Memory effects are studied in the simplest scalar–tensor theory, the Brans–Dicke (BD) theory. To ... more Memory effects are studied in the simplest scalar–tensor theory, the Brans–Dicke (BD) theory. To this end, we introduce, in BD theory, novel Kundt spacetimes (without and with gyratonic terms), which serve as backgrounds for the ensuing analysis on memory. The BD parameter $$\omega $$ ω and the scalar field ($$\phi $$ ϕ ) profile, expectedly, distinguishes between different solutions. Choosing specific localised forms for the free metric functions $$H'(u)$$ H ′ ( u ) (related to the wave profile) and J(u) (the gyraton) we obtain displacement memory effects using both geodesics and geodesic deviation. An interesting and easy-to-understand exactly solvable case arises when $$\omega =-2$$ ω = - 2 (with J(u) absent) which we discuss in detail. For other $$\omega $$ ω (in the presence of J or without), numerically obtained geodesics lead to results on displacement memory which appear to match qualitatively with those found from a deviation analysis. Thus, the issue of how memory effe...
Perturbations of giant magnons and single spikes in a 2 + 1 dimensional R × S background spacetim... more Perturbations of giant magnons and single spikes in a 2 + 1 dimensional R × S background spacetime are analysed. Using the form of the giant magnon solution in the Jevicki-Jin gauge, the well-known Jacobi equation for small normal deformations of an embedded time-like surface are written down. Surprisingly, this equation reduces to a simple wave equation in a Minkowski background. The finiteness of perturbations and the ensuing stability of such giant magnons under small deformations are then discussed. It turns out that only the zero mode has finite deformations and is stable. Thereafter, we move on to explore the single spike solution in the Jevicki-Jin gauge. We obtain and solve the perturbation equation numerically and address stability issues.
We discuss a proposal on how gravitational collapse of a NEC (Null Energy Condition) violating sp... more We discuss a proposal on how gravitational collapse of a NEC (Null Energy Condition) violating spherically symmetric fluid distribution can avoid the formation of a zero proper volume singularity and eventually lead to a Lorentzian wormhole geometry. Our idea is illustrated using a timeevolving wormhole spacetime in which, we show how a collapsing sphere may never reach a zero proper volume end-state. The nature of geodesic congruences in such spacetimes is considered and analysed. Our construction is inspired from a recently proposed static wormhole geometry, the multi-parameter Simpson-Visser line element, which is known to unite wormholes and black holes (regular and singular) in a single framework.
Displacement and velocity memory effects in the exact, vacuum, plane gravitational wave line elem... more Displacement and velocity memory effects in the exact, vacuum, plane gravitational wave line element have been studied recently by looking at the behavior of pairs of geodesics or via geodesic deviation. Instead, one may investigate the evolution of geodesic congruences. In our work here, we obtain the evolution of the kinematic variables which characterize timelike geodesic congruences, using chosen pulse profiles (square and sech squared) in the exact, plane gravitational wave line element. We also analyse the behavior of geodesic congruences in possible physical scenarios describable using derivatives (first, second and third) of one of the chosen pulses. Beginning with a discussion on the generic behaviour of such congruences and consequences thereof, we find exact analytical expressions for shear and expansion with the two chosen pulse profiles. Qualitatively similar numerical results are noted when various derivatives of the sech-squared pulse are used. We conclude that for geodesic congruences, a growth (or decay) of shear causes focusing of an initially parallel congruence, after the departure of the pulse. A correlation between the "focusing time" (or u value, u being the affine parameter) and the amplitude of the pulse (or its derivatives) is found. Such features distinctly suggest a memory effect, named in recent literature as B memory.
We obtain a large class of Lorentzian wormhole spacetimes in scalar-tensor gravity, for which the... more We obtain a large class of Lorentzian wormhole spacetimes in scalar-tensor gravity, for which the matter stress energy does satisfy the weak energy condition. Our constructions have zero Ricci scalar and an everywhere finite, non-zero scalar field profile. Interpreting the scalar-tensor gravity as an effective on-brane theory resulting from a two-brane Randall-Sundrum model of warped extra dimensions, it is possible to link wormhole existence with that of extra dimensions. We study the geometry, matter content, gravitational red-shift and circular orbits in such wormholes and argue that our examples are perhaps among those which may have some observational relevance in astrophysics in future. We also study traversability and find that our wormholes are indeed traversable for values of the metric parameters satisfying the weak energy condition.
Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations alo... more Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations along families of D dimensional surfaces embedded in a background N dimensional spacetime are discussed. These include string worldsheets embedded in four dimensional spacetimes and two dimensional timelike hypersurfaces in a three dimensional curved background. The issue of focussing of families of surfaces is introduced and analysed in some detail.
Branes are embedded surfaces in a given background (bulk) spacetime. Assuming a warped bulk, we i... more Branes are embedded surfaces in a given background (bulk) spacetime. Assuming a warped bulk, we investigate, in analogy with the case for geodesics, the notion of focusing of families of such embedded, extremal 3-branes in a five dimensional background. The essential tool behind our analysis, is the well-known generalised Raychaudhuri equations for surface congruences. In particular, we find explicit solutions of these equations, which seem to show that families of 3branes can focus along lower dimensional submanifolds depending on where the initial expansions are specified. We conclude with comments on the results obtained and possibilities about future work along similar lines.
We explore the existence of Lorentzian wormholes in the context of an effective on-brane, scalart... more We explore the existence of Lorentzian wormholes in the context of an effective on-brane, scalartensor theory of gravity. In such theories, the timelike convergence condition, which is always violated for wormholes, has contributions, via the field equations, from on-brane matter as well as from an effective geometric stress energy generated by a bulk-induced radion field. It is shown that, for a class of wormholes, the required on-brane matter, as seen by an on-brane observer in the Jordan frame, is not exotic and does not violate the Weak Energy Condition. The presence of the effective geometric stress energy in addition to on-brane matter, is largely responsible for creating this intriguing possibility. Thus, if such wormholes are ever found to exist in the Universe, they would clearly provide pointers towards the existence of a warped extra dimension as proposed in the two-brane model of Randall and Sundrum.
We study the kinematics of timelike geodesic congruences, in the spacetime geometry of rotating b... more We study the kinematics of timelike geodesic congruences, in the spacetime geometry of rotating black holes in three (the BTZ) and four (the Kerr) dimensions. The evolution (Raychaudhuri) equations for the expansion, shear and rotation along geodesic flows in the such spacetimes are obtained. For the BTZ case, the equations are solved analytically. The effect of the negative cosmological constant on the evolution of the expansion (θ), for congruences with and without an initial rotation (ω 0) is noted. Subsequently, the evolution equations, in the case of a Kerr black hole in four dimensions are written and solved numerically, for some specific geodesics flows. It turns out that, for the Kerr black hole, there exists a critical value of the initial expansion below (above) which we have focusing (defocusing). We delineate the dependencies of the expansion, on the black hole angular momentum parameter, a, as well as on ω 0. Further, the role of a and ω 0 on the time (affine parameter) of approach to singularity (defocusing/focusing) is studied. While the role of ω 0 on the time to singularity is as expected, the effect of a leads to an interesting new result.
The evolution of timelike geodesic congruences in a spherically symmetric, nonstatic, inhomogeneo... more The evolution of timelike geodesic congruences in a spherically symmetric, nonstatic, inhomogeneous spacetime representing gravitational collapse of a massless scalar field is studied. We delineate how initial values of the expansion, rotation and shear of a congruence, as well as the spacetime curvature, influence the global behavior and focusing properties of a family of trajectories. Under specific conditions, the expansion scalar is shown to exhibit a finite jump (from negative to positive value) before focusing eventually occurs. This nonmonotonic behavior of the expansion, observed in our numerical work, is successfully explained through an analysis of the equation for the expansion. Finally, we bring out the role of the metric parameters (related to nonstaticity and spatial inhomogeneity), in shaping the overall behavior of geodesic congruences.
Exact solutions of the string equations of motion in a specific Lorentzian wormhole background ar... more Exact solutions of the string equations of motion in a specific Lorentzian wormhole background are obtained. These include both closed and open string configurations. Perturbations about some of these configurations are investigated using the manifestly covariant formalism of Larsen and Frolov. Finally, the generalized Raychaudhuri equations for the corresponding string worldsheet deformations are written down and analysed briefly.
We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essentia... more We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essential features of the original article by Raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate non-Riemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion (Einstein-Cartan-Sciama-Kibble theory). Finally, we give an overview of some recent applications of these equations in general relativity, quantum field theory, string theory and the theory of relativisitic membranes. We conclude with a summary and provide our own perspectives on directions of future research.
We demonstrate, by a simple analysis, that cosmological line elements related by scale factor dua... more We demonstrate, by a simple analysis, that cosmological line elements related by scale factor duality also exhibit a duality with respect to the conservation/violation of the Weak Energy Condition (WEC) by the matter that acts as the source in the one-loop β function equations for the metric coupling written explicitly in the form of the Einstein equations. Furthermore, a study of specific pairs of line elements (obtained via O(d,d) transformations) hints at a possible generalisation of the above duality w.r.t. WEC for the case of O(d, d) related spacetimes. Consequences and extensions thereof are also pointed out.
In this article, we explore the kinematics of timelike geodesic congruences in warped five dimens... more In this article, we explore the kinematics of timelike geodesic congruences in warped five dimensional bulk spacetimes, with and without thick or thin branes. Beginning with geodesic flows in the Randall-Sundrum AdS (Anti de Sitter) geometry without and with branes we find analytical expressions for the expansion scalar and comment on the effects of including thin branes on its evolution. Later, we move on to congruences in more general warped bulk geometries with a cosmological thick brane and a time-dependent extra dimensional scale. Using analytical expressions for the velocity field, we interpret the expansion, shear and rotation (ESR) along the flows, as functions of the extra dimensional coordinate. The evolution of a cross-sectional area orthogonal to the congruence, as seen from a local observer's point of view, is also shown graphically. Finally, the Raychaudhuri and geodesic equations in backgrounds with a thick brane are solved numerically in order to figure out the role of initial conditions (prescribed on the ESR) and spacetime curvature on the evolution of the ESR.
We explore the possibilities of constructing bulk spacetimes in five dimensions for warped branew... more We explore the possibilities of constructing bulk spacetimes in five dimensions for warped braneworld models with a spatially flat Friedmann-Robertson-Walker (FRW) line element on the 3-brane and with a time-dependent extra dimension. Our first step in this direction involves looking at the status of energy conditions when such a bulk line element is assumed. We check these conditions by analysing the relevant inequalities, for specific functional forms (chosen to satisfy certain desirable features) of the warp factor, the cosmological scale factor and the extra-dimensional scale factor. Subsequently, we aim at obtaining solutions with different types of bulk matter sources. We begin with a general analysis of the solution space of non-singular Randall-Sundrum type bulk models with an exponential warp factor and a chosen equation of state. Thereafter, we focus on three specific bulk sources-the ordinary scalar field, the Brans-Dicke scalar and the dilaton. In each case, we are able to solve the field equations and obtain desirable solutions for which, we once again check the viability of the energy conditions. We also show how one can place branes in the bulk using the junction conditions. The issue of resolution of the bulk singularities which appear in our solutions, using standard methods, is also presented briefly. In summary, we are able to demonstrate, that it is indeed possible to construct viable bulk spacetimes for warped cosmological braneworlds with a time-varying extra dimension and with bulk matter satisfying the energy conditions.
We study the kinematics of timelike geodesic congruences in two and four dimensions in spacetime ... more We study the kinematics of timelike geodesic congruences in two and four dimensions in spacetime geometries representing stringy black holes. The Raychaudhuri equations for the kinematical quantities (namely, expansion, shear and rotation) characterising such geodesic flows are written down and subsequently solved analytically (in two dimensions) and numerically (in four dimensions) for specific geodesics flows. We compare between geodesic flows in dual (electric and magnetic) stringy black hole backgrounds in four dimensions, by showing the differences that arise in the corresponding evolutions of the kinematic variables. The crucial role of initial conditions and the spacetime curvature on the evolution of the kinematical variables is illustrated. Some novel general conclusions on caustic formation and geodesic focusing are obtained from the analytical and numerical findings. We also propose a new quantifier in terms of the time (affine parameter) of approach to a singularity, which may be used to distinguish between flows in different geometries. In summary, our quantitative findings bring out hitherto unknown features of the kinematics of geodesic flows, which, otherwise, would have remained overlooked, if we confined ourselves to only a qualitative analysis.
Renormalisation Group (RG) flows in theory space (the space of couplings) are generated by a vect... more Renormalisation Group (RG) flows in theory space (the space of couplings) are generated by a vector field-the β function. Using a specific metric ansatz in theory space and certain methods employed largely in the context of General Relativity, we examine the nature of the expansion, shear and rotation of geodesic RG flows. The expansion turns out to be a negative quantity inversely related to the norm of the β function. This implies the focusing of the flows towards the fixed points of a given field theory. The evolution equation for the expansion along geodesic RG flows is written down and analysed. We illustrate the results for a scalar field theory with a jφ coupling and pointers to other areas are briefly mentioned.
Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations alo... more Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations along families of D dimensional surfaces embedded in a background N dimensional spacetime are discussed. These include string worldsheets embedded in four dimensional spacetimes and two dimensional timelike hypersurfaces in a three dimensional curved background. The issue of focussing of families of surfaces is introduced and analysed in some detail.
Schild's null (tensionless) strings are discussed in certain flat and curved backgrounds. We find... more Schild's null (tensionless) strings are discussed in certain flat and curved backgrounds. We find closed, stationary, null strings as natural configurations existing on the horizons of spacetimes which possess such null hypersurfaces. Examples of these are obtained in Schwarzschild and Rindler spacetimes. A dynamic null string is also identified in Rindler spacetime. Furthermore, a general prescription (with explicit examples) is outlined by means of which null string configurations can be obtained in a large class of cosmological backgrounds .
A two-parameter family of spherically symmetric, static Lorentzian wormholes is obtained as the g... more A two-parameter family of spherically symmetric, static Lorentzian wormholes is obtained as the general solution of the equation ρ = ρt = 0, where ρ = Tij u i u j , ρt = (Tij − 1 2 T gij) u i u j , and u i ui = −1. This equation characterizes a class of spacetimes which are "self dual" (in the sense of electrogravity duality). The class includes the Schwarzschild black hole, a family of naked singularities, and a disjoint family of Lorentzian wormholes, all of which have vanishing scalar curvature (R = 0). Properties of these spacetimes are discussed. Using isotropic coordinates we delineate clearly the domains of parameter space for which wormholes, nakedly singular spacetimes and the Schwarzschild black hole can be obtained. A model for the required "exotic" stress-energy is discussed, and the notion of traversability for the wormholes is also examined.
Memory effects are studied in the simplest scalar–tensor theory, the Brans–Dicke (BD) theory. To ... more Memory effects are studied in the simplest scalar–tensor theory, the Brans–Dicke (BD) theory. To this end, we introduce, in BD theory, novel Kundt spacetimes (without and with gyratonic terms), which serve as backgrounds for the ensuing analysis on memory. The BD parameter $$\omega $$ ω and the scalar field ($$\phi $$ ϕ ) profile, expectedly, distinguishes between different solutions. Choosing specific localised forms for the free metric functions $$H'(u)$$ H ′ ( u ) (related to the wave profile) and J(u) (the gyraton) we obtain displacement memory effects using both geodesics and geodesic deviation. An interesting and easy-to-understand exactly solvable case arises when $$\omega =-2$$ ω = - 2 (with J(u) absent) which we discuss in detail. For other $$\omega $$ ω (in the presence of J or without), numerically obtained geodesics lead to results on displacement memory which appear to match qualitatively with those found from a deviation analysis. Thus, the issue of how memory effe...
Perturbations of giant magnons and single spikes in a 2 + 1 dimensional R × S background spacetim... more Perturbations of giant magnons and single spikes in a 2 + 1 dimensional R × S background spacetime are analysed. Using the form of the giant magnon solution in the Jevicki-Jin gauge, the well-known Jacobi equation for small normal deformations of an embedded time-like surface are written down. Surprisingly, this equation reduces to a simple wave equation in a Minkowski background. The finiteness of perturbations and the ensuing stability of such giant magnons under small deformations are then discussed. It turns out that only the zero mode has finite deformations and is stable. Thereafter, we move on to explore the single spike solution in the Jevicki-Jin gauge. We obtain and solve the perturbation equation numerically and address stability issues.
We discuss a proposal on how gravitational collapse of a NEC (Null Energy Condition) violating sp... more We discuss a proposal on how gravitational collapse of a NEC (Null Energy Condition) violating spherically symmetric fluid distribution can avoid the formation of a zero proper volume singularity and eventually lead to a Lorentzian wormhole geometry. Our idea is illustrated using a timeevolving wormhole spacetime in which, we show how a collapsing sphere may never reach a zero proper volume end-state. The nature of geodesic congruences in such spacetimes is considered and analysed. Our construction is inspired from a recently proposed static wormhole geometry, the multi-parameter Simpson-Visser line element, which is known to unite wormholes and black holes (regular and singular) in a single framework.
Displacement and velocity memory effects in the exact, vacuum, plane gravitational wave line elem... more Displacement and velocity memory effects in the exact, vacuum, plane gravitational wave line element have been studied recently by looking at the behavior of pairs of geodesics or via geodesic deviation. Instead, one may investigate the evolution of geodesic congruences. In our work here, we obtain the evolution of the kinematic variables which characterize timelike geodesic congruences, using chosen pulse profiles (square and sech squared) in the exact, plane gravitational wave line element. We also analyse the behavior of geodesic congruences in possible physical scenarios describable using derivatives (first, second and third) of one of the chosen pulses. Beginning with a discussion on the generic behaviour of such congruences and consequences thereof, we find exact analytical expressions for shear and expansion with the two chosen pulse profiles. Qualitatively similar numerical results are noted when various derivatives of the sech-squared pulse are used. We conclude that for geodesic congruences, a growth (or decay) of shear causes focusing of an initially parallel congruence, after the departure of the pulse. A correlation between the "focusing time" (or u value, u being the affine parameter) and the amplitude of the pulse (or its derivatives) is found. Such features distinctly suggest a memory effect, named in recent literature as B memory.
We obtain a large class of Lorentzian wormhole spacetimes in scalar-tensor gravity, for which the... more We obtain a large class of Lorentzian wormhole spacetimes in scalar-tensor gravity, for which the matter stress energy does satisfy the weak energy condition. Our constructions have zero Ricci scalar and an everywhere finite, non-zero scalar field profile. Interpreting the scalar-tensor gravity as an effective on-brane theory resulting from a two-brane Randall-Sundrum model of warped extra dimensions, it is possible to link wormhole existence with that of extra dimensions. We study the geometry, matter content, gravitational red-shift and circular orbits in such wormholes and argue that our examples are perhaps among those which may have some observational relevance in astrophysics in future. We also study traversability and find that our wormholes are indeed traversable for values of the metric parameters satisfying the weak energy condition.
Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations alo... more Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations along families of D dimensional surfaces embedded in a background N dimensional spacetime are discussed. These include string worldsheets embedded in four dimensional spacetimes and two dimensional timelike hypersurfaces in a three dimensional curved background. The issue of focussing of families of surfaces is introduced and analysed in some detail.
Branes are embedded surfaces in a given background (bulk) spacetime. Assuming a warped bulk, we i... more Branes are embedded surfaces in a given background (bulk) spacetime. Assuming a warped bulk, we investigate, in analogy with the case for geodesics, the notion of focusing of families of such embedded, extremal 3-branes in a five dimensional background. The essential tool behind our analysis, is the well-known generalised Raychaudhuri equations for surface congruences. In particular, we find explicit solutions of these equations, which seem to show that families of 3branes can focus along lower dimensional submanifolds depending on where the initial expansions are specified. We conclude with comments on the results obtained and possibilities about future work along similar lines.
We explore the existence of Lorentzian wormholes in the context of an effective on-brane, scalart... more We explore the existence of Lorentzian wormholes in the context of an effective on-brane, scalartensor theory of gravity. In such theories, the timelike convergence condition, which is always violated for wormholes, has contributions, via the field equations, from on-brane matter as well as from an effective geometric stress energy generated by a bulk-induced radion field. It is shown that, for a class of wormholes, the required on-brane matter, as seen by an on-brane observer in the Jordan frame, is not exotic and does not violate the Weak Energy Condition. The presence of the effective geometric stress energy in addition to on-brane matter, is largely responsible for creating this intriguing possibility. Thus, if such wormholes are ever found to exist in the Universe, they would clearly provide pointers towards the existence of a warped extra dimension as proposed in the two-brane model of Randall and Sundrum.
We study the kinematics of timelike geodesic congruences, in the spacetime geometry of rotating b... more We study the kinematics of timelike geodesic congruences, in the spacetime geometry of rotating black holes in three (the BTZ) and four (the Kerr) dimensions. The evolution (Raychaudhuri) equations for the expansion, shear and rotation along geodesic flows in the such spacetimes are obtained. For the BTZ case, the equations are solved analytically. The effect of the negative cosmological constant on the evolution of the expansion (θ), for congruences with and without an initial rotation (ω 0) is noted. Subsequently, the evolution equations, in the case of a Kerr black hole in four dimensions are written and solved numerically, for some specific geodesics flows. It turns out that, for the Kerr black hole, there exists a critical value of the initial expansion below (above) which we have focusing (defocusing). We delineate the dependencies of the expansion, on the black hole angular momentum parameter, a, as well as on ω 0. Further, the role of a and ω 0 on the time (affine parameter) of approach to singularity (defocusing/focusing) is studied. While the role of ω 0 on the time to singularity is as expected, the effect of a leads to an interesting new result.
The evolution of timelike geodesic congruences in a spherically symmetric, nonstatic, inhomogeneo... more The evolution of timelike geodesic congruences in a spherically symmetric, nonstatic, inhomogeneous spacetime representing gravitational collapse of a massless scalar field is studied. We delineate how initial values of the expansion, rotation and shear of a congruence, as well as the spacetime curvature, influence the global behavior and focusing properties of a family of trajectories. Under specific conditions, the expansion scalar is shown to exhibit a finite jump (from negative to positive value) before focusing eventually occurs. This nonmonotonic behavior of the expansion, observed in our numerical work, is successfully explained through an analysis of the equation for the expansion. Finally, we bring out the role of the metric parameters (related to nonstaticity and spatial inhomogeneity), in shaping the overall behavior of geodesic congruences.
Exact solutions of the string equations of motion in a specific Lorentzian wormhole background ar... more Exact solutions of the string equations of motion in a specific Lorentzian wormhole background are obtained. These include both closed and open string configurations. Perturbations about some of these configurations are investigated using the manifestly covariant formalism of Larsen and Frolov. Finally, the generalized Raychaudhuri equations for the corresponding string worldsheet deformations are written down and analysed briefly.
We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essentia... more We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essential features of the original article by Raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate non-Riemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion (Einstein-Cartan-Sciama-Kibble theory). Finally, we give an overview of some recent applications of these equations in general relativity, quantum field theory, string theory and the theory of relativisitic membranes. We conclude with a summary and provide our own perspectives on directions of future research.
We demonstrate, by a simple analysis, that cosmological line elements related by scale factor dua... more We demonstrate, by a simple analysis, that cosmological line elements related by scale factor duality also exhibit a duality with respect to the conservation/violation of the Weak Energy Condition (WEC) by the matter that acts as the source in the one-loop β function equations for the metric coupling written explicitly in the form of the Einstein equations. Furthermore, a study of specific pairs of line elements (obtained via O(d,d) transformations) hints at a possible generalisation of the above duality w.r.t. WEC for the case of O(d, d) related spacetimes. Consequences and extensions thereof are also pointed out.
In this article, we explore the kinematics of timelike geodesic congruences in warped five dimens... more In this article, we explore the kinematics of timelike geodesic congruences in warped five dimensional bulk spacetimes, with and without thick or thin branes. Beginning with geodesic flows in the Randall-Sundrum AdS (Anti de Sitter) geometry without and with branes we find analytical expressions for the expansion scalar and comment on the effects of including thin branes on its evolution. Later, we move on to congruences in more general warped bulk geometries with a cosmological thick brane and a time-dependent extra dimensional scale. Using analytical expressions for the velocity field, we interpret the expansion, shear and rotation (ESR) along the flows, as functions of the extra dimensional coordinate. The evolution of a cross-sectional area orthogonal to the congruence, as seen from a local observer's point of view, is also shown graphically. Finally, the Raychaudhuri and geodesic equations in backgrounds with a thick brane are solved numerically in order to figure out the role of initial conditions (prescribed on the ESR) and spacetime curvature on the evolution of the ESR.
We explore the possibilities of constructing bulk spacetimes in five dimensions for warped branew... more We explore the possibilities of constructing bulk spacetimes in five dimensions for warped braneworld models with a spatially flat Friedmann-Robertson-Walker (FRW) line element on the 3-brane and with a time-dependent extra dimension. Our first step in this direction involves looking at the status of energy conditions when such a bulk line element is assumed. We check these conditions by analysing the relevant inequalities, for specific functional forms (chosen to satisfy certain desirable features) of the warp factor, the cosmological scale factor and the extra-dimensional scale factor. Subsequently, we aim at obtaining solutions with different types of bulk matter sources. We begin with a general analysis of the solution space of non-singular Randall-Sundrum type bulk models with an exponential warp factor and a chosen equation of state. Thereafter, we focus on three specific bulk sources-the ordinary scalar field, the Brans-Dicke scalar and the dilaton. In each case, we are able to solve the field equations and obtain desirable solutions for which, we once again check the viability of the energy conditions. We also show how one can place branes in the bulk using the junction conditions. The issue of resolution of the bulk singularities which appear in our solutions, using standard methods, is also presented briefly. In summary, we are able to demonstrate, that it is indeed possible to construct viable bulk spacetimes for warped cosmological braneworlds with a time-varying extra dimension and with bulk matter satisfying the energy conditions.
We study the kinematics of timelike geodesic congruences in two and four dimensions in spacetime ... more We study the kinematics of timelike geodesic congruences in two and four dimensions in spacetime geometries representing stringy black holes. The Raychaudhuri equations for the kinematical quantities (namely, expansion, shear and rotation) characterising such geodesic flows are written down and subsequently solved analytically (in two dimensions) and numerically (in four dimensions) for specific geodesics flows. We compare between geodesic flows in dual (electric and magnetic) stringy black hole backgrounds in four dimensions, by showing the differences that arise in the corresponding evolutions of the kinematic variables. The crucial role of initial conditions and the spacetime curvature on the evolution of the kinematical variables is illustrated. Some novel general conclusions on caustic formation and geodesic focusing are obtained from the analytical and numerical findings. We also propose a new quantifier in terms of the time (affine parameter) of approach to a singularity, which may be used to distinguish between flows in different geometries. In summary, our quantitative findings bring out hitherto unknown features of the kinematics of geodesic flows, which, otherwise, would have remained overlooked, if we confined ourselves to only a qualitative analysis.
Renormalisation Group (RG) flows in theory space (the space of couplings) are generated by a vect... more Renormalisation Group (RG) flows in theory space (the space of couplings) are generated by a vector field-the β function. Using a specific metric ansatz in theory space and certain methods employed largely in the context of General Relativity, we examine the nature of the expansion, shear and rotation of geodesic RG flows. The expansion turns out to be a negative quantity inversely related to the norm of the β function. This implies the focusing of the flows towards the fixed points of a given field theory. The evolution equation for the expansion along geodesic RG flows is written down and analysed. We illustrate the results for a scalar field theory with a jφ coupling and pointers to other areas are briefly mentioned.
Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations alo... more Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations along families of D dimensional surfaces embedded in a background N dimensional spacetime are discussed. These include string worldsheets embedded in four dimensional spacetimes and two dimensional timelike hypersurfaces in a three dimensional curved background. The issue of focussing of families of surfaces is introduced and analysed in some detail.
Schild's null (tensionless) strings are discussed in certain flat and curved backgrounds. We find... more Schild's null (tensionless) strings are discussed in certain flat and curved backgrounds. We find closed, stationary, null strings as natural configurations existing on the horizons of spacetimes which possess such null hypersurfaces. Examples of these are obtained in Schwarzschild and Rindler spacetimes. A dynamic null string is also identified in Rindler spacetime. Furthermore, a general prescription (with explicit examples) is outlined by means of which null string configurations can be obtained in a large class of cosmological backgrounds .
A two-parameter family of spherically symmetric, static Lorentzian wormholes is obtained as the g... more A two-parameter family of spherically symmetric, static Lorentzian wormholes is obtained as the general solution of the equation ρ = ρt = 0, where ρ = Tij u i u j , ρt = (Tij − 1 2 T gij) u i u j , and u i ui = −1. This equation characterizes a class of spacetimes which are "self dual" (in the sense of electrogravity duality). The class includes the Schwarzschild black hole, a family of naked singularities, and a disjoint family of Lorentzian wormholes, all of which have vanishing scalar curvature (R = 0). Properties of these spacetimes are discussed. Using isotropic coordinates we delineate clearly the domains of parameter space for which wormholes, nakedly singular spacetimes and the Schwarzschild black hole can be obtained. A model for the required "exotic" stress-energy is discussed, and the notion of traversability for the wormholes is also examined.
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Papers by Sayan Kar