A SCALAR GEODESIC DEVIATION EQUATION AND A PHASE THEOREM

A scalar equation is derived for , the distance between two structureless test particles falling freely in a gravitational field:

in general relativity is a physical equation, because it relates the relative acceleration between neighbouring test particles to certain physical components of the Riemann-curvative tensor [7].However, instead of working with the standard tensor form of the geodesic deviation equation, we use the 'scalar form' + (K-C) 0 which is derived in Sec. 2. The reason for adopting this approach is because of the difficulty in formally solving the tensor form of the equation except in the most trivial cases.The 'scalar form', though containing less information regarding the source field can be solved formally relatively easily.Further the analogy with the equation of a time-dependent simple harmonic oscillator can be exploited to give insight into the phenomenon governed by the equation.
For example, the formal solution of the scalar equation shows the motion of the test particles to be 'wave-like', where the amplitude of the wave is a function of the phase.Strictly this means that the motion may not be periodic at all, but as we shall see that the analogy is useful.Also this analogy enables us to define the concepts of phase, amplitude and relative energy associated with the motion.From the solution the existence of the bimodal character of the vibrations can be seen.
The difference in the two modes, i.e., one mode tends to pull together while the other repels simultaneously, can also be seen.Thus the well known fact that the effect of any gravitational source on two test particles is tidal in character, can be seen.The scalar equation also has a constant of the motion associated with it.
In Sec. 4 the phase shift, due to the effect of a perturbation in the source, on the motion of the test particles is deduced.
The geodesic deviation equation is the equation of motion of the space-like i part of the deviation vector between two neighbouring test particles in a gravitational field, namely i Ri u j nku n + jk 0, where i i u.
(2.2) i , Then (0,) are the direction cosines of the deviation vector in the local rest space.
On transvecting (2.3) with i we get + (-) -Rijk umJuk Z and the form of the geodesic deviation equation used in this article is obtained [10].The resemblance of (2.4) to the equation of a time- dependent simple harmonic oscillator should be noted.All the details of the derivation are given in reference [2]. 3.

AMPLITUDE PHASE AND RELATIVE ENERGY.
Let 0 be a particular solution of the differential equation (3.1) which never vanishes [5,8] is a conserved quantity for the equation of motion (2.4).It is usual to call J a e, the Lewis invariant of (2.4).It has the character of an action [2,5].
On introducing the phase variable When K e > 0 we say that the relative motion is in an elliptic phase as is real.
Otherwise the phase is called hyperbolic or parabolic according as K fl < 0 or O.   ].We will restrict our attention to the recurrence of elliptic phases and so we can invert ( Thus we know the phase (s) in terms of p and so in terms of n, i.e. in principle, the observable quantity q determines the phase (s) and the anplitude

A(s) [].
A further reason for the restriction to elliptic phases is that when a hyper- bolic phase arises it always terminates in the onset of a new elliptic phase.
Since J is an exact invariant and has, as noted earlier, the dimensions of "energy/frequency" and as is a frequency we can define an energy for the relative motion to be If we make the adiabatic assumption, namely that -is small, then p and neglected and equation (3.7) gives (3.7) can be 2 + 2 q2 j E :[ where w Kwill hold approximately [5].
The question of phase stability will now be discussed.This will lead to a 'phase shift' criterion.
Using (3.6)If the relative motion of the test particles is to be stable, they must have smooth motions, or motions which can be varied continuously from one configuration to another.In terms of variations, it is possible to define a continuous parameter which distinguishes one particular configuration from another.So (4.2) may be rewritten as, s 61(s",e) s (s,e) -(s,e) fs (s",e) sin {2 s,,(s',e)ds'}ds", O where 0(s,e) 30(s,e) de (s,e)de and we give e the range 0 < e < I. Then on integrating (4.3) with respect to e we get s I 's",e) s A(s) (s,0) (s,1) /0 de Is (s",a) sin{2/s"(s''e)ds'}ds"" O A further integrating of (4.4) with respect to proper time gives for the new left hand side S 0(s) (s) /s o A(s"' as"' which is a phase shift seen in the relative motion of the two test bodies due to a perturbation in the source.For the new right hand side we get after changing the order of the integration variables, [3]. S I (S" ,e) s e sine{ fs"(s' e)ds'}ds".d -so This expression is complicated, but it can be understood in the following way.
Since the system has a time-dependent phase, so the first integral from the right hand side accounts for the phase.The multiplicative factor I (s",e)/(s" e) is expected if one notes that from (4.1) it is the frequency shift of the system, which is also time-dependent.The second integral from the right hand side indicates the times for the observation of the phases.The final integral gives the average over all virtual configurations within that time period.For the time independent simple harmonic oscillator, since the motion is periodic this situation does not arise at all.
In the next section we apply the phase shift theorem to the plane wave metric.

5.
THE GRAVITATIONAL PLANE WAVE.and E(u) an arbitrary differentiable function of the retarded distance u, describes one mode of a plane gravitational wave [7] du2 z where du/ds is a Doppler shift factor and x ct, X I x, x y, x z.
On examination we see that the phase-shift is expressed completely in terms of the observable quantity n.

APPENDIX.
As a gravitational shock wave passes over a pair of freely falling structureless test particles the discontinuity [Rijk] in the curvature tensor causes a discontinuity in the relative acceleration given by [6].
[i] + [R i uJku 0 jk or using (3) we get In the limit as a plane sandwich wave approaches such a shock front, all physical quantities in the coefficient of other than the curvature tensor component K have negligible variation, which is the comment made in section 5.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning INTRODUCTION.It is well known that the geodesic deviation equation .

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation