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Claus Gerhardt

Publications and source records attributed to Claus Gerhardt.

At least 19 recordsLinked to original sources
arXiv2026

Quantizing the exterior region of a Schwarzschild-AdS black hole leads to a resolution of the information paradox on a quantum level

We quantize the exterior region of a Schwarzschild-AdS black hole using our model of quantum gravity. The resulting hyperbolic equation is solved by products of temporal eigenfunctions $w_i$, the eigenvalues of which all have multiplicity one, and spatial eigendistributions $v_{ij}$ having the same eigenvalues but with multiplicities $1\le m_i$, where the $m_i$ could in principle be arbitrarily large. Regarding only the exterior region, there was no guidance how to determine the values of the $m_i$. However, considering also the quantization of the interior region, where the same question did not arise since the $m_i$ could be chosen by maximizing the value, it seemed logical to choose the same values, too, in the exterior case. Since the eigenvalues in the interior are the same because the temporal Hamiltonian is the same in both cases, this choice defined a unitary equivalence between the respective Hilbert spaces and the respective Hamiltonians. Hence, there is no information paradox on a quantum level.

gr-qc↗
arXiv2025

Extending the solutions and the equations of quantum gravity past the big bang singularity

In [8] we recently proved that in our model of quantum gravity the solutions to the quantized version of the full Einstein equations or to the Wheeler-DeWitt equation could be expressed as products of spatial and temporal eigenfunctions, or eigendistributions, of self-adjoint operators acting in corresponding separable Hilbert spaces. Moreover, near the big bang singularity we derived sharp asymptotic estimates for the temporal eigenfunctions. In this paper we show that, by using these estimates, there exists a complete sequence of unitarily equivalent eigenfunctions which can be extended past the singularity by even or odd mirroring as sufficiently smooth functions such that the extended functions are solutions of the appropriately extended equations valid in $\R[]$ in the classical sense. We also use this phenomenon to explain the missing antimatter.

physics.gen-ph↗
arXiv2023

The quantization of gravity: The quantization of the full Einstein equations

We quantized the full Einstein equations in a globally hyperbolic spacetime $N=N^{n+1}$, $n\ge 3$, and found solutions of the resulting hyperbolic equation in a fiber bundle $E$ which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second order ordinary differential equation in $\mathbb{R}_+$. In case $n\ge 17$ and provided the cosmological constant $\Lambda$ is negative the temporal eigenfunctions are eigenfunctions of a self-adjoint operator $\hat H_0$ such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space.

physics.gen-ph↗
arXiv2022

A unified quantization of gravity and other fundamental forces of nature

We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Let us\ann{as} abbreviate the spatial Hamilton functions of the Standard Model by $H_{SM}$ and the Hamilton function of gravity by $H_G$. Working in a fiber bundle $E$ with base space $\socc=\R[n]$, where the fiber elements are Riemannian metrics, we can express the Hamilton functions in the form $H_G+H_{SM}=H_G+t^{-\frac23}\tilde H_{SM}$ if $n=3$, where $\tilde H_{SM}$ depends on metrics $\s_{ij}$ satisfying $\det{\s_{ij}}=1$. In the quantization process, we quantize $H_G$ for general $\s_{ij}$ but $\tilde H_{SM}$ only for $\s_{ij}=\de_{ij}$ by the usual methods of QFT. Let $v$ \resp $\psi$ be the spatial eigendistributions of the respective Hamilton operators, then, the solutions $u$ of the Wheeler-DeWitt equation are given by $u=wv\psi$, where $w$ satisfies an ODE and $u$ is evaluated at $(t,\de_{ij})$ in the fibers.

gr-qc↗
arXiv2021

The quantization of gravity: Quantization of the Hamilton equations

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form $-\D u=0$ in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler-DeWitt metric provided $n\not=4$. Using then separation of variables the solutions $u$ can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space $SL(n,\R[])/SO(n)$. Since one can define a Schwartz space and tempered distributions in $SL(n,\R[])/SO(n)$ as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.

gr-qc↗
arXiv2019

CMC foliations of open spacetimes asymptotic to open Robertson-Walker spacetimes

We consider open globally hyperbolic spacetimes $N$ of dimension $n+1$, $n\ge 3$, which are spatially asymptotic to a Robertson-Walker spacetime or an open Friedmann universe with spatial curvature $\tilde\kappa = 0,-1$ and prove, under reasonable assumptions, that there exists a unique foliation by hypersurfaces of constant mean curvature and that the mean curvature function $\tau$ is a smooth time function if $N$ is smooth. Moreover, among the Friedmann universes which satisfy the necessary conditions are those that reflect the present assumptions of the development of the universe.

math.DG↗
arXiv2017

A partition function for Schwarzschild-AdS and Kerr-AdS black holes and for quantized globally hyperbolic spacetimes with a negative cosmological constant

We apply quantum statistics to our quantized versions of Schwarzschild-AdS and Kerr-AdS black holes and also to the quantized globally hyperbolic spacetimes having an asymptotically Euclidean Cauchy hypersurface by first proving, for the temporal Hamiltonian $H_0$, that $e^{-βH_0}$, $β>0$, is of trace class and then, that this result is also valid for the spatial Hamiltonian $H_1$, which has the same eigenvalues but with larger multiplicities. Since the lowest eigenvalue is strictly positive the extension of $e^{-βH_1}$ to the corresponding symmetric Fock space is also of trace class and we are thus able to define a partition function $Z$, the operator density $ρ$, the entropy $S$, and the average energy $E$. We prove that $S$ and $E$ tend to infinity if the cosmological constant $Λ$ tends to $0$ and vanish if $|Λ|$ tends to infinity. We also conjecture that $E$ is the source of the dark matter and that the dark energy density is a multiple of the eigenvalue of $ρ$ with respect to the vacuum vector which is $Z^{-1}$.

physics.gen-ph↗
arXiv2017

The quantization of a Kerr-AdS black hole

We apply our model of quantum gravity to a Kerr-AdS spacetime of dimension $2 m+1$, $m\ge2$, where all rotational parameters are equal, resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS spacetime can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system the $r$ variable by $\rho=r^2$ such that the extended spacetime has a timelike curvature singularity in $\rho=-a^2$.

gr-qc↗
arXiv2017

The quantum development of an asymptotically Euclidean Cauchy hypersurface

In our model of quantum gravity the quantum development of a Cauchy hypersurface is governed by a wave equation derived as the result of a canonical quantization process. To find physically interesting solutions of the wave equation we employ the separation of variables by considering a temporal eigenvalue problem which has a complete countable set of eigenfunctions with positive eigenvalues and also a spatial eigenvalue problem which has a complete set of eigendistributions. Assuming that the Cauchy hypersurface is asymtotically Euclidean we prove that the temporal eigenvalues are also spatial eigenvalues and the product of corresponding eigenfunctions and eigendistributions, which will be smooth functions with polynomial growth, are the physically interesting solutions of the wave equation. We consider these solutions to describe the quantum development of the Cauchy hypersurface.

math-ph↗
arXiv2016

The quantization of a black hole

We apply our model of quantum gravity to an AdS black hole resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur.

gr-qc↗
arXiv2016

Deriving a complete set of eigendistributions for a gravitational wave equation describing the quantized interaction of gravity with a Yang-Mills field in case the Cauchy hypersurface is non-compact

In a recent paper we quantized the interaction of gravity with a Yang-Mills and Higgs field and obtained as a result a gravitational wave equation in a globally hyperbolic spacetime. Assuming that the Cauchy hypersurfaces are compact we proved a spectral resolution for the wave equation by applying the method of separation of variables. In this paper we extend the results to the case when the Cauchy hypersurfaces are non-compact by considering a Gelfand triplet and applying the nuclear spectral theorem.

gr-qc↗
arXiv2016

A unified field theory II: Gravity interacting with a Yang-Mills and Higgs field

We quantize the interaction of gravity with a Yang-Mills and Higgs field using canonical quantization. Similar to the approach in a previous paper we discard the Wheeler-DeWitt equation and express the Hamilton constraint by the evolution equation of the mean curvature of the hypersurfaces in the foliation defined by the Hamiltonian setting. Expressing the time derivative of the mean curvature with the help of the Poisson brackets the canonical quantization of this equation leads to a wave equation in $Q=(0,\infty)\times \cal{S}_o$, where $\cal{S}_o$ is one of the Cauchy hypersurfaces in the Hamiltonian setting. The wave equation describes the interaction of an arbitrary Riemannian metric in $\cal{S}_o$ and a given Yang-Mills and Higgs field. If the metric is complete $Q$ is globally hyperbolic. In case $\cal{S}_o$ is compact we also prove a spectral resolution of the wave equation and establish sufficient conditions guaranteeing a mass gap.

gr-qc↗
arXiv2015

A unified field theory I: The quantization of gravity

In a former paper we proposed a model for the quantization of gravity by working in a bundle $E$ where we realized the Hamilton constraint as the Wheeler-DeWitt equation. However, the corresponding operator only acts in the fibers and not in the base space. Therefore, we now discard the Wheeler-DeWitt equation and express the Hamilton constraint differently, either with the help of the Hamilton equations or by employing a geometric evolution equation. There are two modifications possible which both are equivalent to the Hamilton constraint and which lead to two new models. In the first model we obtain a hyperbolic operator that acts in the fibers as well as in the base space and we can construct a symplectic vector space and a Weyl system. \nd In the second model the resulting equation is a wave equation in $\so \times (0,\infty)$ valid in points $(x,t,\xi)$ in $E$ and we look for solutions for each fixed $\xi$. This set of equations contains as a special case the equation of a quantized cosmological Friedmann universe without matter but with a cosmological constant, when we look for solutions which only depend on $t$. Moreover, in case $\so$ is compact we prove a spectral resolution of the equation.

gr-qc↗
arXiv2014

Inverse curvature flows in hyperbolic space

We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in $C^\infty$ to a sphere.

math.DG↗
arXiv2014

A unified quantum theory II: gravity interacting with Yang-Mills and spinor fields

We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Using canonical quantization we obtain solutions of the Wheeler-DeWitt equation in a vector bundle and the method of second quantization leads to a symplectic vector space $(V,\om)$ and a corresponding CCR representation for the bosonic components and a CAR representation for the fermionic part. The solution space of the Wheeler-DeWitt equation is invariant under gauge transformations and under isometries in the spacelike base space $\so$ of a given Riemannian metric $ρ_{ij}$. We also define a net of local subalgebras which satisfy four of the Haag-Kastler axioms.

math-ph↗