Correction to: Z. Angew. Math. Phys. https://doi.org/10.1007/s00033-023-02140-4

There is an error in the final equation on page 8 of Hill [2], with the consequence that the solution analysis on page 9 and the first equation on page 10 do not apply. The solution analysis on page 9 is not employed in the remainder of the paper so that the remaining analysis leading up to the final equation on page 8, the subsequent analysis following page 9 and the Appendices of [2] are correct and remain unchanged. Accordingly, the error is strictly confined to page 9, the last equation of page 8 and the first equation of page 10. The purpose of this Corrigendum is to present the correct version of these details, including the correct formal general solution. A lengthy detailed analysis of the general solution is currently under investigation and will be presented in a subsequent publication Magyari and Hill [3].

Fromholz, Poisson and Will [1] use the Landau–Lifshitz formulation of the Einstein field equations of general relativity, to determine the Schwarzschild’s metric and derive the following nonlinear second-order ordinary differential equation for a function W(r), which they were unable to fully solve, thus

$$\begin{aligned} W''(r) - \frac{W'(r)}{r} = C^{*}\frac{W'(r)}{W(r)^2}, \end{aligned}$$
(0.1)

where primes denote derivatives with respect to r, and \(C^{*}\) denotes an arbitrary constant, and for which the Schwarzschild metric arises from the case \(C^{*} = 0\). The correct first integral of this equation is derived in [2] (see equations (4.3)–(4.6) of [2]), but there is an error in the final equation on page 8 of [2] leading to an error in equation (4.7) and the subsequent analysis on page 9 and the first equation of page 10. On making use of (0.1) the correct version of this equation should be

$$\begin{aligned} \frac{d}{dr}\left( W' - \frac{W}{r} + \frac{C^{*}}{W}\right) = \frac{W}{r^2}. \end{aligned}$$

With y as defined in [2], namely

$$\begin{aligned} y^2 = \left( \frac{W}{r}\right) ^2 + k^2 - \frac{2C^{*}}{r}, \end{aligned}$$

where k is the constant arising from the first integral, the correct first-order ordinary differential equation to be solved becomes

$$\begin{aligned} y' = \pm \frac{W}{r^2} = \pm \frac{1}{r} \left( y^2 - k^2 + \frac{2C^{*}}{r}\right) ^{1/2}, \end{aligned}$$

leading to the non-standard first-order ordinary differential equation

$$\begin{aligned} y'^2 - \frac{y^2}{r^2} = \frac{2C^{*}}{r^3}- \frac{k^2}{r^2}. \end{aligned}$$

While this equation appears not to be amenable to a direct solution method, the general solution in parametric form may be deduced by factorizing the left-hand side and splitting the equation into two equations, thus

$$\begin{aligned} y' + \frac{y}{r} = b(r), \quad \quad \quad y' - \frac{y}{r} = \frac{a(r)}{b(r)}, \end{aligned}$$

where \(a(r) = {2C^{*}}/{r^3}- {k^2}/{r^2}\) and b(r) is as yet an unknown function which is determined from the compatibility of the two expressions

$$\begin{aligned} y = \frac{r}{2}\left( b - \frac{a}{b}\right) , \quad \quad \quad y' = \frac{1}{2}\left( b + \frac{a}{b}\right) . \end{aligned}$$

On making use of the condition \(ra' + 2a = -{2C^{*}}/{r^3}\) we may eventually obtain the first-order differential equation for the function b(r), namely

$$\begin{aligned} \left( b + \frac{a}{b}\right) b' = -\frac{2C^{*}}{r^4}. \end{aligned}$$
(0.2)

Using \(\eta = b^2\) as the independent variable, on making the successive substitutions \(x = 1/r\) and \(\xi = x/\eta ^{1/2}\), equation (0.2) may be progressively transformed to the separable ordinary differential equation

$$\begin{aligned} \frac{d\xi }{d\eta } = \frac{1}{4C^{*}\eta ^{3/2}}\left( \frac{1}{\xi ^2} - k^2\right) , \end{aligned}$$

which may be fully integrated with the further substitution \(k\xi = \tanh \phi \), and the final integral becomes

$$\begin{aligned} \tanh ^{-1}(k\xi ) = \frac{1}{2}\log \left( \frac{1 + k\xi }{1 - k\xi }\right) = k\left( \xi - \frac{\lambda }{\eta ^{1/2}}\right) + \lambda C_1, \end{aligned}$$
(0.3)

where \(\lambda = k^2/2C^{*}\) and \(C_1\) denotes the constant of integration. On re-tracing the various substitutions and using the parameter \(z = k\xi = k/rb\) and the logarithmic expression for \(\tanh ^{-1}(k\xi ) = \tanh ^{-1}z\), we may deduce from (0.3) the expression for r in terms of z, thus

$$\begin{aligned} r = \frac{1}{\lambda }\left[ 1 - \frac{1}{2z}\log \left( \frac{1 + z}{1 - z}\right) \right] + \frac{C_1}{z}, \end{aligned}$$

while the parametric expression for W(r) is obtained from

$$\begin{aligned} W(r) = \pm r^2y' = \pm \frac{r^2}{2}\left( b + \frac{a}{b}\right) . \end{aligned}$$

In terms of the parameter z, this equation may be simplified to yield

$$\begin{aligned}&W(r) = \pm \frac{k}{2\lambda }\left[ z + \frac{(1 - z^2)\lambda r}{z}\right] \nonumber \\&= \pm \frac{k}{2\lambda z}\left[ 1 - \frac{(1 - z^2)}{2z}\log \left( \frac{1 + z}{1 - z}\right) + \frac{\lambda C_1(1 - z^2)}{z}\right] . \end{aligned}$$
(0.4)

A detailed investigation of the main physical implications of this general solution is currently in preparation [3].