Problems In Special Relativity Research Articles

A review of the explanations to the twin paradox

The Twin Paradox is a representative problem in special relativity. It proposed a problem that from a spacecraft that is moving at a speed close to light, the earths time will be slower; in contrast, the earth will also consider the time of the spacecraft slower. However, this paradoxs solution can expand beyond the special relativity and Lorentz transformation. Basically, the seeming paradox can be solved using the period when the spacecraft turns around. Therefore, it can be considered a good way to have a better understanding of relativity. In this paper, this paradox is explained in three ways: the Lorentz transformation, the Minkowski geometry, a special geometry including both space and time, which is proposed to have a better explanation of the special relativity and gravitational time dilation in general relativity, and will also expand to other effects in relativity, such as the gravitational redshift. This paper hopes to offer some references for a better understanding of the Twin Paradox.

Using three-dimensional spacetime diagrams in special relativity

We provide three examples of the use of geometric reasoning with three-dimensional spacetime diagrams, rather than algebraic manipulations using three-dimensional Lorentz transformations, to analyze problems in special relativity. The examples are the “rising manhole” paradox, the “moving spotlight” problem, and Einstein's light-clock derivation of time dilation.

Diachronic Representation of Space-Time Applied to Problems in Special Relativity and in Quantum Optics.

Abstract A central observer at Greenwich builds in real time a diachronic representation of 4-D(t) space-time by assigning two distinct times to a given remote event, such as a meteorite impact on Mars: (1) diachronic time read on his Greenwich clock (GMT) at observation time, and (2) synchronic or local Einstein time as directly seen from Greenwich on the Mars clock. In a spherical coordinate system the Greenwich diachronic observer uses local Einstein time relative to diachronic time as a radial coordinate, leaving the role of the fourth coordinate to dia-chronic time. In a 4-D(t) conceptual framework the same diachronic time applies integrally to all directly observed local Einstein times, thus implying an infinite diachronic speed for incoming light. In the 4-D(t) conceptual framework the de-tection of a photon is considered to be an instantaneous transfer of energy and momentum across the time gap between emitter and receiver. The Lorentz trans-formation can be obtained by requiring that history updated now and here be the same for another observer flying by. Some results are (1) the conventional (syn-chronic) speed of light c is identified with the flow of time, (2) the twin paradox is resolved in a twin-symmetric fashion, and (3) a new solution for the Einstein–Podolsky–Rosen paradox maintains harmony between special relativity and quan-tum mechanics. The diachronic approach presents the pedagogical advantage of using a single diagram to display space-time relationships of events in two coor-dinate systems in relative motion.

Complex speeds and special relativity

The quest to find faster-than-light particles has intrigued physicists for decades, though it has yet to turn up any real candidates. Even if a superluminal universe does exist, we have no way to reach it given that we must go through the speed of light, which to the best of our knowledge is impossible. In this paper, I show that by making speed complex, we can go around the speed of light in a manner analogous to the way a car faced with an infinitely tall road block might leave the road to go around that barrier. The treatment is a mathematical device; no known physical interpretation exists for the imaginary part of a complex speed. However, it can provide an entertaining problem in special relativity, one that may encourage students to think about the connections between equations and the physical universe.

The Pauli-Algebra approach to special relativity

The Pauli algebra introduces an associative, invertible multiplication of vectors in 3-dimensional Euclidean space. Some elements of the algebra may be interpreted as 4-vectors, the Minkowski metric for which appears naturally. The algebra gives a compact, covariant formulation for problems in special relativity.

The Pauli algebra approach to special relativity

The Pauli algebra P, in which the usual dot and cross products of 3-space vectors are combined in an associative, invertible, but non-commutative multiplication, provides a simple but powerful approach to problems in special relativity. Even though the Pauli algebra is the Clifford algebra for Euclidean 3-space, Minkowski 4-vectors and their products in the Minkowski metric appear in a natural and covariant way as elements of P. The authors review the algebra and develop a formulation which, although closely tied to elementary vector and functional analysis, nevertheless allows a compact coordinate-free treatment of essentially all problems in special relativity. They derive a number of useful results and show how the elements are related both to traditional Minkowski-space tensors and to elements of the Dirac algebra.

Problem Book in Relativity and Gravitation

Alan P Lightman, William H Press, Richard H Price and Saul A Teukolsky Princeton, NJ: Princeton University Press 1975 pp xiv + 599 price £11.70 (cased), £4.40 (paperback) This book contains some 500 problems in special relativity, general relativity and associated topics such as cosmology and black holes, with their solutions. It will be of great value to someone working in these fields or to students, used in conjunction with a text such as Misner, Thorne and Wheeler's Gravitation, and covers many physically interesting cases.

The electromagnetic two-body problem in special relativity

A system of two point charged particles is considered. Each particle moves in the electromagnetic field created by the other particle according to Maxwell's equations. A scheme of successive approximations is developed to study the field and the motion of the charges. The field (potentials and intensities) are exapanded in powers of c −1 using a retarded time coordinate. The variables of the motion (position vectors, velocities, etc) are expanded in powers of c −1 with coefficients depending on t only. The field is evaluated in the first three approximations. The equations of motion are derived in the same approximations and the corresponding conserved quantities are explicitly given. Thus, the usual assumption of an action-at-a-distance principle is avoided and the original nonlinear integrodifferential equations are reduced to a sequence of linear equations.

The electromagnetic two-body problem in special relativity: S. Persides and J. Pascalis. University of Thessaloniki, Thessaloniki, Greece

The electromagnetic two-body problem in special relativity: S. Persides and J. Pascalis. University of Thessaloniki, Thessaloniki, Greece

Erratum: Time-asymmetric two-body problem in special relativity

Erratum: Time-asymmetric two-body problem in special relativity

The Brehme Diagram in Two Spatial Dimensions

An extension of the Brehme diagram is given which describes two spatial dimensions and one time dimension. The extended diagram is applied to several problems in special relativity: the Lorentz velocity transformations, the transverse Doppler effect, the Michelson-Morley experiment, the transformation of the direction of propagation of light, and the visual appearance of rapidly moving objects.

Re-entrant Motion in Special Relativity

A RECENT communication1 discusses the uniform rotation of a rigid circular disk as a problem in special relativity. Phipps is in error in supposing that there is more than one way to measure the peripheral velocity in a Lorentz frame in which the centre of the disk is at rest. He will see that the velocities v and rω are equal (if it is not at once obvious from the kinematics in S) by drawing a Minkowski diagram showing the approximately parallel world lines of adjacent markers. These move with velocity v in S, and are at rest in S*. S* ascribes a spatial displacement Δx* to the pair of events consisting of successive transits of a point fixed in S. Since the two markers are at rest in S* Δx* may instead be determined from a pair of events in their histories simultaneous in S*. But then Δx* is just the rest displacement, namely, 2πr/n√1−v2/c2. This reflects the well-known fact that in a rigid rotating (but non-holonomic) co-ordinate system the ratio circumference : radius is given by 2π : √1−v2/c2. Substituting the above value for Δx* into the Lorentz formula, we find that the ‘round-trip’ velocity rω and the relative velocity v are in fact equal.