Daniel Soler

Reference frames and rigid motions in relativity

A reference frame consists of a reference space, a time scale and a spatial metric. The geometric structure induced by these objects in spacetime is developed. The existence of a class of spatial metrics which are rigid, have free mobility and can be derived as a slight deformation of the radar metric, is demonstrated.

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric

AbstractAny three-dimensional metric g may be locally obtained from a constant curvature metric, h, by a deformation like

A Note on Interpreting Tool Temperature Measurements from Thermography

g = \sigma h + \in s \otimes s,

A relativistic generalisation of rigid motions

where σ and s are respectively a scalar and a one-form, the sign ∈ = ±1 and a functional relation between σ and the Riemannian norm of s can be arbitrarily prescribed. The general interest of this result in geometry and physics, and the related open problems, are stressed.

Rigid Motions in Relativity: Applications

A semi-Riemannian metric in a n-manifold has n(n − 1)/2 degrees of freedom, i.e., as many as the number of components of a differential 2-form. We prove that any analytic semi-Riemannian metric can be obtained as a deformation of a constant curvature metric, this deformation being parametrized by an analytic 2-form.

Effect of heat treatment conditions on the machinability of Ti64 and Ti54M alloys

Thermography (thermal imaging) is a well-established experimental method for studying cutting tool temperature distributions. In one form, cutting edge temperatures within the chip / tool contact area are deduced from thermal images of tool faces normal to the cutting edge but offset from the contact region. In general practice, the offset is made as small as possible (<< 1 mm) and it is assumed that the observed temperature is the same as that within the contact. In this short communication an approximate analytical model is developed for the influence of the offset on the observed temperature. The predictions from the model are compared with previously unpublished existing results on the machining of Ti alloys (Ti6Al4V and Ti5Al4V) and on steel (AISI 4140). It is shown that ignoring the offset may introduce underestimates of cutting edge temperature of ≈ 30% or more. This is large compared to the usually considered uncertainties of ± 5% from camera and tool emissivity calibration. There is a need for a dedicated study of this effect.

Metal cutting experiments and modelling for improved determination of chip/tool contact temperature by infrared thermography

The concept of rigid reference frame and of constricted spatial metric, given in the previous work [Class. Quantum Grav. 21, 3067 (2004)] are here applied to some specific space-times: in particular, the rigid rotating disc with constant angular velocity in Minkowski space-time is analyzed, a new approach to the Ehrenfest paradox is given as well as a new explanation of the Sagnac effect. Finally the anisotropy of the speed of light and its measurable consequences in a reference frame co-moving with the Earth are discussed.

Remarks on the analysis method for determining diffusion coefficient in ternary mixtures

A weaker substitute for the too restrictive class of Born-rigid motions is proposed, which we call radar-holonomic motions. The definition is expressed as a set of differential equations. Integrability conditions and Cauchy problem are studied. We finally obtain an example of a radar-holonomic congruence containing a given worldline with a given value of the rotation on this line.