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The Thermodynamic Uncertainty Relation for Spacetime

This theorem proposes a fundamental equation of state for the de Sitter vacuum, linking its thermodynamic energy directly to its quantum-gravitational uncertainty. Let the de Sitter cosmological horizon be described by: 1. Its thermodynamic energy \(E_{\text{th}}\), the unique energy that satisfies the first law of thermodynamics, \(dE_{\text{th}} = T\,dS\). 2. Its intrinsic quantum position uncertainty (or "fuzziness"), \(\Delta x\), arising from quantum gravitational effects. The rate of change of the thermodynamic energy with respect to the area of this quantum uncertainty, \(A_{\text{unc}} = \Delta x^2\), is a universal constant equal to the Planck pressure, \(P_{\text{Pl}}\). Mathematically: $$\frac{dE_{\text{th}}}{d(\Delta x^2)} = \frac{\hbar c^7}{G^2} \equiv P_{\text{Pl}}$$   Derivation The theorem is derived by combining three principles: 1. Thermodynamic Energy: The Brown-York energy is the correct thermodynamic potential, given as a function of the de Sitter radius ...
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Blurring the horizon - the quantum width of the cosmic event horizon

A 2021 paper by Zurek applied a random walk argument to a black hole horizon.   Credit, Zurek, 2021 Zurek called this a blurring of the horizon (a fuzzy, or uncertain horizon), and also went through some equivalent derivations, which basically supported the idea this length scale is the quantum uncertainty in the position of the BH horizon, aka a dynamic quantum width of an event horizon (a concept which would therefore also apply to the universe's own CEH). The Bekenstein-Hawking entropy gives the number of quantum degrees of freedom that can fluctuate. Below, we step out our own cosmic de-Sitter derivation of the random walk argument, obtaining the same result as Zurek did, so its certainly correct! \begin{equation} \Delta x^2 =2 DT \end{equation} In this equation, $\Delta x$ is the position uncertainty, $D$ is the Einstein diffusion coefficient and $T$ is the time between measurements (aka the relaxation time).    So, lets look at the Einstein diffusion co...
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Our cosmic event horizon on a string

In this post, we introduced the idea that in the presence of a positive cosmological constant, there is a minimum (local) mass $m_{GR}$   McDormand swinging a cosmic light-like mass-energy $m_{GR}$, with a "cosmic string" of radius $l_{\Lambda}$, giving a centripetal force $F_{local}=m_{GR} \ c^2/l_{\Lambda}$.    Let's think about that string for a bit. In fact, a great number of physicists have spent their entire careers tied up unraveling string theory . For a classical string, which lives in D = 10 dimensions, 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 \prime} \end{equation}$\alpha \prime$ is the Regge slope parameter, set here with dimensions of inverse force.  The wavelength of the stringy mass-energy standing wave (such that it does not interfere with itself), is the circumference of the circle, and we know it moves...
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A cosmological constant solution via the holographic principle

In a previous post we mentioned how the discrepancy in the classical and quantum estimates of the stiffness of space-time was another version of the cosmological constant problem (CCP). 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.  From observations, the cosmological constant $\Lambda \sim 1.3\times10^{-52} \ m^{-2}$. The so-called $10^{121}$ crisis can be expressed as the ratio of the Planck density (from QFT) and the actual observed dark energy density: $\rho_{Planck}/ \rho_{\Lambda}$.  \begin{equation} \rho_{\Lambda} = \frac{\Lambda c^2}{8 \pi G} \ \end{equation}\begin{equation} \rho_{Planck} = \frac{c^5}{\hbar G^2} \end{equation}  The QFT result comes from quantising a 'particle in a box'. However, in 1998 the CKN bound was proposed. CKN realised that if you put particles in a box and heat them, you can only increase their energy so much before the box collapses int...
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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 Youngs Modulus .  Classical answer: Youngs Modulus of space-time $\sim$ 20 orders of magnitude greater than steel.   DALL.E2 depiction of classical space-time. (Credit SR Anderso...
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