A Generalized Uncertainty Principle in Quantum Gravity

We discuss a Gedanken experiment for the measurement of the area of the apparent horizon of a black hole in quantum gravity. Using rather general and model-independent considerations we find a generalized uncertainty principle which agrees with a similar result obtained in the framework of string theories. The result indicates that a minimum length of the order of the Planck length emerges naturally from any quantum theory of gravity, and that the concept of black hole is not operationally defined if the mass is smaller than the Planck mass. 1. In field theories which do not involve gravitation the transition from a classical to a quantum description is performed ’superimposing’ a quantum structure (e.g., commutation relations) to the classical theory; when we attempt to combine gravitation and quantum theory, however, we expect that at distances of the order of the Planck length the very notion of space-time might need a radical revision. Because of this, one cannot exclude a priori that a proper quantum theory of gravity might require a modification of basic quantum principles. In this Letter we examine the uncertainty principle and suggest that in quantum gravity it is indeed modified. This result, in itself, is not new; a generalized uncertainty principle has already been proposed in the context of string theories in refs. [1-2] through an analysis of Gedanken string collisions at planckian energies [2-4] (for a recent review, see [5]), and in [6] through a renormalization group analysis applied to the string. Our new point is that we do not consider strings, but use only general model-independent properties of a quantum theory of gravitation: our main physical ingredient is the Hawking radiation [7]. The functional form of the generalized uncertainty principle that we obtain agrees with the one found in string theory. Let us consider a Reissner-Nordström black hole with mass M and charge Q (the generalization of our considerations to Kerr-Newman black holes does not present conceptual difficulties). The apparent horizon is defined [8] as the outer boundary of a region of closed trapped surfaces; it has spherical topology and, in Boyer-Lindquist coordinates, it is located at r = Rh, Rh = GM [1 + (1− Q GM2 )] . (1) (We set c = 1 but write explicitly h̄ and G in the following). In classical general relativity an observer has no direct access to the apparent horizon: no signal is emitted from the black hole. If an observer wants to obtain the area of the apparent horizon, he should measure the mass and charge of the black hole from the motion of test particles at infinity, and then resort to the theory, which predicts the relation (1) for Rh as a function of M and Q; the area of the horizon is then A = 4πR h. Alternatively, one can perform a scattering experiment and again resort to the theory which predicts the relation between the measured cross section and Rh. For instance, for ultra-relativistic particles impinging on a Schwarzschild black hole, general relativity predicts a capture cross section σ = (27/4)πR h. Instead, it is