_{1}

^{*}

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 M_{TOV}~3 solar mass M_{SUN} 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 ~M_{TOV} 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 m_{A}, m_{B} and m_{C} located at x_{I}, x_{II} and x_{III}, respectively. The quarks interact with each other via scalar strong potentials V_{AB}(x_{I}), V_{AC}(x_{I}), ⋯ 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 χ_{B} and ψ_{B} are linear combinations of the four components in the conventional Dirac bispinor ψ according to [ [_{B} is quark mass, about 2/3 of the proton mass [ [_{BA} and V_{BC} generated by quark A and C, respectively, are governed by [ [_{GB} from ambient matter and energy, neglected in the references, has been included.

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 x_{I}, x_{II} and x_{III} according to [ [_{III} has been merged into quark A at x_{I} to form a diquark at x_{I}. The nearly intractable three body problem is thus reduced to a manageable two body problem.

In [ [^{p}, where p runs from 1 to 6 representing the six quark flavours. The quark mass is recovered as eigenvalue of such an operator. The three flavour functions, one for each quark, are analogously multiplied together and generalized to a nonseparable baryon flavour function. The product of the three mass operators are similarly generalized to a nonseparable baryon mass operator in the flavour space and leads to a mass M_{b} [ [

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 Φ_{b} is a triple product of three strong interaction quark potentials of the type of V_{BA} and V_{BC} 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 χ_{0} and ψ_{0} the wave functions of the doublet baryons.

Since quarks cannot be observed, their coordinate spaces are converted into an observable laboratory space X^{μ} 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, a_{m} is often determined by that X^{μ} is the center of mass of these particles. If these particles have equal mass, a_{m} = 1/2. Such kind of determination cannot be done here because quarks are not observable individually. a_{m} 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” [ [_{m}, leads to that both the diquark at x_{I} and quark at x_{II} become observables, contrary to experience. This can also be seen directly in the first of (3.1) in which the right side members cannot be measured; hence x^{μ} is also “hidden”, independent of a_{m}. It turns out that the bulk of hadron physics in SSI lies in such “hidden” spaces.

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 E_{K} is the energy of the baryon and K its momentum. x^{0} is the relative time and −ω_{K} 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 f_{0}(r) in (3.4).

Consider baryons at rest, K = 0. a_{m} = 1/2 is set as in the meson case [ [_{0} = 0 is also set following 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 Φ_{cd}(x) [ [_{cb}®∞ in [ [_{b}’s are unknown integration constants.

The two coupled third order Equations (4.1) have been converted into six first order equations [ [_{+} values in [ [_{(1)} and w_{(2)} in [ [_{b}_{4} = 0 led to results in near agreement with data [ [_{b} and two w totalling six constants that can be varied to satisfy the six boundary conditions posed by the six first order system.

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 E_{0} and the quark masses obtained from meson spectra given in [ [_{b} according to (2.2c) where m_{A} = m_{C} is the d quarks mass and m_{B} the u quark mass. The six first order equations [ [_{b}_{2} value, the remaining five constants can be freely chosen such that g_{0}(r) and f_{0}(r) converge at large r. This is allowed since this takes place in the “hidden” relative space.

Here, the word “finite” on the first line of the second paragraph on [ [_{b}_{2} can be varied continuously to yield converging g_{0}(r) and f_{0}(r). One so-obtained neutron wave functions are plotted in [ [

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 a_{m} = 1/2 and −ω_{0} = 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 −ω_{0} is “hidden” with respect to strong and electromagnetic but not gravitational interactions [_{BG} in (2.1), which has been neglected in SSI because it is generally small next to the strong V_{BA}, V_{BC} there. In neutron stars and black holes, however, V_{BG} can be large and may not be neglected. Since a_{m},hence also ω_{0}, can take on any real value, from −∞ to +∞, the relative energy can be important is astrophysics.

In [_{0} > 0 is identified as dark energy and −ω_{0} < 0 plays the role of dark matter. Both E_{0} and ω_{0} are observables with respect to gravitational forces. Lorentz invariance leads to that the associated baryon momentum K in (3.3) and the relative momentum represented by ∂ _ in (5.1) are also observables. Thus, the conjugate variables X, the coordinate of the baryon, and x, the relative coordinate in (3.1), are also observables, remembering that x is still a “hidden” variable with respect to strong forces. By (3.1), both the diquark coordinate x_{I} and quark coordinate x_{II} are now observables with respect to gravitation indicated by V_{BG} in (2.1).

Consider the following idealized scenario. A neutron star with a mass M_{NS} = M_{NSa}, 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 _{a} lies in the middle corresponding to a_{m} = 1/2 in (3.1) and (4.1) and hence ω_{0} = 0 by (5.2). This neutron falls from X_{a} to X_{b}. To begin with, the three positions x_{I}_{,a}, X_{a} and x_{I}_{I}_{,a} are simply

shifted to the left to become x_{I}_{,b},X_{b} and x_{I}_{I}_{,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 R_{b} < R_{q} due to gravitational collapse. The incoming neutron is thus similarly compressed to become smaller by a factor of R_{b}/R_{a} so that r_{b} is also reduced to a smaller value r_{bG} 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 x_{II} in (2.1), hence also with that in _{I} in _{b} of the star at position b is taken to be x_{I}_{,b }

If X_{b} were taken to be the star radius R_{b} at position b, then the diquark coordinate x_{I},_{b} in _{b} = 0, contrary to the requirement that a radius ≥ 0. Practically, X_{b} and R_{b} are of the magnitude km while their difference is of magnitude fm; both are nearly the same.

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 E_{0} and the quark masses in M_{b} at position b are unchanged. So does also the confining constant d_{b}_{2} = −0.3202 in _{b} and two w constants mentioned above _{0}(αr) and f_{0}(αr) analogous to the computations that led to _{a}~4α fm.

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 −ω_{0} is the relative energy gained when the external neutron arriving at position a falls to position b. In this fall, the gravitational energy E_{G} 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 R_{Sa} is the Schwarzschild radius of this collapsing star with radius R_{a} in

R a = R S a (7.3)

i.e., when the neutron star in _{NSa} has a radius R_{a} equal to its Schwarzschild radius R_{Sa}. At the star center R_{b} = 0, both energies become infinite but these both singularities cancel; no net mass singularity is created when (7.3) holds. This is consistent with that SSI is a quantum theory not allowing such a singularity. Note that when approaching the star center and R_{b} becomes of the size of the neutron of a few fm. For R_{a} ~10 km; a_{m} = R_{a}/2R_{b} is a huge number ~10^{18}.

In this case, the fall of this external neutron actually gains no energy because E_{G} − ω_{0} = 0 for all R_{b}. 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 R_{a} 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 R_{a} = R_{Sa} = 10.8 km and M_{NSa} = 3.6M_{SUN}. This mass is consistent with the similarly estimated maximum mass 4.3M_{SUN} [ [_{TOV}~3M_{SUN}. cited in Section 1. However, estimates of M_{TOV} are uncertain and vary widely due to that the equations of state for extremely dense matter are not well known and to uncertain effects of neutron-neutron repulsion and of pressure in general relativity [ [

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 ΔM_{NSa}, 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 R_{Sa}_{Δ} > R_{a}_{Δ}, a black hole is created. As a new neutron arrive at the new star surface with radius R_{aΔ}, 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 > R_{Sa}_{Δ}, the back hole is lost and this enlarged star will be a visible neutron star similar to the initial star in _{Sa}_{Δ}, the created black hole persists and one arrives at a situation nearly the same as that on the right part of _{a} replaced by R_{a}_{Δ} and (7.3) holds. This process may be repeated and more shells are added. It can in principle lead to a super-massive black hole. How the neutrons, part of them “weightless”, are distributed inside such a black hole is unknown and needs not be known, inasmuch as they are inside its Schwarzschild radius and cannot be observed any way; there is no conflict with observation.

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 [ [_{0} < 0 between the diquark and quark in nucleon. Here, the role of the dark matter in accounting for this rotation curve will be played by this −ω_{0}.

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 _{c} which lies in the middle between the diquark at x_{1,c} and quark at x_{I}_{1,c}. This corresponds to a_{m} = 1/2 and ω_{0} = 0 in (7.1) for a free baryon at rest, similar to those for the neutron in position a in _{.} The new x_{I}_{,c} and x_{I}_{I}_{,c} are on both sides of X_{d} with the same separation distance.

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 X_{d} 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 x_{I}_{,d} closer to this center than does the lighter quark at x_{I1,d},as is shown in

Applying (3.1) the gravity pulled position d in _{0} given by

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 X_{d} 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 x_{I,d}, x_{1I,d} behind the observable proton coordinate X_{d}in _{0} itself, as can be seen below (2.1) and (6.1); the unobservable quarks experience greater gravitational pull than does the proton containing them. The difference between these two masses is taken up by the strong, confining potential F_{bd}(r) of (4.2) in the “hidden”, relative space.

The required energy is -w_{0}» -5.5E_{0} per proton, when averaged over the whole universe, and is generated for “free” at “no cost”. This −ω_{0} value leads to a_{m} ≈ 6 ( [_{d} − x_{I}_{,d} ≈ 24 fm by (8.1). Further, the proton in a hydrogen atom moves in an orbit around the center of gravity of the electron-proton system. This orbit has a diameter ≈ 60 fm so that X_{d} is uncertain by this amount which is greater than the above 24 fm in

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 −ω_{0} > 0 corresponding to a_{m} < 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 _{m}»1/2 and the pulling energy-w_{0}< 0 approaching -w_{0}» 0. The situation is illustrated in position e of

Let a second hydrogen molecule arrive at position e simultaneously. It contains a second proton with its diquark and quark at the same x_{I}_{,e} and x_{II}_{,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 X_{e}_{+}, 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 _{e}. No appreciable net movement of the quarks is expected; the situation is stable.

If its position is X_{e}_{−}, then (8.1) yields a positive relative energy − ω 0 = − Δ E 0 > 0 . Contrary to the above X_{e}_{+} case, this small energy tends to push the quarks outward, towards the right part of _{e}; Δ increases and a_{m} decreases. The quarks move toward the outer region of the universe but the observable proton coordinate X_{e}_{−} lags behind; this situation is opposite to that for X_{e}_{+} above or to that in position d in _{I}_{,f} and x_{II}_{,f} in _{m} has decreased from 1/2−Δ at position e to <0 at position f so that the positive relative energy −ω_{0} by (8.1) has grown appreciably to >E_{0}/2. This new, larger positive relative energy further pushes the quarks outwards at a greater pace and tends to increase the lag of the proton coordinate X_{e}_{-}_{f} behind the quark coordinates x_{I,f,} x_{II,f} in _{0} > 0 which gives still greater pushing force on the quarks. This is an unstable, “run away” situation which can in principle continue forever. The driving relative energy is “free”, “costs nothing” and is not conserved. If this mechanism is applied to all protons in the outer regions of the universe, an acceleratingly expanding universe emerges.

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 −ω_{0}, (8.1) leads to a_{m} = −13.5 ( [_{I}_{I}_{,f} − X_{e}_{−f} = 58 fm which is close to but still within the error margin ≈ 60 fm for X_{e}_{−f} mentioned at the end of Section 8. This margin is small relative to the size of hydrogen atom and does not alter its behavior as a hydrogen atom.

This scenario, characterized by

As the above “run way” situation continues, the “whole” size of this proton, x_{I}_{I}_{,f} − X_{e}_{−f} in _{e}_{-}_{f} is located at the center of the atom, the quarks will in the same orbit as the electron. These two objects may come to be close to each other and interact via some so far unspecified mechanism. Since the electron is much lighter than a quark, which has a mass of about 2/3 of the proton mass mentioned below (2.1), it will be assumed that the electron gets ejected, noting that the ionization energy of hydrogen atom is negligible relative to the energy scale involved here. This hydrogen atom becomes ionized.

The relative energy corresponding to this case is by (8.1) −ω_{0} ≈ 1.29 × 10^{1}^{3} 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 × 10^{13} 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 −ω_{0} 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 > 10^{13} eV,

For a cosmic ray proton having a high energy of, say, 10^{20} eV, The “whole” size of this proton x_{I}_{I}_{,f} − X_{e}_{−f} in

In Figures 2-4, the diquark is closer to the center of the star, galaxy or universe; x_{I} < x_{II}. 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, x_{1I,f} - X_{e}_{-}_{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 x_{I,f} < x_{II,f} in _{I} > x_{II} farther away. In this scenario, (8.1) shows that a_{m}, hence also −ω_{0}, changes sign; the expansion force switches to contracting force. The expansion will slow down and change to contraction. If such processes continue, an “oscillating” scenario of the outer regions of the universe may emerge.

But even before this scenario, another one may take place. The protons with energy > 10^{13} 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 R_{u} 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 = M_{TOV} 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