Mark D. Roberts

Scalar Field Counterexamples to the Cosmic Censorship Hypothesis

The applications of spherically symmetric solutions of the massless scalar Einstein equations to cosmic censorship are discussed. A new nonstatic solution to these equations is given. The Vaidya form of Wymans solution is constructed and is shown to obey reasonable energy conditions.

A new gravitational energy tensor

The Bel-Robinson tensor is the most used gravitational energy tensor; however, it has the dimensions of energy squared. How to construct tensors with the dimensions of energy by using Lancoz tensors is shown here. The resulting tensors have a large number of arbitrary parameters, frequently have spacelike currents, and frequently do not reduce to familiar pseudo-energy tensors in the weak field limit. Two particular examples of interest are one with well-behaved currents and one which reduces to an energy pseudo-tensor in the weak field limit.

Gain-Scheduled Inverse Optimal Satellite Attitude Control

Despite the theoretical advances in optimal control, satellite attitude control is still typically achieved with linear state feedback controllers, which are less efficient but easier to implement. A switched controller is proposed, based on inverse optimal control theory, which circumvents the complex task of numerically solving online the Hamilton-Jacobi-Bellman (HJB) partial differential equation of the global nonlinear optimal control problem. The inverse optimal controller is designed to minimize the torque consumption pointwise, while imposing the stabilization rate of a linear benchmark controller. The controller is then modified by gain scheduling to achieve a tradeoff enhancement compared with the benchmark controller, while maintaining torque saturation limits. The extent to which performance can be enhanced is shown to be dependent on the controller parameters. A controller tuning analysis shows how a design settling time limit can be achieved, within the problems constraints on the maximum torque and the total integrated torque. The proposed optimization approach is globally stabilizing and presents low implementation complexity, which is highly desirable given the limited resources onboard satellites.

Dimensional reduction and the Lanczos tensor

The Lanczos tensor Hαβγ is a potential for the Weyl tensor. Given the symmetries of these tensors it would be expected that the identification Hαβ5=Fαβ would give a reduction of the five dimensional vacuum field equations into equations related to the Einstein Maxwell equation, it is shown that this does not happen; furthermore it is shown that there is no dimensional reduction scheme involving the Lanczos tensor which agrees with the one devised by Kaluza and Klein in the weak field limit. The covariant derivative of the Weyl tensor can be expressed as a type of non-linear wave equation in the Lanczos tensor, the literature contains two incorrect expressions for this equation, here the correct expression is given for the first time. The expression for the Lanczos tensor in the case of weak fields is generalized. Some remarks are made on other approaches to include electro-magnetic theory into the theory of the Lanczos tensor.

A Scalar Polynomial Singularity Without an Event Horizon.

It is shown that the solution of the field equations for a static spherically symmetric scalar field has a scalar polynomial singularity and no event horizon. The solution does not develop from nonsingular data on any Cauchy surface. The possible existence of a universal scalar field, the conformal diagram and geodesies of the solution, and the energy and momentum of the field present are discussed.

Fractional Derivative Cosmology.

The degree by which a function can be dierentiated need not be restricted to integer values. Usually most of the eld equations of physics are taken to be second order, curiosity asks what happens if this is only approximately the case and the eld equations are nearly second order. For Robertson-Walker cosmology there is a simple fractional modication of the Friedman and conservation equations. In general fractional gravitational equations similar to Einstein’s are hard to dene as this requires fractional derivative geometry. What fractional derivative geometry might entail is briey looked at and it turns out that even asking very simple questions in two dimensions leads to ambiguous or intractable results. A two dimensional line element which depends on the Gamma-function is looked at.

Imploding scalar fields

Static spherically symmetric uncoupled scalar space–times have no event horizon and a divergent Kretschmann singularity at the origin of the coordinates. The singularity is always present so that nonstatic solutions have been sought to see if the singularities can develop from an initially singular free space–time. In flat space–time the Klein–Gordon equation ⧠φ=0 has the nonstatic spherically symmetric solution φ=σ(v)/r, where σ(v) is a once differentiable function of the null coordinate v. In particular, the function σ(v) can be taken to be initially zero and then grow, thus producing a singularity in the scalar field. A similar situation occurs when the scalar field is coupled to gravity via Einstein’s equations; the solution also develops a divergent Kretschmann invariant singularity, but it has no overall energy. To overcome this, Bekenstein’s theorems are applied to give two corresponding conformally coupled solutions. One of these has positive ADM mass and has the following properties: (i) it devel...

Ultrametric Distance in Syntax

Abstract Phrase structure trees have a hierarchical structure. In many subjects, most notably in taxonomy such tree structures have been studied using ultrametrics. Here syntactical hierarchical phrase trees are subject to a similar analysis, which is much simpler as the branching structure is more readily discernible and switched. The ambiguity of which branching height to choose, is resolved by postulating that branching occurs at the lowest height available. An ultrametric produces a measure of the complexity of sentences: presumably the complexity of sentences increases as a language is acquired so that this can be tested. All ultrametric triangles are equilateral or isosceles. Here it is shown that X̅ structure implies that there are no equilateral triangles. Restricting attention to simple syntax a minimum ultrametric distance between lexical categories is calculated. A matrix constructed from this ultrametric distance is shown to be different than the matrix obtained from features. It is shown that the definition of C-COMMAND can be replaced by an equivalent ultrametric definition. The new definition invokes a minimum distance between nodes and this is more aesthetically satisfying than previous varieties of definitions. From the new definition of C-COMMAND follows a new definition of of the central notion in syntax namely GOVERNMENT.

The relative motion of membranes

The relative classical motion of membranes is governed by the equation (wβ cα crβa)a = Rδγβαrgbxδapaγ, where w is the hessian. This is a generalization of the geodesic deviation equation and can be derived from the lagrangian p · ṙ. Quantum mechanically the picture is less clear. Some quantizations of the classical equations are attempted so that the question as to whether the Universe started with a quantum fluctuation can be addressed.

THE STRING DEVIATION EQUATION

It is well known that the relative motion of many particles can be described by the geodesic deviation equation. Less well known is that the geodesic deviation equation can be derived from the second covariant variation of the point particle’s action. Here it is shown that the second covariant variation of the string action leads to a string deviation equation. This equation is a candidate for describing the relative motion of many strings, and can be reduced to the geodesic deviation equation. Like the geodesic deviation equation, the string deviation equation is also expressible in terms of momenta and projecta. It is also shown that a combined action exists, the first variation of which gives the deviation equations. The combined actions allow the deviation equations to be expressed soley in terms of the Riemann tensor, the coordinates, and momenta. In particular geodesic deviation can be expressed as: Π̇ = R .αβγr P β ẋ, and string deviation can be expressed as: Π̇τ + Π ′μ σ = R μ .αβγr (P β τ ẋ α + P β σ x ). Mathematical Review Classification Scheme: 81T30, Physics and Astronomy Classification Scheme: 12.25+e, Keyword Index: String Deviation, Geodesic Deviation.It is well known that the relative motion of many particles can be described by the geodesic deviation equation. Less well known is that the geodesic deviation equation can be derived from the second covariant variation of the point particles action. Here it is shown that the second covariant variation of the string action leads to a string deviation equation. This equation is a candidate for describing the relative motion of many strings, and can be reduced to the geodesic deviation equation. Like the geodesic deviation equation, the string deviation equation can also be expressed in terms of momenta and projecta. It is also shown that a combined action exists, the first variation of which gives the deviation equations. The combined actions allow the deviation equations to be expressed solely in terms of the Riemann tensor, the coordinates and momenta. In particular geodesic deviation can be expressed as: and string deviation can be expressed as: