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- TL;DR Summary
- The Papapetrou transformation. Conditions to be satisfied to achieve requirements of transformation. My conditions don't match Chandrasekhar's conditions.
I'm looking at the Papapetrou transformation in Ch. 6, ##\S 52## of Chandrasekhar's book. He cf's Ch. 2, ##\S##11.I understand Ch. 2, ##\S##11. There he considers a coordinate transformation,
\begin{align*}
{x'}^1 = \phi (x^1,x^2) \qquad \text{and} \qquad {x'}^2 = \psi (x^1,x^2)
\end{align*}
which will reduce the contravariant form of the line element
\begin{align*}
ds^2 = g^{11} (dx_1)^2 + 2 g^{12} dx_1 dx_2 + g^{22} (dx_2)^2
\end{align*}
to diagonal form with equal coefficients for ##(dx_1)^2## and ##(dx_2)^2##. For a transformation to achieve this it is necessary and sufficient that
\begin{align*}
g^{'12} = g^{11} \phi_{,1} \psi_{,1} + 2 g^{12} (\phi_{,1} \psi_{,2} + \phi_{,2} \psi_{,1}) + g^{22} \phi_{,2} \psi_{,2} = 0
\end{align*}
\begin{align*}
g^{'11} - g^{'22} = g^{11} ({\phi_{,1}}^2 - {\psi_{,1}}^2) + 2 g^{12} (\phi_{,1} \phi_{,2} - \psi_{,1} \psi_{,2}) + g^{22} ({\phi_{,2}}^2 - {\psi_{,2}}^2) = 0
\end{align*}
I get all of this.In Ch. 6, ##\S##52, (b) The Papapetrou transformation, he is wanting to perform a coordinate transformation
\begin{align*}
(x^2,x^3) \rightarrow (\rho , z)
\end{align*}
such that
\begin{align*}
e^{2 \mu} [(dx_2)^2 + (dx_3)^2] \rightarrow f (\rho , z) [(d \rho)^2 + (dz)^2]
\end{align*}Regarding the possibility of making such a coordinate transformation, he cf's Ch. 2 ##\S##11. So I was thinking I should write
\begin{align*}
{x'}^2 = \rho (x^2,x^3) \qquad \text{and} \qquad {x'}^3 = z (x^2,x^3)
\end{align*}
where ##\rho## and ##z## are to be chosen so that the metric remains in diagonal form and with equal coefficients for ##(d \rho)^2## and ##(dz)^2##. For a transformation to achieve this it is necessary and sufficient that
\begin{align*}
g^{'23} = g^{22} \rho_{,2} z_{,2} + 2 g^{23} (\rho_{,2} z_{,3} + \rho_{,3} z_{,2}) + g^{33} \rho_{,3} z_{,3} = 0
\end{align*}
\begin{align*}
g^{'22} - g^{'33} = g^{22} ({\rho_{,2}}^2 - {z_{,2}}^2) + 2 g^{23} (\rho_{,2} \rho_{,3} - z_{,2} z_{,3}) + g^{33} ({\rho_{,3}}^2 - {z_{,3}}^2) = 0
\end{align*}
As ##g^{23} = 0## and ##g^{22} = g^{33}##, the first condition requires
\begin{align*}
\rho_{,2} z_{,2} + \rho_{,3} z_{,3} = 0
\end{align*}
As ##g^{23} = 0## and ##g^{22} = g^{33}##, the second condition requires
\begin{align*}
{\rho_{,2}}^2 - {z_{,2}}^2 = - {\rho_{,3}}^2 + {z_{,3}}^2
\end{align*}However, Chandrasekhar gets these conditions instead:
\begin{align*}
{\rho_{,2}}^2 + {z_{,2}}^2 & = {\rho_{,3}}^2 + {z_{,3}}^2
\nonumber \\
\rho_{,2} \rho_{,3} + z_{,2} z_{,3} & = 0
\end{align*}
How does Chandrasekhar arrive at these conditions?Are my conditions not necessary and sufficient conditions for the transformation to achieve the requirements I stated? Does Chandrasekhar have other requirements in mind? Chandrasekhar notes that his conditions are satisfied by ##\rho_{,2} = +z_{,3}## and ##\rho_{,3} = - z_{,2}##. I notice that my conditions are satisfied by these choices as well.
\begin{align*}
{x'}^1 = \phi (x^1,x^2) \qquad \text{and} \qquad {x'}^2 = \psi (x^1,x^2)
\end{align*}
which will reduce the contravariant form of the line element
\begin{align*}
ds^2 = g^{11} (dx_1)^2 + 2 g^{12} dx_1 dx_2 + g^{22} (dx_2)^2
\end{align*}
to diagonal form with equal coefficients for ##(dx_1)^2## and ##(dx_2)^2##. For a transformation to achieve this it is necessary and sufficient that
\begin{align*}
g^{'12} = g^{11} \phi_{,1} \psi_{,1} + 2 g^{12} (\phi_{,1} \psi_{,2} + \phi_{,2} \psi_{,1}) + g^{22} \phi_{,2} \psi_{,2} = 0
\end{align*}
\begin{align*}
g^{'11} - g^{'22} = g^{11} ({\phi_{,1}}^2 - {\psi_{,1}}^2) + 2 g^{12} (\phi_{,1} \phi_{,2} - \psi_{,1} \psi_{,2}) + g^{22} ({\phi_{,2}}^2 - {\psi_{,2}}^2) = 0
\end{align*}
I get all of this.In Ch. 6, ##\S##52, (b) The Papapetrou transformation, he is wanting to perform a coordinate transformation
\begin{align*}
(x^2,x^3) \rightarrow (\rho , z)
\end{align*}
such that
\begin{align*}
e^{2 \mu} [(dx_2)^2 + (dx_3)^2] \rightarrow f (\rho , z) [(d \rho)^2 + (dz)^2]
\end{align*}Regarding the possibility of making such a coordinate transformation, he cf's Ch. 2 ##\S##11. So I was thinking I should write
\begin{align*}
{x'}^2 = \rho (x^2,x^3) \qquad \text{and} \qquad {x'}^3 = z (x^2,x^3)
\end{align*}
where ##\rho## and ##z## are to be chosen so that the metric remains in diagonal form and with equal coefficients for ##(d \rho)^2## and ##(dz)^2##. For a transformation to achieve this it is necessary and sufficient that
\begin{align*}
g^{'23} = g^{22} \rho_{,2} z_{,2} + 2 g^{23} (\rho_{,2} z_{,3} + \rho_{,3} z_{,2}) + g^{33} \rho_{,3} z_{,3} = 0
\end{align*}
\begin{align*}
g^{'22} - g^{'33} = g^{22} ({\rho_{,2}}^2 - {z_{,2}}^2) + 2 g^{23} (\rho_{,2} \rho_{,3} - z_{,2} z_{,3}) + g^{33} ({\rho_{,3}}^2 - {z_{,3}}^2) = 0
\end{align*}
As ##g^{23} = 0## and ##g^{22} = g^{33}##, the first condition requires
\begin{align*}
\rho_{,2} z_{,2} + \rho_{,3} z_{,3} = 0
\end{align*}
As ##g^{23} = 0## and ##g^{22} = g^{33}##, the second condition requires
\begin{align*}
{\rho_{,2}}^2 - {z_{,2}}^2 = - {\rho_{,3}}^2 + {z_{,3}}^2
\end{align*}However, Chandrasekhar gets these conditions instead:
\begin{align*}
{\rho_{,2}}^2 + {z_{,2}}^2 & = {\rho_{,3}}^2 + {z_{,3}}^2
\nonumber \\
\rho_{,2} \rho_{,3} + z_{,2} z_{,3} & = 0
\end{align*}
How does Chandrasekhar arrive at these conditions?Are my conditions not necessary and sufficient conditions for the transformation to achieve the requirements I stated? Does Chandrasekhar have other requirements in mind? Chandrasekhar notes that his conditions are satisfied by ##\rho_{,2} = +z_{,3}## and ##\rho_{,3} = - z_{,2}##. I notice that my conditions are satisfied by these choices as well.
