Skip to main content

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)=...
Post a Comment

The Factor of Two in de Sitter Thermodynamics


At the de Sitter horizon, using the stated Brown–York convention

$E_{\rm BY}=|E_{\rm active}|=2E_{\rm MS}=2TS$ links four distinct descriptions of de Sitter equilibrium. The factor of two is generated by the negative pressure of vacuum energy.

Under standard static-patch conventions,

$$\boxed{ E_{\rm BY} = |E_{\rm active}| = 2E_{\rm MS} = 2k_BT_{\rm dS}S_{\rm dS} = \frac12N_{\rm sur}k_BT_{\rm dS} }$$

where

$$N_{\rm sur}=\frac{A}{\ell_p^2}=4S_{\rm dS}.$$

The key is vacuum pressure. For $p=-\rho c^2$, the active gravitational density is

$$\rho+\frac{3p}{c^2}=-2\rho.$$

Consequently, the magnitude of the active gravitational energy inside the de Sitter horizon is

$$|E_{\rm active}|=2\rho Vc^2=2E_{\rm MS}.$$

With the stated Brown–York subtraction and orientation convention, this is also the horizon-limit Brown–York energy. The factor of two therefore represents the gravitational contribution of negative pressure, not merely an unspecified boundary correction.

The same quantity has a thermodynamic interpretation:

$$E_{\rm BY}=2k_BT_{\rm dS}S_{\rm dS} =\frac12N_{\rm sur}k_BT_{\rm dS}.$$

It is the equipartition energy of the horizon's $A/\ell_p^2$ degrees of freedom.

Meanwhile, the Misner–Sharp energy satisfies

$$S_{\rm dS} = \frac{2\pi E_{\rm MS}\ell}{\hbar c}.$$

This has the algebraic form of Bekenstein saturation, although applying the ordinary Bekenstein bound to a de Sitter horizon requires interpretive care.

The relation extends naturally to spatially flat FLRW cosmology. Defining

$$K_A=(\rho+3p)V_A, \qquad \epsilon=-\frac{\dot H}{H^2},$$

one obtains

$$\boxed{ K_A+2E_{\rm MS} = \frac{\dot S_A}{2\pi} = 2\epsilon E_{\rm MS} }.$$

Exact de Sitter space is therefore the fixed point

$$\dot H=0 \iff \rho+p=0 \iff \dot S_A=0 \iff K_A=-2E_{\rm MS}.$$

Away from that point, $\epsilon$ measures both horizon-entropy evolution and the departure from the active-energy factor-of-two relation.

The result provides a unified picture: Misner–Sharp energy sets the bulk scale, negative vacuum pressure doubles it into the magnitude of the active gravitational energy, Brown–York energy represents that quantity at the boundary, and horizon equipartition supplies its thermodynamic interpretation.

Comments

Post a Comment

Popular posts from this blog

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...
Post a Comment
Read more

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 ...
Post a Comment
Read more

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...
Post a Comment
Read more