Lemaître–Tolman metric

Lemaître–Tolman metric
NASA / WMAP Science Team · Public domain · source
Name	Lemaître–Tolman metric
Type	Exact solution
Field	General relativity
Discoverers	Georges Lemaître, Richard C. Tolman, Howard P. Robertson
First published	1933
Coordinates	comoving, synchronous

Lemaître–Tolman metric is an exact, spherically symmetric, inhomogeneous solution of Einstein field equations for a dust-filled spacetime that generalizes homogeneous Friedmann–Lemaître–Robertson–Walker metric models to allow radial inhomogeneity. It is widely used in studies of cosmological inhomogeneities, gravitational collapse, void models, and observational tests involving Hubble constant, supernova luminosity distances, and cosmic microwave background anisotropies. The metric links work by Georges Lemaître, Richard C. Tolman, and later treatments influenced by Howard P. Robertson, and it underpins many theoretical analyses by researchers such as George F. R. Ellis, Hermann Bondi, and Roy Maartens.

Introduction

The solution describes a pressureless fluid ("dust") in General relativity with spherical symmetry about a central worldline, accommodating arbitrary radial profiles for energy density and expansion. It reduces to the Friedmann–Lemaître–Robertson–Walker metric under special choices of functions and shares mathematical structure with the Schwarzschild metric in vacuum regions. The model has been exploited in analyses by Stephen Hawking, Roger Penrose, Martin Rees, P. J. E. Peebles, and James Peebles for issues ranging from gravitational lensing to backreaction.

Mathematical Formulation

In comoving, synchronous coordinates (t, r, θ, φ) the line element is usually written with two arbitrary functions of r: an energy function E(r) and a mass function M(r), plus a bang-time function t_B(r). The metric components connect to the Einstein field equations via the radial evolution equation analogous to the Friedmann equation: - radial scale R(t,r) satisfies (dR/dt)^2 = 2E(r) + 2M(r)/R. The functions M(r), E(r), and t_B(r) play roles similar to parameters in Friedmann models and determine whether local evolution is hyperbolic, parabolic, or elliptic, mirroring classifications familiar from Karl Schwarzschild and Alexander Friedmann analyses. Junction conditions to exterior vacuum solutions invoke relations used in matching schemes by Werner Israel and applications by Paul S. Wesson.

Physical Interpretation and Properties

Physically, M(r) measures the active gravitational mass within comoving radius r, E(r) parametrizes local energy per unit mass, and t_B(r) encodes non-simultaneous big-bang surfaces that allow shell-dependent initial conditions studied by George F. R. Ellis and Herman Bondi. The dust assumption makes the model suitable for late-time cosmology and stages of gravitational collapse investigated by Subrahmanyan Chandrasekhar and Alan Guth. Key properties include possible shell-crossing singularities, shell-focusing singularities, and local apparent horizons determined by R = 2M, connecting to horizon analyses by Stephen Hawking and Roger Penrose. The metric preserves spherical symmetry like the Reissner–Nordström metric in charged contexts when generalized.

Exact Solutions and Special Cases

Specializations yield known exact solutions: choosing E(r) = const and t_B(r) = const recovers Friedmann–Lemaître–Robertson–Walker metric subclasses studied by Alexander Friedmann and Georges Lemaître; setting M(r)=const outside a radius recovers the Schwarzschild metric exterior to a dust sphere as in Oppenheimer–Snyder collapse. The marginally bound case E(r)=0 gives parabolic evolution used in analytic models by Richard C. Tolman and later exploited in works by George F. R. Ellis and Hermann Bondi. Elliptic (E<0) and hyperbolic (E>0) regimes parallel analyses in the literature by Edward Kolb and Michael Turner for cosmological parameter dependence.

Cosmological and Astrophysical Applications

Researchers have applied the metric to interpret apparent acceleration without dark energy via inhomogeneous void models developed by groups including Timothy Clifton, Chris Clarkson, and Julian Barbour, and to compute observables such as luminosity distance-redshift relations compared against Type Ia supernova surveys led by teams like Saul Perlmutter, Adam Riess, and Brian Schmidt. Applications extend to modeling structure formation, Swiss-cheese models matched to ΛCDM backgrounds used by Pawel Mazur, gravitational lensing studies referenced by Pyne Wilkinson-style analyses, and examinations of cosmic backreaction pursued by Thomas Buchert and Syksy Räsänen. The metric is also a tool in studies of black hole formation and naked singularity criteria engaged by Pankaj S. Joshi and Demetrios Christodoulou.

Historical Development and Attribution

The solution’s roots are in early 20th-century relativistic cosmology: Georges Lemaître developed inhomogeneous cosmologies, while Richard C. Tolman provided comprehensive formulations for spherically symmetric dust in the 1930s; later expositions and rediscoveries involved Hermann Bondi and George F. R. Ellis. Subsequent rigorous mathematical treatments and observational applications were advanced by scholars including Roger Penrose, Stephen Hawking, Pankaj S. Joshi, and Thomas Buchert, consolidating the metric’s role in modern relativistic cosmology and gravitational collapse research.

Category:Exact solutions in general relativity