On the Origin of Gravity and the Laws of Newton

Transcription

1 S.N.Bose National Centre for Basic Sciences,India S.N. Bose National Centre for Basic Sciences, India Dept. of Theoretical Sciences 1 st April, E. Verlinde, arxiv:

2 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.

3 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.

4 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: ), which addressed such topics and found that gravitational force can be thought of as the entropic force.

5 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.

6 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 = 2πk B. Verlinde mimics this fact and also uses these results in addition with the entropic force concept in his analysis!!

7 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 = Ac3 G.

8 Suppose total energy E is distributed among the bits by equipartition law: E = 1 2 Nk BT. Again, if M be the mass emerged in the space-time enclosed by the screen, then: E = Mc 2. N = 2Mc2 k B T. Combining this with holographic principle: T = GM 2πck B R 2.

9 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 : Newton s law of gravitation. R 2 Entropic force is the origin of gravity!!

10 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 = 2πk B mc x. Note: When x = Compton length, Bekenstein s result is recovered. Why proportional to m? One particle (m) m 1 + m Each particle increases entropy for same x. Since entropy and mass both are additive, S m.

11 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 = 1 2 nk BT. Unruh: If a is the local acceleration of this particle then k B T = a 2πc. Combination of these: mc = n a 4πc 2.

12 Substitute in the generalised form of Benensten s result: S n = k B 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.

13 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( ξa ξ a ). 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 = c3 G da. Assume the generalised form of equipartition law: E = 1 2 k B TdN. E = c3 k B 2G A TdA.

14 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. a b : 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 a b = N b b Φ. Generalised Unruh temp.: k B T = a 2πc = Nb b Φ 2πc. Put e Φ (red shift factor) by hand in the numerator, since the temp. is measured at infinity!! k B T = eφ N b b Φ 2πc.

15 Substitute T in E: E = c2 4πG A eφ a ΦdA a. Suppose mass emerged in the space enclosed by the screen is M: E = Mc 2. M = 1 4πG A eφ a ΦdA a. This is the Komar expression for energy. This can be written in another form: M = 1 4πG A aξ b n b da a ; n a : normal to the 3-surface (constant t - surface).

16 Entropic force equation: T S = F a x a ; S = S(x a ). (F a T a S) x a = 0; a S = S x a. If all the directions are independent, then F a = T a S. Use Bekenstein s result and the generalised expression for T: F a = 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.

17 Einstein equations Earlier result: M = 1 4πG A aξ b n b da a ; n a : normal to the 3- surface Σ (constant t- surface). Use Stokes theorem: M = 1 4πG Σ ( a a ξ b )dσ b. Use identity a a ξ b = R ab ξ a : M = 1 4πG Σ R abξ a dσ b. Komar expression for mass: M = 2 Σ (T ab 1 2 Tg ab)ξ b dσ a. Use this 2 Σ (T ab 1 2 Tg ab)ξ b dσ a = 1 4πG Σ R abξ a dσ b. Since it holds for arbitrary screen: R ab = 8πG(T ab 1 2 Tg ab). Einstein equation.

18 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.

19 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.

20 Thank You