Computation of the renormalized stress-energy tensor is the most serious obstacle in studying the dynamical, self-consistent, semiclassical evaporation of a black hole in 4D. The difficulty arises from the delicate regularization procedure for the stress-energy tensor, combined with the fact that in practice the modes of the field need be computed numerically. We have developed a new method for numerical implementation of the point-splitting regularization in 4D, applicable to the renormalized stress-energy tensor as well as to φ 2 ren , namely the renormalized φ 2 . So far we have formulated two variants of this method: t-splitting (aimed for stationary backgrounds) and angular splitting (for spherically-symmetric backgrounds). In this paper we introduce our basic approach, and then focus on the t-splitting variant, which is the simplest of the two (deferring the angular-splitting variant to a forthcoming paper). We then use this variant, as a first stage, to calculate φ 2 ren in Schwarzschild spacetime, for a massless scalar field in the Boulware state. We compare our results to previous ones, obtained by a different method, and find full agreement. We discuss how this approach can be applied (using the angular-splitting variant) to analyze the dynamical self-consistent evaporation of black holes.

We report here on a new method for calculating the renormalized stress-energy tensor (RSET) in black-hole (BH) spacetimes, which should also be applicable to dynamical BHs and to spinning BHs. This new method only requires the spacetime to admit a single symmetry. So far we developed three variants of the method, aimed for stationary, spherically symmetric, or axially symmetric BHs. We used this method to calculate the RSET of a minimally-coupled massless scalar field in Schwarzschild and Reissner-Nordstrom backgrounds, for several quantum states. We present here the results for the RSET in the Schwarzschild case in the Unruh state (the state describing BH evaporation). The RSET is type I at weak field, and becomes type IV at r 2.78M . Then we use the RSET results to explore violation of the weak and null Energy conditions. We find that both conditions are violated all the way from r 4.9M to the horizon. We also find that the averaged weak energy condition is violated by a class of (unstable) circular timelike geodesics. Most remarkably, the circular null geodesic at r = 3M violates the averaged null energy condition.Semiclassical gravity is a theory that describes the interaction of quantum fields with a classical spacetime metric, and their coupled evolution. One of the central goals of semiclassical gravity is to allow detailed understanding of black-hole (BH) evaporation. Since Hawking's discovery in 1974 that BHs emit quantum radiation [1] and evaporate, many efforts have been made to properly formulate and analyze this dynamical process of semiclassical BH evaporation. This phenomenon is directly related to a number of profound issues like the information puzzle and loss of unitarity.To properly address semiclassical BH evaporation one should (at least in principle) solve the semiclassical Einstein equation [36] where G αβ is the Einstein tensor and T αβ ren is the quantum field's renormalized stress-energy tensor (RSET). Both sides depend on the evolving spacetime metric g αβ (x), which is the unknown in this equation. The RSET, which emerges from the field's quantum fluctuations, also depends on the type of matter field as well as on its quantum state. Throughout this paper we shall consider a minimally-coupled massless scalar field (MCMSF) φ(x), satisfying φ = 0.One of the hardest aspects in dealing with Eq. (1) is the computation of the RSET. In this paper we shall mostly address this aspect: computation of T αβ ren (and analysis thereof) for a prescribed spacetime metric g αβ (x).In principle, this computation involves summation (and integration) of the contributions to T αβ from the individual field's modes. The naive mode sum is divergent and requires regularization. This is not surprising, as the naive expectation value diverges already in flat spacetime. In flat spacetime, however, one can use the normal ordering procedure. Unfortunately, this simple procedure is not applicable in curved spacetime.Instead, in curved spacetime one can use the pointsplitting regularization which was developed by ...

The computation of the renormalized stress-energy tensor or φ 2 ren in curved spacetime is a challenging task, at both the conceptual and technical levels. Recently we developed a new approach to compute such renormalized quantities in asymptotically-flat curved spacetimes, based on the point-splitting procedure. Our approach requires the spacetime to admit some symmetry. We already implemented this approach to compute φ 2 ren in a stationary spacetime using t-splitting, namely splitting in the time-translation direction. Here we present the angular-splitting version of this approach, aimed for computing renormalized quantities in a general (possibly dynamical) spherically-symmetric spacetime. To illustrate how the angular-splitting method works, we use it here to compute φ 2 ren for a quantum massless scalar field in Schwarzschild background, in various quantum states (Boulware, Unruh, and Hartle-Hawking states). We find excellent agreement with the results obtained from the t-splitting variant, and also with other methods. Our main goal in pursuing this new mode-sum approach was to enable the computation of the renormalized stressenergy tensor in a dynamical spherically symmetric background, e.g. an evaporating black hole. The angular-splitting variant presented here is most suitable to this purpose.

We derive explicit expressions for the two-point function of a massless scalar field in the interior region of a Reissner-Nordstrom black hole, in both the Unruh and Hartle-Hawking quantum states. The two-point function is expressed in terms of the standard lmω modes of the scalar field (those associated with a spherical harmonic Y lm and a temporal mode e −iωt ), which can be conveniently obtained by solving an ordinary differential equation, the radial equation. These explicit expressions are the internal analogs of the well known results in the external region (originally derived by Christensen and Fulling), in which the two-point function outside the black hole is written in terms of the external lmω modes of the field. They allow the computation of < Φ 2 >ren and the renormalized stress-energy tensor inside the black hole, after the radial equation has been solved (usually numerically). In the second part of the paper, we provide an explicit expression for the trace of the renormalized stress-energy tensor of a minimally-coupled massless scalar field (which is non-conformal), relating it to the d'Alembertian of < Φ 2 >ren. This expression proves itself useful in various calculations of the renormalized stress-energy tensor. I. INTRODUCTIONIn the framework of semiclassical general relativity, the gravitational field is treated classically as a curved spacetime while other fields are treated as quantum fields residing in this background spacetime. The relation between the spacetime geometry and the stress-energy of the quantum fields is described by the semiclassical Einstein equation( 1.1) where G µν is the Einstein tensor of spacetime, and ren is the renormalized stress-energy tensor (RSET), which is the renormalized expectation value of the stress-energy tensor operatorT , associated with the quantum fields. In Eq. (1.1) and throughout this paper we adopt standard geometric units c = G = 1, and signature (− + ++).The main challenge in analyzing the semiclassical Einstein equation is the computation of the RSET. Even when the background geometry is fixed and the corresponding metric is given, performing a procedure of renormalization is not an easy task. The task becomes much more difficult upon trying to solve the full self-consistent problem represented by Eq. (1.1). One reason (besides the obvious numerical challenge) is that Eq. (1.1) admits runaway solutions [1].An example of a quantum field that is often chosen for its simplicity is that of a scalar field, which satisfies the Klein-Gordon equationwhereΦ is the scalar field operator, m denotes the field's mass, and ξ is the coupling constant. As a first stage towards calculating the RSET, it is customary to begin by calculating <Φ 2 > ren , as this quantity is endowed with many of the essential features of the RSET, but is simpler to compute. In other words, <Φ 2 > ren serves as a simple toy model for the RSET. The standard method to calculate quantities which are quadratic in the field and its derivatives is point-splitting (or covariant point ...

In an ongoing effort to explore quantum effects on the interior geometry of black holes, we explicitly compute the semiclassical flux components Tuu ren and Tvv ren (u and v being the standard Eddington coordinates) of the renormalized stress-energy tensor for a minimally-coupled massless quantum scalar field, in the vicinity of the inner horizon (IH) of a Reissner-Nordström black hole. These two flux components turn out to dominate the effect of backreaction in the vicinity of the IH; and furthermore, their regularization procedure reveals remarkable simplicity. We consider the Hartle-Hawking and Unruh quantum states, the latter corresponding to an evaporating black hole. In both quantum states, we compute Tuu ren and Tvv ren in the vicinity of the IH for a wide range of Q/M values. We find that both Tuu ren and Tvv ren attain finite asymptotic values at the IH. These asymptotic values are found to be either positive or negative (or vanishing in-between), depending on the Q/M parameter. Note that having a nonvanishing Tvv ren at the IH implies the formation of a curvature singularity on its ingoing section, the Cauchy horizon. Motivated by these findings, we also take initial steps in the exploration of the backreaction effect of these semiclassical fluxes on the near-IH geometry.