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Showing posts from November, 2022

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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Our cosmic event horizon on a string

A Cosmic Stringy Adventure !   Requiring the reduced Compton wavelength to fit within a cosmic event horizon results in a "minimum mass" (this is not a minimum particle mass) $m_s = \frac{\hbar}{c l_{\Lambda}}$.   In this post  we interpret $m_s$ as a horizon energy gap and model the associated gravitational scale as a  lightlike energy flow along an effective classical string.   McDormand with a cosmic light-like energy flow $m_{s}$ along a string of minimal  radius $2L_p$, giving a centripetal force $F_{max}=c^4/4G$.    Mass-Energy Flux Along the String   Let's think about that string for a bit. In fact, a great number of physicists have spent their entire careers tied up  unravelling  string theory . For a classical string associated with Nambu-Goto action, the the string tension $T_G$ is a local force, or energy per unit length (dimensions $MLT^{-2}$): \begin{equation} \notag T_G = \frac{1}{2\pi \alpha_G \prime} \e...
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The Cosmological Constant Problem Revisited

  You will find some people claiming this is a non-problem , however, the CCP is actually one of the two great naturalness problems in modern physics.   $$\Lambda \simeq 1.3 \times 10^{-52}\ \mathrm{m}^{-2}$$ yet naïve quantum field theory (QFT) estimates of vacuum energy overshoot the observed value by an astonishing factor of order $10^{121}$. This discrepancy is often expressed as the ratio between the Planck energy density and the observed dark energy density: $$\frac{\rho_{\text{Planck}}}{\rho_\Lambda} \sim 10^{121}$$ where $$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G}, \qquad \rho_{\text{Planck}} = \frac{c^5}{\hbar G^2}$$   At face value, this looks like a total failure of theoretical physics. However, this interpretation rests on an assumption that turns out to be wrong :   that vacuum  degrees of freedom  scale with volume .  Where the naive QFT estimate comes from   Now, Rugh and Zinkernagel's   paper from 2000 works out the Q...
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The fabric of space-time: stiffer than steel or weaker than jello?

  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, McDonald quantified the classical stiffness of space-time via an effective  Youngs Modulus .  Classical answer: Youngs Modulus of space-time $\sim$ 20 orders of magnitude greater than steel.   DALL.E2 depiction of classical space-time....
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Zitter at the Edge of Spacetime

 In a previous post we introduced the idea that our current Universe has boundary conditions.     We also showed a diagram similar to Figure 1 below. Except here, we are once again thinking about the future dS state . Figure 1 . For an observer at O inside the cosmic event horizon (CEH) with radius $l_{\Lambda}$, the universe can be divided into two sub-vacuums, $(A)$ inside the CEH, and $(B)$, outside. The horizon surface $\Sigma$ has entanglement entropy $S_{dS}$ and rest energy $E_H$   Figure 2. The maximum entropy of the Universe (credit: Lineweaver ).   Now, a comoving volume of the Universe, when considered together with its associated cosmic event horizon, forms a thermodynamically closed system obeying the generalised second law, $$dS_{\text{bulk}} + dS_{\text{horizon}} \ge 0.$$ The maximum entropy of a closed system, in this case  ( Figure 2 ) with $L=2 \pi l_{\Lambda}$, the circumference of a circle with radius  $l_{\Lambda}$, ...
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The mimumim length-scale: max GR tension smacking the quantum force scale!

In a previous post, we showed how, if the ultimate fate of our Universe is space empty of matter...but not quite....of energy (a de Sitter space), then this future cosmic event horizon (CEH) radius of our current, quantum, Universe set a natural maximum length-scale. Amazingly, the future CEH radius also defines the cosmological constant $\Lambda$.     For an observer at O inside the cosmic event horizon (CEH) with radius $l_{\Lambda}$, the universe can be divided into two sub-vacuums, $(A)$ inside the CEH, and $(B)$, outside. The horizon surface $\Sigma$ has entanglement entropy $S_{dS}$ and rest energy $E_H$ What about a Universal minimum length-scale ?  Quantum localisation plus gravitational collapse gives $\ell_{min}\gtrsim L_p$ (Planck length $L_p$) being a standard minimal length  argument. Now, we can find this scale falls out, without needing to be assumed.     Horizon action   Barrow and Gibbons  proposed , from pure classical GR...
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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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Our cosmic event horizon on a string

A Cosmic Stringy Adventure !   Requiring the reduced Compton wavelength to fit within a cosmic event horizon results in a "minimum mass" (this is not a minimum particle mass) $m_s = \frac{\hbar}{c l_{\Lambda}}$.   In this post  we interpret $m_s$ as a horizon energy gap and model the associated gravitational scale as a  lightlike energy flow along an effective classical string.   McDormand with a cosmic light-like energy flow $m_{s}$ along a string of minimal  radius $2L_p$, giving a centripetal force $F_{max}=c^4/4G$.    Mass-Energy Flux Along the String   Let's think about that string for a bit. In fact, a great number of physicists have spent their entire careers tied up  unravelling  string theory . For a classical string associated with Nambu-Goto action, the the string tension $T_G$ is a local force, or energy per unit length (dimensions $MLT^{-2}$): \begin{equation} \notag T_G = \frac{1}{2\pi \alpha_G \prime} \e...
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