Including gravity in unification (GUT) and Planck scales

v. 5 n. 15

IN AN ACADEMIC CASE OF FORCE UNIFICATION for the approach to a mathematical point for two Planck masses, each with a Planck charge, the Planck scale is encountered given sufficient (impractical) collision energy to overcome like-charge repulsion. Disregarding relativistic effects and taking the ratio of electromagnetic to gravitational forces at this scale,

F_e/F_g = (kqq/r^2) / (Gmm/r^2)

where k = Coulomb force constant

q = (4Π∈_°?c)^1/2 = Planck charge

m = (?c/G)^1/2 = Planck mass;

then

F_e/F_g = 1 ...................................................................................... (0)

so that the opposing electromagnetic and gravitational forces at the Planck scale in this instance are indistinguishable in magnitude.

That the equations for electromagnetism and gravitation are of the same form here might also attest to the former being rooted in the latter, given the derivation of electron mass from the latter framework at the galactic supercluster scale, discussed earlier.

This idealized high energy collision of two particles of the same charge suggests a highly charged "nucleus" marginally held together by gravitation; the particles neither wish to stick together nor part company -- a highly charged stable plasma?

GIVEN RELATIVISTIC EFFECTS the strong force could be

F_g,n = (n/2)^3/2 (0.058) (c^7/Gh)^1/2 m ........................ (1)

where n=1,2 ..., and m is the subatomic mass under consideration. Equations (1) and (2) were derived from Newtonian gravity and special relativity,

m_q^3 = (A/G)(h/c)^2 ................................................................ (2)

where m_q is light quark mass, and acceleration of the universe A~10^-14 m/s^2. *

When n=1 and m=m_q in Equation (1), the Strong force at quark level is predicted.

When n=1 and m is the Planck mass in Equation (1), the Strong force at the Planck scale is predicted.

The unification (GUT) and Planck scales commonly ignore gravity. The Planck scale is three orders of magnitude greater than the conventional unification scale estimate. The Planck mass (scale) is 1.220890x10^19 GeV, and the unification scale is conventionally estimated at 10^16 GeV. **

"... At this [Planck] scale, present descriptions and theories of sub-atomic particle interactions in terms of quantum field theory break down and become inadequate, due to the impact of the apparent non-renormalizability of gravity within current theories.[19] " -- Planck units - Wikipedia

"At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it has been theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown.[22]" -- Planck units - Wikipedia

THE UNIT CHARGE. To substantiate Equation (1) and to derive the unit charge, let

F_g,1 ≈> F_coulomb

for the relativistic form of Equation (0).

In Equation (1) when m is the Planck mass and n=1 for this mass, then the charge q of the above is about the unit charge,

10^-18 coul ≈> q ≈> e.

UNIFICATION SCALE? Note that the derivation of the unit charge from gravity is at the Planck scale, not the conventional hypothesized unification scale. This might suggest that the unification scale estimate is three orders of magnitude too small, especially considering gravity is conventionally omitted from the unification and Planck scales because of lack of renormalizability in conventional theories, as mentioned in the quote.

* (5) How might gravity relate to the other forces? | LinkedIn

** AUTHOR: "What is the value of the strong force in Newtons at the unification energy?"

ChatGPT: "At the unification energy scale, which is hypothesized to be around 10^16 GeV (gigaelectronvolts), the electromagnetic, weak, and strong nuclear forces are believed to merge into a single force. This unified force is often referred to as the Grand Unified Theory (GUT) force.

However, it's difficult to directly quantify the value of the strong force at this energy scale because our current understanding of physics breaks down at such extreme conditions. At energies around the GUT scale, quantum gravity effects become significant, and we would need a theory of quantum gravity to make accurate predictions.

... "

AUTHOR: "How was this [unification] scale arrived at?"

ChatGPT: "... the inclusion of gravity in a unified theory, which would require a theory of quantum gravity, could potentially modify the predicted energy scale of unification."

Image: https://www.quora.com/What-if-we-had-not-Newton-and-Einstein