In physics, the Lemaître–Tolman metric, also known as the Lemaître–Tolman–Bondi metric or the Tolman metric, is a
Lorentzian metric based on an exact solution of
Einstein's field equations; it describes an
isotropic
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
and
expanding (or contracting)
universe
The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
which is not
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
,
and is thus used in
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
as an alternative to the standard
Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker metric (FLRW; ) is a metric that describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form o ...
to model the
expansion of the universe
The expansion of the universe is the increase in proper length, distance between Gravitational binding energy, gravitationally unbound parts of the observable universe with time. It is an intrinsic and extrinsic properties (philosophy), intrins ...
. It has also been used to model a universe which has a
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
distribution of matter to explain the
accelerating expansion of the universe. It was first found by
Georges Lemaître
Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, and mathematician who made major contributions to cosmology and astrophysics. He was the first to argue that the ...
in 1933 and
Richard Tolman
Richard Chace Tolman (March 4, 1881 – September 5, 1948) was an American mathematical physicist and physical chemist who made many contributions to statistical mechanics and theoretical cosmology. He was a professor at the California In ...
in 1934
and later investigated by
Hermann Bondi
Sir Hermann Bondi (1 November 1919 – 10 September 2005) was an Austrian-British people, British mathematician and physical cosmology, cosmologist.
He is best known for developing the steady state model of the universe with Fred Hoyle and Thom ...
in 1947.
Details
In a
synchronous reference system where
and
, the time coordinate
(we set
) is also the
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
and clocks at all points can be synchronized. For a dust-like medium where the pressure is zero, dust particles move freely i.e., along the geodesics and thus the synchronous frame is also a comoving frame wherein the components of four velocity
are
. The solution of the field equations yield
:
where
is the ''radius'' or ''luminosity distance'' in the sense that the surface area of a sphere with radius
is
and
is just interpreted as the
Lagrangian coordinate and
:
subjected to the conditions
and
, where
and
are arbitrary functions,
is the matter density and finally primes denote differentiation with respect to
. We can also assume
and
that excludes cases resulting in crossing of material particles during its motion. To each particle there corresponds a value of
, the function
and its time derivative respectively provides its law of motion and radial velocity. An interesting property of the solution described above is that when
and
are plotted as functions of
, the form of these functions plotted for the range