Lemaître–Tolman metric

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 and expanding (or contracting) universe which is not homogeneous,^[1][2] and is thus used in cosmology as an alternative to the standard Friedmann–Lemaître–Robertson–Walker metric to model the expansion of the universe.^[3][4][5] It has also been used to model a universe which has a fractal distribution of matter to explain the accelerating expansion of the universe.^[6] It was first found by Georges Lemaître in 1933^[7] and Richard Tolman in 1934^[1] and later investigated by Hermann Bondi in 1947.^[8]

Details

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 Schwarzschild solution expressed in Lemaître coordinates.

Gravitational collapse

The gravitational collapse occurs when ${\displaystyle \tau }$ reaches ${\displaystyle \tau _{0}(R)}$ with ${\displaystyle \tau _{0}'>0}$. The moment ${\displaystyle \tau =\tau _{0}(R)}$ corresponds to the arrival of matter denoted by its Lagrangian coordinate ${\displaystyle R}$ to the center. In all three cases, as ${\displaystyle \tau \rightarrow \tau _{0}(R)}$, the asymptotic behaviors are given by

${\displaystyle r\approx \left({\frac {9F}{4}}\right)^{1/3}(\tau _{0}-\tau )^{2/3},\quad e^{\lambda /2}\approx \left({\frac {2F}{3}}\right)^{1/3}{\frac {\tau _{0}'}{\sqrt {1+f}}}(\tau _{0}-\tau )^{-1/3},\quad 4\pi \rho \approx {\frac {F'}{3F\tau _{0}'(\tau _{0}-\tau )}}}$

in which the first two relations indicate that in the comoving frame, all radial distances tend to infinity and tangential distances approaches zero like ${\displaystyle \tau -\tau _{0}}$, whereas the third relation shows that the matter density increases like ${\displaystyle 1/(\tau _{0}-\tau ).}$ In the special case ${\displaystyle \tau _{0}(R)=}$constant where the time of collapse of all the material particle is the same, the asymptotic behaviors are different,

${\displaystyle r\approx \left({\frac {9F}{3}}\right)^{1/3}(\tau _{0}-\tau )^{2/3},\quad e^{\lambda /2}\approx \left({\frac {2}{3}}\right)^{1/3}{\frac {F'}{2F^{2/3}{\sqrt {1+f}}}}(\tau _{0}-\tau )^{2/3},\quad 4\pi \rho \approx {\frac {2}{3(\tau _{0}-\tau )^{2}}}.}$

Here both the tangential and radial distances goes to zero like ${\displaystyle (\tau _{0}-\tau )^{2/3}}$, whereas the matter density increases like ${\displaystyle 1/(\tau _{0}-\tau )^{2}.}$

See also

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

References

1. Tolman, Richard C. (1934). "Effect of Inhomogeneity on Cosmological Models". Proc. Natl. Acad. Sci. 20 (3). National Academy of Sciences of the USA: 169–76. Bibcode:1934PNAS...20..169T. doi:10.1073/pnas.20.3.169. PMC 1076370. PMID 16587869.
2. Krasiński, Andrzej (1997). Inhomogeneous cosmological models (Digitally printed first paperback version (with corrections) ed.). Cambridge ; New York: Cambridge University Press. ISBN 978-0-521-03017-5.
3. Kenworthy, W. D’Arcy; Scolnic, Dan; Riess, Adam (20 April 2019). "The Local Perspective on the Hubble Tension: Local Structure Does Not Impact Measurement of the Hubble Constant". The Astrophysical Journal. 875 (2): 145. arXiv:1901.08681. Bibcode:2019ApJ...875..145K. doi:10.3847/1538-4357/ab0ebf. ISSN 0004-637X.
4. Cai, Rong-Gen; Ding, Jia-Feng; Guo, Zong-Kuan; Wang, Shao-Jiang; Yu, Wang-Wei (2021-06-22). "Do the observational data favor a local void?". Physical Review D. 103 (12): 123539. arXiv:2012.08292. Bibcode:2021PhRvD.103l3539C. doi:10.1103/PhysRevD.103.123539. ISSN 2470-0010. S2CID 229180790.
5. Luković, Vladimir V; Haridasu, Balakrishna S; Vittorio, Nicola (4 November 2019). "Exploring the evidence for a large local void with supernovae Ia data". Monthly Notices of the Royal Astronomical Society. 491 (2). arXiv:1907.11219. doi:10.1093/mnras/stz3070. ISSN 0035-8711.
6. Cosmai, L; Fanizza, G; Sylos Labini, F; Pietronero, L; Tedesco, L (2019-02-21). "Fractal universe and cosmic acceleration in a Lemaître–Tolman–Bondi scenario". Classical and Quantum Gravity. 36 (4): 045007. arXiv:1810.06318. Bibcode:2019CQGra..36d5007C. doi:10.1088/1361-6382/aae8f7. ISSN 0264-9381. S2CID 119517591.
7. Lemaître, G. (1933). "l'Universe en expansion". Annales de la Société Scientifique de Bruxelles. 53: 51–85.
8. Bondi, Hermann (1947). "Spherically symmetrical models in general relativity". Monthly Notices of the Royal Astronomical Society. 107 (5–6): 410–425. Bibcode:1947MNRAS.107..410B. doi:10.1093/mnras/107.5-6.410.
9. Landau, Lev Davidovič; Lifšic, Evgenij M. (2010). The classical theory of fields. Course of theoretical physics / L. D. Landau and E. M. Lifshitz (4. rev. Engl. ed., repr ed.). Amsterdam Heidelberg: Elsevier Butterworth Heinemann. ISBN 978-0-7506-2768-9.
10. Zelʹdovich, I︠A︡ B.; Novikov, I. D. (1996). Stars and relativity. Mineola, N.Y: Dover Publications. ISBN 978-0-486-69424-5.
11. Ruban, V. A. (1969). "Spherically symmetric T-models in the general theory of relativity" (PDF). Soviet Journal of Experimental and Theoretical Physics. 29 (6).