[Paper Review] On the conversion of rest energy in horizon energy

Hristu Culetu|arXiv (Cornell University)|May 10, 2010

Cosmology and Gravitation Theories3 citations

TL;DR

This paper proposes that a particle's rest energy $mc^2$ converts into horizon energy when its acceleration causes the Rindler horizon to lie within its Compton wavelength, leading to entropy increase via the Unruh effect. The key result is that Verlinde's entropy formula $\Delta S = 2\pi k_B \frac{mc}{\hbar} \Delta x$ holds only when $\Delta x = c^2/a$, reconciling entropic gravity with holography and quantum uncertainty.

ABSTRACT

It is shown that the Verlinde formula for the entropy variation of a holographic screen is a consequence of the conversion of the particle energy in horizon energy. The special role played by the particular displacement $Δx = c^{2}/a$ is emphasized, $a$ being the particle acceleration. Using the Heisenberg Principle we show that the energy on the causal horizon (viewed as a holographic screen) of an inertial observer is proportional to its radius, as for a black hole.

Motivation & Objective

• To resolve the apparent contradiction between Verlinde's linear entropy change $\Delta S \propto \Delta x$ and the standard area law $\Delta S \propto \Delta A$.
• To clarify the physical meaning of the displacement $\Delta x = c^2/a$ in Verlinde's entropic gravity framework.
• To establish a mechanism for rest energy $mc^2$ conversion into horizon energy $E_h$ when the particle is within its Compton wavelength of the horizon.
• To show that the holographic screen's energy and temperature are subject to quantum fluctuations via the Heisenberg uncertainty principle.
• To provide a physical basis for the Unruh effect and its role in entropic gravity, grounded in quantum field theory and relativistic kinematics.

Proposed method

• Analyzes the Rindler horizon formed by a uniformly accelerated particle, with horizon distance $c^2/a$.
• Applies the Unruh temperature $T_U = a\hbar / (2\pi c k_B)$ to the causal horizon of the accelerated observer.
• Uses energy conservation: $mc^2 = \Delta E_h = T_U \Delta S_h$, equating rest energy to horizon energy increase.
• Applies the Heisenberg uncertainty principle to derive quantum fluctuations in energy and temperature on the holographic screen.
• Reconciles Verlinde's entropy formula with the area law by restricting $\Delta x$ to $c^2/a$, the horizon distance.
• Considers the Compton wavelength $\lambda_C = \hbar / (mc)$ as the threshold for particle-horizon merging, triggering energy conversion.

Experimental results

Research questions

• RQ1 Under what conditions does a particle's rest energy convert into horizon energy?
• RQ2 Why does Verlinde's entropy formula $\Delta S = 2\pi k_B \frac{mc}{\hbar} \Delta x$ only hold when $\Delta x = c^2/a$?
• RQ3 How does the Unruh effect and the uncertainty principle constrain the energy and entropy on a holographic screen?
• RQ4 What is the physical significance of the distance $c^2/a$ in the context of entropic gravity and horizon thermodynamics?
• RQ5 Can the apparent conflict between $\Delta S \propto \Delta x$ and $\Delta S \propto \Delta A$ be resolved through quantum and relativistic constraints?

Key findings

• The rest energy $mc^2$ of a particle is converted into horizon energy $E_h$ when the particle's acceleration causes its Rindler horizon to lie within its Compton wavelength.
• Verlinde's entropy formula $\Delta S = 2\pi k_B \frac{mc}{\hbar} \Delta x$ is valid only when $\Delta x = c^2/a$, the distance to the Rindler horizon.
• The energy on the holographic screen is proportional to its radius, mimicking black hole thermodynamics, due to the Unruh effect and quantum fluctuations.
• The horizon temperature $T_U = a\hbar / (2\pi c k_B)$ is derived from quantum field theory in accelerated frames, not classical mechanics.
• The surface gravity $\kappa = g$ at the horizon is equivalent to the Schwarzschild black hole's surface gravity, validating the analogy with black hole thermodynamics.
• For a proton, an acceleration of $a \approx 10^{34}~\text{cm/s}^2$ is required for the horizon to lie at its Compton wavelength, yielding a temperature of $1~\text{GeV}$, consistent with RHIC collision conditions.