Holomorphic general coordinate invariant modified measure gravitational theory

The relational framework of canonical quantum gravity with nonultralocal constraints is explored. After demonstrating the absence of anomalies, a spatially discretized version of the relational framework is introduced. This allows the application of Lieb-Robinson bounds to on-shell monotonic gauge flow when there is a continuous external ``time'' parameter. An explicit Lieb-Robinson bound is derived for the differential on-shell evolution of the operator norm of the commutator of discretized Dirac observables, demonstrating how a local light conelike causal structure emerges. Ultralocal constraints do not permit such a structure to arise via Lieb-Robinson bounds. Gauge and ($3+1$)-diffeomorphism invariance of the light cone is discussed along with the issues of quantum fluctuations, the nature of the nonlocalities, the spatial continuum limit, and the possible links to noncommutative geometry.

A nondiagonal tetrad field having six unknown functions plus an angle Φ, which is a function of the radial coordinate r, azimuthal angle θ and the polar angle ϕ, is applied to the charged field equations of modified teleparallel theory of gravity. A special nonvacuum solution is derived with three constants of integration. The tetrad field of this solution is axially symmetric and its scalar torsion is constant. The associated metric of the derived solution gives Kerr–Newman spacetime. We have shown that the derived solution can be described by a local Lorentz transformations plus a diagonal tetrad field that is the square root of the Kerr–Newman metric. We show that any solution of general relativity (GR) can be a solution in f(T) under certain conditions.

Recent cosmological observations of unprecented accuracy, by WMAP in particular, have established a `Standard Model' of cosmology, just as LEP established the Standard Model of particle physics. Both Standard Models raise open questions whose answers are likely to be linked. The most fundamental problems in both particle physics and cosmology will be resolved only within a framework for Quantum Gravity, for which the only game in town is string theory. We discuss novel ways to model cosmological inflation and late acceleration in a non-critical string approach, and discuss possible astrophysical tests.

<sec>The black hole information paradox—a fundamental conflict between quantum unitarity and semi-classical gravity—remains unresolved within the framework of quantum gravity. This work builds a conceptual bridge between this modern dilemma and Zeno’s ancient paradox of motion, emphasizing that they all rely on <i>limit processes</i> to reconcile infinite divisibility with finite physical outcomes. Although previous researches have established information conservation in black hole evaporation by copying wormholes, this study transcends mere unitarity verification to answer a deeper question: <i>How do we quantify the statistical evolution and inherent irreversibility of Hawking radiation</i>?</sec><sec> <b>Methods</b> :</sec><sec> <b>i) Gravitational path integral formalism</b> : We compute Rényi entropies <inline-formula><tex-math id="M1">\begin{document}$ S_n = \dfrac{1}{1-n} \ln \text{Tr}(\rho^n) $\end{document}</tex-math></inline-formula> by analytic continuation <inline-formula><tex-math id="M2">\begin{document}$( n \to 1 )$\end{document}</tex-math></inline-formula>, incorporating both disconnected geometries (Hawking saddles) and connected replica wormhole saddles.</sec><sec> <b>ii) JT gravity model</b> : A two-dimensional Jackiw-Teitelboim (JT) gravity coupled to a conformal bath is analyzed, with End-of-the-World (EOW) branes modeling early Hawking radiation states.</sec><sec> <b>iii) Modular thermodynamics</b> : Modular entropy <inline-formula><tex-math id="M3">\begin{document}$ S_m $\end{document}</tex-math></inline-formula> and entanglement capacity <inline-formula><tex-math id="M4">\begin{document}$ C_n $\end{document}</tex-math></inline-formula> are defined as:</sec><sec> <inline-formula><tex-math id="M5">\begin{document}$ S_m = \log\text{Tr}(\rho^n) - n\partial_n\log\text{Tr}(\rho^n), \quad C_n = n^2\partial_n^2\log\text{Tr}(\rho^n),$\end{document}</tex-math></inline-formula>mirroring thermodynamic entropy and heat capacity.</sec><sec> <b>Key results</b> :</sec><sec> <b>i) Page curve restoration</b> : The von Neumann entropy transitions from linear growth (Hawking phase) to zero (wormhole-dominated phase) at the Page time: <inline-formula><tex-math id="M6">\begin{document}$ t_{\text{Page}} \propto {S_{\text{BH}}}/({n\kappa c}), \quad S_{\text{BH}} = {A}/({4G_N}), $\end{document}</tex-math></inline-formula>indicating faster entropy reduction for higher <i>n</i>.</sec><sec> <b>ii) Replica wormhole mechanism</b> : The gravitational path integral reveals a topological transition:</sec><sec> <inline-formula><tex-math id="M7">\begin{document}$ \text{Tr}(\rho^n) \approx \dfrac{kZ_1^n + k^n Z_n}{(kZ_1)^n} \quad \Rightarrow \quad \text{Dominance shift at } k \sim {\mathrm{e}}^{S_0}$\end{document}</tex-math></inline-formula>.</sec><sec>Quantum non-cloning constraints: A reversibility paradox arises: while <i>n</i>-copies -copies of a <i>known</i> state allow full reconstruction, the non-cloning theorem forbids replication of <i>unknown</i> states. This necessitates the <i>a priori</i> existence of multiple replicas to ensure consistency.</sec><sec> <b>iii) Connection to Zeno’s paradox</b> : The resolution of Zeno’s paradox through Newtonian limits <inline-formula><tex-math id="M9">\begin{document}$( v = \lim\limits_{\Delta t \to 0} \Delta x / \Delta t $\end{document}</tex-math></inline-formula>) parall) parallels the replica trick’s role in black hole physics:</sec><sec> <b>iv) Infinite-to-finite mapping</b> : Both employ limit operations to compress divergent processes (infinite steps or entropy growth) into finite observables.</sec><sec> <b>Discrete <i>vs</i>. continuous</b> : Challenge classical spacetime continuity, hinting at holographic or discrete quantum spacetime.</sec><sec> <b>Irreversibility and generalized second law</b> : The relative entropy <inline-formula><tex-math id="M10">\begin{document}$ S(\rho \| \sigma) $\end{document}</tex-math></inline-formula>, defined as:</sec><sec> <inline-formula><tex-math id="M11">\begin{document}$ S(\rho \| \sigma) = \text{Tr}(\rho \log \rho) - \text{Tr}(\rho \log \sigma),$\end{document}</tex-math></inline-formula>monotonically increases during the Hawking-to-wormhole transition, establishing statistical irreversibility in modular space. This generalized second law persists despite microscopic unitarity, analogous to thermodynamic irreversibility in closed systems with the formula: <inline-formula><tex-math id="Z-20250414151430">\begin{document}$ \left(S_n-S_{\rho}\right)-n\left(\left\langle H\right\rangle_n-\left\langle H\right\rangle_{\rho}\right)\geqslant 0 $\end{document}</tex-math></inline-formula>.</sec><sec> <b>Conclusion</b> :</sec><sec>Our results address the black hole information paradox and the irreversible problem through three pillars:</sec><sec> <b>i) Unitarity</b> : Replica wormholes enforce unitarity via non-perturbative spacetime topology changes.</sec><sec> <b>ii) Modular thermodynamics</b> : A framework linking entanglement entropy, relative entropy, and irreversibility.</sec><sec> <b>iii) Replica priority principle</b> : Multiple radiation replicas exist <i>a priori</i>, circumventing quantum cloning constraints.</sec><sec>This work transcends mere information conservation, advancing into the second layer of black hole physics: Characterizing the irreversible evolution of Hawking radiation in modular space.</sec>

The quest to unify quantum mechanics and general relativity remains one of the greatest challenges The quest to unify quantum mechanics and general relativity remains one of the greatest challenges in theoretical physics.While string theory and loop quantum gravity dominate current research,their reliance on abstract geometries or unobserved dimensions limits experimental accessibility.This paper introduces Membrane-Point Theory(MPsT),a paradigm grounded in spacetime-matter symbiosis,offering testable predictions and bridging quantum and classical regimes. At the same time,this research present mathematical formulations,experimental predictions,and pathways for empirical validation,inviting collaborative efforts to refine this framework.

We study black hole radiation inside black holes within the framework of quantum gravity.First, we review on our previous work of a canonical quantization for a spherically symmetric geometry where one of the spatial coordinates is treated as the time variable, since we think of the interior region of a black hole.Based on this formalism, under physically plausible assumptions, we solve the Wheeler-De Witt equation inside the black hole, and show that the mass-loss rate of an evaporating black hole due to thermal radiation is equivalent to the result obtained by Hawking in his semi-classical approach.A remarkable point is that our assumptions make the momentum constraint coincide with the Hamiltonian constraint up to an irrelevant overall factor.Furthermore, for comparison, we solve the Wheeler-De Witt equation outside the black hole as well, and see that the mass-loss rate of an evaporating black hole has the same expression.The present analysis suggests that the black hole radiation comes from the black hole singularity.We also comment on the Birkhoff theorem in quantum gravity.

Complexity will be more and more essential in high-energy physics. It is naturally extended into the very early universe. Considering the universe as a quantum chaotic system, the curvature perturbation of the scalar field is identified with the two-mode squeezed state. By solving the Schrödinger equation, one can obtain the numerical solutions of the angle parameter and squeezing parameter. The solution of the squeezing parameter mainly determines the evolution of complexity. Our numeric indicates that the complexity of the modified dispersion relation will have a non-linear pattern after the horizon exits. Meanwhile, its corresponding Lyapunov index is also larger compared with the standard case. During the inflationary period, the complexity will irregularly oscillate and its scrambling time is also shorter compared with the standard case. Since the modified dispersion relation can be dubbed as the consequences of various frameworks of quantum gravity, it could be applicable to these frameworks. Finally, one can expect the framework of quantum gravity will lead to the fruitful evolution of complexity, which guides us in distinguishing various inflationary models.

We show that recently proposed generalized Chern-Simons action can be identified with the field theory action of a topological point particle. We find the crucial correspondence which makes it possible to derive the field theory actions from a special version of the generalized Chern-Simons actions. We provide arguments that the general coordinate invariance in the target space and the flat connection condition as a topological field theory can be accommodated in a very natural way. We propose series of new gauge invariant observables.

General relativity with fermions has two independent symmetries: general coordinate invariance and local Lorentz invariance. General coordinate invariance is implemented by the Levi-Civita connection and by Cartan's tetrads both of which have as their action the Einstein-Hilbert action. It is suggested here that local Lorentz invariance is implemented not by a combination of the Levi-Civita connection and Cartan's tetrads known as the spin connection, but by independent Lorentz bosons ${{L}^{ab}}_{i}$ that gauge the Lorentz group, that couple to fermions like Yang-Mills fields, and that have their own Yang-Mills-like action. Because the Lorentz bosons couple to fermion number and not to mass, they generate a static potential that violates the weak equivalence principle. If a Higgs mechanism makes them massive, then the static potential also violates the inverse-square law. Experiments put upper bounds on the strength of such a potential for masses ${m}_{L}<20\text{ }\text{ }\mathrm{eV}$. These upper limits imply that Lorentz bosons, if they exist, are nearly stable and contribute to dark matter.

Space-time intervals are the fundamental components of conscious experience, gravity, and a Theory of Everything. Space-time intervals are relationships that arise naturally between events. They have a general covariance (independence of coordinate systems, scale invariance), a physical constancy, that encompasses all frames of reference. There are three basic types of space-time intervals (light-like, time-like, space-like) which interact to create space-time and its properties. Human conscious experience is a four-dimensional space-time continuum created through the processing of space-time intervals by the brain; space-time intervals are the source of conscious experience (observed physical reality). Human conscious experience is modeled by Einstein’s special theory of relativity, a theory designed specifically from the general covariance of space-time intervals (for inertial frames of reference). General relativity is our most accurate description of gravity. In general relativity, the general covariance of space-time intervals is extended to all frames of reference (inertial and non-inertial), including gravitational reference frames; space-time intervals are the source of gravity in general relativity. The general covariance of space-time intervals is further extended to quantum mechanics; space-time intervals are the source of quantum gravity. The general covariance of space-time intervals seamlessly merges general relativity with quantum field theory (the two grand theories of the universe). Space-time intervals consequently are the basis of a Theory of Everything (a single all-encompassing coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe). This theoretical framework encompasses our observed physical reality (conscious experience) as well; space-time intervals link observed physical reality to actual physical reality. This provides an accurate and reliable match between observed physical reality and the physical universe by which we can carry on our activity. The Minkowski metric, which defines generally covariant space-time intervals, may be considered an axiom (premise, postulate) for the Theory of Everything.

The standard way of deriving Euler–Lagrange (EL) equations given a point particle action is to vary the trajectory and set the first variation of the action to zero. However, if the action is (i) reparameterization invariant, and (ii) generally covariant, I show that one may derive the EL equations by suitably nullifying the variation through a judicious coordinate transformation. The net result of this is that the curve remains fixed, while all other geometrical objects in the action undergo a change, given precisely by the Lie derivatives along the variation vector field. This, then, is the most direct and transparent way to elucidate the connection between general covariance, diffeomorphism invariance, and Lie derivatives, without referring to covariant derivative. I highlight the geometric underpinnings and generality of above ideas by applying them to simplest of field theories, keeping the discussion at a level easily accessible to advanced undergraduates. As non-trivial applications of these ideas, I (i) derive the geodesic deviation equation using first order diffeomorphisms, and (ii) demonstrate how they can highlight the connection between canonical and metric stress–energy tensors in field theories.

A fixed-grid local method for 1-D Stefan problems

Volume 6 of The Collected Papers of Albert Einstein includes Einstein's writings during 1914 - 1917, his first three years in Berlin. Einstein moved to Berlin in April 1914 from Zurich where he had been a Professor at his old undergraduate school, The Federal Institute of Technology, since February 1912. In the spring of 1913, Max Planck and Walther Nernst journeyed from Berlin to visit Einstein in Zurich in order to make him an offer that, in the end, he could not refuse: a professorship with no teaching obligations and the directorship of the fledgling Kaiser Wilhelm Institute for Physics. Soon after their arrival in Berlin, the Einsteins separated. Mileva departed for Zurich with their sons Hans Albert and Eduard. At this point Einstein's relationship with his cousin Elsa Löwenthal deepened further. Thus, as the editors put so well, in the period covered in this volume, `Einstein's life and career entered a new phase'. Amongst the key documents in Volume 6 are publications from 1914 in which Einstein concluded that gravitational field equations `cannot possibly be generally covariant'. After having realized his errors, Einstein returned to the theme of general covariance and, in 1915, published three papers in succession in which he developed the generalized theory of relativity. The third paper, published on 25 November 1915 and entitled `The Field Equations of Gravitation', is the capstone of the trio, correcting as it does errors in the previous two. A week prior to its publication Einstein published a result of his new theory: the calculation from the new generally covariant field equations of Mercury's perihelion motion of 43 seconds of arc per century, in agreement with observation. Some time later, Einstein told his former collaborator Adriaan Fokker that upon seeing the result emerge he had heart palpitations. On 8 February 1917 Einstein essentially started the field of modern cosmology with the publication of `Cosmological Considerations in the General Theory of Relativity'. One of the many points of interest in this pioneering paper is Einstein's modification of his field equations with the so-called `cosmological constant' in order to incorporate, as best he could, Machian ideas on effects of distant rotating masses. Besides tidying up general relativity theory with his review paper of 1916, Einstein made new and far reaching contributions to the quantum theory of radiation. In two papers of 1916 he published his A and B coefficients and went on to conclude that atoms need not emit radiation in spherical waves, but in light quanta with a specific direction and definite momentum. The presence of probability in this theory was to Einstein a `weakness'. The eclectic set of publications in Volume 6 include ones on molecular currents in magnets, book reviews, contributions to Bohr's atomic theory and statements on the war, as well as documents relating to Einstein's participation as an expert witness in a patent dispute between the German firm Anschütz & Co. and the American Sperry Gyroscope Company. In the end the court chose to follow the former patent clerk's report and decided to prohibit Sperry Gyroscope from manufacturing gyrocompasses that incorporated methods patented by Anschütz. In summary there is something for everyone in this fascinating collection of papers published by Einstein during 1914 - 1917 and assembled in what we have come to expect as a meticulous scholarly presentation replete with an up to date listing of secondary studies and informative footnotes. We eagerly await further volumes of this truly historic project.

General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas

A search for narrow resonances decaying to an electron and a muon is presented. The mathrm {e}{mu } mass spectrum is also investigated for non-resonant contributions from the production of quantum black holes (QBHs). The analysis is performed using data corresponding to an integrated luminosity of 19.7~text {fb}^text {-1} collected in proton-proton collisions at a centre-of-mass energy of 8~text {TeV} with the CMS detector at the LHC. With no evidence for physics beyond the standard model in the invariant mass spectrum of selected mathrm {e}mu pairs, upper limits are set at 95 % confidence level on the product of cross section and branching fraction for signals arising in theories with charged lepton flavour violation. In the search for narrow resonances, the resonant production of a mathrm {tau } sneutrino in R-parity violating supersymmetry is considered. The mathrm {tau } sneutrino is excluded for masses below 1.28~text {TeV} for couplings lambda _{132}=lambda _{231}=lambda '_{311}=0.01, and below 2.30~text {TeV} for lambda _{132}=lambda _{231}=0.07 and lambda '_{311}=0.11. These are the most stringent limits to date from direct searches at high-energy colliders. In addition, the resonance searches are interpreted in terms of a model with heavy partners of the {mathrm {Z}} boson and the photon. In a framework of TeV-scale quantum gravity based on a renormalization of Newton’s constant, the search for non-resonant contributions to the mathrm {e}{mu } mass spectrum excludes QBH production below a threshold mass M_{mathrm {th}} of 1.99~text {TeV}. In models that invoke extra dimensions, the bounds range from 2.36~text {TeV} for one extra dimension to 3.63~text {TeV} for six extra dimensions. This is the first search for QBHs decaying into the mathrm {e}{mu } final state.