On the method of modified equations. Analysis of the Timoshenko beam theory is done in two ways. It turns out that for any finite-dimensional Lie algebra this system has a large collection of integrals which are in involution. The equations were introduced by L. Numerical techniques based on the equivalent equation for the Euler forward difference method. In all cases, the main types of motion are translations, rotations, oscillations, or any combinations of these.
Green Lagrange strain tensor is relative to undefined coordinate system. With today's computer, an accurate solution can be obtained rapidly. We spend a 5. In the case of a compressible Newtonian fluid, this yields These equations are at the heart of fluid flow modeling. It is Equations of motion.
Although Navier-Stokes equations only refer to the equations of motion conservation of momentum , it is commonly root finding techniques iteration, bisection, Newton-Raphson for non linear equations in one variable, integration Simpson and Trapezoidal rules and Solution of ordinary differential equations Euler and modified Euler. Comparisons with Euler simulations are performed and the transition from regular to irregular reflection is also discussed. Title: Modified Euler approximation scheme for stochastic differential equations driven by fractional Brownian motions Authors: Yaozhong Hu , Yanghui Liu , David Nualart Submitted on 6 Jun v1 , last revised 4 Mar this version, v2 Looking for Euler equation of motion?
Find out information about Euler equation of motion. The Euler equations can be applied to incompressible and to compressible flow — assuming the flow velocity is a solenoidal field, or using another appropriate energy equation respectively the simplest form for Euler equations being the conservation of the specific entropy. Notice that all of the dependent variables appear in each equation. Combining Newton's two laws, we obtain a system of differential equations which characterize the motion of the particles The EL equation you quote is only valid when the Lagrangian only contains first order derivatives.
These three equations are extensions of the planar Euler—Savary relations for envelopes to spatial relations. The Lagrangian is an object that has to be guessed by making use of symmetry considerations and characterizes the system in question. After reading this chapter, you should be able to.
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Here the Cp and the Cv represent the specific heats of the fluid at constant pressure and constant volume, respectively. This means that elementary solutions cannot be combined to provide the solution for a more complex The Euler's equation for steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure and density of a moving fluid.
The most important equation on modern physics are equations of motions. Torque-free motion of an axisymmetric body. These generalizations possess first integrals which are polynomial in the angular momenta. Jan 11, describes the motion of an inviscid incompressible fluid. Control Moment Gyroscopes CMGs are Collisional motion of a granular material composed of rough inelastic spheres is analysed on the basis of the kinetic Boltzmann—Enskog equation. The most relevant feature of the DG finite element method is that equations are solved not only for the mean flow field, but also for the flow field gradients.
How to Solve Differential Equations. This motion can be visualized by Poinsot's construction. Equations of motion listed as EOM by Omelayan and modified by them to obtain equations of motion for Euler 2. However, except in degenerate cases in very simple geometries such as equations, such as the large scale problems of modeling ocean dynamics, weather systems and even cosmological problems stemming from general relativity. We can follow this procedure to write the second order equation as a first order system. Find the Hamiltonian Equations of Motion in the y dimension. In Chapter 1, we introduce the basic settings and define time-discrete numerical approximation schemes.
The Modified Rodrigues parameter MRP is used for kinematic parameterization and is the only measurable variable at the plant output. The nonlinear term in these equations has primarily only a growth condition assumption. Reading the literature on Euler, one finds he was very interested in the math and physics of "spolling" spin and roll rigid bodies hoops, etc. This results in a very The Euler method is a numerical method that allows solving differential equations ordinary differential equations. The derivation of the equations of motion of damped and driven pendula extends the derivation of the undamped and undriven case.
Extremal energy properties and construction of stable solutions of the Euler equations By G. October 21, 7. Let's denote the time at the nth time-step by t n and the computed solution at the nth time-step by y n, i. Mishchenko, A. Modified Euler method c. Dombre effect of the joint accelerations on the base motion, and the dual effect. Numerical Method In this study, the governing equations used are the two-dimensional Euler equations.
The discretized Euler equations are solved using the modified four-stage Runge-Kutta scheme developed by Jameson et al24, While the motion of a finite set of particles, such as the motion of our solar system can be modeled by a set of ordinary differential equations, partial differential equations are for the motion of a set of uncountable number of particles. Classical dynamics focuses on formulations and solutions of equations of motion for 3D dynamic systems. Finding the initial condition based on the result of approximating with Euler's method.
They provide several serious challenges to obtaining the general solution for the motion of a three-dimensional rigid body. The bearing is modelled with a non-constant stiffness distribution along its length and a non-symmetric centre of gravity. Equations governing the motion of a specific class of singularities of the Euler equation in On a modified streamline curvature method for the Euler equations.
Scale invariant forms of Cauchy, Euler, Navier-Stokes and modified equations of motion are described. It is based on the Newton's Second Law of Motion. Villatoro, F. This course in Kinematics covers four major topic areas: an introduction to particle kinematics, a deep dive into rigid body kinematics in two parts starting with classic descriptions of motion using the directional cosine matrix and Euler angles, and concluding with a review of modern descriptors like quaternions and Classical and Modified The iPhone drawing was modified from an Apple Inc. The Euler and Navier-Stokes equations describe the motion of a fluid in.
The modeling problems of biped robots lie in their varying configurations during locomotion. We can derive these equations using symmetry considerations from the corresponding Lagrangian using the Euler-Lagrange Equations. Fomenko Abstract: In this paper, a special class of dynamical systems is studied-the so-called Euler equations a natural generalization of the classical equations of motion of a rigid body with one fixed point. Edward J.
The sole aim of this page is to share the knowledge of how to implement Python in numerical stochastic modeling to anyone, for free. The approximated Lagrangian generates filtered wave equations and the linear filtered equations of motion. Versions of Maxwell's equations based on the electric and magnetic potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. Equation of motion for a rotating fluid 6. Jun 2, Euler's rotation equations are a vectorial quasilinear first-order I have just reproduced, with some small modifications, Figure III.
Dynamics is general, since momenta, forces and energy of the particles are taken into account. Just as with integration, although one can solve some equations analytically. Global Variables The Navier-Stokes equations govern the motion of fluids and can be seen as Newton's second law of motion for fluids. Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler-Lagrange EL equations. The classical Euler—Poinsot case of the rigid body dynamics admits a class of simple but non-trivial integrable generalizations, which modify the Poisson equations describing the motion of the body in space.
The above equations are known as Euler's equations. Euler method b. Excel was used. The Euler method gave "acceptable" results for our previous problems, but in this case of oscillatory motion the results are not acceptable. A rigorous analysis is presented for the method of modified equations whereby its range of applicability and its shortcomings are delineated. Cromer, Stable solutions using the Euler Approximation, American Journal of Physics, 49, , this simple modification conserves energy for oscillatory problems unlike Euler method which artifactually increases energy of the oscillator with time.
A Baseline 6 Degree of Freedom DOF Mathematical Model of a Generic Missile Executive Summary Computer Simulation Models of many new missile systems will be required in the near In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia.
These generalizations possess first integrals which are polynomial in the angular momenta vibration differential equations of motion of Euler-Bernoulli beams with different boundary conditions and dynamic loads. Consider the differential equation: The first step is to convert the above second-order ode into two first-order ode. The modified equations of motion are:.
Euler in Vortex approximation, Euler's equation, two dimensions, incom- pressible fluid flow calculations. In this instance, sometimes the term refers to the differential equations that the system satisfies e. These equations are to be solved for an unknown velocity vector and pressure, defined for position and time. Solving Nonlinear Euler Equations With Arbitrary Accuracy Glenn Research Center The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation MESA schemes.
The equations of motion for rigid bodies in terms of the unit quaternions can be found, e. They are ubiquitous is science and engineering as well as economics, social science, biology, business, health care, etc. In general a fluid body may be subjected to the following forces namely. We also describe the corresponding general Hamiltonian framework of hydrodynamics Disclaimer This is not an official course offered by Boston University.
Several other mathematics and physics books offer somewhat modified versions. Eulers method requires a starting value. Find the Lagrangian Equations of Motion in the y dimension. Euler's method is a numerical tool for approximating values for solutions of differential equations. UNIT — 4 Descriptive statistics, exploratory data analysis Abstract: Small scale creation in fluid motion is ubiquitous. The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.
Finally, there is always common sense. The equations have application to both analytical and wind tunnel captive trajectory store separation testing. An arbitrary member of the family of fractional Brownian motions can be used in these equations. In this section we focus on Euler's method, a basic numerical method for solving initial value problems. Leu, "Computer generation of robot dynamics equations and the related. Turbulence is most significant macroscopic problem in physics.
I thought an equation of motion was something where you are given a Lagrangian and, using the Euler-Lagrange equation, you then find the equations of motion for that system. Let be the displacement of the center of mass of the disk down the slope, and let be the angle subtended between the pendulum and the downward vertical. This presents some interesting ways of investigating solutions to the Euler equations. MCST due to analyze the. In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia.
These are the original rigid body rotational motion equations, modified to account for reaction wheel torques. The nature of dissipation and central importance of Heisenberg spectral definition of kinematic viscosity and their connections to Planck energy distribution law for equilibrium statistical fields are discussed. It is an explicit method for solving initial value problems IVPs , as described in the wikipedia page. ENO schemes are not yet supported, but the proposed answer below is robust enough to tackle a range of Euler fluid dynamic and MHD problems Euler's Method We have seen how to use a direction field to obtain qualitative information about the solutions to a differential equation.
For the case when the Lagrangian contains higher order derivatives, you need to do additional partial integrations when deriving the corresponding equations of motion. We develop an alternative approach to this theory, using modified Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. Equation of motion for a nonrotating fluid 3.
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Newton-Euler equations of motion. Then we put the Lagrangian into the Euler-Lagrange equation and this gives us the equations of motion of the system. Provide the source term input to the code, 5. Demonstrate that Lagrange's equations of motion for the system are The existence of a martingale solution to 2-dimensional stochastic Euler equations is proved.
Multigrid algorithms 'Flux vector splitting and approximate Newton methods' -- subject s : Euler equations of motion, Flux splitting In this dissertation, we investigate time-discrete numerical approximation schemes for rough differential equations and stochastic differential equations SDE driven by fractional Brownian motions fBm.
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Solution of first-order problems a. Body cone and space cone. Geometric methods and the Euler equations of an Ideal fluid Jonathan Munn Imperial College London November 16, Abstract We formulate the Euler equations for Inviscid ideal fluid flow in terms of quater-nionic and differential geometry. The model describes a modern passenger car rear axle suspension where the compliances in bushing B are taken into account.
In this hands-on course, learn about the history and recent developments in math, mechanics, and computation. Update The 1D Euler equations were modified to match this source. Euler's method starting at x equals zero with the a step size of one gives the approximation that g of two is approximately 4. Explicit methods calculate the state of the system at a later time from the state of the system at the current time without the need to solve algebraic equations. Historically, only the incompressible equations have been derived by With the advent of special relativity and general relativity, the theoretical modifications to spacetime meant the classical equations of motion were also modified to account for the finite speed of light, and curvature of spacetime.
ENO schemes are not yet supported, but the proposed answer below is robust enough to tackle a range of Euler fluid dynamic and MHD problems They complete the issue of the new elastic terms of the enhanced nonlinear 3D Euler-Bernoulli beam. Sometimes, however, we want more detailed information. Differential Equation: Contains an unknown function and its derivatives. The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid. Analytical solutions of the proposed modified restricted Euler equation appear to be difficult to obtain.
Equation of motion is. In Section 3. A corollary of the main result in [ 20 ] concerns the issue of global existence and thus we mention it here. It is shown that a particular class of initial data, which lead to formation of black holes, have the property that the solutions exist for all Schwarzschild time. The initial data consist of two parts: an inner part, which is a static solution of the Einstein-Vlasov system, and an outer part with matter moving inwards. The set-up is shown to preserve the direction of the momenta of the outer part of the matter, and it is also shown that in Schwarzschild time the inner part and the outer part of the matter never interact in Schwarzschild time.
As was mentioned at the end of Section 3. However, it is reasonable to believe that global existence for general data may hold in a polar time gauge or a maximal time gauge, cf. However, there is no proof of this statement for any matter model and it would be very satisfying to provide an answer to this conjecture for the Einstein-Vlasov system.
A proof of global existence in these time coordinates would also be of great importance due to its relation to the weak cosmic censorship conjecture, cf. The methods of proofs in the cases described in Sections 3. In this section we discuss some attempts to treat general initial data. These results are all conditional in the sense that assumptions are made on the solutions, and not only on the initial data. The first study on global existence for general initial data is [ ], which is carried out in Schwarzschild coordinates.
The authors introduce the following variables in the momentum space adapted to spherical symmetry,. A consequence of spherical symmetry is that angular momentum is conserved along the characteristics. The main result in [ ] shows that as long as there is no matter in the ball. Thus, in view of the continuation criterion this can be viewed as a global existence result outside the center of symmetry for initial data with compact support.
This result rules out shell-crossing singularities, which are present when, e. The bound of Q is obtained by estimating each term individually in the characteristic equation associated with the Vlasov equation 48 for the radial momentum. This involves a particular difficulty. The Einstein equations imply that. The method in [ ] makes use of a cancellation property of the radial momenta in T 1 so that outside the center this term is manageable but in general it seems very unpleasant to have to treat point-wise terms of this kind.
In [ ] Rendall shows global existence outside the center in maximal-isotropic coordinates. The bound on Q t is again obtained by estimating each term in the characteristic equation. In this case there are no point-wise terms in contrast to the case with Schwarzschild coordinates.
However, the terms are, in analogy with the Schwarzschild case, strongly singular at the center. A recent work [ 12 ] gives an alternative and simplified proof of the result in [ ]. This results in a combination of terms involving second-order derivatives, which can be substituted for by one of the Einstein equations. In addition, the bound of Q is improved compared to 49 and reads. This bound is sufficient to conclude that global existence outside the center also holds for non-compact initial data.
The method in [ 12 ] also applies to the case of maximal-isotropic coordinates studied in [ ]. There is an improvement concerning the regularity of the terms that need to be estimated to obtain global existence in the general case. A consequence of [ 12 ] is accordingly that the quite different proofs in [ ] and in [ ] are put on the same footing. We point out that the method can also be applied to the case of maximal-areal coordinates.
The results discussed above concern time gauges, which are expected to be singularity avoiding so that the issue of global existence makes sense. The main motivation for studying the system in these coordinates has its origin from the method of proof of the cosmic-censorship conjecture for the Einstein-scalar field system by Christodoulou [ 60 ]. An essential part of his method is based on the understanding of the formation of trapped surfaces [ 58 ]. The results in [ ] and in [ ] are not sufficient for concluding that the hypothesis of the matter needed in the theorem in [ 62 ] is satisfied, since they concern a portion of the maximal development covered by particular coordinates.
Therefore, Dafermos and Rendall [ 64 ] choose double-null coordinates, which cover the maximal development, and they show that the mentioned hypothesis is satisfied for Vlasov matter. The main reason that the question of global existence in certain time coordinates discussed in the previous Section 3. Now there is, in fact, no theorem in the literature, which guarantees that weak cosmic censorship follows from such a global existence result, but there are strong reasons to believe that this is the case, cf.
Hence, if initial data can be constructed, which lead to naked singularities, then either the conjecture that global existence holds generally is false or the viewpoint that global existence implies the absence of naked singularities is wrong. In view of a recent result by Rendall and Velazquez [ ] on self similar dust-like solutions for the massless Einstein-Vlasov system, this issue has much current interest. However, this result is based on a scaling of the density function itself and therefore makes the result less related to the Cauchy problem.
Also, their proof is, in part, based on numerics, which makes it harder to judge the relevance of the result. The main aim of the work [ ] is to establish self-similar solutions of the massive Einstein-Vlasov system and the present result can be viewed as a first step to achieving this. In the set-up, two simplifications are made. First, the authors study the massless case in order to find a scaling group, which leaves the system invariant. More precisely, the massless system is invariant under the scaling. The massless assumption seems not very restrictive since, if a singularity forms, the momenta will be large and therefore the influence of the rest mass of the particles will be negligible, so that asymptotically the solution can be self-similar also in the massive case, cf.
The second simplification is that the possible radial momenta are restricted to two values, which means that the density function is a distribution in this variable. Thus, the solutions can be thought of as intermediate between smooth solutions of the Einstein-Vlasov system and dust. For this simplified system it turns out that the existence question of self-similar solutions can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters.
The proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes. The reason that an ODE system is obtained is due to the assumption on the radial momenta, and if regular initial data is considered, an ODE system is not sufficient and a system of partial differential equations results. The self-similar solution obtained by Rendall and Velazquez has some interesting properties. The solution is not asymptotically flat but there are ideas outlined in [ ] of how this can be overcome.
It should be pointed out here that a similar problem occurs in the work by Christodoulou [ 59 ] for a scalar field, where the naked singularity solutions are obtained by truncating self-similar data. The singularity of the self-similar solution by Rendall and Velazquez is real in the sense that the Kretschmann scalar curvature blows up. The asymptotic structure of the solution is striking in view of the conditional global existence result in [ 12 ].
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Hence, the asymptotic structure of the self-similar solution has properties, which have been shown to be difficult to treat in the search for a proof of global existence. We have previously mentioned that there exist initial data for the spherically-symmetric Einstein-Vlasov system, which lead to formation of black holes. The first result in this direction was obtained by Rendall [ ]. He shows that there exist initial data for the spherically-symmetric Einstein-Vlasov system such that a trapped surface forms in the evolution. The occurrence of a trapped surface signals the formation of an event horizon.
As mentioned above, Dafermos [ 62 ] has proven that, if a spherically-symmetric spacetime contains a trapped surface and the matter model satisfies certain hypotheses, then weak cosmic censorship holds true. In [ 64 ] it was then shown that Vlasov matter does satisfy the required hypotheses. Hence, by combining these results it follows that initial data exist, which lead to gravitational collapse and for which weak cosmic censorship holds. However, the proof in [ ] rests on a continuity argument, and it is not possible to tell whether or not a given initial data set will give rise to a black hole.
Moreover, the mechanism of how trapped surfaces form is not revealed in [ ]. This is in contrast to the result in [ 24 ], where explicit conditions on the initial data are given, which guarantee the formation of trapped surfaces in the evolution. The analysis is carried out in Eddington-Finkelstein coordinates and a central result in [ 24 ] is to control the life span of the solution to ensure that there is sufficient time to form a trapped surface before the solution may break down.
In particular, weak cosmic censorship holds for these initial data. In [ 20 ] the formation of the event horizon in gravitational collapse is analyzed in Schwarzschild coordinates. Note that these coordinates do not admit trapped surfaces. The initial data in [ 20 ] consist of two separate parts of matter. One inner part and one outer part, in which all particles move inward initially.
The reason for the inner part is that it is possible to choose the parameters for the data such that the particles of the outer matter part continue to move inward for all Schwarzschild time as long as the particles do not interact with the inner part. This fact simplifies the analysis since the dynamics is much restricted when the particles keep the direction of their radial momenta.
The main result is that explicit conditions on the initial data with ADM mass M are given such that there is a family of outgoing null geodesics for which the area radius r along each geodesic is bounded by 2 M. It is furthermore shown that if. Hence, spacetime converges asymptotically to the Schwarzschild metric. In [ 23 ] it is shown that for initial data, which are closely related to those in [ 20 ], but such that the radial momenta are unbounded, all the matter do cross the event horizon asymptotically in Schwarzschild time.
This is in contrast to what happens to freely-falling observers in a static Schwarzschild spacetime, since they will never reach the event horizon. The result in [ 20 ] is reconsidered in [ 19 ], where an additional argument is given to match the definition of weak cosmic censorship given in [ 61 ]. It is natural to relate the results of [ 20 , 24 ] to those of Christodoulou on the spherically-symmetric Einstein-scalar field system [ 57 ] and [ 58 ]. In [ 58 ] explicit conditions on the initial data are specified, which guarantee the formation of trapped surfaces. The conditions on the initial data in [ 58 ] allow the ratio of the Hawking mass and the area radius to cover the full range, i.
Hence, it would be desirable to improve the conditions on the initial data in [ 24 ], although the conditions by Christodoulou for a scalar field are not expected to be sufficient in the case of Vlasov matter. In [ ] a numerical study on critical collapse for the Einstein-Vlasov system was initiated. A numerical scheme originally used for the Vlasov-Poisson system was modified to the spherically-symmetric Einstein-Vlasov system. It has been shown by Rein and Rodewis [ ] that the numerical scheme has desirable convergence properties. In the Vlasov-Poisson case, convergence was proven in [ ], see also [ 77 ].
The speculation discussed above that there may be no naked singularities formed for any regular initial data is in part based on the fact that the naked singularities that occur in scalar field collapse appear to be associated with the existence of type II critical collapse, while Vlasov matter is of type I. The primary goal in [ ] was indeed to decide whether Vlasov matter is type I or type II. These different types of matter are defined as follows.
Given small initial data, no black holes form and matter will disperse. For large data, black holes will form and consequently there is a transition regime separating dispersion of matter and formation of black holes. If we introduce a parameter A on the initial data such that for small A dispersion occurs and for large A a black hole is formed, we get a critical value A c separating these regions. For more information on critical collapse we refer to the review paper by Gundlach [ 89 ].
The conclusion of [ ] is that Vlasov matter is of type I. There are two other independent numerical simulations on critical collapse for Vlasov matter [ , 21 ]. In these simulations, maximalareal coordinates are used rather than Schwarzschild coordinates as in [ ]. The conclusion of these studies agrees with the one in [ ]. We end this section with a discussion of the spherically-symmetric Einstein-Vlasov-Maxwell system, i. Whereas the constraint equations in the uncharged case, written in Schwarzschild coordinates, do not involve solving any difficulties once the distribution function is given, the charged case is more challenging.
However, in [ ] it is shown that solutions to the constraint equations do exist for the Einstein-Vlasov-Maxwell system. In [ ] local existence is shown together with a continuation criterion, and global existence for small initial data is shown in [ ]. In this section we discuss the Einstein-Vlasov system for cosmological spacetimes, i.
The main goal is to determine the global properties of the solutions to the Einstein-Vlasov system for initial data given on a compact 3-manifold. In order to do so, a global time coordinate t must be found and the asymptotic behavior of the solutions when t tends to its limiting values has to be analyzed.
This might correspond to approaching a singularity, e. Presently, the aim of most of the studies of the cosmological Cauchy problem has been to show existence for unrestricted initial data and the results that have been obtained are in cases with symmetry see, however, [ 27 ], where to some extent global properties are shown in the case without symmetry. These studies will be reviewed below. This result will be reviewed at the end of this section.
The only spatially-homogeneous spacetimes admitting a compact Cauchy surface are the Bianchi types I, IX and the Kantowski-Sachs model; to allow for cosmological solutions with more general symmetry types, it is enough to replace the condition that the spacetime is spatially homogeneous, with the condition that the universal covering of spacetime is spatially homogeneous.
One of the first studies on the Einstein-Vlasov system for spatially-homogeneous spacetimes is the work [ ] by Rendall. He chooses a Gaussian time coordinate and investigates the maximal range of this time coordinate for solutions evolving from homogeneous data. For Bianchi IX and for Kantowski-Sachs spacetimes he finds that the range is finite and that there is a curvature singularity in both the past and the future time directions. For the other Bianchi types there is a curvature singularity in the past, and to the future spacetime is causally geodesically complete.
In particular, strong cosmic censorship holds in these cases. Although the questions on curvature singularities and geodesic completeness are very important, it is also desirable to have more detailed information on the asymptotic behavior of the solutions, and, in particular, to understand in which situations the choice of matter model is essential for the asymptotics. In recent years several studies on the Einstein-Vlasov system for spatially locally homogeneous spacetimes have been carried out with the goal to obtain a deeper understanding of the asymptotic structure of the solutions.
Roughly, these investigations can be divided into two cases: i studies on non-locally rotationally symmetric non-LRS Bianchi I models and ii studies of LRS Bianchi models. In case i Rendall shows in [ ] that solutions converge to dust solutions for late times. Under the additional assumption of small initial data this result is extended by Nungesser [ ], who gives the rate of convergence of the involved quantities. In [ ] Rendall also raises the question of the existance of solutions with complicated oscillatory behavior towards the initial singularity may exist for Vlasov matter, in contrast to perfect fluid matter.
Note that for a perfect fluid the pressure is isotropic, whereas for Vlasov matter the pressure may be anisotropic, and this fact could be sufficient to drastically change the dynamics. This question is answered in [ 93 ], where the existence of a heteroclinic network is established as a possible asymptotic state. This implies a complicated oscillating behavior, which differs from the dynamics of perfect fluid solutions.
The results in [ 93 ] were then put in a more general context by Calogero and Heinzle [ 46 ], where quite general anisotropic matter models are considered. In case ii the asymptotic behaviour of solutions has been analyzed in [ , , 48 , 47 ]. In [ ], the case of massless particles is considered, whereas the massive case is studied in [ ]. Both the nature of the initial singularity and the phase of unlimited expansion are analyzed. The authors compare their solutions with the solutions to the corresponding perfect fluid models. A general conclusion is that the choice of matter model is very important since, for all symmetry classes studied, there are differences between the collision-less model and a perfect fluid model, both regarding the initial singularity and the expanding phase.
The most striking example is for the Bianchi II models, where they find persistent oscillatory behavior near the singularity, which is quite different from the known behavior of Bianchi type II perfect fluid models. In [ ] it is also shown that solutions for massive particles are asymptotic to solutions with massless particles near the initial singularity.
For Bianchi I and II, it is also proven that solutions with massive particles are asymptotic to dust solutions at late times. It is conjectured that the same also holds true for Bianchi III. This problem is then settled by Rendall in [ ]. The investigation [ 48 ] concerns a large class of anisotropic matter models, and, in particular, it is shown that solutions of the Einstein-Vlasov system with massless particles oscillate in the limit towards the past singularity for Bianchi IX models.
This result is extended to the massive case in [ 47 ]. Before finishing this section we mention two other investigations on homogeneous models with Vlasov matter. In [ ] Lee considers the homogeneous spacetimes with a cosmological constant for all Bianchi models except Bianchi type IX. She shows global existence as well as future causal geodesic completeness.
Anguige [ 28 ] studies the conformal Einstein-Vlasov system for massless particles, which admit an isotropic singularity. He shows that the Cauchy problem is well posed with data consisting of the limiting density function at the singularity. In the spatially homogeneous case the metric can be written in a form that is independent of the spatial variables and this leads to an enormous simplification. Another class of spacetimes that are highly symmetric but require the metric to be spatially dependent are those that admit a group of isometries acting on two-dimensional spacelike orbits, at least after passing to a covering manifold.
In all these cases, the quotient of spacetime by the symmetry group has the structure of a two-dimensional Lorentzian manifold Q. The orbits of the group action or appropriate quotients in the case of a local symmetry are called surfaces of symmetry. Thus, there is a one-to-one correspondence between surfaces of symmetry and points of Q. There is a major difference between the cases where the symmetry group is two- or three-dimensional. In the three-dimensional case no gravitational waves are admitted, in contrast to the two-dimensional case where the evolution part of the Einstein equations are non-linear wave equations.
Three types of time coordinates that have been studied in the inhomogeneous case are CMC, areal, and conformal coordinates. A CMC time coordinate t is one where each hypersurface of constant time has constant mean curvature and on each hypersurface of this kind the value of t is the mean curvature of that slice. In the case of areal coordinates, the time coordinate is a function of the area of the surfaces of symmetry, e. In the case of conformal coordinates, the metric on the quotient manifold Q is conformally flat. The CMC and the areal coordinate foliations are both geometrically-based time foliations.
The advantage with a CMC approach is that the definition of a CMC hypersurface does not depend on any symmetry assumptions and it is possible that CMC foliations will exist for general spacetimes. The areal coordinate foliation, on the other hand, is adapted to the symmetry of spacetime but it has analytical advantages and detailed information about the asymptotics can be derived. The conformal coordinates have mainly served as a useful framework for the analysis to obtain geometrically-based time foliations. Let us now consider spacetimes M , g admitting a three-dimensional group of isometries.
A three-dimensional group G of isometries is assumed to act on. In the case of spherical symmetry the existence of one compact CMC hypersurface implies that the whole spacetime can be covered by a CMC time coordinate that takes all real values [ , 42 ]. The existence of one compact CMC hypersurface in this case was proven by Henkel [ 94 ] using the concept of prescribed mean curvature PMC foliation. Accordingly, this gives a complete picture in the spherically symmetric case regarding CMC foliations.
In the case of areal coordinates, Rein [ ] has shown, under a size restriction on the initial data, that the past of an initial hyper-surface can be covered, and that the Kretschmann scalar blows up. Hence, the initial singularity for the restricted data is both a crushing and a curvature singularity. In the future direction it is shown that areal coordinates break down in finite time. In the case of plane and hyperbolic symmetry, global existence to the past was shown by Rendall [ ] in CMC time.
This implies that the past singularity is a crushing singularity since the mean curvature blows up at the singularity. Also in these cases Rein showed [ ] under a size restriction on the initial data, that global existence to the past in areal time and blow up of the Kretschmann scalar curvature as the singularity is approached. Hence, the singularity is both a crushing and a curvature singularity in these cases too. In both of these works the question of global existence to the future was left open.
This gap was closed in [ 25 ], and global existence to the future was established in both CMC and areal time coordinates. The global existence result for CMC time is a consequence of the global existence theorem in areal coordinates, together with a theorem by Henkel [ 94 ] which shows that there exists at least one hypersurface with negative constant mean curvature. In addition, the past direction is analyzed in [ 25 ] using areal coordinates, and global existence is shown without a size restriction on the data. It is not concluded if the past singularity, without the smallness condition on the data, is a curvature singularity as well.
The issues discussed above have also been studied in the presence of a cosmological constant, cf. In this context we also mention that surface symmetric spacetimes with Vlasov matter and with a Maxwell field have been investigated in [ ]. This question was first resolved by Weaver [ ] for T 2 symmetric spacetimes with Vlasov matter. The important question of strong cosmic censorship for surface-symmetric spacetimes has recently been investigated by neat methods by Dafermos and Rendall [ 67 , 65 ].
The standard strategy to show cosmic censorship is to either show causal geodesic completeness in case there are no singularities, or to show that some curvature invariant blows up along any incomplete causal geodesic. In both cases no causal geodesic can leave the maximal Cauchy development in any extension if we assume that the extension is C 2. In [ 67 , 65 ] two alternative approaches are investigated. Both of the methods rely on the symmetries of the spacetime. The first method is independent of the matter model and exploits a rigidity property of Cauchy horizons inherited from the Killing fields.
The areal time described above is defined in terms of the Killing fields and a consequence of the method by Dafermos and Rendall is that the Killing fields extend continuously to a Cauchy horizon, if one exists. Now, since global existence has been shown in areal time it follows that there cannot be an extension of the maximal hyperbolic development to the future. This method is useful for the expanding future direction. The second method is dependent on Vlasov matter and the idea is to follow the trajectory of a particle, which crosses the Cauchy horizon and shows that the conservation laws for the particle motion associated with the symmetries of the spacetime, such as the angular momentum, lead to a contradiction.
In most of the cases considered in [ 67 ] there is an assumption on the initial data for the Vlasov equation, which implies that the data have non-compact support in the momentum space. It would be desirable to relax this assumption. The results of the studies [ 67 , 65 ] can be summarized as follows. The difficulties to show cosmic censorship in this case are related to possible formation of extremal Schwarzschild-de-Sitter-type black holes.
Although the methods developed in [ 67 , 65 ] provide a lot of information on the asymptotic structure of the solutions, questions on geodesic completeness and curvature blow up are not answered. In a few cases, information on these issues has been obtained. As mentioned above, blow up of the Kretschmann scalar curvature has been shown for restricted initial data [ ].
In the case of hyperbolic symmetry causal future geodesic completeness has been established by Rein [ ] when the initial data are small. The plane and hyperbolic symmetric cases with a positive cosmological constant are analyzed in [ ]. The authors show global existence to the future in areal time, and in particular they show that the spacetimes are future geodesically complete. The positivity of the cosmological constant is crucial for the latter result. A form of the cosmic no-hair conjecture is also obtained in [ ]. The first study of spacetimes admitting a two-dimensional isometry group was carried out by Rendall [ ] in the case of local T 2 symmetry.
For a discussion of the possible topologies of these spacetimes we refer to the original paper. In [ ] CMC coordinates are in fact considered rather than areal coordinates.
Under the hypothesis that there exists at least one CMC hypersurface, Rendall proves for general initial data that the past of the given CMC hypersurface can be globally foliated by CMC hypersurfaces and that the mean curvature of these hypersurfaces blows up at the past singularity. The future direction was left open. The result in [ ] holds for Vlasov matter and for matter described by a wave map. That the choice of matter model is important was shown in [ ], where a non-global existence result for dust is given, which leads to examples of spacetimes [ ] that are not covered by a CMC foliation.
There are several possible subcases to the T 2 symmetric class. Another subcase, which still admits only two Killing fields and which includes plane symmetry as a special case , is Gowdy symmetry. In [ 6 ] Gowdy symmetric spacetimes with Vlasov matter are considered, and it is proven that the entire maximal globally hyperbolic spacetime can be foliated by constant areal time slices for general initial data.
The areal coordinates are used in a direct way for showing global existence to the future, whereas the analysis for the past direction is carried out in conformal coordinates. These coordinates are not fixed to the geometry of spacetime and it is not clear that the entire past has been covered. A chain of geometrical arguments then shows that areal coordinates indeed cover the entire spacetime. The method in [ 6 ] was in turn inspired by the work [ 37 ] for vacuum spacetimes, where the idea of using conformal coordinates in the past direction was introduced.
As pointed out in [ 25 ], the result by Henkel [ 95 ] guarantees the existence of one CMC hypersurface in the Gowdy case and, together with the global areal foliation in [ 6 ], it follows that Gowdy spacetimes with Vlasov matter can be globally covered by CMC hypersurfaces as well. The more general case of T 2 symmetry was considered in [ 26 ], where global CMC and areal time foliations were established for general initial data.
As we pointed out in Section 4. The issue of strong cosmic censorship for T 2 symmetric spacetimes has been studied by Dafermos and Rendall using the methods, which were developed in the surface symmetric case described above. In [ 66 ] strong cosmic censorship is shown under the same restriction on the initial data that was imposed in the surface symmetric case, which implies that the data have non-compact support in the momentum variable.
Their result has been extended to the case with a positive cosmological constant by Smulevici [ ]. The present cosmological observations indicate that the expansion of the universe is accelerating, and this has influenced theoretical studies in the field during the last decade. One way to produce models with accelerated expansion is to choose a positive cosmological constant. Another way is to include a non-linear scalar field among the matter fields, and in this section we review the results for the Einstein-Vlasov system, where a linear or non-linear scalar field have been included into the model.
Lee considers in [ ] the case where a non-linear scalar field is coupled to Vlasov matter. The form of the energy momentum tensor then reads. Under the assumption that V is a non-negative C 2 function, global existence to the future is obtained, and if the potential is restricted to the form. In [ ] the Einstein-Vlasov system with a linear scalar field is analyzed in the case of plane, spherical, and hyperbolic symmetry. Here, the potential V in Equations 53 and 54 is zero. A local existence theorem and a continuation criterion, involving bounds on derivatives of the scalar field in addition to a bound on the support of one of the moment variables, is proven.
For the Einstein scalar field system, i. The work [ ] extends the result in the plane and hyperbolic case to a global result in the future direction. The past time direction is considered in [ ] and global existence is proven. It is also shown that the singularity is crushing and that the Kretschmann scalar diverges uniformly as the singularity is approached. In standard cosmology, the universe is taken to be spatially homogeneous and isotropic. This is a strong assumption leading to severe restrictions of the possible geometries as well as of the topologies of the universe.
Thus, it is natural to ask if small perturbations of an initial data set, which corresponds to an expanding model of the standard type, give rise to solutions that are similar globally to the future? The standard model of the universe is spatially homogeneous and isotropic, has flat spatial hypersurfaces of homogeneity, a positive cosmological constant and the matter content consists of a radiation fluid and dust. Hence, to investigate the question on stability it is natural to consider cosmological solutions with perfect fluid matter and a positive cosmological constant.
Approximating dust with Vlasov matter is straightforward, whereas approximating a radiation fluid is not. The main results in [ ] are stability of expanding, spatially compact, spatially locally homogeneous solutions to the Einstein-Vlasov system with a positive cosmological constant as well as a construction of solutions with arbitrary compact spatial topology. In other words, the assumption of almost spatial homogeneity and isotropy does not seem to impose a restriction on the allowed spatial topologies. Let us mention here some related works although these do not concern the Einstein-Vlasov system.
In [ ] the expansion is exponential and in [ ] it is of power law type. The corresponding problem for a fluid has been treated in [ ] and [ ], and the Newtonian case is investigated in [ ] and [ 40 ] for Vlasov and fluid matter respectively. Equilibrium states in galactic dynamics can be described as static or stationary solutions of the Einstein-Vlasov system, or of the Vlasov-Poisson system in the Newtonian case. Here we consider the relativistic case and we refer to the excellent review paper [ ] for the Newtonian case.
First, we discuss spherically-symmetric solutions for which the structure is quite well understood. On the other hand, almost nothing is known about the stability of the spherically-symmetric static solutions of the Einstein-Vlasov system, which is in sharp contrast to the situation for the Vlasov-Poisson system. At the end of this section a recent result [ 18 ] on axisymmetric static solutions will be presented. Let us assume that spacetime is static and spherically symmetric.
Let the metric have the form. As before, asymptotic flatness is expressed by the boundary conditions. Following the notation in Section 3. Recall that there is an additional Equation 39 of second order, which contains the tangential pressure p T , but we leave it out since it follows from the equations above. The matter quantities are defined as before:.
E is the particle energy and L is the angular momentum squared. If we let. If we furthermore assume that. It was an open question for some time whether or not this was also true for the Einstein-Vlasov system. However, almost all results on static solutions are based on this ansatz.
Existence of solutions to this system was first proven in the case of isotropic pressure in [ ], and extended to anisotropic pressure in [ ]. The main difficulty is to show that the solutions have finite ADM mass and compact support. The argument in these works to obtain a solution of compact support is to perturb a steady state of the Vlasov-Poisson system, which is known to have compact support. Two different types of solutions are constructed, those with a regular centre [ , ], and those with a Schwarzschild singularity in the centre [ ].
This result is obtained in a more direct way and is not based on the perturbation argument used in [ , ]. Their method is inspired by a work on stellar models by Makino [ ], in which he considers steady states of the Euler-Einstein system. In [ ] there is also a discussion about steady states that appear in the astrophysics literature, and it is shown that their result applies to most of these steady states.
An alternative method to obtain steady states with finite radius and finite mass, which is based on a dynamical system analysis, is given in [ 76 ]. In [ ] Rein showed that steady states also exist whose support is a finite, spherically-symmetric shell with a vacuum region in the center. In [ 8 ] it was shown that there are shell solutions, which have an arbitrarily thin thickness. A systematic study of the structure of spherically-symmetric static solutions was carried out mainly by numerical means in [ 22 ] and we now present the conclusions of this investigation.
In this way the cut-of energy disappears as a free parameter of the problem and we thus have the four free parameters k , l , L 0 and y 0. The structure of the static solutions obtained in [ 22 ] is as follows:. The value y 0 determines how compact or relativistic the steady state is, and the smaller values the more relativistic. For moderate values of y 0 the solutions have a distinct inner peak and a tail-like outer peak, and by making y 0 smaller more peaks appear, cf.
In the case of shells there is a similar structure but in this case the peaks can either be separated by vacuum regions or by thin atmospheric regions as in the case of ball configurations. A different feature of the structure of static solutions is the issue of spirals. For a fixed ansatz of the density function f , there is a one-parameter family of static solutions, which are parameterized by y 0. A natural question to ask is how the ADM mass M and the radius of the support R change along such a family.
By plotting for each y 0 the resulting values for R and M a curve is obtained, which reflects how radius and mass are related along such a one-parameter family of steady states. This curve has a spiral form, cf. Another aspect of the structure of steady states investigated numerically in [ 22 ] concerns the Buchdahl inequality. In Buchdahl [ 41 ] extended his result to isotropic solutions for which the energy density is non-increasing outwards and he showed that also in this case. This is sometimes called the Buchdahl inequality. The assumptions made by Buchdahl are very restrictive.
Also for other matter models the assumptions are not satisfying. In addition, there are also several astrophysical models of stars, which are anisotropic. Lemaitre [ ] proposed a model of an anisotropic star already in , and Binney and Tremaine [ 38 ] explicitly allow for an anisotropy coefficient. Moreover, the inequality is sharp and sharpness is obtained uniquely by an infinitely thin shell solution.
On the other hand, the result in [ ] is weaker than the result in [ 10 ] in the sense that the latter method implies that the steady state that saturates the inequality is unique; it is an infinitely thin shell. The studies [ 10 , ] are of general character and in particular it is not shown that solutions exist to the coupled Einstein-matter system, which can saturate the inequality.
This question is given an affirmative answer in [ 8 ], where in particular it is shown that arbitrarily thin shells exist, which are regular solutions of the spherically-symmetric Einstein-Vlasov system. Using the strategy in [ 9 ] it follows that. A Buchdahl type inequality gives a lower bound of the area radius of a static object and this radius is thus often called the critical stability radius.
Note, in particular, that the inequality holds for solutions of the Einstein-Vlasov-Maxwell system, since the conditions above are always satisfied in this case. In [ 78 ] the relevance of an inequality of this kind on aspects in black-hole physics is discussed. In contrast to the case without charge, the saturating solution is not unique. An infinitely thin shell solution does saturate the inequality 65 , but numerical evidence is given in [ 16 ] that there is also another type of solution, which saturates the inequality for which the inner and outer horizon coincide.
The following inequality is derived.
Communications in Mathematical Sciences
In this situation, the question of sharpness is essentially open. In the latter case there is a constant density solution, and the exterior spacetime is the Nariai solution, which saturates the inequality and the saturating solution is thus non-unique. In this case, the cosmological horizon and the black hole horizon coincide, which is in analogy with the charged situation described above where the inner and outer horizons coincide when uniqueness is likely lost. An important problem is the question of the stability of spherically-symmetric steady states. At present, there are almost no theoretical results on the stability of the steady states of the Einstein-Vlasov system.
Wolansky [ ] has applied the energy-Casimir method and obtained some insights, but the theory is much less developed than in the Vlasov-Poisson case and the stability problem is essentially open.
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The situation is very different for the Vlasov-Poisson system, and we refer to [ ] for a review on the results in this case. However, there are numerical studies [ 21 , , ] on the stability of spherically-symmetric steady states for the Einstein-Vlasov system. The latter two studies concern isotropic steady states, whereas the first, in addition, treats anisotropic steady states.
Here we present the conclusions of [ 21 ], emphasizing that these agree with the conclusions in [ , ] for isotropic states. To allow for trapped surfaces, maximal-areal coordinates are used, i. Thus, the radial coordinate r is the area radius. A maximal gauge condition is then imposed, which means that each hypersurface of constant t has vanishing mean curvature. The boundary conditions, which guarantee asymptotic flatness and a regular center, are given by.
Steady states are numerically constructed, and these are then perturbed in order to investigate the stability. More precisely, to construct the steady states the polytropic ansatz is used, cf. Section 5. The distribution function f s of the steady state is then multiplied by an amplitude A , so that a new, perturbed distribution function is obtained.
This is then used as initial datum in the evolution code. We remark that also other types of perturbations are analyzed in [ 21 ]. For k and l fixed each steady state is characterized by its central red shift Z c and its fractional binding energy E b , which are defined by.
M is the ADM mass given by. The central redshift is the redshift of a photon emitted from the center and received at infinity, and the binding energy e b is the difference of the rest mass and the ADM mass. Varying the parameters k , l and L 0 give rise to essentially the same tables, cf. A careful investigation of the perturbed solutions indicates that they oscillate in a periodic way. For larger values of Z c the evolution leads to the formation of trapped surfaces and collapse to black holes. Plotting E b versus Z c with higher resolution, cf.
The crucial quantity in this case is the fractional binding energy E b. The perturbed solution then drifts outwards, turns back and reimplodes, and comes close to its initial state, and then continues to expand and reimplode and thus oscillates, cf. A simple analytic argument is given in [ 21 ], which relates the question, whether a solution disperses or not. As we have seen above, a broad variety of static solutions of the Einstein-Vlasov system has been established, all of which share the restriction that they are spherically symmetric.
The recent investigation [ 18 ] removes this restriction and proves the existence of static solutions of the Einstein-Vlasov system, which are axially symmetric but not spherically symmetric. Before discussing this result, let us mention that similar results have been obtained for two other matter models. In the case of a perfect fluid, Heilig showed the existence of axisymmetric stationary solutions in [ 92 ].
These solutions have non-zero angular momentum since static solutions are necessarily spherically symmetric. In this respect the situation for elastic matter is more similar to Vlasov matter. The existence of static axisymmetric solutions of elastic matter, which are not spherically symmetric, was proven in [ 1 ]. Stationary solutions with rotation were then established in [ 2 ].
Let us now briefly discuss the method of proof in [ 18 ], which relies on an application of the implicit function theorem. Also, the proofs in [ 92 , 1 , 2 ] make use of the implicit function theorem, but apart from this fact the methods are quite different. The set-up of the problem in [ 18 ] follows the work of Bardeen [ 31 ], where the metric is written in the form.
In addition the solutions are required to be locally flat at the axis of symmetry, which implies the condition. Let us now recall from Section 5. Due to the symmetries of the metric 68 the following quantities are constant along geodesics:. Here p a are the canonical momenta.
E can be thought of as a local or particle energy and L is the angular momentum of a particle with respect to the axis of symmetry. It then remains to solve the Einstein equations with this energy momentum tensor as right-hand side. The Newtonian limit of the Einstein-Vlasov system is the Vlasov-Poisson system and the strategy in [ 18 ] is to perturb off spherically symmetric steady states of the Vlasov-Poisson system via the implicit function theorem to obtain axisymmetric solutions.
Since L is not invariant under arbitrary rotations about the origin the solution is not spherically symmetric if f depends on L. An important argument in the proof is indeed to justify that there are steady states of the Vlasov-Poisson system satisfying this condition. It is of course desirable to extend the result in [ 18 ] to stationary solutions with rotation. Moreover, the deviation from spherically symmetry of the solutions in [ 18 ] is small and an interesting open question is the existence of disk-like models of galaxies.
In the Vlasov-Poisson case this has been shown in [ 74 ]. National Center for Biotechnology Information , U. Living Reviews in Relativity. Living Rev Relativ. Published online May Author information Article notes Copyright and License information Disclaimer. Corresponding author. Accepted May 6. This article has been cited by other articles in PMC. Abstract The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system.
Introduction to Kinetic Theory In general relativity, kinetic theory has been used relatively sparsely to model phenomenological matter in comparison to fluid models, although interest has increased in recent years. The relativistic Boltzmann equation Consider a collection of neutral particles in Minkowski spacetime. The Vlasov-Maxwell and Vlasov-Poisson systems Let us consider a collision-less plasma, which is a collection of particles for which collisions are relatively rare and the interaction is through their charges.
The Einstein-Vlasov System In this section we consider a self-gravitating collision-less gas in the framework of general relativity and we present the Einstein-Vlasov system. The Asymptotically-Flat Cauchy Problem: Spherically-Symmetric Solutions In this section, we discuss results on global existence and on the asymptotic structure of solutions of the Cauchy problem in the asymptotically-flat case.
Set up and choice of coordinates The study of the global properties of solutions to the spherically-symmetric Einstein-Vlasov system was initiated two decades ago by Rein and Rendall [ ], cf. Global existence for small initial data In [ ] the authors also consider the problem of global existence in Schwarzschild coordinates for small initial data for massive particles. Global existence for special classes of large initial data In the case of small initial data the resulting spacetime is geodesically complete and no singularities form.
On global existence for general initial data As was mentioned at the end of Section 3. Self-similar solutions The main reason that the question of global existence in certain time coordinates discussed in the previous Section 3. Formation of black holes and trapped surfaces We have previously mentioned that there exist initial data for the spherically-symmetric Einstein-Vlasov system, which lead to formation of black holes. Numerical studies on critical collapse In [ ] a numerical study on critical collapse for the Einstein-Vlasov system was initiated. The charged case We end this section with a discussion of the spherically-symmetric Einstein-Vlasov-Maxwell system, i.
Spatially-homogeneous spacetimes The only spatially-homogeneous spacetimes admitting a compact Cauchy surface are the Bianchi types I, IX and the Kantowski-Sachs model; to allow for cosmological solutions with more general symmetry types, it is enough to replace the condition that the spacetime is spatially homogeneous, with the condition that the universal covering of spacetime is spatially homogeneous. Inhomogeneous models with symmetry In the spatially homogeneous case the metric can be written in a form that is independent of the spatial variables and this leads to an enormous simplification.
Surface symmetric spacetimes Let us now consider spacetimes M , g admitting a three-dimensional group of isometries. Gowdy and T 2 symmetric spacetimes The first study of spacetimes admitting a two-dimensional isometry group was carried out by Rendall [ ] in the case of local T 2 symmetry. Cosmological models with a scalar field The present cosmological observations indicate that the expansion of the universe is accelerating, and this has influenced theoretical studies in the field during the last decade. Stability of some cosmological models In standard cosmology, the universe is taken to be spatially homogeneous and isotropic.
Stationary Asymptotically-Flat Solutions Equilibrium states in galactic dynamics can be described as static or stationary solutions of the Einstein-Vlasov system, or of the Vlasov-Poisson system in the Newtonian case. Existence of spherically-symmetric static solutions Let us assume that spacetime is static and spherically symmetric. Open in a separate window.
Figure 1. Figure 2. Figure 3. Figure 4. Buchdahl-type inequalities Another aspect of the structure of steady states investigated numerically in [ 22 ] concerns the Buchdahl inequality. Stability An important problem is the question of the stability of spherically-symmetric steady states. Figure 5. Figure 6. Figure 7. Existence of axisymmetric static solutions As we have seen above, a broad variety of static solutions of the Einstein-Vlasov system has been established, all of which share the restriction that they are spherically symmetric. Acknowledgements I would like to thank Alan Rendall for helpful suggestions.
Static self-gravitating elastic bodies in Einstein gravity. Pure Appl. Rotating elastic bodies in Einstein gravity. Controlling the propagation of the support for the relativistic Vlasov equation with a selfconsistent Lorentz invariant field.