[Paper Review] Conservation Laws and Formation of Singularities in Relativistic Theories of Extended Objects

Jens Hoppe|arXiv (Cornell University)|Mar 10, 1995

Relativity and Gravitational Theory2 references19 citations

TL;DR

This paper formulates the dynamics of M-dimensional extended objects in M+2-dimensional spacetime using geometric variables—first and second fundamental forms—showing that their equations of motion arise as consistency conditions of a linear system, leading to non-local conserved quantities. The key result is that for M=1 (strings), all motions are necessarily singular, while for M=2 (membranes), an explicit non-symmetric solution in terms of elliptic functions is constructed, yielding a 3-fold periodic spacelike maximal hypersurface in ℝ¹,³.

ABSTRACT

The dynamics of an M-dimensional extended object whose M+1 dimensional world volume in M+2 dimensional space-time has vanishing mean curvature is formulated in term of geometrical variables (the first and second fundamental form of the time-dependent surface $\sum_M$), and simple relations involving the rate of change of the total area of $\sum_M$, the enclosed volume as well as the spatial mean -- and intrinsic scalar curvature, integrated over $\sum_M$, are derived. It is shown that the non-linear equations of motion for $\sum_M(t)$ can be viewed as consistency conditions of an associated linear system that gives rise to the existence of non-local conserved quantities (involving the Christoffel-symbols of the flat M+1 dimensional euclidean submanifold swept out in ${\Bbb R}^{M+1}$). For M=1 one can show that all motions are necessarily singular (the curvature of a closed string in the plane can not be everywhere regular at all times) and for M=2, an explicit solution in terms of elliptic functions is exhibited, which is neither rotationally nor axially symmetric. As a by-product, 3-fold-periodic spacelike maximal hypersurfaces in ${\Bbb R}^{1,3}$ are found.

Motivation & Objective

• To formulate the dynamics of M-dimensional extended objects in M+2-dimensional spacetime using intrinsic geometric variables such as the first and second fundamental forms.
• To derive conservation laws for total area, enclosed volume, and intrinsic curvatures integrated over the worldvolume.
• To show that the nonlinear equations of motion are consistency conditions of an associated linear system, implying existence of non-local conserved quantities.
• To analyze the formation of singularities in relativistic extended objects, particularly for M=1 and M=2.
• To construct explicit solutions for M=2 membranes, including non-rotationally symmetric and periodic spacelike maximal hypersurfaces in ℝ¹,³.

Proposed method

• Parametrize the worldvolume ℳ ⊂ ℝ¹,ᴹ⁺¹ as a time-dependent M-dimensional surface Σᴹ(t), using coordinates (φ⁰=t, φ¹,…,φᴹ) with normal evolution.
• Express the induced metric Gαβ on ℳ using the Minkowski metric ημν and the embedding functions xμ(φ⁰,…,φᴹ), leading to a volume functional S = ∫√|G| dφ.
• Derive the equations of motion from the extremality of the volume functional, resulting in a first-order system where normal velocity ẋ is proportional to sinθ, with θ related to the spatial metric determinant g.
• Introduce the geometric variables gᵣₛ (first fundamental form), Lᵣₛ (second fundamental form), and use the Gauss-Codazzi equations to relate time evolution to curvature.
• Utilize the Weierstrass ℘-function to construct explicit solutions for M=2, particularly for the case where the surface evolves periodically and forms a maximal hypersurface in ℝ¹,³.
• Show that the dynamics are consistent with a linear system whose compatibility condition yields the nonlinear equations of motion, implying non-local conserved quantities via Christoffel symbols in the ambient flat space ℝᴹ⁺¹.

Experimental results

Research questions

• RQ1 Under what conditions do relativistic extended objects in M+2-dimensional spacetime develop singularities in their motion?
• RQ2 How can the dynamics of such objects be reformulated in terms of geometric invariants like the first and second fundamental forms?
• RQ3 What conserved quantities arise from the underlying linear system associated with the nonlinear equations of motion?
• RQ4 Can explicit, non-trivial solutions be constructed for M=2 membranes, particularly those that are not rotationally or axially symmetric?
• RQ5 What is the geometric structure of spacelike maximal hypersurfaces in ℝ¹,³, and how do they emerge from the dynamics of extended objects?

Key findings

• For M=1 (strings), all motions are necessarily singular: the curvature of a closed string in the plane cannot remain regular for all time.
• For M=2, an explicit solution is constructed using the Weierstrass ℘-function, describing a non-rotationally symmetric membrane whose shape evolves periodically in time.
• The solution (73) describes a 3-fold periodic spacelike maximal hypersurface in ℝ¹,³, which is convex and evolves from a flat surface at t=0 to a round point at t=ω.
• The mean curvature H and Gauss curvature K of the surface are derived in terms of the logarithmic derivatives of the ℘-function, with singularities occurring at the corners and edges of the unit cube.
• The solution (74) describes a membrane that evolves symmetrically from the upper and lower faces of a cube, meeting at a point at t=ω with diverging curvature.
• The existence of non-local conserved quantities is established through the compatibility of a linear system, with the conserved densities involving the Christoffel symbols of the flat M+1-dimensional ambient space ℝᴹ⁺¹.