[Paper Review] (1+1)-dimensional formalism and quasi-local conservation equations

Jong Hyuk Yoon|arXiv (Cornell University)|Dec 9, 2002

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper formulates a (1+1)-dimensional reduction of 3+1-dimensional general relativity using fiber bundle geometry, deriving exact quasi-local conservation laws for energy, linear momentum, and angular momentum as invariant two-surface integrals. It shows these laws reduce to Bondi-type fluxes at null infinity and match Price-Thorne stretched horizon equations at the quasi-local horizon, with a novel quasi-local analog of Carter's fourth constant emerging under weaker conditions than Killing symmetries.

ABSTRACT

A set of exact quasi-local conservation equations is obtained in the (1+1)-dimensional description of the Einstein's equations of (3+1)-dimensional spacetimes. These equations are interpreted as quasi-local energy, linear momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that these quasi-local conservation equations reduce to the conservation equations of Bondi energy, linear momentum, and angular momentum, respectively. When restricted to the quasi-local horizon of a generic spacetime, which is defined without referring to the infinity, the quasi-local conservation equations coincide with the conservation equations on the stretched horizon studied by Price and Thorne. All of these quasi-local quantities are expressed as invariant two-surface integrals, and geometrical interpretations in terms of the area of a given two-surface and a pair of null vector fields orthogonal to that surface are given.

Motivation & Objective

• To develop a (1+1)-dimensional formalism for 3+1D Einstein gravity using fiber bundle structures with (u,v) as coordinates and (y^a) as fiber coordinates.
• To derive exact quasi-local conservation equations for energy, momentum, and angular momentum that are geometrically invariant and defined on compact two-surfaces.
• To demonstrate that these quasi-local equations reduce to standard Bondi conservation laws at null infinity in asymptotically flat spacetimes.
• To show equivalence with the stretched horizon conservation laws of Price and Thorne on the quasi-local horizon of generic dynamical spacetimes.
• To identify a quasi-local analog of Carter’s fourth constant, existing under weaker conditions than full Killing symmetries.

Proposed method

• Uses a line element with null coordinates (u,v), fiber coordinates (y^a), and connection 1-forms (A_±^a) to decompose the 3+1D metric into a (1+1) base and 2D fiber.
• Introduces horizontal vector fields (ĥ∂₊, ĥ∂₋) orthogonal to the fiber, with affine parameter v along ĥ∂₋, defining a null hypersurface.
• Defines quasi-local quantities via invariant two-surface integrals involving the area element e^σ and the conformal metric ρ_ab with det ρ_ab = 1.
• Derives conservation equations by analyzing the (1+1) Hamiltonian formalism with v as time, leading to integro-differential equations on compact 2-surfaces.
• Applies the Lie derivative along a vector field ξ^a on the 2-surface to derive flux equations, using covariant derivatives and shear tensors.
• Identifies a new quasi-local invariant analogous to Carter’s fourth constant, arising when ∂₊A₊^a = 0, not requiring full Killing fields.

Experimental results

Research questions

• RQ1 Can quasi-local conservation laws for energy, momentum, and angular momentum be derived in a (1+1)-dimensional reduction of 3+1D general relativity?
• RQ2 Do these quasi-local conservation laws reduce to the standard Bondi fluxes at future null infinity in asymptotically flat spacetimes?
• RQ3 Do they reproduce the conservation equations on the stretched horizon as formulated by Price and Thorne for generic dynamical black holes?
• RQ4 What is the geometric and physical interpretation of the quasi-local quantities in terms of null geometry and surface invariants?
• RQ5 Does a quasi-local analog of Carter’s fourth constant exist, and under what weaker conditions than Killing symmetries?

Key findings

• The quasi-local energy, momentum, and angular momentum are expressed as invariant two-surface integrals involving the area element e^σ and the conformal metric ρ_ab.
• At null infinity, the derived quasi-local conservation laws exactly reproduce the Bondi energy, momentum, and angular momentum fluxes.
• On the quasi-local horizon, the conservation equations match those of the stretched horizon by Price and Thorne, validating the formalism in a dynamical context.
• The formalism yields a new quasi-local invariant analogous to Carter’s fourth constant, existing under the condition ∂₊A₊^a = 0, which is weaker than requiring two commuting Killing vectors.
• The shear tensor σ_Hab and the expansion θ_H appear naturally in the flux equations, with the latter governing the time evolution of the surface geometry.
• The derivation confirms the equivalence of two different expressions for the same flux, validating the consistency of the (1+1) Hamiltonian approach.