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Register a new userUnitarity alternatives in the reduced-action model for gravitational collapse
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Based on the ACV approach to transplanckian energies, the reduced-action model for the gravitational S-matrix predicts a critical impact parameter b_c ~ R = 2 G sqrt{s} such that S-matrix unitarity is satisfied in the perturbative region b > b_c, while it is exponentially suppressed with respect to s in the region b < b_c that we think corresponds to gravitational collapse. Here we definitely confirm this statement by a detailed analysis of both the critical region b ~ b_c and of further possible contributions due to quantum transitions for b < b_c. We point out, however, that the subcritical unitarity suppression is basically due to the boundary condition which insures that the solutions of the model be ultraviolet-safe. As an alternative, relaxing such condition leads to solutions which carry short-distance singularities presumably regularized by the string. We suggest that through such solutions - depending on the detailed dynamics at the string scale - the lost probability may be recovered.
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Starting from the semiclassical reduced-action approach to transplanckian scattering by Amati, Veneziano and one of us and from our previous quantum extension of that model, we investigate the S-matrix expression for inelastic processes by extending to this case the tunneling features previously found in the region of classical gravitational collapse. The resulting model exhibits some non-unitary S-matrix eigenvalues for impact parameters b < b_c, a critical value of the order of the gravitational radius R = 2 G sqrt(s), thus showing that some (inelastic) unitarity defect is generally present, and can be studied quantitatively. We find that S-matrix unitarity for b < b_c is restored only if the rapidity phase-space parameter y is allowed to take values larger than the effective coupling G s / hbar itself. Some features of the resulting unitary model are discussed.
The Reduced Basis Method (RBM) is a model reduction technique used to solve parametric PDEs that relies upon a basis set of solutions to the PDE at specific parameter values. To generate this reduced basis, the set of a small number of parameter values must be strategically chosen. We apply a Metropolis algorithm and a gradient algorithm to find the set of parameters and compare them to the standard greedy algorithm most commonly used in the RBM. We test our methods by using the RBM to solve a simplified version of the governing partial differential equation for hyperspectral diffuse optical tomography (hyDOT). The governing equation for hyDOT is an elliptic PDE parameterized by the wavelength of the laser source. For this one-dimensional problem, we find that both the Metropolis and gradient algorithms are potentially superior alternatives to the greedy algorithm in that they generate a reduced basis which produces solutions with a smaller relative error with respect to solutions found using the finite element method and in less time.
Using the recently introduced ACV reduced-action approach to transplanckian scattering of light particles, we show that the $S$-matrix in the region of classical gravitational collapse is related to a tunneling amplitude in an effective field space. We understand in this way the role of both real and complex field solutions, the choice of the physical ones, the absorption of the elastic channel associated to inelastic multigraviton production and the occurrence of extra absorption below the critical impact parameter. We are also able to compute a class of quantum corrections to the original semiclassical $S$-matrix that we argue to be qualitatively sensible and which, generally speaking, tend to smooth out the semiclassical results.
We generalize the semiclassical treatment of graviton radiation to gravitational scattering at very large energies $sqrt{s}gg m_P$ and finite scattering angles $Theta_s$, so as to approach the collapse regime of impact parameters $b simeq b_c sim Requiv 2Gsqrt{s}$. Our basic tool is the extension of the recently proposed, unified form of radiation to the ACV reduced-action model and to its resummed-eikonal exchange. By superimposing that radiation all-over eikonal scattering, we are able to derive the corresponding (unitary) coherent-state operator. The resulting graviton spectrum, tuned on the gravitational radius $R$, fully agrees with previous calculations for small angles $Theta_sll 1$ but, for sizeable angles $Theta_s(b)leq Theta_c = O(1)$ acquires an exponential cutoff of the large $omega R$ region, due to energy conservation, so as to emit a finite fraction of the total energy. In the approach-to-collapse regime of $bto b_c^+$ we find a radiation enhancement due to large tidal forces, so that the whole energy is radiated off, with a large multiplicity $langle N ranglesim Gs gg 1$ and a well-defined frequency cutoff of order $R^{-1}$. The latter corresponds to the Hawking temperature for a black hole of mass notably smaller than $sqrt{s}$.
We generalize the semiclassical treatment of graviton radiation to gravitational scattering at very large energies $sqrt{s}gg m_P$ and finite scattering angles $Theta_s$, so as to approach the collapse regime of impact parameters $b simeq b_c sim Requiv 2Gsqrt{s}$. Our basic tool is the extension of the recently proposed, unified form of radiation to the ACV reduced-action model and to its resummed-eikonal exchange. By superimposing that radiation all-over eikonal scattering, we are able to derive the corresponding (unitary) coherent-state operator. The resulting graviton spectrum, tuned on the gravitational radius $R$, fully agrees with previous calculations for small angles $Theta_sll 1$ but, for sizeable angles $Theta_s(b)leq Theta_c = O(1)$ acquires an exponential cutoff of the large $omega R$ region, due to energy conservation, so as to emit a finite fraction of the total energy. In the approach-to-collapse regime of $bto b_c^+$ we find a radiation enhancement due to large tidal forces, so that the whole energy is radiated off, with a large multiplicity $langle N ranglesim Gs gg 1$ and a well-defined frequency cutoff of order $R^{-1}$. The latter corresponds to the Hawking temperature for a black hole of mass notably smaller than $sqrt{s}$.
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