The Carter constant, a fourth first integral for Kerr metric geodesics, was discovered in the course of showing that the Hamilton-Jacobi equation is separable for the Kerr metric. This prompts the question: if the direction of the argument is reversed, what conditions does separability of that equation impose on the metric?

For consistency with a previous presentation, begin with a metric of the form

$d{s}^2=(Adt+\epsilon Bd\phi {)}^2-(Cd\phi +\epsilon Ddt{)}^2-E^{2}d{r}^2-F^{2}d{\theta }^2$

where metric coefficients are initially unspecified as to functional dependence. The parameter ε measures deviation from a diagonal metric. The metric tensor is

$g=\left[\begin{array}{cccc}{A}^2-{\epsilon }^2{D}^2& \epsilon (AB-CD)& 0& 0\\ \epsilon (AB-CD)& -C^2+{\epsilon }^2{B}^2& 0& 0\\ 0& 0& -E^2& 0\\ 0& 0& 0& -F^2\end{array}\right]$

and its inverse is

$g^{-1}=\left[\begin{array}{cccc}\frac{C^2-{\epsilon }^2{B}^2}{(AC-{\epsilon }^{2}BD{)}^2}& \frac{\epsilon (AB-CD)}{(AC-{\epsilon }^{2}BD{)}^2}& 0& 0\\ \frac{\epsilon (AB-CD)}{(AC-{\epsilon }^{2}BD{)}^2}& -\frac{A^2-{\epsilon }^2{D}^2}{(AC-{\epsilon }^{2}BD{)}^2}& 0& 0\\ 0& 0& -\frac{1}{E^2}& 0\\ 0& 0& 0& -\frac{1}{F^2}\end{array}\right]$

The Hamilton-Jacobi equation in curved space can be written

$\frac{\partial S}{\partial s}=g^{kl}\frac{\partial S}{\partial x^k}\frac{\partial S}{\partial x^l}$

The equation is often written with an additional factor of one half on the right-hand side, but this can be absorbed into the affine parameter s. Inserting the components of the inverse metric tensor one has

$\begin{array}{l}\frac{\partial S}{\partial s}=\frac{C^2-{\epsilon }^2{B}^2}{(AC-{\epsilon }^{2}BD{)}^2}(\frac{\partial S}{\partial t}{)}^2+\frac{2\epsilon (AB-CD)}{(AC-{\epsilon }^{2}BD{)}^2}\frac{\partial S}{\partial t}\frac{\partial S}{\partial \phi }\\ -\frac{A^2-{\epsilon }^2{D}^2}{(AC-{\epsilon }^{2}BD{)}^2}(\frac{\partial S}{\partial \phi }{)}^2-\frac{1}{E^2}(\frac{\partial S}{\partial r}{)}^2-\frac{1}{F^2}(\frac{\partial S}{\partial \theta }{)}^2\end{array}$

For the form of the metric given the left-hand side of this equation factors easily,

$\begin{array}{l}\frac{\partial S}{\partial s}=\frac{1}{(AC-{\epsilon }^{2}BD{)}^2}(C\frac{\partial S}{\partial t}-\epsilon D\frac{\partial S}{\partial \phi }{)}^2\\ -\frac{1}{(AC-{\epsilon }^{2}BD{)}^2}(A\frac{\partial S}{\partial \phi }-\epsilon B\frac{\partial S}{\partial t}{)}^2\\ -\frac{1}{E^2}(\frac{\partial S}{\partial r}{)}^2-\frac{1}{F^2}(\frac{\partial S}{\partial \theta }{)}^2\end{array}$

where the squared quantities in numerators are proportional to the orthonormal frame field vectors applied to Hamilton’s principal function.

For the most general case of a separable principal function, one can write

$S=qs+S_t\left(t\right)+S_{\phi }\left(\phi \right)+S_r\left(r\right)+S_{\theta }\left(\theta \right)$

where q is a constant determining the nature of geodesics: $q=1$ for timelike geodesics, $q=0$ for null geodesics and $q=-1$ for spacelike geodesics.

For the Kerr metric, t and φ are both cyclic variables with trivial contributions to the principal function:

$S_t\left(t\right)=-E_{\mathrm{gr}}t{S}_{\phi }\left(\phi \right)=L\phi$

where the separation constant corresponding to energy is given a subscript to distinguish it from the metric function. The Hamilton-Jacobi equation becomes

$\begin{array}{l}(\frac{C{E}_{\mathrm{gr}}+\epsilon DL}{AC-{\epsilon }^{2}BD}{)}^2-(\frac{AL+\epsilon B{E}_{\mathrm{gr}}}{AC-{\epsilon }^{2}BD}{)}^2-\frac{1}{E^2}(\frac{d{S}_r}{dr}{)}^2-\frac{1}{F^2}(\frac{d{S}_{\theta }}{d\theta }{)}^2=q\end{array}$

where coefficients are now assumed to be functions of only r and θ. One can see immediately that an additively separable equation is only possible if the ratio of E and F is multiplicatively separable. This implies in general

$E=e\left(r\right)\rho (r,\theta )F=f\left(\theta \right)\rho (r,\theta )$

When these quantities are inserted into the equation and common factors cleared, the square of the function ρ will multiply a constant: for the equation to continue to be additively separable, so must this square. This square will also multiply the first two terms of the equation, so that the combinations

${\rho }^2(\frac{C{E}_{\mathrm{gr}}+\epsilon DL}{AC-{\epsilon }^{2}BD}{)}^2{\rho }^2(\frac{AL+\epsilon B{E}_{\mathrm{gr}}}{AC-{\epsilon }^{2}BD}{)}^2$

will each be a function of either r or θ but not both. Functions that are paired in the frame vectors and coframe forms, A with B and C with D, continue to be paired in the numerators here. Since cancellations between numerator and denominator are expected to achieve expressions in a single variable, the paired functions will share common multiplicatively separable factors. And because these paired functions appear under a square, any remaining functional dependence can only be in one variable or a common factor for all of these metric functions.

In the Schwarzchild limit $\epsilon \to 0$ the metric function A becomes a function of only r, so its radial dependence can be assigned as a common multiplicative factor to B. In the same limit the metric function C is a product of separate functions of r and θ, so it is not unreasonable to assign its angular dependence to D. Including a common factor for all these metric functions, one can reasonably take

$\begin{array}{l}A=a\left(r\right)\sigma (r,\theta )\\ B=a\left(r\right)b\left(\theta \right)\sigma (r,\theta )\\ C=c\left(r\right)d\left(\theta \right)\sigma (r,\theta )\\ D=d\left(\theta \right)\sigma (r,\theta )\end{array}$

With these choices the first two terms of the Hamilton-Jacobi equation are

$\frac{{\rho }^2}{{\sigma }^2}(\frac{c\left(r\right)E_{\mathrm{gr}}+\epsilon L}{a\left(r\right)\left[c\right(r)-{\epsilon }^{2}b(\theta \left)\right]}{)}^2\frac{{\rho }^2}{{\sigma }^2}(\frac{L+\epsilon b\left(\theta \right)E_{\mathrm{gr}}}{d\left(\theta \right)\left[c\right(r)-{\epsilon }^{2}b(\theta \left)\right]}{)}^2$

and both terms can be turned into functions of one variable with the single choice

${\rho }^2=\frac{1}{{\sigma }^2}=c\left(r\right)-{\epsilon }^{2}b\left(\theta \right)$

The metric functions as determined solely by additive separability of the Hamilton-Jacobi equation are

$\begin{array}{c}A=\frac{a\left(r\right)}{\rho }B=\frac{a\left(r\right)b\left(\theta \right)}{\rho }C=\frac{c\left(r\right)d\left(\theta \right)}{\rho }D=\frac{d\left(\theta \right)}{\rho }\\ \\ E=e\left(r\right)\rho F=f\left(\theta \right)\rho \end{array}$

That appears to be the extent of what can be inferred from separability, with the remaining functions evaluated by the usual procedures of differential geometry. There appears to be quite a bit of information here, with the general functional form of ρ and its relation to all metric functions already determined. One would expect such explicit forms to aid in a derivation of the full metric, but unfortunately the higher number of individual factors merely leads to cumbersome expressions not nearly as concise as a general derivation.

Features of the previous presentation, however, stand out rather clearly with these functional forms. The multiplicative separability of the metric function G, which was assumed to motivate the derivation, is now automatic

$G=g\left(r\right)h\left(\theta \right)=AC-{\epsilon }^{2}BD=a\left(r\right)d\left(\theta \right)$

with exact correspondences $g\left(r\right)=a\left(r\right)$ and $h\left(\theta \right)=d\left(\theta \right)$ between the two presentations. The functional form assumed for the ratio of E and F, with an added subscript to distinguish it,

$\frac{F}{E}=f(r{)}_{\mathrm{previous}}=\frac{f\left(\theta \right)}{e\left(r\right)}$

along with the subsequent assumption $f(r{)}_{\mathrm{previous}}=g\left(r\right)$ allows one to conclude here that

$e\left(r\right)=\frac{1}{a\left(r\right)}f\left(\theta \right)=1$

and obviates the need for the final step determining the metric function F.

The process to determine the explicit functional forms of $a\left(r\right)$ and $c\left(r\right)$ unfortunately does not appear to be avoidable even with all of the information available here. Pity.

Part of the motivation for this presentation was to determine if its reasoning could lead to generalizations of the Kerr metric in which metric functions are allowed to depend on all available variables. Inspection of the factored form of the Hamilton-Jacobi equation indicates that it would remain additively separable if the first four metric functions were replaced by

$\begin{array}{c}A\to A\frac{\partial S}{\partial t}\\ D\to D\frac{\partial S}{\partial t}\end{array}\begin{array}{c}B\to B\frac{\partial S}{\partial \phi }\\ C\to C\frac{\partial S}{\partial \phi }\end{array}$

With these replacements the original metric becomes

$d{s}^2=(A\frac{\partial S}{\partial t}dt+\epsilon B\frac{\partial S}{\partial \phi }d\phi {)}^2-(C\frac{\partial S}{\partial \phi }d\phi +\epsilon D\frac{\partial S}{\partial t}dt{)}^2-E^{2}d{r}^2-F^{2}d{\theta }^2$

and it is immediately clear that such an attempted generalization is merely a redefinition of the coordinates t and φ and will not differ from the standard Kerr metric. Back to the drawing board...

Uploaded 2018.04.28 analyticphysics.com