# 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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requested Apr 3

paper authored Apr 1 to astro-ph

The article summary is as follows:

We can establish a mathematical correspondence between the classical Lagrangian approach and geodesic analysis as suggested by the standard general relativity (GR), in finding the nature of planetary orbits:

• (1) 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) Write the potential in kinetic energy form by introducing an additional coordinate and convert the three-dimensional total energy equation into a four-dimensional total energy equation. This will give same results as (1) above, in a four-dimensional Lagrangian analysis.
• (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). However, the impact of such modification on the geometry of space needs to be examined.
• (4) We can then 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. The physical interpretations shall remain different as per the classical and GR approaches.
• (5) However, it is difficult to draw a geometrical picture of the four-dimensional curved space. It is difficult to write expressions for the unit vectors and the unit vector derivatives give absurd results. As it is difficult to write the unit vectors, a proper coordinate system cannot be defined. Therefore, it becomes difficult to define a force vector or any other vector such as the position vector, which are important in a classical approach.
• (6) To overcome this difficulty, it is suggested in standard GR analysis that the space is generally curved but approximately flat in a local inertial frame. This argument may take care of the problem locally, but it is not sufficient to give a proper geometrical picture to the curved space.

The question is: Is the classical flat space three-dimensional Lagrangian analysis geometrically superior to the GR analysis of a four-dimensional curved space, in finding planetary orbits, when we can get similar results in a three-dimensional flat space by modifying potential? We can get this expression for the modified potential by rearranging the curved space four-dimensional metric (or the corresponding total energy equation). The flat space analysis will always have a proper geometric support.

We already know that, the arrangement of charges in the source and speed of the source modifies the potential function, in electrodynamics.

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