טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentAssaf Lanir
SubjectComputation of a Scalar-Field Squared Expectation Value
Inside 4-Dimensional Schwarzschild and
Reissner-Nordstrom Black Holes
DepartmentDepartment of Physics
Supervisor Full Professor Ori Amos
Full Thesis textFull thesis text - English Version


Abstract

The discovery that the classical theory of general relativity admits black hole (BH) solutions led eventually to the study of their internal structure. Endowed with remarkable features, such as Cauchy horizons, bridges between universes, closed time-like curves and naked singularities, BH interiors proved to be an extraordinarily interesting fertile ground for understanding the large scale structure of spacetime.

BHs had yet more in store for us, as realized when quantum effects had begun to be taken into account within the framework of semiclassical general relativity. Particularly, BH evaporation through the emission ofHawking radiation poses the questions of the fate of the evaporating BH and the information therein (a problem known as the “information loss puzzle”). Clearly, quantum effects change considerably the internal structure of BHs, and a straightforward way to investigate the resulting picture within the semiclassical framework is to compute the renormalized stress-energy tensor (RSET) in 4D BH interiors. In this thesis, we make a first step towards that goal and compute the renormalized expectation value of 2(x) > for a minimally-coupled, massless scalar field in the interior of 4D Schwarzschild and Reissner-Nordström (RN) BHs.

The full computation of the renormalized expectation values 2(x) > and the RSET in 4D BH interiors has been a long standing challenge, which has impeded the investigation of quantum effects on the internal structure of BHs for decades. Employing a recently developed mode sum renormalization scheme to numerically implement the point-splitting method, we report here the first computation of 2(x)>ren in Unruh state in the region interior to the event horizon of a 4D Schwarzschild BH. We further present its Hartle-Hawking counterpart, which we calculated using the same method, and obtain a fairly good agreement with previous results attained using an entirely different method by Candelas and Jensen in 1986. Our results further agree upon approaching the event horizon when compared with previous results calculated outside the BH.

We further report here the first computation of 2(x)>ren in both Hartle-Hawking and Unruh states inside a 4D RN BH. Our results agree at the external event horizon with those previously obtained outside the event horizon (A. Levi and A. Ori) and with analytic asymptotic approximations (A. Ottewill). They also agree with our results obtained by a semi-asymptotic approximation at the vicinity of the inner (Cauchy) horizon. Finally, our results are consistent with those of an asymptotic analysis of the expected leading order in δr of 2(x)>ren near the Cauchy horizon which appears to vanish (O. Sela and A. Ori) suggesting a weaker modification of the spacetime metric than previously expected.

This work sets the stage for further explorations of 2(x)>ren and the RSET in 4D BH interiors, an endeavor we have already undertaken, also with other collaborators, and which will be reported in the future.