Calculus of Lengths on Space-Time Manifolds

Selenu Simone

M. I. U. R., Tortolì, Italy.

DOI: 10.4236/jmp.2025.165039                

Abstract

This article reports on the calculation of the lengths of space-time curves $\gamma$ [1] generated by Jacobian transformations of reference systems, which determines the manifold as an observed inner structure.

Keywords

Electron Field, Cosmology, Theoretical Physics

1. Introduction

The article reviews a new approach to the study of metric properties of space-time manifolds. As a space of events [2], it represents a more general description of space-time, through transformations of reference systems that impose an inner structure. This new mathematical model of space-time is employed to measure changes in distances within gravitational fields [3]-[6]. Lenghts between events can be generated from any observed force field [7], by Jacobian transformation matrices $H_J$ of reference systems. A new model of the common space-time [8] is then proposed through a generalization of the tensorial form associated with the space-time coordinates of a new topology of homotopies of space-time. In this new framework of general relativity, the lengths of space-time paths are calculated as integrals of metric tensors, uniquely determined by the existence [9] of groupal derivatives [10] [11] of reference system coordinates. By enablig the calculation of Jacobian mapping of events as a second-order differential form ${\text{d}}^{2}s$ , referred to the Hessian structure [12], is Schwartz’s theorem [13] in not longer a useful mathematical model of the topological space-time. The existence of asymmetric metric tensors introduces new contributions to the eventual measure of space-time distances. The following section will be presented the calculation of space-time lengths and concluded the article.

2. Tensorial Form and Vector Groupal Derivatives

This section is developed a new model of space-time, by utilizing the operation of groupal derivative [10]. Later considered as a new mathematical model for vector transformations of reference systems, are used to calculate space-time lengths as integrals of space-time differential forms [9] $\text{d}s$ . The use of groupal derivatives [10] then allows to perform the calculus of transformed events $x^{\prime }$ as shown in the next equation:

$D{x}^{\prime }={\displaystyle \sum _{l^{\prime }}\frac{\text{d}{x}^{\prime }}{\text{d}{x}_{m^{\prime },l^{\prime }}}}={\displaystyle \sum _k{\displaystyle \sum _{l^{\prime }}{\nabla }_{x_k}{x^{\prime }}_m\left(x_0\right)\frac{\text{d}{x}_{k,l}}{\text{d}{x}_{m^{\prime },l^{\prime }}}}}=\frac{\partial {x^{\prime }}_m}{{\partial }_{x_{m^{\prime }}}}$ (1)

Deriving a new differential form of the Jacobian transformation matrices $H_J$ :

$D{x}^{\prime }=D\left[H_J\otimes x\right]=H_J$ (2)

the inner structure of space-time is then defined, along with the differential form $\text{d}{x}^{\prime }$ which allows for the calculation of infinitesimal displacements:

$\text{d}{x}^{\prime }=\left[H_J\otimes \text{d}x\right]$ (3)

being written the Hessian structure on the following second-order differential form:

${\text{d}}^{2}s=\text{d}x\otimes \text{d}{x}^{\prime }=\left[\text{d}x\otimes H_J\otimes \text{d}x\right]$ (4)

The second rank tensor:

$\frac{{\partial }^{2}s}{\partial x_i\partial x_j}={\left[H_J\right]}_{i,j}$ (5)

does not fulfill Schwartz’s equality, when ${\left[H_J\right]}_{i,j}\ne {\left[H_J\right]}_{j,i}$ , allowing to measure lengths of space-time paths $\gamma$ by the integrability of differential forms $\text{d}s$ [14], firstly considered:

$L={\displaystyle {\int }_{\gamma }\text{d}s}$ (6)

having chosen $k=\nabla s\left(x_0\right)\cdot {\displaystyle {\int }_{\gamma }\text{d}x}$ , as an integration constant, and calculated space-time lengths on integrals of first-order expansions of differential forms $\text{d}s$ [15]:

$L={\displaystyle {\int }_{\gamma }{\displaystyle \sum _i{\nabla }_i{\nabla }_{i}s\text{d}{x}_i\text{d}{x}_i}}+{\displaystyle {\int }_{\gamma }{\displaystyle \sum _{i<j}\left[{\nabla }_i,{\nabla }_j\right]s\text{d}{x}_i\text{d}{x}_j}}+2{\displaystyle {\int }_{\gamma }{\displaystyle \sum _{i<j}{\nabla }_j{\nabla }_{i}s\text{d}{x}_i\text{d}{x}_j}}+k$ (7)

For $g_{i,j}={\left[H_J\right]}_{i,j}$ for any $i\ne j$ and $g_{i,j}=2{\left[H_J\right]}_{i,j}$ for any $i=j$ , taken into account Equation (5), Equation (7) with $k=0$ as a boundary, integrals of the differential forms $\text{d}s$ are written as:

$L={\displaystyle {\int }_{\gamma }{\displaystyle \sum _i{g}_{i,i}\text{d}{x}_i\text{d}{x}_i}}+{\displaystyle {\int }_{\gamma }{\displaystyle \sum _{i<j}\left[g_{i,j}-g_{j,i}\right]\text{d}{x}_i\text{d}{x}_j}}+2{\displaystyle {\int }_{\gamma }{\displaystyle \sum _{i<j}{g}_{j,i}\text{d}{x}_i\text{d}{x}_j}}$ (8)

determining then a space-time metric [16]-[19]. By Jacobian transformations of reference systems, lengths in metric spaces are defined as:

${\text{d}}^{2}s={\displaystyle \sum _{i,j}{g}_{i,j}\text{d}{x}_i\text{d}{x}_j}$ (9)

let to conclude the existence of a metric tensor, onto the evaluation of relative displacements between events of the four-dimensional space-time [20].

3. Conclusion

The article concludes by calculating the lengths of space-time topological paths as integrals of differential forms. This is based on any measure of its Hessian structure, which is determined by the existence of the topological inner structure of space-time manifolds.

Acknowledgements

I thank to acknowledge Prof. A. Rizzo for the reading of this article.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Munkres, J.R. (2000) Topology. 2nd Edition, Prentice Hall.
[2]	Grossmann for the Mathematical Part and Einstein for the Physical Part (1913) Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravita-tion [Outline of a Generalized Theory of Relativity and of a Theory of Gravitation]. Zeitschrift für Mathematik und Physik, 62, 225-261.
[3]	Saadati, R., Valsan, L. and Faraoni, V. (2025) Non-Geodesic Timelike Observers and the Ultralocal Limit. The European Physical Journal C, 85, Article No. 176. https://doi.org/10.1140/epjc/s10052-025-13888-6
[4]	Armstrong, J. and Vollmer, A. (2025) Abundant Superintegrable Systems and Hessian Structures. Journal of Geometry and Physics, 214, Article ID: 105503. https://doi.org/10.1016/j.geomphys.2025.105503
[5]	Rizzo, A. (2025) On a Unified Geometric Field Equation: The General Theory of Singularity. Elsevier.
[6]	Einstein, A. (2011) Relativity—The Special and General Theory. Read Books Ltd., 4.
[7]	Selenu, S. (2023) Forces in Quantum Matter. Research Gate. https://www.researchgate.net/publication/375756410_Forces_in_quantum_matter
[8]	Weil, A. (1946) Foundations of Algebraic Geometry. American Mathematical Society, 1-363.
[9]	Vega, C. (2021) Spacetime Distances: An Exploration. arXiv: 2103.01191
[10]	Selenu, S. (2025) Calculus of the Groupal Derivatives. Research Gate. https://www.researchgate.net/publication/388192452_Calculus_of_the_groupal_derivatives
[11]	Beyer, W.H. (1987) Derivatives. 28th Edition, CRC Press, 229-232.
[12]	Totaro, B. (2004) The Curvature of a Hessian Metric. https://www.math.ucla.edu
[13]	(2006) Young’s Theorem. University of California Berkeley.
[14]	Selenu, S. (2025) Simplectic form Integral of Differential Forms. Research Gate. https://www.researchgate.net/publication/387935615_Simplectic_form_integral_of_differential_forms
[15]	Selenu, S. (2025) Calculus of Integrals of Differential Forms through Jacobian Matrix. Research Gate. https://www.researchgate.net/publication/388192452_Calculus_of_the_groupal_derivatives
[16]	Weinberg, S. (1972) Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley.
[17]	Shima, H. (2007) The Geometry of Hessian Structures. World Scientific. https://doi.org/10.1142/6241
[18]	Selenu, S. (2024) Hydrogen Stability in Quantum Mechanics. Research Gate. https://www.researchgate.net/publication/386106799_Hydrogen_stability_in_quantum_mechanics
[19]	Selenu, S. (2024) Quantum Mechanical Explanation of the Inertial Mass as a Gravitational Mass. Research Gate. https://www.researchgate.net/publication/386245505_Quantum_mechanical_explanation_of_the_inertial_mass_as_a_gravitational_mass
[20]	Davis, D.M. (2021) Geodesic Complexity for Non-Geodesic Spaces. Boletín de la Sociedad Matemática Mexicana, 28, Article No. 8. https://doi.org/10.1007/s40590-021-00400-9