Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpace-time trigonometry and formalization of the Twin Paradox for uniform and accelerated motions
299
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The formal structure of the early Einsteins Special Relativity follows the axiomatic deductive method of Euclidean geometry. In this paper we show the deep-rooted relation between Euclidean and space-time geometries that are both linked to a two-dimensional number system: the complex and hyperbolic numbers, respectively. By studying the properties of these numbers together, pseudo-Euclidean trigonometry has been formalized with an axiomatic deductive method and this allows us to give a complete quantitative formalization of the twin paradox in a familiar Euclidean way for uniform motions as well as for accelerated ones.
rate research
Read More
By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representation, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalize the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry.
Relativistic kinematics is usually considered only as a manifestation of pseudo-Euclidean (Lorentzian) geometry of space-time. However, as it is explicitly stated in General Relativity, the geometry itself depends on dynamics, specifically, on the energy-momentum tensor. We discuss a few examples, which illustrate the dynamical aspect of the length-contraction effect within the framework of Special Relativity. We show some pitfalls associated with direct application of the length contraction formula in cases when an extended object is accelerated. Our analysis reveals intimate connections between length contraction and the dynamics of internal forces within the accelerated system. The developed approach is used to analyze the correlation between two congruent disks - one stationary and one rotating (the Ehrenfest paradox). Specifically, we consider the transition of a disk from the state of rest to a spinning state under the applied forces. It reveals the underlying physical mechanism in the corresponding transition from Euclidean geometry of stationary disk to Lobachevskys (hyperbolic) geometry of the spinning disk in the process of its rotational boost. A conclusion is made that the rest mass of a spinning disk or ring of a fixed radius must contain an additional term representing the potential energy of non-Euclidean circumferential deformation of its material. Possible experimentally observable manifestations of Lobachevskys geometry of rotating systems are discussed.
The twin paradox, which evokes from the the idea that two twins may age differently because of their relative motion, has been studied and explained ever since it was first described in 1906, the year after special relativity was invented. The question can be asked: Is there anything more to say? It seems evident that acceleration has a role to play, however this role has largely been brushed aside since it is not required in calculating, in a preferred reference frame, the relative age difference of the twins. Indeed, if one tries to calculate the age difference from the point of the view of the twin that undergoes the acceleration, then the role of the acceleration is crucial and cannot be dismissed. In the resolution of the twin paradox, the role of the acceleration has been denigrated to the extent that it has been treated as a red-herring. This is a mistake and shows a clear misunderstanding of the twin paradox.
Recently Abramowicz and Bajtlik [ArXiv: 0905.2428 (2009)] have studied the twin paradox in Schwarzschild spacetime. Considering circular motion they showed that the twin with a non-vanishing 4-acceleration is older than his brother at the reunion and argued that in spaces that are asymptotically Minkowskian there exists an absolute standard of rest determining which twin is oldest at the reunion. Here we show that with vertical motion in Schwarzschild spacetime the result is opposite: The twin with a non-vanishing 4-acceleration is younger. We also deduce the existence of a new relativistic time effect, that there is either a time dilation or an increased rate of time associated with a clock moving in a rotating frame. This is in fact a first order effect in the velocity of the clock, and must be taken into account if the situation presented by Abramowicz and Bajtlik is described from the rotating rest frame of one of the twins. Our analysis shows that this effect has a Machian character since the rotating state of a frame depends upon the motion of the cosmic matter due to the inertial dragging it causes. We argue that a consistent formulation and resolution of the twin paradox makes use of the general principle of relativity and requires the introduction of an extended model of the Minkowski spacetime. In the extended model Minkowski spacetime is supplied with a cosmic shell of matter with radius equal to its own Schwarzschild radius, so that there is perfect inertial dragging inside the shell.
We propose an implementation of a twin paradox scenario in superconducting circuits, with velocities as large as a few percent of the speed of light. Ultrafast modulation of the boundary conditions for the electromagnetic field in a microwave cavity simulates a clock moving at relativistic speeds. Since our cavity has a finite length, the setup allows us to investigate the role of clock size as well as interesting quantum effects on time dilation. In particular, our theoretical results show that the time dilation increases for larger cavity lengths and is shifted due to quantum particle creation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
