Energy as a complex number
Equations of gravity and electromagnetism exhibit structural similarities. After a convertion of units, the pattern we see could be mathematically explained in terms of complex variables, this expression ends up capturing some aspects of both fields in a compact notation.
: energy,
: mass,
: speed of light,
: complex unit,
: charge,
: vacuum permittivity,
: universal gravitational constant
Classical mechanics
Given that Coulomb's inverse-square law [1] takes the form of Newton's law of universal gravitation [2] if we write:
It seems reasonable to explore the use of complex numbers, as a data structure, to treat electrostatics in a gravitational context.
Given the assumption:
Mass would be:
A complex gravitational potential defined as:
would hold enouh information to describe the classical static interaction with a different body. Which could provide a framework to justify why we don't seem to messure direct attraction/repulsion between mass and charge, since the representation of this interaction is caught in the imaginary part of:
Special Relativity
Consider a complex Momentum P
: mass energy contribution
: charge energy contribution
To write:
being:
Which suggests that magnetic forces could be expressed in terms of P.
Electromagnetism
The description of Gravitoelectromagnetism [3] already gives us a good picture of the pattern. Converting Coloumbs into Kilograms using the following convertion factor, and allowing variables to be complex:
The full set of equations could be written as:
Subscripts m and e, refer to massive and electric quantities.
Lastly
A complex number seems to be a suitable data structure to hold information about energy for certain calculations. Schrodinger equation, already deals with complex numbers, perhaps it is describing something similar. The energy of a photon would be a complex number too, just as the energy of an electron, making the absortion/emmision of a photon, comprehensible in terms of energy flow.
The study of a complex metric tensor, as the result of expressing it in terms of a complex potential, is left out of this document, and would be the subject of a second document.
Schwarzschild radius would be a complex number, which suggests that a change in the frame of reference could allow us to express the metric in terms of real numbers, forcing us to rotate, in the complex sense, the 4 momentum of a test particle, which would make it look with a different mass/charge distribution, even with negative mass, which in turn suggests that mass could be a relative quantity.
References
3. Wikipedia, Gravitoelectromagnetism
4. Wikipedia, Minkowski metric
5. Wikipedia, Schwarzschild metric
6. Carlos Francisco Romero Madrid, Derivation of the Metric of Reissner-Nordström and Kerr-Newman Black Holes, 2018
7. Guansheng He, Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole Article in Communications in Theoretical Physics · January 2014 261016243
8. Richard A. Hutchin, A Natural Combination of Gravity and Electromagnetism, Journal of Modern Physics, 2015, 6, 749-757 2015.66080
9. Bahram Mashhoon, Gravitoelectromagnetism: A Brief Review 0311030
10. Athanasios Bakopoulos, Gravitoelectromagnetism: Basic principles, novel approaches and their application to Electromagnetism 1610.08357
This equation was first digitally written and cryptographically signed on Sep-04-2022 09:46:26 PM +UTC as shown in "decoded input data" in the "more details" section of the following link:
E = Y*(m + i*Q/(4*pi*e0*G)^(1/2))*C^2