Geodesic equations

An interesting consequence of the existence of 5th dimension is that in the 4 dimensional observations an apparent violation of the equivalence principle must appear as one can show it by writing the geodesic equation in 5 dimensions and then projecting it into 4 dimensions. Without going into details, the sign of p5 causes the appearance of a pseudo-charge q in the 4 dimensional formalism,... 

Geodesic equations can be developed for the vacuum plane wave from the starting point [110]... 

This has the form of a geodesic equation [111], and is obeyed by a plane wave. Similarly, we obtain ... 

The geodesic equations in the space with the metric tensor (6) can be obtained, in the usual way, by defining the Lagrangian density... 

In this example, Euler s equation takes the form of the geodesic equation... 

Now, a theorem in Riemannian geometry tells us that locally any metric (7) with the correct signature can be rewritten as (6) by an appropriate change of coordinates. At different points we use different transformations of coordinates, but always end up with the Lorentz metric in the new coordinates. So the equation (8), when written in terms of the coordinates for which the metric looks like (7), must describe the trajectory in the gravitational field. This is the geodesic equation (sum over / , 7)... 

To get a better feeling for the nature of the frame dragging, let us find the angular velocity Q = dtp/dt of a free particle in a circular orbit r = const, and 0 = tt/2 about the BH. Symmetry tells us such an orbit will not get out of the equatorial plane. The geodesic equation (9) again gives Eq. (18). In view of definition (10) we now have... 

The defining equations for a geodesic in Riemann space becomes... 

The positivity of the right-hand side of Inequality (8.4) implies the positivity of the left-hand side. Thus, in view of Equation (8.1), the curvature 2 of the geodesic -gon is positive. The resulting inequality 2(r + q) — qr > 0 contradicts the assumption made. Hence, a finite non-extensible polycycle cannot have parabolic or hyperbolic parameters (r, q) these parameters are elliptic. ... 

In this theory the equations of motion of an electrified particle are geodesics referred to a line element... 

By dehning the geodesic as a curve in a surface for which the tangent vector remains constant it is found to obey the equations... 

It is remarkable that these are not dependent on

small region, one and only one curve of the system goes through two specified points. The paths are a generalization of the geodesics of Riemannian geometry. 

If (34) is not satisfied the differential equations are not homogeneous in t. By fixing k, each point with given velocity vector completely determines a curve. Only for the geodesic lines are the curves conditioned by the initial point. 

In [208] the authors obtained a numerical scheme and code for estimating the deposition of energy and momentum due to the neutrino pair annihilation (v -f- V e + c+) in the vicinity of an accretion tori around a Kerr black hole. In order to solve the collisional Boltzmann equation in curved space-time, the authors solved approximately the so-called rendering equation along the null geodesics. They used the Runge-Kutta Fehlberg... 

Meshcheryakov obtained some results on exact integration of geodesic flows of metrics PabD simple Lie groups by means of special functions. The functional nature of the solutions of the equations for geodesics is as follows they are either quasi-polynomials or rational functions of the restrictions of the theta functions of compact Riemann surfaces to rectilinear windings of Jacobian tori of these surfaces. These methods rest upon the papers by Novikov and Dubrovin [45]. 

It turns out that one may present an exact integration of the equations for geodesics of the metrics (pahD on the group SL(m,C). Metrics of the type (pabD appeared for the first time in the course of construction of nonlinear differential equations integrable by the inverse scattering method. FVom the paper [38] it readily follows that the Euler equation X = [X,ipahD ) on a classical Lie algebra of series Afn-i serves as a commutativity equation for a pair of operators. 

The matrix elements of the solutions of the equations for geodesics of left-invariant metrics (pabD on the group SL(m, C) with initial data of general position turn out to be rational functions of the exponents and theta-functions of the Riemann surfaces T Q(W X) = 0. More exactly (Meshcheryakov), the solution g t) has the form g t) = e(t)r(t) exp(tZ)(/i)). The matrices e(t),r(t) are expressed here by the following formulae through the data on the Riemann surface T ... 

Theorem 5.4.3 (see [307]). The geodesic Sow in T M, where M is a two-dimensional Riemannian manifold, has an additional integral quadratic in momenta if and only if in any isothermic coordinates x, y the function X setting the metric satisSes equation (4), where R = R - 2iR2 is a holomorphic function of the variable z = X + ty, which is not identical zero and under transition to other isothermic coordinates is transformed in accordance with formula (7). 

Metrics of the form (/(x) + h(y)) geodesic flow has an additional integral quadratic in momenta is of Liouville type. Consequently, the equations for geodesics are integrated in this case by way of separation of variables. 

In the assumption that the geodesic flow has an additional integral quadratic in momenta we find out the form of the function R(z) setting up the basic equation (4) in global coordinates. It can be expounded (we leave the proof to the reader) that in the global coordinates z = x + ty there hold the following asymptotic formulae A(z) = and R z) = (c H- 0(l))z as z — oo, where a and c... 

The next step was made in the end of the last century by Tannery who constructed a "pear all of whose geodesics are closed and have the smallest period 1, except for the equator whose smallest period is equal to 1/2. This pear is an algebraic surface but not a smooth manifold because at one point it has a singularity. 

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