Cosmological redshift as a manifestation of nonlinear critical relativistic quantum wave fields

We explore the hypothesis that both Newtonian and relativistic gravitational phenomena may emerge from critical dynamics in nonlinear relativistic wave fields, in analogy with wave–particle duality in quantum systems. Using a general nonlinear self-adjoint wave Hamiltonian, we construct a set of illustrative toy models in which increasing interaction strength drives a transition from extended, oscillatory waves to localized particle-like excitations. In this framework, gravitational and cosmological effects—including redshift—arise as collective, near-critical wave phenomena rather than as properties of a predefined spacetime geometry. The constancy of the speed of light emerges from nonlinear terms in the dispersion relation, and wave localization provides a common origin for inertial and gravitational mass without requiring a separate equivalence principle. These models also exhibit the absence of singularities (e.g., at the Schwarzschild radius or at $r=0$) and suggest potential connections between local gravitational behavior and large-scale structure. As a proof of concept, we show how critical-wave effects may alleviate the observed tension between Planck CMB inferences and supernova/Cepheid distance measurements without assuming FLRW metrics or $\mathrm{\Lambda }$CDM dynamics; the same mechanism provides an alternative simple explanation for the BAO peak in the galaxy distribution function. We emphasize that these results are not proposed as a replacement for general relativity or standard cosmology, but as a conceptual demonstration that nonlinear wave dynamics can reproduce qualitative gravitational and cosmological features, motivating further investigation into emergent-gravity scenarios.

1 Introduction

Wave-particle duality is a foundational concept in quantum mechanics, demonstrating how quantum entities such as photons and electrons can exhibit both wave-like and particle-like characteristics, depending on the experimental context. This duality challenges the traditional separation of waves and particles in classical physics and has led to profound shifts in our understanding of the quantum world.

The evolution of quantum mechanics is closely linked to the development of wave-particle duality. Initially, light was considered a wave, as demonstrated by interference and diffraction patterns observed in the late 19th and early 20th centuries (Taylor, 1909). However, Albert Einstein’s explanation of the photoelectric effect in 1905 (Einstein, 1905a) revealed that light also behaves as discrete energy packets, or photons, indicating its particle-like nature. Louis de Broglie extended this duality to matter in 1924 (de Broglie, 1923; de Broglie, 1925), proposing that particles such as electrons also exhibit wave-like properties, a hypothesis later confirmed by experiments.

The widely known experiment to observe the wave-particle duality is the double-slit experiment, where particles like electrons create an interference pattern indicative of wave behavior when not observed, but behave like particles when measured. Understanding wave-particle duality is essential for grasping the behavior of quantum systems and has significant implications for quantum mechanics. It led to the creation of the Schrödinger equation (Schrödinger, 1926), which describes the wave function of quantum systems and predicts the probability distribution of particles. The duality also supports the principle of complementarity, which suggests that wave and particle descriptions are both necessary for a complete understanding of quantum phenomena (Bohr, 1928).

Quantum field theory (QFT) offers a framework that explains wave-particle duality by describing particles as excitations of underlying fields (Born et al., 1926; Dirac, 1927; Weinberg, 1995). In this perspective, both particles and forces are manifestations of fields, with properties such as charge and mass arising from interactions between these fields. For instance, photons are excitations of the electromagnetic field, while electrons are excitations of the electron field. The wave-like behavior corresponds to the field’s propagation, whereas particle-like behavior emerges during interactions or measurements. This view shifts the focus from particles as fundamental entities to fields as the primary constituents of reality, with particles being secondary, emergent phenomena. Thus, QFT provides a more unified and comprehensive explanation of wave-particle duality by framing it in terms of field interactions and quantization (Ryder, 1996; Cini, 2003).

In this work, we propose a hypothesis that extends the explanatory power of wave-particle duality (Galinsky and Frank, 2025) to gravity itself. Specifically, we suggest that gravity—both in its classical Newtonian form and as described by general relativity—emerges from the dynamics of nonlinear relativistic waves undergoing critical transitions. These nonlinear waves can form localized, stable structures—akin to wave packets—through a process similar to that which underlies quantum wave-particle duality.

Just as wave-particle duality demonstrates that quantum phenomena arise from the interplay between wave-like propagation and particle-like localization, we hypothesize that gravitational phenomena emerge from the self-organization of nonlinear waves at criticality. In this framework, what we perceive as the gravitational field is not a fundamental entity, but rather a manifestation of collective, emergent behavior arising from the underlying wave dynamics. General relativity, with its description of spacetime curvature, can thus be seen as a crude, but relatively effective, macroscopic theory that captures the large-scale consequences of these microscopic nonlinear processes.

This perspective offers a unified mechanism by which both quantum features and gravity can be understood as emergent from critical transitions in nonlinear wave systems, potentially bridging the conceptual gap between quantum mechanics and general relativity. By reframing gravity as an emergent phenomenon rooted in the same nonlinear wave dynamics that give rise to quantum duality, this hypothesis aims to provide new insights into the origin and nature of gravity, and the emergence of the quantum world itself.

2 Theory

To illustrate this hypothesis we will first consider the simplest case of three dimensional $(\mathit{x}=\{x_i,i=\mathrm{1,2,3}\})$ relativistic wave, i.e., wave propagating with the speed of light c. For clarity we will use the natural units, i.e., set both the reduced Planck constant $\hslash$ and the speed of light $c$ to unity. For scalar function $\mathrm{\Phi }\equiv \mathrm{\Phi }(\mathit{x},t)$ wave propagation is then described by the standard multidimensional wave equation

$\frac{{\partial }^2\mathrm{\Phi }}{\partial t^2}-{\mathbf{\nabla }}^2\mathrm{\Phi }=0.(1)$

If we assume stationarity of sources/sinks both inside the domain and at the boundaries then we can seek the solution in the form $\mathrm{\Phi }(\mathit{x},t)={\mathrm{\Phi }}^{+}(\mathit{x},t)+{\mathrm{\Phi }}^{-}(\mathit{x},t)=\mathcal{M}(\mathit{x}){\mathrm{\Psi }}^{+}(\mathit{x},t)+\mathcal{M}(\mathit{x}){\mathrm{\Psi }}^{-}(\mathit{x},t)$, where $\mathcal{M}(\mathit{x})$ is a solution of Laplace equation with stationary boundary conditions

${\mathbf{\nabla }}^2\mathcal{M}=0,(2)$

and ${\mathrm{\Psi }}^{\pm }(\mathit{x},t)$ are the original and adjoint (its Hermitian conjugate) wave solutions of the remaining time dependent part that can be written in quantum-mechanical bra-ket notation (${\mathrm{\Psi }}^{-}\equiv \langle \mathrm{\Psi }|$, ${\mathrm{\Psi }}^{+}\equiv |\mathrm{\Psi }\rangle$) as

$\langle \mathrm{\Psi }|\square \mathcal{M}=0,(\mathrm{3a})$

${\mathcal{M}}^{†}\square |\mathrm{\Psi }\rangle =0.(\mathrm{3b})$

Here $\square$ denotes the d’Alembertian operator

$\square =-{\widehat{p}}^{\mu }{\widehat{p}}_{\mu }={\partial }^{\mu }{\partial }_{\mu }={\partial }_t^2-{\nabla }^2,(4)$

where in Equation 4

${\widehat{p}}_{\mu }=i{\partial }_{\mu }=\left(i{\partial }_t,-i\nabla \right)(\mathrm{5a})$

${\widehat{p}}^{\mu }=i{\partial }^{\mu }=\left(i{\partial }_t,i\nabla \right)(\mathrm{5b})$

is the 4-momentum operator (Equation 5a) and its adjoint (Equation 5b).

Those equations may be satisfied for any arbitrary unit vector $\widehat{\mathit{a}}$,

$a^{\mu }=\left(1,\widehat{\mathit{a}}\right),\text{with\hspace{0.17em}}\Vert \widehat{\mathit{a}}\Vert =1(6)$

not necessarily along the wave propagation direction, if

$a^{\mu }{\widehat{p}}_{\mu }|\mathrm{\Psi }\rangle =0\Rightarrow \left(-i{\partial }_t+i\widehat{\mathit{a}}\cdot \nabla \right){\mathrm{\Psi }}^{+}=0(\mathrm{7a})$

$\langle \mathrm{\Psi }|{\widehat{p}}^{\mu }{a}_{\mu }=0\Rightarrow \left(i{\partial }_t+i\widehat{\mathit{a}}\cdot \nabla \right){\mathrm{\Psi }}^{-}=0(\mathrm{7b})$

Therefore, ${\mathrm{\Psi }}^{-}(\mathit{x},t)=\mathrm{\Psi }({\mathit{\xi }}^{-})$ and ${\mathrm{\Psi }}^{+}(\mathit{x},t)=\mathrm{\Psi }({\mathit{\xi }}^{+})$, where ${\mathit{\xi }}^{\pm }=\widehat{\mathit{a}}t\pm \mathit{x}$. The total solution $\mathrm{\Phi }(\mathit{x},t)=\mathcal{M}(\mathit{x})\mathrm{\Psi }({\xi }^{+})+\mathcal{M}(\mathit{x})\mathrm{\Psi }({\xi }^{-})$ satisfies the remaining term $2\nabla \mathcal{M}(\mathit{x})\cdot \nabla (\mathrm{\Psi }({\xi }^{+})+\mathrm{\Psi }({\xi }^{-}))$ in the wave Equation 1 as well.

The normalization factor $\mathcal{M}(\mathit{x})$ reduces to a constant in the case of homogeneous boundary conditions with no sources present within the domain. For a single point source located at the origin, $\mathit{x}=0$, and under the assumption of spherically symmetric wave propagation, $\mathcal{M}(\mathit{x})$ can be represented as the harmonic function $1/r\equiv 1/|\mathit{x}|$, which is the solution to the spherically symmetric Laplace equation. Another possible solution of the Laplace equation involving an angular dependence with arbitrary direction $\widehat{\mathit{a}}$ (Equation 6) is $(\widehat{\mathit{a}}\cdot \mathit{x})/(\mathit{x}\cdot \mathit{x})$. More complex boundary conditions or the presence of multiple sources and sinks will generally lead to a more intricate dependence of $\mathcal{M}(\mathit{x})$.

The two factors (Equation 7) in the scalar wave Equation 3 can be viewed as a form of dual (original and adjoint) Schrödinger equations

$i\frac{\partial \mathrm{\Psi }}{\partial t}=\widehat{H}\mathrm{\Psi },-i\frac{\partial {\mathrm{\Psi }}^{*}}{\partial t}={\widehat{H}}^{†}{\mathrm{\Psi }}^{*},(8)$

where the Hamiltonian $\widehat{H}$ is a self-adjoint placeholder operator that is substituted by various expressions for obtaining almost everything that is currently known about quantum world.

Although (Equation 7) describes linear waves using a linear self-adjoint Hamiltonian, $\widehat{H}={\widehat{H}}_0=i\widehat{\mathit{a}}\cdot \nabla$, we propose that restricting the Hamiltonian to be strictly linear is an oversimplification, introduced primarily for mathematical convenience. Instead, we allow the Hamiltonian $\widehat{H}$ to include both linear and nonlinear components, i.e., $\widehat{H}={\widehat{H}}_0+{\widehat{H}}_1$. Accordingly, we assume that the relativistic wave equation can, in general, be nonlinear

$i\frac{\partial \mathrm{\Psi }}{\partial t}={\widehat{H}}_0\mathrm{\Psi }+{\widehat{H}}_1\mathrm{\Psi }(\mathrm{9a})$

$-i\frac{\partial {\mathrm{\Psi }}^{*}}{\partial t}={\widehat{H}}_0^{†}{\mathrm{\Psi }}^{*}+{\widehat{H}}_1^{†}{\mathrm{\Psi }}^{*}(\mathrm{9b})$

where $†$ denotes adjoint.

As the Hamiltonian operator is a Hermitian and self-adjoint operator $({\widehat{H}}_1\equiv {\widehat{H}}_1^{†})$, we can express the self-adjoint form of nonlinear operator ${\widehat{H}}_1$ in a general form as

${\widehat{H}}_1={\displaystyle \sum _n}{\widehat{\alpha }}_n\left[|\mathrm{\Phi }{|}^n\right]+\frac{1}{2}{\displaystyle \sum _{n,m}}{\widehat{\beta }}_{nm}\left[{{\mathrm{\Phi }}^{*}}^n{\mathrm{\Phi }}^m\right],(10)$

where we used the full wave function $\mathrm{\Phi }=\mathcal{M}\mathrm{\Psi }$ and assume that ${\widehat{\alpha }}_n$ and ${\widehat{\beta }}_{nm}$ are operators acting on their nonlinear arguments $|\mathrm{\Phi }{|}^n$ and ${{\mathrm{\Phi }}^{*}}^n{\mathrm{\Phi }}^m$. Hence for ${\widehat{H}}_1$ to be self-adjoint it would be sufficient to guarantee that ${\widehat{\alpha }}_n^{†}={\widehat{\alpha }}_n$ (or ${({\widehat{\alpha }}_n\left[|\mathrm{\Phi }{|}^n\right])}^{*}={\widehat{\alpha }}_n\left[|\mathrm{\Phi }{|}^n\right]$) and ${\widehat{\beta }}_{nm}^{†}={\widehat{\beta }}_{mn}$ (or ${({\widehat{\beta }}_{nm}\left[{{\mathrm{\Phi }}^{*}}^n{\mathrm{\Phi }}^m\right])}^{*}={\widehat{\beta }}_{mn}\left[{{\mathrm{\Phi }}^{*}}^m{\mathrm{\Phi }}^n\right]$).

Strictly speaking, the notion of a self-adjoint operator is defined only within the framework of linear operators acting on Hilbert spaces. When one departs from linearity, the standard spectral and adjoint machinery no longer applies in a straightforward way. In the nonlinear regime, an analogous structure must instead be formulated using tools from nonlinear functional analysis—typically through real-valued generating functionals, effective Hamiltonians, or expectation-valued evolution equations.

In this sense, when we refer to a “self-adjoint nonlinear Hamiltonian operator,” we are speaking informally. What we actually mean is a real-valued energy functional

$H\left[\mathrm{\Phi }\right],$

defined on the space of field configurations $\mathrm{\Phi }$, whose functional derivative

$\frac{\delta H}{\delta \mathrm{\Phi }}$

governs the dynamical evolution of the field. The resulting flow preserves the appropriate norm or probability density, and thus plays the role normally occupied by a Hermitian Hamiltonian in linear quantum theory. This framework should therefore be understood as a phenomenological nonlinear wave model, not as a rigorously constructed quantum field theory in the axiomatic or constructive sense.

In Supplementary Appendix A, we illustrate how an effective “self-adjoint” or Hermitian character of the evolution emerges naturally from the Euler–Lagrange formalism when starting from a nonlinear action. Specifically, we analyze a Lagrangian density of the schematic form

$F_a^b{S}_b^c{F}_c^a,$

and show that its associated variational equations yield a norm-preserving flow consistent with the interpretation above.

To highlight the robustness of this mechanism, Supplementary Appendix A also examines a distinct nonlinear Lagrangian based on a higher-order invariant,

${\left(F_{ab}{F}^{ab}\right)}^2.$

Despite their very different algebraic structures, both Lagrangians generate the same critical dynamics—that is, the same dominant nonlinear evolution at leading order in the regime of interest. The emergence of identical critical behavior from two structurally independent Lagrangians underscores that the resulting dynamics is not an artifact of a particular choice of action.

Moreover, the fact that two inequivalent nonlinear actions yield the same universal dynamics provides a bridge toward a more rigorous interpretation: it suggests that the critical behavior depends only on a small set of structural features (such as symmetries, conserved quantities, and scaling properties), and thus can be characterized in a manner reminiscent of a rigorously defined quantum field theory, where universality classes and renormalization-group structure underlie independence from microscopic details. Although our construction does not capture the full mathematical apparatus of formal QFT, the convergence of critical behavior from multiple nonlinear actions indicates that the underlying dynamical structure admits a mathematically well-defined characterization at an effective-field-theory level.

For the purpose of illustrating of our hypothesis we will view all operators ${\widehat{\alpha }}_n$ and ${\widehat{\beta }}_{nm}$ as just simple functions omitting $\widehat{}$ symbols and we will only keep terms with ${\alpha }_1$, ${\beta }_{01}$ and ${\beta }_{10}={\beta }_{01}^{*}$. Hence

$\begin{array}{ll}\hfill {\widehat{H}}_1& ={\alpha }_1|\mathrm{\Phi }|+\frac{1}{2}\left({\beta }_{01}^{*}{\mathrm{\Phi }}^{*}+{\beta }_{01}\mathrm{\Phi }\right)\hfill \\ \hfill & ={\alpha }_1|\mathcal{M}\Vert \mathrm{\Psi }|+\frac{|{\beta }_{01}\mathcal{M}|}{2}\left(e^{i\delta }{\mathrm{\Psi }}^{*}+e^{-i\delta }\mathrm{\Psi }\right)\hfill \\ \hfill & =\alpha |\mathrm{\Psi }|+\frac{\beta }{2}\left(e^{i\delta }{\mathrm{\Psi }}^{*}+e^{-i\delta }\mathrm{\Psi }\right),\hfill \end{array}(11)$

where we set $\alpha ={\alpha }_1|\mathcal{M}|$, $\beta =|{\beta }_{01}\mathcal{M}|$, ${\beta }_{01}=|{\beta }_{01}|\exp (-i\delta )$, and ${\beta }_{10}=|{\beta }_{01}|\exp (i\delta )$.

The formal solution of (Equation 8) is $\mathrm{\Psi }=\exp (-i\widehat{H}t){\mathrm{\Psi }}_0$, where without the loss of generality of illustration of our hypothesis we can express $\mathrm{\Psi }$ as a relativistic plane wave with wave number $\mathit{k}$ and frequency ${\omega }_0$ (${\omega }_0=k\equiv |\mathit{k}|$ in normal units) that along any arbitrary direction $\pm \widehat{\mathit{a}}$ can be written as a function of ${\mathit{\xi }}^{\pm }$ as

$\mathrm{\Psi }=\exp \left(-i\left(\psi +\mathit{k}\cdot {\mathit{\xi }}^{\pm }-\delta \right)\right),(12)$

where $\psi$ denotes a possible deviation of the wave phase due to the nonlinear part of the Hamiltonian $H_1$. The term $\mathit{k}\cdot {\mathit{\xi }}^{\pm }\equiv \omega t\pm \mathit{k}\cdot \mathit{x}$, with $\omega =\mathit{k}\cdot \widehat{\mathit{a}}={\omega }_0\cos \theta$ describes plane wave spacetime variations evaluated along the arbitrary $\widehat{\mathit{a}}$ direction with angle $\theta =\angle (\mathit{k},\widehat{\mathit{a}})$. The nonlinear part of the Hamiltonian $H_1$ then can be expressed as

${\widehat{H}}_1=\alpha +\beta \cos \left(\psi +\mathit{k}\cdot {\mathit{\xi }}^{\pm }\right).(13)$

Assuming temporal dependence of the deviation of the phase $\psi$, we can obtain an equation for $\psi$ from (Equations 9a, 12, 13) due to the nonlinear part of the Hamiltonian $H_1$ as

$\frac{d\psi }{dt}=\alpha +\beta \cos \left(\psi +\mathit{k}\cdot {\mathit{\xi }}^{\pm }\right),(14)$

that can be simplified using substitution $\varphi \equiv \psi +\mathit{k}\cdot {\mathit{\xi }}^{\pm }$ as

$\frac{d\varphi }{dt}=\omega +\alpha +\beta \cos \varphi .(15)$

Similar equation can be recovered if we assume the spatial rather than temporal dependence of $\psi$ and $\varphi$,

$\widehat{\mathit{a}}\cdot \nabla \varphi =\omega \pm \alpha +\beta \cos \varphi ,(16)$

where $\widehat{\mathit{a}}\cdot \nabla \varphi$ is equivalent to a derivative along $\widehat{\mathit{a}}$ direction.

In Equations 15, 16 we assume that both $\omega$ and $\beta$ can always be made positive, $\omega$ can be made positive by an appropriate choice of signs of $\widehat{\mathit{a}}$ and $\varphi$, and for negative $\beta$ we can always add constant phase shift $\pi$ to $\varphi$ that flips the sign of $\beta$, but $\alpha$ may be both positive and negative. In the spatial Equation 16, we retained the $\pm$ sign for $\alpha$ to emphasize that it may contribute to distinct symmetry conditions for ${\mathit{\xi }}^{+}$ and ${\mathit{\xi }}^{-}$.

The solution $\psi$ can be expressed as

$\begin{array}{ll}\hfill \psi =& -\mathit{k}\cdot {\mathit{\xi }}^{\pm }+2\arctan X,\hfill \\ \hfill X=& \sqrt{\frac{\omega +\alpha +\beta }{\omega +\alpha -\beta }}\tan \left(\frac{1}{2}\sqrt{{\left(\omega +\alpha \right)}^2-{\beta }^2}\left(t+\mathcal{C}\left(\mathit{x}\right)\right)\right),\hfill \end{array}(17)$

where $\mathcal{C}(\mathit{x})$ is an arbitrary function of $\mathit{x}$ that we can choose as $\mp \widehat{\mathit{a}}\cdot \mathit{x}$, expressing $\psi$ in Equation 17 not as a function of $t$ but as a function of ${\mathit{\xi }}^{\pm }$ and hence obtain a solution of (Equation 9a) $\mathrm{\Psi }$ as a function of ${\mathit{\xi }}^{\pm }$

$\begin{array}{ll}\hfill X& =\sqrt{\frac{\omega +\alpha +\beta }{\omega +\alpha -\beta }}\tan \left(\frac{1}{2}\sqrt{{\left(\omega +\alpha \right)}^2-{\beta }^2}\left(\widehat{\mathit{a}}\cdot {\mathit{\xi }}^{\pm }+C\right)\right),\hfill \\ \hfill \mathrm{\Psi }& =\exp \left(-i\left(2\arctan X-\delta \right)\right)\hfill \end{array}(18)$

where we also included an arbitrary constant $C$.

The important part here is that the solution shows critical transition from oscillatory to stationary phase solution for $|\beta |<|\omega +\alpha |$ and $|\beta |\ge |\omega +\alpha |$ such that for the simplest example of $\alpha =0$ and $|\beta |=|\omega |$

$\begin{array}{ll}\hfill \mathrm{\Psi }=& \exp \left(-2i\arctan \left(\omega \widehat{\mathit{a}}\cdot {\mathit{\xi }}^{\pm }+C\right)\right)\hfill \\ \hfill =& \exp \left(-2i\arctan \left(\omega t\pm \left(\widehat{\mathit{a}}\cdot \mathit{k}\right)\left(\widehat{\mathit{a}}\cdot \mathit{x}\right)+C\right)\right).\hfill \end{array}(19)$

For $\widehat{\mathit{a}}\Vert \mathit{k}$ (Equation 19) becomes

$\mathrm{\Psi }=\exp \left(-2i\arctan \left(\mathit{k}\cdot {\mathit{\xi }}^{\pm }+C\right)\right).(20)$

The behavior of wave function $\mathrm{\Psi }$ Equation 18 for several parameter values is shown in Figure 1 clearly indicating an increase in localization and collapse of the wave function to a structure of $T\sim 2\pi /\omega$ duration or $L\sim 2\pi /k$ length.

Figure 1

Figure 1. The plot displays the real (red) and imaginary (blue) components of the wave function $\mathrm{\Psi }$ (as defined in Equation 18), plotted as functions of $(\widehat{\mathit{a}}\cdot \mathit{\xi })$. This visualization demonstrates how the localization of the wave function depends on both the frequency $\omega$ and the nonlinearity parameter $\beta$. As $\omega$ and $\beta$ vary, the nonlinear wave can continue to propagate as a wave along the direction parallel to $\mathit{k}$, but becomes increasingly localized—appearing particle–like—at oblique angles (for example, for $\theta >\pi /3$ given the values of ${\omega }_0$ and $\beta$ used in this plot). This reflects the enhanced localization effect due to nonlinearity, as the wave function’s phase change becomes concentrated in specific regions, particularly at larger angles.

3 Criticality

It is well known that in the vicinity of any critical point a small change in initial conditions can produce drastically different results. From (Equation 14) it is easy to find the period $T^{\prime }$ of the nonlinear wave function in the oscillatory parameter range as

$T^{\prime }=\underset{0}{\overset{2\pi }{\int }}\frac{d\psi }{\omega +\alpha +\beta \cos \left(\psi +\mathit{k}\cdot {\mathit{\xi }}^{\pm }\right)}=\frac{2\pi }{\sqrt{{\left(\omega +\alpha \right)}^2-{\beta }^2}}.(21)$

Therefore, a small change in initial linear wave phase value between 0 and 2$\pi$, i.e., a change of position $\xi$ from 0 to $2\pi /\omega \equiv 2\pi /k$, may result in a collapse of the wave function producing an object localized at any time moment between 0 and $T^{\prime }$, where $T^{\prime }\to \infty$ as $\beta \to \omega +\alpha$.

An expression for the wavelength $L^{\prime }=2\pi /k^{\prime }$ of the nonlinear wave function with ${\omega }^{\prime }\equiv \omega \pm \alpha$ in the oscillatory parameter range from (Equation 16) can be shown to have exactly the same form as (Equation 21), or

$L^{\prime }=\frac{2\pi }{\sqrt{{{\omega }^{\prime }}^2-{\beta }^2}}\Rightarrow {{\omega }^{\prime }}^2={k^{\prime }}^2+{\beta }^2,(22)$

that corresponds to a simple dispersion relation which in a standard units taking $\beta =m_0{c}^2/\hslash$ becomes a relativistic dispersion relation of matter waves ${\omega }^2=k^2{c}^2+{(m_0{c}^2/\hslash )}^2$. Thus, the appearance of observable rest mass for localized waves can be assumed to be attributed to the nonlinear components of the Hamiltonian ansätze (Equation 10). This also provides a new derivation of Einstein’s mass-energy equivalence relationship as the consequence of the critical dynamics of the wave field (Einstein, 1905b).

We should emphasize that the constructed nonlinear illustrative solution has constant probability density $|\mathrm{\Psi }{|}^2=1$, nevertheless expectation values for various observable, e.g., momentum or position operators, that involves derivatives of $\mathrm{\Psi }$ will provide appropriately quantized and localized values. For example, calculating expectation values for a position $(i\partial /\partial \mathit{k})$ and a momentum $(-i\partial /\partial \mathit{x}=-i\nabla )$ along any arbitrary direction $\widehat{\mathit{a}}$ using the wave function at the critical point (Equations 19, 20) results in localized and quantized expressions.

$\begin{array}{c}{⟨\widehat{\mathit{a}}\cdot \mathit{k}⟩}_{\mathrm{\Psi }}=-\underset{\mathbf{\Omega }}{\int }{\mathrm{\Psi }}^{*}i\left(\widehat{\mathit{a}}\cdot \nabla \right)\mathrm{\Psi }ds=-\underset{-\infty }{\overset{\infty }{\int }}{\mathrm{\Psi }}^{*}i\left(\widehat{\mathit{a}}\cdot \nabla \right)\mathrm{\Psi }d\tau \hfill \\ =\underset{-\infty }{\overset{\infty }{\int }}\frac{\mp 2\omega d\tau }{{\left(\omega \left(t_0\pm \tau \right)\pm \left(\widehat{\mathit{a}}\cdot \mathit{k}\right)\left(\widehat{\mathit{a}}\cdot {\mathit{x}}_0\right)+C\right)}^2+1}=\mp 2\pi ,\hfill \end{array}(23)$

$\begin{array}{l}\hfill {⟨\widehat{\mathit{a}}\cdot \mathit{x}⟩}_{\mathrm{\Psi }}=\underset{\mathbf{\Omega }}{\int }{\mathrm{\Psi }}^{*}i\left(\widehat{\mathit{a}}\cdot \frac{\partial }{\partial \mathit{k}}\right)\mathrm{\Psi }ds=\underset{-\infty }{\overset{\infty }{\int }}{\mathrm{\Psi }}^{*}i\left(\widehat{\mathit{a}}\cdot \frac{\partial }{\partial \mathit{k}}\right)\mathrm{\Psi }d\omega \\ \hfill =\underset{-\infty }{\overset{\infty }{\int }}\frac{\pm 2td\omega }{{\left(\left({\omega }_0\pm \omega \right)t\pm \left(\widehat{\mathit{a}}\cdot {\mathit{k}}_{\mathit{0}}\right)\left(\widehat{\mathit{a}}\cdot \mathit{x}\right)+C\right)}^2+1}=\pm 2\pi .\end{array}(24)$

where $ds$ is an invariant measure for the integral of the spacetime path $s$ over the spacetime contour $\mathbf{\Omega }$, and the expectation values are evaluated either at an arbitrary spacetime point $(t_0,{\mathit{x}}_0)$ for the momentum operator (Equation 23), or at an arbitrary energy-momentum $({\omega }_0,{\mathit{k}}_0)$ for the position operator (Equation 24).

The correlation functions that arise between wave functions corresponding to spatially or temporally separated critical nonlinear states exhibit a decay as the separation increases

$\langle {\mathrm{\Psi }}^{*}\left(\mathit{x},0\right)\omega \mathrm{\Psi }\left(\mathit{x},t\right)\rangle =\frac{-8\pi }{{\left(\omega t+2i\right)}^2}\underset{t\to \pm \infty }{\to }0,(25)$

$\langle {\mathrm{\Psi }}^{*}\left(0,t\right)\left(\widehat{\mathit{a}}\cdot \mathit{k}\right)\mathrm{\Psi }\left(\mathit{x},t\right)\rangle =\frac{-8\pi }{{\left(\left(\widehat{\mathit{a}}\cdot \mathit{k}\right)\left(\widehat{\mathit{a}}\cdot \mathit{x}\right)+2i\right)}^2}\underset{\widehat{\mathit{a}}\cdot \mathit{x}\to \infty }{\to }0.(26)$

This behavior eliminates the need for introducing artificial delta function constructs, as used in quantum field theory (QFT), to limit the spatial extent of correlations between plane wave solutions of the linear Schrödinger equation. This principle aligns with the cluster decomposition property (Weinberg, 1995), which ensures that sufficiently distant experiments or systems do not influence each other and that correlations decay over large separations in time (Equation 25) or in space (Equation 26).

Measurement of wave quantities inevitably introduces nonlinearity into the state. This nonlinearity can be added to the Hamiltonian while preserving its Hermitian (self-adjoint) property, as shown in Equation 10.

Our hypothesis is supported by the illustration provided: increasing nonlinear input in the self-adjoint Hamiltonian form collapses a wave function of infinite spatial and/or temporal extent into a localized entity - a wave particle.

This process encapsulates the essence of wave-particle duality. However, from our perspective, it is not a mysterious “duality” at all. Rather, it is a critical point transition from a wave of infinite spatial/temporal extent to a highly localized field resonance in spacetime. In this perspective, the wave-particle duality emerges as a natural consequence of nonlinear interactions in quantum systems, shedding new light on a long-standing enigma in quantum mechanics. This viewpoint carries direct implications for phenomena such as the double-slit experiment, providing a clearer understanding of the underlying quantum behavior (Frank and Galinsky, 2025).

4 Relativity

It is interesting that not just quantum wave-particle duality but the special relativity itself can be viewed as another facet of critical nonlinear relativistic wave behavior and follows directly from the simple model (Equations 8, 9a, 10) if we take ${\widehat{\beta }}_{01}$ to be an operator of the form ${\widehat{\beta }}_{01}=iv\frac{\partial }{\partial x}$ where the derivative is only applied to the wave function $\mathrm{\Psi }$ of the Hamiltonian ${\widehat{H}}_1$ itself, i.e., if we assume that the nonlinear input ${\widehat{H}}_1$ is related to the movement of the entire frame of reference with a constant velocity $v$. As it can easily be seen from (Equation 21) for $\alpha =0$

$T^{\prime }=\frac{2\pi }{\sqrt{{\omega }^2-k^2{v}^2}}=\frac{T}{\sqrt{1-v^2}},(27)$

that in the standard units is simply the well-known time interval or time dilation equation $T^{\prime }=\gamma T$ where $\gamma =1/\sqrt{1-v^2/c^2}$ is the well-known Lorentz factor. Thus, Equation 27 is the temporal part of the Lorentz transformation (where $T=2\pi /\omega$ is the temporal period in the rest frame).

The various derivations of Lorentz transformations typically result in a linear transformation normalized by the Lorentz factor, $\gamma =1/\sqrt{1-v^2/c^2}$ (Rindler, 2006; Steane, 2011). In these derivations, $c$ represents the speed of light in vacuum, which is not obtained through the derivation itself, but postulated as a fundamental constant in special relativity. The invariance of $c$ across all inertial frames is, in fact, one of the two key postulates used to derive the Lorentz transformations, the other being the principle of relativity.

The Postulate of the Constancy of Light Speed, a cornerstone and a fundamental tenet of Special Relativity (Einstein, 1905b), has traditionally been introduced as an axiom–an assumed property of nature not derivable from basic principles. However, our analysis suggests that this postulate may no longer require axiomatic status. Our simple derivation demonstrates that the constancy of light speed across all inertial frames emerges naturally from a nonlinear quantum relativistic wave field framework (Equations 8, 9a, 10). The statement that nothing can move faster than the speed of light in a vacuum–one of the central principles of special relativity–is then just another consequence of the same critical transition of the wave field that occurs when the velocity $v$ approaches a critical point at $v=c$.

5 Gravity

Having demonstrated that the self-adjoint non-linear Hamiltonian theory, even in its simplest form, provides an explanation for the origins of quantum interference and special relativity, we now demonstrate that it may also be employed to explain the outcomes of the classic tests of general relativity proposed by Einstein:

(a) Gravitational redshift of spectral lines,

(b) Deflection of light by the Sun,

(c) Precession of the perihelia of planetary orbits,

As well as other experimental confirmations, without invoking or relying on the gravitationally curved metric of empty spacetime postulated by general relativity (see Supplementary Appendix B for more details on Dispersion Relation, Effective Geometry, and Rays as Null Geodesics).

For these illustrations, consider the simplest scenario: a nonlinear wave (either a massless light wave or a massive matter wave) propagating from a distant boundary toward (or around) a single, massive, spherically symmetric source or sink located at the origin. By applying the nonlinear wave dispersion relation from (Equation 22), while now using the fact that the parameters $\alpha$ and $\beta$ in the nonlinear Hamiltonian (Equation 11) may depend on spatial coordinates (and possibly on wave numbers, but not explicitly on time, so that $\omega$ remains constant), we arrive at a dispersion relation $D(\omega ,\mathit{x},\mathit{k})$. This enables us to derive the corresponding wave trajectory equations as (see Supplementary Appendix C for more details of the derivations of geodesic-like equation and emergence of Parameterized Post-Newtonian (PPN) parameters)

$\frac{d{x}^j}{dt}=-\frac{\partial D}{\partial k_j}/\frac{\partial D}{\partial \omega },\frac{d{k}_j}{dt}=\frac{\partial D}{\partial x^j}/\frac{\partial D}{\partial \omega }.(28)$

For the dispersion relation $D(\omega ,\mathit{x},\mathit{k})$, we use the following form (that follows from (Equation 22)):

$D\left(\omega ,\mathit{x},\mathit{k}\right)={\left(\omega +\alpha \left(\mathit{x},\mathit{k}\right)\right)}^2-|\mathit{k}{|}^2-\beta {\left(\mathit{x},\mathit{k}\right)}^2,(29)$

where we assume that both $\alpha$ and $\beta$ are functions only of $r\equiv |\mathit{x}|$ and $k_r$, so the motion is confined to a plane, e.g., $\alpha (\mathit{x},\mathit{k})\equiv \alpha (r)$ and $\beta (\mathit{x},\mathit{k})\equiv {\beta }_0+k_r{\beta }_1(r)$. The functional forms of $\alpha (r,k)$ and $\beta (r,k)$ are chosen as ansätze to reproduce known results, and should not be interpreted as uniquely predicted by the theory. We employ the standard spherical coordinate system $(r,\theta )$, where $r$ is the radial distance from the origin and $\theta$ is the polar angle measured from the positive $z$-axis. The metric components are $g_{rr}=1$ and $g_{\theta \theta }=r^2$, with no additional curvature introduced by the central mass. Hence

$|\mathit{k}{|}^2=g^{ij}{k}_i{k}_j=g^{rr}{k}_r^2+g^{\theta \theta }{k}_{\theta }^2=k_r^2+\frac{k_{\theta }^2}{r^2}.(30)$

We will further use the previously mentioned fact that $\alpha (r)$ and ${\beta }_1(r)$ can be expressed as functions of $\mathcal{M}(\mathit{x})$. In the spherically symmetric case, $\mathcal{M}(\mathit{x})$ reduces to $\mathcal{M}(r)$, which, as previously discussed, behaves as $\mathcal{M}(r)\sim 1/r$. Note that ${\beta }_0$ can be interpreted as the rest mass. From this, we can obtain an expression for $|\mathit{k}{|}^2$ using Equations 28–30 as follows:

$\frac{dr}{dt}=\frac{k_r\left(1+{\beta }_1^2\right)+{\beta }_0{\beta }_1}{\omega +\alpha },\Rightarrow (31)$

$k_r=\frac{\left(\omega +\alpha \right)}{1+{\beta }_1^2}\frac{dr}{dt}-\frac{{\beta }_0{\beta }_1}{1+{\beta }_1^2},(32)$

$\frac{d{k}_{\theta }}{dt}=0,\Rightarrow k_{\theta }=\omega J=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.(33)$

where $J$ is a conserved angular momentum associated with the rotational symmetry of the system, a consequence of Noether’s Theorem (Noether, 1918). The dispersion relation (Equation 29) using Equations 31–33 then takes the form

$\frac{{\left(1+\tilde{\alpha }\right)}^2}{1+{\beta }_1^2}{\left(\frac{dr}{dt}\right)}^2+\frac{J^2}{r^2}-{\left(1+\tilde{\alpha }\right)}^2=-\frac{{\tilde{\beta }}_0^2}{1+{\beta }_1^2},(34)$

where we set $\tilde{\alpha }=\alpha /\omega$ and ${\tilde{\beta }}_0={\beta }_0/\omega$. Assuming further that ${\beta }_1$ is a small parameter, it can be written as

$A\left(r\right){\left(\frac{dr}{dt}\right)}^2+\frac{J^2}{r^2}-\frac{1}{B\left(r\right)}=-E,(35)$

where

$A\left(r\right)={\left(1+\tilde{\alpha }\right)}^2\left(1-{\beta }_1^2\right),(36)$

$\frac{1}{B\left(r\right)}={\left(1+\tilde{\alpha }\right)}^2+{\tilde{\beta }}_0^2{\beta }_1^2,(37)$

$E={\tilde{\beta }}_0^2.(38)$

It is easy to see that this expression has the form of the expression (8.4.13) from Steven Weinbergs’s “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity” textbook [19, p.186], denoting an integral of the General Equations of Motion (Section 8.4). As it was argued in [19, (8.4.15) and (8.4.16)] $E>0$ (or ${\beta }_0>0$ in our (Equation 34)) corresponds to massive particles and $E=0$ $({\beta }_0=0)$ (Equation 38) describes photons (massless particles).

The expression (8.4.13) has been used as a ground base to prove that the general relativity explains all the above classic Einstein’s suggested tests – “Unbound Orbits: Deflection of Light by the Sun” (Section 8.5), and “Bound Orbits: Precession of Perihelia” (Section 8.6), as well as an additional “Radar Echo Delay” (Section 8.7).

The calculations (as well as an explanation of the gravitational red shift of spectral lines) use Schwarzschild solution (Schwarzschild, 1916) for functions $A(r)$ and $B(r)$ [Weinberg, 1972, (8.1.10) and (8.1.11)]

$A\left(r\right)=B{\left(r\right)}^{-1}={\left[1-\frac{2MG}{r}\right]}^{-1},(39)$

where

$\mathrm{\Theta }\left(r\right)=-\frac{MG}{r}(40)$

is the gravitational potential (i.e., the potential energy per unit mass) in which $M$ is the central mass and $G$ is the Newton’s gravitations constant. In all the above general relativity tests presented in (Weinberg, 1972) the gravitational potential $\mathrm{\Theta }(r)$ is assumed to be a small parameter $(|2\mathrm{\Theta }(r)|\ll 1)$, therefore, the following expansions are used instead of Equations 39, 40 [Weinberg, 1972, p.196]

$A\left(r\right)\approx 1+\mathrm{\Gamma }\frac{2MG}{r}+\dots ,(41)$

$B{\left(r\right)}^{-1}\approx 1+\frac{2MG}{r}+2\left(2-\mathcal{B}+\mathrm{\Gamma }\right)\frac{M^2{G}^2}{r^2}+\dots ,(42)$

where we replaced the dimensionless parameters $\beta$ and $\gamma$ used in (Weinberg, 1972) with capital letters $\mathcal{B}$ and $\mathrm{\Gamma }$ to avoid confusion with our notation. Those parameters in general relativity are both equal to one and the above tests have been used to provide confirmation that both $\mathcal{B}=\mathrm{\Gamma }=1$.

• Deflection of Light by the Sun [Weinberg, 1972, p.190, (8.5.8)]

$\mathrm{\Delta }\varphi =\frac{4MG}{r_0}\left(\frac{1+\mathrm{\Gamma }}{2}\right),(43)$

where $r_0$ is the closest distance.

The expression (Equation 43) has been used to demonstrate that general relativity predicts a deflection of light by gravity that is twice as large as the value predicted by Newtonian mechanics. This doubling arises from general relativity’s inclusion of spatial curvature-represented by the parameter $\mathrm{\Gamma }=1$, in addition to the Newtonian contribution, which only accounts for the curvature of time. Experimental observations, such as the measurement of starlight deflection during solar eclipses, have confirmed this prediction, providing strong evidence for the relativistic effect.

Similarly, radar echo delay experiments (the Shapiro time delay) offer independent confirmation of $\mathrm{\Gamma }=1$. In these experiments, radar signals passing near a massive object like the Sun experience a time delay that matches the predictions of general relativity, again exceeding the delay expected from Newtonian theory alone. The precision of these experiments has allowed for stringent tests, with results consistent with the general relativistic value of unity for the relevant parameter.

• Bound Orbits: Precession of Perihelia [Weinberg, 1972, p.197, (8.6.10)]

$\mathrm{\Delta }\varphi =\frac{6\pi MG}{L}\left(\frac{2+\mathcal{B}-2\mathrm{\Gamma }}{3}\right)(44)$

where $L$ is the semilatus rectum and $\mathrm{\Delta }\varphi$ (Equations 43, 44) is in units of (radians/revolution).

Regarding the precession of perihelia, it has been clearly emphasized that:

“…This is by far the most important experimental verification of general relativity, both by virtue of its high accuracy, and because it alone is sensitive to the coefficient $\mathcal{B}$ [$\beta$ in the original text] appearing in the second-order term in $g_{tt}$ …” [see Ref. [Weinberg, 1972, p. 198]].

This statement highlights not only the exceptional precision of perihelion precession measurements (such as those for Mercury), but also their unique sensitivity to the parameter $\mathcal{B}$, which appears in the second-order expansion of the time-time component of the metric $(g_{tt})$. As a result, these observations provide a crucial and stringent test of general relativity that is not accessible through other classical tests.

However, our nonlinear wave theory reproduces this same result of $\mathcal{B}=\mathrm{\Gamma }=1$ without the need to invoke curved spacetime. From Equations 36, 37, 41, 42],

$\left.\begin{array}{c}\tilde{\alpha }=\frac{MG}{r}+\frac{3}{2}{\left(\frac{MG}{r}\right)}^2+\dots \hfill \\ {\beta }_1=0\hfill \end{array}\right\}\mathcal{B}=\mathrm{\Gamma }=1$

Alternatively,

$\left.\begin{array}{c}\tilde{\alpha }=\frac{MG}{r}\hfill \\ {\tilde{\beta }}_0{\beta }_1=\sqrt{3}\frac{MG}{r}\hfill \end{array}\right\}\mathcal{B}=\mathrm{\Gamma }=1$

And, in fact, the same result can also be achieved by setting all the coefficients to unity:

$\left.\begin{array}{c}\tilde{\alpha }=\frac{MG}{r}+{\left(\frac{MG}{r}\right)}^2+\dots \hfill \\ {\tilde{\beta }}_0{\beta }_1=\frac{MG}{r}+\dots \hfill \end{array}\right\}\mathcal{B}=\mathrm{\Gamma }=1$

Thus, any of these choices fully reproduce all of Einstein’s classic tests of general relativity—remarkably, without any need for the warping or bending of either proper physical space or proper time.

6 Cosmology

The nonlinear effects described by the proposed nonlinear Hamiltonian (Equation 11) — though typically very small for the local detection of electromagnetic radiation due to the massless nature of photons—may accumulate and become significant when light traverses cosmological distances. This speculative concept can be illustrated mathematically using the dispersion relation from the nonlinear critical wave model (Equation 29).

As various scientific instruments (e.g., optical, radio, X-ray telescopes) gather radiation from distant regions of space, it is certain that these electromagnetic waves interact with numerous localized wave packets, both small and large, along their trajectories—from the source and throughout their cosmic journey. This scenario can be analyzed in analogy to the unbound orbit problem discussed previously, but with a key distinction: here, telescopes pointed in different directions across the universe function not as sources, but as sinks that collect incoming radiation.

In the ‘sink’ case, the spherically symmetric normalization solution to the corresponding Laplace equation can be represented by a harmonic function: $\mathcal{M}(\mathit{x})=|\mathit{x}|\equiv r$. This elementary solution captures an essential aspect of Olbers’ Paradox: if the universe is considered as a series of concentric spherical shells centered on the observer, each shell contains more stars as its volume increases, while the light from each individual star is dimmed by the inverse square law. The consequence is that every shell, despite growing larger, contributes the same amount of light to the observer’s sky, implying that the sky should be as bright as a typical stellar surface—contradicting the observed darkness of the night sky.

Then the analysis, analogous to that of the previous section but significantly simpler, clearly demonstrates that the cosmological redshift is not unexpected and can be readily explained by the same nonlinear critical wave mechanism. For the purpose of illustrating this hypothesis, we will start with the simple case of one-dimensional massless wave propagation, using the simplified dispersion relation that follows from (Equation 29) by setting $\alpha (\mathit{x},\mathit{k})\equiv \alpha (r,k)$ and $\beta (\mathit{x},\mathit{k})\equiv \beta (r,k)$, where $k\equiv k_r$:

${\left(\omega +\alpha \left(r,k\right)\right)}^2-k^2-\beta {\left(r,k\right)}^2=0.(45)$

For $\alpha (r,k)$ and $\beta (r,k)$ we will use simple ansätze that involve $\mathcal{M}(\mathit{x})\sim r$, as $\beta (r,k)=-\alpha (r,k)=akr$, where $a$ is a constant. The expression (Equation 45) clearly satisfies the most general properties of light: (1) the absence of photon mass, since $\omega =0$ when $k=0$, and (2) the frequency is proportional to the wave number at any arbitrary position, i.e., $\omega \sim k$ for any $r$. This formulation ensures consistency with fundamental characteristics expected for massless photons and their dispersion relation in the given medium.

The redshift $z$ then directly follows from (Equation 45) as

$1+z=\frac{{\lambda }_{\mathit{obsv}}}{{\lambda }_{\mathit{emit}}}=\frac{\tilde{k}\left(0\right)}{\tilde{k}\left(r\right)}=ar+\sqrt{1+a^2{r}^2},(46)$

where $\tilde{k}=k/\omega$. Hence

$z\sim ar+O\left(r^2\right),r<1/a,(\mathrm{47a})$

$z\sim 2ar+O\left(\frac{1}{r}\right),r>1/a.(\mathrm{47b})$

Of course, for small distances and low redshift values, it is reasonable to consider gravitational effects in addition to the cosmological effects (Equations 46, 47a, b) discussed above, similar to the approach in the previous section. In a simplified form that includes both gravitational and cosmological contributions, we can use the harmonic function solutions of the Laplace equation (cf. Equation 2), namely, terms proportional to $1/r$ representing the Newtonian gravitational potential, and terms proportional to $r$ corresponding to the cosmological linear potential effect. Thus, we propose the following ansätze:

$\alpha \left(r,k\right)=k\left(-ar+\frac{b}{r}\right),(\mathrm{48a})$

$\beta \left(r,k\right)=k\left(ar+\frac{b}{r}\right),(\mathrm{48b})$

where $a$, $b$ are coefficients encoding the strength of the cosmological and the gravitational contributions.

Substituting these forms (Equations 48a, b) into Equation 45, and ignoring the angular dependence and rest mass, we obtain the following dispersion relation:

${\left(1+\left(-ar+\frac{b}{r}\right)\tilde{k}\right)}^2-{\tilde{k}}^2-{\left(\tilde{k}\left(ar+\frac{b}{r}\right)\right)}^2=0.(49)$

One can then derive an expression for the wavelength dependence as a function of the distance $r$, given by

$\frac{\lambda }{2\pi }=\frac{1}{\tilde{k}}=ar-\frac{b}{r}+\sqrt{1+{\left(ar+\frac{b}{r}\right)}^2}.(50)$

What is interesting is that, when both the cosmological and gravitational contributions are taken into account in Equations 49, 50, two distinct linear redshift regimes emerge: one in the small-$z$ (small source distance $r$) limit, and another in the large-distance (and high-redshift) limit:

$z\sim \left(2a+\frac{1}{2b}\right)r+O\left(r^2\right),r<\min \left(\frac{1}{a},b\right),(\mathrm{51a})$

$z\sim 2ar+O\left(\frac{1}{r}\right),r>\frac{1}{a}.(\mathrm{51b})$

Assuming that the distant redshift results obtained from the Planck mission are accurate and correspond to the large-distance, high-redshift limit (Equation 51b), we can conclude that

$H_0^{\hslash }\approx 66.6{\mathrm{k}\mathrm{m}\mathrm{s}}^{-1}{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}\Rightarrow \frac{1}{2a}=\frac{c}{H_0^{\hslash }}\approx 4500\mathrm{M}\mathrm{p}\mathrm{c}.(52)$

For the gravitational input, we will assume the following typical parameters (Aghanim et al., 2020):

Gravitational constant: $G=6.67430\times 10^{-11}{\mathrm{m}}^3{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$

Speed of light: $c=2.99792458\times 10^8\mathrm{m}/\mathrm{s}$

Conversion factor: $1\mathrm{M}\mathrm{p}\mathrm{c}=3.085677581\times 10^{22}\mathrm{m}$

Estimated mass of the visible universe (including dark matter): $M_U=7.7\times 10^{53}\mathrm{k}\mathrm{g},$

Hence we can estimate $b$ as

$b=\frac{G{M}_U}{c^2}\approx 1.85\times 10^4\mathrm{M}\mathrm{p}\mathrm{c}.(53)$

Using this estimate (Equation 53), we obtain the redshift for the nearby Universe (Equation 51a) as

$\begin{array}{ll}\hfill & {\left(2a+\frac{1}{2b}\right)}^{-1}\approx {\left(\frac{1}{4500}+\frac{1}{37000}\right)}^{-1}\approx 4012\mathrm{M}\mathrm{p}\mathrm{c},\hfill \\ \hfill & H_0^{\mathit{Cep}}\approx 74.75{\mathrm{k}\mathrm{m}\mathrm{s}}^{-1}{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}.\hfill \end{array}(54)$

Figure 2 illustrates the relationship between redshift $z$ and distance measured in megaparsecs (Mpc), highlighting different cosmological regimes. The plot includes three distinct curves: (1) The high-redshift/large-distance limit (blue dashed line) represents a linear redshift-distance relationship appropriate for large-scale measurements in the universe (Equations 51b, 52). This curve reflects an extended linear approximation of cosmological parameters valid in that regime. (2) The nearby Universe/small-redshift limit (brown dash-dotted line) also shows a linear relationship between redshift and distance, consistent with the classical Hubble law for small redshifts $z\le 1$ (Equations 51a, 53, 54), where the proportionality between redshift and distance holds accurately. (3) The nonlinear critical wave redshift curve, valid across both large-distance and nearby Universe regimes, accurately captures the transition region where the redshift-distance relation deviates from simple linearity due to nonlinear cosmological effects.

Figure 2

Figure 2. Redshift as a function of distance (in megaparsecs, Mpc) is shown for three regimes: the nearby Universe/small-redshift limit (brown dash-dotted line), the large-distance/high-redshift limit (blue dashed line), and the nonlinear critical-wave redshift (red solid line). The figure demonstrates the transition in the effective Hubble parameter, from $H_0=H_0^{\mathit{Cep}}\approx$ 75 km ${\text{s}}^{-1}$ ${\text{Mpc}}^{-1}$ for $z\le 1$ (see top-left inset, where the red and brown lines coincide) to $H_0=H_0^{\hslash }\approx$ 67 km ${\text{s}}^{-1}$ ${\text{Mpc}}^{-1}$ for $z>1$ (see bottom-right inset, where the red and blue lines coincide). Insets illustrate the agreement between the nonlinear model and the respective limiting cases at low and high redshift.

This figure clearly demonstrates the transition in the measured Hubble constant $H_0=H_0^{\mathit{Cep}}\approx$ 75 km ${\text{s}}^{-1}$ ${\text{Mpc}}^{-1}$ at redshifts below $z=1$, consistent with distances measured using Cepheid variable stars and other local distance indicators, to $H_0=H_0^{\hslash }\approx$ 67 km ${\text{s}}^{-1}$ ${\text{Mpc}}^{-1}$ at redshifts above $z=1$, inferred from cosmological observations such as the cosmic microwave background and large-scale structure.

Both the blue and brown curves maintain linear behavior but at different effective slopes, representing different calibration scales and epochs of measurement. The nonlinear curve between them captures the cosmological evolution where this linear approximation becomes less accurate. As such, it encapsulates the well-known “Hubble tension,” reflecting the differing values of $H_0$ obtained from local and cosmological-scale observations (Verde et al., 2019), and highlights the cosmological complexity in relating redshift to distance across different redshift ranges.

It may be insightful to relate the Hubble tension problem—particularly the relationships described in (Equation 51)—to another key dark-and-cold development in modern cosmology: Baryon Acoustic Oscillations (BAO). These oscillations are currently considered by the cosmological community to represent periodic fluctuations in the density of visible (baryonic) matter, originating from sound waves that propagated through the hot plasma of the early universe before recombination. The physical scale defined by these oscillations serves as a “standard ruler,” approximately 150 megaparsecs in length, imprinted in the large-scale distribution of galaxies. This standard ruler is a crucial tool for probing cosmic expansion and constraining the properties of dark energy and dark matter. Today, BAO measurements form a cornerstone of precision cosmology, providing a powerful connection between the physics of the early universe and the present-day structure and evolution of the cosmos (Eisenstein and Hu, 1998).

The detection of the BAO peak is achieved by measuring the pair-wise galaxy distribution in redshift space through the two-point galaxy correlation function. This procedure involves quantifying the excess probability, relative to a random distribution, of finding pairs of galaxies separated by various distances. The BAO peak appears as a small but distinct excess in the number of galaxy pairs separated by about 150 megaparsecs, marking the imprint of sound waves from the early universe in the large-scale structure of galaxy clustering (Eisenstein et al., 2005).

The galaxy pair correlation function, commonly denoted $\xi (r)$, quantifies the excess probability—relative to a random distribution—of finding two galaxies separated by a distance $r$. It is defined as

$\xi \left(r\right)=\frac{DD\left(r\right)}{RR\left(r\right)}-1,(55)$

where $DD(r)$ is the number of galaxy pairs with separation $r$ observed in the data, and $RR(r)$ is the expected number of such pairs in a random, uniformly distributed galaxy catalog. The correlation function can be estimated by counting galaxy pairs within a spherical shell of radius $r$ and thickness $dr$. Specifically, $\xi (r)$ (Equation 55) is obtained by comparing the observed count $DD(r)$ of galaxy pairs in that shell to the expected count $DR(r)$ for a random distribution.

More formally, the two-point correlation function can also be expressed as the ensemble average of overdensity fluctuations:

$\xi \left(\mathrm{\Delta }\right)=⟨\delta \left(\mathbf{x}\right)\delta \left(\mathbf{x}+\mathrm{\Delta }\right)⟩,$

where $\delta (\mathbf{x})=\frac{\rho (\mathbf{x})-\bar{\rho }}{\bar{\rho }}$ is the fractional overdensity at position $\mathbf{x}$, and $\mathrm{\Delta }=|{\mathbf{x}}_1-{\mathbf{x}}_2|$ is the galaxy pair separation.

The expected number of pairs $dR(r)$ (continuous version of $RR(r)$) in a random distribution is given by the number density of galaxies squared multiplied by the volume of the spherical shell $d{V}_{\text{shell}}$ at radius $r$ with thickness $dr$:

$dR\left(r\right)={\bar{n}}^{2}d{V}_{\text{shell}}={\bar{n}}^{2}4\pi r^{2}dr,$

where $\bar{n}$ denotes the average number density of galaxies. This expression assumes a uniform, unclustered Poisson distribution of galaxies over the survey volume, which corresponds to an exponential distribution in the continuous limit. The volume element $4\pi r^{2}dr$ represents the volume of a thin spherical shell at radius $r$ with thickness $dr$, specifying the region where galaxy pairs are counted for separations between $r$ and $r+dr$.

But what happens to the random exponential distribution if, instead of counting pairs in $r$-space, we use some new coordinate space, say $z\equiv z(r)$? It is straightforward to see that

${\bar{n}}^{2}4\pi r^{2}dr=\left[\frac{r^2}{z^2}\frac{dr}{dz}{\bar{n}}^2\right]4\pi z^{2}dz={\bar{n}}_{\text{bao}}^{2}4\pi z^{2}dz.$

For the new “BAO”-modified exponential distribution in redshift space (denoted by $z$), which is related to the real-space coordinate $r$ through an expression analogous to (51), the transformation behaves as follows:

$\frac{r^2}{z^2}\frac{dr}{dz}\sim {\left(2a+\frac{1}{2b}\right)}^{-3},r<r_0,(\mathrm{56a})$

$\frac{r^2}{z^2}\frac{dr}{dz}\sim {\left(2a\right)}^{-3},r>r_0,(\mathrm{56b})$

Here, $r_0$ denotes a characteristic scale, conventionally set to $150\mathrm{M}\mathrm{p}\mathrm{c}$ to align with experimental observations. Although this scale does not appear explicitly in (Equation 51), it emerges naturally once the relation between the wavelength $\lambda$ and the redshift $z$ is introduced.

Since the constant for smaller $r$ (Equation 56a) is lower than that for larger $r$ (Equation 56b), multiplying a random exponential distribution by such a transformation function inevitably produces a feature resembling a bump in the transition region at the distribution’s tail—a feature that is currently interpreted as one of the proofs of the universe’s expansion.

This framework naturally accounts for puzzling anomalies (Verde et al., 2019) that have arisen in attempts to fit current theories to observations, such as the discrepancy between distant and nearby redshift measurements, offering excellent agreement for the Hubble constant values derived from both the Planck cosmic microwave background observations (Aghanim et al., 2020) and measurements in the nearby Universe using supernovae and Cepheid variables (Riess et al., 2016; Riess et al., 2019): approximately 67 km ${\text{s}}^{-1}$ ${\text{Mpc}}^{-1}$ and 75 km ${\text{s}}^{-1}$ ${\text{Mpc}}^{-1}$, respectively.

These results provide a natural explanation for the puzzling discrepancy highlighted by the Nobel Prize–winning Planck mission (Aghanim et al., 2020): Why does the Hubble constant, measured from early Universe data such as the Planck cosmic microwave background observations, disagree with measurements based on the nearby Universe, including supernovae and Cepheid variables?

We emphasize that these results are obtained as a direct consequence of the nonlinear critical effects emergent from the nonlinear critical relativistic quantum wave field theory posited above to be the fundamental framework for both quantum phenomena and gravity. They are not reliant upon invoking the curvature of spacetime via the Friedmann-Lemaître-Robertson-Walker (FLRW) metric that yields the Friedmann equations as solutions to Einstein’s field equations of general relativity, and form the basis of current “Big Bang” theories, such as the $\mathrm{\Lambda }$CDM model (Riess et al., 1998; Perlmutter et al., 1999; Melia, 2012; Riess et al., 2016; Aghanim et al., 2020).

This framework thus has significant implications for revising and reinterpreting standard cosmological models, including the “Big Bang,” which are based on linear frameworks such as Hubble’s (or the Hubble–Lemaître) law (Lemaître, 1927)—the linear relation stating that galaxies recede from Earth at speeds proportional to their distance. Rather, our theory predicts that the observed redshift can emerge from intrinsic nonlinear wave dynamics rather than from cosmic expansion, challenging prevailing assumptions grounded in linear cosmological theories.

7 Discussion and conclusion

The purpose of this work is not to prove or disprove the quantum field theory or the theory of general relativity. Rather, we present a hypothesis: the same class of nonlinear critical waves, as described by a relatively simple and general nonlinear Hamiltonian operator (see Equation 10), may offer a unified explanation for longstanding questions regarding the emergence and origin of both quantum, relativistic, and cosmological effects. Notably, the sign of the $\tilde{\alpha }$ term in (Equation 10) indicates that the nonlinear interaction may manifest as either repulsion (‘+‘) or attraction (‘-‘), and thus is potentially not limited to gravitational effects alone.

The wave-based framework (see Eqs. Equations 28,29) we illustrate here demonstrates how gravity effects, analogous to those described by general relativity, can emerge. This approach offers several significant advantages over the traditional curved-metric formulation of general relativity.

1. The emergence of both the quantum rest mass of a particle and gravitational effects arises naturally from the same critical wave description. As a result, the requirement to independently prove the equivalence of gravitational and inertial mass is eliminated.

2. The phenomenon of global spacetime curvature attributed to distant masses in general relativity can be more simply interpreted as a consequence of local nonlinear wave dispersion.

3. The constancy of the speed of light is a direct result of the critical nonlinear terms in the wave dispersion relation.

4. The singularities that appear in the spacetime metric solutions of general relativity–such as at the Schwarzschild radius $r_s=2MG$ and at $r=0$ – do not arise in the nonlinear quantum wave gravity framework. Specifically, there are no $r_s$ singularities in (Equation 29) or (Equation 34). For $r=0$, it is straightforward to show from (Equation 29) that when $r$ is small and $\alpha (r)\sim a/r$, ${\beta }_1(r)\sim b/r$, the attractive solution of (Equation 29)

$\omega =-\alpha \left(r\right)+\sqrt{k_r^2+\frac{k_{\theta }^2}{r^2}+{\left({\beta }_0+k_r{\beta }_1\left(r\right)\right)}^2}(57)$

leads to a leading $1/r$ term

$\omega \approx \frac{-a+\sqrt{b^2{k}_r^2+k_{\theta }^2}}{r}.(58)$

Rather than remaining singular and attractive, this term (Equation 58) necessarily approaches zero or can even change sign (becoming repulsive) as the radial wave number (momentum) increases while the wave packet propagates to smaller $r$. This conclusion will not change if both $\alpha (r)$ and $\beta (r)$ in Equation 57 include higher order $1/r^n$ terms (see Supplementary Appendix D for details of a curvature analysis in the corresponding effective metric).

5. This approach offers a unified framework that naturally integrates gravitational phenomena with cosmological effects, providing a common wave-based description connecting local gravity and large-scale structure in the universe. In addition, this approach provides a natural explanation for the ‘bump’ in the distribution function of the galaxy pair correlations. In the current literature, this feature is considered as ‘proof’ of the expansion of the universe. However, it emerges naturally in our model as the consequence of a simple coordinate transformation of the probability distribution in the appropriate redshift space.

The self-adjoint form of the Hamiltonian operator (see Equation 10) is versatile in its ability to describe a range of nonlinear conditions, particularly when the parameters ${\alpha }_n$ and ${\beta }_{nm}$ are expressed as general functions. However, it does not fully capture the diversity and complexity of nonlinear operational forms that may involve multiple operators acting on different components. The primary objective of this brief communication is not to provide an exhaustive analysis of this diversity, but rather to present a concise yet sufficiently general illustration in support of our hypothesis regarding the critical origin of quantum wave-particle duality and the emergence of both relativity and gravity.

We should also mention that if in the above illustrative example we assume that the nonlinear input in (Equation 10) contains only higher order terms ${\beta }_{nm}$ and ${\beta }_{mn}^{*}$ (with $m>n$), then the same conclusion about critical transition and emergence of quantized wave particles at $|\omega +\alpha |=|\beta |$ can be easily found by a simple substitution of $\psi \to \psi /(m-n)$ and $(\omega ,\alpha ,\beta )\to (m-n)(\omega ,\alpha ,\beta )$. A complex case that involves multiple wave modes (and not just a simple relativistic plane wave modes described by system Equations 9a, b) with multiple nonlinear terms present in the Hamiltonian (Equation 10) may result in the appearance of multiple critical points supporting critical transitions between different regimes. Indeed, based on our simple illustrations above, we hypothesize that currently known different types of particles can be attributed to critical transitions that occur through nonlinear wave propagation in these complex situations.

We have stated a hypothesis, and illustrated it with very straightforward mathematical details, showing how relativistic plane waves turn into localized/quantized particle-like activation with just a change of nonlinearity-frequency ratio, giving the same effective dispersion relation as matter waves have and at the same time showing that it can not propagate faster that the phase speed of the linear wave, hence, providing a suggestion for the roots of the postulate of the speed of light constancy.

We also have provided an illustration that uses wave-functions to describe actual fields, where their collapse and subsequent spatiotemporal localization arise naturally as a result of a critical transition. This approach eliminates the need to invoke artificial statistical ‘ensembles,’ thereby avoiding ambiguous questions regarding the ‘reality’ of wave-functions.

And finally, noting that the nonlinear effects described by a nonlinear Hamiltonian, though negligible for local electromagnetic detection due to the massless nature of photons, may accumulate over cosmological distances and become significant, we explored the implications for effects on the cosmological scale. This led to the conclusion that the cosmological redshift might potentially stem from the same nonlinear critical phenomena hypothesized here as the underlying cause of quantum effects and gravity. If true, this would have profound implications for revising and reinterpreting standard models such as the “Big Bang,” which are based not only on linear frameworks like Hubble’s (or Hubble–Lemaître’s) law (Lemaître, 1927; Lemaître, 1931)—a linear relation stating that galaxies recede from Earth at speeds proportional to their distance—but also fundamentally rely on the curved spacetime metric (Melia, 2012) described by Einstein’s general relativity. The Big Bang model emerges as a solution to Einstein’s field equations, where the curvature of spacetime, shaped by matter and energy content, governs cosmic evolution, making general relativity essential to its formulation and interpretation.

In this paper, we have advanced the hypothesis that gravitational phenomena are not fundamental in nature but rather emergent, arising from the self-organization of nonlinear waves at criticality. Analogous to wave–particle duality—where quantum behavior reflects both localized excitations and wave-like propagation—we propose that the gravitational field represents a collective, emergent manifestation of underlying wave dynamics. Within this framework, the classical effects ascribed to gravity in general relativity can be reproduced across all currently tested regimes, while avoiding the singularities intrinsic to Einstein’s formulation. Solutions such as black holes and the Big Bang singularity are therefore not interpreted as literal singular physical entities, but instead as effective descriptions whose breakdown signals the limits of the underlying theoretical model. In particular, the Big Bang provides a paradigmatic example of such a singular solution that we regard as non-physical in itself, pointing instead to the need for a more complete underlying framework.

The aim of this work is not to reject the standard cosmological model in its entirety, but rather to establish a theoretical basis from which it may be systematically reassessed. The conventional Big Bang paradigm is founded on three principal observational “pillars”: (1) Hubble’s law and galactic redshifts, (2) the Cosmic Microwave Background (CMB), and (3) the primordial abundances of light elements predicted by Big Bang Nucleosynthesis (BBN). Of these, our framework naturally accounts for the first two. It provides a self-consistent description of galactic redshifts that can reproduce a Hubble-tension–like behavior under a specific nonlinear ansätze, and it ensures that the emergence of a CMB-like spectrum arises as a necessary consequence of the underlying dynamics. Specifically, the redshifting of a blackbody spectrum of starlight rigorously preserves its blackbody character, with the emission unavoidably translated to longer wavelengths and expressed as a correspondingly lower effective temperature.

We emphasize that our framework is not a variant of “tired-light” models. It successfully reproduces all tests of general relativity, including those involving time dilation. Consequently, Type Ia supernova time-dilation measurements are also accurately described by our approach. Furthermore, the relatively recent detection of a peak in the galaxy pair distribution function—and its speculative attribution to so-called “Baryonic Acoustic Oscillations”—is likewise well accounted for within the same byproduct mechanisms that resolve the Hubble tension (as disscussed in Section 6).

It must be acknowledged that a wide array of observational results are commonly cited in favor of the expanding-universe paradigm. Yet, among these, only the galactic redshift is a direct and unambiguous empirical fact. The remaining evidence is often indirect, incomplete, or internally inconsistent, as has been widely noted.

For example, the Tolman surface brightness test was first proposed by Richard C. Tolman in 1930 as a cosmological test to determine if the universe is expanding or static. It measures how the surface brightness of galaxies changes with redshift. In a static, non-expanding universe, surface brightness is expected to remain constant regardless of distance or redshift. However, in an expanding universe, surface brightness decreases sharply with redshift, proportional to ${(1+z)}^{-3}$ or ${(1+z)}^{-4}$ depending on units, due to photon energy loss and time dilation effects. Over the decades, the Tolman test has been applied to observational data, with some studies initially supporting the expanding universe. Yet, newer analyses have found inconsistencies with the expected brightness decline, particularly when accounting for galaxy evolution and telescope resolution. Some results favor a static Euclidean universe model, where surface brightness remains constant with redshift, over the standard expanding universe predictions.

In this context, a 2018 study (Lerner, 2018) found that observed galaxy sizes and surface brightness contradict expanding universe predictions but fit well with a static universe model, revisiting the implications of Tolman’s test using modern galaxy data and resolution considerations. This challenges the widespread acceptance of expansion based solely on surface brightness tests.

Recent high-resolution results from the James Webb Space Telescope (JWST) increasingly highlight tensions within the standard model rather than reinforcing it. For example, observations reveal galaxies with unexpectedly mature, inside-out growth as early as 700 Myr after the presumed Big Bang (Baker et al., 2025), unexpectedly bright hydrogen emission at 330 Myr (Witstok et al., 2025), and luminous, massive early galaxies whose properties significantly exceed $\mathrm{\Lambda }$CDM expectations (Jacak, 2025). Moreover, surveys have identified hundreds of anomalously bright objects in the early universe that resist standard cosmological interpretation. Perhaps most strikingly, kinematic measurements from the JWST Advanced Deep Extragalactic Survey (JADES) reveal a strong asymmetry in galaxy rotation orientations, with nearly two-thirds rotating clockwise—contradicting the isotropy assumption of standard cosmology (Sun and Yan, 2025). Collectively, such findings pose challenges that deepen rather than resolve theoretical uncertainties.

The third pillar of Big Bang cosmology—nucleosynthesis of light elements—remains the least secure. Classical BBN theory predicts deuterium (D), helium-3 (³He), helium-4 (⁴He), and lithium-7 (⁷Li) abundances, with robust consistency in the ${\text{predicted}}^4$He mass fraction (25%) and hydrogen (75%). However, discrepancies emerge for the lighter isotopes. Deuterium is predicted at $\sim 2.5\times 10^{-5}$ relative to ${\text{hydrogen,}}^3$He at $\sim 10^{-5}$, ${\text{and}}^7$Li at $\sim 10^{-10}$ (Wu, 2023). Observational data, however, show persistent mismatches. The long-recognized “cosmological lithium problem” persists, with ${\text{measured}}^7$Li abundances in Population II stars falling below predictions by factors of 2.4–4.3 (Fields, 2011; Deal and Martins, 2021). Similarly, deuterium ${\text{and}}^3$He data exhibit significant scatter: D/H measurements vary widely, and the theoretically expected anticorrelation between D/H and metallicity remains absent (Friedman et al., 2023). Such patterns challenge the assumption that deuterium must be exclusively primordial in origin.

The case of helium-3 is particularly challenging. In situ measurements from the Ulysses mission, spanning 7 years, ${\text{indicated}}^3$ ${\text{He/}}^4$He number ratios of approximately $2.5\times 10^{-4}$ in the local interstellar cloud—substantially higher than primordial estimates. Even larger ratios were observed in the solar wind, with values of $\sim 4.1\times 10^{-4}$ in slow streams and $\sim 3.3\times 10^{-4}$ in fast streams (Gloeckler and Geiss, 1998). Because a large fraction of these solar wind ions are likely of stellar origin, the anomalously ${\text{elevated}}^3$He abundances imply that significant local production is taking place, thereby challenging the assumption that helium-3 is derived exclusively from primordial nucleosynthesis.

Taken together, these considerations indicate that while the Big Bang model remains the prevailing cosmological paradigm, cumulative tensions—both observational and theoretical—underscore the need for alternative conceptual approaches. The emergent gravity perspective advanced here provides such a foundation: a framework grounded in well-defined dynamical principles, capable of reproducing classical gravitational phenomena, and free of non-physical singularities.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

VG: Writing – original draft, Writing – review and editing. LF: Writing – original draft, Writing – review and editing.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/10.3389/fspas.2025.1731374/full#supplementary-material

References

Baker, W. M., Tacchella, S., Johnson, B. D., Nelson, E., Suess, K. A., D’Eugenio, F., et al. (2025). A core in a star-forming disc as evidence of inside-out growth in the early universe. Nat. Astron. 9, 141–154. doi:10.1038/s41550-024-02384-8

CrossRef Full Text | Google Scholar

Bohr, N. (1928). The quantum postulate and the recent development of atomic theory. Nature 121, 580–590. doi:10.1038/121580a0

CrossRef Full Text | Google Scholar

Born, M., Heisenberg, W., and Jordan, P. (1926). Zur Quantenmechanik II. Z. für Phys. 35, 557–615. doi:10.1007/BF01379806

CrossRef Full Text | Google Scholar

Cini, M. (2003). Field quantization and wave particle duality. Ann. Phys. 305, 83–95. doi:10.1016/S0003-4916(03)00042-4

CrossRef Full Text | Google Scholar

de Broglie, L. (1923). Waves and quanta. Nature 112, 540. doi:10.1038/112540a0

CrossRef Full Text | Google Scholar

de Broglie, L. (1925). Recherches sur la théorie des Quanta. Ann. Phys. Paris. 10, 22–128. doi:10.1051/anphys/192510030022

CrossRef Full Text | Google Scholar

Deal, M., and Martins, CJAP (2021). Primordial nucleosynthesis with varying fundamental constants: solutions to the lithium problem and the deuterium discrepancy. Astronomy and Astrophysics 653, 13. doi:10.1051/0004-6361/202140725

CrossRef Full Text | Google Scholar

Dirac, P. A. M. (1927). The quantum theory of the emission and absorption of radiation. Proc. R. Soc. A 114, 243–265. doi:10.1098/rspa.1927.0039

CrossRef Full Text | Google Scholar

Einstein, A. (1905a). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Ann. Phys. 322, 132–148. doi:10.1002/andp.19053220607

CrossRef Full Text | Google Scholar

Einstein, A. (1905b). Zur Elektrodynamik bewegter Körper. Ann. Phys. 322, 891–921. doi:10.1002/andp.19053221004

CrossRef Full Text | Google Scholar

Eisenstein, D., and Hu, W. (1998). Baryonic features in the matter transfer function. Astrophysical J. 496, 605–614. doi:10.1086/305424

CrossRef Full Text | Google Scholar

Eisenstein, D. J., Zehavi, I., Hogg, D. W., Scoccimarro, R., Blanton, M. R., Nichol, R. C., et al. (2005). Detection of the baryon acoustic peak in the large-scale correlation function of sdss luminous red galaxies. Astrophysical J. 633, 560–574. doi:10.1086/466512

CrossRef Full Text | Google Scholar

Fields, B. D. (2011). The primordial lithium problem. Annu. Rev. Nucl. Part. Sci. 61, 47–68. doi:10.1146/annurev-nucl-102010-130445

CrossRef Full Text | Google Scholar

Frank, L. R., and Galinsky, V. L. (2025). Quantum wave-particle duality effects in the double-slit experiment as a manifestation of criticality of relativistic nonlinear wave fields. doi:10.2139/ssrn.5349255

CrossRef Full Text | Google Scholar

Friedman, S. D., Chayer, P., Jenkins, E. B., Tripp, T. M., Williger, G. M., Hébrard, G., et al. (2023). A high-precision survey of the D/H ratio in the nearby interstellar medium. Astrophys. J. 946, 34. doi:10.3847/1538-4357/acbcbf

CrossRef Full Text | Google Scholar

Galinsky, V. L., and Frank, L. R. (2025). Emergence of quantum wave-particle duality from critical transition of relativistic nonlinear wave fields. doi:10.2139/ssrn.5209094

CrossRef Full Text | Google Scholar

Galinsky, V. L., and Frank, L. R. (2025). Gravity as a nonlinear critical property of relativistic quantum wave fields. doi:10.2139/ssrn.5349260

CrossRef Full Text | Google Scholar

Gloeckler, G., and Geiss, J. (1998). Measurement of the abundance of Helium-3 in the sun and in the local interstellar cloud with SWICS on ulysses. Space Sci. Rev. 84, 275–284. doi:10.1023/A:1005095907503

CrossRef Full Text | Google Scholar

Jacak, J. E. (2025). Possible nonstellar explanation for the unexpected brightness of the earliest galaxies observed by the james webb space telescope. Sci. Rep. 15, 19204. doi:10.103Y8/s41598-025-04418-1

PubMed Abstract | CrossRef Full Text | Google Scholar

Lemaître, G. (1927). Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques. Ann. Société Sci. Brux. 47, 49–59. doi:10.1093/mnras/91.5.483

CrossRef Full Text | Google Scholar

Lemaître, G. (1931). Reprinted in English as “A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extragalactic nebulae”. Mon. Notices R. Astronomical Soc. 91, 483–490. doi:10.1093/mnras/91.5.483

CrossRef Full Text | Google Scholar

Lerner, E. J. (2018). Observations contradict galaxy size and surface brightness predictions that are based on the expanding universe hypothesis. Mon. Notices R. Astronomical Soc. 477, 3185–3196. doi:10.1093/mnras/sty728

CrossRef Full Text | Google Scholar

Melia, F. (2012). Cosmological redshift in friedmann–robertson–walker metrics with constant space–time curvature. Mon. Notices R. Astronomical Soc. 422, 1418–1424. doi:10.1111/j.1365-2966.2012.20714.x

CrossRef Full Text | Google Scholar

Noether, E. (1918). Invariante variationsprobleme. Nachrichten Ges. Wiss. Göttingen, Mathematisch-Physikalische Kl., 235–257.

Google Scholar

Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A., Nugent, P., Castro, P. G., et al. (1999). Measurements of Ω and Λ from 42 high-redshift supernovae. Astrophysical J. 517, 565–586. doi:10.1086/307221

CrossRef Full Text | Google Scholar

Planck CollaborationAghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., et al. (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy and Astrophys. 641, A6. doi:10.1051/0004-6361/201833910

CrossRef Full Text | Google Scholar

Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P. M., et al. (1998). Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astronomical J. 116, 1009–1038. doi:10.1086/300499

CrossRef Full Text | Google Scholar

Riess, A. G., Macri, L. M., Hoffmann, S. L., Scolnic, D., Casertano, S., Filippenko, A. V., et al. (2016). A 2.4% determination of the local value of the hubble constant. Astrophysical J. 826, 56. doi:10.3847/0004-637X/826/1/56

CrossRef Full Text | Google Scholar

Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., and Scolnic, D. (2019). Large Magellanic Cloud cepheid standards provide a 1% foundation for the determination of the hubble constant and stronger evidence for physics beyond ΛCDM. Astrophysical J. 876, 85. doi:10.3847/1538-4357/ab1422

CrossRef Full Text | Google Scholar

Rindler, W. (2006). Relativity: special, general, and cosmological. Oxford, UK: Oxford University Press. doi:10.1093/acprof:oso/9780198567324.001.0001

CrossRef Full Text | Google Scholar

Ryder, L. H. (1996). Quantum field theory. 2 edn. Cambridge, UK: Cambridge University Press. doi:10.1017/CBO9780511813900

CrossRef Full Text | Google Scholar

Schrödinger, E. (1926). An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28, 1049–1070. doi:10.1103/PhysRev.28.1049

CrossRef Full Text | Google Scholar

Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theorie. Sitzungsberichte Königlich-Preussischen Akad. Wiss., 189–196.

Google Scholar

Steane, A. M. (2011). Relativity made relatively easy. Oxford University Press. doi:10.1093/acprof:oso/9780199662852.001.0001

CrossRef Full Text | Google Scholar

Sun, B., and Yan, H. (2025). On the very bright dropouts selected using the james webb space telescope NIRCam instrument. Astrophys. J. 987, 60–619. doi:10.3847/1538-4357/addbe0

CrossRef Full Text | Google Scholar

Taylor, G. I. (1909). Interference fringes with feeble light. Proc. Camb. Philosophical Soc. 15, 114–115.

Google Scholar

Verde, L., Treu, T., and Riess, A. G. (2019). Tensions between the early and late universe. Nat. Astron. 3, 891–895. doi:10.1038/s41550-019-0902-0

CrossRef Full Text | Google Scholar

Weinberg, S. (1972). Gravitation and cosmology: principles and applications of the general theory of relativity. New York: John Wiley and Sons.

Google Scholar

Weinberg, S. (1995). The quantum theory of fields, volume 1: foundations, 1. Cambridge, UK: Cambridge University Press. doi:10.1017/CBO9781139644167

CrossRef Full Text | Google Scholar

Witstok, J., Jakobsen, P., Maiolino, R., Helton, J. M., Johnson, B. D., Robertson, B. E., et al. (2025). Witnessing the onset of reionization through Lyman-α emission at redshift 13. Nature 639, 897–901. doi:10.1038/s41586-025-08779-5

PubMed Abstract | CrossRef Full Text | Google Scholar

Wu, C. (2023). The big bang nucleosynthesis abundances of the light elements using improved thermonuclear reaction rates. General Relativ. Gravit. 55, 48. doi:10.1007/s10714-023-03096-6

CrossRef Full Text | Google Scholar