In Einstein's "Relativity" Appendix 2 he mentions meters=i*c*seconds where "c" is simply the number 3E8 without any units. Plugging this into the "speed" of light, it gives c to have the "unit" "i". By not using this "unit", or rather, by not making it explicit that c is unitless like the fine structure constant, many other physics equations are made to contain a -1 or "i" error. For example, F=ma would become F = - ma which directly shows force is equal and opposite without having to specify Newton's 3rd law. Also instead of E=mc^2 we would have E = - mc^2 which is what cosmologists say. I prefer the statements to be written as F+ma=0 and E+mc^2 = 0.
To see how this can generate ideas, here a cosmological possibility. Black hole entropy would have to be S+A/4=0 instead of S=A/4 in order to get the relativistic units correct. This implies black holes do not "have" entropy (there's an on-going debate about what their entropy means) but that it is offset by a decrease in the area caused by the presence of mass:
dS+dA/4=0
Using Feynman's excess radius idea of the effect gravity has on space, the change in the surface area of the space due to the presence of mass is:
dA= - 4*pi*(GM/3/c^2)^2
I view gravity as decreasing the space/time ratio, i.e. that c^2 is reduced due to G*M. This is possible because c is unitless and thereby does not need to affect any other constants. This is simply is a different way of expressing relativistic effects. Using the equation above, I get
dS+d(1/c^4) = 0
i.e. as mass increases, S2<S1 is offset by (1/c2)^4 > (1/c1)^4. There are some missing constants of proportionality, but they should remain relativistically unitless (maintaining the "i" system). So as entropy is emitted, it reduces the contraction of space-time caused by gravity. If comoving volume expansion follows black hole expansion, then the red shift of both are from decreasing c and the increasing Hubble constant would be from:
dH/dt*T+d(c)/dt = 0
Where T is age of universe. c can change without affecting other physical constants or anything else we might notice because it is unitless. We only notice that it changes in the presence of mass, acceleration, and in the red shift of the expansion. Again, there are missing constants of proportionality but the relativistic units would cancel as these do.
Entropy would thereby be a constant in a comoving space-time volume even as galaxies emit entropy to make up for the reduction in the comoving surface area caused by gravity, and thereby not increase entropy on a large-scale comoving basis.
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You mean using the units like I suggest would result sinh(x) instead of i*sin? Well, i*sin(x) is not correct, it's supposed to be -i*sin(i*x). The derivatives of the inverse hyberbolics do not give indeterminate +/- solutions. e^x seems easier to work with than forcing e^(i*x) by throwing away the required i from the units. Even worse than that it's supposed to be -i*e^(i*x) not e^(i*x) if you do throw away the relativistic i. "Accurate units are harder to work with because we are used to sin instead of sinh" does not seem like a good reason for the graduate level. Experimenters should be smart enough to talk in i/meters instead of seconds or i*seconds to replace meters. Keeping them separate has been a disaster. The mass of physicists not using the correct units blocks good ideas. It forces units where they do not exist. c is more fundamental than h, like alpha. Relativity can be expressed and viewed more simply if you let acceleration and gravity change the ratio c compared to other frames of reference. All else follows from this and it prevents double-talk like "the speed of the same photon is constant for all reference frames as long as you don't complain its energy is different". Not using the correct units adds complexity to relativity. It violates Occam's razor which makes new theories less falsifiable. It prevents people from realizing c changing based on reference frame is a simpler view of relativity.
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