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An Account on Links Between Finsler and Lorentz Geometries for Riemannian Geometers

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New Trends in Geometric Analysis

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Abstract

Some links between Lorentz and Finsler geometries have been developed in the last years, with applications even to the Riemannian case. Our purpose is to give a brief description of them, which may serve as an introduction to recent references. As a motivating example, we start with Zermelo navigation problem, where its known Finslerian description permits a Lorentzian picture which allows for a full geometric understanding of the original problem. Then, we develop some issues including (a) the accurate description of the Lorentzian causality using Finsler elements, (b) the non-singular description of some Finsler elements (such as Kropina metrics or complete extensions of Randers ones with constant flag curvature), (c) the natural relation between the Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler setting, and (d) practical applications to the propagation of waves and firefronts.

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Notes

  1. 1.

    These metrics are well-known in the Finsler literature since the original article [57].

  2. 2.

    Notice that, whenever \( \lambda < 0 \), h has signature \((+,-,\dots , -)\). Anyway, h is well defined as a signature changing metric on the whole M, so that it can be used to determine the admissible curves between any two points \(x_0, y_0\in M\). Comparing with \(\tilde h\) in (7) (which was defined for \(\lambda >0\)), one has \(\tilde h=h/\lambda ^2\).

  3. 3.

    See [52] for a full development of this condition.

  4. 4.

    It is worth emphasizing that this last case appears in the same footing as the others from the spacetime viewpoint. However, in the Finslerian description, it corresponds with the limit of the geodesics of F and \(F_l\) (indeed, as a limit, \(F(\dot x)=F_l(\dot x)\equiv 1\)). Moreover, x might start at the region of \(M_l\), arrive at the region of critical wind \(M_{crit}\), and come back to \(M_l\). On \(M_l\), x becomes a pregeodesic of \(\Lambda g_0+ \omega \otimes \omega \) consistent with (17) (see the case (iii) (b) in the aforementioned [23, Theorem 5.5]).

  5. 5.

    See [23, Definition 2.8 and Example 2.16] for a subtlety about smoothness applicable here.

  6. 6.

    In the sense of [55, Sect. 3.2] (thus, positive and two-homogeneous) up to the fact that it may be non-smooth on \(\partial _t\). This is not relevant here, because we are only interested in the directions of \(\mathcal {C}\) (anyway, G can be smoothen on \(\partial _t\); see [55, Theorem 5.6].)

  7. 7.

    The convention for the zero vector depends on the reference, we follow [65].

  8. 8.

    Sometimes Cauchy hypersurfaces and the initial value problem are allowed to be more general so that non-smooth data or data on degenerate hypersurfaces are permitted, but we will not go into these issues.

  9. 9.

    The natural time-orientation chosen here and in the remainder is the one such that the timelike vector \(\partial _\tau \) becomes future-pointing.

  10. 10.

    See [78] for background, especially Chap. 12 and Sects. 12.3, 12.5.

  11. 11.

    In particular, it is continuous outside the diagonal [23, Theorem 4.5] (this property does not hold for any conic Finsler metric [52, Proposition 3.9]).

  12. 12.

    Here pseudo-Finsler means that the fundamental tensor (1) is non-degenerate, and the domain is conic, not necessarily the whole TM; in particular, so are the Finsler metric F and the Lorentz-Finsler one \(F_l\).

  13. 13.

    Among the subtleties appearing here, it is worth mentioning the step causal continuity (i.e., the spacetime is causal and the causal future and past vary continuously with the point) in the ladder of spacetimes. This is more restrictive than stable causality, and it is satisfied by all the standard stationary ones, as well as whenever \(\Lambda \geq 0\); however, it is not satisfied by all the SSTK spacetimes.

  14. 14.

    Hadamard manifolds (which are complete, simply connected and with non-positive sectional curvature) become diffeomorphic to \(\mathbb R^n\) by Cartan-Hadamard theorem. Eberlein and O’Neill’s boundary becomes a topological sphere \(S^{n-1}\) located at infinity. Indeed, the Busemann functions yield a quotient in the space of rays so that each function can be regarded as a direction at infinity; then, the set of all these directions can be regarded as the sphere \(S^{n-1}\).

  15. 15.

    Notice that, for a smooth curve c in such a space, the role of its Riemannian norm \(g_R(\dot c(s),\dot c(s))\) at each s is played by the local dilatation there; moreover, notions such as geodesic or cut point have a natural sense (see [45, Chap. 1] for background).

  16. 16.

    There is a non-equivalent way to define forward (and backward) Cauchy sequences; however, it would yield the same forward Cauchy completion (see [40, Sect. 3.2.2]).

  17. 17.

    Its consistency relies on a non-symmetric version of Arzela theorem [40, Theorem 5.12].

  18. 18.

    Intuitively, it does not admit “almost closed” causal curves. A formalization of this property is that each point \(p\in M\) has a neighborhood U such that any inextendible causal curve starting at U will leave U at some point so that it will not return to U.

  19. 19.

    Noticeably, they agree in the class of globally hyperbolic spacetimes-with-timelike-boundary, see [1].

  20. 20.

    A subtlety is that the Busemann functions which can be constructed with the restriction \(F(\dot c)\leq 1\) coincide with those constructed with \(F(\dot c)< 1\).

  21. 21.

    This is not a restriction in any realistic situation (see [50, Remark 3.2]).

  22. 22.

    Note that all the usual concepts about causality can be directly translated from the Lorentzian case to the Lorentz-Finsler metrics and the more general setting of cone structures (see, for example, [55, 64]). However, due to the non-reversibility of the Finsler metrics, usually only future directions are considered.

  23. 23.

    Alternatively and more generally, we can consider \( \partial _t \) as an observer’s vector field co-moving with the medium in which the wave propagates, as suggested at the end of Sect. 2.

  24. 24.

    For simplicity, \( \mathcal {S} \) will be assumed to be a hypersurface of \( M \), although the results we present here can be generalized to any submanifold (see [50]).

  25. 25.

    Nevertheless, the results we present here directly apply to the other wavefront simply by replacing “outward” with “inward.”

  26. 26.

    Richards’ equations are equivalent to the ODE system (28) when \( \Sigma _t \) is an ellipse (see [50, Sect. 5.2]).

  27. 27.

    In its usual version, Matsumoto metrics effectively measure the travel time for a walker on a slope, favoring the downward direction (see [75]).

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Acknowledgements

MAJ and EPR were partially supported by the projects PGC2018-097046-B-I00 and PID2021-124157NB-I00, funded by MCIN/AEI/10.13039/501100011033/ “ERDF A way of making Europe” and also by Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos (Comunidad Autónoma de la Región de Murcia), included in the Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2022) of the Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia, REF. 21899/PI/22. EPR and MS were partially supported by the project PID2020-116126GB-I00 funded by MCIN/AEI/10.13039/501100011033 and P20-01391 (PAIDI 2020, Junta de Andalucía), as well as the framework IMAG-María de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. EPR was also supported by Ayudas para la Formación de Profesorado Universitario (FPU) from the Spanish Government.

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  1. Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain

    Miguel Ángel Javaloyes & Enrique Pendás-Recondo

  2. Departamento de Geometría y Topología, Facultad de Ciencias & IMAG (Centro de Excelencia María de Maeztu), Universidad de Granada, Granada, Spain

    Miguel Sánchez

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  1. Miguel Ángel Javaloyes
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  1. Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

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  3. Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

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Javaloyes, M.Á., Pendás-Recondo, E., Sánchez, M. (2023). An Account on Links Between Finsler and Lorentz Geometries for Riemannian Geometers. In: Alarcón, A., Palmer, V., Rosales, C. (eds) New Trends in Geometric Analysis. RSME Springer Series, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-031-39916-9_10

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