Einstein's General Theory of Relativity, With Modern Applications in Cosmology
Abstract
Many of us have experienced the same; fallen and broken something. Yet supposedly, gravity is the weakest of the fundamental forces; it is claimed to be 10-15 times weaker than electromagnetism. Still, every one of us has more or less had a personal relationship with gravity. Einstein's General Theory of Relativity: With Modern Applications in Cosmology by Oyvind Gron and Sigbjorn Hervik is about gravity and the concept of gravity as Albert Einstein saw it- curved spaces, four-dimensional manifolds and geodesics. The book starts with the 1st principals of relativity and an introduction to Einstein's field equations. Next up are the three classical tests of the relativity theory and an introduction to black holes. The book contains several topics not found in other textbooks, such as Kaluza-Klein theory, anisotropic models of the universe, and new developments involving brane cosmology. Gron and Hervik have included a part in the book called "Advanced Topics." These topics range from the very edge of research to older, accepted ideas. In particular, the last two chapters deal with Einstein gravity in five dimensions, which has been a hot topic of research in recent years. On one final note, cosmology has also proven to be a very important testing arena for the general theory of relativity and a large part of the book is therefore devoted to this subject. © 2007 Springer Science+Business Media, LLC. All rights reserved.
Chapters (18)
To obtain a mathematical description of physical phenomena, it is advantageous to introduce a reference frame in order to keep track of the position of events in space and time. The choice of reference frame has historically depended upon the view of human beings and their position in the Universe.
In this chapter we shall give a short introduction to the fundamental principles of the special theory of relativity, and deduce some of the consequences of the theory.
We shall present the theory of differential forms in a way so that the structure of the theory appears as clearly as possibly. In later chapters this formalism will be used to give a mathematical formulation of the fundamental principles of the general theory of relativity. It will also be employed to give an invariant formulation of Maxwell’s equations so that the equations can be applied with reference to an arbitrary basis in curved spacetime.
In this chapter we are going to introduce the basic concepts necessary to grasp the geometrical significance of the metric tensor.
In this chapter we shall consider some consequences of the formalism developed so far, by studying the relativistic kinematics in two types of non-inertial reference frames: the rotating reference frame and the uniformly accelerating reference frame.
Forms prove to be a powerful tool in differential geometry and in physics. They have many wonderful properties that we shall explore further in this chapter. We know that in physics and mathematics, integration and differentiation are important, if not essential, operations that appear in almost all physical theories. In this chapter we will explore differentiation on curved manifolds and reveal several interesting properties.
We have seen, for example in rotating reference frames, that the geometry in a space with non-vanishing acceleration of gravity, may be non-Euclidean. It is easy to visualize curves and surfaces in three-dimensional space but it is difficult to grasp visually what curvature means in three-dimensional space, or worse still, in four-dimensional space-time. However the curvature of such spacesmay be discussed using the lower dimensional analogues of curves and surfaces. It is therefore important to have a good knowledge of the differential geometry of surfaces. Also the formalism used in describing surfaces may be taken over with minor modifications, when we are going to describe the geometric properties of curved space-time.
Einstein’s field equations are the relativistic generalization of Newton’s law of gravitation. Einstein’s vision, based on the equality of inertial and gravitational masses, was that there is no gravitational force at all. What is said to be “particle motion under the influence of the gravitational force” in Newtonian theory, is according to the general theory of relativity, free motion along geodesic curves in a curved space-time.
Einstein’s theory of general relativity leads to Newtonian gravity in the limit when the gravitational field is weak and static and the particles in the gravitational field moves slowly compared to the velocity of light. In the case of mass distributions of limited extension the field is weak at distances much larger than the Schwarzschild radius of the mass (see Chapter 10). At such distances the absolute value of the gravitational potential is much less than 1, and there is approximately Minkowski spacetime.
We have now established the Einstein field equations and explained their contents. In this chapter we will explore the first known non-trivial solution to these equations. The solution is due to the astronomer Karl Schwarzschild, and in his honour the solution is referred to as the Schwarzschild solution for empty space. This solution represents a spacetime outside a non-rotating black hole. The Kerr solution representing spacetime outside a rotating black hole will also be deduced. Finally, interior solutions will be investigated.
One of the most successful and useful applications of Einstein’s General Theory of Relativity is within the field of cosmology. Newton’s theory of gravitation, involves attraction between celestial bodies. However, very little is said of the evolution of the universe itself. The universe was believed to be static, and its evolution was beyond any physical theory. But after the year 1917, things were different. Within two years after the birth of the General Theory of Relativity, Einstein realized that this theory actually could say something about the universe and constructed a static universemodel as a solution of the relativistic field equations. The era of modern cosmology had begun, which would revolutionize our view of the universe.
Soon after Einstein had introduced the cosmological constant he withdrew it and called it “the biggest blunder” of his life. However, there has been developments in the last decades that have given new life to the cosmological constant. Firstly, the idea of inflation gave cosmology a whole new view upon the first split second of our universe. A key ingredient in the inflationary model is the behaviour of models that have a cosmological constant-like behaviour. Secondly, recent observations may indicate that we live in an accelerated universe. The inclusion of a cosmological constant can give rise to such behaviour as we will show in this chapter. We will first start with the static solution that Einstein found and was the reason that Einstein introduced the cosmological constant in the first place.
In this chapter we will investigate anisotropic and inhomogeneous universe models. If we relax the cosmological principles a bit we can get new and interesting models of our universe. Actually, one of the main goals of cosmology today is to explain the isotropy and homogeneity the universe has and in order to explain a certain property of the universe one has to consider sufficiently generalmodels that need not have this property. In this chapter we will investigate the Bianchi type I universe model and the inhomogeneous Lemaître-Tolman-Bondi (LTB) universe models. The Bianchi type I model is the simplest of the spatially homogeneous models which allows for anisotropy and the LTB-models are inhomogeneous universe models with spherically symmetric three-space.
In this chapter we will perform a 3+1 decomposition of the spacetime. This decomposition is very useful for various applications, in particular, we will use the 3+1 decomposition to derive a Lagrangian and Hamiltonian formalismof general relativity. We will also see howthe singularity theoremcan be described in this framework.
In this section we will explore the concept of symmetries even further. We introduced some of the basics in chapter 6, and we will pursue the ideas further here. In doing so, we will generalise the FRW models to the Bianchi models which are in general spatially homogeneous but not necessarily isotropic.
A question that often arises in gravitational theory is what happens to the geometry of spacewhen there is a jump discontinuity in the energy-momentum tensor along a surface. For example, what is the connection between the curvature properties for the interior Schwarzschild solution and the exterior Schwarzschild solution? Here, along the boundary of some surface, the energy density experiences a jump discontinuity. Another analogous scenario is for example a shock wave propagating outwards from an exploding star. In models of such shock waves the density can be infinite.
In 1999, Lisa Randall and Raman Sundrumpresented a five-dimensional model for our universe [RS99b, RS99a]. They imagined our four-dimensional world as a brane-world or a surface layer in a five-dimensional bulk. This bulk may be infinite in size, but due to the special properties of the bulk the gravitational fields are effectively localised to the brane. The other standard model fields are confined to the brane; only gravity is allowed to propagate in the fifth dimension.
Already in 1914 – before Einstein had fulfilled the construction of the general theory of relativity – Gunnar Nordström1 had published a five-dimensional scalar-tensor theory of gravitation in an effort to unify gravitation and electromagnetism. Since it was based upon his own theory of gravitation which was soon surpassed by Einstein’s theory, this work was neglected for several decades.
... And having assumed that u = m d=1 u d , we have that in (4 ...
... Where n is the dimension of the variety, g ij is the element of the inverse matrix of the metric tensor g ij and |g| is the modulus of the determinant of the metric tensor matrix; for further study of the topics Riemannian and pseudo-Riemannian varieties with related metrics and the Laplace-Beltrami operator, one can consult the text [4] in the bibliography. However, (4.17) has already been used in the Riemannian variety of the Poincairè semiplane in Example 8; this is precisely the hyperbolic Laplacian ∆ H , and it is from a generalization of (4.13) in terms of metric tensors that we will develop this topic. ...
... We will not go into purely physical matters for example near the singularities of the metric tensor where these are static black holes and the event horizon is a spherical surface and without charge. For further discussion see ch 10 [4]. Also in (4.20) we will treat t as a spatial coordinate, which means that we will have another time variable if we want to consider it that way, or a parameter called τ . ...
In this article we study generalizations of the inhomogeneous Burgers equation. First at the operator level, in the sense that we replace classical differential derivations by operators with certain properties, and then we increase the spatial dimensions of the Burgers equation, which is usually studied in one spatial dimension. This allows us, in one dimension, to find mathematical relationships between solutions of hyperbolic Brownian motion and the Burgers equations, which usually study the behaviour of mechanical fluids, and also, through appropriate transformations, to obtain in some cases exact solutions that depend on Hermite polynomials composed of appropriate functions. In the multi-dimensional case, this generalization allows us, by means of the method of invariant spaces, to find exact solutions on Riemannian and pseudo-Riemannian varieties, such as Schwarzschild and Ricci Solitons space, with time dictated by fractional derivatives, such as a Caputo-type operator of fractional evolution.
... That generalization of this equation was obtained in several forms in some applications for example for time like and null geodesics and for spinning and hot gravitating fluids and in strings and membranes and in different vectors [5,11]. The generalized Raychaudhuri equation substituted in absence of an extra force [3] and it calculated in f (R) modified gravity with non minimal coupling constant [4]. We will obtain this generalizing equation by calculating the extra force in f (G) modified gravity with non minimal coupling constant in imperfect fluid in a compared with f (R) modified gravity for perfect fluid. ...
... The Raychaudhuri equation can be written in the presence of an extra force as follows [11,3]θ ...
... The vorticity ω µν satisfies the equation (see [3,11]) ...
The equation of motion is the important equation for obtain the extra force and Raychaudhuri equation. By considering an explicitly coupling between an arbitrary function of the scalar Gauss-Bonnet, G and the Lagrangian density of matter, it is shown that an extra force normal to their four-velocities arises. In this paper, we obtain the extra force and the generalized Raychaudhuri equation in F(G) modified theory of gravity in an imperfect fluid for the massive particle by divergence of energy momentum tensor so we earn extra force an Raychaudhuri equation in a compared with f(R) modified gravity for perfect fluid this conclusion giving the evolution of the kinematical quantities and describing the relative accelerations of nearby particles .
... However, local times at different positions will not combine into one physically acceptable global time (cf. Section 2 in [16], Sections 5.2-5.3 in [17]). According to Einstein, an acceptable global time should ensure that physical laws, particularly the law governing the propagation of light, do not depend explicitly on time. ...
... (There are exceptions. For example, in [17], p. 22, coordinates are associated with coordinate clocks, and the local clocks in a static gravitational field are said to run slow, because they lag behind coordinate clocks at the same locations [17], p. 219). Only after a solution of the field equations has been obtained, and an expression for ds 2 in terms of coordinates has been found, does it make sense to ask about the meaning of the coordinates in terms of measured length and time intervals. ...
... (There are exceptions. For example, in [17], p. 22, coordinates are associated with coordinate clocks, and the local clocks in a static gravitational field are said to run slow, because they lag behind coordinate clocks at the same locations [17], p. 219). Only after a solution of the field equations has been obtained, and an expression for ds 2 in terms of coordinates has been found, does it make sense to ask about the meaning of the coordinates in terms of measured length and time intervals. ...
In his groundbreaking 1905 paper on special relativity, Einstein distinguished between local and global time in inertial systems, introducing his famous definition of distant simultaneity to give physical content to the notion of global time. Over the following decade, Einstein attempted to generalize this analysis of relativistic time to include accelerated frames of reference, which, according to the principle of equivalence, should also account for time in the presence of gravity. Characteristically, Einstein’s methodology during this period focused on simple, intuitively accessible physical situations, exhibiting a high degree of symmetry. However, in the final general theory of relativity, the a priori existence of such global symmetries cannot be assumed. Despite this, Einstein repeated some of his early reasoning patterns even in his 1916 review paper on general relativity and in later writings. Modern commentators have criticized these arguments as confused, invalid, and inconsistent. Here, we defend Einstein in the specific context of his use of global time and his derivations of the gravitational redshift formula. We argue that a detailed examination of Einstein’s early work clarifies his later reasoning and demonstrates its consistency and validity.
... [13,15] In other words, one of the significant implications of general relativity in conventional cosmology is that what causes celestial bodies to orbit in curved paths is not a force called gravity per se, but rather that these bodies follow the nearest mass in a curved spacetime where gravity is a consequence of this curvature, [16,18] and this movement takes place on a direct path known as a geodesic. [12,14,16,19] A geodesic generalizes the concept of a straight line in three-dimensional space to curved spacetime and, according to the definition, is the shortest or longest path between two adjacent points. [16,17,19] (Figure 5) ...
... [12,14,16,19] A geodesic generalizes the concept of a straight line in three-dimensional space to curved spacetime and, according to the definition, is the shortest or longest path between two adjacent points. [16,17,19] (Figure 5) ...
... This means that this speed is the same for all observers, regardless of their state of motion. [1,6,8,19] Another significant consequence of special relativity, one can refer to time dilation and length contraction as well. [10,19] According to this theory, in addition to the mass of an object, the passage of time and subsequently the length of an object are considered relativistic quantities, and they depend on the speed of the object, which itself depends on the observer. ...
Space and time, from Newton's point of view, were considered as two separate and absolute concepts, and gravity was also defined as a force that varies according to the mass of two objects and their distance from each other. However, Hermann Minkowski, using Einstein's theory of relativity, offered a geometric interpretation of special relativity, integrating time and the three dimensions of space into a unified four-dimensional model now known as Minkowski space. In the theory of general relativity, gravity is also defined as a geometric function and a consequence of the curvature of space-time, which itself arises from the uneven distribution of mass and energy. One of the key outcomes of the theory of general relativity is that the motion of celestial bodies orbiting around other bodies in curved paths is not driven by a force known as gravity. Instead, bodies are actually following the closest objects to them through curved space-time, a bending that gravity is a result of. This movement happens along the shortest path, known as a geodesic. In contrast to Newton's view, the theory of relativity has shown that the speed at which gravity acts is limited to the speed of light. Modern physics proposes that this transmission of gravity might be carried out by hypothetical particles called gravitons. Another prediction of this theory is the concept of a gravitational potential well, which explains why the frequency of light increases or decreases as it falls into or escapes from this well, due to the curvature of space-time caused by a massive object. Time dilation near a massive object is also a phenomenon that occurs due to the strong gravitational field of that object. In this regard, Einstein's theory of special relativity explains how the laws of physics are the same for all observers moving at the same velocity and how the speed of light is constant in a vacuum. One of the consequences of this theory is that mass and energy are considered equivalent and interchangeable according to the famous E=mc2 equation. However, in the theories of T-Consciousness Cosmology, space is conceptualized independently and as a principle, in the form of a mesh, and time is described as an entropic force that acts opposite to gravity to break down objects that cause space to contract. In other words, the entropic force of time is introduced as a force that arises from mass to release space from stress, not as a fourth dimension perpendicular to the dimensions of space. Furthermore, this perspective considers varying viscosities for the space surrounding celestial objects. It introduces gravity as a force that is equivalent to the viscosity of space, rather than as a consequence of curved geometry, as typically presented in relativity. Gravity functions in accordance with the structure of space mesh. Consequently, phenomena such as gravitational redshift or blueshift are interpreted as outcomes of the viscosity of space. This perspective states that as light enters black holes, the mass-energy equivalence principle is violated because of the energetic resonance of the wave during its gravitational blackshift; implying that black holes are matter production factories. Furthermore, the mass-energy equivalence principle and the law of conservation of matter and energy do not hold true in the Cosmic Black Hole, which is the beginning of the cosmos, or after the final stage of space Rebound.
... then conditionally energy-momentum tensor obeys weak energy condition for all time-like vectors u µ . Here the energy density being positive and is satisfied by most fluids including vacuum fluid [36]. ...
... The author in [36] considers 3 + 1 split of the space-time M with the threedimensional spatial sections. This is the ADM formalism. ...
... Variation with respect to the variables (N, N i , h ij ) immediately yields the constants of motion and this fact becomes more apparent in the Hamiltonian formulation of GR [36]. We know total lagrangian density coincides with the lagrangian density for pure gravity L G in an otherwise empty universe. ...
We study the wave function of the Universe. We apply the results from General Relativity and Quantum Field Theory In Curved Space-time to understand the dynamics of the real scalar field. We propose diffeomorphism between tensor fields is responsible for the quantum evolution of the universe based on the No Boundary Proposal.
... It opined that Λ supports the accelerationä/a and opposes barion pressure so it is anti gravity and produces negative pressure. New cosmology is given the name Λ CDM ( [27] − [29]). Under this, it is proposed that nearly 27% of the total content of the universe is CDM and 70% is dark energy which is interpreted as Λ energy. ...
... The distance modulus µ(z) is defined by [29] ...
... The literature describes that the absolute magnitudes of a standard candles are more or less same. It is found [29] that the absolute magnitude of a supernova is obtained as M = −19.09, so we can convert all the apparent magnitude m b data's into distant modulus data's and vice-versa by adding or subtracting 19.09 into them. ...
In this work, we have developed an FLRW type model of a universe which displays transition from deceleration in the past to the acceleration at the present. For this, we have considered field equations of f(R,T) gravity and have taken , being an arbitrary constant. We have estimated the parameter in such a way that the transition red shift is found similar in the deceleration parameter, pressure and the equation of state parameter . The present value of Hubble parameter is estimated on the basis of the three types of observational data set: latest compilation of 46 Hubble data set, SNe Ia 580 data sets of distance modulus and 66 Pantheon data set of apparent magnitude which comprised of 40 SN Ia binned and 26 high redshift data's in the range . These data are compared with theoretical results through the statistical test. Interestingly, the model satisfies all the three weak, strong and dominant energy conditions. The model fits well with observational findings. We have discussed some of the physical aspects of the model, in particular the age of the universe.
... For the present purpose, Einstein's equations are most easily solved by using Cartan formalism [34]. In this context, a useful set of orthonormal basis one-forms are given by ...
... Noting that Ω r Q is also a function of ω s , we may invert this relation to obtain ω s as a function of ω s . A straightforward substitution for ω s in terms of ω s in Equation (38) then yields Equation (34). Finally, as a rather vivid illustration of the arbitrariness of the numerical value of Ω(r), we may now consider the case, shown in Figure 3, for which there is perfect dragging, (R = r S ), the angular velocity of the shell vanishes, ω s = 0, and we keep r Q → ∞ as above (but let Ω Q be arbitrary). ...
... This observation may at first appear completely trivial, as it follows directly from a simple transformation to a rotating coordinate system. However, as was discussed below Equations (34) and (38), this particular freedom of choice of coordinate system is inherently linked to the observation that Einstein's equations determine only differences in angular velocities of ZAMOs, implying that only relative angular velocities are meaningful concepts. The angular velocity (40) obtained in the Boyer-Lindquist coordinates appears to be a consequence of imposing the asymptotic boundary condition lim r→∞ Ω(r) = 0 at the outset of the derivation [35], potentially leading to the misconception that the inertial frames at asymptotic infinity single out a global standard of non-rotation. ...
In axistationary, asymptotically flat spacetimes, zero angular momentum observers (ZAMOs) define an absolute standard of non-rotation locally, as can be verified by the absence of any Sagnac effect for these observers. Nevertheless, we argue that on a global scale the only physically meaningful concept is that of relative rotation. The argument is substantiated by solving Einstein’s equations for an approximate thin shell model, where we maintain a degree of freedom, by relaxing the natural assumption of vanishing rotation at asymptotic infinity, at the outset of the analysis. The solution reveals that Einstein’s equations only determine differences in the rotation rate of ZAMOs, thereby establishing the concept of relative rotation globally. The interpretation of rotation as relative in a global context is inherently linked to the freedom to transform between coordinate systems rotating relative to each other, implying that an arbitrary ZAMO located at any radius may claim to be the one that is non-rotating on a global scale, and that the notion of an asymptotic Lorentz frame relative to which one may measure absolute rotation is devoid of any meaning. The concept of rotation in Kerr spacetime is then briefly discussed in the context of this interpretation.
... Recall that the r −2 term in Reissner-Nordström metric is due to the energy-momentum tensor of the electromagnetic field [23], ...
... We accordingly postulate that dynamics of spacetime (as a pseudo-Riemannian manifold) is completely determined by its metric g µν and its energy-momentum tensor T µν , which are 'independent' of one another, thereby requiring all dynamics of the manifold to be expressed solely in terms of g µν and T µν . This postulate resembles that of ADM formulation of GR [23,29], although key differences will be seen. Indeed the coherent structure and relative successes of ADM theory are strong motivations and support for this postulate. ...
... The task of ensuring that the total energy of spacetime as a whole is conserved, is undertaken by another equation, to which we shall soon arrive. Assuming (23), immediately the analogue of the second Hamilton equation is expected to be 19 The word counterpart is vital. This cannot be a 'generalization' of classical Hamiltonian dynamics in any sense. ...
There is no formal difference between particles and black holes. This formal similarity lies in the intersection of gravity and quantum theory; quantum gravity. Motivated by this similarity, `wave-black hole duality' is proposed, which requires having a proper energy-momentum tensor of spacetime itself. Such a tensor is then found as a consequence of `principle of minimum gravitational potential'; a principle that corrects the Schwarzschild metric and predicts extra periods in orbits of the planets. In search of the equation that governs changes of observables of spacetime, a novel Hamiltonian dynamics of a Pseudo-Riemannian manifold based on a vector Hamiltonian is adumbrated. The new Hamiltonian dynamics is then seen to be characterized by a new `tensor bracket' which enables one to finally find the analogue of Heisenberg equation for a `tensor observable' of spacetime.
... The Principle of least action demands δS E−H δgµν = 0 and equation (17) reduces to: ...
... then conditionally energy-momentum tensor obeys weak energy condition for all time-like vectors u µ . Here the energy density being positive and is satisfied by most fluids including vacuum fluid [17]. ...
... The author in [17] considers (3 + 1) split of the space-time M with the three-dimensional spatial sections. This is the ADM formalism. ...
In this article key results in general relativity are studied and the dynamics of quantum fields in curved space-time is r eviewed. It has b een shown geometrically how dark energy breaks strong energy condition. We study tan-hyperbolic inflation f or an open FRWL universe and show that over half and full period scalar field r emains in it's initial state.
... In particular, the Einstein-de Sitter space with the scale factor a(t) = t 2/3 is modeling the expanding matter dominated universe, if a(t) = t 1/2 is radiation dominated universe (see, e.g., [17,18]). The scale factor a(t) = t describes the Milne model [6,11,22]. The scalar curvature of the space with a(t) = a 0 t is R(t) = −6 (1 − 2 ) t −2 . ...
... (see, e.g., [2]), where ∂ 0 = ∂/∂t, ∂ k = ∂/∂x k , andȧ (t) a(t) is the Hubble parameter (see, e.g., [6,Ch.8], [11,Sec.11.4]), while ...
... We consider the Eq. (1.9) in the proper time (see, e.g., [11,17,18]) ...
We prove the existence of global in time solution with the small initial data for the semilinear equation of the spin-12 particles in the Friedmann–Lemaître–Robertson–Walker spacetime. Moreover, we also prove that if the initial function satisfies the Lochak–Majorana condition, then the global solution exists for arbitrary large initial value. The solution scatters to free solution for large time. The mass term is assumed to be complex-valued. The conditions on the imaginary part of mass are discussed by proving nonexistence of the global solutions if certain relation between scale function and the mass are fulfilled.
... But Einstein did 9 There are exceptions. For example, in [16] (p. 22) coordinates are associated with coordinate clocks, and the local clocks in a static gravitational field are said to run slow, as they lag behind coordinate clocks at the same locations [16] (p. ...
... For example, in [16] (p. 22) coordinates are associated with coordinate clocks, and the local clocks in a static gravitational field are said to run slow, as they lag behind coordinate clocks at the same locations [16] (p. 219) 12 of 14 not approach the subject from this modern perspective. ...
.In his groundbreaking 1905 paper on special relativity, Einstein distinguished between local and global time in inertial systems, introducing his famous definition of distant simultaneity to give physical content to the notion of global time. Over the following decade, Einstein attempted to generalize this analysis of relativistic time to include accelerated frames of reference, which, according to the principle of equivalence, should also account for time in the presence of gravity. Characteristically, Einstein's methodology during this period focused on simple, intuitively accessible physical situations, exhibiting a high degree of symmetry. However, in the final general theory of relativity, the a priori existence of such global symmetries cannot be assumed. Despite this, Einstein repeated some of his early reasoning patterns even in his 1916 review paper on general relativity and in later writings. Modern commentators have criticized these arguments as confused, invalid, and inconsistent. Here, we defend Einstein in the specific context of his use of global time and his derivations of the gravitational redshift formula. We argue that a detailed examination of Einstein's early work clarifies his later reasoning and demonstrates its consistency and validity.
... In the co-moving coordinate system, the velocity components are u k ¼ ð0; 0; 0; 1Þ which obey the condition u k u k ¼ 0: By utilizing Eqs. (11) and (15), we can depict the¯eld equations (14) corresponding to the metric Eq. (16) as ...
... They provide us with valuable insights into the overall structure and dynamics of the universe as described by cosmological theories. Mean Hubble parameter (H), As we know 11 that ...
This study includes the cosmic evolution and the potential periodicity of the universe. It employs a periodic varying deceleration parameter (PVDP) within the framework of [Formula: see text] theory of gravity, with a specific focus on the Bianchi-II model. We explore the dynamic nature of the universe, with physical and geometrical properties within this theoretical framework. We have also analyzed the cosmographic parameters, including jerk, snap, and lerk, for deeper insights into the universe’s evolution and behavior. Utilizing state finder diagrams and the Om diagnostic (which illustrates the variation of [Formula: see text] with redshift) signifying a shift from matter dominance to a stronger influence of dark energy (DE), we construct a comprehensive map of the universe’s trajectory and behavior. By employing the Bianchi-II model within the [Formula: see text] theory, our proposed model helps us in understanding the universe’s oscillatory patterns and underlying mechanisms. This research significantly contributes to our understanding of cosmic evolution and periodicity within the [Formula: see text] theory.
... where, in general, it is possible to have a = b = c = a, being all of them the different scale factors for each anisotropic direction of the Universe (here µ, ν = 0, 1, 2, 3). Different exact solutions have been found for these kind of cosmologies [9]. Therefore, any of these solutions can be used as a cosmological background for the supersymmetric dynamics of massless scalar and electromagnetic fields that are studied below. ...
... In this sense, field ϕ + (φ + ) is the superpartner of ϕ − (φ − ), which is explicitly displayed by Eqs. (9). They do not evolve in the same Universe, but rather in different cosmological scenarios. ...
It is shown that any cosmological anisotropic model produces supersymmetric theories for both massless scalar and electromagnetic fields. This supersymmetric theory is the time-domain analogue of a supersymmetric quantum mechanical theory. In this case, the variations of the anisotropic scale factors of the Universe are responsible for triggering the supersymmetry. For scalar fields, the superpartner fields evolve in two different cosmological scenarios (Universes). On the other hand, for propagating electromagnetic fields, supersymmetry is manifested through its polarization degrees of freedom in one Universe. In this case, polarization degrees of freedom of electromagnetic waves, which are orthogonal to its propagation direction, become superpartners from each other. This behavior can be measured, for example, through the rotation of the plane of polarization of cosmological light.
... where k , K are the (null) 1-forms (23). To keep the duality of bases we must also change ...
... (see for instance equation (6.176) in Ref. [23]). In the basis {n 0 ,n 1 , n 2 , n 3 , n 4 } the structure functions are such that ...
Five-dimensional Einstein–Maxwell–Chern–Simons equations are investigated in the framework of an extended Kerr–Schild strategy to search for black holes solutions. The fulfillment of Einstein equations constrains the Chern–Simons coupling constant to a value determined by the trace of the energy-momentum tensor of the electromagnetic configuration.
... Disformal transformation [1] is a metric transformation introduced by Bekenstein in 1992 as a generalisation of the more commonly known conformal transformation [2,3]. The aim was to relate Finsler and Riemann geometries in a single gravitational theory. ...
... The immediately preceding section saw the calculation of φ using a direct 'expansion' involving the Christoffel symbol. In this section, we present a slightly different pathway, used in Ref. [28] for the usual disformal transformation, to the conditions leading to the symmetry of the massless Klein-Gordon equation under the general disformal transformation given by (3). The process of calculation is mainly a mathematical exercise and a sort of verification of the previous section's results. ...
The Klein-Gordon equation, one of the most fundamental equations in field theory, is known to be not invariant under conformal transformation. However, its massless limit exhibits symmetry under Bekenstein's disformal transformation, subject to some conditions on the disformal part of the metric variation. In this study, we explore the symmetry of the Klein-Gordon equation under the general disformal transformation encompassing that of Bekenstein and a hierarchy of `sub-generalisations' explored in the literature (within the context of inflationary cosmology and scalar-tensor theories). We find that the symmetry of the massless limit can be extended under this generalisation provided that the disformal factors are bound by what we call an orthogonality condition involving them, and that they have special functional dependency on the conformal factor. Upon settling the effective extension of symmetry, we also tackle the invertibility of the general disformal transformation to avoid propagating non-physical degrees of freedom upon changing the metric. In the end, we derive the inverse transformation and the accompanying restrictions that make this inverse possible.
... where A = πR is the area of the disk and λ is the wavelength of the beam. According to [21], the speed in both directions will be v = r dθ dt = −rω ± c. ...
The study of spinning systems plays a question of interest in several research branches in physics. It allows the understanding of simple classical mechanical systems but also provides us with tools to investigate a wide range of phenomena, from condensed matter physics to gravitation and cosmology. In this contribution, we review some remarkable theoretical aspects involving the description of spinning quantum systems. We explore the nonrelativistic and relativistic domains and their respective applications in fields such as graphene physics and topological defects in gravitation.
... These distinct phases are well supported by observational evidence and are essential components of the standard cosmological model [5]. These dynamics can be achieved by introducing various material contents into Einstein's theory of general relativity [6,7] or by appropriately modifying the theory itself [8]. ...
We apply a combined study in order to investigate the dynamics of cosmological models incorporating nonlinear electrodynamics (NLED). The study is based on the simultaneous investigation of such fundamental aspects as stability and causality, complemented with a dynamical systems investigation of the involved models, as well as Bayesian inference for parameter estimation. We explore two specific NLED models: the power-law and the rational Lagrangian. We present the theoretical framework of NLED coupled with general relativity, followed by an analysis of the stability and causality of the various NLED Lagrangians. We then perform a detailed dynamical analysis to identify the ranges where these models are stable and causal. Our results show that the power-law Lagrangian model transitions through various cosmological phases, evolving from a Maxwell radiation-dominated state to a matter-dominated state. For the rational Lagrangian model, including the Maxwell term, stable and causal behavior is observed within specific parameter ranges, with critical points indicating the evolutionary pathways of the universe. To validate our theoretical findings, we perform Bayesian parameter estimation using a comprehensive set of observational data, including cosmic chronometers, baryon acoustic oscillation (BAO) measurements, and supernovae type Ia (SNeIa). The estimated parameters for both models align with the expected values for the current universe, particularly the matter density Ω m and the Hubble parameter h. However, the parameters of the models are not tightly constrained within the prior ranges. Our combined studies approach rules out the mentioned models as an appropriate description of the cosmos. Our results highlight the need for further refinement and exploration of NLED-based cosmological models to fully integrate them into the standard cosmological framework.
... Nonetheless, under specific conditions and with a metric adhering to the cosmological principle, a global time for the Universe can be defined. It is achieved by foliating spacetime into a series of non-intersecting space-like 3D surfaces [40][41][42][43][44][45][46][47][48][49]. ...
Cosmography, as an integral branch of cosmology, strives to characterize the Universe without relying on pre-determined cosmological models. This model-independent approach utilizes Taylor series expansions around the current epoch, providing a direct correlation with cosmological observations and the potential to constrain theoretical models. Cosmologists can describe many measurable aspects of cosmology by using various combinations of cosmographic parameters. The varying speed of light model can be naturally implemented, provided that we do not make any further assumptions from the Robertson-Walker metric for cosmological time dilation. Therefore, we apply cosmography to the so-called minimally extended varying speed of light model. In this case, other cosmographic parameters can be used to construct the Hubble parameter for both the standard model and the varying speed-of-light model. On the other hand, distinct combinations of cosmographic values for the luminosity distance indicate the two models. Hence, luminosity distance might provide a method to constrain the parameters in varying speed-of-light models.
... Nonetheless, under specific conditions and with a metric adhering to the cosmological principle, a global time for the Universe can be defined. It is achieved by foliating spacetime into a series of non-intersecting space-like 3D surfaces [40][41][42][43][44][45][46][47][48][49]. ...
Cosmography, as an integral branch of cosmology, strives to characterize the Universe without relying on pre-determined cosmological models. This model-independent approach utilizes Taylor series expansions around the current epoch, providing a direct correlation with cosmological observations and the potential to constrain theoretical models. Various observable quantities in cosmology can be described as different combinations of cosmographic parameters. Furthermore, one can apply cosmography to models with a varying speed of light. In this case, the Hubble parameter can be expressed by the same combination of cosmographic parameters for both the standard model and varying speed-of-light models. However, for the luminosity distance, the two models are represented by different combinations of cosmographic parameters. Hence, luminosity distance might provide a method to constrain the parameters in varying speed-of-light models.
... The ΛCDM model stands as the conventional cosmological model, recognized for its simplicity and effective alignment with contemporary cosmic observations. A positive cosmological constant, denoted by Λ, can be explored to delve into the Cold Dark Matter (CDM) model, a subject previously investigated by Sahni et al. [25] and Gron and Hervik [26]. The kinematic parameters for the ΛCDM model are provided below. ...
This communication aims to discuss the interplay between cosmic acceleration and deceleration in the context of the f(R, T ) theory, along with different forms of deceleration parameters. The study encompasses the evolution of the universe's expansion rate concerning cosmic time, denoted as t . We have also derived solutions for the Einstein field equations (EFE) within the context of this study. The significant implications of the f(R, T ) theory concerning cosmic acceleration and deceleration have been explored. Furthermore, we have presented pictorial representations of key cosmological parameters. Additionally, we have tabulated the values of all cosmographic parameters for Model-I and Model-II. In the end, a comprehensive analysis, along with a tabulated form of all important cosmological parameters under the studied cases along with the standard ΛCDM model-have been presented.
... The former as in Regge calculus, and the latter as in the Israel formalism (Grøn & Hervik, 2007). The mean curvature term on the boundary is sometimes included in the Einstein-Hilbert action (Kiefer, 2012). ...
Regge calculus defines a curvature for piecewise constant metrics on simplicial complexes subject to a partial continuity requirement. We prove that if a part of the complex is embedded in a Euclidean space, by a piecewise affine map, and we perform smoothing by convolution there, then the smoothed metrics have a densitized scalar curvature that converges, in the sense of measures, to that defined by Regge, as the smoothing parameter goes to zero.
... Einstein's theory of relativity is a pillar of modern physics. Not only did it predict novel phenomena, it also gave rise to modern cosmology, providing us with great tools for exploring the cosmos [1,2,3]. Powerful and successful as the theory is, standard General Relativity (hereafter GR) has not been able to explain certain observational data by itself, most notably the phenomena attributed to Dark matter [5,6]. ...
Null geodesics of a spacetime are a key factor in determining dynamics of particles. In this paper, it is argued that, within the scope of validity of Cosmological Principle, expansion of the Universe causes the null geodesics to accelerate, providing us with a universal acceleration scale cH_0/e. Since acceleration of null rays of spacetime corresponds to null rays of velocity space, demanding the invariance of acceleration of light yields a new metric for the velocity space, which introduces time as a dimension of the velocity space. Being part of the configuration space, modification of distance measurements in velocity space alters the Euler-Lagrange equation and from there the equation of motion, Newton's Second Law. It is then seen that the resulting modification eliminates the need for Dark matter, yielding MOND as an approximation, fixing Milgrom's constant to the unprecedentedly exact value of a_0=cH_0/2e, without suffering MOND's theoretical shortcomings.
... We assume that the fluid 4−velocity u i is equal to the unit normal of the homogeneous spatial hyper-surfaces. And, thus, the propagation equations of these quantities are given by [87] ...
We study a class of homogeneous and anisotropic geometries with affine equation of state (EoS) for different physically plausible scenarios of the universe evolution using dynamical system technique. We analyze the locally rotationally symmetric Bianchi I (LRS BI), Bianchi III (LRS BIII) and Bianchi V (LRS BV) geometry for the exhibition of the effects of affine EoS in the model. The model exhibits stable attractor which is also isotropic and thus, it may explain the late-time accelerated expansion of the universe. The model also possess stiff matter-, radiation- and matter-dominated phases prior to the dark energy assisted accelerating phase which are confirmed by the behaviours of effective equation of state and deceleration parameters. We use the statefinder diagnostic which is a geometrical diagnostic to explore model independent features of the cosmological dynamical system. The LRS BI, BIII and BV geometry based dynamical systems exhibit r=1,s=0 r = 1 , s = 0 ( Λ cold dark matter model) at late-times, which is compatible with the observations. The dynamical system for the Kantowski–Sachs model yields synchronous bounce on the basis of the model parameters. It also yields a late-time attractor which may explain the accelerated expansion of the universe in the model. The qualitative differences between LRS BIII and BV cosmological dynamical systems have also been discussed.
... It is worthwhile to note the FRW models of the Universe with absence of Λ exhibits a model that describes decelerated expansion phase of the Universe while these models have gained acceleration for some specific value of Λ. Later on, it has been observed that the ΛCDM model is in good agreement with recent astrophysical observations [8][9][10][11][12]. Further, we note that Wilkinson Microwave Anisotropy Probe (WMAP) [6] have nailed down the curvature of space and ordinary matter up to 0.4% and 4.6% respectively. ...
In this paper, we investigate an exact Universe which is observational viable and filled with binary mixture of perfect fluid and cosmological constant . Owning the non-uniform expansion of cosmos, we have considered redshift drift and performed statistical test to obtain the best fit value of model parameters of derived Universe with its observed values. Here and z denote the present value of Hubble constant and redshift respectively. We estimate the best fit values of Hubble constant and density parameters are km/s/Mpc, and by bounding the derived model with latest observational Hubble data (OHD) while with joint Pantheon data and OHD, its values are km/s/Mpc, and . The analysis of deceleration parameter and jerk parameter show that the Universe in derived model is compatible with CDM model.
... These observers are constantly accelerated in Minkowski space (their worldlines are the blue hyperbolas in Fig. 17). Inside the lightcone we introduce similar coordinates (τ,ρ) through t = τ coshρ, z = τ sinhρ, which give the so-called Milne spacetime (see, e.g., [81]). We have τ 2 = t 2 − z 2 and tanhρ = z/t, so the field is given by ϕ(τ,ρ) =φ(τ ). ...
This is the first in a series of papers where we study the dynamics of a bubble wall beyond usual approximations, such as the assumptions of spherical bubbles and infinitely thin walls. In this paper, we consider a vacuum phase transition. Thus, we describe a bubble as a configuration of a scalar field whose equation of motion depends only on the effective potential. The thin-wall approximation allows obtaining both an effective equation of motion for the wall position and a simplified equation for the field profile inside the wall. Several different assumptions are involved in this approximation. We discuss the conditions for the validity of each of them. In particular, the minima of the effective potential must have approximately the same energy, and we discuss the correct implementation of this approximation. We consider different improvements to the basic thin-wall approximation, such as an iterative method for finding the wall profile and a perturbative calculation in powers of the wall width. We calculate the leading-order corrections. Besides, we derive an equation of motion for the wall without any assumptions about its shape. We present a suitable method to describe arbitrarily deformed walls from the spherical shape. We consider concrete examples and compare our approximations with numerical solutions. In subsequent papers, we shall consider higher-order finite-width corrections, and we shall take into account the presence of the fluid.
... In the modern age of the Universe, any effective cosmological model in accordance with CDM exhibits similar kinematics. In an approach to studying the CDM model, Sahni and Starobinsky (2000) and Grøn and Hervik (2007) have researched further, as a positive cosmological constant ( (t) = as a constant quantity). The kinematical parameters for the CDM model are given below: ...
Through the analysis of the linearly varying deceleration parameter (LVDP) (q) in the framework of general relativity (GR) and spatially anisotropic Bianchi type (BT) II, VIII and IX Universes, the Tsallis holographic dark energy (THDE) has been examined (Tavayef et al. in Phys. Lett. B 781:195, 2018). As a result of the phase change from the slow expansion of the cosmos to the accelerated expansion of the cosmos, the deceleration parameter(DP) must flip its signatures at the transition redshift, where the DP’s values are constrained as (Cunha in Phys. Rev. D 79:047301, 2009) and transition redshift (i.e., ). The proposed THDE model predicts that the Universe is an expanding one based on the equation of state parameter (EoS) () and DP (q), along with the plane (the ′ shows the differentiation based on ) and also supports the recent observational data. The analysis confirms that the quintessence, phantom dark energy regions, and cold dark matter (CDM) limit, exist in our model. We have observed the evolution of (r,s) and (r,q) trajectories in order to study different phases of the Universe and also cosmological quantities such as jerk (j), snap(s), lerk(l) parameters, Om diagnostic, luminosity distance , angular diameter and distance modulus have been discussed. In addition, we compare the behavior of scale factor (a), Hubble (H), deceleration (q), jerk (j), and snap (s) parameters of CDM and LVDP models and discover that their behavior has been almost identical over the observed Universe’s history.
... Dark energy, which is believed to cause this acceleration is most often represented by the very simple concordance ΛCDM model. This model fits well on observational grounds [27]− [29], despite certain weaknesses [30] that it suffers with fine tuning and the cosmic coincidence problems. To solve these, scalar field dominated tracker field quintessence and phantom dark energy models were proposed [31]− [34]. ...
In this paper, an attempt is made to construct a Friedmann–Lemaitre–Robertson–Walker model in [Formula: see text] gravity with a perfect fluid that yields acceleration at late times. We take [Formula: see text] as [Formula: see text]. As in the [Formula: see text]CDM model, we take the matter to consist of two components, viz., [Formula: see text] and [Formula: see text] such that [Formula: see text]. The parameter [Formula: see text] is the matter density (baryons [Formula: see text] dark matter), and [Formula: see text] is the density associated with the Ricci scalar [Formula: see text] and the trace [Formula: see text] of the energy–momentum tensor, which we shall call dominant matter. We find that at present [Formula: see text] is dominant over [Formula: see text], and that the two are in the ratio 3:1–3:2 according to the three data sets: (i) 77 Hubble OHD data set, (ii) 580 SNIa supernova distance modulus data set and (iii) 66 pantheon SNIa data which include high red shift data in the range [Formula: see text]. We have also calculated the pressures and densities associated with the two matter densities, viz., [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It is also found that at present, [Formula: see text] is greater than [Formula: see text]. The negative dominant matter pressure [Formula: see text] creates acceleration in the universe. Our deceleration and snap parameters show a change from negative to positive, whereas the jerk parameter is always positive. This means that the universe is at present accelerating and in the past it was decelerating. State finder diagnostics indicate that our model is at present a dark energy quintessence model. The various other physical and geometric properties of the model are also discussed.
... These observers are constantly accelerated in Minkowski space (their worldlines are the blue hyperbolas in Fig. 17). Inside the lightcone we introduce similar coordinates (τ,ρ) through t = τ coshρ, z = τ sinhρ, which give the so-called Milne spacetime (see, e.g., [81]). We have τ 2 = t 2 − z 2 and tanhρ = z/t, so the field is given by ϕ(τ,ρ) =φ(τ ). ...
This is the first in a series of papers where we study the dynamics of a bubble wall beyond usual approximations, such as the assumptions of spherical bubbles and infinitely thin walls. In this paper, we consider a vacuum phase transition. Thus, we describe a bubble as a configuration of a scalar field whose equation of motion depends only on the effective potential. The thin-wall approximation allows obtaining both an effective equation of motion for the wall position and a simplified equation for the field profile inside the wall. Several different assumptions are involved in this approximation. We discuss the conditions for the validity of each of them. In particular, the minima of the effective potential must have approximately the same energy, and we discuss the correct implementation of this approximation. We consider different improvements to the basic thin-wall approximation, such as an iterative method for finding the wall profile and a perturbative calculation in powers of the wall width. We calculate the leading-order corrections. Besides, we derive an equation of motion for the wall without any assumptions about its shape. We present a suitable method to describe arbitrarily deformed walls from the spherical shape. We consider concrete examples and compare our approximations with numerical solutions. In subsequent papers, we shall consider higher-order finite-width corrections, and we shall take into account the presence of the fluid.
... Nevertheless, one can define a global time for the Universe when a set of requirements is satisfied and a metric embodying the cosmological principle meets these requirements. Then, one can define a global time by a foliation of spacetime as a sequence of non-intersecting spacelike 3D surfaces [10][11][12][13][14][15][16][17][18][19]. ...
The Robertson-Walker (RW) metric allows us to apply general relativity to model the behavior of the Universe as a whole (\textit{i.e.}, cosmology). We can properly interpret various cosmological observations, like the cosmological redshift, the Hubble parameter, geometrical distances, and so on, if we identify fundamental observers with individual galaxies. That is to say that the interpretation of observations of modern cosmology relies on the RW metric. The RW model satisfies the cosmological principle in which the 3-space always remains isotropic and homogeneous. One can derive the cosmological redshift relation from this condition. We show that it is still possible for us to obtain consistent results in a specific time-varying speed of light model without spoiling the success of the standard model. The validity of this model needs to be determined by observations.
We analyze finite element discretizations of scalar curvature in dimension N ≥ 2 N \ge 2 . Our analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric g g on a simplicial triangulation of a polyhedral domain Ω ⊂ R N \Omega \subset \mathbb {R}^N having maximum element diameter h h . We show that if such an interpolant g h g_h has polynomial degree r ≥ 0 r \ge 0 and possesses single-valued tangential-tangential components on codimension-1 simplices, then it admits a natural notion of (densitized) scalar curvature that converges in the H − 2 ( Ω ) H^{-2}(\Omega ) -norm to the (densitized) scalar curvature of g g at a rate of O ( h r + 1 ) O(h^{r+1}) as h → 0 h \to 0 , provided that either N = 2 N = 2 or r ≥ 1 r \ge 1 . As a special case, our result implies the convergence in H − 2 ( Ω ) H^{-2}(\Omega ) of the widely used “angle defect” approximation of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on the interpolated metric g h g_h . We present numerical experiments that indicate that our analytical estimates are sharp.
This research focuses on exploring the dynamics of a cosmos model under the influence of Ricci dark energy within the framework of modified Lyra geometry. The modified field equations of Einstein for Lyra’s geometry are introduced, and specific solutions are obtained for a Big Rip cosmos scenario. Additionally, various physical and geometrical aspects of the cosmos are comprehensively investigated and discussed.
Edited Book titled Trends in Contemporary Mathematics & Applications provides an excellent international platform for academicians, and researchers around the world to publish their research and to exchange ideas on recent developments in Contemporary Mathematics & Its Applications. Its goal is to support researchers through the dissemination of information, research findings, and practice. This book facilitates the publication of original articles, reports of professional experience, comments, and reviews.
فضــا و زمــان، از دیــدگاه نیوتــن بــه عنــوان دو مفهــوم مجــزا و مطلــق در نظــر گرفتــه می شــد و گرانــش نیــز بــه عنــوان نیرویــی تعریــف شــده بــود کــه متناســب بــا میــزان جــرمِ دو جســم و فاصلــۀ آنهــا از یکدیگــر تغییــر می کنــد. امــا هرمــان مینکوفســکی بــا اســتفاده از نظریــۀ نســبیت انیشــتین، یــک تفســیر هندســی از نســبیت خــاص ارائــه داد کــه زمــان و ســه بُعــد فضــا را در یــک مــدل چهــار بُعــدیِ واحــد کــه اکنــون بــه عنــوان فضــای مینکوفســکی شــناخته می شــود ادغــام کــرد. گرانــش نیــز در نظریــۀ نســبیت عــام، بــه عنــوان تابعــی هندســی و پیامــدی از انحنــای فضا-زمــان تعریــف شــد کــه خــود ایــن انحنــا از توزیــع نامتــوازن جــرم و انــرژی منتــج می گــردد. از تبعــات مهــم نظریــۀ نســبیت عــام، ایــن اســت کــه آنچــه منجــر بــه گــردش اجــرام ســماوی حــول ســایر اجــرام در مدارهــای منحنــی می شــود، نیرویــی موســوم بــه گرانــش نیســت، بلکــه اجــرام در فضا-زمــان خمیــده کــه گرانــش هــم پیامــدی از آن خمــش اســت، صرفــاً نزدیکتریــن جســم نســبت بــه خــود را دنبــال می کننــد کــه ایــن حرکــت، در مســیری مســتقیم بــه نــام ژئودزیــک صــورت می گیــرد. از طرفــی برخــلاف دیــدگاه نیوتــن، در نظریــۀ نســبیت نشــان داده می شــود کــه ســرعت انتقــال گرانــش بــه ســرعت نــور محــدود می شــود کــه از دیــدگاه فیزیــک مــدرن، ایــن انتقــال توســط ذرات فرضــی بــه نــام گرویتــون صــورت می گیــرد. از پیش بینی هــای دیگــر ایــن نظریــه، چــاه پتانســیل گرانشــی اســت کــه افزایــش یــا کاهــش فرکانــس نــورِ در حــالِ ســقوط یــا فــرار از آن چــاه، بــه خاطــر خمــش فضا-زمانــی اســت کــه توســط جســمِ ثقیــل ایجــاد شــده اســت. اتســاع زمــان نیــز در نزدیکــی یــک جســمِ پرجــرم، پدیــده ای اســت کــه بــه دلیــل میــدان گرانشــی قــوی آن رخ می دهــد. در ایــن راســتا، نســبیت خــاصِ انیشــتین نیــز نظریــه ای اســت کــه توضیــح می دهــد چگونــه قوانیــن فیزیــک بــرای همــۀ ناظرانــی کــه در حــال حرکــتِ یکنواخــت هســتند، یکســان بــوده و چگونــه ســرعت نــور در محیــط خــلاء ثابــت می باشــد. یکــی از پیامدهــای ایــن نظریــه ایــن اســت کــه جــرم و انــرژی، معــادل و هــم ارز در نظــر گرفتــه می شــوند و می تــوان آن هــا را مطابــق معادلــۀ معــروف E=mc2 بــه یکدیگــر تبدیــل کــرد. امــا در نظریــات کیهان شناســی شــعوری، فضــا بــه صــورت توری هــای مجــزا بــه عنــوان اصــل در کیهــان و زمــان بــه عنــوان یــک نیــروی آنتروپیایــی کــه در جهــت عکــس گرانــش بــرای اضمحــلال اجرامــی کــه عامــل انقبــاض فضــا شــده اســت معرفــی می شــوند. بــه عبارتــی نیــروی آنتروپیایــی زمــان، منتــج از جــرم بــرای رهایــی فضــا از اســترس اســت، نــه بــه عنــوان بُعــد چهارمــی کــه عمــود بــر ابعــاد فضاســت. همچنیــن ایــن دیــدگاه ویســکوزیته های متفاوتــی بــرای فضــای پیرامــون اجــرام در نظــر می گیــرد و گرانــش را نیــز بــه عنــوان یــک نیــرو، معــادل ویســکوزیتۀ فضــا بــه جــای هندســۀ خمیــده در نســبیت معرفــی می کنــد و عملکــرد ایــن نیــرو را نیــز بــه ســاختار توری هــای فضــا نســبت می دهــد؛ درنتیجــه انتقــال بــه ســرخ یــا آبــی گرانشــی حاصــل ویســکوزیتۀ فضــا اســت. ایــن دیــدگاه بیــان می کنــد کــه بــا ورود نــور بــه درون ســیاهچاله ها بــه خاطــر رزنانــس انرژیایــی مــوج در طــی انتقــال بــه ســیاه گرانشــی، اصــل هــم ارزی مــاده و انــرژی نقــض می شــود. بــه عبارتــی ســیاهچاله ها کارخانــۀ تولیــد مــاده می باشــند. ازطرفــی ایــن اصــل و پایســتگی مــاده و انــرژی در ســیاهچالۀ کیهانــی کــه آغــاز کیهــان اســت و یــا بعــد از واگــرد نهایــی فضــا نیــز صــادق نیســتند.
A thin dust shell contracting from infinity to near its gravitational radius r +, in a spacetime AdS 3 is analyzed; its equation of motion is determined and the solution R(t) as seen by a FIDO observer is estimated. It is concluded that this Shell's exterior looks like a BTZ black hole with similar properties. Based on the Thermo Field Dynamics technique, a scalar field Φ in the proximity of a non-rotating BTZ (2 + 1) black hole is studied. From the corresponding Killing-Boulware 0KB* and Hartle-Hawking 0HH* vacuum states, the associated Wightman function Wx,x′HH*−Wx,x′KB* is determined and based on it, the time component of the momentum-energy tensor of the system ∂0∂0′WHH*−WKB*(x,x′)≈T00(x,x′)=σ(r) is calculated. Which allows establishing the origin and location of the degrees of freedom responsible for the entropy that describes a source for the Bekenstein-Hawking S BH entropy. The thermal environment described by this model manifests itself with a well-defined and concentrated energy density near the event horizon, according to a FIDO observer.
Regular black hole spacetimes are obtained from an effective Lagrangian for quantum Einstein gravity. The interior matter is modeled as a dust fluid, which interacts with the geometry through a multiplicative coupling function denoted as χ. The specific functional form of χ is deduced from asymptotically safe gravity, under the key assumption that the Reuter fixed point remains minimally affected by the presence of matter. As a consequence the gravitational coupling vanishes at high energies. The static exterior geometry of the black hole is entirely determined by the junction conditions at the boundary surface. Consequently, the resulting global spacetime geometry remains devoid of singularities at all times. This outcome offers a new perspective on how regular black holes are formed through gravitational collapse.
Both circular and epicyclic motion of test particles along equatorial circular orbits in the revisited Kerr–de Sitter spacetime are analyzed. We present relations for specific energy, specific angular momentum, and Keplerian angular velocity of particles on equatorial circular orbits, and discuss criteria for the existence and stability of such orbits giving limits on spacetime parameters. Finally, we discuss the epicyclic motion along equatorial circular orbits obtaining relations for radial and vertical epicyclic frequencies. The results are compared with those for standard Kerr–de Sitter geometry.
The revolution in our conceptions of time and space from Einstein’s theory of relativity is described, with an overview of how the special theory of relativity was developed from a standpoint of preserving the laws of physics and electromagnetism for all observers. The speed of light, provided as a fundamental constant in Maxwell’s equation, inspires Einstein’s theory, which can be explained using the notion of a light clock, with subsequent relative effects on time for different observers. The notions of spacetime and the light cone are explained based on a standpoint of “world lines” in which time is a dimension of space. Broadening Einstein’s theory to apply toward accelerating observers or those near large masses gives rise to the general theory of relativity, and the nature of curved space and its manifestations are described in a historical context. Eddington’s work in measuring the deflection of light, solutions to Einstein’s general relativity by Karl Schwarzschild, and the predicted shifts in the perihelion of Mercury’s orbit are described. The resulting new perspectives from Einstein’s theory provides the basis for new models of time and space developed by Lemaitre and Friedman, that explains the expansion of the universe.KeywordsEinsteinSpecial relativityGeneral relativityCosmologyCurved spaceGravitational lensing
The Robertson–Walker (RW) metric allows us to apply general relativity to model the behavior of the Universe as a whole (i.e., cosmology). We can properly interpret various cosmological observations, like the cosmological redshift, the Hubble parameter, geometrical distances, and so on, if we identify fundamental observers with individual galaxies. That is to say that the interpretation of observations of modern cosmology relies on the RW metric. The RW model satisfies the cosmological principle in which the 3-space always remains isotropic and homogeneous. One can derive the cosmological redshift relation from this condition. We show that it is still possible for us to obtain consistent results in a specific time-varying speed-of-light model without spoiling the success of the standard model. The validity of this model needs to be determined by observations.
It is known that the prediction of classical mechanics for velocity-distance relation fails to account for the rotation curves of galaxies, implying that either the Universe is pervaded by some yet-unseen novel form of matter --dark matter-- or the dynamical laws that lead to this result require revision --MOND approach--.
In this note the latter approach is adopted and origins of the successes of MOND are sought.
It is shown that the universal acceleration scale that has appeared in MOND is the acceleration of null rays of FLRW spacetime. Since acceleration of null rays of spacetime corresponds to null rays of velocity space, demanding the invariance of acceleration of light yields a new metric for the velocity space which introduces time as a dimension of the velocity space. Being part of the configuration space, modification of distance measurements in velocity space alters the Euler-Lagrange equation and from there the equation of motion (Newton's Second Law). In this light therefore, modification of the dynamical law of motion is seen as a necessary consequence of the expansion of the Universe.
The Klein–Gordon equation, one of the most fundamental equations in field theory, is known to be not invariant under conformal transformation. However, its massless limit exhibits symmetry under Bekenstein’s disformal transformation, subject to some conditions on the disformal part of the metric variation. In this study, we explore the symmetry of the Klein–Gordon equation under the general disformal transformation encompassing that of Bekenstein and a hierarchy of “sub-generalizations” explored in the literature (within the context of inflationary cosmology and scalar–tensor theories). We find that the symmetry in the massless limit can be extended under this generalization provided that the disformal factors take a special form in relation to the conformal factor. Upon settling the effective extension of symmetry, we investigate the invertibility of the general disformal transformation to avoid propagating nonphysical degrees of freedom upon changing the metric. We derive the inverse transformation and the accompanying restrictions that make this inverse possible.
Humanity’s understanding of the universe has evolved significantly over time. During the major part of History, Earth was regarded as the center of all things, with planets and stars orbiting around it. A profound shift in the understanding of the cosmos was made by Nicolaus Copernicus who suggested that Earth and the other planets in the solar system in fact orbited around the sun. The legend said he released the final version of his book in 1543 on the day of his death [1]. Despite waves of opposition, the heliocentric view of the solar system was finally accepted after the year 1687 when Isaac Newton formulated the law of gravitation to unify terrestrial and celestial mechanics [2]. An important step in the history of astronomy was made by Charles Messier with the release in 1781 of a catalog of 100 nebulas and star clusters visible from Paris [3]. During the next century, in 1864, the catalog got enlarged to 5000 objects including the ones visible from the south hemisphere, by Wiliams Hershell, his sister Carolina and his son John [4]. At that time the study of the universe was restricted to the observation and description of astrophysical objects. Any conjecture on the fundamental origin of those structures was speculative and metaphysical.
We attempt to model a present time accelerating universe, in the framework of FLRW space-time using field equations of f(R,T) gravity and taking , being an arbitrary constant. For this, terms containing in the field equation are assumed as a source of energy producing negative pressure. Our model is a novel one in the sense that the parameter develops a fluid whose equation of state is parameterized. The model parameters, present values of density, Hubble and deceleration parameters are estimated statistically to arrive at physically viable cosmology. We consider three types of observational data set: 46 Hubble parameter data set, SNe Ia 715 data sets of distance modulus and apparent magnitude and 66 Pantheon data set (the latest compilation of SN Ia 40 bined plus 26 high redshift apparent magnitude data set in the redshift range . These data are compared with theoretical results through the statistical test. The universe model exhibits phase transition from decelerating to accelerating one. We have calculated transional red shifts and time for the data sets. Our estimated results for the present values of various model parameters such Hubble , deceleration etc. are found as per expectations and surveys. We get a very interesting result from estimations that at present, the value of density is . The critical density is estimated as in the literature. The higher value of present density is attributed to the presence of dark matter and dark energy in the universe. We have also examined the behaviour of pressure in our model. It is negative and is dominant over density .
In this paper, we have investigated an exact solution of Einstein’s field equation of isotropic and homogeneous universe. We have performed test to obtain the best fit value of model parameters of derived model with its observed values. It is obtained that the best fit values of Hubble constant and density parameters are , and by bounding the derived model with latest H(z) data while with Pantheon data, its values are , and . The dynamics of deceleration parameter shows that in the derived model, the universe was in decelerating phase for the transition red-shift . At , the present universe has entered in accelerating phase of expansion. The age of current universe is obtained as Gyrs which is in good consistency with its value observed by Plank collaboration results and WMAP observations.PACS number: 98.80.-k, 04.20.Jb
This chapter deploys evidence from the physical sciences in favour of the temporality of determinacy. It introduces research in the philosophy of cosmology which provides a framework for the evolution of physical parameters in time. It challenges the conjecture that time variables can be straightforwardly expunged from physical laws in certain physical theories, such as statistical mechanics, by virtue of characteristics such as probabilistic irrelevance. Turning to quantum mechanics, the chapter explores a number of debates whose outcomes have implications for the atemporality conjecture, including Huw Price’s discussions of retrocausal behaviour, transactional theories originating in the research of Feynman and Wheeler, as well as recent innovations by Pusey and Leifer.
In this article, we explore the heuristic power of the theoretical distinction between framework and interaction theories applied to the case of General Relativity. According to the distinction, theories and theoretical elements can be classified into two different groups, each with clear ontological, epistemic and functional content. Being so, to identify the group to which a theory belongs would suffice to know a priori its prospects and limitations in these areas without going into a detailed technical analysis. We make the exercise here with General Relativity, anticipate its ontological, epistemic and functional content and show afterwords that such expectations are justified in this case, being consistent with formal issues of General Relativity. With this, we attempt to make a case for the use of the distinction as a powerful tool for scientific and philosophical analysis.
In this article we give sufficient and necessary conditions for the validity of the Huygens' principle for the Dirac equation in the non-constant curvature spacetime of the Friedmann-Lemaître-Robertson-Walker models of cosmology. The Huygens' principle discussed for the equation of a field with mass m=0 as well as a massive spin-12 field undergoing a red shift of its wavelength as the universe expands.
In this paper, we use four-dimensional quaternionic algebra to describe space-time geodesics in curvature form. The transformation relations of a quaternionic variables are established with the help of basis-transformations of quaternion algebra. We deduce the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and the geodesic equation are also discussed to describe the quaternionic line element in curved space-time. We examine a quaternionic metric tensor equation for the Riemannian Christoffel curvature tensor. We present the quaternionic Einstein’s field-like equation, which indicates that quaternionic matter and geometry are equivalent. Relevance of the work:In recent decades, hypercomplex algebra, viz., quaternion and octonions, has been widely used to explain various branches of physics. In this way, we have investigated quaternionic transformations and field equations in curved space-time. The present novel work will help to explain the characteristics of the curved space-time universe in terms of quaternion algebra. It can also be used to describe quaternionic gravitational waves, the black hole formulation, and so on.
Using the language of differential forms, the Kaluza–Klein theory in 4+1 dimensions is derived. This theory unifies electromagnetic and gravitational interactions in four dimensions when the extra space dimension is compactified. Without any ad hoc assumptions about the five-dimensional metric, the theory also contains a scalar field coupled to the other fields. By a conformal transformation the theory is transformed from the Jordan frame to the Einstein frame where the physical content is more manifest. Including a cosmological constant in the five-dimensional formulation, it is seen to result in an exponential potential for the scalar field in four dimensions. A similar potential is also found from the Casimir energy in the compact dimension. The resulting scalar field dynamics mimics realistic models recently proposed for cosmological quintessence.
Algebraically special solutions of Einstein's empty-space field ; equations that are characterized by the existence of a geodesic and shear-free ; ray congruence are considered. A class of solutions is presented for which the ; congruence is diverging and is not necessarily hypersurface orthogonal. (C.E.S.);
Recent observational indications of an accelerating universe enhance the interest in studying models with a cosmological constant. We investigate cosmological expan- sion (FRW metric) with Lambda > 0 for a general linear equation of state p = w\rho, w > −1, so that the interplay between cosmological vacuum and quintessence is allowed, as well. Four closed-form solutions (flat universe with any w, and w = 1/3, −1/3, −2/3) are given, in a proper compact representation. Various estimates of the expansion are presented in a general case when no closed-form solutions are available. For the open universe a simple relation between solutions with different parameters is established: it turns out that a solution with some w and (properly scaled) � is expressed algebraically via another solution with special different values of these parameters. The expansion becomes exponential at large times, and the amplitude at the exponent depends on the parameters. We study this dependence in detail, deriving various representations for the amplitude in terms of integrals and series. The closed- form solutions serve as benchmarks, and the solution transformation property noted above serves as a useful tool. Among the results obtained, one is that for the open universe with relatively small cosmological constant the amplitude is independent of the equation of state. Also, estimates of the cosmic age through the observable ratio �/M and parameter w are given; when inverted, they provide an estimate of w, i. e., the state equation, through the known �/M and age of the universe.
We address the localization of gravity on the Friedmann-Robertson-Walker type brane embedded in either AdS5 or dS5 bulk space, and derive two definite limits between which the value of the bulk cosmological constant has to lie in order to localize the graviton on the brane. The lower limit implies that the brane should be either dS4 or 4D Minkowski in the AdS5 bulk. The positive upper limit indicates that the gravity can be trapped also on a curved brane in the dS5 bulk space. Some implications for recent cosmological scenarios are also discussed.
In this paper we investigate the canonical structure of diffeomorphism-invariant phase spaces for spatially locally homogeneous
spacetimes with 3-dimensional compact closed spaces. After giving a general algorithm to express the diffeomorphism-invariant
phase space and the canonical structure of a locally homogeneous system in terms of those of a homogeneous system on a covering
space and a moduli space, we determine completely the symplectic structures and the Hamiltonians of locally homogeneous pure
gravity systems on orientable, compact closed 3-spaces of the Thurston-type E3, Nil and Sol for all possible space topologies and invariance groups. We point out that in many cases the symplectic structure
of the phase space becomes degenerate in the moduli sector. This implies that locally homogeneous systems are not canonically
closed in general in the full diffeomorphism-invariant phase space of generic spacetimes with compact closed spaces.
This textbook provides advanced undergraduate and graduate students with a complete introduction to modern cosmology. It successfully bridges the gap between undergraduate and advanced graduate texts by discussing topics of current research, starting from first principles. Throughout this authoritative volume, emphasis is given to the simplest, most intuitive explanation for key equations used by researchers. The first third of the book carefully develops the necessary background in general relativity and quantum fields. The rest of the book then provides self-contained accounts of all the key topics in contemporary cosmology, including inflation, topological defects, gravitational lensing, galaxy formation, large-scale structure and the distance scale. To aid understanding, the book is well illustrated with helpful figures and includes outline solutions to nearly 100 problems. All necessary astronomical jargon is clearly explained, ensuring the book is self-contained for any student with undergraduate physics.
The kinematical properties of the de Sitter space-time are reviewed and investigated. The properties of the static sections are clarified. A deduction of the analytic extension, analogous to that of Kruskal and Szekeres for the Schwarzschild space-time, of the static section to the region outside the horizon is given. The representation of the de Sitter space-time as a four-dimensional hyperboloid in Minkowskian five-dimensional spacetime is reviewed. Coordinate transformations between different sections of the de Sitter space-time are found. By means of the transformation formulae the different sections are mapped onto each other in space-time diagrams. These mappings are interpreted kinematically. We have aimed at providing, whenever possible, an intuitive understanding of the kinematical properties of the different sections, and how they are interrelated. Among others we present real coordinate transformations between the static and the three Robertson-Walker sections of the de Sitter space-time on one hand and the vacuum dominated Bianchi type-III model on the other hand. These transformations are used to map the path of a typical Bianchi type-III reference particle into the static and the Robertson-Walker sections.
Introduction to Cosmology provides a rare combination of a solid
foundation of the core physical concepts of cosmology and the most
recent astronomical observations. The text is designed for advanced
undergraduates or beginning graduate students and assumes no prior
knowledge of general relativity. An emphasis is placed on developing the
students' physical insight rather than losing them with complex math. An
approachable writing style and wealth of fresh and imaginative analogies
from "everyday" physics are used to make the concepts of cosmology more
accessible.
Starting with the idea of an event and finishing with a description of
the standard big-bang model of the universe, this textbook provides a
clear, concise and up-to-date introduction to the theory of relativity.
Throughout, the emphasis is on the geometric structure of spacetime,
rather than the traditional coordinate-dependent approach. Topics
covered include flat spacetime (special relativity), Maxwell fields, the
energy-momentum tensor, spacetime curvature and gravity, Schwarzschild
and Kerr spacetimes, black holes and singularities, and cosmology.
Exercises are provided at the end of each chapter and key ideas in the
text are illustrated with worked examples.
Geometry and physics: an overview; 1. The background manifold structure;
2. Differentiation; 3. The curvature; 4. Space-time and tetrad
formalism; 5. Spinors and the classification of the Weyl tensor; 6.
Coupling between fields and geometry; 7. Dynamics on curved manifolds;
8. Geometry of congruences; 9. Physical measurements in space-time; 10.
Spherically symmetric solutions; 11. Axially symmetric solutions;
References; Notation; Index.
There are a number of similarities between black-hole physics and thermodynamics. Most striking is the similarity in the behaviors of black-hole area and of entropy: Both quantities tend to increase irreversibly. In this paper we make this similarity the basis of a thermodynamic approach to black-hole physics. After a brief review of the elements of the theory of information, we discuss black-hole physics from the point of view of information theory. We show that it is natural to introduce the concept of black-hole entropy as the measure of information about a black-hole interior which is inaccessible to an exterior observer. Considerations of simplicity and consistency, and dimensional arguments indicate that the black-hole entropy is equal to the ratio of the black-hole area to the square of the Planck length times a dimensionless constant of order unity. A different approach making use of the specific properties of Kerr black holes and of concepts from information theory leads to the same conclusion, and suggests a definite value for the constant. The physical content of the concept of black-hole entropy derives from the following generalized version of the second law: When common entropy goes down a black hole, the common entropy in the black-hole exterior plus the black-hole entropy never decreases. The validity of this version of the second law is supported by an argument from information theory as well as by several examples.
The relation of the special and the general principle of relativity to the principle of covariance, the principle of equivalence and Mach's principle, is discussed. In particular, the connection between Lorentz covariance and the special principle of relativity is illustrated by giving Lorentz covariant formulations of laws that violate the special principle of relativity: Ohm's law and what we call “Aristotle's first and second laws.” An “Aristotelian” universe in which all motion is relative to “absolute space” is considered. The first law: a free particle is at rest. The second law: force is proportional to velocity. Ohm's law: the current density is proportional to the electrical field strength. Neither of these laws fulfills the principle of relativity. The examples illustrate, in the context of Lorentz covariance and special relativity, Kretschmann's critique of founding Einstein's general principle of relativity on the principle of general covariance. A modification of the principle of covariance is suggested, which may serve as a restricted criterium for a physical law to satisfy Einstein's general principle of relativity. Other objections that have been raised to the validity of Einstein's general principle of relativity are based upon the preferred state of inertial frames in the general, as well as in the special theory, the existence of tidal effects in “true” gravitational fields, doubts as to the validity of Mach's principle, whether electromagnetic phenomena obey the principle, and, finally, the anisotropy of the cosmic background radiation. These objections are reviewed and discussed.
The relativistic equation of motion of a radiating charge is discussed with special emphasis upon a clarification of the significance of the Schott energy for the energy-momentum conservation of the charge and the field it produces. In particular hyperbolic motion is studied. The case that a charge with constant velocity enters and leaves a region with hyperbolic motion is analysed. We find that the Schott energy is increased as the particle enters the region and that the energy it radiates while the charge moves hyperbolically comes from the Schott energy. A result of our analysis is that this energy is localized in the field close to the charge.
The question whether a hyperbolically moving charge emits radiation is discussed. In order to arrive at an unambiguous answer the question is considered from several points of view. The power spectrum of the electromagnetic field due to a hyperbolically moving charge is deduced. Also we analyse the field in order to clarify whether it contains a radiation zone. Our conclusion is that a charge with constant proper acceleration emits radiation in spite of the fact that the Poynting vector of the field vanishes everywhere in the instantaneous inertial rest frame of the charge.
The electromagnetic field of a charged particle with hyperbolic motion is described. The field of a particle with eternal hyperbolic motion is found by taking the limit of a situation with limited hyperbolic motion. The flow of energy in the electromagnetic field is discussed. The question whether the charged particle emits radiation is briefly touched upon in this first article in a series on the electrodynamics of radiating particles and their fields.
This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, the deformation tensors of elasticity, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields, including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers, quarks, and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space; consequently, the book should be of interest also to mathematics students. Ideal for graduate and advanced undergraduate students of physics, engineering and mathematics as a course text or for self study.
An analysis is given of the stress-energy tensor and geometry produced by slowly rotating bodies. The geometrized mass GM/c2 of the body is allowed to be comparable to its radius. The geometry is treated as a perturbation of the Schwarzschild geometry, which leads to considerable simplification of Einstein's equations. The rotation of the intertial frame induced by a rotating massive shell is calculated and discussed with particular attention to two limiting cases: (1) For small masses it reduces to Thirring's well-known result; (2) for large masses, whose Schwarzschild radius approaches the shell radius, the induced rotation approaches the rotation of the shell. These and the corresponding results for an expanding and recollapsing dust cloud are examined for their consistency with particular interpretations of Mach's principle. The analytic extension of the rotating exterior metric is a completely source-free rotating solution. It describes a slowly rotating, expanding, and recontracting Einstein-Rosen bridge which can be taken as a geometrodynamic model for a slowly rotating body.
The author describes the modern approach to quantum cosmology, as initiated by Hartle and Hawking, Linde, Vilenkin and others. The primary aim is to explain how one determines the consequences for the late universe of a given quantum theory of cosmological initial or boundary conditions.
This book provides a concise introduction to the mathematical aspects of the origin, structure and evolution of the universe. The book begins with a brief overview of observational and theoretical cosmology, along with a short introduction of general relativity. It then goes on to discuss Friedmann models, the Hubble constant and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the distant future of the universe. This new edition contains a rigorous derivation of the Robertson-Walker metric. It also discusses the limits to the parameter space through various theoretical and observational constraints, and presents a new inflationary solution for a sixth degree potential. This book is suitable as a textbook for advanced undergraduates and beginning graduate students. It will also be of interest to cosmologists, astrophysicists, applied mathematicians and mathematical physicists.
In the world about us, the past is distinctly different from the future.
More precisely, we say that the processes going on in the world about us
are asymmetric in time or display an arrow of time. Yet this manifest
fact of our experience is particularly difficult to explain in terms of
the fundamental laws of physics. Newton's laws, quantum mechanics,
electromagnetism, Einstein's theory of gravity, etc., make no
distinction between past and future - they are time-symmetric.
Reconciliation of these profoundly conflicting facts is the topic of
this volume. It is an interdisciplinary survey of the variety of
interconnected phenomena defining arrows of time, and their possible
explanations in terms of underlying time-symmetric laws of physics.
Contents: 1. Information, computation and complexity. 2. Statistical origins of irreversibility. 3. Decoherence. 4. Time asymmetry and quantum mechanics. 5. Quantum cosmology and initial conditions. 6. Time and irreversibility in gravitational physics.
An experimental search for the existence of particles that propagate with velocities that are always greater than the velocity of light in a vacuum has been performed. Subject to some assumptions regarding the behavior of such particles, it is found that the photoproduction cross section for such particles in lead is less than 3×10-30 cm2. This upper limit does not depend strongly on the "rest mass" of the particles and is approximately the same for faster-than-light particles of any "rest mass." This search was sensitive to particles having charges between 0.1 and 2 electron charges.
An experiment was performed to demonstrate the relativisilc time dilanon ; as a large effect, using only comparatively simple equipment. Mesons( mu ) were ; selected to have speeds in the range between 0.9960 c and 0.9954 c. The number of ; these that reached sea level was measured. The number expected without time ; dilation was calculated from the distribution of decay times of these mesons( mu ; ) and from the known distance of descent. The observed time dilation factor is ; 8.8 plus or minus 0.8 to be compared with the effective time dilation factor ; calculated for mesons of these speeds in the detection geometry. (auth);
The effect of gravitational potential on the apparent frequency of ; electro-magnetic radiation was measured by using the sharply defined energy of ; recoil-free 14.4kev gamma rays emitted by Coâµâ· and absorbed in Feâµâ·. ; The mean gravitational shift in frequency was measured to be (-17.6 plus or ; minus 0.6) x l0¹âµ. (C.J.G.);
This book is a broad and elementary introduction to the science of modern cosmology, with emphasis on its historical origins. This second edition of "Cosmology" updates and extends the first edition, published in 1981 (see Abstr. 30.003.070). The additional chapters discuss early scientific cosmology, Cartesian and Newtonian world systems, cosmology after Newton and before Einstein, observational cosmology, inflation, and creation of the universe.
This textbook fills a gap in the existing literature on general relativity by providing the advanced student with practical tools for the computation of many physically interesting quantities. The context is provided by the mathematical theory of black holes, one of the most successful and relevant applications of general relativity. Topics covered include congruences of timelike and null geodesics, the embedding of spacelike, timelike and null hypersurfaces in spacetime, and the Lagrangian and Hamiltonian formulations of general relativity.
We discuss the asymptotic dynamical evolution of spatially homogeneous brane-world cosmological models close to the initial singularity. We find that generically the cosmological singularity is isotropic in Bianchi type IX brane-world models and consequently these models do not exhibit Mixmaster or chaotic-like behaviour close to the initial singularity. We argue that this is typical of more general cosmological models in the brane-world scenario. In particular, we show that an isotropic singularity is a past-attractor in all orthogonal Bianchi models and is a local past-attractor in a class of inhomogeneous brane-world models.
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We show that a simple solution to the vacuum field equations of general relativity in 4+1 space-time dimensions leads to a cosmology which at the present epoch has 3+1 observable dimensions in which the Einstein-Maxwell equations are obeyed. The large ratio of the electromagnetic to gravitational forces is a consequence of the age of the Universe, in agreement with Dirac's large-number hypothesis.
Some solutions of the Einstein field equations for a dust source are given in explicit form. They are spatially homogeneous, irrotational, and anisotropic. They can be characterized as those spatially homogeneous expanding models that do not permit a simply transitive three‐parameter group of motions. The models are compared in detail with observations and with the Friedmann models. In a few instances slightly longer time scales are obtained with the present models than from the corresponding Friedmann models.
The quantum state of a spatially closed universe can be described by a wave function which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. The wave function obeys the Wheeler-DeWitt second-order functional differential equation. We put forward a proposal for the wave function of the "ground state" or state of minimum excitation: the ground-state amplitude for a three-geometry is given by a path integral over all compact positive-definite four-geometries which have the three-geometry as a boundary. The requirement that the Hamiltonian be Hermitian then defines the boundary conditions for the Wheeler-DeWitt equation and the spectrum of possible excited states. To illustrate the above, we calculate the ground and excited states in a simple minisuperspace model in which the scale factor is the only gravitational degree of freedom, a conformally invariant scalar field is the only matter degree of freedom and Λ>0. The ground state corresponds to de Sitter space in the classical limit. There are excited states which represent universes which expand from zero volume, reach a maximum size, and then recollapse but which have a finite (though very small) probability of tunneling through a potential barrier to a de Sitter-type state of continual expansion. The path-integral approach allows us to handle situations in which the topology of the three-manifold changes. We estimate the probability that the ground state in our minisuperspace model contains more than one connected component of the spacelike surface.