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The Riemann Hypothesis as a Stability Principle

  The Riemann Hypothesis can be reformulated as an infinite sequence of explicit positivity tests. This is not a proof. It turns a statement about complex zeros into inequalities that can be calculated, falsified at finite order, and compared with operator or physical models.   The completed zeta function is $$\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s),$$ with $$\Xi(t)=\xi\left(\frac12+it\right).$$ The function $\Xi$ is even and real on the real axis. A nontrivial zero $\rho=\beta+i\gamma$ corresponds to $$t_\rho=\gamma-i\left(\beta-\frac12\right).$$ Therefore $$t_\rho\in\mathbb R \quad\Longleftrightarrow\quad \beta=\frac12,$$ and hence $$\boxed{\mathrm{RH}\quad\Longleftrightarrow\quad\text{every zero of }\Xi\text{ is real}.}$$ Folding the Zeros Because $\Xi$ is even, write $$\Xi(t)=\sum_{n\ge0}c_nt^{2n}.$$ Now define $$G(z)=\frac{\Xi(i\sqrt z)}{\Xi(0)}.$$ Although this contains $\sqrt z$, the even expansion makes $G$ entire: $$G(z)=...
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De Sitter Horizon: Surface Tension, Laplace Pressure, and What They Actually Mean

  In the static patch of de Sitter spacetime , the cosmological horizon behaves thermodynamically in ways closely analogous to a physical interface. One can assign it entropy, temperature, and even an effective surface tension . Remarkably, the familiar Young–Laplace pressure relation from surface physics appears naturally at the horizon.      In the static patch of de Sitter spacetime, the cosmological horizon admits a useful surface-thermodynamic description. It has an effective surface tension and pressure satisfying a Young–Laplace-like relation. Use $G=c=\hbar=k_B=1$. The metric is $$ds^2=-f(r),dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2, \qquad f(r)=1-\frac{r^2}{L^2}, \qquad L=\sqrt{\frac3\Lambda}.$$ The horizon is at $r=L$. 1. Horizon Data The area, entropy, and temperature are $$A=4\pi L^2, \qquad S=\frac A4=\pi L^2, \qquad T=\frac{1}{2\pi L}.$$ The Misner–Sharp energy is $M(r)=\frac r2(1-f)=\frac{r^3}{2L^2}$, so at the horizon $M(L)=L/2$. This equals both th...
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Riemann Zeros from the Edge of Spacetime

This not a proof of the Riemann Hypothesis (RH) .  Its purpose is to explore a simple idea: that horizons supply a natural quantum clock. Black-hole horizons and the horizon of de Sitter space share a simple property: they stretch a light-ray coordinate exponentially. That stretching supplies a natural quantum clock. If the clock is prepared in one special state, its return amplitude is the Riemann $\Xi$-function. The real zeros of $\Xi$ then mark exact times when the state has become orthogonal to where it began. This idea does not turn the zeros into energy levels. It gives them a different physical meaning: they are horizon times.  A Horizon That Stretches Light Let $V>0$ be an affine coordinate along a horizon light ray, and let $\kappa>0$ be the surface gravity. Horizon time acts as $$V(t)=e^{\kappa t}V.$$ A wavefunction $\psi(V)\in L^2(\mathbb R_+ dV)$ has norm $\int_0^\infty |\psi(V)|^2 dV$. Probability conservation requires $$(U_t\psi)(V) = e^{-\kappa t/2}\p...
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Blurring the horizon - the quantum width of the cosmic event horizon

A  paper by Zurek applied a random walk argument to a black hole horizon. Credit, Zurek, 2022  Zurek  ( Snowmass White Paper: Observational Signatures of Quantum Gravity )  called this a blurring of the horizon: a fuzzy, or uncertain horizon, and went through derivations supporting the idea that this length scale is the quantum uncertainty in the position of the black hole horizon: a dynamic quantum width of an event horizon. This is a concept which fundamentally applies to the Universe's own Cosmic Event Horizon (CEH).   Below, we step out our own derivation of the random walk argument, giving us the phenomenological signature Zurek found: $$\Delta x^2 = 2DT$$ In this equation, $\Delta x$ is the position uncertainty, $D$ is the Einstein diffusion coefficient, and $T$ is the time between measurements (the relaxation time). Einstein Diffusion Coefficient So, let's look at the Einstein diffusion coefficient. $\mu = v_d / F$ is the mobility and $v_d$ is...
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The Cosmic Strange Metal?

        Strange metals have no long-lived particle-like excitations. They relax near the fastest rate allowed by temperature and can scramble information rapidly. Space itself is not a strange metal. However, the de Sitter stretched horizon may share its Planckian transport pattern: $$\boxed{\text{The cosmic horizon may be a Planckian quantum fluid.}}$$ Horizon temperature For de Sitter space, $$R_H=\frac{c}{H}, \qquad T_{dS}=\frac{\hbar H}{2\pi k_B}.$$ Two related thermal timescales are $$\tau_T=\frac{\hbar}{k_BT_{dS}}=\frac{2\pi}{H}, \qquad \tau_{\rm ch}=\frac{\hbar}{2\pi k_BT_{dS}}=\frac1H.$$ Thus the horizon has no macroscopic clock other than $H^{-1}$, up to numerical factors. Einstein gravity dissipation In the membrane description, $$\eta_{\rm mem}=\frac{c^3}{16\pi G}, \qquad s_{\rm mem}=\frac{k_Bc^3}{4G\hbar}.$$ Therefore $$\boxed{ \frac{\eta_{\rm mem}}{s_{\rm mem}} =\frac{\hbar}{4\pi k_B}. }$$ This is the Einstein-gravity membrane value, also ...
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The fabric of space-time: stiffer than steel or weaker than jello?

  The smart money is on cosmic jello, not comic steel    The notorious rubber sheet analogy of spacetime teaches one concept and once concept only: Mass-energy causes curvature of space-time.   When a gravitating mass recedes from a region of space-time the curvature diminishes. The field equations of General Relativity don’t have an explicit term for this elastic property, but the framework as a whole does have that property. As very large mass-energies are required to generate gravitational waves (ripples in space-time), the elastic property of space-time is generally regarded as very stiff One other interesting consideration here is that elasticity is an emergent phenomenon. There is a great deal of interest in the idea that gravity is similarly emergent .   In 2018, Kirk T.  McDonald quantified the classical stiffness of space-time via an effective  Youngs Modulus  experienced by a gravitational wave frequency of 100 Hz. $$ Y_{\mathrm{GW}} =...
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