[Paper Review] Oberwolfach Report: Spacetime Intrinsic Flat Convergence

Christina Sormani|arXiv (Cornell University)|May 22, 2018

Cosmology and Gravitation Theories11 references3 citations

TL;DR

This paper introduces spacetime intrinsic flat (SF) convergence by defining a null distance function using a cosmological time function, enabling the application of intrinsic flat convergence to Lorentzian spacetimes. The method ensures causality is preserved in the limit, allowing compactness theorems and stability results for big bang and black hole spacetimes.

ABSTRACT

This Oberwolfach report describes published work with Carlos Vega introducing the null distance on a spacetime, and announces unpublished applications of this work jointly with Carlos Vega and with Anna Sakovich applying the null distance to define a spacetime intrinsic flat convergence (SF convergence). The announced work with Vega concerns convergence of sequences of big bang spacetimes and the announced work with Anna Sakovich concerns convergence of future maximal developments of initial data sets. Both were originally announced at the 2018 JMM. This report lists some open questions that will be discussed at the Oberwolfach meeting this coming August.

Motivation & Objective

• To develop a notion of convergence for spacetimes analogous to intrinsic flat convergence in Riemannian geometry, suitable for sequences with long thin gravity wells or singularities.
• To address the challenge of applying intrinsic flat convergence to Lorentzian manifolds, which lack a metric space structure and cannot use standard isometric embeddings.
• To define a spacetime intrinsic flat distance using a null distance function derived from a regular cosmological time function, ensuring causality is preserved in the limit.
• To establish conditions under which the null distance defines a metric space that encodes causality, enabling the use of integral current spaces in Lorentzian geometry.
• To prove compactness theorems for sequences of big bang and maximal development spacetimes under SF convergence, with limits that are integral current spaces with causal structure.

Proposed method

• Define a null distance function $\hat{d}_{\tau}(p,q) = \inf_{\beta} \sum_{i=1}^{k} |\tau(\beta(t_i)) - \tau(\beta(t_{i+1}))|$, where $\beta$ ranges over piecewise causal curves from $p$ to $q$, using a time function $\tau$.
• Use a regular cosmological time function $\tau$ (finite on $M$, converging to 0 along past inextensible curves) to ensure $\hat{d}_{\tau}$ defines a metric space.
• Prove that $\hat{d}_{\tau}$ encodes causality: $p$ is in the future of $q$ iff $\hat{d}_{\tau}(p,q) = \tau(p) - \tau(q)$, which implies definiteness and metric structure.
• Construct a metric space $ (X, \hat{d}_{\tau}) $ from a spacetime with a regular $\tau$, enabling the definition of an integral current space structure.
• Define the spacetime intrinsic flat distance $ d_{\mathcal{SF}} $ as the $\mathcal{F} $-distance between the resulting integral current spaces, using isometric embeddings into a common metric space $Z$.
• Apply the theory to big bang spacetimes (e.g., FLRW) and future maximal developments, proving that the big bang point $p_{BB}$ exists in the metric completion and $\tau(p) = \hat{d}_{\tau}(p_{BB}, p)$.

Experimental results

Research questions

• RQ1 Which Lorentzian manifolds admit a regular cosmological time function that allows the null distance $\hat{d}_{\tau}$ to define a metric space?
• RQ2 Under what conditions does the null distance $\hat{d}_{\tau}$ encode causality, ensuring $\hat{d}_{\tau}(p,q) = \tau(p) - \tau(q)$ iff $p$ is in the future of $q$?
• RQ3 Do future maximal developments of initial data sets have a cosmological time function that vanishes on the initial Cauchy surface and yields a null distance that encodes causality?
• RQ4 Can the $\mathcal{SF}$ convergence framework be used to prove compactness theorems for sequences of black hole spacetimes with mass tending to zero?
• RQ5 Are there alternative canonical time functions—such as those based on maximal slicing or other geometric flows—that are better suited for studying future maximal developments of the Einstein equations?

Key findings

• The null distance $\hat{d}_{\tau}$ defines a metric space when $\tau$ is a regular cosmological time function, as defined by Anderson-Howard-Galloway.
• When $\hat{d}_{\tau}$ encodes causality, it is definite and thus defines a proper metric, ensuring the space is suitable for intrinsic flat convergence.
• In FLRW big bang spacetimes with $\tau = t$, the metric completion $\bar{X}$ contains a unique big bang point $p_{BB}$ such that $\tau(p) = \hat{d}_{\tau}(p_{BB}, p)$.
• For such spacetimes, the pointed $\mathcal{F}$-convergence based at $p_{BB}$ is well-defined, and the limit is an integral current space with a causal structure encoded by the null distance.
• The method enables the proof of compactness theorems for sequences of big bang spacetimes under $\mathcal{SF}$ convergence, with limits that are integral current spaces.
• Sequences of black hole spacetimes with mass tending to zero are conjectured to converge in the $\mathcal{SF}$ sense, with the null distance preserving causality and enabling stability results for the positive mass theorem and Penrose inequality.