Papapetrou transformation: Conditions to be satisfied to achieve transformation

In summary, the Papapetrou transformation is a coordinate transformation that reduces the contravariant form of the line element to diagonal form with equal coefficients for (dx_1)^2 and (dx_2)^2. This transformation is necessary and sufficient for achieving a desired metric, which is met by choosing ##\rho## and ##z## so that ##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}
  • #1
julian
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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.
 
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  • #2
julian said:
[...] Chandrasekhar's book. [...]
Which book? He's written quite a few.

Edit: Oh, I guess you mean "The Mathematical Theory of Black Holes" (1983).
Geez, I hate Chandrasekhar's conventions for chapter/section numbering.... (sigh)

Alas, I don't have time to give a detailed answer right now. I'll try later this week if no one else jumps in first. :oldfrown:
 
Last edited:

FAQ: Papapetrou transformation: Conditions to be satisfied to achieve transformation

What is the Papapetrou transformation?

The Papapetrou transformation is a mathematical technique used in general relativity to transform the metric tensor in a way that simplifies the equations of motion for certain types of spacetimes, particularly those involving rotating bodies.

What are the key conditions that need to be satisfied for a Papapetrou transformation?

The key conditions for a Papapetrou transformation typically include the existence of a Killing vector field, which represents a symmetry of the spacetime, and the spacetime being stationary and axisymmetric. Additionally, the metric must be expressible in a form that allows the transformation to decouple the equations of motion.

How does the Papapetrou transformation affect the metric tensor?

The Papapetrou transformation modifies the metric tensor by introducing new coordinates or potentials that exploit the symmetries of the spacetime. This often results in a simplified form of the metric, making it easier to solve the Einstein field equations in the presence of rotating bodies.

Can the Papapetrou transformation be applied to any spacetime?

No, the Papapetrou transformation cannot be applied to any spacetime. It is specifically useful for spacetimes that are stationary and axisymmetric, and where a Killing vector field exists. These conditions are not met in all spacetimes.

What are the practical applications of the Papapetrou transformation in physics?

The Papapetrou transformation is particularly useful in the study of rotating black holes and other astrophysical objects with rotational symmetry. It helps in deriving solutions to the Einstein field equations that describe the behavior of such objects, including their gravitational fields and the motion of particles around them.

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