They derived a new gravitational wave equation from the general relativistic
Hamilton-Jacobi equation for a test particle of mass [mu] as given by Landau and Lifshitz:
Dispersion Equation Versus
Hamilton-Jacobi EquationIn this condition the
Hamilton-Jacobi equation of the problem can be written as
The first equation can be taken as the classical
Hamilton-Jacobi equation with one extra term.
Neglecting the self-gravitation, this method assumes that the action of an emitted particle satisfies the relativistic
Hamilton-Jacobi equation. Taking the symmetries of the metric into account, one can adopt an appropriate ansatz for the form of the action.
Details about the derivation of QCM from the general relativistic
Hamilton-Jacobi equation and its applications to orbiting bodies in the Schwarzschild metric approximation and to the Universe in the the interior metric can be found in our original 2003 paper [2] titled "Exploring Large-scale Gravitational Quantization without [??] in Planetary Systems, Galaxies, and the Universe".
In this section, considering the effect of GUP, applying with the corrected
Hamilton-Jacobi equation, we will focus on investigating the scalar particle's tunneling radiation from a Rutz-Schwarzschild black hole.
For the derivation of QCM from the general relativistic
Hamilton-Jacobi equation, see the published articles online [2,4].
These articles, which are written with research chemists in mind, discuss such trends as new tools for treating intermolecular interactions, calculating quantum dot structures and using the
Hamilton-Jacobi Equation in relativistic quantum chemistry.
First of all, we analyzed the modified
Hamilton-Jacobi equation by resolving the modified Lagrangian equation utilized for the magnetized particles in the space-time.
Our original article [1] contains the derivation of QCM from the general relativistic
Hamilton-Jacobi equation and its new gravitational wave equation for any metric.