​On the Origin of Gravity and the Laws of Newton

Erik Verlinde

Abstract

Starting from 1st principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force caused by changes in the information associated with the positions of ma- terial bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic.

Basic Idea

• Laws of Newton lead to Black Hole Thermodynamics and the holographic principle
• Holographic principle must lead to Newton's laws
• Newton's law of Gravity (and Gravity in general) is an emergent phenomena much like Hooke's law

Gravity as an entropic force

From the canonical partition function

(T,F)=\int dE dx \Omega(E,x) e^{-\left(E+Fx)/k_BT\right)

we can derive

frac{F}{T}=\frac{\partial S}{\partial x} = -C

= -C T

Black hole entropy

Proportional to surface area of even horizon

_{BH}=\frac{kA}{4l_p^2

where

_p=\sqrt{G\hbar/c^3

(BH does not stand for 'black hole', but for Bekenstein-Hawking)

Entropy is stored on surface ' holographic principle'

Black holes also have a Hawking Temperature

_H=\frac{\hbar c^3}{8 \pi GMk_B

If we assume that the work done by gravity must equal the increase in the Bekenstein-Hawking entropy, then we have

=T_H\frac{\partial S_{BH}}{\partial x

Temperature of accelerated objects

Assume that Delta S = 2\pi k_

when

Delta x = \frac{\hbar}{mc

then

Delta S = 2 \pi k_B \frac{mc}{\hbar}\Delta

and as Unruh showed, an observer in an accelerated frame experiences a temperature

_BT = \frac{1}{2\pi} \frac{\hbar a}{c

If we take this temperature and our entropy formula, and calculate the entropic force we recover

=m

What about gravitational force

If we assume that the number of degrees of freedom is proportional to the area of the enclosed space

=\frac{Ac^3}{G\hbar

and that the temperature is determined by the equipartition rule

=\frac{1}{2}Nk_B

and that the total energy is just the enclosed rest mass

=Mc^

we recover =G\frac{Mm}{R^2