Gravity as the Sole Fundamental Force

We had explained electromagnetism by gravity before a recent publication in this Journal, in which we further incorporated the nuclear strong force in the framework of gravity. This paper, summarizing our cumulative results, continues to integrate the nuclear weak force with gravity, where we go by the following line of logic: Planck’s formula shows energy E = frequency = probability = wave; hence quantum waves have energies and the Universe is a diagonal spacetime manifold containing {(particle pi, electromagnetic wave λ (pi))}. By Feynman’s analysis on electromagnetic mass, we assume that the distribution of E over (p, λ (p)) is ( 3 4 , 1 4 ) E. Then Newton’s gravitational acceleration formula yields E = 1.6 × the observed energy o f p, so that p exists only for a duration of 8 λ c over the cycle [ 0, λc ] , such as evidenced in quantum tunneling, opening the possibility for λ (p) to be combined with other waves forming new particle(s) for t > 8 λ c . By the time ratios of two frames in General Relativity we deduce neutron’s lifetime, and by the Higgs mechanism we show neutron’s decay products.


Unfulfilled Goal of Unifying the Four Forces
The quest for a unified field theory has been an ongoing endeavor in Physics for nearly a century (for a recent study, see Kamada, 2018).Previously we explained electromagnetism by an extended (set of) Einstein Field Equations and we invalidated the electroweak unification by noting the fact that in the Weinberg linear transformation an interchange of the two entries in (Higgs ϕ, 0) T would fail the sought objective of distributing ϕ entirely to the Z boson to leave photon without a rest mass.

Hypothetical Background of Our Unification of the Four Forces
We present a general description here (for analytical details, refer to Light, 2019): Before the Big Bang there had been M [2] of electromagnetic waves (EMW's) without their particle presentations (now recognized as "dark energy") along with their collided standing masses (now recognized as "dark matter").Because of the gravitational constant G [2] ≈ 10 85 G [3]  (the Newton constant), a cosmic black hole B formed in M [2] .The interior of B had (has) a quotient topology due to g [2]  11 < 0. Since the center of mass of B had to be averaged over one common equivalence class of spacetime and the least class had length 10 −63 meters < our assumed least indivisible unit of distance 10 −35 meters, B blew up at its center (the Big Bang), creating (particle p i , EMW (p i )) existing in a diagonal spacetime manifold M [3] = M [1] × B = the known Universe with linearization (t + it, x + iy, y + iz, z + ix), where p i = electron e − , up quark u, down quark d, or neutrino ν along with its anti particle of the same handedness; the pair-creation came about through a pair of (photon γ, EMW (γ)) ′ s intersecting each other at 90, 60, 30, or 0 degrees respectively.For example, a left-handed electron e − L has its EMW (5) A left-handed proton of opposite angular momentum to ν.
(Without loss of generality, the following analysis will be presented by the left-handed convention.)

Method and Results
To make this paper self-contained yet without undue redundancy to our previous publication (Light, 2019), we will present the key logical elements of our integration of gravity with electromagnetism in Section 2.1 and that with the nuclear strong force in Section 2.2; we will then show our new results of integrating gravity with the nuclear weak force in Section 2.3.Our method will be that of mathematical proofs with conclusions followed as results.

Integration of Electromagnetism with Gravity
Einstein Field Equations ("EFE") begins with a Lorentzian metric g, which defines the associated covariant derivative ∇ X Y by the Koszul formula, then in turn the Riemann-Christoffel curvature (1, 3) − tensor R (U, V) W, in turn to the Ricci curvature (0, 2) − tensor R µν and finally to the Ricci curvature (0, 0)−tensor R.That is, the entire left-hand-side E := Ric− 1 2 gR, recognized as the Einstein-Hilbert tensor, is dependent on g and is essentially the derivative of the total scalar curvature of a bounded manifold with respect to g. E by its own nature is already an energy-momentum tensor T , for which the constant of proportionality was (is) determined by Newton's gravitational law.The reason why using flat spacetime to determine physical constants is valid is owing to the algebraic homomorphism of tensors where A * is any pull-back operator of any tensor T that operates on vectors in a tangent space and α is any numerical constant; A * corresponds to any arbitrary frame transformation in physics, which includes in particular a frame situated in a flat spacetime.As such, c, G, and h are universal, and all weak-field "approximations" (thus a misnomer) are to yield exact physical relationships.
Consider now the dynamics of a proton Q (≡ P as denoted above) at (0, 0, 0, 0) ∈ U ⊂ R 1+3 (flat spacetime with Minkowski metric η := diag ) ) that attracts an electron q (≡ e − as denoted above) at (0, x, y, z) ∈ U; set r ≡ √ ( Proposition 1 Let g be a local metric of a spacetime manifold M 4 and express g as a matrix in the basis of B ≡ , where f : (t, x, y, z) ∈ U → ( t, x, ỹ, z) ∈ M 4 is a local parametrization.Assume that M 4 is near flat; then a neighborhood Ũ of any point x = ( t, x, ỹ, z) ∈ M 4 can be identified with the tangent space T x Ũ, i.e., Ũ ≡ T x Ũ with x = (i) a point in Ũ or (ii) a vector in T x Ũ (to be distinguished by a subscript "o" for clarity of the underlying context, such as a proper time to , which is the projection of the vector ( t, x, ỹ, z) onto the vector ( t, 0, 0, 0 ) , but as a 1-dimensional value to := t − 0 ≡ t).Set the Lorentz transformation L = f ; then the derivative of proper time to with respect to proper time t o is: ) is the velocity of q.
Proof.Without loss of generality, consider transforming frame S to S , and calculate where ∆ to is the proper time of S by definition) analogous to the above Equation for L).
Remark 1 The meaning of and corresponds to the least action , by Equation ( 15) , , by Proposition 1. I.e., (1, 0, 0, 0 Remark 3 (1) dt o dt o = the effect of the velocity V (t o ) of the electron q in the electromagnetic field as established by q and the proton Q on q′s proper time.( 2 the universal time over U as synchronized with t o =t * of Q (γ accounts for time dilation). where is an electrodynamic adjustment factor of the electrostatic potential; i.e., the point charge Q travels to the boundary of its wave ball r o , or equivalently, Q is a stationary proton.Thus, taking into account the effect of Special Relativity we have Proposition 3 where A (t) := the vector potential, or curl A (t) = the magnetic field B (t).
we have Proposition 4 For any weakly attractive or repulsive electromagnetic field, the metric g att; rep em has the following matrix representation in the basis of B (for generalization, we now include the repulsive case, shown as the second or lower index): Proof.First, we note that besides being symmetric, g att; rep em Third, by Proposition 1 we have thus, by Equation ( 17) we see that g att; rep em = g B , satisfying the least-action energy condition, and the theorem is proved.
Lemma 1 Assume that the energy-momentum tensor T ≈ 0 for M 4 ; then where r K > 0 denotes the radius of curvature of the space-time. Proof.
) ∈ R 4 } be a parameter domain of M 4 ; adopt the Cartesian coordinate system on U; let (e i ) 4 i=1 be the standard basis of R 4 ; define metric I to be the standard inner product of R 4 , i.e., i=1 .Since T ≈ 0, the space-time is asymptotically flat and thus any Lorentzian metric g on M 4 is such that g ≈ I. Consequently, , where they fall into two categories: (t, x) and (x, x), i.e., Thus, Since metric I is equivalent to the Minkowski metric ) we thus have Since η measures space-time distance by ds and we have for each space-time sectional curvature ≈ K < 0; denoting the radius of curvature by r K > 0, we then have Proposition 5 The Einstein tensor Proof.
(where r K ≡ the radius of sectional curvatures).Thus, substituting Equation ( 30) into , we arrive at the conclusion.

Proposition 6 Set
T att; rep em,11 • c 2 and (49) where as derived below: First g 11,grav is the gravitation effect of q on the spacetime metric g, but by Remark 3(2) As such, continuing on Equation (57) Theorem 1 ) c 5 T att; rep µν,em . (64) Proof.(By the above Proposition 7.) Vol. 11, No. 3;2019 Remark 4 A distinct feature of gravity is the existence of the principle of equivalence between inertial masses and gravitational masses, so that the two cancel out and the size of the inertial mass does not need to be addressed explicitly, but electromagnetism lacks the same principle and we solved this problem via ) , the greater the inertial mass of the particle being acted upon, the smaller the g 11,grav and hence the greater the denominator of the constant of proportionality.In this connection, we also made a distinct identification of T att;rep 11,em with the of the Poynting vector S, and as a result, the derived geodesics correspond exactly to the least action by Feynman.As the above showed that a Poynting vector on the right-hand-side of EFE is in direct correspondence with a minimization of the integral of kinetic energy minus potential energy over all trajectories on the left, we see the reasons why any other identifications of T µν,em have resulted in difficulties in geometrizing electromagnetism.In this regard, our T att;rep 11,em has unit joule/ ) , representing energy flows in a specific direction across an area of square meter per second, and yet the common identification of T 11,em with the energy densities has unit joule/ ( meter 3 ) , representing stationary energies.
Theorem 2 The set of Einstein Field Equations has solutions: and g µν = w grav • g µν,grav ± w em ) ; we see that the operation of E µν,grav ± E att;rep µν,em is valid if and only if g µν is forminvariant with respect to measuring geodesics, possessing the same energy interpretations as g grav and g att;rep em .Here we have: (1, 0, 0, 0) where T repul 11,em and (1, 0, 0, 0) where Consequently, g µν = w grav • g µν,grav ± w em • g att;repul µν,em is form-invariant in measuring geodesics, with identical interpretations of energies to that of g µν,grav and g att;rep µν,em .I.e., T att;repul em results in a metric g µν that renders

Integration of the Nuclear Strong Force with Gravity
Most recently we showed that (Light, 2019): (1) quarks combine themselves by superpositions of coincidental fields, hence inseparable, and (2) proton P has a mini black hole of radius 10 −15 meters, attracting a neutron n with g 11 ≈ 10 −20 by the extended EFE, so that G • 10 10 m P • 10 10 m n ( 10 As such, the nuclear strong force is a manifestation of gravity.For the purpose of this paper we will give a sketch of the proof of g 11 ≈ 10 −20 by the extended EFE.

Integration of the Nuclear Weak Force with Gravity
In the above analysis we noticed that n, not possessing a mini black hole due to having no electric charge, is fundamentally different from P (cf.Bigazzi, & Niro, 2018, for isospin-breaking); for example, while the radius of P is 10 −15 e meters, that of n is 10 −15 meters, smaller.Since in our hypothesized diagonal 4-manifold M [3] of the Universe (cf.Brandenberger, et al., 2018, andSengupta, 2017, for other constructs of a dual universe not based on the particle-wave duality), black holes are intersections of the visible particle universe M [1] and the invisible wave universe M [2] , P being embedded in its own mini black hole does not vanish from M [1] , but n without a mini black hole vanishes from M [1] (cf.Barducci, 2018, for "disappearing into the darkness"), which motivates the following analysis.

Lifetime of Neutron
Proposition 8 Assume that the distribution of energy E over (particle p, EMW (p)) is ) . Then E = 1.6 Ê, where Ê ≡ the laboratory-measured energy of p.
Proof.Previously we derived the metric tensor g [3] of M [3] to be g [3]   11 = G [2] G [1] + G [2] g [1]  11 + [2]  11 , with (86) Therefore, , where (89) Proof Remark 5 The laboratory-measured energy of a single-cycle particle p is Ê owing to the existence of (p, λ (p)).Since wave is probability of no energy in the quantum formulation (which contains the formula Ê = hν), p must carry the entire energy E = 1.6 Ê that would enjoy a higher frequency 1.6ν, but the measured frequency is (still) ν; therefore, p exists (only) in λ.That is, the fact that p carries a wave length λ (p) does not imply p exists throughout [0, 1] λ, as evidenced in, say, quantum tunneling.Equivalently, if p lasted for [0, 1] λ, then one would have Remark 6 Single-cycle particles include electron, neutrino, proton, neutron, photon, and any hadrons that are made up of u and/or d.
Proposition 9 Assume that in the motion of EMW (n): W → N → E → N → W, the stopping time at W and E is 10 −24 seconds; then n in isolation has a lifetime about 900 seconds.
Proof.By the above Remark 3 (2) the ratio of two proper times is the eigenvalue λ of the Lorentz transformation, Set the proper time t [2]  0 to be that of EMW (n) and t [1]  0 to be that of the laboratory frame.Then Since by Corollary 1 93) (in analogy with m 0 = mγ −1 , but the relation here is that of λ = γ , not of the Lorentz factor γ), one has Substituting the rest mass m 0 implying that v c and this reduction of speed in EMW (n) by a factor of 6 × 10 −27 implies a prolongation of n to finish 5 8 λ by (about) the same factor (as in (1 − ϵ) −1 ≈ 1 + ϵ).Thus in the laboratory frame, EMW (n) stops at the intersection points W and E for about 900 × 6 × 10 −27 = 5.4 × 10 −24 seconds. (99) Remark 7 The lifetime of n in its own frame is then 900 √ 3 × 10 −27 ≈ 5 × 10 −11 seconds (cf.Leontaris, & Vergados, 2019, for recent interest in the neutron lifetime puzzle).

Neutron Decay
Proposition 10 Assume that the Higgs particle h has a representation h = 2γ L + γ R + γ iL , where  Proof.We index the flows to keep track of the re-combination of waves: (105) Remark 8 Weak decay has many practical implications, for example, nuclear reactor operations (Gebre, & Surukuchi, 2018).Our graph-theory approach may serve as an added tool in quantum field theories.

Discussion
(1) Every single-cycle particle (p, λ (p)) exists in M [3] for a duration of 5 8 λ c ; then it becomes (0, λ (p)) in B ⊂ M [2] for the remaining 3 8 λ c .This process repeats itself if p has a mini black hole; otherwise, λ (p) can combine with other waves to form new particle(s) (for recent studies on beta decay, see, e.g., Deppisch, et al., 2018).
(2) Both the strong and the weak nuclear forces can be explained by General Relativity via the metric tensor g 11 which relates two proper spacetimes.
(3) As EFE also explains electromagnetism, gravity remains as the sole fundamental force as based on our geometry of diagonal 4-manifold.We remark that the basic spacetime hypotheses determine the rest of Physics (cf.Minazzoli, 2018).

) 1
Corollary Any particle p that has its EMW (p) a union of two semi-circular rotations presents itself as (p, EMW (p)) for a duration of 5 8 λ c and (0, EMW (p)) for a duration of 3 8 λ c , where λ ≡ the wave length of EMW (p) (to define such particles as "single-cycle particles"). and