By taking into account the relative energy between the diquark and the quark in nucleons, the gravitational singularity in a black hole created from a collapsing neutron star can be removed; compatibility with quantum mechanics is restored. This black hole becomes a “black” neutron star. The negative relative energy identified as dark matter in the previous paper can account for the galaxy rotation curve. The positive relative energy identified as dark energy in the previous paper can explain the accelerating expansion of the universe. A possible scenario for cosmic ray generation is given.
A neutron star with mass greater that the Tolman-Oppenheimer-Volkoff (TOV) limit MTOV~3 solar mass MSUN becomes a black hole in which the star core collapses to a gravitational mass singularity in general relativity [ [
Dark matter needed to account for many observation, e. g., the galaxy rotation curve, and the dark energy required to drive the rapidly expanding universe remain hypothetical as they cannot be observed.
Recently, it was shown [
In this paper, it is shown that, as a neutron on the surface of a neutron star with mass ~MTOV falls toward the star center, not only gravitational energy is released, but a negative relative energy also emerges simultaneously. These two energies can cancel each other and prevent the collapse of the star and formation of a mass singularity. The negative relative energy generated in an expanding galaxy can play the role of dark matter and account for the galaxy rotation curve. The positive relative energy generated in outer regions of the universe can play the role of dark energy, leads to accelerating expansion of the universe and may eventually give rise to cosmic rays.
The relevant parts of SSI are outlined in Sections 2-5. Creation of negative relative energy applied to neutron star collapse is considered in Sections 6-7. In Section 8, such negative relative energy reinforces the existing gravitational potential in an expanding galaxy to account for the galaxy rotation curve qualitatively. The accelerating expansion of the universe by means of the positive relative energy created in the outer parts of the universe is qualitatively outlined in Section 9. Some scenarios of the outer regions of the universe including cosmic ray generation are shown in Section 10. The so-obtained scenario of the universe is summarized in Section 11.
The starting point is the Dirac equations for three quarks A, B and C with masses mA, mB and mC located at xI, xII and xIII, respectively. The quarks interact with each other via scalar strong potentials VAB(xI), VAC(xI), ⋯ in [ [
∂ I I d e ˙ χ B e ˙ ( x I I ) − i ( V B C ( x I I ) + V B A ( x I I ) + V B G ( x I I ) ) ψ B d ( x I I ) = i m B ψ B d ( x I I ) (2.1.a)
∂ I I e ˙ f ψ B f ( x I I ) − i ( V B C ( x I I ) + V B A ( x I I ) + V B G ( x I I ) ) χ B e ˙ ( x I I ) = i m B χ B e ˙ ( x I I ) (2.1.b)
where ∂ I = ∂ / ∂ x I , ⋯ . The spinor indices run from 1 to 2 and the quark spinors χB and ψB are linear combinations of the four components in the conventional Dirac bispinor ψ according to [ [
To construct baryon wave functions, the three pairs of quark equations, one of them being (2.1), together with the cited strong potential equations, are multiplied together and the products of the quark spinors and those of the strong potentials are generalized to baryon wave functions and baryon potentials nonseparable in xI, xII and xIII according to [ [
In [ [
The baryon flavour function can now be removed leaving behind the ground state baryon wave equation [ [
∂ I a b ˙ ∂ I g h ˙ ∂ I I e ˙ f χ { b ˙ h ˙ } f ( x I , x I I ) = − i ( M b 3 + Φ b ( x I , x I I ) ) ψ e ˙ { a g } ( x I , x I I ) (2.2a)
∂ I b ˙ c ∂ I h ˙ k ∂ I I d e ˙ ψ e ˙ { c k } ( x I , x I I ) = − i ( M b 3 + Φ b ( x I , x I I ) ) { b ˙ h ˙ } d ( x I , x I I ) (2.2b)
M b = ( m A + m B + m C ) / 2 (2.2c)
The interbaryon strong potential Φb is a triple product of three strong interaction quark potentials of the type of VBA and VBC in (2.1) and is governed by [ [
∂ I a b ˙ ∂ I I f e ˙ ∂ I e f ˙ χ 0 b ˙ ( x I , x I I ) = − i 2 ( M b 3 + Φ b ( x I , x I I ) ) ψ 0 a ( x I , x I I ) (2.3a)
∂ I b ˙ c ∂ I I e ˙ h ∂ I h ˙ e ψ 0 c ( x I , x I I ) = − i 2 ( M b 3 + Φ b ( x I , x I I ) ) χ 0 b ˙ ( x I , x I I ) (2.3b)
where χ0 and ψ0 the wave functions of the doublet baryons.
Since quarks cannot be observed, their coordinate spaces are converted into an observable laboratory space Xμ for the baryon and a relative space x between the diquark and the quark via the linear transformation given above [ [
x μ = x I I μ − x I μ , X μ = ( 1 − a m ) x I μ + a m x I I μ (3.1)
For observable particles, am is often determined by that Xμ is the center of mass of these particles. If these particles have equal mass, am = 1/2. Such kind of determination cannot be done here because quarks are not observable individually. am has been taken to be an arbitrary real constant and represents a new degree of freedom. The relative space x = ( x 0 , x _ ) is “hidden” [ [
The baryon wave functions in (2.1) have been factorized into the form of [ [
χ 0 b ˙ ( x I , x I I ) = χ 0 b ˙ ( x _ ) exp ( − i K μ X μ + i ω K x 0 ) ψ 0 a ( x I , x I I ) = ψ 0 a ( x _ ) exp ( − i K μ X μ + i ω K x 0 ) (3.2)
K μ = ( E K , − K _ ) [ [
where EK is the energy of the baryon and K its momentum. x0 is the relative time and −ωK the associated relative energy in the “hidden” relative space and can also not be observed within SSI.
In spherical coordinates, x _ = ( r , θ , ϕ ) [ [
ψ 0 1 ( x _ ) = g 0 ( r ) Y 00 ( θ , ϕ ) + i f 0 ( r ) 1 3 Y 10 ( θ , ϕ ) , ψ 0 2 ( x _ ) = i f 0 ( r ) 2 3 Y 11 ( θ , ϕ ) (3.4)
where the Y’s are the usual spherical harmonics. χ 0 a ˙ is found by changing the signs of f0(r) in (3.4).
Consider baryons at rest, K = 0. am = 1/2 is set as in the meson case [ [
[ E 0 3 8 + M b 3 + Φ b d ( r ) + E 0 2 Δ 0 ] g 0 ( r ) + ( E 0 2 4 + Δ 0 ) ( ∂ ∂ r + 2 r ) f 0 ( r ) = 0 (4.1a)
[ E 0 3 8 − M b 3 − Φ b d ( r ) + E 0 2 Δ 1 ] f 0 ( r ) − ( E 0 2 4 + Δ 1 ) ∂ ∂ r g 0 ( r ) = 0 (4.1b)
where the subscript d denotes doublet and [ [
Φ b d ( r ) = d b r + d b 0 + d b 1 r + d b 2 r 2 + d b 4 r 4 (4.2)
here, the nonlinear potential Φcd(x) [ [
The two coupled third order Equations (4.1) have been converted into six first order equations [ [
Due to the large number of unknown constants, (4.1-2) could not be solved as a conventional eigenvalue problem. A less ambitious approach has been adopted. The known mass of the neutron is used as input for the eigenvalue E0 and the quark masses obtained from meson spectra given in [ [
Here, the word “finite” on the first line of the second paragraph on [ [
These wave functions have led to the nearly correct predictions of the neutron life and the electron asymmetry parameter A or the neutrino asymmetry parameter B [ [
If am = 1/2 and −ω0 = 0 used in (4.1) were not assumed, a derivative in (2.3) operated on (3.2) will, in terms of (3.1) [see [
∂ I a b ˙ = ( 1 − a m ) ∂ X a b ˙ − ∂ a b ˙ = ( 1 − a m ) ( − δ a b ˙ ∂ X 0 − σ _ a b ˙ ∂ X _ 2 ) + δ a b ˙ ∂ 0 + σ _ a b ˙ ∂ _ = δ a b ˙ i 1 2 E 0 + σ _ a b ˙ ∂ _ → δ a b ˙ i ( ( 1 − a m ) E 0 + ω 0 ) + σ _ a b ˙ ∂ _ (5.1)
The expressions on both sides of the arrow will be equal if [ [
a m = 1 2 + ω 0 E 0 (5.2)
which keeps (4.1) invariant. The relative energy −ω0 is “hidden” with respect to strong and electromagnetic but not gravitational interactions [
In [
Consider the following idealized scenario. A neutron star with a mass MNS = MNSa, a critical mass, such that the quantum degeneracy pressure of the neutrons and the strong neutron-neutron repulsion precisely balance off the gravitational pressure. Let an external neutron arrive at the surface of this star. The star now becomes too heavy and a gravitational collapse is anticipated to start. Conventionally, this collapse gives rise to a black hole and continues until the star ends up in a mass singularity at its center; the quark structure of the neutron is ignored. Actually, in neutron stars and black holes, the gravitational forces are strong and their gradients large that the diquark-quark structure of the neutron may need be taken into account. This scenario is illustrated in
In
shifted to the left to become xI,b,Xb and xII,b, respectively. The diquark-quark distance r b = x I I , b − x I , b = r a remains unchanged. In the same fall, however, the neutron star radius at position b is reduced to Rb < Rq due to gravitational collapse. The incoming neutron is thus similarly compressed to become smaller by a factor of Rb/Ra so that rb is also reduced to a smaller value rbG by the same factor;
r b = x I I , b − x I , b → r b G = x I I , b G − x I , b = r a α , α = R b / R a < 1 (6.1)
As was mentioned at the end of Section 5, gravity interacts directly with the quark at xII in (2.1), hence also with that in
If Xb were taken to be the star radius Rb at position b, then the diquark coordinate xI,b in
The radial wave Equations (4.1-2) hold for the arriving neutron in position a. At position b, (6.1) shows that the diquark-quark distance or radius r in (4.1) is reduced by a factor of α. Equations (4.1-2) remain invariant under the transformations r → r α , E 0 → E 0 / α , M b → M b / α , d b 2 → d b 2 / α 5 , ⋯ . The neutron mass E0 and the quark masses in Mb at position b are unchanged. So does also the confining constant db2 = −0.3202 in
Application of (3.1), (5.2) and (6.1) to position b in
a m = 1 2 + ω 0 E 0 = 1 + X b − x I I , b G x I I , b G − x I , b = 1 2 r a r b G = R a 2 R b , − ω 0 E 0 = − R a 2 ( 1 R b − 1 R a ) (7.1)
where −ω0 is the relative energy gained when the external neutron arriving at position a falls to position b. In this fall, the gravitational energy EG released is given by
E G E 0 = R S a 2 ( 1 R b − 1 R a ) , R S a = 2 G M N S a (7.2)
where G is the gravitational constant and RSa is the Schwarzschild radius of this collapsing star with radius Ra in
R a = R S a (7.3)
i.e., when the neutron star in
In this case, the fall of this external neutron actually gains no energy because EG − ω0 = 0 for all Rb. This neutron therefore remains at position a and becomes “weightless”. The anticipated collapse of the neutron star does not start and no mass singularity is created.
The radius Ra of the initial neutron star has been estimated by equating gravitational pressure to the pressure of the degenerate neutrons [ [
R a = 0.0026 R E a r t h ( M S U N M N S a ) 1 / 3 (7.4)
which together with (7.2-3) yields Ra = RSa = 10.8 km and MNSa = 3.6MSUN. This mass is consistent with the similarly estimated maximum mass 4.3MSUN [ [
A possible scenario is as follows. As additional neutrons arrive at the star surface, a thin shell of weightless neutrons is added to the star at first. Let the mass of this shell be ΔMNSa, the new Schwarzschild radius will be R S a Δ = ( 1 + Δ ) R S a and the new star radius will be R a Δ = ( 1 + Δ ) 1 / 3 R a ,assuming that the neutron density in the shell is the same as that in the star.
Since RSaΔ > RaΔ, a black hole is created. As a new neutron arrive at the new star surface with radius RaΔ, it will tend to fall inwards. This time, the gravitational energy (7.2) will slightly exceed the relative energy loss (7.1) and this new neutron tends to fall slowly. However, the pressure of the degenerate neutron gas in the shell is now unopposed by the weightless neutrons and this gas will therefore expand accompanied by a reduced density. If the expansion reaches a radius > RSaΔ, the back hole is lost and this enlarged star will be a visible neutron star similar to the initial star in
In this scenario, the neutrons in such a black hole fill it up to its Schwarzschild radius. Such a black hole may more suitably be called a “black” neutron star. The absence of mass singularity here is consistent with the conjecture that an eventual future quantum gravity theory will not contain any singularity.
There is a large body of data that require the presence of dark matter. The first one is the galaxy rotation curve [ [
Consider again an idealized scenario as follows. A spiral galaxy with an appreciable part of its mass consisting of hydrogen gas was in its earlier stage of development. In that stage, this galaxy was smaller, denser and hotter according to the big bang model and the gravitational potential resisting such an expansion was insufficient. It therefore expanded. The situation is analogous to a violation of Jeans criterion for a gas cloud.
To illustrate the mechanism, let us turn off the gravity for a moment. Follow now the movement of a proton, denoted by c, in a hydrogen molecule. In this thermal expansion, this molecule will collide with other molecules, exchange energy and momentum with them and end up in a new position, labeled d, farther out from the galaxy center. In this journey, the forces involved are all Coulomb forces, between the proton and the orbiting electron and between the orbiting electrons in other molecules. The protons get dragged along; their quark structure is not involved in the expansion.
The situation is illustrated in
In
Turn now on gravity, the situation is reversed. Gravitational effects on electrons are negligible due to their small mass. Gravitational forces now act on this proton at Xd and try to pull it back together with other protons acted upon. However, they turn out to be too small to account for the galaxy rotation curve and prevent the escape of the outer stars. Here, such escapes are prevented by including the negative energies generated by differentiated gravitational pull on quarks and proton.
Just like the neutron star case mentioned below (6.1), gravity also acts directly on the quarks and tends to pull them towards the galaxy center, with the heavier diquark at xI,d closer to this center than does the lighter quark at xI1,d,as is shown in
Applying (3.1) the gravity pulled position d in
a m = 1 2 + ω 0 E 0 = X d − x I , d x I I , d − x I , d , − ω 0 E 0 = 1 2 − X d − x I , d r a , r a = x I I , d − x I , d ≈ 4 fm (8.1)
This energy has the same negative sign as the gravitational potential energy produced by matter inside Xd and hence reinforces it to become large enough to keep the outer stars of the galaxy from escaping. It is due to the lag of the “hidden” quark coordinates xI,d, x1I,d behind the observable proton coordinate Xdin
The required energy is -w0» -5.5E0 per proton, when averaged over the whole universe, and is generated for “free” at “no cost”. This −ω0 value leads to am ≈ 6 ( [
This scenario, characterized by
The observed accelerating expansion of the universe is currently considered to be due to assumed dark energy in the outer regions of the universe. This hypothetical energy may here be replaced by the positive relative energy −ω0 > 0 corresponding to am < 1/2 in (8.1).
As in Sections 6 and 8, consider the following idealized scenario. The expansion mechanism in Section 8 applied to a galaxy can analogously be used in some later stage of the development of the universe. In the outer part of the universe, its expansion leads to that the hydrogen gas density decreases and the gas temperature drops there. The expansion nearly comes to a halt. In this region, the gas is tenuous, cold and experiences very weak gravitational force.
Consider a proton in a hydrogen molecule of this gas. The configuration of this proton in position d of
Let a second hydrogen molecule arrive at position e simultaneously. It contains a second proton with its diquark and quark at the same xI,e and xII,e. Due to differences in the paths of these both molecules before they reach position e, this second proton may end up at nearby X e − = X e − Δ ( x I I , e − x I , e ) or X e + = X e + Δ ( x I I , e − x 1 , e ) where Δ ≪ 1 .
If its position is Xe+, then (8.1) yields a negative relative energy − ω 0 = − Δ E 0 < 0 . Analogous to the situation below (8.1), this small energy tends to pull the quarks toward the left in
If its position is Xe−, then (8.1) yields a positive relative energy − ω 0 = − Δ E 0 > 0 . Contrary to the above Xe+ case, this small energy tends to push the quarks outward, towards the right part of
Current data show that dark energy exceeds the energy of ordinary matter by a factor of ≈14 when averaged over the universe. Identifying this energy with −ω0, (8.1) leads to am = −13.5 ( [
This scenario, characterized by
As the above “run way” situation continues, the “whole” size of this proton, xII,f − Xe−f in
The relative energy corresponding to this case is by (8.1) −ω0 ≈ 1.29 × 1013 eV. In this scenario, hydrogen atoms in the expanding outer regions of the universe having this energy or greater turn into plasma.
According to 10.1., the hydrogen gas in outer regions of the universe can expand until an average atom acquires an energy ~1.29 × 1013 eV, beyond which the atom becomes ionized. For atoms carrying higher energies, the gas becomes a tenuous plasma in intergalactic space. In this plasma, the protons continue to gain energy by the increasing −ω0 but the ejected electrons lag behind, inasmuch as gravity is absent in Dirac’s equation for an electron in a Coulomb field. This leads to a proton current flowing radially outwards which in its turn generates transverse magnetic field. As there are inhomogeneities in gas distribution in this region, the magnetic field will also vary in space.
As the “run away” expansion in Section 9 continues, some protons become very energetic. The magnetic field will cause them to move perpendicular to their path and to the direction of the magnetic field. Some of these high energy protons may be moved by magnetic fields such that they return to the inner part of the universe. Such protons can be a source of cosmic rays with energy > 1013 eV,
For a cosmic ray proton having a high energy of, say, 1020 eV, The “whole” size of this proton xII,f − Xe−f in
In Figures 2-4, the diquark is closer to the center of the star, galaxy or universe; xI < xII. This is due to that the heavier diquark experiences a stronger pull towards the centers than does the lighter quark. This pulling force becomes very weak far out in the universe where the positive relative energy increases by the above run away instability pushing the quarks outwards with ever increasing proton energy and greater proton-quark separation or the “whole” size of the proton, x1I,f - Xe-f in
This may in principle go on forever. This pushing force is stronger on the heavier diquark than that on the quark and may eventually push the diquark past the quark farther out and ”flip” their positions from xI,f < xII,f in
But even before this scenario, another one may take place. The protons with energy > 1013 eV are part of a plasma with transverse magnetic field in §10.2. Some of them will eventually move in a direction perpendicular to the direction of Ru in
The present results lead to the following scenario for the universe.
1) The universe contains neither dark matter nor dark energy.
2) A heavy neutron star with mass = MTOV and radius = its Schwarzschild radius does not collapse into a mass singularity and may be called a “black” neutron star. If every neutron star gets heavier, becomes a black hole and passes through this stage in its development, there will be no mass singularity in the universe, in agreement with that quantum mechanics does not allow such a singularity.
3) The run away instability in Section 9 provides a mechanism for an accelerating expansion of the universe. The driving positive relative energy is “free” and “costs nothing”. The density and temperature of the universe will decrease with time. This scenario may in principle go on forever.
4) In outer parts of the universe, fast expanding hydrogen gas may turn into plasma and part of it may become cosmic rays. There may also exist a scenario in which the above free expansion can halt and eventually reverts to contraction.
The above results are derived phenomenologically by joining SSI to aspects of general relativity. A formal integration of these both theories is beyond reach; no quantum gravity theory exists presently.
The author declares no conflicts of interest regarding the publication of this paper.
Hoh, F.C. (2019) Cosmic Applications of Relative Energy between Quarks in Nucleons. Journal of Modern Physics, 10, 1645-1658. https://doi.org/10.4236/jmp.2019.1014108