Fundamental Constants: Uniting Gravity and the Accelerated Expansion of the Universe through the Higgs Boson

Fundamental Constants: Uniting Gravity and the Accelerated Expansion of the Universe through the Higgs Boson M. Makraini ([email protected]) Relativistic and Fundamental Physics Espa˜na (Spain) August 15, 2023 Abstract This article presents a new reinterpretation of the curved geometry of spacetime, where it is considered that spacetime undergoes longitudinal contraction. This effect is manifested in changes in the spacetime metric that determine how distances and temporal intervals are measured in that region. In other words, a variation in the scale, size, or apparent length of spacetime. This reinterpretation is compatible with Einstein’s field equations and Maxwell’s equations. The universal gravitational constant of Newton, GN, the Hubble constant for the accelerated expansion of the universe, H(0), and the cosmological constant associated with hypothetical dark energy, Λ±, can be obtained and approximated using this new approach, where the mass of the Higgs boson with its unique and privileged characteristics plays a crucial role in addressing numerous open questions in physics and modern cosmology. The reinterpretation of curved geometry through spacetime contraction provides a new framework for better understanding gravity. By obtaining very close values of the universal gravitational constant, it is possible to determine the inverse force to gravity responsible for the accelerated expansion of the universe. This is achievable through Gauss’s divergence theorem, where the charge distribution determined by the Coulomb constant within the framework of multipolar expansion defined by electromagnetism constitutes a quite solid analogy, being inversely proportional to gravity. This allows for the precise calculation of the value of the Hubble constant, H(0). The cosmological constant Λ±, considered as a potential dark energy driving the accelerated expansion of the universe, can also be obtained and explained through this new approach. The reinterpretation of the curved geometry of gravity as spacetime contraction would affect the properties of spacetime expansion, where the interpretation of the universe’s contraction described by General Relativity must be reinterpreted, understood, and accepted as gravity itself at any scale.. Key words and phrases: Universal constants, Higgs boson, General Relativity, Hubble stress, cosmological constant. 1 Introduction The longitudinal contraction of space-time emerges as a bold proposal in the field of theoretical physics and cosmology, challenging conventional concepts of space-time geometry that try to describe gravity. And the proposal that is postulated in this work, describes a revolutionary fundamental reinterpretation of space-time, in which, the slightest presence of mass and energy, gives rise to a space-time contraction around this mass and energy, both at astronomical levels and from its nature at the quantum level. In this context, a strong connection and similarity arises, with the longitudinal contraction experienced by objects in situations of high speeds close to light, which is inversely proportional and analogous to the spatiotemporal contraction proposed in this work. We know very well that the theory of Special Relativity masterfully proposed by A. Einstein [1], establishes that as an object approaches at speeds close to that of light, its length in the direction of movement experiences a contraction relative to an observer at rest. This contraction known as “Lorentz Contraction” [2], is in fact analogous and inversely proportional to the space-time contraction that bodies generate at any quantum or astronomical level to create gravity. The similarity 1 between the contraction of Lorentz and the longitudinal space-time contraction proposed, opens the doors to a deep exploration of the physical and geometric implications associated with external phenomena of movement, speed and energy that describe the gravitational field. In this work, the convergence between the longitudinal contraction of space-time and the Lorentz contraction will be shown in detail, to mathematically approximate the values of the three most fundamental constants from a technical and rigorous perspective. The implications of this similarity in the conceptual framework of fundamental physics will be analyzed, from which its culmination arises in the global and relevant theoretical set that experimentally demonstrates the standard model of particles, by being able to consider the foundations of space-time geometry, as a result in its vehicular relationship through the interaction of the Higgs mechanism [3] and the percentage difference between the mass of the proton and its constituents, the quarks, proposed in this study supported by experimental results, as the most suitable candidates. solid, responsible for gravity. The reasoning behind the concepts proposed here emerges from a new perspective in the Michelson-Morley experiment [4]. An unprecedented historical milestone, which once again opens the doors to a deeper understanding of the intrinsic properties between matter, energy, space-time and their connection with the fundamental laws and constants of nature. From which, through this analysis, it is hoped to shed light on the essential nature of space-time and the possibility of a solid fundamental reinterpretation of geometry, at the extreme limits of physics. 2 Curvature scalar: When the line element for a Rindler-Minkowski spacetime depends on the mass. In a uniformly accelerated reference frame in Rindler-Minkowski space-time [5], it is possible to obtain an expression for the mass-dependent Laplacian. In this context, the relationship between the gravitational field and the mass, completes in a different way, what is found in General Relativity, where the slightest presence of mass and energy should describe the spacetime contraction (gravity) of proportional to any level, then this contraction may be at quantum levels as well as astronomical levels. That is, whatever the mass and minimum energy levels, Newton’s universal constant must arise logically and naturally. Rindler-Minkowski spacetime, where spacetime is supposed to be flat and not curved, is used to describe a uniformly accelerated reference frame in Special Relativity. In this accelerated system, the concept of a gravitational field can be approximated and modeled through a uniform acceleration. But it is not a real gravitational field generated by the mass distribution, rather it is a solid analogy of the true meaning of the weak equivalence principle. Taking into account a Rindler-Minkowski space-time, we can make an approximation within another approximation, expressing it by making the Newtonian limit by this line element: [6] ds2=−1+2ˆ Φc2dt2+dx2+dy2+dz2(2.1) Where ˆ Φ = Φ mc2and proposing that Φ = m2 ϕϕ2, we can calculate the Ricci tensor of the line element, but first, it is necessary to compute the Christoffel symbols for the following metric: 2 g00 =−1−2mϕϕ2 c2g11 = 1 g22 = 1 g33 = 1 (2.2) Now, using the formula: Γα βγ =1 2gαµ (∂γgβµ +∂βgγµ −∂µgβγ) (2.3) Where, for the term Γα 00, we have: Γα 00 =1 2gαµ (∂0g0µ+∂0g0µ−∂µg00) =1 2gαµ∂µ−1−2mϕϕ2 c2=1 2gαµ∂µ 2mϕϕ2 c2(2.4) So, we simplify and rearrange to get the Christoffel symbol for Γα 00, as: Γα 00 =mϕ c2gβµ∂µϕ2(2.5) Calculating the same for the term Γ0 β0, we have that: Γ0 β0=1 2g0µ(∂0gβµ +∂βg0µ−∂µgβ0) =1 2g00 (∂βg00 −∂0gβ0) = 1 2g00∂µg00 =1 2 1 g00 ∂β−1−2mϕϕ2 c2=−mϕ c2 1 g00 ∂βϕ2(2.6) So, substituting g00, we have: Γ0 β0=mϕ c2 1 1 + 2mϕϕ2 c2 ∂βϕ2(2.7) 3 Now, we can calculate the Ricci tensor: Rσν =∂µΓµ νσ −∂νΓµ µσ + ΓΓ −ΓΓ (2.8) In the approximation for a weak field in Rindler-Minkowski spacetime, we can dispense with the components ΓΓ −ΓΓ. Being the Ricci tensor, for weak fields, in this way: Rσν ≃∂µΓµ νσ −∂νΓµ µσ (2.9) Then, to calculate the component R00 of the Ricci tensor, we have: R00 =∂βΓµ 00 =∂µmϕ c2gβµ∂µϕ2(2.10) Where: R00 =mϕ c2∂0g0µ∂µϕ2+∂1g1µ∂µϕ2+∂2g2µ∂µϕ2+∂3g3µ∂µϕ2 =mϕ c2∂1g11∂1ϕ2+∂2g22∂2ϕ2+∂3g33∂3ϕ2 =mϕ c2∂2 1ϕ2+∂2 2ϕ2+∂2 3ϕ2(2.11) And finally, we have that the result for R00 of the Ricci tensor is: R00 =mϕ c2∇2ϕ2(2.12) Here, the Laplacian scalar ϕ2in Rindler-Minkowski spacetime can effectively depend on mass via the term mϕ c2for a very large gravitational field. weak or practically none. However, at quantum scales, the intensity of the gravitational field will be proportional to the energy or mass of the particle. In this context, the Higgs boson (without spin and without charge) could be the perfect candidate to explain gravity. 4 It is important to note that this idea is related to a specific model of approximation and uniform acceleration in Special Relativity. By incorporating even the smallest amount of mass and energy, a strong relationship emerges. By considering the mass of the Higgs boson as a solution of Einstein’s field equations and incorporating the tensor of the electromagnetic field, it is possible to approximate the values of the three most fundamental constants. 2.1 Chiral symmetry breaking It is an example of spontaneous symmetry breaking that affects the chiral symmetry of strong interactions in particle physics. It is a property of quantum chromodynamics, the quantum field theory that describes these interactions, being responsible for most of the mass (more than 99%) of nucleons, and therefore of all ordinary matter, since which converts very light quarks that are bound as constituents into 100 times heavier among the [7] baryons. As a consequence, the effective theory of QCD bound states, such as baryons, must now include a mass term for these states, ostensibly prohibited by unbroken chiral symmetry. Therefore, chiral symmetry breaking induces most of the mass of baryons, such as nucleons, and explains the origin of most of all the mass that makes up visible matter [8]. 3 Electromagnetic energy density: Momentum energy tensor of the electromagnetic field Consider the energy-momentum tensor Tµν in an electromagnetic field. In the static and uniform case, the only nonzero components of the tensor are T00 and Tij, but this time we will focus only on the T00 component. For a static and uniform electromagnetic field, the energy-momentum tensor takes the following form: T00 =1 8πE2+c2B2(3.1) In a vacuum, with no free charges or currents (ρ= 0 and J = 0), Maxwell’s equations are further simplified: ∇·E= 0; ∇·B= 0; ∇×E= 0; ∇×B= 0; (3.2) Given that ∇× E= 0 and ∇× B= 0, we can conclude that the electric and magnetic fields are conservative, that is, they can be expressed as the gradient of some scalar potential ϕand vector A: E=−∇ϕB=∇×A(3.3) 5 Gauss’s law for the electric field ∇·E= 0 implies that the Laplacian of the electric potential ϕis zero: ∇2ϕ= 0 (3.4) Now, using the vector potential A, the Amp`ere-Maxwell law ∇×B= 0 becomes: ∇×∇×A= 0 (3.5) Applying the vector identity ∇×∇×A=∇(∇·A)−∇2A, and given that ∇2A= 0 due to ∇·B= 0, we get: ∇2A=−∇(∇·A) = 0 (3.6) Therefore, we can also state that the vector potential Asatisfies Laplace’s equation ∇2A= 0. In the static and uniform case, we can assume that the electric and magnetic fields do not depend on time and are constant in space. If we take the limit as c→ ∞, the terms proportional to c2in the energy-momentum tensor T00 become negligible. Then, the energymomentum tensor simplifies to: T00 =1 8πE2(3.7) In the static case ∇·E= 0, we can ensure that the scalar potential ϕis simply a constant. By convention, we can take ϕ= 0, which leads to: E=−∇ϕ= 0 (3.8) This implies that the static and uniform electric field is zero. Now, let’s consider a specific case. Suppose there is a point charge qlocated at the origin of the coordinate system. The charge density ρassociated with the point charge is: E(r) = qδ3(r) (3.9) 6 Where δ3(r) is the Dirac delta in three dimensions. The electric field Edue to a point charge is given by Coulomb’s law: E(r) = q 4πϵ0 r r3(3.10) To calculate the energy-momentum tensor in this case, we need to calculate the square of the electric field E2. Taking the coordinate system as (x,y,z) and the position vector r= (x, y, z), we obtain: E2= (E)2=q 4πϵ02(r)2 r6(3.11) Since there is spherical symmetry in the system, we can write xixi=r2, and the square of the electric field simplifies to: E2=q 4πϵ02r2 r6=q 4πϵ021 r4(3.12) Substituting the component T00 of the energy-momentum tensor, we obtain: T00 =1 8πE2=1 8πq 4πϵ021 r4=q2 32π3ϵ2 0 1 r4(3.13) So, always considering the case of a static and uniform electromagnetic field for a point charge at the origin of the coordinate system, and so that this equation maintains its form and physical meaning in the framework of special relativity, we include the term c0to be able to express the electromagnetic energy-momentum tensor in this way: T00 =−1 32π3ϵ2 0c0 (3.14) Where, we can highlight that this equation has fundamental applications in electromagnetic theory and particle physics. 7 4 Variational principle of the actions of coupled gravitational and electromagnetic theory. Einstein’s equations are an improvement on Newton’s equations. So, we can appreciate this difference in a very notable way, when gravity is intense and for speeds comparable to light. But when gravity is not strong and speeds are not very high, we clearly recover Newton’s equations. Now, if we accept that gravity exists in an infinitesimal region for a Rindler-Minkowski space-time, where the intensity of the gravitational field is very weak, being practically negligible, then, gravity being an effect of deformation and contraction of space -time, and although the intensity is very negligible at quantum levels, gravity exists proportionally to the mass and energy of the particle. And if the deformation due to space-time contraction is proportional at quantum levels, the only candidate particle due to its special characteristics with a solid connection so that energy and mass proportionally can have a close relationship to deform space-time, is the Higgs boson. And if we consider that at quantum levels where the gravitational field is infinitely weak or practically very negligible, then the same equations that are derived from the actions of the gravitational field must perfectly describe the approximate values of the three most fundamental constants of physics. . So, to achieve this, we are going to calculate the variation of the Hilbert-Einstein action and the action of matter given by the energy-momentum tensor, but coupling in between, the action of the electromagnetic field in this way: S[g] = Zd4x√−gR +Zd4x√−gLem +Zd4x√−gLmatt (4.1) We multiply the first two terms of the action by an unknown constant λ: S[g] = λZd4x√−gR +Zd4x√−gLem+Zd4x√−gLmatt (4.2) Next, we first perform the variation with respect to the metric gµν : δS [g] gµν =λδ δgµν Zd4x√−gR +δ δgµν Zd4x√−gLem+δ δgµν Zd4x√−gLmatt (4.3) The variation of the metric in the first integral affects the curvature R, considering at all times an infinitely weak or practically negligible gravitational field, and in the second integral, it affects the electromagnetic tensor Fµν Fµν . Continuing with the variation with respect to gµν , we obtain: δ δgµν Zd4x√−gR =√−gGµν (4.4) 8 Where we know perfectly well that Gµν is the Einstein tensor: Gµν ≡Rµν −1 2gµν R(4.5) The variation of the electromagnetic term is as follows: δ δgµν Zd4x√−gLem =−1 4√−gFµν Fµν (4.6) And the variation of the matter term that we know perfectly well is given by the energy-momentum tensor: δ δgµν Zd4x√−gLmatt =1 2√−gTµν (4.7) Putting all the contributions together and setting the variance equal to zero, we can rewrite the action like this: δS [g] = λRµν −1 2gµν R−1 4gµν Fµν Fµν +1 2Tµν = 0 (4.8) Finally we rearrange terms to arrive at the following equation: Rµν −1 2gµν R−1 4gµν Fµν Fµν =−1 2λTµν (4.9) With these equations, we can represent the coupled gravitational and electromagnetic field equations. 4.1 Calculation of the trace for the coupling between the gravitational and electromagnetic field. To calculate the trace of the equation [4.9], we do the following: 9 with an uncertainty of 2.4%. The WFC3 (Wide Field Camera 3) camera of NASA’s Hubble telescope was used for the measurement. The importance of this value differs by 3 sigmas from that obtained thanks to the microwave background:[12]. H(0) = 67,6±0,6Km/s/Mpc (PlanckΠ + LowP +BA0) Results obtained in 2015, but subsequent estimates obtained by SPT-3G, allow us to estimate the Hubble constant at: H(0) = 67,24 ±0,54 Km/s/Mpc Now, considering the accelerated expansion of the universe inversely proportional to gravity, and taking into account said expansion in all spatial directions, then we write the result as: Gµν −Fµν = 8πϵ0cΛ3 +=2c ke Λ3 +=2c(100)3 1/4πϵ0≃66,71281903495602 Km/s/Mpc (7.1) To give consistency to these results and taking into account that there is a close inversely proportional relationship between Hubble’s constant and Newton’s universal constant, we can equate the term 1 16πϵ0cwith the magnetic constant of the Biot-Savart law in this way: −1 16πϵ0c=−µ0 4π=⇒Gµν −Fµν = 2µ0ϵ0cΛ−=2 (0.01) c≃GNewton (7.2) 16 8 Conclusions The gravitational field equations, which are described by General Relativity, are presented as highly complex nonlinear differential equations both in the mathematical and physical fields. This historically unprecedented complexity poses a fundamental challenge: the unification of gravity with quantum mechanics. The exploration that we have presented in this article opens doors of immense dimensions to shed light on the most transcendental questions of modern physics. Our approach opens the possibility of interpreting dark energy as a manifestation of the universe in a multidirectional free fall in space-time. In addition, we propose that dark matter, although difficult to develop a theory consistent with observations, could be explained by the constant compression of space-time from the center to the outside of galaxies. This space-time compression, a phenomenon described by General Relativity, could be driving the movement of matter in galaxies, eliminating the need to resort to inert and invisible dark matter. The presence of black holes in the nuclei of most galaxies becomes relevant in this context. The rotation of these black holes may be compressing space-time in a way that simulates the existence of dark matter, thus offering an alternative explanation for the constant rotation curve observed in galaxies. This interpretation stands in stark contrast to the fruitless search for elusive dark matter. It is crucial to recognize that the rotations of planetary systems, such as our own, are not adequate analogues to justify the existence of dark matter. Rotation patterns in planetary systems differ significantly from galaxies, where space-time compression is a dominant factor. We encourage the scientific community to explore and develop equations that support this innovative perspective. Our unwavering commitment to this new research direction drives us to seek a deeper understanding of physics, including the puzzle of quantum entanglement. We encourage collaboration and discussion around these ideas, in the hope that our work will inspire significant advances in our understanding of the fundamental nature of the universe. In short, our research presents a provocative approach that invites a reconsideration of the nature of energy and dark matter, challenging conventional assumptions and opening new avenues for scientific exploration at the intersection of gravity, quantum mechanics, and General Relativity. 17 References [1] Albert Einstein. On the electrodynamics of moving bodies. Annalen der physik, 17(10) 1905. [2] Einstein, Albert (1905a), ≪Zur Elektrodynamik bewegter K¨orper≫, Annalen der Physik 322 (10): 891-921 [3] The Higgs boson, The Brout-Englert-Higgs mechanism, CERN [4] Michelson, Albert A.; Morley, Edward W. (1887). ≪On the Relative Motion of the Earth and the Luminiferous Ether≫ [5] Rindler, Wolfgang (2001). Relativity: Special, General and Cosmological. [6] Javier Garc´ıa “Curso de Relatividad General”.https://youtu.be/HI3m80zLo24 [7] Wikipedia Chiral Spontaneous Symmetry Breaking.https://en.wikipedia.org/wiki/Chiral_symmetry_breaking [8] Wikipedia Chirality of baryonic matter https://en.wikipedia.org/wiki/Chirality_(physics) [9] CMS Experiment, CERN, July 4, 2012 “Observation of a new particle with a mass of 125 GeV”. [10] A. A. Michelson and E.W. Morley, Philos. Mag (1887)http://www.aip.org/history/gap/PDF/michelson.pdf. [11] Paul Davies The Accidental Universe (1984) p. 107. [12] June 2016. Adam G. Riess, Lucas M. Macri, Samantha L. Hoffmann, Dan Scolnic, Stefano Casertano, Alexei V. Filippenko, Brad E. Tucker, Mark J. Reid, David O. Jones, Jeffrey M. Silverman, Ryan Chornock, Peter Challis, Wenlong Yuan, Peter J. Brown, and Ryan J. Foley A 2.4%Determination of the Local Value of the Hubble Constant https://arxiv.org/pdf/1604.01424.pdf Fundamental Constants: Uniting Gravity and the Accelerated Expansion of the Universe through the Higgs Boson © 2023 by MOHAMED MAKRAINI HAMED MUSTAFA is licensed under CC BY-ND 4.0. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/4.0/ 18