[Paper Review] Natural Coordinate System in Curved Space-time

Ying-Qiu Gu|arXiv (Cornell University)|Dec 28, 2006

Cosmology and Gravitation Theories10 references3 citations

TL;DR

This paper introduces a 'natural coordinate system' (NCS) in curved spacetime that synchronizes the time coordinate with the evolving, hypersurface-orthogonal simultaneity of the physical world. By constructing a globally valid coordinate system where the time coordinate is orthogonal to spacelike hypersurfaces representing real-time simultaneity, the method simplifies the Einstein field equations and enables a clear geometric and physical interpretation of dynamics, particularly in weak-field and low-speed limits.

ABSTRACT

In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be preseted according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it "natural coordinate system". The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In NCS we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein's equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is a nice coordinate system to describe the realistic curved space-time, and is helpful to understand the nature of space-time.

Motivation & Objective

• To establish a globally valid coordinate system in curved spacetime that respects the physical evolution of the universe by aligning the time coordinate with evolving simultaneity hypersurfaces.
• To ensure the time coordinate is hypersurface-orthogonal, enabling a clean Hamiltonian formulation and simplifying the calculation of Noether charges.
• To provide a constructive method for transforming any given spacetime metric into this natural coordinate system through regular spatial coordinate transformations.
• To demonstrate the physical and geometric advantages of NCS in the weak-field and low-velocity limits, particularly in recovering Newtonian mechanics and simplifying the Einstein equations.
• To clarify the role of coordinate conditions in general relativity by showing they emerge naturally from dynamics and initial/boundary conditions in NCS.

Proposed method

• Constructs a coordinate transformation $ x^k = x^k(t, y^l) $ that eliminates cross terms $ dt hinspace dy^k $, ensuring $ dt $ is orthogonal to $ dy^k $, via solving a first-order ODE system for $ rac{ ext{d}x^k}{ ext{d}t} = -ar{g}^{kl} A_l $.
• Uses the existence of a unique, globally defined simultaneity hypersurface $ f(x^ u) = C $, with $ f $ satisfying $ abla_ u f > 0 $ and sufficient smoothness, to redefine the time coordinate as $ t = f(x^ u) $.
• Applies the ADM decomposition to the metric, reducing it to $ ds^2 = g_{tt} dt^2 - g_{kl} dy^k dy^l $, and derives the connection coefficients $ ilde{oldsymbol{ ho}}^ u $ and $ ilde{oldsymbol{ ho}}^ u $ from the Ricci tensor components.
• Derives the wave equation for metric perturbations $ h_{kl} $ by solving $ G^{00} $ and $ G^{0k} $ equations, which yield $ rac{1}{2} abla^2 ilde{ ho} = rac{ ilde{ ho}}{2} $ and $ rac{1}{2}rac{ ext{d}}{ ext{d}t}( ilde{oldsymbol{ ho}}^k - abla^k ilde{ ho}) = ext{source terms} $.
• Performs classical and weak-field approximations of the spinor equation and Einstein’s equations, showing that Newtonian gravity emerges naturally in NCS.
• Demonstrates that in NCS, the gravitomagnetic force term vanishes in the particle motion equation, unlike in other coordinate systems, due to the absence of $ ilde{oldsymbol{ ho}}^k $-dependent terms in the geodesic equation.

Experimental results

Research questions

• RQ1 Can a globally valid coordinate system be constructed in curved spacetime such that the time coordinate is orthogonal to spacelike hypersurfaces that represent the physical simultaneity of the evolving universe?
• RQ2 How can such a coordinate system be systematically constructed from a given spacetime metric, and what are the necessary conditions on the metric and coordinate transformations?
• RQ3 What are the implications of this coordinate system for the Hamiltonian formulation of general relativity and the calculation of conserved charges?
• RQ4 How does the natural coordinate system simplify the weak-field and low-velocity limits of general relativity, and what is the resulting form of the particle motion equation?
• RQ5 Why does the gravitomagnetic force vanish in this coordinate system, and how does this compare to its presence in standard coordinate systems?

Key findings

• A globally valid natural coordinate system (NCS) exists in curved spacetime where the time coordinate is orthogonal to the evolving simultaneity hypersurfaces $ f(x^ u) = C $, and this system can be constructed via a regular spatial coordinate transformation.
• The construction is based on solving a first-order ODE system $ rac{ ext{d}x^k}{ ext{d}t} = -ar{g}^{kl} A_l $, which ensures orthogonality between $ dt $ and $ dy^k $, and is guaranteed to have a unique solution for given initial conditions.
• In NCS, the Einstein field equations simplify significantly: the $ G^{00} $ and $ G^{0k} $ components yield independent equations for $ ilde{ ho} $ and $ ilde{oldsymbol{ ho}}^k $, which are then used to derive the wave equation for $ h_{kl} $.
• The classical approximation of the spinor equation in NCS reproduces Newtonian mechanics, demonstrating a clear geometric and dynamical link between quantum mechanics, classical mechanics, and general relativity.
• In the weak-field and low-speed limit, the particle motion equation in NCS takes the form $ rac{d}{dt}v^k = - abla_k ilde{ ho} + v^k rac{ ext{d}}{ ext{d}t} ilde{ ho} + v^n rac{ ext{d}}{ ext{d}t}h_{kn} $, with the gravitomagnetic force term absent.
• The coordinate condition in NCS is not imposed a priori but emerges from the dynamics, initial conditions, and the evolving nature of the simultaneity hypersurface, making it physically natural and self-consistent.