To start off, the gravitational parameter is given as
[1]
This is pretty standard, now I derived this form for the standard gravitational parameter
[2]
Seeing the energy in terms of quantum mechanics we can have
[3]
I don't think this has ever been written in this form and it occurred to me that simply solving for
gives
[4]
The gravitational redshift is
[5]
where
is the Schwarzschild radius. Plugging equation 4 into equation 5 yields
[6]
Interestingly, the quantity on the right hand side has yielded the gravitational fine structure, or also known as the gravitational coupling constant giving
[7]
Because
[8]
So how does someone interpret the gravitational time dilation formula with the gravitational coupling constant? Perhaps this is by no means a mistake it has appeared since the two undoubtedly have characteristics which play off each other. The Schwartzschild radius is sometimes set equal to the Compton wavelength, and loosely speaking the Compton wavelength describes when quantum effects take role in the presence of gravity... so as I speculated perhaps the way it appeared in the equation is not by a mistake.
If this is taking about quantum effects, since the use of the gravitational coupling constant has appeared in the equations and since the Schwarzschild radius is the radius of the gravitational field then perhaps the gravitational coupling constant is some kind of limit as well?
I have looked everywhere for a relation that is similar... and even equations that have some sort of Schwarzschild radius on one side and the constant on another but I cannot find anything in the literature anywhere. Also, there is very little literature to go on with the constant anyway in general, according to wiki. A similar relationship can be derived:
[9]
Cancel out the
on both sides gives
[10]
rearrange
[11]
substitute again
for equation 4,
[12]
Multiply
on both sides gives
[12]
And so we have retrieved the constant again. So any idea's how to interpret this? This one is a little different because it is not concerned with a gravitational red shift.
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Originally Posted by Aethelwulf
is dimensionless.
was dimensionless (which it is most obviously is now after checking the units), then 
is the ratio 
.
