In Chap. 18 we explore some of the geometrical properties of smooth, non-null curves that are not necessarily defined as integral curves of any prescribed particular vector field on a manifold. Although in this book we are primarily concerned with parametrised curves in 4-dimensional spacetime it proves useful to establish a scheme for constructing a unique basis of vectors on a non-null curve (if it exists) immersed in an n-dimensional ( \(n\geq 2\) ) manifold \(\mathcal {M}\) with a metric tensor field \(\mathtt {g}\in \Gamma T^{2}_{0}\mathcal {M}\) of arbitrary signature \((p,q)\) (see Sect. 2.2 ). In Sects. 18.1 and 18.2 the general theory for constructing a matrix ODE system for such bases (generalising the original work of Frenet and Serret) is presented using the Gram-Schmidt orthogonalisation process for arbitrarily parametrised non-null curves. Explicit matrix ODE systems are displayed for \(2\leq n\leq 4\) in terms of scaled absolute covariant derivatives of n-frame vectors and “Frenet-Serret curvatures” of the immersed curves. Section 18.3 considers simple examples of plane curves in Euclidean \(\mathbb {R}^{2}\) in a local plane polar coordinate chart. Section 18.4 consider spacecurves in Euclidean \(\mathbb {R}^{3}\) and emphasises the advantages of using non-arclength parametrised curves in the analysis. Section 18.5 is devoted to a computation of Frenet-Serret curvatures of constant 4-acceleration timelike curves in flat Minkowski spacetime and discusses the reduction of such curves to the rectilinear motion of a massive particle with constant Newtonian 3-acceleration. It also relates the relativistic hyperbolic motion to that of an electrically charged massive particle accelerated by an external uniform, constant background electric field. In Sect. 18.6 we analyse the full relativistic motion of a massive particle in Minkowski spacetime that takes place in a particular flat hyperplane. A solution to the Frenet-Serret metric system is then found that gives rise to a right-circular helix in spacetime that projects to a circular orbit in the hyperplane. The solution is analysed in a Newtonian approximation to leading order in terms of a “Newtonian linear speed” and a “Newtonian centrifugal acceleration”. Finally we explore particular features of a 1-parameter family of non-geodesic timelike Frenet-Serret curves as a model for a family of 1-parameter accelerating observers immersed in a background 4-dimensional spacetime Einstein-de Sitter metric. By selecting a particular member of the observer family, in an open domain that excludes the Einstein-de Sitter geometric singularity, the first Frenet-Serret curvature is explicitly calculated and found to be non-zero whilst the second is calculated to be zero. This is a reflection of the fact that the Frenet-Serret curve lies in a spacetime hypersurface (that is explicitly calculated). Consequently for this particular case, there does not exist a unique timelike Frenet-Serret 4-frame on the immersed curve in the specified domain.

The General Frenet-Serret Equations on a Manifold

章节
发表日期：

2025年

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Robin W. Tucker,
Timothy J. Walton

摘要

In Chap. 18 we explore some of the geometrical properties of smooth, non-null curves that are not necessarily defined as integral curves of any prescribed particular vector field on a manifold. Although in this book we are primarily concerned with parametrised curves in 4-dimensional spacetime it proves useful to establish a scheme for constructing a unique basis of vectors on a non-null curve (if it exists) immersed in an n-dimensional ( \(n\geq 2\) ) manifold \(\mathcal {M}\) with a metric tensor field \(\mathtt {g}\in \Gamma T^{2}_{0}\mathcal {M}\) of arbitrary signature \((p,q)\) (see Sect. 2.2 ). In Sects. 18.1 and 18.2 the general theory for constructing a matrix ODE system for such bases (generalising the original work of Frenet and Serret) is presented using the Gram-Schmidt orthogonalisation process for arbitrarily parametrised non-null curves. Explicit matrix ODE systems are displayed for \(2\leq n\leq 4\) in terms of scaled absolute covariant derivatives of n-frame vectors and “Frenet-Serret curvatures” of the immersed curves. Section 18.3 considers simple examples of plane curves in Euclidean \(\mathbb {R}^{2}\) in a local plane polar coordinate chart. Section 18.4 consider spacecurves in Euclidean \(\mathbb {R}^{3}\) and emphasises the advantages of using non-arclength parametrised curves in the analysis. Section 18.5 is devoted to a computation of Frenet-Serret curvatures of constant 4-acceleration timelike curves in flat Minkowski spacetime and discusses the reduction of such curves to the rectilinear motion of a massive particle with constant Newtonian 3-acceleration. It also relates the relativistic hyperbolic motion to that of an electrically charged massive particle accelerated by an external uniform, constant background electric field. In Sect. 18.6 we analyse the full relativistic motion of a massive particle in Minkowski spacetime that takes place in a particular flat hyperplane. A solution to the Frenet-Serret metric system is then found that gives rise to a right-circular helix in spacetime that projects to a circular orbit in the hyperplane. The solution is analysed in a Newtonian approximation to leading order in terms of a “Newtonian linear speed” and a “Newtonian centrifugal acceleration”. Finally we explore particular features of a 1-parameter family of non-geodesic timelike Frenet-Serret curves as a model for a family of 1-parameter accelerating observers immersed in a background 4-dimensional spacetime Einstein-de Sitter metric. By selecting a particular member of the observer family, in an open domain that excludes the Einstein-de Sitter geometric singularity, the first Frenet-Serret curvature is explicitly calculated and found to be non-zero whilst the second is calculated to be zero. This is a reflection of the fact that the Frenet-Serret curve lies in a spacetime hypersurface (that is explicitly calculated). Consequently for this particular case, there does not exist a unique timelike Frenet-Serret 4-frame on the immersed curve in the specified domain.