Undergrad An alternative derivation of the equation of motion in General Relativity

Thread starter Bishal Banjara
Start date Mar 3, 2025
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The discussion explores a unique derivation of the equation of motion in General Relativity, highlighting a general form not typically covered in standard literature. It notes that while charged particles' motion is addressed in existing texts, mass motion lacks similar treatment. The author references a Wikipedia entry on relativistic Lagrangian mechanics, suggesting that an additional interaction term can be incorporated into the Lagrangian. This leads to an extra term in the equation of motion representing forces beyond gravity. The author seeks validation of this approach and requests authentic references to support or critique the derivation.

Mar 3, 2025

Bishal Banjara

TL;DR: I need a standard reference/s for an alternative form of the equation of motion in general relativity that I saw elsewhere.

I am extremely engaged in studying the different ways of derivation of the equation of motion in General Relativity. On the way, I found a very general form of the equation of motion that no standard books have done (to my knowledge).

Although the process is implemented in deriving the equation of motion for charged particles in standard books, it is done by none for the case of mass. Although it is not an authentic reference we can see the interaction term ##L_I## involved (added) to generalize the Lagrangian than that for the free particle somewhere in Wikipedia: https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechanics under the heading "Lagrangian formulation in general relativity."

This finally leads an extra term to the equation of motion ##f_{\alpha}## representing the additional source of force beside the gravitational force of the mass in reference by varying the lagrangians with the position of the particle. This is done without quoting any reference. I want to know its authenticity. Will anybody provide me with any authentic standard reference to this type of derivation? Or, is this the wrong way to do the job?

Last edited by a moderator: Mar 3, 2025

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