A classical scrutiny of the Schwarzschild solution

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Referee this paper: DOI 10.13140/RG.2.2.34735.87207 by Nitin Gadre

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It is a well-known fact that the GR has a massive experimental support. The doubts raised in the article, https://www.researchgate.net/publication/350546500_A_classical_scrutiny_of_the_Schwarzschild_solution are regarding the geometrical aspects underlining the theory. The analysis goes as follows:

(1) We first write a classical total energy equation (Kinetic energy + Potential energy) for a three- dimensional flat space, in a conservative field. For the classical potential (-M/r) created by a static source, we get the Newton’s force equations from the three-dimensional Lagrangian analysis.

(2) We can then write the potential in kinetic energy form by introducing an additional coordinate and write a four-dimensional total energy equation. This will give same results as (1) above, in a four-dimensional Lagrangian analysis. We can also write a corresponding four-dimensional metric.

(3) We can then multiply the four-dimensional total energy equation in (2) by a suitable function to write the curved space four-dimensional modified total energy equation. Appropriate choice of the function will give us desired results in a four-dimensional Lagrangian analysis which can match the experimental observations (such as Mercury orbits).

The Lagrangian, Christoffel symbols and geodesic equations require the space to satisfy certain geometric properties such as symmetry of basis vector. A three-dimensional flat space satisfies these geometrical requirements. Hence, in the corresponding physical picture, the impact of such modification on the geometry of curved space needs to be examined.

(4) We can write a curved space-time metric (such as Schwarzschild metric) corresponding to the four-dimensional modified total energy equation in (3). A geodesic analysis of this metric will give same results for planetary orbits as the Lagrangian analysis of (3).

We can therefore establish a correspondence between the classical Lagrangian analysis and the geodesic equations analysis of GR, based on the mathematical similarities. It seems that, a very similar mathematical analysis has two different physical interpretations as per the classical Lagrangian and the GR approaches. In the Lagrangian analysis, we have to suitably modify the potential to get similar results for planetary orbits as the curved space geodesic analysis.

(5) However, it is difficult to draw a geometrical picture of the four-dimensional curved space. As it is difficult to write the unit vectors, even a proper coordinate system for the curved space cannot be defined. Thus, it becomes difficult to write a force vector or any other vector.

(6) To overcome this difficulty, it is suggested in standard GR that the space is generally curved but approximately flat in a local inertial frame. This arrangement can take care of the problem of defining vectors locally, but it is not sufficient to satisfy the necessary condition of symmetry of basis vectors.

The objection to the GR approach can be now stated: In a classical approach, the mathematics essentially suggests a model which gives same results in a theoretical analysis as the reality in the experiments. The model will be closer to reality if we can draw a corresponding geometrical picture. In GR, the mathematics itself is supposed to describe the reality. But if we are not able to define a coordinate system corresponding to a metric, then it becomes a mathematical expression without proper geometrical support, even if the results match the experimental observations. In the same manner, we may write many such metric expressions (adopting method in point (3)) which may not have geometrical support, but give us desired results in a mathematical analysis, matching various experimental observations. A classical three-dimensional flat space Largrangian analysis with a modified potential function can also give desired results for planetary orbits. In electrodynamics, we know that arrangements of charges in a source and its velocity can modify the potential function. This classical approach cannot give a cause and effect relationship for time dilation due to our limited knowledge of physical nature of entities such as the field. Still, the classical approach will be geometrically correct.