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

Remarks on the cosmological constant and minimal acceleration

As Lineweaver explained, because our Universe is expanding at an accelerating rate, our Universe has a cosmic event horizon (CEH).  

Its an event horizon Jim, but not as we know it...Image credit: DALL.E2 by SR Anderson

Events beyond the CEH will never be observed. The CEH is also the source of de Sitter radiation, which has a specific temperature $T_{dS}$. It is the minimum possible temperature of the Universe, and, it is not absolute zero (zero Kelvins). Numerically $T_{dS} \approx 2.4 \times 10^{-30}K$, the universal minimum (black body) `absolute cold' local temperature of the future dS state. As a comparison, in 2020 the NASA Cold Atom lab was able to cool an atom to a record low $\sim 2 \times 10^{-7}K$. 

Now, in any theory one may think of temperature as an energy, and from the semi-classical Unruh relationship, temperature $\sim$ acceleration. Therefore:  

\begin{equation}
\notag
E_{dS} = T_{ds} \ k_B= \frac {\hslash a}{2\pi c}
\end{equation}

What we get from Unruh is $a=cH$, the universal background (local) minimum acceleration. $H$ is the future Hubble constant (as you probably know, this constant is not actually...constant). You could also express this as saying that our present Universe has a de Sitter attractor in our infinite far future.

Gibbons taught us that a theory with a minimum length should have a maximal acceleration and a maximal temperature. Now, we all know that in classical General Relativity (GR) there is no such thing as a minimum length. However, in most approaches to quantum gravity, which includes semi-classical approximations such as Hawking's black hole temperature and the Unruh relationship, there is such a beast. 

 

(Lambda, the symbol used for the cosmological constant) 

Here of course, we are talking about the local minimum universal acceleration, which implies the existence of a maximum length scale, being the de Sitter characteristic length $l_{\Lambda}$ the future cosmic event horizon radius. Numerically $l_{\Lambda} \sim 16 \ Gly$. You can also write $l_{\Lambda} =\sqrt{ {3}/{\Lambda}}$, showing that the cosmological constant is the only parameter in dS space.  

Generated image 

The de Sitter horizon

A future de Sitter universe has a cosmic event horizon (CEH):

$
l_\Lambda = \sqrt{\frac{3}{\Lambda}} \sim 16 , \text{Gly}.
$

  • The associated Gibbons-Hawking temperature is:

$
T_{dS} = \frac{\hbar H}{2 \pi k_B}, \quad H = \sqrt{\frac{\Lambda}{3}}.
$

  • The corresponding minimum acceleration:

$
a_{\rm min} = c H.
$ Yes, this is similar to the MOND acceleration, but no-one is quite sure what this coincidence means. 

CEH energy scale

The energy enclosed in the de Sitter horizon (using the Misner-Sharp energy in spherical symmetry) is approximately:

$
E_{\Lambda} = \frac{c^4}{2G} l_\Lambda.
$

  • This comes from the analogy with the Schwarzschild radius: $(R_s = 2 G M / c^2 \implies M = c^2 R_s / (2G))$.

  • Here $(R_s \to l_\Lambda)$ gives the “mass-energy” associated with the CEH.

So numerically:

$
E_\Lambda = \frac{c^4}{2 G}  l_\Lambda.
$

Cosmic maximum power

If we imagine emission at the fastest rate allowed (light crossing time across the CEH):

$
t_\Lambda = \frac{l_\Lambda}{c}.
$

Then the average power is:

$
P_{\rm max}^{\rm CEH} = \frac{E_\Lambda}{t_\Lambda} = \frac{\frac{c^4}{2 G} l_\Lambda}{l_\Lambda / c} = \frac{c^5}{2 G}.
$

It’s exactly the same as the black-hole horizon maximum power! The maximum luminosity allowed by causality in GR is set by horizon formation, whether it’s a black hole or the cosmic event horizon.
 

 Now consider - Start with three horizon quantities:

  • Horizon entropy: $S=\dfrac{\pi k_Bc^3R^2}{G\hbar}$
  • de Sitter temperature: $T_{\rm dS}=\dfrac{\hbar c}{2\pi k_BR}$
  • Maximum entropy throughput, meaning one horizon's information per radial light-crossing time: $\dot S_{\max}\equiv\dfrac{S}{R/c} =\dfrac{\pi k_Bc^4R}{G\hbar}$

Therefore,

$$\boxed{ P_{\max}^{\rm dS} =T_{\rm dS}\dot S_{\max} =\frac{c^5}{2G} }$$

For a same-radius black hole, $T_{\rm BH}=\frac12T_{\rm dS}$, because its surface gravity is $\kappa_{BH}=\frac{c^2}{2R}$, while for de Sitter, $\kappa_{dS}=\frac{c^2}{R}$.  The Schwarzschild Smarr relation supplies the missing factor of two: $E_{\rm BH}=2T_{\rm BH}S_{\rm BH}$. Hence,

$$\boxed{ P_{\max}^{\rm BH} =2T_{\rm BH}\dot S_{\max} =T_{\rm dS}\dot S_{\max} =\frac{c^5}{2G} }.$$

The black hole is half as hot, but both horizons have the same maximum radial-throughput power.  

The underlying relation $P=T\dot S$ is the rate form of the Clausius identity $\delta Q=T dS$ used in Jacobson's thermodynamic derivation of Einstein's equation. Saturating it with the horizon entropy and radial causal-time bounds gives $c^5/(2G)$. Schiller's related maximum-force formulation instead uses the full diameter and obtains $c^5/(4G)$.

Finally, each radial thermal channel obeys

$$P_{\rm ch}\propto T^2, \qquad \dot S_{\rm ch}\propto T \quad\Longrightarrow\quad \dot S_{\rm ch}\propto\sqrt{P_{\rm ch}}.$$

Thus the CEH's radiative entropy flow is one-dimensional in the same channel-theoretic sense identified by Bekenstein and Mayo, 2001  Black holes are one-dimensional.  








 

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