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The '''Hartle-Thorne metric''' is a [[spacetime metric]] in [[General Relativity]] describes the exterior of a slowly and rigidly rotating, stationary and axially symmetric body.<ref>https://ift.tt/2HVQHyC> It is an approximate solution of the vacuum [[Einstein equations]].<ref name=revistas>https://ift.tt/2uE5eqN>
The metric was found by [[James Hartle]] and [[Kip Thorne]].
==Metric==
Up to second order in the angular momentum <math>J</math>, mass <math>M</math> and quadrupole moment <math>q</math>, the
metric 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>
==References==
The metric was found by [[James Hartle]] and [[Kip Thorne]].
==Metric==
Up to second order in the angular momentum <math>J</math>, mass <math>M</math> and quadrupole moment <math>q</math>, the
metric 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>
==References==
from Wikipedia - New pages [en] https://ift.tt/2I0bzEU
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