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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
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
With y as defined in [2], namely
where k is the constant arising from the first integral, the correct first-order ordinary differential equation to be solved becomes
leading to the non-standard first-order ordinary differential equation
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
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
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
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
which may be fully integrated with the further substitution \(k\xi = \tanh \phi \), and the final integral becomes
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
while the parametric expression for W(r) is obtained from
In terms of the parameter z, this equation may be simplified to yield
A detailed investigation of the main physical implications of this general solution is currently in preparation [3].
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References
Fromholz, P., Poisson, E., Will, C.M.: The Schwarzschild metric: It’s the coordinates, stupid! Am. J. Phys. 82, 295–300 (2014)
Hill, J.M.: Schwarzschild’s metric and the Landau–Lifshitz formulation of general relativity. Z. Angew. Math. Phys. 74, 246 (2023)
Magyari, E., Hill, J.M.: A note on Schwarzschild’s metric arising from the Landau–Lifshitz formulation. In preparation (2024)
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Hill, J.M., Magyari, E. Correction to: Schwarzschild’s metric and the Landau–Lifshitz formulation of general relativity. Z. Angew. Math. Phys. 75, 157 (2024). https://doi.org/10.1007/s00033-024-02244-5
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DOI: https://doi.org/10.1007/s00033-024-02244-5