|
|
| Line 1: |
Line 1: |
| {{general relativity}}
| | Andrew Berryhill is what his spouse enjoys to contact him and he totally digs that name. To perform lacross is one of the things she loves most. Distributing manufacturing is exactly where her primary income arrives from. Alaska is exactly where I've usually been residing.<br><br>My homepage ... free psychic reading ([http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 aseandate.com]) |
| {{multiple issues|
| |
| {{no footnotes|date=January 2012}}
| |
| {{Technical|date=August 2011|reason=This article is great if one is a mathematician. This is an interesting metric that really needs an expert's touch to explain it. P. Ellsworth, ed.}}
| |
| }}
| |
| | |
| In mathematical physics, the '''Lemaître–Tolman metric''' is the spherically symmetric dust solution of [[Einstein's field equations]] was first found by [[Georges Lemaître|Lemaître]] in 1933 and then [[Richard Tolman|Tolman]] in 1934. It was later investigated by [[Hermann Bondi|Bondi]] in 1947. This solution describes a spherical cloud of dust (finite or infinite) that is expanding or collapsing under gravity. It is also known as the '''Lemaître-Tolman-Bondi metric''' and the '''Tolman metric'''.
| |
| | |
| The metric is:
| |
| : <math>\mathrm{d}s^{2} = \mathrm{d}t^2 - \frac{(R')^2}{1 + 2 E} \mathrm{d}r^2 - R^2 \, \mathrm{d}\Omega^2</math>
| |
| | |
| where: | |
| : <math>\mathrm{d}\Omega^2 = \mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2</math></center>
| |
| : <math>R = R(t,r)~,~~~~~~~~ R' = \partial R / \partial r~,~~~~~~~~ E = E(r)</math>
| |
| | |
| The matter is comoving, which means its 4-velocity is:
| |
| : <math>u^a = \delta^a_0 = (1, 0, 0, 0)</math>
| |
| so the spatial coordinates <math>(r, \theta, \phi)</math> are attached to the particles of dust.
| |
| | |
| The pressure is zero (hence ''dust''), the density is
| |
| : <math>8 \pi \rho = \frac{2 M'}{R^2 \, R'}</math>
| |
| and the evolution equation is
| |
| : <math>\dot{R}^2 = \frac{2 M}{R} + 2 E</math>
| |
| where | |
| : <math>\dot{R} = \partial R / \partial t</math>
| |
| | |
| The evolution equation has three solutions, depending on the sign of <math>E</math>,
| |
| : <math>E > 0:~~~~~~~~ R = \frac{M}{2 E} (\cosh\eta - 1)~,~~~~~~~~ (\sinh\eta - \eta) = \frac{(2 E)^{3/2} (t - t_B)}{M}~;</math>
| |
| : <math>E = 0:~~~~~~~~ R = \left( \frac{9 M (t - t_B)^2}{2} \right)^{1/3}~;</math>
| |
| : <math>E < 0:~~~~~~~~ R = \frac{M}{2 E} (1 - \cos\eta)~,~~~~~~~~ (\eta - \sin\eta) = \frac{(-2 E)^{3/2} (t - t_B)}{M}~;</math>
| |
| which are known as ''hyperbolic'', ''parabolic'', and ''elliptic'' evolutions respectively.
| |
| | |
| The meanings of the three arbitrary functions, which depend on <math>r</math> only, are:
| |
| * <math>E(r)</math> – both a local geometry parameter, and the energy per unit mass of the dust particles at comoving coordinate radius <math>r</math>,
| |
| * <math>M(r)</math> – the gravitational mass within the comoving sphere at radius <math>r</math>,
| |
| * <math>t_B(r)</math> – the time of the big bang for worldlines at radius <math>r</math>.
| |
| | |
| Special cases are the [[Schwarzschild metric]] in [[geodesic coordinate]]s
| |
| <math>M =</math> constant, and the [[Friedmann–Lemaître–Robertson–Walker metric]], e.g. <math>E = 0~,~~ t_B =</math> constant for the flat case. | |
| | |
| ==See also==
| |
| | |
| *[[Lemaître coordinates]]
| |
| *[[Introduction to the mathematics of general relativity]]
| |
| *[[Stress–energy tensor]]
| |
| *[[Metric tensor (general relativity)]]
| |
| *[[Relativistic angular momentum]]
| |
| | |
| == References ==
| |
| | |
| * {{cite journal | last =Bondi | first =Hermann | authorlink =Hermann Bondi | coauthors = | title =Spherically symmetrical models in general relativity | journal =Monthly Notices of the Royal Astronomical Society | volume =107 | issue = | pages =410 | publisher = | location = | year =1947 | issn = | doi = | id = | archiveurl= | archivedate= | bibcode =1947MNRAS.107..410B }}
| |
| * Krasinski, A., ''Inhomogeneous Cosmological Models'', (1997) Cambridge UP, ISBN 0-521-48180-5
| |
| * Lemaitre, G., Ann. Soc. Sci. Bruxelles, A53, 51 (1933).
| |
| * {{cite journal | last =Tolman | first =Richard C. | authorlink =Richard C. Tolman | coauthors = | title =Effect of Inhomogeneity on Cosmological Models | journal =Proc. Natl. Acad. Sci. | volume =20 | issue =3 | pages =169 | publisher =National Academy of Sciences of the USA | location = | year =1934 | url =http://www.pnas.org/content/20/3/169.full.pdf | issn = | doi = | id = | accessdate =2011-01-27 | archiveurl=http://www.webcitation.org/5w35fIoge | archivedate=2011-01-27 | pmid =16587869 | pmc =1076370 }}
| |
| | |
| {{relativity-stub}}
| |
| | |
| {{DEFAULTSORT:Lemaitre-Tolman metric}}
| |
| | |
| [[Category:Metric tensors]]
| |
| [[Category:Spacetime]]
| |
| [[Category:Coordinate charts in general relativity]]
| |
| [[Category:General relativity]]
| |
| [[Category:Gravitation]]
| |
| [[Category:Exact solutions in general relativity]]
| |
Andrew Berryhill is what his spouse enjoys to contact him and he totally digs that name. To perform lacross is one of the things she loves most. Distributing manufacturing is exactly where her primary income arrives from. Alaska is exactly where I've usually been residing.
My homepage ... free psychic reading (aseandate.com)