On the Origin of Gravity and the Laws of Newton
Bibhas Ranjan Majhi
S.N.Bose National Centre for Basic Sciences,India
S.N. Bose National Centre for Basic Sciences, India
Dept. of Theoretical Sciences
1st April, 2010.
E. Verlinde, arXiv:1001.0785
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
PLAN OF THE TALK
(i) Why is gravity so special?
(ii) Brief introduction: (a) Entropic force;
(b) Bekenstein’s result.
(iii) Newton’s law of gravitation from entropic force.
(iv) What is inertia in this frame work?
(v) General theory of relativity from entropic force.
(vi) Final remarks.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
Why is gravity so special?
◮
Gravity influences and is influenced by everything that carries
an energy.
◮
It is connected with the structure of space-time.
◮
Universal nature: the basic equations of gravity closely
resemble the law of thermodynamics.
◮
Gravity is considerably harder to combine with QM.
⇒ The quest of unification of it with other forces of nature,
at microscopic level, may therefore not be the right approach.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
◮
People are then trying to find alternative ways to study gravity.
◮
One interesting thought is Gravity may not be the
fundamental force.
Rather Gravity and space-time geometry are emergent.
Such a prediction comes from String theory: AdS/CFT
correspond - a duality between theories that contain gravity
and those that don’t.
⇒ Gravity can emerge from a microscopic description that
doesn’t know about its existence.
Here I shall review a paper by E. P. Verlinde
(arXiv:1001.0785), which addressed such topics and found
that gravitational force can be thought of as the entropic
force.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
Brief Introduction
Entropic force
◮
An effective macroscopic force that originates in a system with
many d.o.f. by statistical tendency to increase its entropy.
◮
Force equation is expressed in terms of entropy differences and
is independent of the details of the microscopic dynamics.
◮
Example: A single polymer molecule made of many monomers
of fixed length immersed into a heat bath of temp. T .
It is then stretched (say by ∆x) to a configuration which is
favored for entropy increase (say by ∆S). Here the
microscopic changes translates to a macroscopic force - the
entropic force (say F ).
The force equation: F ∆x = T ∆S.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
Bekenstein’s result
◮
◮
Consider a particle of mass m of size ∆x is dropped to a black
hole.
~
When ∆x = mc
(one Compton length) = distance of the
particle from the horizon;
it is part of the BH.
Increase in entropy of the BH: ∆S = 2πkB .
◮
Verlinde mimics this fact and also uses these results in
addition with the entropic force concept in his analysis !!
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
Newton’s law of gravitation from entropic force
◮
Consider a spherical holographic screen of radius R with
temp. T .
Area of the screen: A = 4πR 2 .
◮
AdS/CFT: the boundary of the screen can be thought as a
storage device for information.
◮
Let us consider N = total no. of bits on this boundary which
encodes all informations about it.
◮
Holographic principle: N ∝ A;
N=
Bibhas Ranjan Majhi
Ac 3
G~ .
On the Origin of Gravity and the Laws of Newton
◮
Suppose total energy ‘E ’ is distributed among the bits by
equipartition law:
E = 21 NkB T .
◮
Again, if ‘M’ be the mass emerged in the space-time enclosed
by the screen, then:
E = Mc 2 .
⇒N=
◮
2Mc 2
kB T .
Combining this with holographic principle: T =
Bibhas Ranjan Majhi
~GM
.
2πckB R 2
On the Origin of Gravity and the Laws of Newton
◮
◮
Consider a particle of mass m is at a ∆x distance from the
surface of the screen.
Now use this along with Bekenstein’s result in F = T ∆S
∆x :
⇒F =
GMm
:
R2
Newton’s law of gravitation.
Entropic force is the origin of gravity !!
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
What is inertia in this frame work?
◮
Rewrite Bekenstein’s result in the slightly more general form
by assuming that the change in entropy near the screen is
linear in displacement ∆x:
∆S = 2πkB mc
~ ∆x.
Note: When ∆x = Compton length, Bekenstein’s result is
recovered.
◮
Why proportional to m?
One particle (m) ≡ m1 + m2 + ....
Each particle increases entropy for same ∆x.
Since entropy and mass both are additive, ∆S ∝ m.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
◮
Consider a flat screen with temp. T , whose all informations
are encoded with n no. of bits according to equipartition law
of energy.
◮
A particle of mass m is approaching towards the screen and
ultimately absorbed among the bits. Hence mc 2 = 12 nkB T .
◮
Unruh: If a is the local acceleration of this particle then
~a
kB T = 2πc
.
◮
Combination of these:
mc =
n~a
.
4πc 2
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
◮
Substitute in the generalised form of Benensten’s result:
∆S
n
.
= kB a∆x
2c 2
⇒ Acceleration is related to the entropy gradient.
If there is no entropy gradient, then the particle in rest (or
uniform velocity), will remain in rest (or uniform velocity).:
Inertia.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
General theory of relativity from entropic force
◮
Consider a static background with global time-like Killing
vector ξ a .
◮
Generalised Newton’s potential: Φ =
1
2
ln(−ξ a ξa ).
e Φ : the redshift factor; relates the local time coordinate to
that at a reference pt. with Φ = 0.
◮
Put a holographic screen at const. Φ.
◮
From the earlier argument: dN =
◮
Assume the
R generalised form of equipartition law:
1
E = 2 kB TdN.
◮
E=
c 3 kB
2G ~
R
c3
G ~ dA.
A TdA.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
◮
Consider a particle of mass m and size ∆x on the elementary
surface dA.
◮
u a : 4-velocity of the particle = e −Φ ξ a .
ab : 4-acceleration = u a ∇a u b = e −2Φ ξ a ∇a ξ b = ∇a Φ.
a = acceleration opposite to N a (outward normal to dA)
= N b ab = N b ∇b Φ.
~a
2πc
~N b ∇b Φ
2πc .
◮
Generalised Unruh temp.: kB T =
◮
Put e Φ (red shift factor) by hand in the numerator, since the
temp. is measured at infinity !!
kB T =
=
~e Φ N b ∇b Φ
.
2πc
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
◮
Substitute T in E :
R Φ
c2
a
E = 4πG
A e ∇a ΦdA .
◮
Suppose mass emerged in the space enclosed by the screen is
M:
⇒ E = Mc 2 .
◮
M=
1
4πG
R
A
e Φ ∇a ΦdAa .
This is the Komar expression for energy.
This can be written in another form:
R
1
b
a
a
M = − 4πG
A ∇a ξb n dA ; n : normal to the 3-surface
(constant t - surface).
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
◮
Entropic force equation: T ∆S = Fa ∆x a ; S = S(x a ).
⇒ (Fa − T ∇a S)∆x a = 0; ∇a S =
∂S
∂x a .
If all the directions are independent, then
Fa = T ∇a S.
◮
Use Bekenstein’s result and the generalised expression for T :
Fa = −me Φ ∇a Φ.
This is the generalised expression for the entropic force. It is
required to keep the particle at fixed position near the screen
as measured at infinity.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
Einstein equations
◮
R
1
b
a
a
Earlier result: M = − 4πG
A ∇a ξb n dA ; n : normal to the 3surface Σ (constant t- surface).
◮
1
Use Stokes’ theorem: M = − 4πG
R
Σ (∇
◮
Use identity ∇a ∇a ξb = −Rab ξ a :
R
1
a
b
M = 4πG
Σ Rab ξ dΣ .
◮
Komar expression for mass: M = 2
◮
Use this ⇒ 2
R
Σ (Tab
R
a∇
a ξb )dΣ
Σ (Tab
− 21 Tgab )ξ b dΣa =
b.
− 12 Tgab )ξ b dΣa .
1
4πG
R
Σ Rab ξ
a dΣb .
Since it holds for arbitrary screen: Rab = 8πG (Tab − 12 Tgab ).
Einstein equation.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
FINAL REMARKS
◮
Gravity may not be fundamental. Rather it is an emergent
phenomenon.
◮
Then question is: What is the origin of gravity?
◮
This analysis shows that entropic force may a candidate to
describe gravity.
◮
Using the entropic force concept and Bekenstein’s argument
the Newton’s law of gravity was derived. It reveled that the
gravitation force is the entropic force.
◮
Also the inertia can be described within this approach. Here
inertia was defined as the absence of entropic gradient.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
◮
The general theory of relativity was also discussed within this
approach. First the expression for Komar mass was derived.
◮
An expression for the generalised entropic force was also given.
◮
It was shown that Einstein equation can be obtained.
◮
As a final comment I want to mention the advantage of this
approach over the other approaches.
It shed some light on the origin of gravity.
Also there is another advantage which is not mentioned in the
paper that one can obtain the geodesic equation which is
necessary to describe the causal structure of space-time.
In absence of entropic force: ∇a Φ = 0 ⇒ u a ∇a u b = 0. This
is the geodesic equation.
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton
Thank You
Bibhas Ranjan Majhi
On the Origin of Gravity and the Laws of Newton