Extended uncertainty principle for rindler and cosmological horizons

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.

On the uncertainty principle in Rindler and Friedmann spacetimes

“…In contrast to the result in [1], the acceleration does not explicitly occur in the invariant representation (4). More precisely, let us discuss the statement (18) of Dabrowski and Wagner in [1]. Therein, it is proposed to express the position uncertainty by the coordinate ∆x in the direction of acceleration and the associated 1-dimensional domain of position uncertainty is taken to be the interval I = [l 0 − ∆x, l 0 + ∆x], with l 0 = 2c 2 /α.…”

“…with acceleration α describing a boost in the x-direction as applied to the Minkowski space, c the speed of light, and y and z denoting the components of the metric perpendicular to the x direction in Rindler space [1]. Without loss of generality, we have chosen the boost of acceleration in the direction of x. Let…”

“…1. For instance, the vertical distance between the center (dot) and the north pole is different from the coordinate distance of (0, 0, 0) to the center, although the geodesic distance of both is identical to r. So, if one wanted to express the position uncertainty relative to the x-direction (as has been done in [1]), then one must take into account that the vertical coordinate distance is dependent on which position the circle is located in this direction. Actually, such a dependency is somewhat cumbersome and hard to handle by the observer.…”

“…Therein, it is proposed to express the position uncertainty by the coordinate ∆x in the direction of acceleration and the associated 1-dimensional domain of position uncertainty is taken to be the interval I = [l 0 − ∆x, l 0 + ∆x], with l 0 = 2c 2 /α. At this point we want to mention that the corresponding (coordinate) distance d x := 2∆x in this direction is not equal to the geodesic diameter which is d = 2 r. Our starting point to understand the inequality (18) of [1] is to express the geodesic radius r of (4) in terms of ∆x. Therefore, we first define the north pole and the south pole x ± = l 0 ± ∆x of the ball in Rindler space.…”

“…Obviously, this is because they are not written in an invariant representation, but are related to a special coordinate system. In the following two sections, we will discuss the statements made in [1] and how they can be derived from (4) and (5) by coordinate transformation. We show that their non-zero minimum standard deviations of the momentum can be removed by standard arguments of geodesic completion.…”