Dhairya Shah

We formulate a condition for the admissibility of parametrized trajectories in relativistic spacetime based on compatibility between worldline ordering and causal structure. A trajectory represents the history of a single physical system only if its parameter ordering agrees with the causal ordering defined by the spacetime metric. It follows that any trajectory for which this condition fails cannot be realized as a future directed causal worldline in a spacetime. Such trajectories therefore do not admit an inter pretation as physically consistent evolutions of a single system, but correspond to a failure of representation within a causal manifold. We further indicate how this condition extends to quantum descriptions, where it acts as a selection rule on admissible histories contributing to single-system amplitudes

We propose a covariant extension of General Relativity in which the growth of spacetime curvature dynamically activates an additional scalar degree of freedom. The activation is generated by a curvature–dependent coupling term of the form α R φ², which modifies the effective mass of the scalar field and causes it to become dynamical only when the curvature approaches a critical scale. As a result, the scalar field backreacts on the Einstein equations and produces a self–regulating behavior that prevents the divergence of curvature invariants. The theory remains local, covariant, and second order in derivatives, and therefore avoids the higher–derivative instabilities that occur in many modified gravity models. We derive the field equations from a variational principle, analyze conservation laws, and study the behavior of the theory in cosmology, gravitational collapse, and static spherically symmetric spacetimes. In homogeneous cosmology the model allows finite–curvature bounce solutions, while in gravitational collapse the scalar field generates an effective pressure that can prevent the formation of curvature singularities. Black hole interiors may approach a regular finite–curvature core while the weak–field limit remains consistent with General Relativity. The curvature–activated mechanism provides a minimal classical framework in which singularities are avoided through dynamical backreaction rather than through higher–derivative corrections or nonlocal modifications. The model can be interpreted as an effective theory valid below a finite curvature scale and may be relevant for the study of the strong–gravity regime and the conceptual foundations of spacetime singularities.

We present a curvature-activated effective extension of Einstein gravity where nonlinear geometric corrections regulate high-curvature regimes while preserving a consistent General Relativity (GR) limit. Formulated within a covariant effective field theory framework, the model introduces a scalar sector coupled to a bounded activation function $F(R)$, ensuring that additional gravitational dynamics decouple in low-curvature environments. We derive the complete gravitational field equations from the action principle and establish mathematical consistency through the contracted Bianchi identities.A rigorous 3+1 ADM decomposition is performed, and the Hamiltonian constraint structure is analyzed to confirm the closure of the constraint algebra and the stability of the dynamical degrees of freedom. We explicitly derive the linear perturbation equations for scalar, vector, and tensor sectors, demonstrating stable propagation of gravitational waves on cosmological backgrounds. Furthermore, we examine black-hole interior geometries, showing how the activation mechanism regulates the growth of curvature invariants to potentially resolve classical singularities. Our results suggest that curvature-activated dynamics provide a robust, ghost-free framework for addressing the high-curvature limits of Einstein gravity within an effective field theory paradigm.

This research presents a revolutionary solution to the "Big Bang" problem by proving that the universe did not begin with a point of infinite density, but rather with a stable, non-singular "Shiva-Bounce." By evolving the theory of Mimetic Gravity with a specialized protective potential, we establish a framework where the early universe is immune to the mathematical breakdowns and instabilities that plague standard Einsteinian physics. Central to this discovery is the concept of a 5D Tesseract Embedding. We demonstrate that our 4D universe is the boundary of a higher-dimensional 5D shape. This geometric relationship naturally explains the "missing" energy in the cosmos: we derive a sequestering constant showing that roughly 99.3% of the manifold's energy is tucked away in the 5D core. This provides a purely geometric origin for Dark Matter, removing the need for undiscovered particles. Furthermore, because this 5D manifold is spatially closed and finite, the theory predicts a strictly limited number of galaxies—approximately 2 trillion—offering a final resolution to the age-old problem of why the night sky is dark and why the universe appears so uniform. This work provides a complete, stable, and finite alternative to the theory of Cosmic Inflation.

We introduce a cosmological framework termed Hexagonal Phase Cosmology, developed as a dynamical extension of World Consistency Theory. In this approach the vacuum of the universe is modeled by six interacting scalar phase fields forming a discrete hexagonal phase lattice that governs the large-scale evolution of spacetime. The phase variables evolve under a periodic potential with multiple degenerate minima, producing a resonance structure in the vacuum configuration space. The corresponding field equations are derived within a modified gravitational framework incorporating mimetic gravity and Born–Infeld curvature corrections. In a homogeneous Friedmann–Lemaˆıtre–Robertson– Walker cosmological background the phase lattice contributes an effective vacuum sector whose energy density influences the expansion dynamics of the universe. The coupled phase dynamics exhibits a global Lyapunov attractor that drives the system toward progressive phase synchronization over extremely long cosmological timescales. The discrete phase structure generates a finite configuration space containing N = 66 = 46656 resonance states. Using the present Hubble expansion rate as the characteristic phase rotation scale, the complete cosmic resonance cycle is estimated to be Tcycle ≈ 4.3 × 1015 years. At the terminal alignment state the vacuum potential approaches its global minimum, potentially halting cosmic expansion and triggering a nonsingular cosmological bounce regulated by Born–Infeld curvature dynamics. The phase lattice naturally produces an effective vacuum energy component capable of mimicking dark-energy behavior, while small oscillations of the phase variables may generate dark-matter-like excitations. Possible observational implications include modifications to the late-time expansion history and the presence of an ultra–low-frequency stochastic gravitational-wave background. Hexagonal Phase Cosmology therefore provides a geometric realization of global consistency prin ciples in cosmology and offers a novel framework for cyclic cosmic evolution driven by vacuum phase

We construct a minimal covariant extension of Einstein gravity in which extreme curvature growth is dynamically regulated by a curvature-activated scalar field. The additional field remains inactive in weakly curved regimes, ensuring exact recovery of General Relativity, but becomes dynamically relevant when curvature approaches high values. The field couples directly to the Ricci scalar and is sourced by a local scalar constructed from matter degrees of freedom. We derive the field equations from a variational principle and demonstrate, using a homogeneous collapse toy model, that the cou pled system admits solutions with bounded curvature invariants. The framework preserves locality, causality, and stress-energy conservation, providing an effective classical mechanism for avoiding singular curvature divergences.

The nature of time, causality, and temporal displacement remains a foundational problem in modern physics. While relativity treats time as a geometric dimension and quantum interpretations introduce branching histories, the question of whether a physical system can access its own future without violating causal structure remains unresolved. We propose Worldline Consistency Theory (WCT), a principle asserting that any forward-proper-time displacement of a physical system that bypasses causal continuity cannot occur within a single spacetime history. Such a displacement necessitates a transition between distinct universes, where the arrival event is already embedded in the past of the destination universe. This framework preserves local causality, eliminates temporal paradoxes, and reinterprets “time travel” as inter-universal relocation rather than intra-universal motion. The theory is formulated using standard relativistic causal structure and does not posit a mechanism for temporal displacement, focusing solely on consistency conditions. “We present a causal consistency condition on worldlines in Lorentzian spacetime, without introducing new dynamics.” ________________________________________.