View source for Hartle–Thorne metric

{{Short description|Approximate solution to Einstein's field equations}}
{{general relativity|expanded=solutions}}
The '''Hartle–Thorne metric''' is an approximate solution of the vacuum [[Einstein field equations]] of general relativity<ref name="revistas">{{Cite journal |last1=Frutos Alfaro |first1=Francisco |last2=Soffel |first2=Michael |year=2017 |title=On the Post-Linear Quadrupole-Quadrupole Metric |journal=Revista de Matemática: Teoría y Aplicaciones |volume=24 |issue=2 |pages=239 |arxiv=1507.04264 |doi=10.15517/rmta.v24i2.29856 |s2cid=119159263}}</ref> that describes the exterior of a slowly and rigidly rotating, stationary and [[axially symmetric]] body.<ref>{{Cite journal |doi = 10.1086/149707|title = Slowly Rotating Relativistic Stars. II. Models for Neutron Stars and Supermassive Stars|journal = The Astrophysical Journal|volume = 153|pages = 807|year = 1968|last1 = Hartle|first1 = James B.|last2 = Thorne|first2 = Kip S.|bibcode = 1968ApJ...153..807H|doi-access = free}}</ref>

The metric was found by [[James Hartle]] and [[Kip Thorne]] in the 1960s to study the spacetime outside [[neutron stars]], [[white dwarfs]] and [[supermassive stars]]. It can be shown that it is an approximation to the [[Kerr metric]] (which describes a rotating black hole) when the quadrupole moment is set as <math>q=-a^2aM^3</math>, which is the correct value for a black hole but not, in general, for other astrophysical objects.

==Metric==
Up to second order in the [[angular momentum]] <math>J</math>, mass <math>M</math> and [[quadrupole moment]] <math>q</math>, the
metric in [[spherical coordinates]] is given by<ref name=revistas/>
:<math>\begin{align}g_{tt} &= - \left(1-\frac{2M}{r}+\frac{2q}{r^3} P_2 +\frac{2Mq}{r^4} P_2 +\frac{2q^2}{r^6} P^2_2
-\frac{2}{3} \frac{J^2}{r^4} (2P_2+1)\right), \\
g_{t\phi} &= -\frac{2J}{r}\sin^2\theta, \\
g_{rr} &= 1 + \frac{2M}{r} +\frac{4M^2}{r^2} -\frac{2qP_2}{r^3} -\frac{10MqP_2}{r^4}
+ \frac{1}{12} \frac{q^2\left(8P_2^2-16P_2+77\right)}{r^6} +\frac{2J^2(8P_2-1)}{r^4},\\
g_{\theta\theta} &=r^2 \left(1-\frac{2qP_2}{r^3} -\frac{5MqP_2}{r^4} +\frac{1}{36}\frac{q^2\left(44P_2^2 +8P_2 -43\right)}{r^6}
+\frac{J^2P_2}{r^4}\right),\\
g_{\phi\phi}&=r^2\sin^2\theta\left(1-\frac{2qP_2}{r^3} -\frac{5MqP_2}{r^4} +\frac{1}{36}\frac{q^2\left(44P_2^2 +8P_2 -43\right)}{r^6}
+\frac{J^2P_2}{r^4}\right),
\end{align} </math>

where
<math>P_2=\frac{3\cos^2\theta-1}{2}.</math>

==See also==
{{Portal|Astronomy|Physics}}
* [[Kerr metric]]
{{clear}}

==References==
{{reflist}}



{{DEFAULTSORT:Hartle-Thorne metric}}
[[Category:General relativity]]
[[Category:Metric tensors]]


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