zonde said:
Speaking about degeneracy and density dependence. To claim that the two are varying proportionally we have to assume that there is some cut-off distance for quantum level occupancy, meaning that particles don't compete for quantum state given sufficient distance. However we can assume that this "quantum level occupancy" effect drops as inverse square law. And in this case density factor does not exactly determine degeneracy level and it is more related to number of particles and distance to them.
And assuming this PAllen's scenario is still subject to questions about degeneracy levels as number of particles is much higher even so the distances are bigger as well.
I was hoping to see a more complete response to PAllen's response to this. You apparently assume a cut-off distance, but then assume it still applies if the density falls short of forcing the particles within this cut-off distance, per PAllen's description. In fact, it seems to me that, PAllen's description is tailor made to reject this very kind of argument, and your last sentence merely hand waves it away. It's as if when you said degeneracy and density dependence that density was a globally defined property such that local densities, particle separations, within it need not be effected. I can't wrap my head around that without adding some extra unspoken conditions.
I can possibly see some kind of argument against the presumed singularity at the center of a BH based on something similar. However, the event horizon would persist. Hence, even if so, it is still technically a BH.
zonde said:
Yes, I reject general covariance. So I can say that I reject GR and exactly why.
This sentence from wikipedia is in essence what is unacceptable for me:
General covariance: "The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws."
Essential idea is that physical laws do not exist a priori in nature. "Map is not the territory."
What bridges theory (or coordinate system) with nature is
coordinate unit.
This harks back to the original issue of coordinates and observer frames. When it says "do not exist a priori in nature" it is not the same as saying "do not exist". Now mathematically general covariance takes on a form to deal with accelerated motion. Which is observer independent much like the spacetime interval. To illustrate why general covariance is required I'll skip the mathematics and describe one of the things I did in kindergarten with rocks in the back seat of the car. Then repeat the above justification given that scenario. It Galilean character doesn't change its essential character.
If, sitting in a car, you toss a rock straight up it comes straight back down into your hand. Now you look out at the fence post along the road and notice the rock arcs up at one fence post and over till it lands at the next fence post. This arc, I now know, is of course a parabola. You can also consider Earth's motion and view the trajectory as one that angled off to the left or right. The question is, is this straight up and down trajectory "really" the exact same path as the parabola? Well of course it is, the rock didn't take a quantum superposition of paths. General covariance, at its fundamental core, is nothing more than an axiomatization of this sameness, with the added provisions that simultaneity and global geometry vary per perspective in exactly the same manner as the path of our rock.
The conceptual difficulties arise because our description does not specify a path as such, per the stated condition that nature doesn't uniquely specify it. Yet any observer is by definition stuck with observing reality from a certain perspective. Rejecting general covariance is tantamount to claiming the rock either took multiple paths, or that all but one of the possible observable paths is an illusion, such that only one real path remains. This is exactly the error of reasoning that lead to the failures of the classical ether theory. It doesn't even mean something resembling an ether doesn't exist. It just means not only that any such ether model cannot be used to uniquely specify a coordinate choice, but also that any coordinate choice we do make must covary with any relative variances of the supposed ether.
When you say the "map is not the territory" is valid but often misleading. In effect, by rejecting general covariance, you are attempting to force fit all coordinate choices single coordinate choice, while failing to recognize that general covariance is fully justified on the foundational grounds that all these coordinate choices are describing the exact same "set of paths" (states) to begin with. This rejection, in turn, falsely implies the "real" territory (like ether theory) is a singular coordinate choice, as if the choice between using metric or English had some real physical meaning.
Anyway, that seems to me to be the logical consequences of your issues with coordinates and observer perspectives. Just remember that general covariance simply entails that all the different paths observers might describe your rock to take in the car is the exact same path, plus time and geometry.
