Rigid Motion in Special Relativity

Found Phys, 2006

The concept of rigid reference frame and of constricted spatial metric, given in the previous work [\emph{Class. Quantum Grav.} {\bf 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.

International Journal of Geometric Methods in Modern Physics 16 , 1950015, 2019

Riemann's principle ``force equals geometry" provided the basis for Einstein's General Relativity - the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The geometry of spacetime of a moving object is described by a metric obtained from the potential of the force field acting on it. We introduce a generalization of Newton's First Law - the Generalized Principle of Inertia stating that: An inanimate object moves inertially, that is, with constant velocity, in its own spacetime whose geometry is determined by the forces affecting it}. Classical Newtonian dynamics is treated within this framework, using a properly defined Newtonian metric with respect to an inertial lab frame. We reveal a physical deficiency of this metric (responsible for the inability of Newtonian dynamics to account for relativistic behavior), and remove it. The dynamics defined by the corrected Newtonian metric leads to a new \emph{Relativistic Newtonian Dynamics} for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity, as well as the post-Keplerian precession of binaries.

Computers & mathematics with applications, 2005

We add physical appeal to Einstein velocity addition law of relativistically admissible velocities, thereby gaining new analogies with classical mechanics and invoking new insights into the special theory of relativity. We place Einstein velocity addition in the foundations of both special relativity and its underlying hyperbolic geometry, enabling us to present special relativity in full three space dimensions rather than the usual one-dimensional space, using three-geometry instead of four-geometry. Doing so we uncover unexpected analogies with classical results, enabling readers to understand the modern and unfamiliar in terms of the classical and familiar. In particular, we show that while the relativistic mass does not mesh up with the four-geometry, it meshes extraordinarily well with the three-geometry, providing unexpected insights that are not easy to come by, by other means.

Classical and Quantum Gravity, 2009

In this first of a series of papers we will introduce the notion of a rigid quasilocal frame (RQF) as a geometrically natural way to define a "system" in the context of the dynamical spacetime of general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary (topologically R × S 2) of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines is expansion-free (the "size" of the system is not changing) and shear-free (the "shape" of the system is not changing). This definition of a system is anticipated to yield simple, exact geometrical insights into the problem of motion in general relativity. It begins by answering, in a precise way, the questions what is in motion (a rigid two-dimensional system boundary with topology S 2 , and whatever matter and/or radiation it happens to contain at the moment), and what motions of this rigid boundary are possible. Nearly a century ago Herglotz and Noether showed that a three-parameter family of timelike worldlines in Minkowski space satisfying Born's 1909 rigidity conditions does not have the six degrees of freedom we are familiar with from Newtonian mechanics, but a smaller number-essentially only three. This result curtailed, to a large extent, subsequent study of rigid motion in special and (later) general relativity. We will argue that in fact we can implement Born's notion of rigid motion in both flat spacetime (this paper) and arbitrary curved spacetimes containing sources (subsequent papers)-with precisely the expected three translational and three rotational degrees of freedom (with arbitrary time dependence)-provided the system is defined quasilocally as the two-dimensional set of points comprising the boundary of a finite spatial volume, rather than the three-dimensional set of points within the volume.

Annales De L Institut Henri Poincare-physique Theorique, 1979

A systematic and geometrical analysis of shock structures in a Riemannian manifold is developed. The jump, the infinitesimal jump and the covariant derivative jump of a tensor are defined globally. By means of derivation laws induced on the shock hypersurface, physically significant operators are defined. As physical applications, the charged fluid electromagnetic and gravitational interacting fields are considered. INTRODUCTION Several authors have developed the shock waves from different points of view, under both mathematical and physical aspects. In General Relativity shock waves assume a peculiar theoretical role. In fact they constitute one of the few strictly covariant signals occurring in the space-time manifolds, where the usual way, to describe waves (as plane waves, Fourier series, etc.) are globally meaningless. Of course shock may be considered as a mathematical abstraction that approximates more realistic physical phenomena. A very large bibliography on shock waves in ...

2003

In 1909 Born studied the "relativistic undeformable body" but made the mistake of calling it "rigid". The "rigid body" as one can find in Relativity books is, in fact, this Born "undeformable body". In Relativity it is necessary to distinguish between "rigid" and "undeformable". The "undeformable" body (in the sense of the most rigid possible) must be the "deformable" body where schock waves propagate with maximum speed c. We present in this text the elastic laws for rigid bodies. We think that these laws, which are ignored by the majority of relativists, should be taught in the elementary relativistic courses. With the approach of 2005, the centenary year of Relativity, we should like to appeal to all those who have some influence on these matters to avoid this mistake of repeatedly calling "rigid" to the "undeformable body".

Archive for Rational Mechanics and Analysis, 2019

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the Riemann-flat condition. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether regularity singularities (points where the curvature is in L ∞ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to C 1,1 by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to C 1,1 locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme.

arXiv (Cornell University), 2016

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the Riemann-flat condition. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether regularity singularities (points where the curvature is in L ∞ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to C 1,1 by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to C 1,1 locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme. 1 The space C 0,1 denotes the space of Lipschitz continuous functions, and C 1,1 the space of functions with Lipschitz continuous derivatives. A function is bounded in C 0,1 if and only if the function and its weak derivatives are bounded in L ∞ , c.f. [7], Chapter 5.8. 2 In geometry, the regularity of a Riemannian or Lorentzian metric is defined to be the regularity of the metric components in each coordinate system of a given atlas. This definition neglects the possibility that the metric might be more regular in particular coordinate systems of this atlas. 3 Since the writing of this paper the Riemann-flat condition has become the starting point for authors' further developments. In fact, we abandoned the Nash embedding idea and used the Riemann-flat condition to derive the Regularity Transformation equations, an elliptic system of PDE's equivalent to the Riemann-flat condition, c.f. .

Classical and Quantum Gravity, 2004

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 that are rigid, have free mobility and can be derived as a slight deformation of the radar metric, is shown.

Mathematics

The projectile motion (PP) in a vacuum is re-examined in this paper, taking into account the relativistic mass in special relativity (SR). In the literature, the mass of the projectile was considered as a constant during motion. However, the mass of a projectile varies with velocity according to Einstein’s famous equation m=m01−v2/c2, where m0 is the rest mass of the projectile and c is the speed of light. The governing system consists of two-coupled nonlinear ordinary differential equations (NODEs) with prescribed initial conditions. An analytical approach is suggested to treat the current model. Explicit formulas are determined for the main characteristics of the relativistic projectile (RP) such as time of flight, time of maximum height, range, maximum height, and the trajectory. The relativistic results reduce to the corresponding ones of the non-relativistic projectile (NRP) in Newtonian mechanics, when the initial velocity is not comparable to c. It is revealed that the mass o...