Extended uncertainty principle for rindler and cosmological horizons

Eur. Phys. J. C (2019) 79:716 https://doi.org/10.1140/epjc/s10052-019-7232-3 Regular Article - Theoretical Physics Extended uncertainty principle for rindler and cosmological horizons Mariusz P. Da̧browski1,2,3,a , Fabian Wagner1,b 1 Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland 2 National Centre for Nuclear Research, Andrzeja Sołtana 7, 05-400 Otwock, Poland 3 Copernicus Center for Interdisciplinary Studies, Szczepańska 1/5, 31-011 Kraków, Poland Received: 14 July 2019 / Accepted: 14 August 2019 / Published online: 27 August 2019 © The Author(s) 2019 Abstract We find exact formulas for the Extended Uncertainty Principle (EUP) for the Rindler and Friedmann horizons and show that they can be expanded to obtain asymptotic forms known from the previous literature. We calculate the corrections to Hawking temperature and Bekenstein entropy of a black hole in the universe due to Rindler and Friedmann horizons. The effect of the EUP is similar to the canonical corrections of thermal fluctuations and so it rises the entropy signalling further loss of information. 1 Introduction The Heisenberg Uncertainty Principle (HUP) constitutes a cornerstone of the quantum physics and is rooted in the quantisation of electromagnetic radiation resulting in photons. It introduces the Planck fundamental constant h̄ which together with two other fundamental constants – the speed of light c, and Newton’s gravitational constant G – form the Planck (natural) scale in physics.One of the units of this scale is the Planck length l p = G h̄/c3 . In fact, the HUP does neither take into account quantum gravity effects of the photon interaction nor the curvature of space-time. However, widely spread by the considerations of superstring theory [1–5] and loop quantum gravity [6,7], these two phenomena have been gradually taken into account resulting in the Generalised Uncertainty Principle (GUP) [8–20] and the Extended Uncertainty Principle (EUP) [21–24], or were even put together as the Generalised Extended Uncertainty Principle (GEUP) [25–29]. Yet, a different approach to the problem using the so-called qmetric was also considered [30,31]. a e-mail: b e-mail: In terms of the standard deviations of position and momentum σx2 = x̂ 2  − x̂2 (1) σ p2 =  p̂ 2  −  p̂2 (2) and in the context of space-times with external horizons, the most general asymptotic form of the GEUP which includes both the GUP and the EUP can be formulated as [25,26]   α0 l 2p 2 h̄ β0 2 σx σ p ≥ (3) 1 + 2 σ p + 2 σx , 2 rhor h̄ where x is the position, p the momentum, l p plays the role of the minimum length, rhor is the radius of the horizon which is introduced by the background space-time, and α0 , β0 are dimensionless parameters. An interesting property of (3) which relates to superstring theory [32] is the invariance of it under the (duality) transformations √ √ α0 l p β0 h̄ lH −1 σp , σx ↔ √ σx−1 , (4) σp ↔ √ α0 l p lH h̄ β0 just for the GUP sector (β0 = 0) and the EUP sector (α0 = 0) respectively, and √ √ α0 l p β0 σp ↔ σx (5) lH h̄ for both sectors simultaneously. It is interesting to note some general relations between black hole and cosmological horizons [33]. There have been various derivations of the GUP which account for the gravitational part of the interaction between an electron and a photon including simple Newtonian arguments [25]. The changes caused by classical gravity could in principle have a great deal of implications. An example is the disappearance of the Chandrasekhar limit [34] under 123 716 Page 2 of 8 Eur. Phys. J. C (2019) 79:716 the GUP and its recovery under the application of the EUP [35]. In fact, it emerges that the curvature effect is missing in the GUP and once the EUP is applied, it helps to recover the limit which is an observational fact. However, the most important consequence of the GUP is its influence on the Hawking temperature [36,37] and Bekenstein entropy [38]. In fact, it modifies the black hole evaporation process which ceases under GUP conditions leaving a remnant which stores information [39,40] giving a possible solution to the information puzzle [41]. There have been some attempts to bound the GUP parameter α0 in (3) observationally[42–46] including the issue of its positivity or negativity [14,42,43,43,44,47–50]. In fact, the microcanonical corrections reduce the entropy and so the parameter α0 seems to be negative while for the canonical corrections it should be positive [18,19,25,48]. It is also worth mentioning that there is some analogy between the GUP in particle physics and the solid state phenomena in graphene which could pave the path to experimentally support this idea [51]. A very nice, quite rigorous derivation of the EUP based on space-times of constant curvature was presented in Ref. [52] and it directly shows that even classical gravity alters the uncertainty relation. This had been suggested earlier in the context of geometry and topology [53–55]. In particular, if we make any measurement, we are certain that the particle we measure is located inside its own universe which thus restricts the uncertainty. The paper is organized as follows. In Sect. 2 we present the method of derivation of the EUP from geometrical arguments. In Sects. 3 and 4 we present the application of the method for Rindler space and Friedmann universes sliced in a way that the cosmological horizon appears manifestly. Section 5 describes a way to interpret some of the results as manifestations of Hawking radiation. In Section 6 we discuss the influence of cosmological horizons onto the Hawking radiation and Bekenstein entropy of a local black hole. 2 Background geometry determined EUP The underlying idea of our approach is that the measurement of momentum depends on a given space-time background [52,54,55]. In order to measure the momentum one needs to consider a compact domain D with boundary ∂ D characterised by the geodesic length x around the location of the measurement with Dirichlet boundary conditions. Thus the wavefunction is confined to D. Note that D lies on a spacelike hypersurface. Thus like other quantum gravity effects this method is observer dependent. The method then reduces to the solution of an eigenvalue problem for the wave function ψ: ˆ + λψ = 0 ψ 123 (6) inside D with the requirement that ψ = 0 on the boundary ˆ is the Laplace-Beltrami ∂ D, λ denotes the eigenvalue, and  operator. As we can choose ψ to be real (the eigenvalue problem is the same for the real and the imaginary part), the Dirichlet boundary conditions assure that  p̂ = 0, and so one can obtain the uncertainty of a momentum p̂ = −i h̄∂i measurement as σp =   p̂ 2  = h̄   ˆ −ψ||ψ ≥ h̄ λ1 (7) where λ1 denotes the first eigenvalue. Multiplying by x, the uncertainty relation corresponding to this momentum measurement is obtained. It was found for Riemannian 3manifolds of constant curvature K that [52]  σ p x ≥ π h̄ 1 − K (x (...truncated)