Null Distance and Temporal Functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912581160009728 |
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| author | Nigri, Andrea |
| author_facet | Nigri, Andrea |
| contents | The notion of null distance was introduced by Sormani and Vega as part of a broader program to develop a theory of metric convergence adapted to Lorentzian geometry. Given a time function $τ$ on a spacetime $(M,g)$, the associated null distance $\hat{d}_τ$ is constructed from and closely related to the causal structure of $M$. While generally only a semi-metric, $\hat{d}_τ$ becomes a metric when $τ$ satisfies the local anti-Lipschitz condition.
In this work, we focus on temporal functions, that is, differentiable functions whose gradient is everywhere past-directed timelike. Sormani and Vega showed that the class of $C^1$ temporal functions coincides with that of $C^1$ locally anti-Lipschitz time functions. When a temporal function $f$ is smooth, its level sets $M_t = f^{-1}(t)$ are spacelike hypersurfaces and thus Riemannian manifolds endowed with the induced metric $h_t$. Our main result establishes that, on any level set $M_t$ where the gradient $\nabla f$ has constant norm, the null distance $\hat{d}_f$ is bounded above by a constant multiple of the Riemannian distance $d_{h_t}$.
Applying this result to a smooth regular cosmological time function $τ_g$ -- as introduced by Andersson, Galloway, and Howard -- we prove a theorem confirming a conjecture of Sakovich and Sormani (arXiv:2410.16800, 2025): if the diameters of the level sets $M_t = τ_g^{-1}(t)$ shrink to zero as $t \to 0$, then the spacetime exhibits a Big Bang singularity, as defined in their work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_07158 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Null Distance and Temporal Functions Nigri, Andrea Differential Geometry General Relativity and Quantum Cosmology 53C50 (Primary) 53Z05, 83C75, 83F05 (Secondary) The notion of null distance was introduced by Sormani and Vega as part of a broader program to develop a theory of metric convergence adapted to Lorentzian geometry. Given a time function $τ$ on a spacetime $(M,g)$, the associated null distance $\hat{d}_τ$ is constructed from and closely related to the causal structure of $M$. While generally only a semi-metric, $\hat{d}_τ$ becomes a metric when $τ$ satisfies the local anti-Lipschitz condition. In this work, we focus on temporal functions, that is, differentiable functions whose gradient is everywhere past-directed timelike. Sormani and Vega showed that the class of $C^1$ temporal functions coincides with that of $C^1$ locally anti-Lipschitz time functions. When a temporal function $f$ is smooth, its level sets $M_t = f^{-1}(t)$ are spacelike hypersurfaces and thus Riemannian manifolds endowed with the induced metric $h_t$. Our main result establishes that, on any level set $M_t$ where the gradient $\nabla f$ has constant norm, the null distance $\hat{d}_f$ is bounded above by a constant multiple of the Riemannian distance $d_{h_t}$. Applying this result to a smooth regular cosmological time function $τ_g$ -- as introduced by Andersson, Galloway, and Howard -- we prove a theorem confirming a conjecture of Sakovich and Sormani (arXiv:2410.16800, 2025): if the diameters of the level sets $M_t = τ_g^{-1}(t)$ shrink to zero as $t \to 0$, then the spacetime exhibits a Big Bang singularity, as defined in their work. |
| title | Null Distance and Temporal Functions |
| topic | Differential Geometry General Relativity and Quantum Cosmology 53C50 (Primary) 53Z05, 83C75, 83F05 (Secondary) |
| url | https://arxiv.org/abs/2507.07158 |