T. Matolcsi

Can material time derivative be objective

The concept of objectivity in classical field theories is traditionally based on time dependent Euclidean transformations. In this Letter we treat objectivity in a four-dimensional setting, calculate Christoffel symbols of the spacetime transformations, and give covariant and material time derivatives. The usual objective time derivatives are investigated.

Absolute time derivatives

A four dimensional treatment of nonrelativistic spacetime gives a natural frame to deal with objective time derivatives. In this framework some well known objective time derivatives of continuum mechanics appear as Lie derivatives. Their coordinatized forms depend on the tensorial properties of the relevant physical quantities. We calculate the particular forms of objective time derivatives for scalars, vectors, covectors, and different second order tensors from the point of view of a rotating observer. The relation of substantial, material, and objective time derivatives is treated.

Spacetime without Reference Frames: An Application to the Velocity Addition Paradox

Abstract Much conceptualisation in contemporary physics is bogged down by unnecessary assumptions concerning a specific choice of coordinates which often leads to misunderstandings and paradoxes. Considering an absolute (coordinate-free) formulation of special relativistic spacetime, we show clearly that the velocity addition paradox emerged because the use of coordinates obscures that the space of relativistic observers is ‘more relative’ than the space of non-relativistic observers.

Spacetime without reference frames: An application to the kinetic theory

Spacetime structures defined in the language of manifolds admit an absolute formulation, i.e., a formulation which does not refer to observers (reference frames). We consider an affine structure for Galilean spacetime. As an application the Chapman-Enskog iteration for the solution of the Boltzmann equation is given in an absolute form. As a consequence, the second approximations of the stress tensor and the heat flux are obtained in a form independent of observers, which throws new light on material frame indifference.

On the classification of phase transitions

A new classification of phase transitions is presented, based on an exact mathematical model of phase, on rigorous definitions and results regarding phase transitions which are classified to be of zeroth, second and first order. Then the relation between the known classifications due to Ehrenfest and Tisza, respectively, becomes clear.

Thomas rotation and Thomas precession

Exact and simple calculation of Thomas rotation and Thomas precessions along a circular world line is presented in an absolute (coordinate-free) formulation of special relativity. A straightforward derivation of the Fermi–Walker equation is also given. Besides the simplicity of calculations the absolute treatment of spacetime allows us to make a clear conceptual distinction between the phenomena of Thomas rotation and Thomas precession.

On the mathematical structure of thermodynamics

A mathematically exact dynamical theory of classical thermodynamics of homogeneous bodies is presented in which processes are time-dependent functions, governed by an ordinary differential equation. The fundamental objects of the mathematical structure of a thermodynamical system are the dynamical law, the thermodynamical force, and the constraints; all the other usual notions, too, such as substances, bodies, linear approximation by Onsager, etc. have got a mathematical definition. Equilibria are the constant processes; their stability is investigated by Lyapunov’s method.

Spacetime without Reference Frames: An Application to Synchronizations on a Rotating Disk

Nonstandard synchronizations of inertial observers in special relativity and synchronizations with respect to a uniformly rotating observer are investigated in a setting which avoids coordinates and transformation rules and so removes some misunderstandings.

On the dynamics of phase transitions

The dynamics of phase transitions is described in ordinary thermodynamics i.e. the bodies are considered to be homogeneous. Stability properties of phase transitions are investigated by Lyapunovs method.

Coordinate Time and Proper Time in the GPS.

The global positioning system (GPS) provides an excellent educational example of how the theory of general relativity is put into practice and becomes part of our everyday life. This paper gives a short and instructive derivation of an important formula used in the GPS, and is aimed at graduate students and general physicists. The theoretical background of the GPS (see [1]) uses the Schwarzschild spacetime to deduce the approximate formula, , for the relation between the proper time rate s of a satellite clock and the coordinate time rate t. Here V is the gravitational potential at the position of the satellite and is its velocity (with light-speed being normalized as c = 1). In this paper we give a different derivation of this formula, without using approximations, to arrive at , where is the normal vector pointing outwards from the centre of Earth to the satellite. In particular, if the satellite moves along a circular orbit then the formula simplifies to . We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice.