On a Schwarzschild like metric

In this short Note we would like to bring into the attention of people working in General Relativity a Schwarzschild like metric found by Professor Cleopatra Mociu\c{t}chi in sixties. It was obtained by the A. Sommerfeld reasoning from his treatise “Elektrodynamik” but using instead of the energy conserving law from the classical Physics, the relativistic energy conserving law.

💡 Research Summary

The paper under review revives a little‑known alternative to the Schwarzschild solution that was proposed in the 1960s by Professor Cleopatra Mociuțchi. The authors first give a pedagogical overview of the foundations of Newtonian mechanics, the Galilean principle of relativity, and the two postulates of special relativity (the invariance of physical laws in inertial frames and the constancy of the speed of light). From these postulates they derive the Lorentz transformation and the invariant line element
(ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}).
They then introduce Einstein’s equivalence principle in its various formulations (PE1–PE3) and the principle of general covariance, which together lead to the Einstein field equations
(R_{ik}-\frac12Rg_{ik}= \kappa T_{ik}).
In vacuum ((T_{ik}=0)) the field equations reduce to (R_{ik}=0), whose simplest exact solution is the Schwarzschild metric. The authors recount the standard derivation of the Schwarzschild line element in spherical coordinates, emphasizing that the product of the metric functions (A(r)B(r)=1) and that the integration constant is fixed by the asymptotic flatness condition, yielding
(A(r)=1-\lambda/r,; B(r)=(1-\lambda/r)^{-1}) with (\lambda=2GM/c^{2}).

The paper then revisits Arnold Sommerfeld’s textbook derivation of the same metric. Sommerfeld assumes a test particle moving radially in the static, spherically symmetric field of a central mass (M). He uses the classical energy conservation law
(\frac12 mv^{2}+V(r)=\text{const.}) with (V(r)=-GMm/r) to obtain the radial dependence of the velocity, (v^{2}/c^{2}=\lambda/r). Substituting this into the Lorentz‑contraction/ time‑dilation relations for an observer at a fixed radius reproduces the Schwarzschild line element.

Mociuțchi’s key innovation is to replace the classical energy law with the relativistic one derived from special relativity:
((m-m_{0})c^{2}-\frac{GMm}{r}=0,\qquad m=\frac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}).
Solving for the velocity yields (1-v^{2}/c^{2}=1-\mu/r) where (\mu=GM/c^{2}). Inserting this expression into the same Lorentz‑contraction formulas leads to a new spherically symmetric line element, which the authors call the Mociuțchi metric (MM):
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