Quantitative Finance This
paper describes a semi analytic approach to implying Default
Probabilites from vanilla CDS spread rates for CDS's having a range of
maturities. A substantial part of the analysis presented in the paper
will be placed in the context of the reference paper 1. In particular
we will maintain the same termonology as described on page 13 of
reference 1. Implied Default Probabilities
Theoretical Physics
In physics the application of Riemann manifolds and the corresponding analytical apparatus is for the most part identified with the General Theory of Relativity and the light speed preserving space-time coordinate systems which place restrictions on the form of the metric, i.e. only transformations which preserve the speed of light are considered, and the spaces considered are 4D. 3 space and one time. In Non-relativistic theories i.e., classical or Newtonian mechanics, Riemann manifolds are confined to 3 space dimensions and the time is considered separate. The cited reason for this is that in general an arbitrary space time transformation from one coordinate system to another will not preserve the speed of light. This maybe the case, but it should not prevent a non light speed preserving metric from expressing an equation of motion, for what is a geodesic in one coordinate system is also a geodesic in any other coordinate system, even those which do not preserve the speed of light. Realizing this key point allows the full power of the Riemann metric to be exploited. In Relativity Theory the metric is restricted so that the speed of light is always the same, and only transformations between such metrics are allowable. In this paper no such restrictions are imposed, an arbitrary transformation from one reference system to another gives rise to a new metric in which perfectly valid equations of motion are obtained. The requirement of Lorentz invariance is only motivated as the theory is developed, especially with the invocation of the concept of a fluid path to describe an infinite set of variables and for the need to remove ambiguity from the field equations which inevitably emerge. More precisely It is the adoption of the requirement that the Harmonic Gauge be satisfied which replaces and subsumes the role of the hypothesis of Lorentz invariance in relativity theory.
The Riemann Foundations of Mechanics and Matter
In this paper it will be shown that Electromagnetism can be feasibly interpreted as having an underlying space time representation, with the underlying metric being directly identified with and constructed from the components of the four potential. The approach taken is basically that which is taken in the linearized version of the Einstein Field Equations (EFE) with the attached Harmonic Gauge constraint, to produce what is known as linearized gravity. However what is different in this analysis lies primarily in the choice and moreover the interpretation of the energy momentum tensor, and in particular the interpretation of mass, from the components of the metric tensor, and its relationship to electric charge. The overall intended viewpoint which this presentation will attempt to convey is that matter is not a distinct entity existing within space time but instead it is to be considered as simply attributable to the non Euclidean nature of space time, or in other words the mass density is simply a component of the underlying metric tensor in the same way that for example in an analogous fashion, the electric potential is a component of the four potential.
Speculations on The Riemann Origins of Electromagnetism
It is here demonstrated that by constraining the Einstein Field Equations (EFE) with the Harmonic Gauge condition results in not only the well established implication that the EFE are nothing more than wave equations, but also in the fact of the existence of a global conservation of energy and momentum law comprised of both matter and fields, with the fields being defined in terms of the underlying metric. It is also speculated, but not elaborated upon here, that the various pseudo tensor approaches to construct global conservation equations are equivalent to the EFE with a corresponding attached gauge constraint, but which may not necessarily in each case correspond to a wave like equation, unlike the case for the Harmonic Gauge
Observations on the Equivalence of the Harmonic Gauge Constraint and Energy-Momentum Conservation in
General Relativity
In this paper it will be shown how the Weber Force law emerges as an averaged effect between two electrically neutral closed current loops under the assumption that the force between current elements is given by the Lorentz Force operating on the four potential as prescribed by the full retarded form of Maxwell’s equations. That is it will be shown that the total force given by the Weber Force between two neutral closed current loops is experimentally indistinguishable from the total force computed by the Lorentz interaction. As a corollary it will be shown how only under the conditions of non neutrality and open circuits, how one force principle might diverge from the other in their respective predictions of the interaction characteristics in such a configuration, and hence how to potentially through experimental means, validate one or the other.
The Weber Force
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