Dark Energy, Black Hole Entropy, and the First Precision Measurement in Quantum Gravity

arXiv:1003.4811v1 [hep-th] 25 Mar 2010 Dark Energy, Black Hole Entropy, and the First Precision Measurement in Quantum Gravity Niayesh Afshordi∗ Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5,Canada and Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada (Dated: November 13, 2021) The two apparently distinct phenomena of dark energy (or late-time cosmic acceleration) and quantum gravity dominate physics on extremely low, and extremely high energies, but do not seem to have any apparent empirical connection. Nevertheless, the two have a theoretical con- nection, through the cosmological constant problem. I argue that the finite temperature quantum gravitational corrections to black hole entropy yields a pressure for the gravitational vacuum (or gravitational aether). Assuming that the relative corrections are linear in horizon temperature (i.e. are suppressed by one power of Planck energy), the pressure is comparable to that of dark energy for astrophysical black holes. This implies that the observation of late-time cosmic acceleration may have provided us with the first precision measurement of quantum gravity, i.e. that of black hole entropy. The discovery of the late-time acceleration of cosmic expansion at the turn of the century [1, 2], and its inter- pretation as being due to a mysterious dark energy com- ponent, sent shock waves through the theoretical physics community, and is arguably the most puzzling aspect of modern cosmology. What puts this problem at the heart of modern physics and cosmology is that the simplest form of dark energy, or a cosmological constant, is predicted in the standard model of particle physics, but with a value that is some sixty orders of magnitude larger than the observed dark energy density. This is known as the cosmological con- stant problem [3], which suggests a yet-unknown connec- tion between the largest and smallest physical scales ever probed, and thus its resolution could revolutionize our understanding of fundamental physics. In this letter, I will argue that such UV-IR connection may indeed exist through horizons of black holes. To see this though, we should first review black hole thermodynamics. A mysterious discovery of the 20th century was that the classical general relativistic black holes appear to have a thermodynamic entropy propor- tional to their horizon area [4, 5]. In fact, one can write analogs of all the laws of thermodynamics for the evolu- tion of black holes [6]. In particular, the first law for a general Kerr-Newman black hole takes the form: dm = THdS + ΩdJ + ΦdQ, (1) where m is the black hole mass (or ADM energy), while TH = κ 2π, S = A 4 (2) are horizon temperature and entropy, which in turn de- pend on surface gravity, κ, and area A of the black hole horizon. Moreover, Ω, J, Φ, and Q are the black hole angular frequency, angular momentum, electrostatic po- tential, and charge respectively. For simplicity, we will focus on Schwarzschild black holes, for which we have: TH = 1 8πm, S = 1 16πT 2 H , (3) while all the other constants vanish. However, note that the results below are only expected to change by dimen- sionless factors of order unity, if we relax this assumption. Also, note that throughout this letter, unless mentioned otherwise, we use Planck units: ¯h = c = G = 1. Interestingly, Jacobson has even argued that Einstein’s equations can be derived from the first law of thermody- namics for horizon areas, suggesting that the full Gen- eral Relativity (GR), and not just black holes, might be a thermodynamic description of a more fundamental the- ory [7]. More recently, Verlinde provided a less technical account of this result for Newtonian gravity [8], which was the intellectual motivation for this letter. One thermodynamic quantity that is notably missing from the first law (Eq. 1) is pressure. The reason is that the asymptotic space-time of Kerr-Newman black holes is Minkowski, which implies zero pressure, if one uses Einstein’s equations. However, as I argue below, there might be reasons to think that this may not be accurate in a UV-complete quantum gravitational framework. An intriguing approach to quantum gravity, known as emergent gravity, postulates that rather than being a fun- damental symmetry of nature, Lorentz symmetry is an emergent phenomenon at low energies, and the funda- mental theory does not have this symmetry [24]. Some examples of this construction are [9, 10]. Also see [11, 12] for experimental/astrophysical bounds on implications of such theories in particle physics. Breaking Lorentz symmetry introduces a preferred frame of reference for the laws of physics. A covariant description of emergent gravity would promote this pre- ferred frame into a fluid, which acts as a modern-day version of gravitational aether [13]. For example, the 2 Horava-Lifshitz construction of emergent gravity [9] re- duces to GR plus an in

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