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S-Matrix formalism to describe the transition of matter falling into a black hole into Hawking radiation?

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Can the S-Matrix formalism (or an extension of it) describe the transition of matter falling into a black hole into Hawking radiation?

What I mean is start with an initial state $| i> = | \psi_i(-\infty)>$ that describes the infalling matter far away from the black hole and assume a final state $| f> = | \psi_f(\infty)>$ where the black hole is evaporated and only Hawking radiation is left.

The transition probability would as usually be $w_{i\rightarrow f} = | M_{if}|^2$ where $M_{if} = \langle f |\hat{S} |i \rangle$ defines an element of the S-Matrix

\[\hat{S} = \hat{T}\exp\left({-\frac{i}{\hbar}} \int\limits _{-\infty}^{\infty} dt \hat{W}_{IP}(t)\right )\]

$\hat{T}$ is the time ordering operator and $\hat{W}_{IP}$ describes perturbation caused by the black hole in the interaction picture.

If the S-Matrix formalism can be applied to this situation, how would one in particular construct $\hat{W}_{IP}$?

asked Mar 15, 2015 in Theoretical Physics by Dilaton (4,295 points) [ revision history ]

I doubt, atleast you cannot do that in Semi-classical GR I think.

I think the point of the hawking derivation is that underlying unitary nature is hidden until a full quantum theory of gravity is found.

Although in principle, if you take into account all the matter, including the matter that made the black hole, then it would be possible to be find such a unitary description. Though it would be far form a simple perturbation theory.

You cannot treat the perturbation introduced by the black hole as a small perturbation, It has singularities, therefore it is anything but small.

It would be good if some one who works on the subject would write an answer.

Something probably relevant: http://arxiv.org/abs/gr-qc/9607022

On TRF, Dejan Stojkovic said that when only interested in the in and out states it should be posssible to construct such a formalism. The difficulty in doing this quantum mechanically would be to find some kind of a path integral like formalism to integrate over all possible geometries. As I understand him, such methods do not yet exist (?).

@Dilaton: When doing S-matrix formalism, you don't need to integrate over geometries directly, the asymptotic states are complete, so you can generate any geometry by varying the asymptotic states. This is what makes it perfect for quantum gravity--- the impossible considerations of the integral over geometries are replaced by the perfectly feasable considerations of the integral over asymptotic scattering states. This is what happens in string theory, this is why string theory is a theory of quantum gravity even though there is no integral over geometries explicitly in the theory (it emerges from the integral over asymptotic states, which is converted in the formalism to an integral over worldsheets, in cases where strings are the only asymptotic S-matrix excitations, or to a dynamics of D0 branes in the case where D0 branes are the only light excitations on a light-front).

The argument from S-matrix is explcitly used to construct both the original string theory of 1969-1974, the string theory that becomes 1980s superstring theory when made correctly supersymmetric, and also the BFSS matrix model is constructed by matching the dynamics of the relevant asymptotic states on the 11-d circle-compactified light-cone (the D0 branes emerge this way).

This idea, to use S-matrix, it designed to circumvent geometrical path integrals, and it works, but the answer to this question is all of modern string theory, that's the S-matrix formalism for black-hole formation/annihilation.

Ah yes,  this makes at least intuitively complete sense to me, thanks for these nice explanations @RonMaimon :-)

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