Lemaître–Tolman metric

In physics, the Lemaître–Tolman metric, also known as the Lemaître–Tolman–Bondi metric or the Tolman metric, is a

Details

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational red shift Gravitational time dilation Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Formalisms ADM BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kerr–Newman–de Sitter Kasner Lemaître–Tolman Taub–NUT Milne Robertson–Walker Oppenheimer–Snyder pp-wave van Stockum dust Weyl−Lewis−Papapetrou
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedman Milne Zwicky Lemaître Oppenheimer Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Raychaudhuri Taylor Hulse van Stockum Taub Newman Yau Thorne others
Physics portal  Category
v t e

In a synchronous reference system where ${\displaystyle g_{00}=1}$ and ${\displaystyle g_{0\alpha }=0}$, the time coordinate ${\displaystyle x^{0}=t}$ (we set ${\displaystyle G=c=1}$) is also the proper time ${\displaystyle \tau ={\sqrt {g_{00}}}x^{0}}$ 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 ${\displaystyle u^{i}=dx^{i}/ds}$ are ${\displaystyle u^{0}=1,\,u^{\alpha }=0}$. The solution of the field equations yield^[9]

${\displaystyle ds^{2}=d\tau ^{2}-e^{\lambda (\tau ,R)}dR^{2}-r^{2}(\tau ,R)(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})}$

where ${\displaystyle r}$ is the radius or luminosity distance in the sense that the surface area of a sphere with radius ${\displaystyle r}$ is ${\displaystyle 4\pi r^{2}}$ and ${\displaystyle R}$ is just interpreted as the Lagrangian coordinate and

${\displaystyle e^{\lambda }={\frac {r'^{2}}{1+f(R)}},\quad \left({\frac {\partial r}{\partial \tau }}\right)^{2}=f(R)+{\frac {F(R)}{r}},\quad 4\pi r^{2}\rho ={\frac {F'(R)}{2r'}}}$

subjected to the conditions ${\displaystyle 1+f>0}$ and ${\displaystyle F>0}$, where ${\displaystyle f(R)}$ and ${\displaystyle F(R)}$ are arbitrary functions, ${\displaystyle \rho }$ is the matter density and finally primes denote differentiation with respect to ${\displaystyle R}$. We can also assume ${\displaystyle F'>0}$ and ${\displaystyle r'>0}$ that excludes cases resulting in crossing of material particles during its motion. To each particle there corresponds a value of ${\displaystyle R}$, the function ${\displaystyle r(\tau ,R)}$ and its time derivative respectively provides its law of motion and radial velocity. An interesting property of the solution described above is that when ${\displaystyle f(R)}$ and ${\displaystyle F(R)}$ are plotted as functions of ${\displaystyle R}$, the form of these functions plotted for the range ${\displaystyle R\in [0,R_{0}]}$ is independent of how these functions will be plotted for ${\displaystyle R>R_{0}}$. This prediction is evidently similar to the Newtonian theory. The total mass within the sphere ${\displaystyle R=R_{0}}$ is given by

${\displaystyle m=4\pi \int _{0}^{r(\tau ,R_{0})}\rho r^{2}dr=4\pi \int _{0}^{R_{0}}\rho r'r^{2}dR={\frac {F(R_{0})}{2}}}$

which implies that Schwarzschild radius is given by ${\displaystyle r_{s}=2m=F(R_{0})}$.

The function ${\displaystyle r(\tau ,R)}$ can be obtained upon integration and is given in a parametric form with a parameter ${\displaystyle \eta }$ with three possibilities,

${\displaystyle f>0:~~~~~~~~r={\frac {F}{2f}}(\cosh \eta -1),\quad \tau _{0}-\tau ={\frac {F}{2f^{3/2}}}(\sinh \eta -\eta ),}$

${\displaystyle f<0:~~~~~~~~r={\frac {F}{-2f}}(1-\cosh \eta ),\quad \tau _{0}-\tau ={\frac {F}{2(-f)^{3/2}}}(\eta -\sinh \eta )}$

${\displaystyle f=0:~~~~~~~~r=\left({\frac {9F}{4}}\right)^{1/3}(\tau _{0}-\tau )^{2/3}.}$

where ${\displaystyle \tau _{0}(R)}$ 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 ${\displaystyle f,F,\tau _{0}}$, only two are independent. In fact, since no particular selection has been made for the Lagrangian coordinate ${\displaystyle R}$ yet that can be subjected to arbitrary transformation, we can see that only two functions are arbitrary.^[10] For the dust-like medium, there exists another solution where ${\displaystyle r=r(\tau )}$ and independent of ${\displaystyle R}$, although such solution does not correspond to collapse of a finite body of matter.^[11]

Schwarzschild solution

When ${\displaystyle F=r_{s}=}$const., ${\displaystyle \rho =0}$ and therefore the solution corresponds to empty space with a point mass located at the center. Further by setting ${\displaystyle f=0}$ and ${\displaystyle \tau _{0}=R}$, the solution reduces to

• Lemaître coordinates
• Introduction to the mathematics of general relativity
• Stress–energy tensor
• Metric tensor (general relativity)
• Relativistic angular momentum
• inhomogeneous cosmology

References