Geometric approaches to proving the Riemann hypothesis

Introduction: The Riemann Hypothesis and its Significance

The Riemann Hypothesis, first posited by Bernhard Riemann in his seminal 1859 paper, stands as one of the most profound and challenging unsolved problems in mathematics.[1, 2] At its core, the hypothesis makes a precise claim about the location of the nontrivial zeros of the Riemann zeta function, asserting that they all lie on the critical line in the complex plane where the real part of the complex variable s is equal to one-half.[3, 4, 5, 6] This seemingly simple statement has far-reaching consequences, particularly for our understanding of the distribution of prime numbers, those fundamental building blocks of arithmetic that cannot be expressed as the product of two smaller integers.[1, 3, 4, 5, 6, 7, 8, 9, 10] The hypothesis provides a crucial refinement to the Prime Number Theorem, which gives an asymptotic estimate of the density of primes. If the Riemann Hypothesis is true, it would establish a precise bound on the error term in this estimate, granting us a much deeper insight into the seemingly irregular pattern of prime numbers.[8, 11]

Landau-Lifshitz, Synge, and Fock pseudotensors are different mathematically and physically

This article delves into the intricacies of pseudotensor analysis, focusing on the derivation of Synge and Fock pseudotensors and their comparison with the Landau-Lifshitz pseudotensor. It underscores the differences in the derivation approaches between Synge and Landau-Lifshitz, particularly highlighting the use of covariant derivatives and the simplification of the Riemann tensor. The author further explores the symmetries of the tensors involved and the implications these have on the derived pseudotensors. The article concludes with a comparative analysis of the canonical forms of the Synge and Landau-Lifshitz pseudotensors, noting the differences in terms and signs.

Proof of a tensor trace inequality using Frobenius norm

The text is a formal proof of an inequality involving Frobenius norm, which is a matrix norm defined as the square root of the sum of the absolute squares of its elements. The inequality states that the product of two tensors is greater than or equal to one-third times the square of their trace. The proof uses matrix notation and properties, Cauchy-Schwarz inequality for Frobenius inner product, and operator norms induced by vector norms. The proof is detailed and rigorous, and provides references and explanations for each step. The text also gives some examples and applications of the inequality in physics and geometry.

The inequality in question is

$$\varkappa_{\beta}^{\alpha}\varkappa_{\alpha}^{\beta} \geq \frac{1}{3}\left(\varkappa_{\alpha}^{\alpha}\right)^2$$ where $\varkappa_{\beta}^{\alpha} = \frac{\partial \gamma_{\beta}^{\alpha}}{\partial t}$ is the time derivative of the three-dimensional metric tensor $\gamma_{\beta}^{\alpha}$.

Landau-Lifshitz stress-energy pseudotensor

This Mathematica notebook explains the derivation of the Landau-Lifshitz pseudotensor, a mathematical object used to describe the energy and momentum of a gravitational field in general relativity. Although not a true tensor, as it depends on the choice of coordinates and cannot be localized, the pseudotensor has useful properties such as being symmetric, having zero divergence in flat spacetime, and being conserved in asymptotically flat spacetime. The concept of the stress-energy-momentum (SEM) tensor is introduced, which is a true tensor that combines energy, mass, momentum, and their fluxes and stresses. Derived from the action principle, the SEM tensor satisfies conservation law in flat spacetime. However, in curved spacetime, it is not conserved due to the presence of a gravitational field. A pseudotensor that locally has zero divergence is then found and corrected for curvature effects. The pseudotensor can be obtained in terms of second derivatives of the metric tensor, which characterizes the geometry of spacetime. It is constructed from the Einstein tensor, related to the Ricci tensor and Ricci scalar, which are in turn related to the Riemann tensor containing all information about spacetime curvature. A computer algebra system called xTensor is used to perform calculations and simplify expressions. Another way to obtain the pseudotensor is in terms of first derivatives of the metric tensor. This form is more convenient for some applications such as calculating gravitational radiation emitted by a system. The pseudotensor can also be obtained in terms of first derivatives of the metric density, a modified version of the metric tensor that includes a factor of $\sqrt{-g}$ where $g$ is the determinant of the metric tensor. Preferred by some authors because it makes integration over curved spacetime easier, this factor is related to the Jacobian determinant of coordinate transformations and affects the volume element. Another way to obtain the pseudotensor is in terms of Christoffel symbols, which describe spacetime curvature. Related to first derivatives of the metric tensor, they are used to define covariant derivatives invariant under coordinate transformations. xTensor is used to perform calculations and compare them with EinS. The notebook concludes with remarks on the physical meaning and interpretation of the pseudotensor and mentions alternative definitions and approaches for finding a pseudotensor that can describe energy and momentum of a gravitational field in general relativity.