arXiv:1606.07210v4 [gr-qc] 23 May 2019 eiso h nih htEnti’ edeutosdrcl ml that imply directly equations quantu field understanding Einstein’s to that insight approach the geometrical on the relies of heart The Introduction 1 atce ssprsiltoso pctm with spacetime of superoscillations as Particles h rtcino pctm tefadisgoercstruct geometric destr its will and which itself remain hole spacetime it of black Therefore, protection a area. the create particular that will at v it mass-density small that a i in spacetime full princip energy/momentum getting in concentrate uncertainty for should the that one sense to it the about Due in hidden is metric. variable i global hidden variable determinis the hidden is This of model variable. density The hidden precondit nonlocal a model. a give contains ontic we spacetime, this in under point nonlocality every on a impact introducing an By has spacetime. ar of particles functions no that possibility of superoscillatory the assumption examine the we under particular, In spacetime g of curvatures have dyn as thi we relativistic ticles In non-general can derive universe? to so, the approach an if in suggest matter And every to matter? foundations metrical creates spacetime curved does isenfil qain hwhwmte uv pctm,bu spacetime, curve matter how show equations field Einstein e-uinUiest fteNegev the of University Ben-Gurion { uvtr fspacetime of Curvature olclmetric? nonlocal a oe Shushi Tomer a 4 2019 24, May Abstract 1 ⇐⇒ Matter } bandby obtained e mc fpar- of amics ure. idnby hidden s ercthat metric nformation h mass the s oe we note, s nlocality. i and tic e this le, o for ion ythe oy olume physics m eo- t, (1) which means that as matter forms curvatures of spacetime, curved space- time also forms matter. This profound insight implies that matter can be originated from the geometry of spacetime. The problem of such equivalence (1) comes when we consider particles. In that case, although it is straightforward that matter curves spacetime, it is not trivial that curva- ture of spacetime creates matter. Therefore, many attempts were made to reconcile curvatures of spacetime with particle physics [1-11]. The spacetime metric in general relativity is defined by a symmetric met- ric gµν that describes the spacetime manifold. Many nonlocal hidden vari- ables theories attempt to describe the quantum world with an ontological interpretation of the wavefunction [13-17]. For example, in [17], a nonlocal global random variable is defined in order to derive the non-relativistic -0 Schr¨odinger equation. Let Ωµν be a spacetime metric, called the global metric that is nonlo- cal with respect to t = x0 and (x1, x2, x3) , the time and the three spatial dimensions, respectively. It represents a small perturbation of spacetime (in Planck scale). We assume that the global metric satisfies the following properties: (P1) Global-nonlocality. We define the concept of nonlocality in the following way: Changing the global metric in one region in spacetime would simultaneously change other separate regions. (P2) The existence of a global mass density. We assume the existence of a global mass density, ρΩ (t, x), associated with the spacetime curvature of Ωµν , that is conserved in the sense that ρΩ (t, x) dx = λ is a constant for each t. (P3) The scalar Ω curvature (Ricci scalar),R (ρΩ), of the global metric satisfies the homo- Ω R Ω geneity property a (a ρΩ) = (ρΩ) for any constant a R. Now, let g where is· R the set· of spacetimeR pseudo-Riemannian∈ metrics that µν ∈ G G creates very small perturbations of spacetime, i.e., gµν = ηµν + ǫµν where ηµν is the Minkowski metric with signature ( 1, 1, 1, 1) and ǫ contains ǫ small − µν − (almost vanished) components. Following our assumptions about the exis- tence of the global metric Ωµν , in order to define an object with gµν we will add it the global metric Ω , which will be denoted by , µν Zµν = g + Ω . (2) Zµν µν µν Taking the summation of two metrics is the standard way to describe the influence of two metrics in spacetime. For example, it is common to define a metric that has a small spacetime perturbation by ηµν + hµν where ηµν is the flat metric (Minkowski metric), and hµν is some small deviation from ηµν .

2 The above metric will play an essential role in the sequel, and as so, we are interested in its fundamental properties. The metric of (2) is given by

2 ds = µν dxµdxν Z 2 2 = dsgµν + dsΩµν .

2 2 where dsgµν = gµνdxµdxν and dsΩµν = Ωµν dxµdxν describe the spacetime curvature with respect to gµν and Ωµν , respectively. The Christoffel symbols are then 1 ∂ ∂ ∂ Γρ = ρα Zαµ + Zαν + Zµν , (3) µν 2Z  ∂xν ∂xµ ∂xα  with the standard notions of the Ricci tensor and scalar curvature (Ricci scalar) that take the following forms

= ∂ Γα ∂ Γα +Γα Γρ Γα Γρ , Rµν α νµ − ν αµ αρ νµ − νρ αµ µν and = µν , respectively, and the Einstein-Hilbert (EH) action with respectR to Z Ris given by S [ ]= 1 det ( )dtd3x. Zµν Zµν 16π R − Zµν For the sequel, we define two types ofR thep classical limit. We denote (rel- ativistic) classical limit as the limit from a relativistic into a non-relativistic theory, and (quantum) classical limit as the limit from to .

2 The Main Idea

In Fourier analysis, it was believed that band-limited signals in space and time could not oscillate locally arbitrarily faster than the highest component in their spectrum. However, it was discovered that there is a unique type of signals that violate such a feature; these are the superoscillatory functions [18-30]. Superoscillations show that a superposition of small Fourier compo- nents that have bounded Fourier spectrum has the possibility to have a shift that is well outside the spectrum. These functions appear in both classical and quantum systems. For example, such functions have been observed and investigated in classical optics, image processing, matter waves, electric and magnetic fields, and even in engineering [19,20,24,27-29]. Superoscillations are defined in the following way [20]:

3 Definition 1 A superposition function n C (n, q) eiSn(j)x,q,α R , x R, j ∈ + ∈ Xj=0 is called a superoscillatory function if it satisfies the following requirements: (S1) S (k) <α for all n and k N 0 . | n | ∈ ∪{ } (S2) There exist a compact subset C of R, on which ϕn converges uni- formly to eiS(q)x where S is a continuous real-valued function such that S(q) >α. | | In their seminal paper, Aharonov et al. (1990) [31] have shown that suitable superposition of time evolutions due to the action of weak forces is equivalent to a time evolution resulting from the action of a strong force, with a new action functional. This startling result is the heart of our work. In another work, Aharonov et al. (2018) [32] present a novel renormalization technique based on a superposition of Lagrangians, which leads to a new form of Lagrangian that is renormalized. As defined earlier, the surprising nature of superoscillations shows that band-limited functions can have oscillations arbitrarily faster than the fastest Fourier component of the function. However, the inverse is also true, and provide a startling fact: A wave with arbitrary large oscillation can be de- scribed by a superposition of waves with much lower oscillations that are independent on both the phase and the amplitude of the original wave. Our goal is to model non-GR particles, i.e., particles in the microscopic scale that their dynamics do not follow the GR . We suggest using super- oscillations to obtain it. Let us consider a with some dynamics, given by, ΨZ = S A(x )ei [xµ]/h, where h R is some real-valued variable that, in the sequel, µ ∈ will be fixed to a constant. This wave function is not restricted to any specific dynamics. However, we define it in accordance with GR. (a) We assume that it represents the dynamics of the system with symmetric time. (b) ΨZ is a continuous function. We also consider [xµ] > α. Then, without loss of generality, using superoscillations, for a|S constant| 1/ h < m (for suitable | | m> 0) one can convert ΨZ into a superposition of gravitational waves with bounded wavenumbers and specific amplitudes that change in xµ n n µ iS[x ]/h ρΩ ik x /h ΨZ = A (x ) e µ (x ; n) e j µ = ϕZ (n) (4) µ ≈ Aj µ j Xj=0 Xj=0

4 ρΩ where each ϕZ represents a gravitational wave with amplitude (x ; n). Aj µ Since the phase of ϕZj (n) does not depend on A nor , it is clear that the ρΩ S µ phase and amplitude of ΨZ are encoded in Aj (xµ; n). kj is a suitable wave µ vector of the j-th gravitational wave, for xµ such that kj xµ <α . When taking the limit n , we have that | | →∞ n ΨZ ϕZ (n) 0 as n . (5) | − j |→ →∞ Xj=0 We conclude out-of-the-spectra dynamical systems where the evolution is outside the scope of the GR evolutions, seemingly showing a new type of dynamics that evolve from the GR one. At first glance, superoscillations are seemed to be a very unique form of waves, and thus, one may assume that they should be very rare in nature. However, it turns out that even for generic random fields (e.g., Gaussian-like random fields) superoscillations [38] may exist in about 1/3 fraction area of the random field. Thus, it is possible that superoscillations of spacetime can be generated frequently in nature, and do not need special environment in order to evolve. The Madelung equations provide a hydrodynamic interpretation of quan- tum mechanics [33-37]. These coupled differential equations are derived from the Schr¨odinger equation, where the first one describes a classical continu- ity equation, while the latter equation describes the classical Euler equation with the addition of a nonlocal potential, called the [35]. In the Madelung equations, the quantum potential is given by 1 2√ρ Q = ∇ , (6) −2m √ρ

S where ρ is the mass density of the quantum wavefunction ψ = √ρei /h in the polar representation, with the units of Planck constant h = ~ = 1, and m is the mass of the particle. The quantum potential (6) depends on the curvature of the density of the quantum particle, and it gives a precondition for the existence of nonlocality, since multiplying the wave function by some constant (which changes its magnitude) will not change the quantum metric (see, for instance, [35]). We now show how the global metric provides a precondition for nonlo- cality. This will be represented by the Ricci scalar for each j-th metric µν (j). One can observed that Z (j; ρ ) can be decomposed to the sum ofZ Ricci R Ω 5 scalar of Ωµν and other scalar curvature that depends on both Ωµν and gµν(j), j =1, 2,...,n,

Z Ω (j; ρ )= (ρ )+ Z (j; ρ ), (7) R Ω R Ω U Ω Ω where (ρΩ) is the Ricci scalar of the global metric, and Z (j; ρΩ). UsingR (7) and from the property (P3) of the global metric,U we clearly see that for any real constant a,

Z Ω a ρ (j; a ρ )= ρ (ρ )+ a ρ Z (j; a ρ ). (8) · ΩR · Ω ΩR Ω · ΩU · Ω Ω The precondition for nonlocality of (ρΩ) comes from the fact that chang- ing the magnitude of the mass densityR ρ by some constant a R will not Ω ∈ change the scalar curvature of Ω (ρ ) . Therefore, Ω (ρ ) is independent R Ω R Ω on the field intensity of Ωµν . Thus, the Ricci scalar of the j-th state is the Ω sum of local scalar Z (j; ρ ) and a nonlocal scalar curvature (ρ ). U Ω R Ω The theory contains a hidden variable, which is the mass density of the global metric. Due to the , the mass density of the metric in a Planck volume is hidden in the sense that for getting full information about it one should concentrate energy/momentum in such small volume in spacetime, which will create a black hole that will destroy such a metric at that particular area. Thus, this hidden variable remains hidden by the protection of spacetime itself and its geometrical structure. An implication of nonlocality in a relativistic causal model is that it evolves the uncertainty principle. This comes from the fact that nonlocality and relativistic causality give the uncertainty principle, and thus, every relativistic nonlocal model contains the uncertainty principle [39]. The classical limit is obtained due to the mass-density of an object that creates its own curvature, that is much larger than the one created by Ωµν which becomes negligible including its nonlocality, and the uncertainty principle disappears. In string theory, the spin of the particles comes from the rotation of the string that represents the particle. In our model, the particle is an oscillation of spacetime that is built from spacetime metrics, which allows, conceptually, to understand the spin of the particles by inner rotations of these sub-metrics µν (j). The Global metric as a conformal transformationZ of flat space- time. From properties (P1)-(P3) we observe that Ωµν depends on the global mass density, and it satisfies the homogeneity property shown in (P3). In the paper of Delphenich [34], it was proved that by taking a conformal trans- formation of the Minkowski metric ηµν we can build a nonlocal quantum

6 potential. We now use this example in favor of our model. Suppose that the global metric is given by a conformal transformation of the flat spacetime 2σ 2σ Ωµν = ηµν e , where e =: ρΩ is the global mass density. Moreover, assume (j) that each amplitude of the j th state ϕZ is given by j (t, x)= √ρΩ aj (t, x) − A · where aj (t, x) is some positive and smooth function. Then, a straightforward calculation of the scalar curvature yields to

Z √ρΩ ρΩ (j; ρΩ)= + ρΩ Z (j; ρΩ), (9) R − √ρΩ U

 2 2 2 2 where = ∂t + ∂x1 + ∂x2 + ∂x3 is the d’alembert . In fact, eq. (8) shows that− the Ricci scalar, Z , coupled to the mass density, ρ , have a R Ω contribution that does represent a ”quantum correction” to the model (see, again, [34]). The nature of the global metric hidden variable. Since the cur- vature of the global metric is very small, in particular, it is a Planck length perturbation of spacetime. Due to the uncertainty principle, when we mea- sure such a metric we must concentrate high energy/momentum in such a small volume in order to get information about the metric, a black hole im- mediately emerges and destroy this particular area. Therefore, one physically cannot know the exact behavior of the particle, which is defined explicitly by ϕZj .

3 Discussion

The quest for geometrical foundations to non-GR dynamics has been exten- sively studied since Einstein’s attempt to reconcile Maxwell’s equations and the GR field equations. We have examined the possibility that non-GR dy- namics emerge from superoscillations of spacetime. This leads to the main features of the theory: (a) Superoscillation of weak gravitational forces is equivalent to a non-GR force, with a new action different from the EH ac- tion. (b) The theory contains a nonlocal hidden variable that is represented by a density function of the global metric.

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