Action and Energy of the Gravitational Field

We present a detailed examination of the variational principle for metric general relativity as applied to a ``quasilocal'' spacetime region \M (that is, a region that is both spatially and temporally bounded). Our analysis relies on the Hamiltonian formulation of general relativity, and thereby assumes a foliation of \M into spacelike hypersurfaces $\Sigma$. We allow for near complete generality in the choice of foliation. Using a field--theoretic generalization of Hamilton--Jacobi theory, we define the quasilocal stress-energy-momentum of the gravitational field by varying the action with respect to the metric on the boundary \partial\M. The gravitational stress-energy-momentum is defined for a two--surface $B$ spanned by a spacelike hypersurface in spacetime. We examine the behavior of the gravitational stress-energy-momentum under boosts of the spanning hypersurface. The boost relations are derived from the geometrical and invariance properties of the gravitational action and Hamiltonian. Finally, we present several new examples of quasilocal energy--momentum, including a novel discussion of quasilocal energy--momentum in the large-sphere limit towards spatial infinity.Comment: To be published in Annals of Physics. This final version includes two new sections, one giving examples of quasilocal energy and the other containing a discussion of energy at spatial infinity. References have been added to papers by Bose and Dadhich, Anco and Tun

Canonical Quasilocal Energy and Small Spheres

Consider the definition E of quasilocal energy stemming from the Hamilton-Jacobi method as applied to the canonical form of the gravitational action. We examine E in the standard "small-sphere limit," first considered by Horowitz and Schmidt in their examination of Hawking's quasilocal mass. By the term "small sphere" we mean a cut S(r), level in an affine radius r, of the lightcone belonging to a generic spacetime point. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero-point of the energy. For the small-sphere limit, we argue that the correct zero-point is obtained via a "lightcone reference," which stems from a certain isometric embedding of S(r) into a genuine lightcone of Minkowski spacetime. Choosing this zero-point, we find agreement with Hawking's quasilocal mass expression, up to and including the first non-trivial order in the affine radius. The vacuum limit relates the quasilocal energy directly to the Bel-Robinson tensor.Comment: revtex, 22 p, uses amssymb option (can be removed

Positivity of Entropy in the Semi-Classical Theory of Black Holes and Radiation

Quantum stress-energy tensors of fields renormalized on a Schwarzschild background violate the classical energy conditions near the black hole. Nevertheless, the associated equilibrium thermodynamical entropy $\Delta S$ by which such fields augment the usual black hole entropy is found to be positive. More precisely, the derivative of $\Delta S$ with respect to radius, at fixed black hole mass, is found to vanish at the horizon for {\it all} regular renormalized stress-energy quantum tensors. For the cases of conformal scalar fields and U(1) gauge fields, the corresponding second derivative is positive, indicating that $\Delta S$ has a local minimum there. Explicit calculation shows that indeed $\Delta S$ increases monotonically for increasing radius and is positive. (The same conclusions hold for a massless spin 1/2 field, but the accuracy of the stress-energy tensor we employ has not been confirmed, in contrast to the scalar and vector cases). None of these results would hold if the back-reaction of the radiation on the spacetime geometry were ignored; consequently, one must regard $\Delta S$ as arising from both the radiation fields and their effects on the gravitational field. The back-reaction, no matter how "small",Comment: 19 pages, RevTe

Slice Stretching Effects for Maximal Slicing of a Schwarzschild Black Hole

Slice stretching effects such as slice sucking and slice wrapping arise when foliating the extended Schwarzschild spacetime with maximal slices. For arbitrary spatial coordinates these effects can be quantified in the context of boundary conditions where the lapse arises as a linear combination of odd and even lapse. Favorable boundary conditions are then derived which make the overall slice stretching occur late in numerical simulations. Allowing the lapse to become negative, this requirement leads to lapse functions which approach at late times the odd lapse corresponding to the static Schwarzschild metric. Demanding in addition that a numerically favorable lapse remains non-negative, as result the average of odd and even lapse is obtained. At late times the lapse with zero gradient at the puncture arising for the puncture evolution is precisely of this form. Finally, analytic arguments are given on how slice stretching effects can be avoided. Here the excision technique and the working mechanism of the shift function are studied in detail.Comment: 16 pages, 4 figures, revised version including a study on how slice stretching can be avoided by using excision and/or shift

Local and global properties of conformally flat initial data for black hole collisions

We study physical properties of conformal initial value data for single and binary black hole configurations obtained using conformal-imaging and conformal-puncture methods. We investigate how the total mass M_tot of a dataset with two black holes depends on the configuration of linear or angular momentum and separation of the holes. The asymptotic behavior of M_tot with increasing separation allows us to make conclusions about an unphysical ``junk'' gravitation field introduced in the solutions by the conformal approaches. We also calculate the spatial distribution of scalar invariants of the Riemann tensor which determine the gravitational tidal forces. For single black hole configurations, these are compared to known analytical solutions. Spatial distribution of the invariants allows us to make certain conclusions about the local distribution of the additional field in the numerical datasets

Detecting cold gas at intermediate redshifts: GMRT survey using Mg II systems

Intervening HI 21-cm absorption systems at z > 1.0 are very rare and only 4 confirmed detections have been reported in the literature. Despite their scarcity, they provide interesting and unique insights into the physical conditions in the interstellar medium of high-z galaxies. Moreover, they can provide independent constraints on the variation of fundamental constants. We report 3 new detections based on our ongoing Giant Metrewave Radio Telescope (GMRT) survey for 21-cm absorbers at 1.10< z_abs< 1.45 from candidate damped Lyman_alpha systems. The 21-cm lines are narrow for the z_abs = 1.3710 system towards SDSS J0108-0037 and z_abs = 1.1726 system toward SDSS J2358-1020. Based on line full-width at half maximum, the kinetic temperatures are <= 5200 K and <=800 K, respectively. The 21-cm absorption profile of the third system, z_abs =1.1908 system towards SDSS J0804+3012, is shallow, broad and complex, extending up to 100 km/s. The centroids of the 21-cm lines are found to be shifted with respect to the corresponding centroids of the metal lines derived from SDSS spectra. This may mean that the 21-cm absorption is not associated with the strongest metal line component.Comment: 13 pages with 5 figures. Accepted for publication in ApJ

Freire re-viewed

The work of Paulo Freire is associated with themes of oppression and liberation, and his critical pedagogy is visionary in its attempts to bring about social transformation. Freire has created a theory of education that embeds these issues within social relations that center around both ideological and material domination. In this review essay, Sue Jackson explores three books: Freire’s final work Pedagogy of Indignation; Cesar Augusto Rossatto’s Engaging Paulo Freire’s Pedagogy of Possibility, which attempts to engage Freire’s pedagogy of possibility; and C.A. Bowers and Frederique Apffel-Marglin’s edited collection Re-thinking Freire, which asks readers to reconsider Freire’s work in light of globalization and environmental crises. Jackson questions the extent to which Freire’s pedagogical approaches are useful to educators as well as to “the oppressed,” and whether challenges to re-think Freire can lead to new kinds of critical pedagogies

High-Redshift Superclustering of QSO Absorption Line Systems on 100 Mpc Scales

We have analyzed the clustering of C IV absorption line systems in an extensive new catalog of heavy element QSO absorbers. The catalog permits exploration of clustering over a large range in both scale (from about 1 to over 300 Mpc) and redshift (z from 1.2 to 4.5). We find significant evidence (5.0 sigma) that C IV absorbers are clustered on comoving scales of 100 Mpc and less --- similar to the size of voids and walls found in galaxy redshift surveys of the local universe --- with a mean correlation function $\xi = 0.42 \pm 0.10$ over these scales. We find, on these scales, that the mean correlation function at low (z=1.7), medium (z=2.4), and high redshift (z=3.0) is $\xi=0.40 \pm 0.17$, $0.32 \pm 0.14$, and $0.72 \pm 0.25$, respectively. Thus, the superclustering is present even at high redshift; furthermore, it does not appear that the superclustering scale, in comoving coordinates, has changed significantly since then. We find 7 QSOs with rich groups of absorbers (potential superclusters) that account for a significant portion of the clustering signal, with 2 at redshift $z\sim 2.8$. We find that the superclustering is just as evident if we take $q_0=0.1$ instead of 0.5; however, the inferred scale of clustering is then 240 Mpc , which is larger than the largest scales of clustering known at present. This discrepancy may be indicative of a larger value of $q_0$, and hence $\Omega_0$. The evolution of the correlation function on 50 Mpc scales is consistent with that expected in cosmologies with density parameter ranging from $\Omega_0 =$ 0.1 to 1. Finally, we find no evidence for clustering on scales greater than 100 Mpc ($q_0=0.5$) or 240 Mpc ($q_0=0.1$).Comment: 16 LaTeX pages with 3 encapsulated Postscript figures included, uses AASTeX (v. 4.0) available at ftp://ftp.aas.org/pubs/ , to appear in The Astrophysical Journal Letter