​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 material 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

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

=T\frac{\partial S}{\partial x

which for polymers gives Hooke's law

= -C T

Background

Black hole "thermodynamics" ​Bardeen, Cater, and Hawking 1973

	black hole mass
	black hole surface area
	black hole surface gravity
Omeg	horizon angular velocity
	black hole angular momentum
ph	electric potential
	electric charge

Correspondence between Temperature and Entropy with surface gravity and area

Zeroth Law	constant over horizon for stationary black hole	is constant for a body in thermal equilibrium
First Law	M=\frac{g}{8\pi}dA+\Omega dJ+\phi d	U = TdS - pdV + \mu d
Second Law	delta A >=	S >=
Third Law	>	>

Black hole 'entropy' should be proportional to area. Dimensional analysis gives:

_{BH} = \frac{k_B A}{4 l_p^2

where

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

and S_BH stands for Bekenstein-Hawking entropy (not Black Hole).

Bekenstein's thought experiment… When lowering a particle into a black hole, information about the particle is lost, so entropy of black hole should increase. This corresponds to an minimal increase in area of

A >= 8 \pi \frac{G m r}{c^2}

and an entropy increase of

S >= 2 \pi k_B \frac{m r c}{\hbar

If we use the reduced compton radius for the particle we get

S >= 2 \pi k_

1 possible microstate is lost to system as the particle merges with the black hole

If we assume that this change in energy occurs linearly over the particle's compton radius then we have

S >= 2 \pi k_B \frac{mc}{\hbar} d x

and if we calculate the entropic force:

= T \frac{dS}{dx

we have = k_B T 2 \pi \frac{mc}{\hbar

For a black hole this entropy is presumably stored on the horizon - 1 bit for every square planck length - holographic principle

Hawking radiation

= \frac{\hbar g}{1\pi k_B c

where

= \frac{GM}{r_s^2

so we have = \frac{mc^4}{4GM} = m

Unruh effect: an observer in an accelerated frame experiences a non zero vacuum temperature

= \frac{\hbar a}{2\pi k_B c

which leads to an entropic force = k_B T 2 \pi \frac{mc}{\hbar} = m

Newton's 2nd law!

What about gravitational force around a non-black hole?

What should be the temperature of the screen?

If we assume that the number of degrees of freedom is proportional to the area of the enclosed space like an event horizon

=\frac{A}{l_p^2

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 get

=\frac{2 l_p^2 Mc^2}{k_B A} = \frac{G \hbar M}{k_B 2\pi R^2 c

and

=G\frac{Mm}{R^2