7 | | 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 |
8 | | is emergent through a holographic scenario. Gravity is explained as an entropic |
9 | | force caused by changes in the information associated with the positions of ma- |
10 | | terial bodies. A relativistic generalization of the presented arguments directly |
11 | | leads to the Einstein equations. When space is emergent even Newton's law of |
12 | | inertia needs to be explained. The equivalence principle leads us to conclude |
13 | | that it is actually this law of inertia whose origin is entropic. |
| 7 | 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. |
39 | | [[latex(S_{BH}=\frac{kA}{4l_p^2})]] |
| 33 | == Background == |
| 34 | Black hole "thermodynamics" |
| 35 | [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103858973 Bardeen, Cater, and Hawking 1973] |
| 36 | |
| 37 | || [[latex(M)]] || black hole mass || |
| 38 | || [[latex(A)]] || black hole surface area || |
| 39 | || [[latex(g)]] || black hole surface gravity || |
| 40 | || [[latex(\Omega_H)]] || horizon angular velocity || |
| 41 | || [[latex(J_H)]] || black hole angular momentum || |
| 42 | || [[latex(\bar{\mu})]] || red shifted chemical potential || |
| 43 | || [[latex(\bar{\theta})]] || red shifted temperature || |
| 44 | |
| 45 | Correspondence between Temperature and Entropy with surface gravity and area |
| 46 | || Zeroth Law || [[latex(g)]] constant over horizon for stationary black hole || [[latex(T)]] is constant for a body in thermal equilibrium || |
| 47 | || First Law || [[latex(d M=\frac{g}{8\pi}dA+\Omega dJ+\phi dQ)]] || [[latex(dU = TdS - pdV + \mu dN)]] || |
| 48 | || Second Law || [[latex(\delta A >= 0)]] || [[latex(dS >= 0)]] || |
| 49 | || Third Law || [[latex(g > 0)]] || [[latex(T > 0)]] || |
| 50 | |
| 51 | Black hole 'entropy' |
| 52 | [[latex(S = \frac{k_B A}{4 l_p^2})]] |
58 | | == Temperature of accelerated objects == |
| 60 | Bekenstein's thought experiment... |
| 61 | 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 increase in area of |
| 62 | |
| 63 | [[latex(dA >= 8 \pi \frac{G m r}{c^2} )]] |
| 64 | |
| 65 | and an entropy increase of |
| 66 | |
| 67 | [[latex(d S >= 2 \pi k_B \frac{m r c}{\hbar})]] |
| 68 | |
| 69 | If we use the reduced compton radius for the particle we get |
| 70 | |
| 71 | [[latex(d S >= 2 \pi k_B)]] |
| 72 | |
| 73 | ''1 bit of information lost as the particle merges with the black hole'' |
| 74 | |
| 75 | If we assume that this change in energy occurs linearly over the particle's compton radius then we have |
| 76 | |
| 77 | [[latex(dS >= 2 \pi k_B \frac{mc}{\hbar} d x )]] |
| 78 | |
| 79 | and if we calculate the entropic force: |
| 80 | |
| 81 | [[latex(F = T \frac{dS}{dx})]] |
| 82 | |
| 83 | we have [[latex(F = k_B T 2 \pi \frac{mc}{\hbar})]] |
| 84 | |
| 85 | |
| 86 | ''' For a black hole this entropy is presumably stored on the horizon - 1 bit for every square planck length - holographic principle''' |
| 87 | |
| 88 | |
| 89 | === Hawking radiation === |
| 90 | |
| 91 | [[latex(T = \frac{\hbar g}{1\pi k_B c})]] |
| 92 | |
| 93 | where |
| 94 | |
| 95 | [[latex(g = \frac{GM}{r_s^2})]] |
| 96 | |
| 97 | so we have [[latex(F = \frac{mc^4}{4GM} = mg)]] |
| 98 | |
| 99 | |
| 100 | === Unruh effect: an observer in an accelerated frame experiences a non zero vacuum temperature === |
| 101 | |
| 102 | [[latex(T = \frac{\hbar a}{2\pi k_B c})]] |
| 103 | |
| 104 | Verlinde supposes that the entropy is stored on a holographic screen bounding emerged part of space... |