(1) There exists a (theoretically) maximal energy density and pressure.

(2) There exists a mass-dependent (theoretically) maximal acceleration given by mc

^{3}/(h bar) if m < m

_{p}and by c

^{4}/Gm if m > m

_{p}. This is of the order of Milgrom's acceleration a

_{0}for ultra-light particles (m approximately H

_{0}) that could be associated to the Dark Energy fluid. This suggests models in which modified gravity in galaxies is driven by the Dark Energy field, via the maximal acceleration principle. It follows trivially from the existence of a maximal acceleration that there also exists a mass dependent maximal force and power.

(3) Any system must have a size greater than the Planck length, in the sense that there exists a minimal area (but without implying for quanta a minimal Planckian wavelength in large enough boxes).

(4) Physical systems must obey the Holographic Principle. Holographic bounds can only be saturated by systems with m > m

_{p}; systems lying on the "Compton line" "l" approximately equal to 1/m are fundamental objects without substructures

Bruenton's conjectures are driven by observations about the relationships of the Planck length, mass, and time which are derived from the speed of light c, the gravitational force constant G, and the reduced Planck's constant h bar, the Schwarzchild solution for the event horizon of a Black Hole in General Relativity reformulated in a generalized and manifestly covariant way, observations about the Kerr-Newman family of black holes, an alternate derivation of the Heisenberg uncertainty principle, the notion of a Compton length, and a few other established relationships.

Bruenton presents his conclusions as heuristic conjectures for any quantum gravity theory that displays a minimum set of commonly hypothesized features, rather than rigorously proven scientific laws.

I have omitted some of his more technical observations and consolidated others.

Bruenton acknowledges that these observations may fail in the case certain theoretically possible exotic "hairy" black holes (while implying that they probably don't exist for some non-obvious reason). He equivocates on the question of whether Lorentz symmetry violations near the Planck scale are possible, reasoning that an absence of a minimal Planckian wavelength could rescue Lorentz symmetry from quantum gravity effects.

I find his suggestion that there is a maximal energy density and pressure particularly notable because of the remarkable coincidence between the maximum density observed by astronomers in Black Holes and neutron stars on one hand, and the maximum observed density of an atomic nucleus on the other.

His suggestion that the Planck scale my denote the line between systems that are "fundamental objects without substructures" and "physical systems" is also shrewd.

## 2 comments:

"the remarkable coincidence between the maximum density observed by astronomers in Black Holes and neutron stars on one hand, and the maximum observed density of an atomic nucleus on the other"

Please explain!

See, e.g., posts The Black Hole Density Coincidence and here.

"The density of black holes (measured by its mass divided by the volume included in the event horizon) increases as black hole mass decreases. At about three solar masses (6*10^30 kg), the mass of a black hole divided by its event horizon volume is comparable to the density of an atomic nucleus, and larger black holes are less dense than an atomic nucleus.

Smaller black holes are theoretically possible in the equations of general relativity, but they aren't formed by gravitational collapse and have never been observed[.] . . . The most dense stars we are pretty sure exist (about 2,000 of them that have been observed) are neutron stars, which are estimated to be on the order of 12-15 km or less in radius:

A typical neutron star has a mass between 1.35 and about 2.0 solar masses, with a corresponding radius of about 12 km . . . . In contrast, the Sun's radius is about 60,000 times that.

Neutron stars have overall densities . . . of 3.7×10^17 to 5.9×10^17 kg/m3 (2.6×10^14 to 4.1×10^14 times the density of the Sun), which compares with the approximate density of an atomic nucleus of 3×10^17 kg/m3. The neutron star's density varies from below 1×10^9 kg/m3 in the crust, increasing with depth to above 6×10^17 or 8×10^17 kg/m3 deeper inside (denser than an atomic nucleus). . . . No quark stars have ever been definitively observed, although three quark star candidates (a minuscule number relative to the 2,000 observed neutron stars) have been identified for further investigation. And, somewhat surprisingly, models of quark stars suggest that a quark star with 2.5 solar masses would have more volume than a neutron star with 2.0 solar masses, so the difference in mass density between the two varieties of post-supernovae stars would be modest. . . .

Suppose that fundamental particles had mass densities comparable to that of atomic nuclei. How big would they be?

Atomic nuclei densities provides an estimate of the size of the neutron and proton: "The diameter of the nucleus is in the range of 1.75 fm (femtometer) (1.75×10−15 m) for hydrogen (the diameter of a single proton) to about 15 fm for the heaviest atoms, such as uranium. These dimensions are much smaller than the diameter of the atom itself (nucleus + electronic cloud), by a factor of about 23,000 (uranium) to about 145,000 (hydrogen)." A nuclear radius is roughly the cube root of the number of protons and neutrons combined in the atom times 1.25 × 10−15 m +/- 0.2 fm (this +/- factor varies from atom to atom), and the shape is approximately spherical.

Protons and neutrons, of course, are made up of three quarks each bound by gluons. Bare quarks make up about 1% of the mass of a proton or neutron, so a first generation quark is about 0.3% of the mass of a proton, with the rest of its mass coming from the "glue" and thus quarks would be 14% of the size of a proton or neutron if they had an equivalent mass density, i.e. about 0.2 fm (2*10^-16 meters).

Interestingly, this coincides rather neatly with the apparent charge distribution within a neutron (which has one up quark of charge +2/3 that appears to be centrally located and two down quarks of charge -1/3 that appear to orbit the up quark): "The neutron has a positively charged core of radius ≈ 0.3 fm surrounded by a compensating negative charge of radius between 0.3 fm and 2 fm. The proton has an approximately exponentially decaying positive charge distribution with a mean square radius of about 0.8 fm." But, a 10^-16 meter order of magnitude for up and down quarks would be much larger than the Planck length that conventional wisdom would ordinarily expect."

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