Gravity

We propose an approach to the quantization of the interior of a Schwarzschild black hole, represented by a Kantowski-Sachs (KS) framework, by requiring its covariance under a notion of residual diffeomorphisms. We solve for the family of Hamiltonian constraint operators satisfying the associated covariance condition, in addition to parity covariance, preservation of the Bohr Hilbert space of Loop Quantum KS and a correct (naïve) classical limit. We further explore imposing minimality of the number of terms, and compare the solution with other Hamiltonian constraints proposed for Loop Quantum KS in the literature, with special attention to a most recent case. In addition, we discuss a lapse commonly chosen to decouple the evolution of the two degrees of freedom of the model, yielding exact solubility of the model, and we show that such choice can indeed be quantized as an operator densely defined on the Bohr Hilbert space, but must include an infinite number of shift operators. Also, we show the reasons why we call the classical limit “naïve”, and point this out as a reason for one limitation of some present prescriptions.

Through the variational principle, we review the gravitational field equations in Einstein gravity and modified f(R) gravity theories. Metric and Palatini formalisms are two different approaches that are employed to obtain the field equations in the context of f(R) theory of gravity. In this framework, we attempt to investigate the energy conditions in Friedmann-Lemaitre-Robertson-Walker (FLRW) metric using the Raychaudhuri equation. Then, we focus on wormhole geometries and their thermodynamics behavior in Palatini and metric versions of modified f(R) gravity, separately. To violate the null and the weak energy conditions, wormhole spacetimes need an exotic matter. It has been shown that in f(R) gravity the matter threading the wormholes serves the energy conditions, and it is the derivative terms of the higher order curvature that may be explained as a gravitational fluid, that supports these geometries. Therefore, in f(R) gravity theory it is not required to introduce exotic matter in order to have traversable wormholes. In the framework of metric and Palatini f(R) gravity, we investigate the thermodynamic properties of evolving wormholes. We obtain an expression for the variation of the total entropy to discuss the thermodynamic behavior of wormhole spacetimes. The investigation has been extended to the apparent and event horizons. Eventually, we apply the radius of these horizons to determine the validity of the generalized second law of thermodynamics. This law states that the rate of change of total entropy is positive.

The study was designed to determine the effect of an organized, conceptually-oriented set of lessons involving the concept of gravity on the immediate and delayed achievement of fourth-grade students. Two demographically similar schools were selected for the study, with two "Treatment" classes in one school and two "Comparison" classes in the other school. The sample included 91 fourth-grade students in the four classes who were taught the concept of gravity for two weeks involving three 30-minute lessons per week. The Treatment Group teachers taught the concept of gravity using six prepared lessons. Within the same time frame, Comparison Group teachers used the text book as a guide to teach six lessons on gravity. At the end of the two weeks of lessons on gravity, a 20-question posttest was administered to each class. After another three weeks, a delayed posttest was given. Results of the Cronbach's alpha procedure showed that the two tests were reliable and appropriate for use with fourth grade students. Results of a student opinion question concerning how much the students thought they learned about gravity during the two week period showed that the students in the Treatment Group had more positive answers concerning their learning than the Comparison Group students. The ANOVA results confirmed there was no significant interaction between gender, socio-economic status, reading ability, or ethnicity and the treatment. An independent t-test found that the mean differences between the Treatment Group and Comparison Group scores on both the posttest and delayed posttest were large (4.6 and 5.2). The Treatment Group results were higher, which points to a greater retention of the concept of gravity over time by students. Cohen's Effect Size resulted in a large effect for the posttest and delayed posttest (1.8 and 2.3), suggesting that the intervention may have had a significant effect on student learning of the concept of gravity. This study suggested that having well organized, conceptually-oriented instructional lessons may assist the teacher in improving student learning of scientific concepts and aid in the retention of the concepts over time.