Istituto Nazionale di Fisica Nucleare
Sezione di Trieste

Gravity and Strings

Astrophysical mass-size black holes are well modeled by classic solutions of Einstein's equations. Theoretical predictions such as the existence of accretion disks and plasma jets along the rotation axis of spinning black holes have been confirmed by the amazing photos taken by orbiting telescopes. Very different is the case of the microscopic-sized "quantum black holes'', whose description calls for a consistent theory of quantum gravity. String theory is presently the only framework providing a finite, anomaly-free, perturbative formulation of gravity at the quantum level. Even within this framework, however, black holes are treated as classic solutions, in the bosonic sector, of certain Super-Gravity field theories which emerge as "point-like limits" from corresponding Super-String models. This approach is powerful enough to shed some light on the microscopic origins of black hole entropy (at least in certain cases) and to introduce multi-dimensional objects, such as D-branes, through solitonic solutions of field equations.

However, there are many answers are still to be found, such as the actual form of the mass spectrum of a quantum black hole, a satisfactory solution to the Information Paradox, an explanation of horizon surface quantization in elementary Planck cells, and so on. All these problems are currently the subject of thorough investigation. Even more pressin is the need for a satisfactory solution to the "singularity problem'': in a theory of extended objects, the very concepts of (point-like) curvature singularity should be meaningless. In other words, one would like to show that in a quantum theory of gravity, there should not be any curvature singularity, whether "naked'' or hidden by an event horizon, but this problem cannot be treated in any kind of field theory, including Super-Gravity, where the fundamental objects are basically point-like. At first glance, this kind of physics could appear purely speculative and detached from any experimental verification. This may be true. However, "TeV quantum gravity" is an intriguing spin-off of non-perturbative String Theory, where all four interactions, including gravity, are unified at an energy scale much lower than Planck energy and, presumably, not too far away from the Large Hadron Collider (LHC) peak energy, i.e. 14 TeV. The coming into operation of the LHC actually makes it possible to test "new physics'' beyond the Standard Model, and hopefully to see some signs of "quantum gravity'' phenomena.

With this background in mind, we have proposed an effective approach to the singularity problem, in which (semi)classical Einstein equations are used to determine black hole solutions which "keep the memory'' of their quantum nature. A common feature of all the candidate theories of quantum gravity is the existence of a fundamental length scale where the very concept of space-time as a classic manifold breaks down. Speaking of arbitrarily small distances becomes meaningless and the concept of "minimum length'' emerges as a new fundamental constant of Nature on the same footing as the speed of light and Planck's quantum of action. String Theory, Loop Quantum Gravity, non-commutative geometry, the Generalized Uncertainty Principle, Path Integral Duality, and so on, all share this common feature. Taking this as our starting-point, in a recent series of papers we have proposed an effective way of "improving" the Einstein equations by inserting this information into an appropriately-shaped energy momentum tensor. In the presence of a minimum length, even an "elementary'' particle cannot be described by a dimensionless matter point. According to the tenets of quantum mechanics, the best-localized quantum particle is described by a minimum width Gaussian distribution of energy and momentum. Starting from this, we constructed an appropriate Gaussian source for the Einstein equations and solved them in a variety of cases describing black holes of a regular type (i.e. those without curvature singularity), and also neutral, charged, rotating and even multi-dimensional black holes.

It is strange to think that solutions of the Einstein equations sourced by Gaussian distributions of energy and momentum have been ignored for such a long time.

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