Manual Integral equations with fixed singularities

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Numerical Methods First Online: 19 November This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Hadamard, J. Google Scholar. Bart, G. Keiz, K. Bzhikhatlov, Kh. Gabbasov, N. Nauk , , vol. MathSciNet Google Scholar. In both models, the Universe is not standing still, it is constantly either expanding becoming larger or contracting shrinking, becoming smaller.

This was confirmed by Edwin Hubble who established the Hubble redshift of receding galaxies.


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The present consensus is that the isotropic model , in general, gives an adequate description of the present state of the Universe; however, isotropy of the present Universe by itself is not a reason to expect that it is adequate for describing the early stages of Universe evolution. At the same time, it is obvious that in the real world homogeneity is, at best, only an approximation. Even if one can speak about a homogeneous distribution of matter density at distances that are large compared to the intergalactic space, this homogeneity vanishes at smaller scales.

On the other hand, the homogeneity assumption goes very far in a mathematical aspect: it makes the solution highly symmetric which can impart specific properties that disappear when considering a more general case. Another important property of the isotropic model is the inevitable existence of a time singularity : time flow is not continuous, but stops or reverses after time reaches some very large or very small value.

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Between singularities, time flows in one direction: away from the singularity arrow of time. In the open model, there is one time singularity so time is limited at one end but unlimited at the other, while in the closed model there are two singularities that limit time at both ends the Big Bang and Big Crunch. The only physically interesting properties of spacetimes such as singularities are those which are stable , i. It is possible for a singularity to be stable and yet be of no physical interest: stability is a necessary but not a sufficient condition for physical relevance.

For example, a singularity could be stable only in a neighbourhood of initial data sets corresponding to highly anisotropic universes. Since the actual universe is now apparently almost isotropic such a singularity could not occur in our universe. A sufficient condition for a stable singularity to be of physical interest is the requirement that the singularity be generic or general. Roughly speaking, a stable singularity is generic if it occurs near every set of initial conditions and the non-gravitational fields are restricted in some specified way to "physically realistic" fields so that the Einstein equations, various equations of state, etc.

It might happen that a singularity is stable under small variations of the true gravitational degrees of freedom , and yet it is not generic because the singularity depends in some way on the coordinate system , or rather on the choice of the initial hypersurface from which the spacetime is evolved. For a system of non-linear differential equations , such as the Einstein equations , a general solution is not unambiguously defined.

In principle, there may be multiple general integrals , and each of those may contain only a finite subset of all possible initial conditions. Each of those integrals may contain all required independent functions which, however, may be subject to some conditions e. Existence of a general solution with a singularity, therefore, does not preclude the existence of other additional general solutions that do not contain a singularity. For example, there is no reason to doubt the existence of a general solution without a singularity that describes an isolated body with a relatively small mass.

It is impossible to find a general integral for all space and for all time. However, this is not necessary for resolving the problem: it is sufficient to study the solution near the singularity. This would also resolve another aspect of the problem: the characteristics of spacetime metric evolution in the general solution when it reaches the physical singularity, understood as a point where matter density and invariants of the Riemann curvature tensor become infinite. One of the principal problems studied by the Landau group to which BKL belong was whether relativistic cosmological models necessarily contain a time singularity or whether the time singularity is an artifact of the assumptions used to simplify these models.

The independence of the singularity on symmetry assumptions would mean that time singularities exist not only in the special, but also in the general solutions of the Einstein equations. It is reasonable to suggest that if a singularity is present in the general solution, there must be some indications that are based only on the most general properties of the Einstein equations, although those indications by themselves might be insufficient for characterizing the singularity.

A criterion for generality of solutions is the number of independent space coordinate functions that they contain. These include only the "physically independent" functions whose number cannot be reduced by any choice of reference frame. In the general solution, the number of such functions must be enough to fully define the initial conditions distribution and movement of matter, distribution of gravitational field at some moment of time chosen as initial. Previous work by the Landau group [11] [12] [13] reviewed in [9] led to the conclusion that the general solution does not contain a physical singularity.

This search for a broader class of solutions with a singularity has been done, essentially, by a trial-and-error method, since a systematic approach to the study of the Einstein equations was lacking. A negative result, obtained in this way, is not convincing by itself; a solution with the necessary degree of generality would invalidate it, and at the same time would confirm any positive results related to the specific solution.

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This indication, however, was dropped after it became clear that it is linked with a specific geometric property of the synchronous frame: the crossing of time line coordinates. This crossing takes place on some encircling hypersurfaces which are four-dimensional analogs of the caustic surfaces in geometrical optics ; g becomes zero exactly at this crossing. This, apparently, removed the incentive among the researchers for further investigations along these lines.

However, the interest in this problem waxed again in the s after Penrose published his theorems [15] that linked the existence of a singularity of unknown character with some very general assumptions that did not have anything in common with a choice of reference frame. Other similar theorems were found later on by Hawking [16] [17] and Geroch [18] see Penrose—Hawking singularity theorems. This revived interest in the search for singular solutions. In a space that is both homogeneous and isotropic the metric is determined completely leaving free only the sign of the curvature.

Assuming only space homogeneity with no additional symmetry such as isotropy leaves considerably more freedom in choosing the metric. The following pertains to the space part of the metric at a given instant of time t assuming a synchronous spacetime reference system so that t is the same synchronized time for the whole space. Homogeneity implies identical metric properties at all points of the space.


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An exact definition of this concept involves considering sets of coordinate transformations that transform the space into itself, i. For a more theoretical and coordinate-independent definition of homogeneous space see homogeneous space. A space is homogeneous if it admits a set of transformations a group of motions that brings any given point to the position of any other point. Since space is three-dimensional the different transformations of the group are labelled by three independent parameters.

In Euclidean space the homogeneity of space is expressed by the invariance of the metric under parallel displacements translations of the Cartesian coordinate system. Each translation is determined by three parameters — the components of the displacement vector of the coordinate origin. All these transformations leave invariant the three independent differentials dx , dy , dz from which the line element is constructed.

In the general case of a non-Euclidean homogeneous space, the transformations of its group of motions again leave invariant three independent linear differential forms , which do not, however, reduce to total differentials of any coordinate functions. The Greek letters label the three space-like curvilinear coordinates.

A spatial metric invariant is constructed under the given group of motions with the use of the above forms:. In the three-dimensional case, the relation between the two vector triples can be written explicitly. The determinant of the metric tensor eq. In order to be integrable, these equations must satisfy identically the conditions. Since x and x' are arbitrary, these expression must reduce to constants to obtain eq. Using eq. The structure constants are antisymmetric in their lower indices as seen from their definition eq.

Another condition on the structure constants can be obtained by noting that eq. In the mathematical theory of continuous groups Lie groups the operators X a satisfying conditions of the form eq. The condition mentioned above follows from the Jacobi identity. With these constants the commutation relations eq. The antisymmetry property is already taken into account in the definition eq. They can be subjected to any linear transformation with constant coefficients:.


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  • The conditions eq. But among the constants admissible by these conditions, there are equivalent sets, in the sense that their difference is related to a transformation of the type eq. The question of the classification of homogeneous spaces reduces to determining all nonequivalent sets of structure constants. This can be done, using the "tensor" properties of the quantities C ab , by the following simple method C. Behr, The unsymmetric "tensor" C ab can be resolved into a symmetric and an antisymmetric part.

    The first is denoted by n ab , and the second is expressed in terms of its " dual vector " a c :. Substitution of this expression in eq. By means of the transformations eq. Equation 6p shows that the "vector" a b if it exists lies along one of the principal directions of the "tensor" n ab , the one corresponding to the eigenvalue zero. Then eq. The Jacobi identities take the form:. The only remaining freedom is a change of sign of the operators X a and arbitrary scale transformations of them multiplication by constants.

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    This permits us simultaneously to change the sign of all the n a and also to make the quantity a positive if it is different from zero. Thus one arrives at the Bianchi classification listing the possible types of homogeneous spaces classified by the values of a , n 1 , n 2 , n 3 which is graphically presented in Fig. This has since been known as the BKL conjecture and implies that Einstein's partial differential equations PDE are well approximated by ordinary differential equations ODEs , whence the dynamics of general relativity effectively become local and oscillatory.

    The time evolution of fields at each spatial point is well approximated by the homogeneous cosmologies in the Bianchi classification. By separating the time and space derivatives in the Einstein equations, for example, in the way used above for the classification of homogeneous spaces, and then setting the terms containing space derivatives equal to zero, one can define the so-called truncated theory of the system truncated equations.

    Weak conjecture : As the singularity is approached the terms containing space derivatives in the Einstein equations are negligible in comparison to the terms containing time derivatives. Thus, as the singularity is approached the Einstein equations approach those found by setting derivative terms to zero. Thus, the weak conjecture says that the Einstein equations can be well approximated by the truncated equations in the vicinity of the singularity. Note that this does not imply that the solutions of the full equations of motion will approach the solutions to the truncated equations as the singularity is approached.

    This additional condition is captured in the strong version as follows. Strong conjecture : As the singularity is approached the Einstein equations approach those of the truncated theory and in addition the solutions to the full equations are well approximated by solutions to the truncated equations. In the beginning, the BKL conjecture seemed to be coordinate-dependent and rather implausible. Barrow and Tipler, [20] [21] for example, among the ten criticisms of BKL studies, include the inappropriate according to them choice of synchronous frame as a means to separate time and space derivatives.

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    The BKL conjecture was sometimes rephrased in the literature as a statement that near the singularity only the time derivatives are important. Such a statement, taken at face value, is wrong or at best misleading since, as shown in the BKL analysis itself, spacelike gradients of the metric tensor cannot be neglected for generic solutions of pure Einstein gravity in four spacetime dimensions, and in fact play a crucial role in the appearance of the oscillatory regime. However, there exist reformulations of Einstein theory in terms of new variables involving the relevant gradients, for example in Ashtekar-like variables, for which the statement about the dominant role of the time derivatives is correct.

    Subsequent analysis by a large number of authors has shown that the BKL conjecture can be made precise and by now there is an impressive body of numerical and analytical evidence in its support. But there has been outstanding progress in simpler models. In particular, Berger, Garfinkle, Moncrief, Isenberg, Weaver, and others showed that, in a class of models, as the singularity is approached the solutions to the full Einstein field equations approach the "velocity term dominated" truncated ones obtained by neglecting spatial derivatives.

    These results were generalized to also include p-form gauge fields.

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    In the general case, the strongest evidence to date comes from numerical evolutions. Berger and Moncrief began a program to analyze generic cosmological singularities. Finally, additional support for the conjecture has come from a numerical study of the behavior of test fields near the singularity of a Schwarzschild black hole. The Einstein equations for a universe with a homogeneous space can reduce to a system of ordinary differential equations containing only functions of time with the help of a frame field.

    To do this one must resolve the spatial components of four-vectors and four-tensors along the triad of basis vectors of the space:. The Einstein equations in vacuum in synchronous reference frame are [9] [10]. Using triads, for eq. Taking into account the transformations of covariant derivatives for arbitrary four-vectors A i and four-tensors A ik.

    It should be emphasized that in setting up the Einstein equations there is thus no need to use explicit expressions for the basis vectors as functions of the coordinates. Much more general solutions are obtained by a generalization of an exact particular solution derived by Edward Kasner [35] for a field in vacuum, in which the space is homogeneous and has a Euclidean metric that depends on time according to the Kasner metric. Here, p 1 , p 2 , p 3 are any 3 numbers that satisfy the following Kasner conditions.

    Because of these relations, only 1 of the 3 numbers is independent 2 equations with 3 unknowns. This is partially proved by squaring both sides of the first condition eq. This is possible if at least one of the p 1 , p 2 , p 3 is negative. The Kasner metric eq. Can you specify the nature of f? TroyHaskin TroyHaskin 8, 3 3 gold badges 18 18 silver badges 22 22 bronze badges. George Yes, I meant gamma2min apologies.

    What you do you mena "it should be scalar"?

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    Maybe I misunderstood the equation you posted. According to that equation, the integration region of gamma2 runs from 0 to infinity independent of gamma1 and gamma1 is integrated from some lower limit to infinity. If this is the case, all bounds should be numbers. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.

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