[Paper Review] Space and Time models

Ll. Bel|arXiv (Cornell University)|Mar 19, 2011

Noncommutative and Quantum Gravity Theories2 references3 citations

TL;DR

This paper re-evaluates Helmholtz's free mobility postulate within general relativity to define consistent space-time models, deriving conditions under which 3D spatial geometries admit rigid motion and time models (chorodesic, ephemeris, universal) are compatible. It establishes that constant curvature spatial metrics satisfy free mobility, and shows that adapted coordinate systems and connection symbols transform consistently, enabling a geometric framework for frames of reference in relativistic spacetimes with explicit transformation laws for metric and connection components.

ABSTRACT

We derive line-element's templates of space-time models with Space models complying with Helmholtz's Free mobility postulate, and discuss some of the Time models compatible with them.

Motivation & Objective

• To re-interpret Helmholtz's empiricist free mobility postulate—requiring rigid body motion without deformation—in the context of general relativity.
• To define space-time models where 3D spatial geometry is Riemannian with constant curvature, ensuring free mobility.
• To classify and analyze compatible time models (chorodesic, ephemeris, universal) within such space-time frameworks.
• To derive transformation laws for metric and connection components under adapted coordinate changes, preserving geometric structure.
• To establish conditions under which spherically symmetric models satisfy free mobility and admit consistent time models.

Proposed method

• Derives a line-element in adapted coordinates where time-like congruence is defined by a unit vector field, separating time and space components via $ A(t,x^i) $, $ f_i(t,x^i) $, and $ \bar{g}_{ij} $.
• Introduces a subordinate vector $ L^i $ that transforms as a quotient manifold vector, ensuring geometric consistency under adapted transformations.
• Defines a time-connection $ \bar{\Gamma}^i_{jk} $ using time-derivative operators $ \bar{\partial}_k $, which remain invariant under adapted time transformations.
• Applies the free mobility postulate to spherically symmetric models by requiring constant curvature $ k $ in the spatial metric $ d\bar{s}^2 $, ensuring isometric rigid motion.
• Derives transformation laws for $ A $, $ f_i $, $ B $, $ C $, and $ \bar{g}_{ij} $ under adapted time transformations $ t = t(t', r) $, showing invariance of $ \bar{\Gamma}^i_{jk} $.
• Imposes the condition $ \partial_t f_i + f_j \partial_j f_i = 0 $ to ensure $ f_i = 0 $ in chorodesic time, simplifying the model to $ f^\prime = 0 $.

Experimental results

Research questions

• RQ1 How can Helmholtz’s free mobility postulate be consistently implemented in general relativity to define spatial models with rigid motion?
• RQ2 What are the geometric and transformational conditions under which time models (chorodesic, ephemeris, universal) are compatible with such space models?
• RQ3 How do the metric components and spatial connections transform under adapted coordinate changes, and what invariants emerge?
• RQ4 What constraints must spherically symmetric space-time models satisfy to preserve free mobility and allow consistent time frameworks?
• RQ5 Can a time-connection $ \bar{\Gamma}^i_{jk} $ be defined that remains invariant under adapted time transformations, and what is its physical significance?

Key findings

• The spatial metric $ \bar{g}_{ij} $ must have constant curvature $ k $ to satisfy the free mobility postulate, ensuring isometric rigid motion.
• The time-connection $ \bar{\Gamma}^i_{jk} $, defined via $ \bar{\partial}_k = \partial_{x^k} + f_k \partial_t $, is invariant under adapted time transformations, providing a geometrically consistent framework.
• Under adapted time transformations $ t = t(t', r) $, the quantities $ A $, $ B $, $ C $ scale by $ \partial t / \partial t' $, while $ f $ transforms as $ f' = (\partial t / \partial t')^{-1}(f - \partial t / \partial r) $, preserving the line-element structure.
• The condition $ f^\prime = 0 $ in the chorodesic time model is equivalent to requiring radial curves $ t' = \text{const} $ to be geodesics, simplifying the model and ensuring consistency.
• For stationary frames, the ephemeris time model is compatible with the chorodesic model and satisfies $ \partial_t A = \partial_t f_i = \partial_t \bar{g}_{ij} = 0 $, with time reparametrization freedom $ t'' = \lambda t' + \psi(x^i) $.
• The free mobility condition leads to a differential equation (74) or (75) for $ B $, $ C $, and $ f $, which must be satisfied for spherically symmetric models to preserve rigid spatial geometry.