Brainliest Users please help derive the dark matter mass equation MDM(r) enclosed in r
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Explanation:
MDM and Gravitational Thermodynamics Recall the work of Jacobson: Start with the thermodynamic relation de = TdS (for energy E, temperature T and entropy S) in Rindler spacetime. E denotes the integral of the energy momentum tensor (T αβ ) of matter. For T, use the Unruh temperature associated with the local accelerating (Rindler) observer T = a 2πck B. For S, the holographic principle gives S = c3 A 4G Rindler horizon., where A is the area of the Jacobson shows: LHS (of thermodynamic relation) E T αβ ; and S R αβ (the Ricci tensor) such that RHS G αβ, yielding Einstein s equations. We generalize Jacobson s treatment with a consistent modification of the energy momentum tensor so that the fundamental acceleration ( a c introduced by hand in MOND) emerges naturally. We assume (1) the validity of Einstein s theory of gravity; (2) a standard energy-momentum tensor.
10 (1) requires that we preserve the holographic scaling of the area. Then (2), in conjunction with the form of the thermodynamic relation, demands that we change the temperature while preserving the entropy. Our model is given by the thermodynamic relation dẽ = TdS. Note: Since the Unruh temperature knows the inertial properties and is fixed by the background, the additional part of the energy-momentum tensor (coming from a modified temperature) will also know the inertial properties and the background. Consider a local observer with local acceleration a in de Sitter space where a 0 = c 2 Λ/3 = ch 0 like our expanding Universe. The Unruh temperature experienced by this observer (Ref.: Deser & Levin) is T a0 +a = a2 +a 2 0 2πck. B Define the effective (normalized) temperature ( ) T T a0 +a T a0 = 2πck a2 B +a 2 0 a 0 ã 2πck B. Our proposal is the generalization (from Λ = 0 = a 0 to Λ 0 a 0 case): T T, hence a ã; E Ẽ, (hence later: M M).
11 MDM and Entropic Gravity Recall Verlinde s recipe : Verlinde derives (I) Newton s 2nd law F = m a, by using (1) First law of thermodynamics entropic force F entropic = T S x, and invoking Bekenstein s original arguments concerning the entropy S of black mc holes: S = 2πk B x. (2) The formula for the Unruh temperature, k B T = a 2πc, associated with a uniformly accelerating (Rindler) observer. Will generalize the T a relation to T ã. (II) Newton s law of gravity a = GM/r 2 by considering an imaginary quasi-local (spherical) holographic screen of area A = 4πr 2 with temperature T, and using (1) Equipartition of energy E = 1 2 Nk BT with N = Ac 3 /(G ) being the total number of degrees of freedom (bits) on the screen; (2) The Unruh temperature formula and the fact that E = Mc 2. Will generalize the T M relation to T M.