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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 g_=1 and g_=0, the time coordinate x^0=t (we set G=c=1) 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 ...
\tau=\sqrt x^0 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 u^i=dx^i/ds are u^0=1,\,u^\alpha=0. The solution of the field equations yield : ds^2 = d\tau^2 - e^ dR^2 - r^2(\tau,R) (d\theta^2 + \sin^2\theta d\phi^2) where r is the ''radius'' or ''luminosity distance'' in the sense that the surface area of a sphere with radius r is 4\pi r^2 and R is just interpreted as the Lagrangian coordinate and :e^\lambda = \frac, \quad \left(\frac\right)^2 = f(R) + \frac, \quad 4\pi r^2\rho = \frac subjected to the conditions 1+f>0 and F>0, where f(R) and F(R) are arbitrary functions, \rho is the matter density and finally primes denote differentiation with respect to R. We can also assume F'>0 and r'>0 that excludes cases resulting in crossing of material particles during its motion. To each particle there corresponds a value of R, the function r(\tau,R) and its time derivative respectively provides its law of motion and radial velocity. An interesting property of the solution described above is that when f(R) and F(R) are plotted as functions of R, the form of these functions plotted for the range R\in ,R_0/math> is independent of how these functions will be plotted for R>R_0. This prediction is evidently similar to the Newtonian theory. The total mass within the sphere R=R_0 is given by :m = 4\pi \int_0^ \rho r^2 dr=4\pi \int_0^ \rho r' r^2 dR= \frac which implies that Schwarzschild radius is given by r_s=2m=F(R_0). The function r(\tau,R) can be obtained upon integration and is given in a parametric form with a parameter \eta with three possibilities, : f > 0:~~~~~~~~ r = \frac(\cosh\eta-1), \quad \tau_0 -\tau = \frac(\sinh\eta-\eta), : f < 0:~~~~~~~~ r = \frac(1-\cosh\eta), \quad \tau_0 -\tau = \frac(\eta-\sinh\eta) : f = 0:~~~~~~~~ r = \left(\frac\right)^(\tau_0-\tau)^. where \tau_0(R) emerges as another arbitrary function. However, we know that centrally symmetric matter distribution can be described by at most two functions, namely their density distribution and the radial velocity of the matter. This means that of the three functions f,F,\tau_0, only two are independent. In fact, since no particular selection has been made for the Lagrangian coordinate R yet that can be subjected to arbitrary transformation, we can see that only two functions are arbitrary. For the dust-like medium, there exists another solution where r=r(\tau) and independent of R, although such solution does not correspond to collapse of a finite body of matter.


Schwarzschild solution

When F=r_s=const., \rho=0 and therefore the solution corresponds to empty space with a point mass located at the center. Further by setting f=0 and \tau_0=R, the solution reduces to Schwarzschild solution expressed in Lemaître coordinates.


Gravitational collapse

The gravitational collapse occurs when \tau reaches \tau_0(R) with \tau_0'>0. The moment \tau=\tau_0(R) corresponds to the arrival of matter denoted by its Lagrangian coordinate R to the center. In all three cases, as \tau\rightarrow \tau_0(R), the asymptotic behaviors are given by :r \approx \left(\frac\right)^(\tau_0-\tau)^, \quad e^ \approx \left(\frac\right)^ \frac (\tau_0-\tau)^, \quad 4\pi \rho \approx \frac in which the first two relations indicate that in the comoving frame, all radial distances tend to infinity and tangential distances approaches zero like \tau-\tau_0, whereas the third relation shows that the matter density increases like 1/(\tau_0-\tau). In the special case \tau_0(R)=constant where the time of collapse of all the material particle is the same, the asymptotic behaviors are different, :r \approx \left(\frac\right)^(\tau_0-\tau)^, \quad e^ \approx \left(\frac\right)^ \frac (\tau_0-\tau)^, \quad 4\pi \rho \approx \frac. Here both the tangential and radial distances goes to zero like (\tau_0-\tau)^, whereas the matter density increases like 1/(\tau_0-\tau)^2.


See also

* Lemaître coordinates *
Introduction to the mathematics of general relativity The mathematics of general relativity is complicated. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be s ...
*
Stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
*
Metric tensor (general relativity) In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as t ...
*
Relativistic angular momentum In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thre ...
* inhomogeneous cosmology


References

{{DEFAULTSORT:Lemaître-Tolman metric Physical cosmology Metric tensors Spacetime Coordinate charts in general relativity General relativity Gravity Exact solutions in general relativity