The luminosity distance can be used to determine the properties of large scale structure around the observer. To this purpose we develop a new inversion method to map luminosity distance to a LTB metric based on the use of the exact analytical solution for Einstein equations. The main advantages of this approach are an improved numerical accuracy and stability, an exact analytical setting of the initial conditions for the differential equations which need to be solved and the validity for any sign of the functions determining the LTB geometry. Given the fully analytical form of the differential equations, this method also simplifies the calculation of the red-shift expansion around the apparent horizon point where the numerical solution becomes unstable. We test the method by inverting the supernovae Ia luminosity distance function corresponding to the the best fit ΛCDM model. We find that only a limited range of initial conditions is compatible with observations, or a transition from red to blue shift can occur at relatively low redshift. Despite LTB solutions without a cosmological constant have been shown not to be compatible with all different set of available observational data, those studies normally fit data assuming a special functional ansatz for the inhomogeneity profile, which often depend only on few parameters. Inversion methods on the contrary are able to fully explore the freedom in fixing the functions which determine a LTB solution. For this reason this inversion method could be applied to explore more exhaustively the compatibility with observations. Another important possible application is not about LTB solutions as cosmological models, but rather as tools to study the effects on the observations made by a generic observer located in an inhomogeneous region of the Universe where a fully non perturbative treatment involving exact solutions of Einstein equations is required.2

No abstract

Motivated by both concepts of R.J. Adler's recent work on utilizing Clifford algebra as the linear line element ds = γµ dX µ , and the fermionization of the cylindrical worldsheet Polyakov action, we introduce a new type of spacetime quantization that is fully covariant. The theory is based on the reinterpretation of Adler's linear line element as ds = γµ λγ µ , where λ is the characteristic length of the theory. We name this new operator as "spacetime interval operator", and argue that it can be regarded as a natural extension to the one-forms in the U (su(2)) non-commutative geometry. By treating Fourier momentum as the particle momentum, the generalized uncertainty principle of the U (su(2)) non-commutative geometry, as an approximation to the generalized uncertainty principle of our theory, is derived, and is shown to have a lowest order correction term of the order p 2 similar to that of Snyder's. The holography nature of the theory is demonstrated, and the predicted fuzziness of the geodesic is shown to be much smaller than conceivable astrophysical bounds.

One interesting proposal to solve the black hole information loss paradox without modifying either general relativity or quantum field theory, is the soft hair, a diffeomorphism charge that records the anisotropic radiation in the asymptotic region. This proposal, however, has been challenged, given that away from the source the soft hair behaves as a coordinate transformation that forms an Abelian group, thus unable to store any information. To maintain the spirit of the soft hair but circumvent these obstacles, we consider Hawking radiation as a probe sensitive to the entire history of the black hole evaporation, where the soft hairs on the horizon are induced by the absorption of a null anisotropic flow, generalizing the shock wave considered in [1, 2]. To do so we introduce two different time-dependent extensions of the diffeomorphism associated with the soft hair, where one is the backreaction of the anisotropic null flow, and the other is a coordinate transformation that produces the Unruh effect and a Doppler shift to the Hawking spectrum. Together, they form an exact BMS charge generator on the entire manifold that allows the nonperturbative analysis of the black hole horizon, whose surface gravity, i.e. the Hawking temperature, is found to be modified. The modification depends on an exponential average of the anisotropy of the null flow with a decay rate of 4M, suggesting the emergence of a new 2-D degree of freedom on the horizon, which could be a way out of the information loss paradox.