Comments on "On the Origin of Gravity and the Laws of Newton", by Erik Verlinde

TL;DR

This paper critiques Verlinde's entropic gravity proposal, clarifying the roles of acceleration and entropy, and aligns the energy on a holographic screen with black hole physics, providing a nuanced perspective on emergent gravity.

Contribution

It offers a critical analysis of Verlinde's framework, correcting misconceptions about temperature and entropy, and connects holographic screens to black hole energy relations.

Findings

Unruh temperature cannot be directly associated with the bits on the screen as Verlinde proposed.

The shift in the holographic screen approaches c^2/a, aligning entropy variations with Gao's calculations.

Energy on the causal horizon scales with its radius, similar to black hole properties.

Abstract

We argue that the relativistic Unruh temperature cannot be associated with the bits on the screen, in the form considered by Verlinde. The acceleration $a$ is a scalar quantity (the modulus of the acceleration four vecor) and not a vector. When the mass $m$ approaches the holographic screen, viewed as a stretched horizon, the shift $\Delta x$ from Verlinde's Eq. (3.15) becomes $c^{2}/a$ and the entropy variation equals $(1/2) k_{B} \Delta N$, in accordance with Gao's calculations. 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.