)$ so that the resulting geodesics are tangent to the curve, i.e.\ so~\eqref{eq:TVA1} and~\eqref{eq:TVA2} are satisfied.\footnote{Note that in the case of $E=0$ the geodesic is restricted to lie on a constant-time slice. In this case the problem is identical to that studied in section~2.3 of~\cite{Headrick-ml-2014eia} (in fact, for any $z$). This geodesic will also be useful later in sections~\ref{sec:causalwedges} and~\ref{sec:entanglementwedge}.} The matching point between the two portions of the geodesic is where it touches the bulk curve, at $u=u_{\star}$. From~\eqref{eq:TVA1} we have \begin{align} t_-(u_{\star}) &= t_+(u_{\star}) = T(\lambda) \,, \label{eq:tmatching}\\ x_-(u_{\star}) &= x_+(u_{\star}) = \xi\, \lambda \,. \label{eq:xmatching} \end{align} This is also the turning point of the geodesic, where \begin{equation}\label{eq:generalturningpoint} \dot{u} |_{u= u_{\star}} = 0 \quad \Rightarrow \quad V_{\textrm{eff}}^{(\kappa=1)}(u_{\star}) = 0 \,. \end{equation} The ratio $\dot{t}/\dot{x}$ is fixed in terms of the geodesic's conserved quantities $E$ and $P$ via~\eqref{eq:Edef} and~\eqref{eq:Pdef}: \begin{equation}\label{eq:tdotoverxdot} \frac{\dot{t}}{\dot{x}} \bigg|_{u= u_{\star}} = \frac{E\, u_{\star}^{2(z-1)}}{P} = \frac{T'(\lambda)}{\xi} \,, \end{equation} where we have applied the second tangency condition~\eqref{eq:TVA2} for the right hand equality. We also note that the spacelike condition~(\ref{eq:spacelikecondition}) on the bulk curve can now be rewritten as \begin{equation}\label{eq:spacelikePE} u_\star ^{2(z-1)} < \frac{P^2}{E^2} \,. \end{equation} That is, a bulk curve at constant radius $u_\star$ with a tangent geodesic having conserved quantities $E,P$ is spacelike as long as the bulk curve is located at a radius smaller then the maximum in~(\ref{eq:spacelikePE}). Since the bulk curve's tangent geodesics could have different conserved quantities at different tangent points, the maximum possible radius is set by the smallest $P/E$ attained along the entire bulk curve. Moving forward, we evaluate $E,P$ using the chain rule and taking the limit $u\to u_{\star}$: \begin{align} E &= \beta\, \frac{T'(\lambda)}{u_{\star}^z \sqrt{\xi^2\, u_{\star}^{2(z-1)}-T'(\lambda)^2}} \,, \label{eq:Eoflambda}\\ P &= \beta\, \frac{\xi\, u_{\star}^{z-2}}{\sqrt{\xi^2\, u_{\star}^{2(z-1)}-T'(\lambda)^2}} \,, \label{eq:poflambda} \end{align} where $\beta=\pm 1$. Comparing this result with~\eqref{eq:TVA2} we identify \begin{equation} \alpha(\lambda) \equiv \beta\, \frac{u_{\star}^{z}}{\sqrt{\xi^2\, u_{\star}^{2(z-1)}-T'(\lambda)^2}} \,. \end{equation} We will choose $\beta=1$ so that the orientation of the two tangent vectors $\dot\Gamma$ and $\gamma_B'$ agree at the point $u=u_\star$.\footnote{The opposite choice is perfectly valid. In that case the differential entropy computes the `signed length' of the curve: \eqref{eq:gravitationalentropy} supplemented with $\operatorname{sgn} \alpha$~\cite{Headrick-ml-2014eia}.} From here on we restrict to $z=2$ for which we can solve the equations~\eqref{eq:gengeodesics},~\eqref{eq:Edef} and~\eqref{eq:Pdef} analytically. The turning point in this case satisfies \begin{equation}\label{eq:ustar} 1-P^2 u_{\star}^2 +E^2 u_{\star}^{4} = 0 \quad \Rightarrow \quad u_{\star} = \frac{\sqrt{P^2 -\sqrt{P^4-4 E^2}}}{\sqrt{2}E} \,. \end{equation} Here we have chosen the smallest positive root because we require boundary-anchored geodesics, as we will explicitly demonstrate below. This root is real as long as the bulk curve's tangent geodesics have $P^4 -4 E^2 \geq 0$. However, choosing the smallest positive root does have a consequence: the maximum turning radius we can produce from this smallest root is \begin{equation}\label{eq:umax} u_{\star,\textrm{max}}^2 = \frac{P^2}{2 E^2} \,. \end{equation} The astute reader will notice that this maximum radius is smaller than the maximum in~\eqref{eq:spacelikePE}. Regardless, we now impose the smaller maximum, and show that reconstruction works for bulk curves whose tangent geodesics and radius satisfy $u_\star \leq u_{\star,\textrm{max}}$. We will return to the case of larger radius in section~\ref{sec:badcurves} below. The solutions that satisfy the matching conditions~\eqref{eq:tmatching} and~\eqref{eq:xmatching} are given by \begin{align} t_{\pm}(u;\lambda) &= T(\lambda) \pm \Delta t \pm \bigg[\frac{1}{2E} \big(1-\sqrt{1-P^2 u^2 +E^2 u^{4}}\big) \nonumber\\ &\hphantom{= T(\lambda) \pm \Delta t \pm \bigg[} +\frac{P^2}{4 E^2} \log \bigg(\frac{P^2-2E}{P^2-2E(E u^2 +\sqrt{1-P^2 u^2 +E^2 u^{4}})}\bigg)\!\bigg] \,, ~~\\ x_{\pm}(u;\lambda) &= \xi\, \lambda \pm \Delta x \pm \frac{P}{2 E} \log \bigg(\frac{P^2-2E}{P^2-2E(E u^2 +\sqrt{1-P^2 u^2 +E^2 u^{4}})}\bigg) \,. \end{align} where we have defined \begin{align} \Delta t &\equiv -\frac{1}{2E} -\frac{P^2}{4E^2} \log \bigg(\frac{P^2-2 E}{\sqrt{P^4-4 E^2}}\bigg) \,, \label{eq:Deltat}\\ \Delta x &\equiv -\frac{P}{2E} \log \bigg(\frac{P^2-2 E}{\sqrt{P^4-4 E^2}}\bigg) \,. \label{eq:Deltax} \end{align} As we can see from these explicit expressions, these geodesics do indeed reach the boundary $u=0$. In fact, from the endpoints at $u=0$ we can read off the family of boundary intervals necessary to reconstruct the bulk curve: \begin{equation}\label{eq:endpoint} \gamma_\pm^{\m} = \{ T(\lambda) \pm \Delta t , \xi\, \lambda \pm \Delta x\} \,. \end{equation} As in an asymptotically AdS spacetime, the length of such a boundary-anchored geodesic diverges in Lifshitz spacetime. We introduce a simple radial cut-off at $u=\varepsilon$ to regulate this divergence: \begin{align} {\cal L} &\equiv 2 \int_{\varepsilon}^{u_{\star}} \frac{du}{u \sqrt{1-P^2 u^2 +E^2 u^{4}}} \label{eq:lengthresult}\\ &= \log \bigg(\frac{u^2}{2-P^2 u^2 +2 \sqrt{1-P^2 u^2 +E^2 u^4}}\bigg) \bigg|_{\varepsilon}^{u_{\star}} \nonumber\\ &= \log \bigg(\frac{4}{\sqrt{P^4 -4 E^2}\, \varepsilon^2}\bigg) +O(\varepsilon^2) \,. \label{eq:lengthresultEp} \end{align} where we used~\eqref{eq:ustar} to obtain the final line. It is not clear how to interpret the length of this geodesic in the dual field theory. As mentioned earlier, when $E=0$ we simply recover the AdS result, so it is tempting to identify this as the entanglement entropy \emph{\`a la} Ryu-Takayanagi. This independence from $z$ is a bit surprising in itself. There is some supporting evidence from field theory calculations featuring Lifshitz symmetry: both~\cite{Fradkin-ml-2006mb} and~\cite{Solodukhin-ml-2009sk} recover an area law, though the former observes an additional sub-leading divergence. However, these two different setups both have more symmetry than we do. Even aside from this possible concern, we are left with a further question: when the interval~\eqref{eq:endpoint} does not lie on a constant-time slice, what are we computing? We do not have a physical understanding of this boundary quantity. For both of these reasons, it is still unclear what is the meaningful entanglement entropy calculation in a Lifshitz field theory. For now we simply assume that the length of this geodesic computes the entanglement entropy according to $S\equiv {\cal L}/(4 G_N)$. We will see that the regulatory cut-off drops out in this construction. We now have all the ingredients we need to evaluate the differential entropy, defined in~\eqref{eq:differentialentropy}. It is useful to rewrite this as \begin{equation}\label{eq:differentialentropy2} \int_0^1 d\lambda \frac{\partial S[\gamma_-(\lambda),\gamma_+(\lambda')]}{\partial \lambda'} \bigg|_{\lambda'=\lambda} = \int_0^1 d\lambda\, \frac{\partial S(\gamma_-,\gamma_+)}{\partial \gamma_+^{\m}}\, \frac{d \gamma_+^{\m}}{d\lambda} \,. \end{equation} We know $S(E,P)$, but would need to invert~\eqref{eq:Deltat} and~\eqref{eq:Deltax} to obtain $S(\gamma_-,\gamma_+)$. However, since we only need its derivative in~\eqref{eq:differentialentropy2}, we can use the chain rule and implicit differentiation. From translation invariance in $t$ and $x$ we can express the length as $S(\Delta t,\Delta x)$. We evaluate the partial derivatives the hard way using~\eqref{eq:lengthresultEp},~\eqref{eq:Deltat} and~\eqref{eq:Deltax}: \begin{equation} \frac{1}{2}\, \frac{\partial S}{\partial \Delta t} = -\frac{E}{4 G_N} \quad \text{and} \quad \frac{1}{2}\, \frac{\partial S}{\partial \Delta x} = \frac{P}{4 G_N} \,. \end{equation} Our results seem surprisingly simple at first. However, as pointed out in the proof given by~\cite{Headrick-ml-2014eia}, the simple explanation is that these are the Hamilton-Jacobi equations from $S= {\cal L}/(4 G_N)$. We now use~\eqref{eq:Eoflambda} and~\eqref{eq:poflambda} to express our derivative as a function of $\lambda$: \begin{align} \frac{\partial S(\gamma_-,\gamma_+)}{\partial \gamma_+^{\m}}\, \frac{d \gamma_+^{\m}}{d\lambda} &= -\frac{E}{4 G_N}\, \frac{d \gamma_+^{t}}{d\lambda} +\frac{P}{4 G_N}\, \frac{d \gamma_+^{x}}{d\lambda} \nonumber\\ &= \frac{1}{4 G_N}\, \frac{1}{u_{\star}^2} \bigg[\sqrt{\xi^2 u_{\star}^2-T'(\lambda)^2} +\frac{1}{2}\, \frac{d}{d\lambda} \log \bigg(\frac{\xi^2 u_{\star}^2-T'(\lambda)^2}{\xi^2 u_{\star}^2-2T'(\lambda)^2}\bigg)\!\bigg] \,. ~~\end{align} The total derivative term drops out of the integral in~\eqref{eq:differentialentropy2} due to the periodic boundary conditions~\eqref{eq:periodicBCgammaB} on $\gamma_B(\lambda)$. Thus, we do indeed recover the length, or gravitational entropy~\eqref{eq:gravitationalentropy2}, of the bulk curve. In conclusion, the differential entropy construction worked just as in asymptotically AdS spacetime, despite the lack of field theory interpretation for the length of a geodesic with endpoints at different times. Furthermore, the cut-off we introduced dropped out. Whilst we considered the simplest non-trivial example of a time-dependent curve, the general result of~\cite{Headrick-ml-2014eia} will still hold for some more general bulk curves. However, in the following section we will see that there exist curves for which it does not. In~\cite{Headrick-ml-2014eia} it is also claimed that the bulk curve emerges from the intersection of neighboring entanglement wedges. We will comment on this in the context of Lifshitz spacetime in section~\ref{sec:Discussion}. ]]>

0$ at a turning point, then instead the geodesics return to larger radii. To find $\ddot{u}$ at a generic turning point, we take the derivative of~(\ref{eq:gengeodesics}) with respect to the affine parameter, finding \begin{equation} \ddot{u} = (z+1) E^2 u^{2z+1}+u-2 u^3 P^2 , \end{equation} where we have restored general $z$. Next, we use the turning point equation $\dot{u}=0$ to eliminate either $E$ or $P$ from the expression $\ddot u<0$. We find \begin{align} u_\star^2 &< \frac{z}{(z-1) P^2} \\ u_\star^{2z} &< \frac{1}{(z-1) E^2} \,. \end{align} Of course these are actually the same maximum possible $u_{\star,\textrm{max}}$, just expressed in a different way. Consequently, although earlier we only exhibited the mismatch for $z=2$, we now see the maximum radius for boundary-anchored spacelike geodesics for general $z$ obeys \begin{equation}\label{eq:umaxgenz} u_{\star,\textrm{max}}^{2(z-1)} = \frac{P^2}{z E^2} \,. \end{equation} Again, spacelike curves with $E,P$ exist up to $u_\star$ satisfying~(\ref{eq:spacelikePE}), but for these larger radii their tangent geodesics never reach the boundary. This failure of the differential entropy reconstruction is reminiscent of that shown in~\cite{Engelhardt-ml-2015dta}. The authors proved that spacetimes with certain types of `holographic screens' contain bulk surfaces that cannot be reconstructed via this hole-olography approach (even though extremal surfaces can reach the bulk surfaces in question). As we exhibit in appendix~\ref{appendix:screens}, Lifshitz spacetime contains surfaces similar to holographic screens; these surfaces, caused because light rays with non-zero momentum turn around in Lifshitz spacetimes, are in fact indicative of the explicit failure we have just shown. Our situation is actually slightly worse than that shown in the horizon-having spacetimes of~\cite{Engelhardt-ml-2015dta}. In Lifshitz spacetime, by choosing a spacelike curve whose tangent somewhere has arbitrarily small $P/E$, we can draw a bulk spacelike curve arbitrarily close to the boundary whose tangent geodesics will never reach the boundary. There are non-reconstructible constant radius bulk curves everywhere in the spacetime. ]]>

1$. As mentioned in section~\ref{sec:intro}, such spacetimes do not have a conformal boundary in the usual sense. Our prescription is to cut off the spacetime at a slice of constant radius $\{u=\varepsilon\}$. We can define a domain of dependence on this cut-off surface for any non-zero $\varepsilon$ and we make the minimal modifications necessary to all other definitions. The purpose of this section is to study the resulting causal wedge and explore what happens as we remove the cut-off. We continue to focus on an interval $\mathcal{A} = \{ (t,x) \,|\, t = 0, |x|\leq a \}$, but now we define a regulated boundary domain of dependence $\domd^{\varepsilon}$ at $u=\varepsilon$ via \begin{equation} \domd^\varepsilon = \big\{ (t,x) \,\big|\, |t| \leq \varepsilon^{z-1} (a-x), x \in [0,a] \big\} \cup \big\{ (t,x) \,\big|\, |t| \leq \varepsilon^{z-1} (a+x), x \in [-a,0] \big\} \,. ~~ \end{equation} The boundaries of this region are null geodesics of the induced metric on $\{u=\varepsilon\}$.\footnote{By definition, these null geodesics satisfy $\dot u \equiv 0$.} The future- and past-most tips are located at $(t,x) = (\pm a\, \varepsilon^{z-1},0)$, respectively. For fixed interval width $2a$, note that $\domd^\varepsilon$ flattens as $\varepsilon$ is lowered, as demonstrated in figure~\ref{fig:BDDshrinkage}. This is a consequence of the non-relativistic causal structure at the boundary, wherein the causal past of a point (or spatial region) includes everything in its past. \begin{figure}[t] \centering \includegraphics[width=0.5\textwidth]{BDDshrinkage} \setlength{\unitlength}{0.1\columnwidth} \begin{picture}(0.3,0.4)(0,0) \put(-3.4,4.7){\makebox(0,0){$t$}} \put(0.3,2.5){\makebox(0,0){$x$}} \end{picture} ]]>

\varepsilon^{z-1}$ are classically forbidden and therefore cannot be included. In the AdS case this bound is $|\ell|=1$ and such geodesics sent from the tips run along the boundary of $\domd$. This is not the case for Lifshitz spacetimes: geodesics with the maximum $|\ell| = |\ell_{\star}| \equiv \varepsilon^{z-1}$ sent from the tips of $\domd^\varepsilon $ bend away from the cut-off surface into the bulk.\footnote{Such geodesics have $\dot u = 0$, $\ddot u >0$ at $u=\varepsilon$, unlike those forming $\domd^\varepsilon$, which have $\dot u \equiv 0$.} This leads us to suspect that the surfaces built from null geodesics sent from these tips will not close at the edges at $\{u=\varepsilon\}$. One could be concerned at this stage that the causal past or future of these tips is ill-defined. However, the allowed geodesics sent from these tips in the cut-off Lifshitz spacetime do indeed form (the curved surfaces of) half-cones \emph{locally}. Unlike the AdS case, this is not true away from the tips. In order to build the full boundaries of the causal wedge we therefore need to include the null geodesics from the rest of the boundary of $\domd^\varepsilon$. Specifically, the causal wedge boundary is built from two types of null geodesic: \begin{description} \item \emph{Type I}: null geodesics sent from the future- and past-most tips of $\domd^\varepsilon$ with $0 \leq |\ell| \leq |\ell_{\star}|$. \item \emph{Type II}: null geodesics sent from other points on the boundary of $\domd^\varepsilon$ with $|\ell| = |\ell_{\star}|$. \end{description} (Again, in the AdS case the latter type run along the boundary of $\domd$.) Finally we are ready to present the causal wedge $\cwedge^\varepsilon$ in the cut-off Lifshitz spacetime. In figure~\ref{fig:causalwedge} we plot an example with $z=2$. This looks qualitatively similar to the wedge for Poincar\'e AdS$_3$ presented in figure~\ref{fig:causalwedgeAdS}, besides its height being rescaled by a factor of $\varepsilon^{z-1}$. However, it is clear that Type~II geodesics are required in order to form a closed co-dimension zero wedge of the bulk. \begin{figure}[t] \centering \includegraphics[width=0.45\textwidth]{LifangledCWedge} \setlength{\unitlength}{0.1\columnwidth} \begin{picture}(0.3,0.4)(0,0) \put(-3.5,3.7){\makebox(0,0){$\domd^\varepsilon $}} \put(-3.25,3.45){\vector(1,-2){0.2}} \put(0.3,4.3){\makebox(0,0){Type I: sent from the tips}} \put(-1.2,4){\vector(-1,-2){0.4}} \put(0.4,1.7){\makebox(0,0){Type II: sent from the edges}} \put(-1.2,2){\vector(0,1){1.3}} \put(-1.2,2){\vector(-3,1){1.6}} \end{picture} ]]>

u_\Xi$ ($\varepsilon=1/3$, right). Black dots mark $u_\Xi$ using~\eqref{eq:radialextent} and $u_\star$ using~\eqref{eq:newustar}. ]]>

1$, any geodesic with nonzero $\ell$ will have a minimum radius of $\ell^{1/(z-1)}$. There are only three future-directed rays in the light-sheet that have $\ell=0$: one leaving from $u_s=a$, $x=0$ and two from $u_s=0$, $x=\pm a$. Thus we might worry that the entanglement `wedge' in Lifshitz does not close; that is, the light-sheet itself may not reach the boundary, so there is no finite subregion of the bulk bounded by the past and future light-sheets of ${\cal E}_{\cal A}$ alone. This is exactly the behaviour we observe below. In order to present explicit solutions, we now specify to the case $z=2$. Next, we solve~(\ref{eq:nullgeodesics}) with this $\ell$ as in~(\ref{eq:ellnormalz}) and insist the geodesics pass through $u=u_s$, $t=0$, $x=-a\sqrt{1-u_s^2/a^2}$. The future light sheet is then described by the paths \begin{align} x &= \sqrt{1-u_s^2/a^2} \bigg(\pm u_s \log \bigg[\frac{au-\sqrt{a^2u^2-a^2 u_s^2+u_s^4}}{a u_s \mp u_s^2}\bigg] -a\bigg) \,, \label{eq:xlightsheet}\\ t &= \frac{1}{2a^2} \bigg(u_s^3 a \mp au \sqrt{a^2u^2-a^2u_s^2+u_s^4} \pm a^2u_s^2 \log \bigg[\frac{au-\sqrt{a^2u^2-a^2u_s^2+u_s^4}}{au_s \mp u_s^2}\bigg] \label{eq:tlightsheet}\\ &\hphantom{= \frac{1}{2a^2} \bigg(} +u_s^4 \log \bigg[\frac{au\pm\sqrt{a^2u^2-a^2u_s^2+u_s^4}}{au_s+u_s^2}\bigg]\bigg) \,. \nonumber \end{align} Here the top signs indicate paths before they reach their turning points, and the bottom signs indicate rays after the turning points, continuing in towards the bulk. Before we present a plot of the light-sheets, we note that the (future) light-sheets only continue as long as their expansion $\theta\leq0$; that is, as soon as the light rays form a caustic, the sheet stops~\cite{Bousso-ml-2002ju}. Caustics occur when neighbouring rays intersect (that is, when $\theta=-\infty$). Beyond a caustic, $\theta$ becomes positive; the light rays have crossed and are now going away from each other. We can calculate the location of the caustics by finding intersections between rays starting at $u_s$ and $u_s+\delta$, then taking $\delta$ to zero. Equivalently, we fix $u$, take the derivative of both~(\ref{eq:xlightsheet}) and~(\ref{eq:tlightsheet}) with respect to $u_s$, and then find the caustic radius $u=u_c$ where both derivatives vanish. The caustic radius $u_c$ will be a function of the starting point $u_s$; it tells us how far we should continue the ray that started~at~$u_s$. As the right hand plot in figure~\ref{fig:AdSentwedge} shows (see also figure~\ref{fig:moreLifwedge}), indeed the light-sheets built from the spacelike surface ${\cal E}_{\cal A}$ alone do not form the boundary of a bulk subregion. Although we have only shown the algebra above for the case when the boundary region ${\cal A}$ is on a constant time slice, extending to the non-constant time slice case (at a cut-off radius) does not improve the situation, nor does considering regions of different sizes. Since the sheets do not come close to the boundary, a cut-off radius alone also cannot help. \begin{figure}[t] \centering \hskip.85em\includegraphics[width=0.45\textwidth]{AdSEntanglementWedge} \hskip1em \includegraphics[width=0.45\textwidth]{LifEntanglementWedge} \setlength{\unitlength}{0.1\columnwidth} \begin{picture}(0.3,0.4)(0,0) \put(-9.6,2.5){\makebox(0,0){$t$}} \put(-9,0.2){\makebox(0,0){$u$}} \put(-3.9,0.2){\makebox(0,0){$u$}} \put(-8.2,5.4){\makebox(0,0){$x$}} \put(-3.,5.4){\makebox(0,0){$x$}} \end{picture} ]]>

1$ as well as the non-relativistic boundary behavior both serve to alter the reconstructibility of bulk spacelike curves, as well as the identification of the bulk reconstructible region via either the causal wedge or entanglement wedge. Constructions that work simply in AdS may encounter complications in Lifshitz systems. The differential entropy approach to building spacelike bulk curves succeeds for constant radius curves, when the bulk curve has tangents satisfying~\eqref{eq:constructible}. Indeed, even the cut-off necessary to define the differential entropy drops out of the final result. If the curve takes on a larger value for $T'(\lambda)$ anywhere along its path at $u=u_\star$, then we cannot reconstruct it from a series of boundary-anchored curves, for the simple reason that the tangent extremal curve is not boundary-anchored. Crucially, it is possible to a pick a curve that is entirely spacelike, but cannot be reconstructed, for any given radius. \begin{figure}[t] \centering \includegraphics[width=0.35\textwidth]{blackjustSigmaSide4} \setlength{\unitlength}{0.1\columnwidth} \begin{picture}(0.3,0.4)(0,0) \put(0.1,2.7){\makebox(0,0){$t$}} \put(-3.7,1.){\makebox(0,0){$x$}} \put(-3.3,2.7){\makebox(0,0){${\cal A}$}} \put(-1.6,0.0){\makebox(0,0){$u$}} \end{picture} ]]>

0$. In our case below, however, we
will find $\theta_k$ and $\theta_l$ vanish at the same location.
Even though the theorems in~\cite{Engelhardt-ml-2015dta} are not
applicable because both $\theta_k$ and $\theta_l$ vanish at the same
location, we can still use lemma~2 (conveniently applicable in 2+1
dimensions!)\ to show that geodesics with turning radii
between~(\ref{eq:umaxgenz}) and the spacelike limit $(P/E)^{1/(1-z)}$
cannot reach the boundary. Lemma~2 shows that in a region of a null
congruence with $\theta_k <0$, for a given leaf $N$ of the associated
foliation, any spacelike geodesic $X$ that is tangent to the leaf $N$
at a point $p$ has some neighborhood ${\cal O}_p$ such that
$X \cap {\cal O}_p$ is nowhere to the past of~$N$.
Let us now be explicit for the case of Lifshitz. We consider the
foliation built from the null generator
\begin{equation}\label{eq:keqn}
k^\mu = \big\{ u^{2z}, u \sqrt{u^{2z}-u^2 \rho^2}, \rho u^2 \big\} \,,
\end{equation}
with $\rho$ a positive constant. Individual leaves of this foliation
solve
\begin{equation}
t = \rho x +\int du k_u +c_N \,,
\end{equation}
where the constant $c_N$ specifies the leaf in the foliation, and we
assume $\rho>0$ for simplicity. This foliation is spacetime filling
for $u>\rho^{1/(z-1)}$.
It is easiest to compute $\theta_k$ by choosing another non-aligned
null vector; the choice of null vector is immaterial but we will
choose one suited to the $\sigma$ above. Specifically, we choose
\begin{equation}
l^\mu = \big\{ u^{2z}, -u \sqrt{u^{2z}-u^2 \rho^2}, \rho u^2 \big\} \,.
\end{equation}
Note that we do not have $k\cdot l = -1$, but this could be acheived
by a trivial rescaling of $l^\mu$ if needed. We will not need it
here.
Instead, we now compute $\theta_k$ by first defining the reduced
induced metric $h^{\mu\nu}$:
\begin{equation}
h^{\mu\nu} \equiv g^{\mu\nu} +k^\mu l^\nu +l^\mu k^\nu ,
\end{equation}
and then computing
\begin{equation}
h^{\mu\nu} \nabla_\mu k_\nu \equiv \theta_k =
\frac{\rho^2 u^2 z -u^{2z}}{\sqrt{u^{2z} -\rho^2 u^2}} \,.
\end{equation}
Similarly for the expansion of $l^\mu$ we find
\begin{equation}
h^{\mu\nu} \nabla_\mu l_\nu \equiv \theta_l =
-\frac{\rho^2 u^2 z -u^{2z}}{\sqrt{u^{2z} -\rho^2 u^2}} \,.
\end{equation}
In both cases, the $\sigma_N$, where $\theta=0$, occur at constant
radius $u_N$ given by
\begin{equation}\label{eq:uNeqn}
u_N = (\rho^2 z)^{\frac{1}{2(z-1)}} .
\end{equation}
Note for $k$, the expansion $\theta_k$ becomes negative for $u>u_N$.
This makes sense as $k$ is future-directed and headed towards larger
radius at larger $t$, so indeed $\theta_k$ can only decrease (or stay
constant) as $u$ gets larger. It is also positive for $u

u_N$ must be to the future of, or above, the leaf, so they head in towards the boundary. ]]>

u_N$, then the curve can be constructed. Rewriting using~\eqref{eq:findrho}, we find the curve is constructible as long as \begin{equation}\label{eq:constructible} u_\star^{2(z-1)} > \bigg(\frac{T'(\lambda)}{\xi}\bigg)^{\!2} z \,. \end{equation} ]]>