October 02, 2025
How should one define metric space notions of convergence for sequences of spacetimes? Since a Lorentzian manifold does not define a metric space directly, the uniform convergence, Gromov-Hausdorff (GH) convergence, and Sormani-Wenger Intrinsic Flat (SWIF) convergence does not extend automatically. One approach is to define a metric space structure, which is compatible with the Lorentzian structure, so that the usual notions of convergence apply. This approach was taken by C. Sormani and C. Vega SV? when defining the null distance. In this paper, we study sequences of static spacetimes equipped with the null distance under uniform, GH, and SWIF convergence, as well as Hölder bounds. We use the results of the Volume Above Distance Below (VADB) theorem of the author, R. Perales, and C. Sormani Allen-Perales-Sormani? to prove an analog of the VADB theorem for sequences of static spacetimes with the null distance. We also give a conjecture of what the VADB theorem should be in the case of sequences of globally hyperbolic spacetimes with the null distance.
How should one define metric space notions of convergence for sequences of spacetimes? Since a Lorentzian manifold does not define a metric space directly, the uniform convergence, Gromov-Hausdorff (GH) convergence, and Sormani-Wenger Intrinsic Flat (SWIF) convergence does not extend automatically. One approach is to define a metric space structure, which is compatible with the Lorentzian structure, so that the usual notions of convergence apply. This approach was taken up by C. Sormani and C. Vega SV? by defining the null distance. Notions of spacetime intrinsic flat convergence employing the null distance have been further developed by A. Sakovich and C. Sormani SS2?, pointing out important nuances which need to be appreciated when extending metric notions of convergence to spacetimes with the null distance. The problem of defining Gromov-Hausdorff convergence for sequences of spacetimes was initially taken up by J. Noldus N?, and has been more recently addressed by E. Minguzzi and S. Suhr MS?, MS-LorentzCompactness?, and by O. Müller M?. Recently, measured Lorentzian Gromov-Hausdorff convergence has been studied by M. Braun and C. Sämann BraunSamann?. These authors take the alternative approach of redefining GH convergence with the spacetime structure in mind instead of defining a metric space structure on the spacetime. It would be great to see examples computed which specifically compare and contrast these different approaches to spacetime metric convergence.
The first exploration of convergence of sequences of spacetimes with the null distance was initiated by the author and A. Burtscher AB? in the case of warped products whose warping functions converge uniformly. Later, the author Allen-Null? extended this work to optimal conditions on a sequence of warping functions which imply uniform and GH convergence. Given the lack of tools for estimating the SWIF distance between spacetimes with the null distance, SWIF convergence was not handled in these previous works. In this paper, we give the first tools for estimating the SWIF distance between globally hyperbolic spacetimes equipped with the null distance (See section 4), using ideas first introduced by the author, R. Perales, and C. Sormani Allen-Perales-Sormani? for estimating the SWIF distance between Riemannian manifolds. Furthermore, we are able to use tools developed by the author and R. Perales Allen-Perales? and the author and E. Bryden Allen-Bryden? to prove a SWIF convergence result (See Theorem 4) under optimal conditions on a sequence of static spacetimes equipped with the null distance. By taking advantage of the work of the author Allen-Holder? for proving Hölder bounds for sequences of Riemannian manifolds, we are also able to provide optimal conditions which establish Hölder bounds for a sequence of static spacetimes with the null distance which imply uniform and GH convergence of a subsequence.
Theorem 1. Let \(0<C, C'<\infty\), \(M\) a compact, connected, and oriented manifold, \((M,\sigma_1)\) a continuous Riemannian manifold, \((M,\sigma_0)\) a smooth Riemannian manifold, \(L=[t_0,t_1]\times M\), \(h_i:M \rightarrow (0,\infty)\), \(g_i=-h_i^2dt^2+\sigma_j\), \(i \in \{0,1\}\). If \(p > n\), \[\begin{align} \int_M\left|\frac{\sigma_1}{h_1}\right|_{\sigma_{0}}^{\frac{p}{2}}dV_{\sigma_{0}} &\le C,\label{eq-L94pBound} \end{align}\qquad{(1)}\] then then for all \(x,y \in L\) we find \[\begin{align} \; \hat{d}_{t,g_1}(x,y) \le C' \hat{d}_{t,g_{0}}(x,y)^{\frac{p-n}{p}}. \end{align}\]
By applying the observation of Theorem 1 to a sequence of static spacetimes with additional necessary hypotheses we are able to imply uniform and GH convergence.
Theorem 2. Let \(0<A,C, C'<\infty\), \(M\) a compact, connected, oriented manifold, \((M,\sigma_j)\) be a sequence of continuous Riemannian manifolds, \((M,\sigma_{\infty})\) a smooth Riemannian manifold, \(L=[t_0,t_1]\times M\), \(h_j:M \rightarrow (0,\infty)\), \(g_j=-h_j^2dt^2+\sigma_j\), \(j \in \mathbb{N}\cup\{\infty\}\), with \(h_j\) continuous for \(j \in \mathbb{N}\) and smooth for \(j =\infty\). If \(p > n\), \[\begin{align} \int_M\left|\frac{\sigma_j}{h_j}\right|_{\sigma_{\infty}}^{\frac{p}{2}}dV_{\sigma_{\infty}} &\le C,\label{zgtdmuli} \\ \frac{\sigma_j(v,v)}{h_j^2} &\ge \left( 1 -\frac{1}{j} \right)\frac{\sigma_{\infty}(v,v)}{h_{\infty}^2}, \quad \forall x \in M, v \in T_xM \label{eq-Space32Component32lower32bound} \\\mathop{\mathrm{Vol}}\left(M,\frac{\sigma_j}{h_j}\right)&\rightarrow \mathop{\mathrm{Vol}}\left(M,\frac{\sigma_{\infty}}{h_{\infty}}\right),\label{eq-VolumeBound} \\ \mathop{\mathrm{Area}}\left(\partial M,\frac{\sigma_j}{h_j}\right) &\le A,\label{eq-AreaBound} \end{align}\] {#eq: sublabel=eq:zgtdmuli,eq:eq-Space32Component32lower32bound,eq:eq-VolumeBound,eq:eq-AreaBound} then then for all \(x,y \in L\) we find \[\begin{align} \; \hat{d}_{t,\tilde{g}_j}(x,y) \le C' \hat{d}_{t,g_{\infty}}(x,y)^{\frac{p-n}{p}}, \end{align}\] and \[\begin{align} (L,\hat{d}_{t,g_j})\rightarrow (L,\hat{d}_{t,g_{\infty}}), \end{align}\] in the uniform and Gromov-Hausdorff sense.
Remark 3. It is natural to want to replace conditions like ?? , ?? , and ?? with conditions on \(h_j\) and \(\sigma_j\) separately. In this remark we discuss several such sufficient conditions which will also apply to the theorem and conjecture which follows. By Hölder’s inequality with \(\frac{1}{q_1}+\frac{1}{q_2}=1\) we know that \[\begin{align} \int_M\left|\frac{\sigma_j}{h_j}\right|_{\sigma_{\infty}}^{\frac{p}{2}}dV_{\sigma_{\infty}} \le \left(\int_M\left|\sigma_j\right|_{\sigma_{\infty}}^{\frac{pq_1}{2}}dV_{\sigma_{\infty}}\right)^{\frac{1}{q_1}}\left(\int_M\left|h_j\right|_{\sigma_{\infty}}^{-\frac{pq_2}{2}}dV_{\sigma_{\infty}}\right)^{\frac{1}{q_2}}, \end{align}\] which clears up what is sufficient for ?? . Now we also observe by the determinant trace inequality that \[\begin{align} \mathop{\mathrm{Vol}}\left(M,\frac{\sigma_j}{h_j}\right)&\le \int_M\left|\frac{\sigma_j}{h_j}\right|_{\sigma_{\infty}}^{\frac{n}{2}}dV_{\sigma_{\infty}},\label{eq-VolumeBoundedByL94n} \\\mathop{\mathrm{Area}}\left(\partial M,\frac{\sigma_j}{h_j}\right)&\le \int_M\left|\frac{\sigma_j}{h_j}\right|_{\sigma_{\infty}}^{\frac{n-1}{2}}dV_{\sigma_{\infty}}, \end{align}\qquad{(2)}\] and so one can again use Hölder’s inequality to find sufficient conditions on \(h_j\) and \(\sigma_j\) separately which imply ?? . To imply ?? , one can use Lemma 4.3 of Allen-Sormani-2? combined with ?? and Hölder’s inequality.
We now prove a similar Theorem which provides conditions which imply Sormani-Wenger intrinsic flat convergence of a sequence of static spacetimes equipped with the null distance metric space. We point out that Example 17 shows that the relaxed conditions of Theorem 4 are not enough to imply uniform or Gromov-Hausdorff convergence and hence Sormani-Wenger Intrinsic Flat convergence is the correct conclusion for this theorem.
Theorem 4. Let \(0<A<\infty\), \(M\) compact, connected, and oriented manifold, \((M,\sigma_j)\) be a sequence of continuous Riemannian manifolds, \((M,\sigma_{\infty})\) a smooth Riemannian manifold, \(L=[t_0,t_1]\times M\), \(h_j:M \rightarrow (0,\infty)\), \(g_j=-h_j^2dt^2+\sigma_j\), \(j \in \mathbb{N}\cup\{\infty\}\), with \(h_j\) continuous for \(j \in \mathbb{N}\) and smooth for \(j =\infty\). If \[\begin{align} \frac{\sigma_j(v,v)}{h_j^2} &\ge \left( 1 -\frac{1}{j} \right)\frac{\sigma_{\infty}(v,v)}{h_{\infty}^2} \quad \forall x \in M, v \in T_xM, \label{mbhtkpil} \\\mathop{\mathrm{Vol}}\left(M,\frac{\sigma_j}{h_j}\right)&\rightarrow \mathop{\mathrm{Vol}}\left(M,\frac{\sigma_{\infty}}{h_{\infty}}\right), \\ \mathop{\mathrm{Area}}\left(\partial M,\frac{\sigma_j}{h_j}\right) &\le A, \end{align}\qquad{(3)}\] then \[\begin{align} (L,\hat{d}_{t,g_j})\rightarrow (L,\hat{d}_{t,g_{\infty}}), \end{align}\] in the Sormani-Wenger intrinsic flat sense.
Remark 5. One should note that both Theorem 2 and Theorem 4 can be applied to non-compact manifolds with boundary where \(h_j \rightarrow \infty\) as one approaches the boundary or \(\sigma_j\) becomes degenerate. The theorems are applied by taking a precompact exhaustion of the non-compact manifold with boundary and considering convergence on compact subsets. So the results of this paper apply to a sequence of Schwarzschild manifolds with vanishing mass as well as similar sequences of static manifolds which one does not expect to converge smoothly on compact subsets.
Remark 6. If \(h_j=1\) and \(\sigma_j\) is given as a graph of a function \(f_j:M\rightarrow (0,\infty)\) so that \(\sigma_j=\sigma_{\infty}+df_j^2\) then we see that ?? is automatically satisfied. This shows that the stability results of L.-H. Huang, D. Lee and C. Sormani HLS?, L.-H. Huang, D. Lee and R. Perales HLP?, and A. J. Cabrera Pacheco, M. Graf and R. Perales PGPHyp? in the time symmetric case can be extended to stability results for the associated static manifolds equipped with the null distance by Theorem 4. This is because the volume convergence and area bound hypotheses were also obtained in the previous works on the graphical case of stability of the positive mass theorem.
Bernal and Sánchez Bernal-Sanchez? show that a globally hyperbolic spacetime \((M,g)\) can be written in the form \(g=-h^2dt^2+\hat{g}(t)\) where \(M=\mathbb{R}\times M\) for some manifold \(\Sigma\), \(h: \mathbb{R}\times M\rightarrow (0,\infty)\), and \(\hat{g}(t)\) is a Riemannian metric on \(M\) for each \(t \in \mathbb{R}\). So it is natural to ask for a generalization of Theorem 2 and Theorem 4 for sequences of globally hyperbolic spacetimes. In a similar way as in the warped product and static cases, since the null distance is conformally invariant, i.e \(\hat{d}_{\tau, \lambda^2 g} = \hat{d}_{\tau, g}\) where \(\lambda: \mathbb{R}\times M \rightarrow (0,\infty)\), we can equivalently study \(h^{-2}g=-dt^2+h^{-2}\hat{g}(t)\) with respect to the null distance, as we will see in the following conjecture.
Conjecture 7. Let \(0<A<\infty\) and \(M\) be a compact, connected, and oriented manifold, \(L=[t_0,t_1]\times M\), \(h_j: [t_0,t_1]\times M \rightarrow (0,\infty)\), \(g_j=-h_j^2dt^2+\sigma_j(t)\), \(j \in \mathbb{N}\cup\{\infty\}\), \(\sigma_j(t)\) is a Reimannian manifold for each \(t \in [t_0,t_1]\), continuous for \(j \in \mathbb{N}\) and smooth when \(j=\infty\). If \[\begin{align} \frac{\sigma_j(v,v)}{h_j^2} &\ge \left( 1 -\frac{1}{j} \right)\frac{\sigma_{\infty}(v,v)}{h_{\infty}^2} \quad \forall x \in M, v \in T_xM, \label{zmreauxq} \\\int_{t_0}^{t_1}\mathop{\mathrm{Vol}}\left(M,\frac{\sigma_j(t)}{h_j(t)}\right)dt&\rightarrow \int_{t_0}^{t_1}\mathop{\mathrm{Vol}}\left(M,\frac{\sigma_{\infty}(t)}{h_{\infty}(t)}\right)dt, \\ \max_{t \in [t_0,t_1]}\mathop{\mathrm{Area}}\left(\partial M,\frac{\sigma_j(t)}{h_j(t)}\right) &\le A, \end{align}\qquad{(4)}\] then \[\begin{align} (L,\hat{d}_{t,g_j})\rightarrow (L,\hat{d}_{t,g_{\infty}}), \end{align}\] in the Sormani-Wenger intrinsic flat sense.
One should note that the tools of section 4 still apply to Conjecture 7 but the arguments used in section 5 are not available in this case. Hence a new way to obtain pointwise convergence of the distance function \(\hat{d}_{t,g_j}\) almost everywhere needs to be developed in order to take advantage of the tools of section 4. This will not be a straightforward adaption of the work of Allen-Perales-Sormani? or of this paper since the Lorentzian structure and the definition of null distance will have to be taken head on with a new insight in this case. That being said, the examples explored and main theorems in AB?, Allen-Null?, as well as this paper, provide strong evidence for Conjecture 7.
In section 2, we review the definition of the null distance, properties which have been established for this distance, the definition of SWIF convergence, as well as the work in Allen-Perales-Sormani?, Allen-Perales?, Allen-Holder?, and Allen-Bryden? which will be needed to prove the main theorems of this paper.
In section 3, we review examples given in AB? and Allen-Null? which shed light on the necessity of the various hypotheses in the main theorems.
In section 4, we adapt a technique for estimating the SWIF distance, first developed for seqeunces of Riemannian manifolds in Allen-Perales-Sormani?, to the setting of globally hyerpbolic spacetimes equipped with the null distance.
In section 5, we combine the tools reviewed in section 2 with estimation techniques of section 4 to prove the main theorems of this paper.
A spacetime is a time oriented Lorentzian manifold \((L,g)\). We define \(J^{\pm}_g(p)\) to be the set of all points which are in the causal future or past of \(p\), with respect to \(g\). Let \(\tau:L \rightarrow \mathbb{R}\) be a time function which is a continuous function that is strictly increasing along all future directed causal curves. We now define the null length and null distance introduced by Sormani and Vega SV?. Let \(\beta :[a,b] \rightarrow L\) be a piecewise causal curve, i.e. a piecewise smooth curve that is either future-directed or past-directed causal on its pieces \(a=s_0 < s_1 < \ldots < s_k =b\) (see figure [fig:AdmissableCurves]). The null length of \(\beta\) is given by \[\begin{align} \hat{L}_{\tau,g} (\beta) = \sum_{i=1}^k |\tau(\beta(s_i))-\tau(\beta(s_{i-1}))|. \end{align}\]
In the case where \(\tau\) is differentiable we can compute the length of any piecewise causal curve \(\beta\) by \[\begin{align} \hat{L}_{\tau,g} (\beta) = \int_a^b |(\tau \circ \beta)'(s)| ds. \end{align}\] We will take advantage of this integral formula when estimating the SWIF distance between static spacetimes in section 4. For any \(p,q \in M\), the null distance is given by \[\begin{align} \hat{d}_{\tau,g} (p,q) = \inf \{ \hat{L}_{\tau,g} (\beta) : \beta \text{ is a piecewise causal curve from } p \text{ to } q \}. \end{align}\]
It is important to note that the null distance is compatible with weak notions of spacetimes such as the Lorentzian length spaces of M. Kunzinger and C. Sämann KCS?. For further properties of null distance see the work by the author and A. Burtscher AB?, A. Burtscher and L. García-Heveling BG?, BG2?, G. Galloway GG?, M. Graff and C. Sormani GS?, M. Kunzinger and R. Steinbauer KS?, B. Meco, A. Sakovich and C. Sormani MSS?, A. Sakovich and C. Sormani SS?, SS2?, C. Sormani and C. Vega SV?, and C. Vega V?. One particularly important property which we will exploit several times in this paper is the conformal invariance of the null distance which was noticed in SV? and follows from the fact that the set of causal curves on a spacetime is a conformal invariant.
When investigating the null distance it is desirable to choose a canonical time function to fix the metric space structure on a spacetime. C. Sormani and C. Vega SV? suggested to use the cosmological time function of L. Anderson, G. Galloway, and R. Howard AGH?. It was later shown by A. Sakovich and C. Sormani SS?, and A. Burtscher and L. García-Heveling BG2? to be a good choice for a canonical time function when defining the null distance on a spacetime. See the work of M. Hoseini, N. Ebrahimi, and M. Vatandoost MNM? for a recent study of the cosmological time function on Lorentzian length spaces. The cosmological time function of a Lorentzian manifold \((L,g)\) is defined to be \[\begin{align} \tau_{AGH}(p)=\sup\{L_g(\gamma):\text{ future timelike } \gamma:[0,1]\rightarrow L, \gamma(1)=p\}, \end{align}\] where the Lorentzian length is defined by \[\begin{align} L_g(\gamma)=\int_0^1\sqrt{|g(\gamma',\gamma')|}ds. \end{align}\] One says that the time function is regular if \(\tau_{AGH}<\infty\) and \(\tau_{AGH} \rightarrow 0\) along every past inextensible causal curve. We now show that for every spacetime in this paper the \(t\)-coordinate will be the cosmological time function. This is not too surprising for static metrics and globally hyperbolic metrics but we provide a simple proof here for convenience.
Lemma 8. Let \((L=[t_0,t_1]\times \Sigma,g=-h(t,x)^2dt^2+f(t,x)^2\sigma)\) be a Lorentzian manifold where \((\Sigma,\sigma)\) is a compact, connected Riemannian manifold, \(h,f: [t_0,t_1]\times M \rightarrow (0,\infty)\), then \(t\) is the cosmological time function.
Proof. First we notice that since for any timelike curve \(\gamma\) we know \(dt(\gamma') \not = 0\) we can reparametrize in the form \(\gamma(t)=(t,\alpha(t))\) where \(t \in [t_0',t_1']\). Now by scaling and shifting the time parameter we can parameterize any timelike curve \(\gamma:[0,\int_{t_0'}^{t_1'}h(s,\alpha(s))ds]\rightarrow M\) in the form \[\begin{align} \gamma(s)=\left(\frac{s(t_1'-t_0')}{\int_{t_0'}^{t_1'}h(s,\alpha(s))ds}+t_1',\alpha(s)\right) \end{align}\] and note that it will be timelike if \[\begin{align} g(\gamma',\gamma')< 0 \quad \Rightarrow \quad-\frac{h^2(t_1'-t_0')^2}{\left(\int_{t_0'}^{t_1'}h(s,\alpha(s))ds\right)^2}+f^2|\alpha'|_{\sigma}^2<0 . \end{align}\]
If we calculate the length of \(\gamma\) we find \[\begin{align} L_g(\gamma)=\int_{t_0'}^{t_1'} \sqrt{\frac{h^2(t_1'-t_0')^2}{\left(\int_{t_0'}^{t_1'}h(s,\alpha(s))ds\right)^2}-f^2|\alpha'|_{\sigma}^2}ds \le t_1'-t_0'. \end{align}\]
In particular, we see that the supremum in the definition of the cosmological time function is achieved by \(\gamma(s)=\left(\frac{s(t_1-t_0)}{\int_{t_0}^{t_1}h(s)ds},p_{\Sigma}\right)\) and hence \(t\) is the cosmological time function. ◻
In order to prove Theorem 2 we will first establish pointwise convergence of the null distance functions. This will be done by first establishing pointwise convergence of the distance functions with respect to the Riemannian manifold which is part of the static spacetime structure. To this end we will apply the following theorem of the author and E. Bryden AB? which extends the work in Allen-Perales-Sormani?, Allen-Perales? where similar theorems were established under less general hypotheses.
Theorem 9 (Theorem 4.5 of Allen-Bryden?). Let \((M,\partial{}M,g_{0})\) be a Riemannian manifold with boundary, and let \(g_{i}\) be a sequence of Riemannian metrics so that \[\begin{align} \mathop{\mathrm{Diam}}(M,g_j) &\le D, \\ \mathop{\mathrm{Vol}}(M,g_j) &\rightarrow \mathop{\mathrm{Vol}}(M,g_0) \\ \mathop{\mathrm{Area}}(\partial M,g_j) &\le A, \\ g_j(v,v) &\ge ( 1 - C_j)g_0(v,v), \quad \forall p \in M, v \in T_pM, \quad C_j \searrow 0, \end{align}\] Then \(d_{g_{j}}\) converges to \(d_{g_{0}}\) pointwsie almost everywhere on \(M\times{}M\) with respect to \(dV_{g_{0}}\otimes{}dV_{g_{0}}\).
After we establish pointwise convergence of the sequence of null distance functions to the desired limiting null distance function, we will apply a compactness theorem to conclude uniform convergence. Since uniform convergence implies Gromov-Hausdorff convergence we will then obtain Theorem 2. The key ingredient in this part of the proof is a Hölder bound for Riemannian metrics, first obtained by the author in Allen-Holder?, which we review for the reader below. Since we will need this result for compact manifolds with boundary we also give a simple proof which extends the original result to manifolds with boundary.
Theorem 10 (Theorem 1.1 of Allen-Holder?). Let \(M^n\) be a compact, connected, and oriented manifold, \(M_0=(M,\sigma_0)\) a smooth Riemannian manifold, and \(M_1=(M,\sigma_1)\) a continuous Riemannian manifold. If \[\begin{align} \exists p > n, \quad \|g_1\|_{L_{g_0}^{\frac{p}{2}}(M)} \le C \end{align}\] then \[\begin{align} d_{\sigma_1}(q_1,q_2) \le C'(M,\sigma_0) d_{\sigma_0}(q_1,q_2)^{\frac{p-n}{p}},\quad \forall q_1,q_2 \in M. \end{align}\]
Proof. If \(M\) is closed then we are done. If \(M\) has boundary, then let \(N\) be the double of \(N\) which induces a natural continuous Riemannian metrics \(N_0=(N,\sigma_0)\), and \(N_1=(N,\sigma_1)\). Since we need \(N_0\) to be smooth, we can run Ricci flow for a short period of time by Theorem 1.1 of P. Burkhardt-Guim PBG? to obtain a smooth Riemannian metric \(\hat{\sigma}_0\) which is \(C^0\) close to \(\sigma_0\). Then since the corresponding distance functions will satisfy a Hölder bound \[\begin{align} \label{eq-Intermediate32Holder32Bound} d_{\hat{\sigma}_0}(q_1,q_2) \le\bar{C}(M,\sigma_0) d_{\sigma_0}(q_1,q_2)^{\frac{p-n}{p}},\quad \forall q_1,q_2 \in M, \end{align}\tag{1}\] we may apply the result of Allen-Holder? to \(N_1\) and \((N,\hat{\sigma}_0)\) and combine with 1 to obtain the desired result on the original manifolds with boundary. ◻
In this section we remind the reader how to obtain distance controls on a closed and measurable good set \(W\) of almost full volume. This was first established by the author, R. Perales, and C. Sormani Allen-Perales-Sormani? and was used to establish the Volume Above Distance Below theorem, which is the inspiration for the main theorems of this paper as well as Conjecture 7. By combining Lemma 4.3 of Allen-Perales? with Theorem 9 of Allen-Bryden? one obtains the following theorem which will be used to prove Theorem 4.
Theorem 11. Let \(M\) be a compact, connected, and oriented manifold, \((M,\sigma_j)\) be a sequence of continuous Riemannian manifolds and \((M,\sigma_0)\) a smooth Riemannian manifold such that \[\begin{align} \sigma_0(v,v) \le \sigma_j(v,v), \quad \forall p \in M, v \in T_pM, \end{align}\] \[\begin{align} \operatorname{Diam}(M,\sigma_j) \le D_0, \end{align}\] and \[\begin{align} \mathop{\mathrm{Vol}}_j(M_j) \rightarrow \mathop{\mathrm{Vol}}_0(M_0). \end{align}\] Then for any \(\lambda \in (0, \operatorname{Diam}(M_0))\) and \(\kappa >1\), there exists a measurable set \(W_{\lambda,\kappa} \subset M\) and a constant \(\delta_{\lambda, \kappa, j}>0\) such that for all \(p_1, p_2 \in W_{\lambda, \kappa}\) \[\begin{align} |d_{\sigma_j}(p_1,p_2)-d_{\sigma_0}(p_1,p_2)| < 2 \lambda + 2\delta_{\lambda,\kappa,j}, \end{align}\] where \(\delta_{\lambda,\kappa,j} \to 0\) as \(j \to \infty\), and \[\begin{align} \mathop{\mathrm{Vol}}(M \setminus W_{\lambda, \kappa},\sigma_j) \le \frac{1}{\kappa}\mathop{\mathrm{Vol}}(M,\sigma_0)+|\mathop{\mathrm{Vol}}(M,\sigma_j)-\mathop{\mathrm{Vol}}(M,\sigma_0)|. \end{align}\]
Given two metric spaces \((X,d_1)\) and \((X,d_2)\), defined on the same set \(X\), we can define the uniform distance between them to be \[\begin{align} \label{def-UniformDist} d_{unif}(d_0,d_1)=\sup_{x_1,x_2\in X}|d_1(x_1,x_2)-d_2(x_1,x_2)| \end{align}\tag{2}\] Given a sequence of metric spaces \((X,d_j)\) we can define the uniform convergence of \(d_j\) to a limiting metric space \((X,d_{\infty})\) by \[\begin{align} \label{def-UniformConvergence} d_{unif}(d_j,d_{\infty}) \rightarrow 0. \end{align}\tag{3}\]
The author and A. Burtscher AB? showed that warped product spacetimes, and more generally globally hyperbolic spacetimes, are integral current spaces (see the definition below) using bi-Lipschitz estimates. Since these are the type of metric spaces which Sormani-Wenger Intrinsic Flat (SWIF) distance is defined on, we are justified in applying the SWIF notion of distance to static spacetimes equipped with the null distance. In this subsection we will remind the reader of some important definitions related to the SWIF distance which will be crucial for estimates we make in section 4.
We start by reviewing the definition of the flat distance of H. Federer and W. H. Fleming FF? which was extended to arbitrary metric spaces by L. Ambrosio and B. Kircheim AK?. Let \((Z,d)\) be a complete metric space, \(\operatorname{Lip}(Z)\) the set of real valued Lipschitz functions on \(Z\), and \(\operatorname{Lip}_b(Z)\) the bounded ones. An \(n\)-dimensional current \(T\) on \(Z\) is a multilinear map \(T: \operatorname{Lip}_b(Z) \times [\operatorname{Lip}(Z)]^n \to \mathbb{R}\) that satisfies properties which can be found in Definition 3.1 of AK?. From the definition of \(T\) we know there exists a finite Borel measure on \(Z\), \(\|T\|\), called the mass measure of \(T\). Then the mass of \(T\) is defined as \({\mathbf{M}}(T)=\|T\|(Z)\). The boundary of \(T\), \(\partial T: \operatorname{Lip}_b(Z) \times [\operatorname{Lip}(Z)]^{n-1} \to \mathbb{R}\) is the linear functional given by \[\begin{align} \partial T(f, \pi) = T(1, (f, \pi)), \end{align}\] and for any Lipschitz function \(\varphi: Z \to Y\) the push forward of \(T\), \({\varphi}_{\sharp} T : \operatorname{Lip}_b(Y) \times [\operatorname{Lip}(Y)]^{m} \to \mathbb{R}\) is the \(n\)-dimensional current given by \[\begin{align} {\varphi}_{\sharp} T (f, \pi) = T( f\circ \varphi, \pi \circ \varphi ). \end{align}\] Furthermore, the following inequality holds \[\label{eq-pushMeasure} \| \varphi_\sharp T\| \leq \operatorname{Lip}(\varphi)^n \varphi_\sharp \|T\|.\tag{4}\]
More generally, an \(n\) dimensional integer rectifiable current, \(T\), can be parametrized by a countable collection of biLipschitz charts, \(\varphi_i: A_i \to \varphi_i(A_i)\subset Z\) where \(A_i\) are Borel in \({\mathbb{R}}^n\) with pairwise disjoint images and integer weights \(\theta_i\in {\mathbb{Z}}\) such that \[\begin{align} T(f, \pi_1,...\pi_n) = \sum_{i=1}^\infty \theta_i \int_{A_i} (f\circ \varphi_i)\, d(\pi_1 \circ \varphi_i)\wedge \cdots \wedge d(\pi_n \circ \varphi_i) \end{align}\] has finite mass, \({\mathbf{M}}(T)=\|T\|(Z)\).
An \(n\)-dimensional integral current in \(Z\) is an \(n\)-dimensional current that can be written as a countable sum of terms, \[\begin{align} T= \sum_{i=1}^\infty \varphi_{i\sharp} [[\theta_i]], \end{align}\] with \(\theta_i \in L^1(A_i, \mathbb{R})\) integer constant functions, such that \(\partial T\) is a current. The class that contains all \(n\)-dimensional integral currents of \(Z\) is denoted by \({\mathbf{I}}_n(Z)\). For \(T\in I_n(Z)\), L. Ambrosio and B. Kirchheim proved that the subset \[\begin{align} \textrm{set}(T)= \left\{ z \in Z \, | \, \liminf_{r \downarrow 0} \frac{\|T\|(B_r(z))}{ r^n }> 0 \right\} \end{align}\] is \(\mathcal{H}^n\)-countably recitifiable. That is, \(\textrm{set}(T)\) can be covered by images of Lipschitz maps from \(\mathbb{R}^n\) to \(Z\) up to a set of zero \(\mathcal{H}^n\)-measure.
Since all the examples of this paper will be static spacestimes, or more generally globally hyerbolic spacetimes, we now look at the example of globally hyerbolic spacetimes as integral current spaces as observed in AB?.
Example 12. For an \(n+1\)-dimensional compact globally hyperbolic spacetime \((L^{n+1},\bar{g}=-h^2dt^2+g)\), the triple \((L,\hat{d}_{t,\bar{g}}, [[L]])\) given as follows is an \(n+1\)-dimensional integral current space. Here \([[L]] : \operatorname{Lip}_b(L) \times [\operatorname{Lip}(L)]^{n+1} \to \mathbb{R}\) is given by \[\begin{align} \label{eq-canonicalT} [[L]] = & \sum_{i,k} {\psi_i}_\sharp [[1_{A_{ik}}]] \end{align}\qquad{(5)}\] where we have chosen the smooth locally finite atlas \(\{(U_i, \psi_i)\}_{i \in \mathbb{N}}\) of \(L\) given in AB? consisting of positively oriented Lipschitz charts, \(\psi_i : U_i \subset \mathbb{R}^n \to L\) and \(A_{ik}\) are precompact Borel sets such that \(\psi_i(A_{ik})\) have disjoint images for all \(i\) and \(k\), and cover \(L\) \(\mathcal{H}^n\)-almost everywhere. In this case, \(\|[[L]]\|= dV_{\hat{g}}\) where \(\hat{g}=h^2dt^2+g\).
In AK?, Ambrosio-Kirchheim prove that for rectifiable currents \[\begin{align} \|T\|= \lambda \theta \mathcal{H}^n \end{align}\] where \(\theta\) is an integer valued function and the area factor \(\lambda: \textrm{set}(T) \to \mathbb{R}\) is a measurable function bounded above by \[\begin{align} \label{C95n} C_n=2^n/\omega_n \textrm{ where } \omega_n=\mathop{\mathrm{Vol}}_{{\mathbb{E}}^n}(B_0(1)). \end{align}\tag{5}\] So that \[\begin{align} {\mathbf{M}}(T) \le C_n \sum_{i=1}^\infty |\theta_i| \mathcal{H}^n( \varphi_i(A_i)) < \infty. \end{align}\]
We will make frequent use of this in section 4 in order to trade mass estimates of integral current spaces for Hausdorff measure estimates of metric spaces. In particular, we will use \(\mathcal{H}^n_d\) to denote the Hausdorff measure with respect to the metric \(d\) and we will use the estimate, which follows from the observations above, that \[\begin{align} \|T\|_d(Z) \le C_n \mathcal{H}^n_d(Z), \end{align}\] where \((Z,d,T)\) has multiplicity \(1\).
The flat distance between two integral currents \(T_1, T_2 \in {\mathbf{I}}_{n} (Z)\) is defined as \[\begin{align} \begin{aligned} d_{F}^Z( T_1, T_2)=\inf\Bigl\{ {\mathbf{M}}(U)+ {\mathbf{M}}(V)\, |& \, \, U \in {\mathbf{I}}_{n}(Z), \, V \in {\mathbf{I}}_{n+1} (Z), \\ & \, \, T_2 -T_1 =U + \partial V \Bigr\}. \end{aligned} \end{align}\]
With the definition of flat convergence on a general metric space in hand we are ready to define integral current spaces which are the spaces for which Sormani-Wenger intrinsic flat distance is defined. One should see C. Sormani and S. Wenger Sormani-Wenger? for more details. An \(n\)-dimensional integral current space \((X, d, T)\) consists of a metric space \((X, d)\) and an \(n\)-dimensional integral current defined on the completion of \(X\), \(T\in I_n(\bar{X})\), such that \(\textrm{set}(T)=X\).
We say that an integral current space \((X,d,T)\) is precompact if \(X\) is precompact with respect to \(d\). Given two \(n\)-dimensional integral current spaces, \((X_1, d_1, T_1)\) and \((X_2, d_2, T_2)\), a current preserving isometry between them is a metric isometry \(\varphi: X_1 \to X_2\) such that \(\varphi_\sharp T_1=T_2\). We are now ready to state the definition of the SWIF distance between integral current spaces.
Definition 13 (Sormani-Wenger Sormani-Wenger?). Given two \(n\)-dimensional precompact integral current spaces \((X_1, d_1, T_1)\) and \((X_2, d_2,T_2)\), the Sormani-Wenger Intrinsic Flat distance between them is defined as \[\begin{align} \begin{aligned} d_{\mathcal{F}}&\left( (X_1, d_1, T_1), (X_2, d_2, T_2)\right) \\&=\inf \Bigl\{d_F^Z(\varphi_{1\sharp}T_1, \varphi_{2\sharp}T_2)| \, (Z,d_Z) \text{ complete} ,\, \varphi_j: X_j \to Z \text{ isometries}\Bigr\}. \end{aligned} \end{align}\]
The function \(d_{\mathcal{F}}\) is a distance up to current preserving isometries. We say that a sequence \((X_j,d_j,T_j)\) of \(n\)-dimensional integral current spaces converges in the volume preserving Sormani-Wenger Intrinsic Flat sense, \(\mathcal{VF}\), to \((X,d,T)\) if the sequence converges with respect to the intrinsic flat distance to \((X,d,T)\) and the masses \({\mathbf{M}}(T_j)\) converge to \({\mathbf{M}}(T)\).
In this section we review examples which were explored in AB?, Allen-Null? which demonstrate the importance of the hypotheses in Theorem 2 and Theorem 4 and build intuition for uniform convergence of the null distance function in the static case. Throughout this section \((\mathbb{D}^n,\sigma)\) stands for the closed flat unit disk which is chosen for notational convenience. It should be noted that the conclusions of the examples does not rely heavily on this choice and similar examples will hold for a compact, connected Riemannian manifold \((\Sigma^n,\sigma)\) replacing \((\mathbb{D}^n, \sigma)\).
The first example demonstrates the necessity of the assumption on \(\sigma_j \ge \sigma_{\infty}\) in the main theorems. We will see that without this assumption one can have a sequence of static metrics which converge to a metric space which is not a spacetime.
Example 14. Let \((\mathbb{D}^n,\sigma)\), \(n \ge 2\) be a flat disk, \(f_j:\mathbb{D}^n \rightarrow (0,\infty)\), \(j \ge 2\) a sequence of continuous functions defined radially on \(\mathbb{D}^n\) where \(s\) is the distance in \((\mathbb{D}^n,\sigma)\) to the boundary by \[\begin{align} f_j(s)= \begin{cases} \frac{1}{j} & s\in \left[0,\frac{1}{j}\right] \\k_j(s)& s \in \left[\frac{1}{j},\frac{3}{2j}\right] \\1& \text{ otherwise } \end{cases} \end{align}\] where \(k_j\) is any increasing continuous function so that \(k_j(\frac{1}{j})=\frac{1}{j}\), and \(k_j(\frac{3}{2j})=1\). If \(g_j=-dt^2+f_j^2\sigma\) and \(g_0=-dt^2+\sigma\) then we find \(\hat{d}_{t,g_j} \not\rightarrow \hat{d}_{t,g_0}\). Furthermore, we let \(L=([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_0})\) and let \[\begin{align} F: [0,1]\times \partial \mathbb{D}^n \subset L \rightarrow [0,1] \end{align}\] be the map defined by projection onto to the time factor of \(L\). Then if \((P,d_0)=(L, \hat{d}_{t,g_0}/\sim)\) where we identify points by the map \(F\) we can conclude that \[\begin{align} ([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_j}) \rightarrow (P,d_0), \end{align}\] in the Gromov-Hausdorff sense.
The next example demonstrates the necessity of assuming \(h_{\infty} \ge h_j\) since we will see that without this assumption we can have a sequences of static spacetimes converge to a metric space which is not a spacetime. One should note that the next example is equivalent to the previous example as far as null distance is concerned since the null distance is conformally invariant. When we combine these two examples we see the necessity of assuming \(\frac{\sigma_j}{h_j} \ge \frac{\sigma_{\infty}}{h_{\infty}}\) in the main theorems.
Example 15. Let \((\mathbb{D}^n,\sigma)\), \(n \ge 2\) be a flat disk, \(h_j:[0,1]\rightarrow (0,\infty)\), \(j \ge 2\) a sequence of continuous functions defined radially on \(\mathbb{D}^n\) where \(s\) is the distance in \((\mathbb{D}^n,\sigma)\) to the boundary by \[\begin{align} h_j(s)= \begin{cases} j & s\in \left[0,\frac{1}{j}\right] \\\bar{k}_j(s)& s \in \left[\frac{1}{j},\frac{3}{2j}\right] \\1& \text{ otherwise } \end{cases} \end{align}\] where \(\bar{k}_j\) is any decreasing continuous function so that \(\bar{k}_j(\frac{1}{j})=j\) and \(\bar{k}_j(\frac{3}{2j})=1\). If \(g_j=-h_j^2dt^2+\sigma\) and \(g_0=-dt^2+\sigma\) then we find \(\hat{d}_{t,g_j} \not\rightarrow \hat{d}_{t,g_0}\). Furthermore, we let \(L=([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_0})\) and let \[\begin{align} F: [0,1]\times \partial \mathbb{D}^n \subset L \rightarrow [0,1] \end{align}\] be the map defined by projection onto to the time factor of \(L\). Then if \((P,d_0)=(L, \hat{d}_{t,g_0}/\sim)\) where we identify points by the map \(F\) in order to conclude that \[\begin{align} ([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_j}) \rightarrow (P,d_0), \end{align}\] in the Gromov-Hausdorff sense.
Proof. By the conformal invariance of the null distance, if we define \(f_j = \frac{1}{h_j}\) then \(f_j^2g_j = -dt^2+f_j^2\sigma\) is the metric of Example 14. Hence the uniform convergence of this example follows from the uniform convergence of Example 15, which is proved in Allen-Null?. ◻
In the next two examples we will see that the rate of blow up of the sequence is crucial to understanding the geometry of the limit. In particular, in the next example we see that if we have a volume bound but no volume convergence then we should not expect the main theorems of this paper to hold.
Example 16. Let \((\mathbb{D}^n,\sigma)\), \(n \ge 2\) be a flat disk, \(f_j: \mathbb{D}^n \rightarrow (0,\infty)\), \(j \ge 2\) a sequence of continuous functions defined radially on \(\mathbb{D}^n\) by \[\begin{align} f_j(r) \begin{cases} j & r\in \left[0,\frac{1}{j}\right] \\h_j(r)& r \in \left[\frac{1}{j},\frac{3}{2j}\right] \\1& \text{ otherwise } \end{cases} \end{align}\] where \(h_j\) is any decreasing continuous function so that \(h_j(\frac{1}{j})=j\), \(h_j(\frac{3}{2j})=1\), and \(\displaystyle \int_{\frac{1}{j}}^{\frac{3}{2j}} h_jdr\le \frac{C}{j}\). If \(g_j=-dt^2+f_j^2\sigma\) and \(g_0=-dt^2+\sigma\) then we find \(\hat{d}_{t,g_j} \not\rightarrow \hat{d}_{t,g_0}\). Furthermore, we let \(N_1=([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_0})\), \(N_2=([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_0})\), and \[\begin{align} F:[0,1] \times \partial \mathbb{D}^2\subset N_1 \rightarrow [0,1]\times\{0\}\subset N_2 \end{align}\] defined by projection of the second factor. Then if \((N_1\sqcup N_2, \hat{d}_{t,g_0})\) is the disjoint union of \(N_1\) and \(N_2\) with the null distance then we can define the metric space \((P,d_0)=(N_1 \sqcup N_2, \hat{d}_{t,g_0}/\sim)\) where we identify points by the map \(F\) in order to conclude that \[\begin{align} ([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_j}) \rightarrow (P,d_0), \end{align}\] in the Gromov-Hausdorff sense.
In the next example we see that for a sequence of functions blowing up at a critical rate the limit will be Minkowski space with a taxi metric defined on \([0,1]\times [0,1]\) attached to the \(t\)-axis. This example illustrates the difference between Theorem 2 and Theorem 4. Without the \(L^{\frac{p}{2}}\), \(p > n\) bound, but with volume convergence (which is equivalent to \(L^n\) convergence of \(f_j\) below), we see that the GH limit will consist of a metric space which is not a spacetime but the SWIF limit will consist of a spacetime.
Example 17. Let \((\mathbb{D}^n,\sigma)\), \(n \ge 2\) be a flat disk, \(f_j:[0,1]\times \mathbb{D}^n\) a sequence of continuous functions defined radially on \(\mathbb{D}^n\) by \[\begin{align} f_j(r) \begin{cases} \frac{j^{\lambda}}{1+\lambda \ln(j)} & r \in \left[0,\frac{1}{j^{\lambda}}\right] \\ \frac{1}{r(1-\ln(r))} & r \in \left[\frac{1}{j^{\lambda}},\frac{1}{j}\right] \\h_j(r) & r \in \left[\frac{1}{j},\frac{3}{2j}\right] \\1& \text{ otherwise } \end{cases} \end{align}\] where \(\lambda >1\) and \(h_j\) is any decreasing, continuous function so that \(h_j(\frac{1}{j})= \frac{j}{1+\ln(j)}\), \(h_j(\frac{3}{2j})=1\), and \(\displaystyle \int_{\frac{1}{j}}^{\frac{3}{2j}} h_jdr\le \frac{C}{j}\). If \(g_j=-dt^2+f_j^2\sigma\), \(g_0=-dt^2+\sigma\) then we find \(\hat{d}_{t,g_j}\not\rightarrow \hat{d}_{t,g_0}\). Furthermore, we let \(N=([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_0})\), \(L=([0,1]\times [0,1],d_{\text{taxi}}^{\lambda})\), \[\begin{align} d_{\text{taxi}}^{\lambda}((s_1,r_1),(s_2,r_2))=|s_1-s_2|+\lambda|r_1-r_2|, \end{align}\] and \[\begin{align} F: [0,1]\times \{1\}\subset L \rightarrow [0,1] \times \{0\}\subset N \end{align}\] so that \(F(t,1)=(t,0)\). Then if \((N\sqcup L, \bar{d})\) is the disjoint union of \(N\) and \(L\) we can define the metric space \((P,d_0)=(N \sqcup L, \bar{d}/ \sim)\) where we identify points by the map \(F\) in order to conclude that \[\begin{align} ([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_j}) \rightarrow (P,d_0), \end{align}\] in the Gromov-Hausdorff sense. Furthermore, we find \[\begin{align} ([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_j}) \rightarrow ([0,1]\times \mathbb{D}^n,\hat{d}_{t,g_0}), \end{align}\] in the Sormani-Wenger Intrinsic Flat sense.
Proof. The proof of the Gromov-Hausdorff convergence was given in Example 3.6 of Allen-Null?. Here we will verify the Sormani-Wenger Intrinsic Flat convergence by applying Theorem 4. The volume convergence, area bound, and diameter bound of \(f_j^2\sigma\) was established in Example 3.7 of Allen-Sormani-2? and hence we may apply Theorem 4 to conclude SWIF convergence. ◻
Now we explain how to construct a metric space \((Z,d_Z)\) in which we can isometrically embed two spacetimes equipped with the null distance. We will use this metric space to calculate the flat distance between the isometric images of two spacetimes \(L_1\) and \(L_2\). This will give us an upper bound on the intrinsic flat distance \(d_{\mathcal{F}} (L_1, L_2)\). One should notice that the results of this section work for any globally hyperbolic spacetime and any time function which is differentiable with respect to \(t\). Hence we are providing the theoretical tools necessary to estimate the intrinsic flat distance required to address Conjecture 7.
Definition 18. Let \(M\) be a compact connected manifold, \(L=M \times [t_0,t_1]\), and \(W \subset L\) a closed set. Let \[\begin{align} Z : = \left( L \times [0,H] \right) \sqcup (L\times \{H+1\})|_\sim \end{align}\] where we identify \((x,H) \sim (x,H+1)\) for all \(x \in W\). Let \(g_1=\sigma_1-h_1^2dt^2\), \(g_2=\sigma_2-h_2^2dt^2\) be two Lorentzian metrics where \(\sigma_1,\sigma_2\) are Riemannian metrics for each time \(t \in [t_0,t_1]\).
Now we define a special class of curves on \(Z\), denoted by \(\mathcal{C}\), so that the curve \(\gamma:[0,T]\rightarrow Z\) is in \(\mathcal{C}\) if for \(z_1=(\ell_1,h_1),z_2=(\ell_2,h_2) \in Z\) we find \(\gamma(s)=(\alpha(s),h(s))\), \(s \in [0,T]\) where \(h(s)\in[0,H]\cup\{H+1\}\), \(h(0)=h_1\), \(h(T)=h_2\), and \(\alpha(s)\in L\) is any piecewise smooth curve joining \(\ell_1\) to \(\ell_2\) so that \(\alpha(s)\) is a piecewise causal curve with respect to \(g_1\) if \(h(s)=0\) and \(g_2\) if \(h(s) \in (0,H]\cup \{H+1\}\).
If we let \(\tau:L \rightarrow \mathbb{R}\) be a differentiable time function then we can define the length function for any \(\gamma \in \mathcal{C}\) \[\begin{align} L_Z(\gamma)=\int_0^T |d\tau(\alpha'(s))|+|h'(s)|ds, \end{align}\] and the corresponding distance function \(d_Z: Z \times Z \to [0, \infty)\) by \[\begin{align} d_Z(z_1, z_2) = \inf \{L_Z(\gamma): \gamma \in \mathcal{C}, \, \gamma(0)=z_1,\, \gamma(1)=z_2\}. \end{align}\]
Lastly, define functions \(\varphi_1: L \to Z\) and \(\varphi_2: L \to Z\) by \[\begin{align} \varphi_1(x) = & (x, 0) \\ \varphi_2(x) = & \begin{cases} (x,H+1) & x \notin W \\ (x, H) & \textrm{otherwise.} \end{cases} \end{align}\]
Now we give some estimates on the metric space \((Z,d_Z)\) which will allow us to show that \(\varphi_1\) and \(\varphi_2\) isometrically embed \((L,\hat{d}_{t,g_1})\) and \((L,\hat{d}_{t,g_2})\) into \(Z\), respectively.
Lemma 19. For \((Z, d_Z), \tau\) as in Definition 18, \((Z,d_Z)\) is a complete metric space and for all \((\ell,h),(\ell',h') \in L\times[0,H] \subset Z\), \[\begin{align} \label{dZEstimatetog950} d_Z((\ell,h),(\ell',h'))\ge \hat{d}_{\tau,g_1}(\ell,\ell') +|h-h'| \end{align}\qquad{(6)}\] and \[\begin{align} \label{region-dist-dec-to-Z} d_Z((\ell,h),(\ell',h')) \le \hat{d}_{\tau,g_2}(\ell,\ell')+|h-h'|. \end{align}\qquad{(7)}\] Furthermore, if \(\operatorname{Diam}(L,\hat{d}_{\tau,g_2}) \le D\), \[\begin{align} \label{Z-Metric-Inequality-Assumption321} \frac{\sigma_2(v,v)}{h_2^2} &\ge \frac{\sigma_{1}(v,v)}{h_{1}^2}, \quad \forall (p,t) \in L, v \in T_pM, \end{align}\qquad{(8)}\] and for all \(x,y \in W\), \(W \subset L\) closed, we assume that \[\begin{align} \label{eq-distCond0} \hat{d}_{\tau,g_2}(x,y) \le \hat{d}_{\tau,g_1}(x,y) +2 \delta \end{align}\qquad{(9)}\] for some \(\delta>0\) and we choose \(H \ge \delta\), then \(\varphi_1: (L,\hat{d}_{\tau,g_1}) \to (Z,d_Z)\) and \(\varphi_2: (L,\hat{d}_{\tau,g_2}) \to (Z,d_Z)\) of Definition 18 are distance preserving.
We note that \(H\) is chosen to prevent having shorter paths between pairs of points either in \((L,\hat{d}_{\tau,g_1})\) or \((L,\hat{d}_{\tau,g_2})\) seen as subsets of \(Z\) than the ones in \((L,\hat{d}_{\tau,g_1})\) or \((L,\hat{d}_{\tau,g_2})\), themselves.
Proof. Let \(C(s)=(\gamma(s),h(s))\), \(s \in [0,1]\), be a curve connecting \((\ell,h),(\ell',h') \in L \times [0,H]\subset Z\) where it is enough to assume that \(C(s) \subset L \times [0,H]\). If we let \(\mathcal{C}_1\) be the set of all curves \(C\) so that \(\gamma\) is piecewise causal with respect to \(g_1\), \(\mathcal{C}_2\) be the set of all curves \(C\) so that \(\gamma\) is piecewise causal with respect to \(g_2\), and \(\mathcal{C}\) be the set of all curves \(C\) so that \(\gamma\) is piecewise causal with respect to \(g_1\) if \(h(s)=0\) and \(\gamma\) is piecewise causal with respect to \(g_2\) if \(h(s) \in (0,H]\) then by ?? we see that \[\begin{align} \label{eq-Curve32Class32Comparison} \mathcal{C}_2 \subset \mathcal{C} \subset \mathcal{C}_1. \end{align}\tag{6}\] Hence by taking infimums over these three classes of curves with respect to the length \(L_Z\) defined in Definition 18 we find ?? and ?? .
By ?? we see that \(\varphi_1(L)\) is isometrically embedded in \((Z,d_Z)\) since by ?? it will always be shorter to consider a curve connecting points in \(\varphi_1(L)\) which lies completely inside \(\varphi_1(L)\).
Now we would like to show that \(\varphi_2(L)\) is isometrically embedded in \((Z,d_Z)\). By choosing \(H\ge \delta\) we will show that for points in \(W \times \{H\}\) it is never more efficient to take advantage of shortcuts in \(L \times \{0\}\subset Z\). To see this consider \(C\) connecting \((p,H)\) to \((p',0)\) to \((q',0)\) and then to \((q,H)\). We see that by ?? , ?? , and 6 that \[\begin{align} L_Z(C)&\ge \hat{d}_{\tau,g_2}(p,p')+H+\hat{d}_{\tau,g_1}(p',q')+\hat{d}_{\tau,g_2}(q',q)+H \\&\ge \hat{d}_{\tau,g_1}(p,p')+\hat{d}_{\tau,g_1}(p',q')+\hat{d}_{\tau,g_1}(q',q)+2H \\&\ge \hat{d}_{\tau,g_1}(p,q)+2H \\&\ge\hat{d}_{\tau,g_1}(p,q)+2\delta \ge \hat{d}_{\tau,g_2}(p,q),\label{eq-Last32Distance32Estimate32Z} \end{align}\tag{7}\] where we used ?? in 7 . Hence for any curve \(C\) connecting points \((p,H),(q,H) \in W \times \{H\}\) we find \[\begin{align} L_{Z}(C) \ge \hat{d}_{\tau,g_2}(p,q).\label{LengthDistanceInequality1} \end{align}\tag{8}\]
Then by the fact that points in \((L\setminus W,H+1)\) are not glued to \(L \times \{H\}\), and hence must enter \(W\times \{H\}\) before attempting to take advantage of shortcuts in \(L \times\{0\}\subset Z\), we are able to conclude that for points \((p,H+1),(q,H+1) \in (L \setminus W,H+1)\) or \((p,H+1) \in (L \setminus W,H+1)\), \((q,H) \in (W,H)\) or \((p,H) \in (W,H+1)\), \((q,H+1) \in (L \setminus W,H+1)\) and curves \(C\) connecting them \[\begin{align} L_{Z}(C) \ge \hat{d}_{\tau,g_2}(p,q).\label{LengthDistanceInequality2} \end{align}\tag{9}\]
Now for \(p,q \in L\), by taking a curve \(C \subset \varphi_2(L)\) connecting \(\varphi_2(p),\varphi_2(q)\) whose length is within \(\varepsilon>0\) of the distance \(\hat{d}_{\tau,g_2}(p,q)\) and then taking \(\varepsilon\rightarrow 0\) we can combine with 8 and 9 to find \[\begin{align} d_Z( \varphi_2(p),\varphi_2(q) ) = \hat{d}_{\tau,g_2}(p,q). \end{align}\] Hence we can conclude that \(\varphi_2(L)\) is isometrically embedded in \((Z,d_Z)\), as desired. ◻
Now we can calculate the flat distance between \(\varphi_{1_\sharp}[[L]]\) and \(\varphi_{2_\sharp}[[L]]\).
Theorem 20. Let \(M\) be a compact, connected, and oriented manifold, \(L=M \times [t_0,t_1]\), and \(\tau:L\rightarrow \mathbb{R}\) a time function which is differentiable with respect to time. Let \(g_1=\sigma_1-h_1^2dt^2\) and \(g_2=\sigma_2-h_2^2dt^2\) be two Lorentzian metrics so that \(\operatorname{Diam}(L,\hat{d}_{\tau,g_2}) \le D\), \[\begin{align} \label{Z-Metric-Inequality-Assumption} \frac{\sigma_2(v,v)}{h_2^2} &\ge \frac{\sigma_1(v,v)}{h_{1}^2}, \quad \forall (p,t) \in L, v \in T_pM. \end{align}\qquad{(10)}\] If \(\mathcal{H}_{\hat{d}_{\tau,g_2}}^{n+1}(L)\le V\), \(\mathcal{H}_{\hat{d}_{\tau,g_2}}^{n}(\partial L) \leq A\), \(W \subset L\) is a closed set, \[\begin{align} \label{eq-volCond} \mathcal{H}_{\hat{d}_{\tau,g_2}}^{n+1}( M \setminus W) \le V' \end{align}\qquad{(11)}\] and assume that there exists a \(\delta > 0\) so that for all \(x,y \in W\), \[\begin{align} \label{eq-distCond} \hat{d}_{\tau,g_2}(x,y) \le \hat{d}_{\tau,g_1}( x, y) +2 \delta \end{align}\qquad{(12)}\] and that \(H \ge \delta\). Then \[\begin{align} \label{Fest} d^Z_{F}( \varphi_1(L), \varphi_2(L)) \le 2C_{n+1}V' + C_{n+2}H V + C_{n+1} H A \end{align}\qquad{(13)}\] where \(\varphi_1,\varphi_2\) are the maps, and \(Z\) is the metric space described in Definition 18.
Proof. Apply Lemma 18 with \(H \ge \delta\) to get a metric space \((Z,d_Z)\) and distance preserving maps \(\varphi_1: L \to Z\) and \(\varphi_2 : L \to Z\). Our goal will be to define integral currents \(T \in {\mathbf{I}}_{m+1}(Z)\) and \(T' \in {\mathbf{I}}_m(Z)\) such that \[\begin{align} \varphi_{2\#}[[L]]- \varphi_{1\#}[[ L]] & = \partial T + T', \\ {\mathbf{M}}(T) & \leq C_{n+2}H V, \\ {\mathbf{M}}(T') & \leq C_{n+1}(2V' +H A). \end{align}\] Then by the definition of flat convergence \[\begin{align} d^Z_{F}( \varphi_2(L), \varphi_1( L)) \le & {\mathbf{M}}(T) + {\mathbf{M}}(T') \end{align}\] and by the mass estimates we will find (?? ).
To this end, since \(\varphi_1\) and \(\varphi_2\) are distance preserving maps, \[\begin{align} \varphi_{1\#}[[ L_1]] & = & [[L \times \{0\}]], \\ \varphi_{2\#}[[ L_2]] & = & [[W \times \{H \} ]] + [[L \setminus W\times\{H+1\}]], \end{align}\] where we are using the notation for a current on a globally hyperbolic spacetime as introduced in Example [ex-Globally32Hyperbolic32Integral32Current32Space]. Then define \[\begin{align} T= & [[ \,L \times [0,H] \,]] \in {\mathbf{I}}_{m+1}(Z),\tag{10} \\ T' = & [[L \setminus W \times \{H+1\}]] - [[ (L \setminus W ) \times \{H \} ]] ,\tag{11} \\&\quad- [ [\, \partial L\times [0,H] \,]] \in {\mathbf{I}}_{m}(Z), \tag{12} \end{align}\] where we are using standard notation for the product of integral current spaces. If we compute the boundary \[\begin{align} \label{eq-DefofBoundaryofT} \partial T = & [[ L \times \{H \} ]] - [[L\times \{0\}]] + [ [\, \partial L \times [0,H] \,]]. \end{align}\tag{13}\] then we can conclude by 10 , 11 , 12 , and 13 that \[\begin{align} \varphi_{2\#}[[L]]- \varphi_{1\#}[[ L]] = \partial T + T'. \end{align}\] Now by the definition of \(T\) and \(T'\), if we define for \((x,h),(x',h') \in Z\) the metric \[\begin{align} d_{Z'}((x,h),(x',h'))= \hat{d}_{\tau,g_2}(x,x') +|h-h'| \end{align}\] then we can calculate \[\begin{align} {\mathbf{M}}(T) \leq & C_{n+2} \mathcal{H}_{d_Z}^{n+2}(L\times [0,H])\\ \leq & C_{n+2} \mathcal{H}_{d_{Z'}}^{n+2}(L\times [0,H])\\ \leq & C_{n+2} H\mathcal{H}_{\hat{d}_{\tau,g_2}}^{n+1}(L), \end{align}\] \[\begin{align} {\mathbf{M}}(T') &\leq C_{n+1} \mathcal{H}_{d_Z}^{n+1}(L \setminus W \times \{H+1\}) \\&\qquad+C_{n+1} \mathcal{H}_{d_Z}^{n+1}((L \setminus W ) \times \{H \}) \\&\qquad +C_{n+1} \mathcal{H}_{d_Z}^{n+1}(\partial L\times [0,H]) \\&\leq C_{n+1} \mathcal{H}_{d_{Z'}}^{n+1}(L \setminus W \times \{H+1\}) \\&\qquad+C_{n+1} \mathcal{H}_{d_{Z'}}^{n+1}((L \setminus W ) \times \{H \}) \\&\qquad +C_{n+1} \mathcal{H}_{d_{Z'}}^{n+1}(\partial L\times [0,H]) \\&\le 2C_{n+1}\mathcal{H}_{\hat{d}_{\tau,g_2}}^{n+1}(L \setminus W )+C_{n+1} H\mathcal{H}_{\hat{d}_{\tau,g_2}}^{n}(\partial L). \end{align}\] The desired estimate then follows by using the assumed Hausdorff measure bounds. ◻
In this section we provide the proofs of the main theorems. We start with a theorem which establishes uniform convergence from below. One should note that the proof given here is similar to the proof given for the corresponding result for warped products, Theorem 4.1 of Allen-Null?.
Theorem 21. Let \(M^n\) be a compact, connected manifold, \(\sigma_j\) a sequence of continuous Riemannian manifolds on \(M\), \(L=[t_0,t_1]\times \Sigma\), \(h_j:M\rightarrow (0,\infty)\), and \(g_j=-h_j^2dt^2+\sigma_j\), \(j \in \mathbb{N}\cup\{\infty\}\). Then if \[\begin{align} \frac{\sigma_j(x)(v,v)}{h_j^2} &\ge \left(1-\frac{1}{j}\right) \frac{\sigma_{\infty}(x)(v,v)}{h_{\infty}^2},\quad \forall x \in M, v \in T_xM. \label{eq-Lower32Bound32Assumption} \end{align}\qquad{(14)}\] then \[\begin{align} \hat{d}_{t,g_j} \ge \hat{d}_{t,g_{\infty}}-C(j), \end{align}\] where \(C(j) \ge 0\) and \(C(j) \rightarrow 0\) as \(j \rightarrow \infty\).
Proof. First we notice that if we define \(\bar{g}_j=-dt^2+\frac{\sigma_j}{h_j^2}\) then by the conformal invariance of the null distance we see that \[\begin{align} \label{eq-Conformal32Invariance32Consequence} \hat{d}_{t,g_j}=\hat{d}_{t,\bar{g}_j}. \end{align}\tag{14}\]
Define the metric \(\tilde{g}_{j,\infty}=-dt^2+\left(1-\frac{1}{j}\right) \frac{\sigma_{\infty}}{h_{\infty}}\) and notice that by ?? every piecewise causal curve with respect to \(\bar{g}_j\) is a piecewise causal curve with respect to \(\tilde{g}_{j,\infty}\). Hence, since the null distance is defined as the infimum of the length of all piecewise causal curves and the length only depends on the time function we find that \[\begin{align} \hat{d}_{t,\bar{g}_j}(p,q) \ge \hat{d}_{t,\tilde{g}_{j,\infty}}(p,q), \quad \forall p,q \in L. \end{align}\]
Then we notice that \(\tilde{g}_{j,\infty} \rightarrow \bar{g}_{\infty}\) uniformly so by Theorem 1.4 of AB?, we see that \[\begin{align} \hat{d}_{t,\tilde{g}_{j,\infty}} \rightarrow \hat{d}_{t,\bar{g}_{\infty}}, \end{align}\] uniformly, which implies the desired result by the conformal invariance observation in 14 ◻
The following lemma was proved in AB? except for the fact that there the time coordinate was in all of \(\mathbb{R}\). Here we just notice that restricting \(t \in [t_0,t_1]\) does not change the conclusion and we slightly change the expression for the metric to highlight the distance function with respect to \(\sigma\).
Lemma 22 (Lemma 4.4 of AB?). Let \(M^n\) be a compact, connected manifold, \(\sigma\) a continuous Riemannian manifold on \(M\), \(L=[t_0,t_1]\times M\), and \(g=-dt^2+\sigma\). Then if \(x,y \in M\), \(t,s \in [t_0,t_1]\) we find \[\begin{align} \hat{d}_{t,g}((t,x),(s,y))=d_{\sigma}(x,y)+\max\{0,|t-s|-d_{\sigma}(x,y)\}. \end{align}\]
Proof. Fix \(p=(t,x),q=(s,y) \in L\) and let \(\gamma\) be a piecewise smooth curve connecting \(x\) to \(y\), parameterized with respect to \(\sigma\) arc length. Without loss of generality we may assume that \(t \le s\). Consider the null curve with respect to \(g\) \[\begin{align} \alpha(\tau)= \left(t+\int_0^{\tau} |\gamma'(\eta)|_{\sigma}d\eta, \gamma(\tau)\right), \end{align}\] on \(\mathbb{R}\times M\) where we note that \(\alpha\) may not fit inside of \(L\). Let \(\tau' \in [0,\infty)\) be such that \(\alpha(\tau')=(t',y)\) for some \(t'\in[t_0,\infty)\) and notice that \(\tau'\ge d_{\sigma}(x,y)\).
If \(t'\le s\) then \(q \in I^+(p)\) and we can build a new curve \[\begin{align} \tilde{\alpha}(\tau)= \left(t+\frac{|s-t|}{|t'-t|}\int_0^{\tau} |\gamma'(\eta)|_{\sigma}d\eta, \gamma(\tau)\right), \end{align}\] so that \[\begin{align} \hat{d}_{t,g}(p,q)&\le \hat{L}_{t,g}(\tilde{\alpha}) = |t-s| = d_{\sigma}(x,y)+\max\{0, |t-s|-d_{\sigma}(x,y)\}. \end{align}\] If \(t' > s\) and \(\alpha\) fits inside of \(L\) then we we can modify \(\alpha\) by breaking into a piecewise null curve with two pieces. If \(\alpha\) does not fit inside \(L\) then we can modify \(\alpha\) to be a piecewise null curve \(\tilde{\alpha}\), with \(k\) pieces. In either case this can be done so that on each piece defined on \([\tau_i,\tau_{i+1}]\) where \(0\le\tau_i<\tau_{i+1}\le\tau'\) and \(1\le i \le k\) where \(\tilde{\alpha}\) is a null curve it is equal to \[\begin{align} \tilde{\alpha}(\tau)= \left(\tau_i\pm\int_0^{\tau} |\gamma'(\eta)|_{\sigma}d\eta, \gamma(\tau)\right). \end{align}\] To complete the proof we can follow the rest of the proof of Lemma 4.4 of AB?, possibly breaking up curves which do not fit into \(L\) as was done above. ◻
We now prove a simple observation comparing the distance between points in \(M\) to points in \(L\) which will be useful for proving the main theorems of this paper.
Lemma 23. Let \(M^n\) be a compact, connected manifold, \(\sigma_1,\sigma_2\) continuous Riemannian manifolds on \(M\), \(L=[t_0,t_1]\times M\), and \(g_i=-dt^2+\sigma_i\), \(i \in \{1,2\}\). Then if \(x,y \in M\), \(t,s \in [t_0,t_1]\) so that \[\begin{align} |d_{\sigma_1}(x,y)-d_{\sigma_2}(x,y)| <K, \quad \forall x, y \in W \subset M, \end{align}\] then \[\begin{align} |\hat{d}_{t,\sigma_1}((x,t),(y,s))-\hat{d}_{t,\sigma_2}((x,t),(y,s))| <2K, \end{align}\] \(\forall (x,t), (y,s) \in W \times [t_0,t_1]\).
Proof. We can calculate by Lemma 22 \[\begin{align} |\hat{d}_{t,\sigma_1}&((x,t),(y,s))-\hat{d}_{t,\sigma_2}((x,t),(y,s))| \\&\le |d_{\sigma_1}(x,y)-d_{\sigma_2}(x,y)| \\& \quad + |\max\{0, |t-s|-d_{\sigma_1}(x,y)\}-\max\{0, |t-s|-d_{\sigma_2}(x,y)\}|, \end{align}\] and hence we need to bound the second term.
If both \(d_{\sigma_1}(x,y),d_{\sigma_2}(x,y)\ge |t-s|\) then the second term is \(0\) and we are done. If both \(d_{\sigma_1}(x,y),d_{\sigma_2}(x,y)< |t-s|\) then \[\begin{align} |\max&\{0, |t-s|-d_{\sigma_1}(x,y)\}-\max\{0, |t-s|-d_{\sigma_2}(x,y)\}| \\&= |d_{\sigma_1}(x,y)-d_{\sigma_2}(x,y)| \le K, \end{align}\] and we are done. If \(d_{\sigma_1}(x,y)< |t-s|\le d_{\sigma_2}(x,y)\) or \(d_{\sigma_2}(x,y)< |t-s|\le d_{\sigma_1}(x,y)\) then for \(i\in\{1,2\}\) equal to the index of the distance which is smaller we find \[\begin{align} |\max&\{0, |t-s|-d_{\sigma_1}(x,y)\}-\max\{0, |t-s|-d_{\sigma_2}(x,y)\}| \\&= ||t-s|-d_{\sigma_i}(x,y)| \\&\le |d_{\sigma_1}(x,y)-d_{\sigma_2}(x,y)|\le K. \end{align}\] ◻
We now make an observation that a Hölder bound between two Riemannian manifolds extends to a Hölder bound between the corresponding static spacetimes.
Lemma 24. Let \(0<C\le C'\), \(M^n\) be a compact, connected manifold, \(\sigma_1,\sigma_2\) continuous Riemannian manifolds on \(M\), \(L=[t_0,t_1]\times M\), and \(g_i=-dt^2+\sigma_i\), \(i \in \{1,2\}\). Then if \(x,y \in M\), \(t,s \in [t_0,t_1]\), \(\alpha \in (0,1]\) so that \[\begin{align} d_{\sigma_1}(x,y) <C d_{\sigma_2}(x,y)^{\alpha}, \quad \forall x, y \in M, \end{align}\] then \[\begin{align} \hat{d}_{t,\sigma_1}((t,x),(s,y)) <C' \hat{d}_{t,\sigma_2}((t,x),(s,y))^{\alpha}, \end{align}\] \(\forall (t,x), (s,y) \in [t_0,t_1]\times M\).
Proof. If both \(d_{\sigma_1}(x,y),d_{\sigma_2}(x,y)\ge |t-s|\) then the second term in the distance estimate of Lemma 22 is \(0\) and we are done. If both \(d_{\sigma_1}(x,y),d_{\sigma_2}(x,y)< |t-s|\) then by Lemma 22 we find \[\begin{align} \hat{d}_{t,\sigma_1}((t,x),(s,y))&=|t-s| \\\hat{d}_{t,\sigma_2}((t,x),(s,y))^{\alpha}&=|t-s|^{\alpha} \end{align}\] and since \(|t-s|\le |t_0-t_1|\) we can pick a constant \(C' \ge C\) so that the desired claim is true in this case. If \(d_{\sigma_1}(x,y)< |t-s|\le d_{\sigma_2}(x,y)\) then by Lemma 22 we find \[\begin{align} \hat{d}_{t,\sigma_1}((t,x),(s,y))&= |t-s| \\& \le C' |t-s|^{\alpha} \\&=C' d_{\sigma_2}(x,y){\alpha} = C'\hat{d}_{t,\sigma_2}((t,x),(s,y))^{\alpha}. \end{align}\] Lastly, if \(d_{\sigma_2}(x,y)< |t-s|\le d_{\sigma_1}(x,y)\) then by Lemma 22 we find \[\begin{align} \hat{d}_{t,\sigma_1}((t,x),(s,y))&=d_{\sigma_1}(x,y) \\& \le Cd_{\sigma_2}(x,y)^{\alpha} \\& \le C|t-s|^{\alpha} = C\hat{d}_{t,\sigma_2}((t,x),(s,y))^{\alpha}. \end{align}\] ◻
In order to complete the proof of Theorem 2 we will need to show pointwise convergence of the sequence of null distances as well as a Hölder distance bound from above. When combined with Lemmas 22, 23, and 24 this will then imply uniform convergence of the null distances.
Proof of Theorem 2. First notice that if \(g_j=-h_{\alpha}^2 dt^2+\sigma_j\) and \(\tilde{g}_j=-dt^2+\tilde{\sigma}_j\) where \(\tilde{\sigma}_j=\frac{\sigma_j}{h_j^2}\) then by the conformal invariance of the null distance we have that \[\begin{align} \hat{d}_{t,g_j}=\hat{d}_{t,\tilde{g}_j}, \end{align}\] and hence we are justified in restricting to the case of \(\tilde{g}_j\) for the rest of the argument.
Now we will show that the assumptions of Theorem 2 imply compactness of the sequence of null distance functions of \(\tilde{g}_j\) in the uniform topology. The first assumption of Theorem 2 implies \[\begin{align} \int_M|\tilde{\sigma}_j|_{\sigma_{\infty}}^{\frac{p}{2}}dV_{\sigma_{\infty}} \le C, \end{align}\] and hence by Theorem 10 we find \[\begin{align} d_{\tilde{\sigma}_j}(q_1,q_2) \le C(M,\sigma_{\infty}) d_{\sigma_{\infty}}(q_1,q_2)^{\frac{p-n}{p}}, \end{align}\] for all \(q_1,q_2 \in M\). Notice that this implies that there exists a \(D \in (0,\infty)\) so that \[\begin{align} \mathop{\mathrm{Diam}}(M,\tilde{\sigma}_j) \le D. \end{align}\] Now by Lemma 24 this implies there exists a \(C' \ge C(M,\sigma_{\infty})\) so that \[\begin{align} \label{eq-Upper32Distance32Bound} \hat{d}_{t,\tilde{g}_j}((s_1,q_1),(s_2,q_2)) \le C' \hat{d}_{t,g_{\infty}}((s_1,q_1),(s_2,q_2))^{\frac{p-n}{p}}, \end{align}\tag{15}\] for all \((s_1,q_1),(s_2,q_2) \in L\). Then by combining 15 with Theorem 21 we see by the Arzela-Ascoli Theorem that a subsequence must converge uniformly to a metric \(d_{\infty}\) on \(L\). We will denote the subsequence in the same way as the sequence. Our goal is to show that \(d_{\infty}=\hat{d}_{t,\tilde{g}_{\infty}}\).
Next we will show that the assumptions of Theorem 2 imply pointwise almost everywhere convergence of \((M,\tilde{\sigma}_j)\). By assumption we know that \[\begin{align} \tilde{\sigma}_j &\ge \left(1-\frac{1}{j}\right)\tilde{\sigma}_{\infty}, \\ \mathop{\mathrm{Vol}}(M, \tilde{\sigma}_j) &\rightarrow \mathop{\mathrm{Vol}}(M,\tilde{\sigma}_{\infty}), \\\mathop{\mathrm{Area}}(\partial M, \tilde{\sigma}_j) &\le A, \\ \mathop{\mathrm{Diam}}(M, \tilde{\sigma}_j)& \le D, \end{align}\] and hence by Theorem 9 we see that \[\begin{align} d_{\tilde{\sigma}_j}(p,q) \rightarrow d_{\tilde{\sigma}_{\infty}}(p,q), \end{align}\] for almost every \(p,q \in M\) with respect to \(\sigma_{\infty}\).
Consider \(p,q \in M\) so that \[\begin{align} \label{Eq-Pointwise32Distance32Convergence} d_{\tilde{\sigma}_j}(p,q)\rightarrow d_{\tilde{\sigma}_{\infty}}(p,q). \end{align}\tag{16}\] Then by Lemma 22 we see that \[\begin{align} \hat{d}_{t,\tilde{g}_j}((s_1,p),(s_2,q))&= d_{\tilde{\sigma}_j}(p,q)+\max\{0,|s_1-s_2|-d_{\tilde{\sigma}_j}(p,q)\}, \\\hat{d}_{t,\tilde{g}_{\infty}}((s_1,p),(s_2,q))&= d_{\tilde{\sigma}_{\infty}}(p,q)+\max\{0,|s_1-s_2|-d_{\tilde{\sigma}_{\infty}}(p,q)\}, \end{align}\] and hence 16 implies that \[\begin{align} \hat{d}_{t,\tilde{g}_j}((s_1,p),(s_2,q))\rightarrow \hat{d}_{t,\tilde{g}_{\infty}}((s_1,p),(s_2,q)). \end{align}\] Since the assumptions of Theorem 2 imply pointwise convergence of \(d_{\tilde{\sigma}_j}\) to \(d_{\tilde{\sigma}_{\infty}}\) for almost every \(p,q \in M\) we see by 16 that \(\hat{d}_{t,\tilde{g}_j}\) converges to \(\hat{d}_{t,\tilde{g}_{\infty}}\) for almost all \((s_1,p),(s_2,q) \in L\).
When we combine the pointwise almost everywhere convergence of \(\hat{d}_{t,\tilde{g}_j}\) to \(\hat{d}_{t,\tilde{g}_{\infty}}\) with the uniform convergence of \(\hat{d}_{t,\tilde{g}_j}\) to \(d_{\infty}\) we see that the subsequence \(\hat{d}_{t,\tilde{g}_j}\) uniformly converges to \(\hat{d}_{t,\tilde{g}_{\infty}}\). Since this is true of every subsequence obtained by compactness we see that the original sequence must convergence uniformly, as desired. ◻
We now give the proof of Theorem 4 which will apply the Sormani-Wenger Intrinsic Flat estimates of Theorem 20, the previous estimates for Riemannian manifolds of Theorem 11, and Lemmas 22, 23, and 24
Proof of Theorem 4. First notice that if \(g_j=-h_j^2 dt^2+\sigma_j\) and \(\tilde{g}_j=-dt^2+\tilde{\sigma}_j\) where \(\tilde{\sigma}_j=\frac{\sigma_j}{h_j^2}\) then by the conformal invariance of the null distance we have that \[\begin{align} \hat{d}_{t,g_j}=\hat{d}_{t,\tilde{g}_j}, \end{align}\] and hence we are justified in restricting to the case of \(\tilde{g}_j\) for the rest of the argument. By assumption we know that if we rescale \(\hat{\sigma}_j = \left(1-\frac{1}{j}\right)^{-1} \tilde{\sigma}_j\) then we find \[\begin{align} \hat{\sigma}_j &\ge\tilde{\sigma}_{\infty}, \\ \mathop{\mathrm{Vol}}(M, \hat{\sigma}_j) &\rightarrow \mathop{\mathrm{Vol}}(M,\tilde{\sigma}_{\infty}), \\ \mathop{\mathrm{Diam}}(M, \hat{\sigma}_j)& \le D, \end{align}\] and hence by Theorem 11 that for any \(\lambda \in (0, \operatorname{Diam}(M, \tilde{\sigma}_{\infty}))\) and \(\kappa >1\), there exists a measurable set \(W_{\lambda,\kappa} \subset M\) and a constant \(\delta_{\lambda, \kappa,j}>0\) such that for all \(p_1, p_2 \in W_{\lambda, \kappa}\) \[\begin{align} \label{eq-Riemannian32Distance32Estimate} |d_{\hat{\sigma}_j}(p_1,p_2)-d_{\tilde{\sigma}_0}(p_1,p_2)| < 2 \lambda + 2\delta_{\lambda,\kappa,j}, \end{align}\tag{17}\] where \(\delta_{\lambda,\kappa,j} \to 0\) as \(j \to \infty\), and \[\begin{align} \label{eq-Riemannian32Volume32Estimate} \mathop{\mathrm{Vol}}(M \setminus W_{\lambda, \kappa},\hat{\sigma}_j) \le \frac{1}{\kappa}\mathop{\mathrm{Vol}}(M,\tilde{\sigma}_0)+|\mathop{\mathrm{Vol}}(M,\hat{\sigma}_j)-\mathop{\mathrm{Vol}}(M,\tilde{\sigma}_0)|. \end{align}\tag{18}\]
By Lemma 23 we see that 17 implies
\[\begin{align} \label{eq-Riemannian32Distance32Estimate322} |\hat{d}_{t,\hat{g}_j}((s_1,p_1),(s_2,p_2))-\hat{d}_{\tilde{g}_0}((s_1,p_1),(s_2,p_2))| < 4 \lambda + 4\delta_{\lambda,\kappa,j}, \end{align}\tag{19}\] for all \((s_1, p_1),(s_2, p_2) \in [t_0,t_1]\times W_{\lambda, \kappa}\) where \(\hat{g}_j = -dt^2+\hat{\sigma}_j\).
Now notice by Lemma 22 we see that \[\begin{align} \label{eq-Euclidean32Overestimate} \hat{d}_{t,\hat{g}_j}((s_1, p_1),(s_2, p_2)) &\le \sqrt{d_{\hat{\sigma}_j}(p_1,p_2)^2+|s_1-s_2|^2} \\&:= d_{\mathbb{E}_{\hat{\sigma}_j}} ((s_1, p_1),(s_2, p_2)). \end{align}\tag{20}\] So by combining 20 with 18 we find \[\begin{align} \mathcal{H}^{n+1}_{\hat{d}_{t,\hat{g}_j}}([t_0,t_1]\times M\setminus W_{\lambda, \kappa}) &\le \mathcal{H}^{n+1}_{d_{\mathbb{E}_{\hat{\sigma}_j}}}([t_0,t_1]\times M\setminus W_{\lambda, \kappa}) \\& \le \mathcal{H}^{n}_{d_{\hat{\sigma}_j}}(M\setminus W_{\lambda, \kappa})\mathcal{H}^{1}([t_0,t_1]) \\&\le \mathop{\mathrm{Vol}}(M\setminus W_{\lambda, \kappa},\hat{\sigma}_j)|t_1-t_0| \\&\le \left(\frac{\mathop{\mathrm{Vol}}(M,\tilde{\sigma}_j)}{\kappa}+C_j\right)|t_1-t_0|, \end{align}\] where \(C_j \searrow 0\) as \(j \rightarrow \infty\). Similarly we can compute the bounds \[\begin{align} \mathcal{H}^{n+1}(L)&\le \mathcal{H}^{n+1}_{d_{\mathbb{E}_{\hat{\sigma}_j}}}([t_0,t_1]\times M ) \\&\le \mathcal{H}^{n}_{d_{\hat{\sigma}_j}}(M)\mathcal{H}^{1}([t_0,t_1]) \le C\mathop{\mathrm{Vol}}(M,\tilde{\sigma}_{\infty}) |t_1-t_0|:=\bar{V}, \\\mathcal{H}^{n}(\partial L)&=\mathcal{H}^{n}_{d_{\mathbb{E}_{\hat{\sigma}_j}}}( [t_0,t_1]\times \partial M )+ \mathcal{H}^{n}_{d_{\mathbb{E}_{\hat{\sigma}_j}}}( \{t_0,t_1\}\times M ) \\&\le \mathcal{H}^{n-1}_{d_{\hat{\sigma}_j}}(\partial M)\mathcal{H}^{1}([t_0,t_1]) + 2\mathcal{H}^{n}_{d_{\hat{\sigma}_j}}(M) \\&\le A |t_1-t_0|+2\mathop{\mathrm{Vol}}(M,\tilde{\sigma}_{\infty})C:=\bar{A}. \end{align}\] This implies for \(H = 4 \lambda + 4\delta_{\lambda,\kappa,j}\) that by applying Theorem 20 with the previous estimates we find \[\begin{align} \label{Flat32Distance32Estimate} d^Z_{F}( \varphi_0(L), \varphi_j(L)) &\le 2C_{n+1}\left(\frac{\mathop{\mathrm{Vol}}(M,\tilde{\sigma}_0)}{\kappa}+C_j\right)|t_1-t_0| \\&\quad +\left(4 \lambda + 4\delta_{\lambda,\kappa,j}\right) \left(C_{n+2} \bar{V} + C_{n+1} \bar{A} \right) \end{align}\tag{21}\] where \(\varphi_j,\varphi_0\) are the maps, and \(Z\) is the metric space described in Definition 18. First by taking the limsup on both sides we find \[\begin{align} \label{cyawoner} \limsup_{j\rightarrow \infty}d^Z_{F}( \varphi_0(L), \varphi_j(L)) &\le 2C_{n+1}\frac{\mathop{\mathrm{Vol}}(M,\tilde{\sigma}_0)}{\kappa}|t_1-t_0| \\&\quad +4 \lambda \left(C_{n+2}\bar{V} + C_{n+1} \bar{A} \right). \end{align}\tag{22}\] Now by taking \(\lambda\rightarrow 0\) and \(\kappa \rightarrow \infty\) we see that \[\begin{align} \label{bicjyznp} 0\le \limsup_{j\rightarrow \infty}d^Z_{F}( \varphi_0(L), \varphi_j(L)) &\le 0, \end{align}\tag{23}\] which implies that \[\begin{align} (L,\hat{d}_{t,\hat{g}_j}) \rightarrow (L,\hat{d}_{t,\tilde{g}_0}), \end{align}\] in the Sormani-Wenger Intrinsic Flat sense.
Now we notice by Lemma 23 that \[\begin{align} | \hat{d}_{t,\hat{g}_j}((s_1, p_1),(s_2, p_2))-\hat{d}_{t,\tilde{g}_j}((s_1, p_1),(s_2, p_2))|&\le \frac{4}{j}, \end{align}\] and by definition \(\hat{\sigma}_j \ge \tilde{\sigma}_j\) and hence by Theorem 20 we see that \[\begin{align} d_{\mathcal{F}}((L,\hat{d}_{t,\hat{g}_j}),(L,\hat{d}_{t,\tilde{g}_j}))&\le C_j \end{align}\] where \(C_j \searrow 0\) as \(j \rightarrow \infty\). Now by the triangle inequality for the Sormani-Wenger intrinsic flat distance we find \[\begin{align} d_{\mathcal{F}}((L,\hat{d}_{t,\hat{g}_j}),(L,\hat{d}_{t,\tilde{g}_0}))&\le d_{\mathcal{F}}((L,\hat{d}_{t,\tilde{g}_j}),(L,\hat{d}_{t,\hat{g}_j})) \\&\quad +d_{\mathcal{F}}((L,\hat{d}_{t,\hat{g}_j}),(L,\hat{d}_{t,\tilde{g}_0})), \end{align}\] which implies that \[\begin{align} (L,\hat{d}_{t,\tilde{g}_j}) \rightarrow (L,\hat{d}_{t,\tilde{g}_0}), \end{align}\] in the Sormani-Wenger Intrinsic Flat sense, as desired. ◻