Condições de energia de hawking e ellis e a equação de raychaudhuri

In the Einstein s theory of General Relativity the field equations relate the geometry of space-time with the content of matter and energy, sources of the gravitational field. This content is described by a second order tensor, known as energy-momentum tensor. On the other hand, the energy-momentum tensors that have physical meaning are not specified by this theory. In the 700s, Hawking and Ellis set a couple of conditions, considered feasible from a physical point of view, in order to limit the arbitrariness of these tensors. These conditions, which became known as Hawking-Ellis energy conditions, play important roles in the gravitation scenario. They are widely used as powerful tools for analysis; from the demonstration of important theorems concerning to the behavior of gravitational fields and geometries associated, the gravity quantum behavior, to the analysis of cosmological models. In this dissertation we present a rigorous deduction of the several energy conditions currently in vogue in the scientific literature, such as: the Null Energy Condition (NEC), Weak Energy Condition (WEC), the Strong Energy Condition (SEC), the Dominant Energy Condition (DEC) and Null Dominant Energy Condition (NDEC). Bearing in mind the most trivial applications in Cosmology and Gravitation, the deductions were initially made for an energy-momentum tensor of a generalized perfect fluid and then extended to scalar fields with minimal and non-minimal coupling to the gravitational field. We also present a study about the possible violations of some of these energy conditions. Aiming the study of the single nature of some exact solutions of Einstein s General Relativity, in 1955 the Indian physicist Raychaudhuri derived an equation that is today considered fundamental to the study of the gravitational attraction of matter, which became known as the Raychaudhuri equation. This famous equation is fundamental for to understanding of gravitational attraction in Astrophysics and Cosmology and for the comprehension of the singularity theorems, such as, the Hawking and Penrose theorem about the singularity of the gravitational collapse. In this dissertation we derive the Raychaudhuri equation, the Frobenius theorem and the Focusing theorem for congruences time-like and null congruences of a pseudo-riemannian manifold. We discuss the geometric and physical meaning of this equation, its connections with the energy conditions, and some of its several aplications.

Estruturas tectônicas cenozóicas na porção leste da Bacia Potiguar - RN

This thesis encompasses the integration of geological, geophysical, and seismological data in the east part of the Potiguar basin, northeastern Brazil. The northeastern region is located in South American passive margin, which exhibits important areas that present neotectonic activity. The definition of the chronology of events, geometry of structures generated by these events, and definition of which structures have been reactivated is a necessary task in the region. The aims of this thesis are the following: (1) to identify the geometry and kinematics of neotectonic faults in the east part of the Potiguar basin; (2) to date the tectonic events related to these structures and related them to paleoseismicity in the region; (3) to present evolutional models that could explain evolution of Neogene structures; (4) and to investigate the origin of the reactivation process, mainly the type of related structure associated with faulting. The main type of data used comprised structural field data, well and resistivity data, remote sensing imagery, chronology of sediments, morphotectonic analysis, x-ray analysis, seismological and aeromagnetic data. Paleostress analysis indicates that at least two tectonic stress fields occurred in the study area: NSoriented compression and EW-oriented extension from the late Campanian to the early Miocene and EW-oriented compression and NS-oriented extension from the early Miocene to the Holocene. These stress fields reactivated NE-SW- and NW-SE-trending faults. Both set of faults exhibit right-lateral strike-slip kinematics, associated with a minor normal component. It was possible to determine the en echelon geometry of the Samambaia fault, which is ~63 km long, 13 km deep, presents NE-SW trend and strong dip to NW. Sedimentfilled faults in granite rocks yielded Optically Stimulated Luminescence (OSL) and Single-Aliquot Regeneration (SAR) ages at 8.000 - 9.000, 11.000 - 15.000, 16.000 - 24.000, 37.000 - 45.500, 53.609 - 67.959 e 83.000 - 84.000 yr BP. The analysis of the ductile fabric in the João Câmara area indicate that the regional foliation is NE-SW-oriented (032o - 042o), which coincides with the orientation of the epicenters and Si-rich veins. The collective evidence points to reactivation of preexisting structures. Paleoseismological data suggest paleoseismic activity much higher than the one indicated by the short historical and instrumental record

Mecanismos de alojamento de magmas granitóides :exemplo do plúton de Japi(RN)

This dissertation describes the igneous suites of the Japi granitoid pluton, intrusive in the Paleoproterozoic gneiss-migmatite complex of the eastern domain of the Seridó Belt, northeastern Brazil. Field relations show that the pluton is affected by strong deformation associated to the Brasiliano orogeny (known as the D3 phase) , with a NW-trending extensionalleft-hand senestral shear zone (the Japi Shear Zone, JSZ) bordering the intrusive body to the west. Four plutonic suites are found in the main pluton and as satellyte intrusions, besides Iate pegmatite and pink leucogranites. An alkaline granitoid suite, dominated by syenogranites bearing sodic augite (and subordinate hornblende), define a main elliptical intrusion. In its northern part, this intrusion is made up by concentric sheets, contrasting with a smaller rounded stock to the south. These granites display a pervasive solid-state S>L fabric developed under high T conditions, characterized by plastic deformation of quartz and feldspar. It is especially, developed along the border of the pluton, with inward dips. A regular magmatic layering is present sometimes, parallel to the tectonic foliation. The syntectonic emplacement as regards to the Brasiliano (D3) event is indicated by the common occurrence of dykes and sheets along transtensional or extensional sites of the major structure. Field relations attest to the early emplacement of the alkaline granites as regards to the other suites. A basic-to-intermediate suite occurs as a western satellyte body and occupying the southern tail of the main alkaline pluton. It comprises a wide variety of compositional terms, including primitive gabbros and gabbro-norites, differentiated to monzonitic intermediate facies containing amphibole and biotite as their main mafic phases. These rocks display transitional high-K calc-alkaline to shoshonitic affinities. Porphyritic monzogranite suítes commonly occur as dykes and minor intrusives, isolated or associated with the basic-tointermediate rocks. In the latter case, magma mingling and mixing features attest that these are contemporaneous igneous suites. These granites show K-feldspar phenocrysts and a hornblende+biotite+titanite assemblage, displaying subalkaline/monzonitic geochemical affinities. Both suites exhibit SL magmatic fabrics overprinting or transitional to solid-state D3 deformation related to the JSI. Chemical data clearly show that they are related to different parental magmas. Finally, a microgranite suite occurs along a few topographic ridges paralell to the JSI. It comprises dominantly granodiorites with a mineralogy similar to the one of the porphyritic granitoids. However, discriminant diagrams show their distinct calc-alkaline affinity. The granodiorites display an essencially magmatic fabric, even though an incipient D3 solid-state structure may be developed along the JSI. Intrusion relationships with the previous suites, as well as regards to the D3 structures, point to their Iate emplacement. All these suites are intrusive in a Paleoproterozoic, high-grade gneiss-migmatite complex affected by two previous deformation phases (D1, D2). The fabrics associated with these earlier events are folded and overprinted by the younger D3 structures along the JSZ. The younger deformation is characterized by NE-dipping foliations and N/NE-plunging stretching lineations. In the JSZ northern termination the foliation acquires an ENE orientation, containing a stretching lineation plunging to the south. Symmetric kinematic cri teria developed at this site confirms the transpressional termination of the JSZ, as also shown by orthorrombic quartz c-axis patterns. E-W-trending d extra I shear zones developed in the central part of the JSZ are interpreted as antithetic structures associated to the transtensional deformation along the JSZ. This is consistent with its extensional-transcurrent kinematics and a flat-and-ramp geometry at depth, as shown by gravimetric data. The lateral displacement of the negative residual Bouguer anomalies, as regards to the main outcropping alkaline pluton, may be modelized by other deeper-seated granite bodies. Based on numerical modelling it was possible to infer two distinct intrusion styles for the alkaline pluton. The calculated model values are consistent with an emplacement by sheeting for the northern body, as already suggested by satellyte imagery and field mapping. On the other hand, the results point to a transition towards a diapir-related style associated to the smaller. southern stock. This difference in intrusion styles may relate to intensity variations and transtensional sites of the shear deformation along the JSZ. Trace element and Sr and Nd isotopes of the alkaline granites are compatible with their derivation trom a more basic crustal source, as compared to the presently outcropping highgrade gneisses, with participation (or alternatively dominated by) of an enriched lithospheric mantle component. Like other igneous suites in the Seridó Belt, the high LlL contents and fractionated REE patterns of the basic rocks also point to an enriched mantle as the source for this kind of magmatism. Geochemical and isotope data are compatible with a lower crustal origin for the porphyritic granites. On the basis of the strong control of the JSZ on the emplacement of lower crustal (porphyritic and alkaline granites) or lithospheric mantle (basic rocks, alkaline granites or a component of them) magmas, one may infer a deep root for this structure, bearing an important role in magma extraction, transport and emplacement in the Japi region, eastern domain of the Seridó Belt

Estudos sedimentológicos e o contexto estrutural da formação Serra do Martins, nos platôs de Portalegre, Martins e Santana/RN

This dissertation deals with sedimentological and structural framework of the siliciclastic rock of the Serra do Martins Formation (FSM) in the Portalegre, Martins and Santana plateau, located to the south of Potiguar Basin, in the southwest and central Rio Grande do Norte state. This formation, regarded as of Oligo-Miocene age based on intrusive relations of the Miocene Macau volcanics, has a still disputable age due to the lack of appropriate bio and/or chronostratigraphic markers. The FSSM deposits crop out along 650 to 750 m high plateau, as a remanescent sedimentary cover directly overlying topographically uplifted pre-cambrian crystalline rocks. During the last decades, these deposits were interpreted according to a Tertiary paleoclimatic evolutionary model, associated to pedogenetic processes. The sedimentological characterization of the FSM was done through a detailed study of its facies, petrography and diagenetic features. The facies study was based on description of field relations, textures and structures, the piling up of the strata and their lateral variations. The FSM was deposited by an anastomosing to coarse-meandering fluvial system, including deposits of lag, cannel-fill, ouver-bank and flood plain. The petrographic composition of the sediments, coupled to their facies and paleocurrent directions, suggest a rather distal sourcearea, to the south of the present plateau. The diagenetic study identified an incipient grain mechanical compaction, pronounced dissolution of the framework, matrix and/or cement components, intense precipitation of kaolinite, silic and, eventually, iron oxides, besides mechanical infiltration of the clays. Most of these events, regarded in the literature as associated to near-surface conditions (eo or telodiagenesis), indicate the FSM sediments were never deeply buried. Topographic relations along longitudinal and transversal sections reaching the Potiguar Basin to the north identified regional dips that allow to discuss stratigraphic correlations between the FSM and the basin formations. The sedimentological features of the different units and the intrusive relations of the Macau volcanics were also considered in these correlations,which support the Oligo-Miocene age previously accepted for the FSM. Concerning the tectonic framework of the FSM, this work investigated the pre-cambrian to cretaceous heritage and the cenozoic deformation, allowing the recognition of pre-, sin and post-FSM structures. The crystalline basement, belonging to the Seridó Belt, displays NE and WNW foliation trends related to the Brasiliano-age ductile shear zones. In this terrain, brittle-ductile and brittle NE- and NW-trending structures, associated with extensional joints filled with pegmatites and quartz veins, are related to an E-W compression by the end of Brasiliano Cycle. The E-W joints and NE-trending fractures were reactivated by N-S to N-S to NW extension during late Jurassic to Cretaceous times, controlling the emplacement of the Rio Ceará-Mirim basic dyke swarm and the opening of the Potiguar rift basin

Estruturas de acumulação de água subterrânea em rochas cristalinas: estudo geofísico e geológico de casos no Estado do Rio Grande do Norte

Geological and geophysical studies (resistivity, self potential and VLF) were undertaken in the Tararaca and Santa Rita farms, respectively close to the Santo Antônio and Santa Cruz villages, eastern Rio Grande do Norte State, NE Brazil. Their aim was to characterize water acummulation structures in crystalline rocks. Based on geological and geophysical data, two models were characterized, the fracture-stream and the eluvio-alluvial through, in part already described in the literature. In the Tararaca Farm, a water well was located in a NW-trending streamlet; surrounding outcrops display fractures with the same orientation. Apparent resistivity sections, accross the stream channel, confirm fracturing at depth. The VLF profiles systematically display an alignment of equivalent current density anomalies, coinciding with the stream. Based on such data, the classical fracture-stream model seems to be well characterized at this place. In the Santa Rita Farm, a NE-trending stream display a metric-thick eluvioregolith-alluvial cover. The outcropping bedrock do not present fractures paralell to the stream direction, although the latter coincides with the trend of the gneiss foliation, which dips to the south. Geophysical data confirm the absence of a fracture zone at this place, but delineate the borders of a through-shaped structure filled with sediments (alluvium and regolith). The southern border of this structure dips steeper compared to the northern one. This water acummulation structure corresponds to an alternative model as regards to the classical fracture-stream, being named as the eluvio-alluvial trough. Its local controls are the drainage and relief, coupled with the bedrock weathering preferentially following foliation planes, generating the asymmetry of the through

Correlação entre contexto morfoestrutural e sismicidade nas regiões de João Câmara e São Rafael (RN)

This MSc thesis describes brittle deformation in two seismic zones located in north-eastern Brazil: João Câmara and São Rafael, Rio Grande do Norte State. Both areas show seismogenic faults, Samambaia and São Rafael, indicated by narrow zones of epicentres with a strike of 040o, a lenght of 30 km and 4 km, and a depth of 1-12 and 0,5-4 km, respectively. The first seismological and geological studies suggested blind faults or faults that were still in the beginning of the nucleation process. The region is under E-W-oriented compression and is underlain by Precambrian terrains, deformed by one or more orogenic cycles, which generated shear zones generally marked by strong pervasive foliation and sigmoidal shapes. The crystalline basement is capped by the Cretaceous Potiguar basin, which is also locally capped by Pliocene continental siliciclastic deposits (Barreiras Formation), and Quaternary alluvium. The main aim of this study was to map epicentral areas and find whether there are any surface geological or morphotectonic expression related to the seismogenic faults. A detailed geological map was carried out in both seismic areas in order to identify brittle structures and fault-related drainage/topographic features. Geological and morphotectonic evidence indicate that both seismogenic faults take place along dormant structures. They either cut Cenozoic rocks or show topographic expression, i.e., are related to topographic heights or depressions and straight river channels. Faults rocks in the Samambaia and São Rafael faults are cataclasite, fault breccia, fault gouge, pseudotachylyte, and quartz veins, which point to reactivation processes in different crustal levels. The age of the first Samambaia and the São Rafael faulting movement possibly ranges from late Precambrian to late Cretaceous. Both fault cut across Precambrian fabric. They also show evidence of brittle processes which took place between 4 and 12 km deep, which probably have not occurred in Cenozoic times. The findings are of great importance for regional seismic hazard. They indicate that fault zones are longer than previously suggested by seismogenic studies. According to the results, the methodology used during this thesis may also be useful in other neotectonic investigation in intraplate areas

Petrologia e geologia estrutural do plutão granítico Barcelona, província Borborema, NE do Brasil

The Dissertation aimed to advance the geological knowledge of the Barcelona Granitic Pluton (BGP). This body is located in the eastern portion of the Rio Grande do Norte Domain (RND), within the São José do Campestre subdomain (SJC), NE of the Borborema Province. The main goal was to understand the geological evolution of the rocks of the pluton and the tectonic setting of magma generation and its emplacement. The BGP has an assumed Ediacaran age and outcropping area of approximately 260 km2, being composed of three varied petrographic/textural facies: (a) porphyritic biotite monzogranite; (b) dykes and sheets of biotite microgranite; (c) dioritic to quartz-dioritic enclaves. The rocks of the BGP have the following structures: (i) a NE-SW and NW-SE directed magmatic fabric (Sγ), accompanied by a magmatic lineation (Lγ) with gentle dip to NE-SW and NW-SE. In the southern portion, there is the concentric pattern of this foliation with medium to high dip, and (ii) a solid state foliation, in part mylonitic (S3+), mainly on the eastern edge with slightly plunging to west. The integration of structural and gravity data permitted to interpret the emplacement of the BGP as controlled by the transcurrent shear zones systems Lajes Pintadas (LPSZ) and Sítio Novo (SNSZ), both of dextral strike-slip kinematics. Mineral chemistry data show that the amphibole form the porphyritic biotite monzogranite facies is hastingsite with moderate Mg / (Mg + Fe) ratios, indicating crystallization under moderate to high ƒO2 and cristallization pressure of around 5.0-6.0 kbar. The biotite tends to be slightly richer in annite molecule and plots in the transitional field from primary biotite to reequilibrated biotite. In discriminant diagrams of magmatic series, the biotite behave like those of subalkaline affinity, consistent with the potassium calc-alkaline / sub-alkaline geochemical affinity of the hosting rock. The opaque minerals are primarily magnetite, with some crystals martitized to hematite indicating relatively oxidizing conditions during magma evolution that originated the BGP. Zoning in plagioclase, K-feldspar and allanite crystals suggest fractional crystallization process. Lithogeochemical data suggest that the facies described for the BGP have similar magma source, usually plotting in the fields and trends of the subalkaline / high potassium calc-alkaline series.

Petrologia do Plutão Bom Jardim de Goiás (PBJG): implicação na evolução neoproterozoica da Província Tocantins

The Bom Jardim de Goiás Pluton (PBJG) is a semi-circular body, located in the central portion of the Tocantins Province, intrusive into orthogneisses and metassupracrustals of the Arenópolis Magmatic Arc. These metasupracrustals present a low to moderate dipping banding or schistosity, have a low to moderate angle of banding / foliation, defined by mica, andalusite and sillimanite and cordierite, which characterize an amphibolite facies metamorphism. This structure is crosscut by the emplacement of the PBJG rocks. The abrupt nature of the contacts and the absence of ductile structures indicate that the intrusion took place in a relatively cold crust. Under petrographic grounds, the pluton consists mainly of monzodiorites, tonalite and granodiorite, following the low to medium-K calk-alkaline alkaline trend. Rocks of the PBJG have hornblende and biotite as the main mafic phases, besides subordinate clinopyroxene, titanite, epidote and opaque. Late dikes of leucogranite contain only mineral biotite as relevant accessory mineral. One U-Pb zircon dating of a monzodiorite yielded an age of 550 ± 12 Ma (MSWD = 1.06). Whole-rock and mineral chemistry suggest that the studied rocks are calc-alkaline, having evolved by fractional crystallization of Ca- and Fe-Mg minerals under high oxygen fugacity. Using the amphibole-plagioclase geothermometer and the Al-in amphibole geobarometer, we calculate temperatures and pressures of, respectively, 692-791 °C e 2.4-5.0 kbar for the intrusion of the PBJG, which is corroborated by previous metamorphic assemblages in the country rocks. The geological, geochemical and geochronological features of PBJG demonstrate their post-tectonic or post-collisional nature, with emplacement into an already uplifted and relatively cool crust at the end of brasiliano orogeny in this portion of the Tocantins Province.

Petrologia do Plutão Bom Jardim de Goiás (PBJG): implicação na evolução neoproterozoica da Província Tocantins

The Bom Jardim de Goiás Pluton (PBJG) is a semi-circular body, located in the central portion of the Tocantins Province, intrusive into orthogneisses and metassupracrustals of the Arenópolis Magmatic Arc. These metasupracrustals present a low to moderate dipping banding or schistosity, have a low to moderate angle of banding / foliation, defined by mica, andalusite and sillimanite and cordierite, which characterize an amphibolite facies metamorphism. This structure is crosscut by the emplacement of the PBJG rocks. The abrupt nature of the contacts and the absence of ductile structures indicate that the intrusion took place in a relatively cold crust. Under petrographic grounds, the pluton consists mainly of monzodiorites, tonalite and granodiorite, following the low to medium-K calk-alkaline alkaline trend. Rocks of the PBJG have hornblende and biotite as the main mafic phases, besides subordinate clinopyroxene, titanite, epidote and opaque. Late dikes of leucogranite contain only mineral biotite as relevant accessory mineral. One U-Pb zircon dating of a monzodiorite yielded an age of 550 ± 12 Ma (MSWD = 1.06). Whole-rock and mineral chemistry suggest that the studied rocks are calc-alkaline, having evolved by fractional crystallization of Ca- and Fe-Mg minerals under high oxygen fugacity. Using the amphibole-plagioclase geothermometer and the Al-in amphibole geobarometer, we calculate temperatures and pressures of, respectively, 692-791 °C e 2.4-5.0 kbar for the intrusion of the PBJG, which is corroborated by previous metamorphic assemblages in the country rocks. The geological, geochemical and geochronological features of PBJG demonstrate their post-tectonic or post-collisional nature, with emplacement into an already uplifted and relatively cool crust at the end of brasiliano orogeny in this portion of the Tocantins Province.