\section{Introduction}\label{Introduction}
In \cite{PSW}, the authors prove a positive mass theorem using properties of the causal structure of spacetime. The proof has the advantage of relying specifically on properties of spacetime which one would expect to be characteristic of positive mass, in particular the focusing and retarding of null geodesics which pass near the source (as well as the corresponding relative time advancement of geodesics passing far from the source). Conversely, null geodesics which pass far from a negative mass source will be delayed relative to those which pass nearby. This is used to show that a certain class of negative mass spacetimes must contain a null line as defined by Galloway \cite{Galloway}:
\begin{definition}
A null line is an inextendible null geodesic which is achronal. In particular, null lines cannot contain conjugate points. 
\end{definition}

On the other hand, under certain conditions (see for example Theorem \ref{thm:Borde} in section \ref{A Focusing Theorem in Higher Dimensions}) it is possible to prove that such a null line must exist. This allows us to prove a positive mass theorem among spacetimes satisfying these conditions (Corollary \ref{cor}). If the Einstein equations are assumed, these conditions usually amount to demanding some sort of local positivity of energy. 

The proof in \cite{PSW} concerns only $3+1$ dimensional spacetimes and relies upon the fact that null geodesics become infinitely delayed as we let their distance of closest approach to a negative mass source tend to infinity (see \cite{Penrose} for further discussion). As discussed in \cite{PCMD} for the specific example of the Schwarzschild metric, this effect becomes vanishingly small in higher dimensions (i.e. in $D+1$ dimensions with $D\geq4$), owing to the faster decay at infinity of the leading order non-Minkowski terms in the metric. Indeed, \cite{PSW} contains a brief discussion of this point and notes that this prevents the proof given from immediately generalising to higher dimensions.

In \cite{poorman2}, the result is generalised to include higher dimensions (albeit with a more restrictive class of metrics than those considered in \cite{PSW}). The assumptions made in this paper are weaker than those of \cite{poorman2} - namely we will require the metric to be \textit{asymptotically Schwarzschildean} (Definition \ref{defn:asymptoticallyschwarzschildean}) rather than the more restrictive assumption that it is \textit{strongly uniformly Schwarzschildean} (\cite{poorman2} Section 1). We will also drop the assumption in \cite{poorman2} that the metric is stationary. On the other hand, unlike in \cite{poorman2}, we will be unable to obtain a rigidity result concerning the zero mass case. Our proof will also apply only to higher dimensions. We hope that this method of proof will prove enlightening as it is closer in spirit to the methods employed in \cite{PSW}, namely the construction of a null line as a \textit{fastest causal curve} (Section \ref{Constructing a Fastest Causal Curve}) between two generators of past and future null infinity.

The theorem we will prove is the following (see later sections for definitions of various terms):
\begin{thm}\label{thm:nullline}
Let $(M,g)$ be an asymptotically Schwarzschildean spacetime in $D+1$ dimensions with ADM mass $m_{ADM}<0$ such that $\mathcal{D}\cup\mathcal{I}$ is globally hyperbolic as a subset of $\tilde{M}$. Then $(M,g)$ contains a null line.
\end{thm}

For $3+1$ dimensional spacetimes, the proof in \cite{PSW} proceeds by constructing a ``fastest causal curve'' between a generator, $\Lambda^-$, of past null infinity (denoted $\mathcal{I}^-$) and a generator, $\Lambda^+$, of future null infinity (denoted $\mathcal{I}^+$). We say that a causal (i.e. nowhere spacelike) curve is ``faster'' than some other if it departs $\mathcal{I}^-$ no earlier and arrives at $\mathcal{I}^+$ no later. A fastest causal curve (if it exists) is defined to be such that no other causal curve is faster. 

In order to specify the generator $\Lambda^+$, we specify some radial, outgoing, future pointing, Minkowski-null direction\footnote{We follow Penrose's ``abstract index notation'' where Latin indices label the rank of a tensor, while Greek indices denote the components of a tensor in some co-ordinate basis.} $k^a$. We then define $\Lambda^+$ to be the intersection of $\mathcal{I}^+$ with the set of null geodesics whose tangent vector asymptotes towards this direction at future null infinity. We similarly define $\Lambda^-$ as the intersection of $\mathcal{I}^-$ with the set of null geodesics with tangent vector asymptoting to $-k^a$ at past null infinity. In this paper we will restrict attention to a quasi-Cartesian frame (Definition \ref{defn:quasicart}) and choose co-ordinates $(x_0,x_1,...,x_D)$ such that $k^a$ has components $k^\mu=(1,0,...,0,1)$ in this frame.  

The key Lemma in \cite{PSW} is Lemma III.2.1. This gives an estimate of the ``time of flight" along a curve comprised of the union of two null geodesics joined at a point\footnote{Note that our choice of co-ordinates such that $k^\mu=(1,0,...,0,1)$ ensures our curve must pass through such a point. We are using the ``mostly plus'' signature (-,+,...,+).} with $x_D=0$ and with endpoints on $\Lambda^\pm$. The time of flight is defined to be the retarded time of arrival on $\Lambda^+$ minus the advanced time of departure from $\Lambda^-$ (where these are defined using the Hamilton-Jacobi functions describing null geodesics with endpoints on $\mathcal{I}^+$ and $\mathcal{I}^-$ respectively - see \cite{PSW}). This quantity is shown to behave asymptotically like 
\begin{equation}\label{eqn:timeofflightestimate}
    4P\cdot k\log(b_0/R_0)
\end{equation}
where $P^a$ is the ADM 4-momentum of the spacetime \cite{ADM}, $b_0$ is the value of $r=\sqrt{\sum\limits_{i=1}^{D}x_i^2}$ (defined using quasi-Cartesian co-ordinates) at the point where the null geodesics join, and $R_0$ is a positive constant. 

Suppose $P\cdot k>0$, i.e. the 4-momentum of the spacetime is not future causal. The aim is to show that there must then exist a fastest causal curve from $\Lambda^-$ to $\Lambda^+$ which enters the interior of the spacetime. If this is the case then the curve must lie on the boundary of the causal future of some point $p\in\Lambda^-$, denoted $\partial J^+(p)$ which is equal to $\partial I^+(p)$. As a result this curve must lie entirely in the boundary of the future null cone from $p$ and hence must be a null geodesic (\cite{Wald} Corollary after Theorem 8.1.2) without conjugate points (\cite{HawkingEllis} Proposition 4.5.12).

The construction begins by finding points $p\in\Lambda^-$, $q\in\Lambda^+$ such that there is no causal curve which departs $\Lambda^-$ later than $p$ and arrives at $\Lambda^+$ earlier than $q$. These will be the endpoints of the fastest causal curve, $\gamma$, to be constructed. Consider a sequence of causal curves, $(\gamma_i)_{i=0}^\infty$, with endpoints on $\Lambda^\pm$ which tend towards $p$ and $q$ and with $\gamma_i$ faster than $\gamma_j$ for $i>j$. The curve $\gamma$ is defined to be the ``limit'' of this sequence of causal curves, where we use the fact that, in a globally hyperbolic set, the space of causal curves between two compact sets is compact (\cite{SorkinWoolgar} Theorem 23). 

It remains to check that $\gamma$ does in fact enter the interior of the spacetime and hence define a null line.  We can modify the curves $(\gamma_i)_{i=0}^\infty$ (possibly making them faster) so that they consist of two null geodesics joined at a point with $x_D=0$, $r=b_0$. This means that the estimate (\ref{eqn:timeofflightestimate}) now applies. If we were to have $b_0\rightarrow\infty$ as $i\rightarrow\infty$ then this estimate tells us that the time of flight along $\gamma_i$ would also diverge to $+\infty$ as $i\rightarrow\infty$. In particular, the sequence ($\gamma_i)^\infty_{i=0}$ would eventually become slower than $\gamma_0$. This contradicts the definition of the sequence, so we conclude that $b_0$ must not diverge along the sequence and hence, possibly restricting to a subsequence, all of the $\gamma_i$ must enter the compact set defined by $r\leq B_0$ for some fixed $B_0>0$. Once again using the compactness result of \cite{SorkinWoolgar}, we conclude that $\gamma$ must also enter this set, and hence must enter the interior of the spacetime. We therefore conclude that $\gamma$ is a null line. 

This proof does not generalise to higher dimensions because the time of flight of curves restricted to arbitrarily large values of $r$ no longer diverges. Instead, the time of flight estimate (\ref{eqn:timeofflightestimate}) is replaced by Lemma \ref{lemma:timeofflight}. This says that the time of flight tends to 0 as we let $b\rightarrow\infty$, where $b$ denotes the impact parameter, defined to be the minimal value of $r$ along the curve.

However, this result combined with ideas from \cite{PCMD} turns out to be sufficient to construct a null line for negative mass spacetimes in higher dimensions (with some slightly modified assumptions). In particular, we generalise a result of \cite{PCMD} to show that, by a comparison argument involving a Minkowski metric on a neighbourhood of conformal infinity, the presence of negative mass allows us to construct a Minkowski-null curve from $\Lambda^-$ to $\Lambda^+$ which is timelike with respect to the physical metric $g$. To do this, we show that for such spacetimes the Minkowski null cones at sufficiently large $r$ are contained inside the $g$-null cones. Defining retarded and advanced time co-ordinates as in Equation \ref{eqn:retadv}, it is a straightforward calculation to show that the time of flight along a Minkowski null geodesic is exactly zero. As a result, if we choose a $\eta$-null geodesic restricted to sufficiently large values of $r$, then this curve must be timelike with respect to the metric $g$. Then since the timelike future of any point is an open set, it must be possible to modify this curve slightly and obtain a $g$-timelike curve, $\gamma_0$, between $\Lambda^-$ and $\Lambda^+$ which has time of flight strictly less than zero. This allows us to use a similar construction as was used in $3+1$ dimensions based on a sequence of faster and faster causal curves $(\gamma)_{i=0}^\infty$ from $\Lambda^-$ to $\Lambda^+$. If $b\rightarrow\infty$ along this sequence then we will eventually get a time of flight larger than that of $\gamma_0$. As in 3+1 dimensions, this allows us to conclude that a fastest causal curve from $\Lambda^-$ to $\Lambda^+$ does exist. This completes the proof of Theorem \ref{thm:nullline}.

\section{Definitions and Assumptions}\label{Definitions and Assumptions}
In order to carry out the comparison with Minkowski spacetime mentioned in the previous section, it will be necessary to impose stronger conditions than those used in \cite{PSW}. These assumptions will be more similar to the ones used in \cite{poorman} and \cite{poorman2}. In this section we outline the various assumptions used.
\begin{definition}\label{defn:spacetime}
A \textit{spacetime} $(M,g)$ is a connected manifold $M$ of dimension $D+1$, where $D\geq3$, equipped with a $C^1$ Lorentzian metric $g$ of signature $(D,1)$.
\end{definition}
For the purposes of Theorem \ref{thm:nullline}, requiring the metric to be $C^1$ will be sufficient. In order to obtain a focusing result in Section \ref{A Focusing Theorem in Higher Dimensions}, it may be necessary to make stronger assumptions. For example, in Theorem \ref{thm:Borde} we assume that the metric is $C^2$.
\begin{definition}\label{defn:quasicart}
A spacetime $(M,g)$ admits \textit{quasi-Cartesian co-ordinates} if there are co-ordinates, defined on some subset of $M$ diffeomorphic to $\mathbbm{R}\times\mathbbm{R}^D\setminus B$ (where $B$ denotes a closed ball in $\mathbbm{R}^3$ centred on the origin), with respect to which the components of the metric, $g$, take the form
\begin{equation}
    g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}
\end{equation}
where for the purposes of Lemma \ref{lemma:timeofflight} it will be sufficient to assume that $h_{\mu\nu}=O\left(r^{1+\delta}\right)$ for some $\delta>0$.
\end{definition}

We will show in Section \ref{A Focusing Theorem in Higher Dimensions} that such spacetimes are weakly asymptotically empty and simple as defined in \cite{PCMD}. In particular, they admit a conformal compactification $(\tilde{M},\tilde{g})$ defined on the same manifold as a conformal compactification of Minkowski spacetime. As in $\cite{PSW}$, we actually assume that $\tilde{M}$ extends slightly past infinity so that it is indeed a manifold. 

Requiring that a spacetime admits quasi-Cartesian co-ordinates will allow us to prove Lemma \ref{lemma:timeofflight}, however in order to construct a null line we will need to consider a more restrictive class of metrics. Recall that the $D+1$ dimensional Schwarzschild metric with ADM mass $m_{ADM}\in\mathbbm{R}$, which we denote $g_m$, has line element
\begin{equation}\label{eqn:Schwarzschild}
    ds_m^2=-\left(1-\frac{2m}{r^{D-2}}\right)dt^2+\frac{dr^2}{1-\frac{2m}{r^{D-2}}}+r^2d\omega^2_{D-1}
\end{equation}
where $d\omega^2_{D-1}$ is the round line element on the unit $(D-1)$-sphere, $S^{D-1}$, and the \textit{mass parameter}, $m$, is related to the ADM mass $m_{ADM}$ by\footnote{We work in units in which $c=G=1$.}
\begin{equation}\label{eqn:mass}
    m_{ADM}=\frac{(D-1)Area(S^{D-1})}{8\pi}m
\end{equation}
\begin{definition}\label{defn:asymptoticallyschwarzschildean}
We say that a metric $g$ on $\mathbbm{R}\times\left(\mathbbm{R}^D\setminus B\right)$, $R^{D-2}>m/2$, is \textit{asymptotically Schwarzschildean} with ADM mass $m_{ADM}\neq0$ (related to the parameter $m$ by Equation \ref{eqn:mass}) if, in the co-ordinates of \ref{eqn:Schwarzschild} (or equivalently in the co-ordinates of eqn (1) of \cite{poorman}):
\begin{equation}
    g-g_m=o\left(|m||x|^{-(D-2)}\right)
\end{equation}
We will say that a metric is asymptotically Schwarzschildean with mass $0$ if it is flat for $r>R$, for some $R\in\mathbbm{R}_{\geq0}$.\footnote{This definition will not be important since we will be unable to comment on the $m=0$ case.}
\end{definition}
Note that the conditions imposed on the spacetime are less restrictive than the ones used in \cite{poorman2}, where the spacetime is assumed to be \textit{strongly uniformly Schwarzschildean} (although as discussed in section \ref{Conclusion}, the result we obtain is weaker since we are unable to comment on the $m=0$ case). Physically this assumption seems reasonable. It says that, near infinity, a localised matter distribution will resemble a point mass to first approximation.

We will also refer to the following sets:
\begin{equation}
    \begin{split}
        J^+(p)&=\{q\in\tilde{M}|\exists\text{ a smooth future-directed causal curve from $p$ to $q$}\}\\
        J^-(p)&=\{q\in\tilde{M}|p\in J^+(q)\}\\
        I^+(p)&=\{q\in\tilde{M}|\exists\text{ a smooth future-directed timelike curve from $p$ to $q$}\}\\
        I^-(p)&=\{q\in\tilde{M}|p\in I^+(q)\}\\
    \end{split}
\end{equation}
For non-compact spacetimes, $(M,g)$, we can define the same sets as subsets of $M$ rather than as subsets of $\tilde{M}$.

In Theorem \ref{thm:nullline}, we will require that $\mathcal{D}\cup\mathcal{I}$ be globally hyperbolic as a subset of $\tilde{M}$, where $\mathcal{D}$ denotes the \textit{domain of outer communications} $\mathcal{D}=I^-(\mathcal{I}^+)\cap I^+(\mathcal{I}^-)$ and $\mathcal{I}=\mathcal{I}^+\cup\mathcal{I}^-$. By this we mean that $\mathcal{D}\cup\mathcal{I}$ is strongly causal and contains $J^+(p)\cap J^-(q)$ as a compact subset for each $p,q\in\mathcal{D}\cup\mathcal{I}$ (\cite{HawkingEllis} Section 6.6).

\section{Time of Flight Estimate in Higher Dimensions}\label{time of flight Estimate in Higher Dimensions}
In this section we will derive a higher dimensional analogue of Lemma III.2.1 in \cite{PSW} which gives an estimate for the time of flight of causal curves near infinity with endpoints on $\Lambda^\pm$. In \cite{PSW} it was found that the time of flight diverged logarithmically as we considered curves restricted to increasingly large values of $r$. In higher dimensions, we show that the time of flight tends to $0$ as we let the impact parameter approach infinity. The difference between the two estimates is the reason the 3+1 dimensional proof given in \cite{PSW} could not be generalised to higher dimensions. In \cite{PSW}, the authors make the point that, to first order, the ``phase delay'' along a $g$-null geodesic consists of two contributions. These consist of the phase delay due to the perturbation to the background metric as well as the contribution due to the subsequent bending of geodesics. The authors only consider the first of these effects, arguing that the second will be sub-leading. Since we will require the phase delay along a geodesic to tend to zero as this geodesic is restricted to arbitrarily large values of $r$, we will need to consider both effects together. 

\begin{lemma}(\textbf{Time of Flight Estimate}) \label{lemma:timeofflight}
Let $(M,g)$ be a spacetime in $D+1$ dimensions which admits quasi-Cartesian co-ordinates and let $\gamma$ be a causal curve which connects $\Lambda^-$ to $\Lambda^+$. Let $b$ denote the impact parameter (the minimum value of $r$ along $\gamma$). Then the time of flight along $\gamma$ tends to zero as $b\rightarrow\infty$.
\end{lemma}

\textbf{Proof}: We begin by showing that, for spacetimes admitting quasi-Cartesian co-ordinates as in Definition \ref{defn:quasicart}, the co-ordinates 
\begin{equation}\label{eqn:retadv}
    \begin{split}
        u=t-r\\
        v=t+r
    \end{split}
\end{equation}
are good co-ordinates on $\mathcal{I}^+$ and $\mathcal{I}^-$ respectively. Recall that for Schwarzschild in $3+1$ dimensions, we use the retarded and advanced time co-ordinates
\begin{equation}
    \begin{split}
        u_s&=t-r-2m\log(r/2m-1)\\
        v_s&=t+r+2m\log(r/2m-1)
    \end{split}
\end{equation}
where $m$ is the mass of the spacetime and the quantities $t-r$ and $t+r$ diverge along outgoing and ingoing radial null geodesics respectively. This divergence means that $u$ and $v$ are not good co-ordinates on $\mathcal{I}^+$ and $\mathcal{I}^-$ in this spacetime.

Another way of saying this is that the Hamilton-Jacobi functions $S^\pm$ defined for $(M,g)$ as in \cite{PSW} agree with $u$ and $v$ at infinity. We will define the (possibly negative) time of flight along an endless curve as $u_\infty-v_\infty$, where $u_\infty$ denotes the value of $u$ at the future endpoint of the curve and $v_\infty$ denotes the value of $v$ at its past endpoint. 

Consider a future endless $g$-null curve whose direction approaches $k^\mu$ as the affine parameter $\tau\rightarrow\infty$. We see that this curve becomes asymptotically radial in the far future. For a $g$-null curve, we have
\begin{equation}
    \begin{split}
    0&=\left(\frac{ds}{d\tau}\right)^2=-\dot{u}\dot{v}+r^2\dot{\omega}_{D-1}^2+h_{\mu\nu}\dot{x}^\mu\dot{x}^\nu\\
        \implies\dot{u}&=\frac{1}{\dot{v}}\left(r^2\dot{\omega}_{D-1}^2+h_{\mu\nu}\dot{x}^\mu\dot{x}^\nu\right)
    \end{split}
\end{equation}
where $\dot{\omega}_{D-1}^2$ denotes $\frac{d\boldsymbol{\omega}^2_{D-1}}{d\tau^2}$ and $d\boldsymbol{\omega}_{D-1}$ is an infinitesimal displacement on the unit $(D-1)$-sphere.

In Minkowski spacetime we have
\begin{equation}
    r^2\dot{\omega}_{D-1}^2=\frac{l^2}{r^2}
\end{equation}
for some constant $l$. For now we are only interested in what happens in the infinite future along a $g$-null curve. We see that
\begin{equation}
    \begin{split}
        r^2\dot{\omega}_{D-1}^2=O\left(1/r^2\right)
    \end{split}
\end{equation}
We also note that $\dot{x}^\mu=O(1)$ and hence that $\dot{v},\dot{r}=O(1)$. From this we deduce (assuming $\delta\leq1$, otherwise replace $\delta$ with 1 in what follows) that
\begin{equation}
    \begin{split}
        \dot{u}&=O\left(1/r^{1+\delta}\right)\\
        \implies \frac{du}{dr}&=O\left(1/r^{1+\delta}\right)\\
        \implies \Delta u&=O(1/r_*^\delta)
    \end{split}
\end{equation}
where $\Delta u$ denotes the change in $u$ along some future endless portion of $\gamma$ with past endpoint $p$ at which $r=r_*$ and along which $r$ is strictly increasing. Similarly, the time of flight along some past endless portion of $\gamma$ with future endpoint at $q$ and along which $r$ is decreasing must also be finite. 
\begin{figure} 
    \centering
    \includegraphics[scale=0.3]{gamma.png}
    \caption{The time of flight along both the future endless portion of $\gamma$ from $p$ and the past endless portion of $\gamma$ from $q$ (shown as unbroken lines) are both finite. The time of flight from $q$ to $p$ must be finite since these points both lie in the interior of the spacetime and therefore have finite values of the co-ordinates $t$ and $r$.}
    \label{fig:gamma}
\end{figure}
We therefore conclude that defining the time of flight along $\gamma$ as $u_\infty-v_\infty$, where $u_\infty$ denotes the value of $u$ on $\Lambda^+$ and $v_\infty$ denotes the value of $v$ at $\Lambda^-$, gives a finite quantity (note that the time of flight from $p$ to $q$ must be finite since both these points have finite values of $t$ and $r$ co-ordinates). This is shown in figure \ref{fig:gamma}. This shows that $g$-null geodesics have endpoints on $\mathcal{I}^\pm$ defined using the co-ordinates $u$ and $v$. In other words, we have shown that if $(M,g)$ admits quasi-Cartesian co-ordinates, then it can be conformally compactified using the same co-ordinate transformations and conformal factor as is used in the standard conformal compactification of Minkowski spacetime (see e.g. \cite{PCMD} Section 3).

From this we see that if we restrict our curve to arbitrarily large values of $r$, the time of flight along it must tend to $0$ (which is the value for a Minkowski-null curve). Indeed, suppose we define
\begin{equation}
\begin{split}
   \epsilon&=\frac{1}{b^{\delta/2}}\\ \Tilde{h}_{\mu\nu}&=r^{\delta/2} h_{\mu\nu}
   \end{split}
\end{equation}
Let $\tilde{\gamma}$ be a curve with endpoints on $\Lambda^\pm$ and which is comprised of two curves which are null with respect to the metric $\eta+\epsilon\Tilde{h}$. Note that $\Tilde{h}_{\mu\nu}=O(1/r^{1+\delta/2})$ (regarding $b$ as fixed), so by the above argument, the time of flight along $\Tilde{\gamma}$ is finite. It therefore makes sense to say that the time of flight along $\tilde{\gamma}$ is an $O(\epsilon)$ quantity. We can now consider the limit as $b\rightarrow\infty$. In this limit, $\epsilon\rightarrow0$ which means the time of flight along $\Tilde{\gamma}$ must tend to $0$. However we also note that 
\begin{equation}
    |\epsilon\Tilde{h}_{\mu\nu}|\geq |h_{\mu\nu}|
\end{equation}
for each $\mu,\nu\in\{0,1,...,D\}$, so we conclude that the time of flight along $\gamma$ must also tend to $0$ as $b\rightarrow\infty$. \qedsymbol

Note that it would not have been sufficient to simply carry out the calculation to first order in $\epsilon$. This is because, before taking the limit as $b\rightarrow\infty$, we need to take another limit as the future endpoint of the curve tends to null infinity. It is not obvious that each of the terms in any expansion in powers of $\epsilon$ will be finite in this limit. The argument above fails for Schwarzschild in $3+1$ dimensions because $u_\infty-v_\infty$ is infinite along future endless null curves. 

The above proof also shows that spacetimes which admit quasi-Cartesian co-ordinates are weakly asymptotically simple and empty as defined in \cite{PCMD} Definition 2.3. The compactification map is given by defining the compactified co-ordinates $T$ and $\chi$ as in \cite{PCMD} Equation 3.6. By showing that $u$ and $v$ as defined in Equation \ref{eqn:retadv} are finite in the infinite future and infinite past respectively of null geodesics, we have shown that null geodesics which do not enter black or white holes have endpoints on $\mathcal{I}^\pm$. This means it makes sense for us to talk about the compactification $(\tilde{M},\tilde{g})$ of the spacetime. 
\section{Constructing a Fastest Causal Curve}\label{Constructing a Fastest Causal Curve}
As discussed in section \ref{Introduction}, we will begin by constructing a causal curve from $\Lambda^-$ to $\Lambda^+$ which has negative time of flight. To do this we use a comparison argument based on Minkowski spacetime, similar to the ones used in \cite{PCMD}. This argument relates to the compactified spacetimes. 

\begin{lemma}\label{lemma:negativetimeofflight}
Let $(M,g)$ be an asymptotically Schwarzschildean spacetime in $D+1$ dimensions with ADM mass $m_{ADM}<0$. Then there exists an endless timelike curve from $\Lambda^-$ to $\Lambda^+$ which has negative time of flight $u_\infty-v_\infty$.
\end{lemma}

\textbf{Proof}: The metric is asymptotically Schwarzschildean to first order, so the line element can be written in some neighbourhood of conformal infinity as
\begin{equation}\label{eqn:lineeleement}
    \begin{split}
        ds^2=ds^2_{Mink}+\frac{2m}{r^{D-2}}\left(dt^2+dr^2\right)+o\left(r^{-(D-2)}\right)
    \end{split}
\end{equation}
where by $o\left(r^{-(D-2)}\right)$ we mean that this term is equal to $g'_{\mu\nu}dx^\mu dx^\nu$ for some $g'_{\mu\nu}=o\left(r^{-(D-2)}\right)$.

As shown in the proof of Lemma \ref{lemma:timeofflight}, the Minkowski space retarded and advanced time co-ordinates, $u$ and $v$, are finite along $g$-null geodesics. This means we can describe the compactified spacetime $(\Tilde{M},\Tilde{g})$ (or strictly speaking the submanifold diffeomorphic to $\mathbbm{R}\times\left(\mathbbm{R}^D\setminus B\right)$ of Minkowski where the quasi-Cartesian co-ordinates are defined) using compactified Minkowski co-ordinates. Furthermore, since the two compactifications are defined on the same manifold (i.e. neighbourhood of infinity), we can consider a curve in Minkowski spacetime (provided it lies entirely in this neighbourhood) as also defining a curve in the spacetime $(M,g)$. 

Consider a Minkowski-null geodesic restricted to sufficiently large values of $r$ to ensure that the quasi-Cartesian co-ordinates are valid. It is clear from \ref{eqn:lineeleement} that for sufficiently large $r$, the term proportional to $m$ will dominate the other non-Minkowski terms. This means that if $m_{ADM}<0$ (or equivalently $m<0$) there exists some $r_0$ such that 
\begin{equation}\label{eqn:bound}
    \begin{split}
        ds^2<ds^2_{Mink}
    \end{split}
\end{equation}
for all $r>r_0$.

Consider a Minkowski null geodesic with past endpoint at $v=0$ on $\Lambda^-$, future endpoint at $u=0$ on $\Lambda^+$ and impact parameter $b>r_0$. So this curve is restricted to the region where $r> r_0$ and hence inequality (\ref{eqn:bound}) guarantees that it is a $g$-timelike curve between the same two endpoints in $(\Tilde{M},\Tilde{g})$.\footnote{This argument only works in higher dimensions. In $3+1$ dimensions, although inequality (\ref{eqn:bound}) still holds, $g$-null geodesics are infinitely delayed or advanced compared to $\eta$-null geodesics (for $m\neq0$). As a result, $g$-null geodesics have endpoints at past and future timelike infinity in compactified Minkowski spacetime (for $m>0$) or at spatial infinity (for $m<0$). See \cite{Penrose} or \cite{PCMD} for more details.}

Now since $I^\pm(p)$ are open sets for any point $p\in \tilde{M}$, we must be able to modify this curve slightly so that it remains $g$-timelike but has past endpoint at $v>0$ on $\Lambda^-$ and future endpoint at $u<0$ on $\Lambda^+$. This ensures that the curve has negative time of flight (see Figure \ref{fig:negativeToF}).  \qedsymbol

We will use this curve as the starting point for our sequence of faster and faster causal curves used in the proof of Theorem \ref{thm:nullline} to construct a fastest causal curve from $\Lambda^-$ to $\Lambda^+$.

\textbf{Theorem \ref{thm:nullline}}. Let $(M,g)$ be an asymptotically Schwarzschildean spacetime in $D+1$ dimensions with ADM mass $m_{ADM}<0$ such that $\mathcal{D}\cup\mathcal{I}$ is globally hyperbolic as a subset of $\tilde{M}$. Then $(M,g)$ contains a null line.

\textbf{Proof:} Using Lemma \ref{lemma:negativetimeofflight}, we construct a causal curve from $\Lambda^-$ to $\Lambda^+$ which has negative time of flight. Denote this curve by $\gamma_0$ and label its past and future endpoints $p_0$ and $q_0$ respectively. 
\begin{figure} 
    \centering
    \includegraphics[scale=0.3]{NegativeToF.png}
    \caption{The Minkowski null path between $\Lambda^-$ and $\Lambda^+$ with endpoints at $u=0$ and $v=0$ respectively and impact parameter $b>r_0$ is shown as a dotted line. If the metric $g$ is asymptotically Schwarzschildean with negative mass then this path can be modified so that it still connects $\Lambda^-$ to $\Lambda^+$ but is $g$-timelike and has strictly negative time of flight.}
    \label{fig:negativeToF}
\end{figure}

Following \cite{PSW}, we introduce a partial ordering $\leq$ on $\left(\left(J^+(p_0)\cap \Lambda^-\right)\cup i^0\right)\times\left(\left(J^-(q_0)\cap \Lambda^+\right)\cup i^0\right)$. We say that $(p',q')\leq (p'',q'')$ if $p'\in J^+(p'')$ and $q'\in J^-(q'')$. Define the set $F$ to contain all pairs of points in $\left(J^+(p_0)\cap \Lambda^+\right)\times\left(J^-(q_0)\cap \Lambda^-\right)$ which can be connected by a causal curve through $\mathcal{D}$ and choose $(p,q)$ to be any element of the closure of this set which is minimal with respect to the partial ordering $\leq$ (so in principle one or both of these points could lie at $i^0$).

We can then define sequences of points $\{p_i\}_{i=0}^\infty$ along $\Lambda^-$ and $\{q_i\}_{i=0}^\infty$ along $\Lambda^+$ such that
\begin{itemize}
    \item $(p_i,q_i)\leq(p_j,q_j)$ for $i>j$
    \item $p_i$ and $q_i$ are connected by a causal curve $\gamma_i$
    \item $p_i\rightarrow p$ and $q_i\rightarrow q$ as $i\rightarrow\infty$
\end{itemize}
The first condition here ensures that the time of flight along $\gamma$ is less than or equal to the time of flight along $\gamma_j$ for $i>j$.

Next we use a result of Sorkin and Woolgar (\cite{SorkinWoolgar} Theorem 23)\footnote{As stated, this theorem only applies to open subsets of spacetime. However, as mentioned in Section \ref{Definitions and Assumptions}, we assume that $\tilde{M}$ extends slightly beyond infinity so that the set $\mathcal{O}:=\mathcal{D}\cup\mathcal{I}$ is open. Alternatively, one can define $p'\in\Lambda^-\cap J^-(p)$ and $q'\in\Lambda^+\cap J^+(q)$. Then, omitting finite subsequences if necessary, we may assume that all the $p_n$ lie in $K^+(p')$ and all the $q_n$ lie in $K^-(q')$, where $K^+$ is defined as in Definition 8 of \cite{SorkinWoolgar} and $K^-$ is defined similarly by replacing $I^+$ with $I^-$. From here on the proof continues unchanged.} to deduce that the sequence $(\gamma_i)_{i=0}^\infty$ tends towards a causal curve $\gamma$ with endpoints at $p$ and $q$. This convergence will be defined in the \textit{Vietoris topology} (see Appendix or \cite{SorkinWoolgar}). By definition, the curve $\gamma$ is a fastest causal curve from $\Lambda^-$ to $\Lambda^+$. It will also define a null line provided it enters the interior of the spacetime. 

Suppose that $\gamma$ does not enter the interior of the spacetime. Following Lemma \ref{lemma:timeofflight}, we can choose $b$ sufficiently large that any causal curve from $\Lambda^-$ to $\Lambda^+$ with $r>b$ along its entire length has time of flight strictly greater than the time of flight of $\gamma_0$. But this contradicts the definition of the sequence $(\gamma_i)_{i=0}^\infty$ as consisting of faster and faster causal curves. We therefore conclude that, restricting to a subsequence if necessary, all of the causal curves $\gamma_i$ enter the ball $B_b=\{r\leq b\}$. But this set is compact, so from the definition of the Vietoris topology \cite{SorkinWoolgar}, the limit curve $\gamma$ must also enter this set.

We therefore conclude that a fastest causal curve from $\Lambda^-$ to $\Lambda^+$ which enters the interior of the spacetime does exist. \qedsymbol
\section{A Focusing Theorem in Higher Dimensions}\label{A Focusing Theorem in Higher Dimensions}
Theorem \ref{thm:nullline} shows that certain classes of higher dimensional, negative mass spacetimes must contain a null line.  This means negativity of mass would be impossible if we also impose conditions on the spacetime which forbid the existence of such a curve.

We will require conditions which ensure that endless null geodesics encounter sufficient regions of positive focusing to guarantee that conjugate points will occur. This focusing of geodesics is consistent with the sort of behaviour we would expect to be caused by regions of positive mass (assuming the Einstein equations hold). In this sense, Corollary \ref{cor} agrees with our prior understanding of positivity of mass.

In \cite{PSW}, the conditions imposed are those stated in Borde's Focusing Theorem \cite{Borde} (where we now assume that the metric is $C^2$): 
\begin{thm}(\textbf{Focusing Theorem 2 \cite{Borde}})\label{thm:Borde}
Let $\gamma$ be a complete, affinely parametrized, causal geodesic with tangent $T^a$ such that $T_{[a}R_{b]cd[e}T_{f]}T^cT^d\neq0$ at some point on $\gamma$. Suppose that for any $\epsilon>0$ and any $t_1<t_2$, there exists $\delta>0$ and intervals $I_1$ and $I_2$ of length $\geq\delta$ with endpoints $<t_1$ and $>t_2$ respectively such that  
\begin{equation}
    \int_{t'}^{t''}R_{ab}T^aT^bdt\geq-\epsilon \quad\quad\forall t'\in I_1,\quad \forall t''\in I_2
\end{equation}
Then $\gamma$ contains a pair of conjugate points.
\end{thm}
Theorem \ref{thm:Borde} was stated in $3+1$ dimensions, although the proof does not rely on this and so also holds in higher dimensions. For this reason, we may use the conditions of this theorem in our higher dimensional generalisation of the positive mass theorem.

It may be the case that we can impose weaker conditions than those used in Borde's theorem and still rule out the existence of null lines. For example, Theorem \ref{thm:nullline} only required the metric to be $C^1$, so a focusing result for metrics which fail to be $C^2$ would provide a more general result. This is mentioned at the end of Section II.2 in \cite{PSW} and in \cite{Perspectives}. The important thing is merely that we impose sufficiently strong conditions to ensure that such geodesics are ruled out. As is pointed out in \cite{PSW}, if we impose the Einstein equations on our spacetime, then the conditions of Borde's theorem can equivalently be expressed in terms of the energy-momentum tensor. The condition that $T_{[a}R_{b]cd[e}T_{f]}T^cT^d\neq0$ at some point on every geodesic is called the \textit{generic condition} and is satisfied for all but a very special class of spacetimes \cite{PenHaw}. As discussed in \cite{PSW}, it does not appear to be a particularly restrictive assumption. This is because if it is not satisfied by our spacetime, then we would expect it to be satisfied by some other spacetime which is ``nearby'' in some sense and whose 4-momentum differs by an arbitrarily small amount. As a result, imposing this assumption does not appear to weaken the positivity of mass result obtained in this paper or in \cite{PSW}. 

Using Theorems \ref{thm:nullline} and \ref{thm:Borde}, we derive the following corollary, which is our version of the positive mass theorem in higher dimensions.
\begin{cor}\label{cor}
Let $(M,g)$ be an asymptotically Schwarzschildean spacetime in $D+1$ dimensions which satisfies the conditions of Borde's theorem and is such that $\mathcal{D}\cup\mathcal{I}$ is globally hyperbolic as a subset of $\tilde{M}$. Then $m_{ADM}\geq0$.
\end{cor}

\section{Conclusion}\label{Conclusion}
We have proved a version of the positive mass theorem for higher dimensional asymptotically Schwarzschildean spacetimes. As mentioned in section \ref{Definitions and Assumptions}, the assumptions made about the spacetime are weaker than those of \cite{poorman2}. In particular, we only require the metric to be asymptotically Schwarzschildean, whereas in \cite{poorman2} the more restrictive requirement that the metric be \textit{strongly uniformly Schwarzschildean} is made. We also drop the assumption of stationarity. However our result is less satisfactory in that we fail to prove anything specifically regarding the $m=0$ case (a similar problem is encountered in \cite{PSW}). This is because the method used in section \ref{Constructing a Fastest Causal Curve} to construct the timelike curve $\gamma_0$ with strictly negative time of flight is no longer guaranteed to work. Its success will be determined by the higher order terms in the metric. 

The assumptions made in this paper are also fundamentally different to those used in the proofs by Witten \cite{Witten} and Schoen and Yau \cite{SchoenYau} (extended to all dimensions up to seven by Eichmeir, Huang, Lee and Schoen \cite{EHLS}). The proof given here relies on properties of the spacetime in a neighbourhood of conformal infinity whereas these other methods impose conditions only on some initial data surface. However, unlike these approaches, the proof contained here relies more clearly on properties we expect to hold in positive mass spacetimes. In particular, we show that null geodesic focusing is compatible only with spacetimes of non-negative mass. In this sense, the proof presented here also acts as evidence that it is consistent to think of the ADM mass defined in higher dimensions as describing the total mass contained in spacetime. 
\subsection*{Acknowledgements} I am grateful to Maciej Dunajski, Greg Galloway, Roger Penrose, Harvey Reall, Paul Tod and Claude Warnick for useful discussions. I am also grateful to St. John's College, Cambridge for their support through a College Scholarship from the Todd/Goddard Fund. Some of this work was carried out while I was visiting Institut Mittag-Leffler in Djursholm, Sweden, where I was supported by the Swedish Research Council under grant no. 2016-06596.
\begin{appendices}
\section{Compactness of Causal Curves Following \cite{SorkinWoolgar}}
In this appendix we summarise how the convergence of the sequence of causal curves $(\gamma_i)_{i=0}^\infty$ is defined. We define $\gamma$ to be the limit of this sequence in the Vietoris topology \cite{SorkinWoolgar}. To understand this convergence, we follow \cite{Convergence} and make the following definitions:
\begin{definition}
The sequence of causal curves $(\gamma_i)_{i=0}^\infty$ converges to $\gamma$ in the \textit{upper topology} iff the $\gamma_i$ are eventually included in every open set which includes $\gamma$.
\end{definition}
\begin{definition}
The sequence of causal curves $(\gamma_i)_{i=0}^\infty$ converges to $\gamma$ in the \textit{lower topology} iff the $\gamma_i$ eventually meet every open set which meets $\gamma$.
\end{definition}
\begin{definition}
The sequence of causal curves $(\gamma_i)_{i=0}^\infty$ converges to $\gamma$ in the \textit{Vietoris topology} iff it converges in both the upper and lower topologies.
\end{definition}
Since we have assumed that $\mathcal{D}\cup\mathcal{I}$ is globally hyperbolic, if we were to  assume that the metric were $C^2$ (as in Theorem \ref{thm:Borde}) then these three definitions would coincide (\cite{SorkinWoolgar} Alternative Definition 26 and comments after proof of Lemma 28). 
\end{appendices}































































