We are all familiar with the fact that black holes have what are called event horizons which to outside observers define the size of the black hole. Information that is inside this horizon cannot escape to outside this horizon and beyond because on matter or energy can travel faster than the speed of light.

Mathematically, however, if you were to look at what is happening to space-time at the event horizon, you would discover that the curvature of space is not 'singular' there. The only pathological location is at r=0 in the center of the black hole. This is where the classical theory of general relativity breaks down. If you were an astronaut falling into a black hole, according to your clocks and meter sticks, you would cross the event horizon and not experience anything odd in terms of your 'comoving' coordinate system.

A non-quantum mechanical explanation takes advantage of a relativistic effect of matter falling into a black hole to form the hole in the first place. A distance outside observer would see the surface of the mass slow down and stop as it reached the horizon, and then rapidly fade from view. Some physicists still call black holes by an earlier term 'frozen star'. The point is that time dilation effects viewed from far away make the black hole look like a star whose surface is dark, but whose size is still slightly larger than the black hole radius for its mass. A classical Newtonian explanation for the gravity from this object would then just say that the gravity we feel is the gravity that is produced by the mass of the star that is just outside the horizon. So, the gravity we feel is not produced by the matter inside the horizon. This explanation, though not so bad for our intuitions to get a handle on, is not completely accurate in terms of general relativity.

General relativity identifies gravity as a curvature of space (space-time actually) and the mathematical machine you use to measure curvature is the Riemann Tensor, or its cousin the Ricci Tensor. When the geometry of spacetime is expressed in tensors, which can be ported from one coordinate system to another without scrambling the physics, you discover that at the event horizon, the curvature of spacetime changes smoothly. There are no sudden barriers or changes. Gravity is not seen in general relativity as something that travels through space like a pulse of light. Gravity is a property of space itself. Because of this, it isnt necessary for gravity to 'escape' a black hole at all. Gravity doesnt work that way as a physical phenomenon. It would be like saying that space has to escape a black hole or that time has to 'escape' a black hole. In general relativity, space, time and gravity are placed on an equal footing as physical things. The quantum description of gravity, however, changes this a bit.

To explain how gravitons escape from a black hole in order to cause the gravitational field we see at great distances, you have to accept the fact that the event horizon is not an infinitly high 'potential barrier' that would impede the wave function of a quantum system. Because general relativity shows that, in the appropriately selected coordinate system, the black hole only has ONE curvature singularity at its core, NOT at the event horizon as well, the event horizon is just a region of space-time which has a particular gravitational potential. This can now be translated into a problem in quantum mechanics where you are asking what the penetration or 'tunneling' probability is for quantum particles like gravitons, electrons etc. Outgoing quanta can therefore pass across the potential barrier at the spatial distance of the event horizon, and tunnel across it. The event horizon is poorly localized to quantum mechanical particles, and is just like any other potential barriers across which quantum mechanical particles can tunnel like a leaky membrane.

According to physicists such as Nobel lauriat Abdus Salam, along with Stephen Hawking and M.J. Duff it is possible to start from a purely quantum mechanical description of gravity in terms of gravitons. By 'summing infinite sets of tree diagrams' describing the quantum self-interaction of gravitons, you can recover the famous Schwarzschild solution for black holes, complete with an event horizon, and a description of its gravitational field. Clearly, the appearance of the famous event horizon which bedevils distant observers when studying black holes from afar, is not incompatible with the quantum behavior of gravitons that build-up the behavior of the black hole space-time through their millions and trillions of self-interactions.

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