Light Inextensible Strings (Thread) Under Tension in the Schwarzschild Geometry

2021 ◽  
Vol 03 (02) ◽  
pp. 2150005
Author(s):  
Robin K. S. Hankin
Keyword(s):  
Free Fall ◽  
Computer Algebra Systems ◽  
Lagrange Equations ◽  
Schwarzschild Metric ◽  
Schwarzschild Geometry ◽  
Homework Problem ◽  
Straight Line ◽  
Taut Strings ◽  
Proper Length ◽  
Insight Into

Light inextensible string under tension is a stalwart feature of elementary physics. Here I show how considering such a string in the vicinity of a black hole, with the help of computer algebra systems, can generate insight into the Schwarzschild geometry in the context of an undergraduate homework problem. Light taut strings minimize their proper length, given by integrating the spatial component of the Schwarzschild metric along the string. The path itself is given by straightforward numerical solution to the Euler–Lagrange equations. If the string is entirely outside the event horizon, its closest approach to the singularity is tangential. At this point the string is visibly curved, surely a memorable and informative insight. The geometry of the Schwarzschild metric induces some interesting nonlocal phenomena: if the distance of closest approach is less than about [Formula: see text], the string self-intersects, even though it is everywhere under tension. Light taut strings furnish a third interpretation of the concept “straight line”, the other two being null geodesics and free-fall world lines. All the software used is available under the GPL.1

The Schwarzschild black hole

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Black Hole ◽  
Gravitational Field ◽  
Equilibrium Configuration ◽  
Original Form ◽  
Schwarzschild Black Hole ◽  
Schwarzschild Metric ◽  
Schwarzschild Geometry

This chapter discusses the Schwarzschild black hole. It demonstrates how, by a judicious change of coordinates, it is possible to eliminate the singularity of the Schwarzschild metric and reveal a spacetime that is much larger, like that of a black hole. At the end of its thermonuclear evolution, a star collapses and, if it is sufficiently massive, does not become stabilized in a new equilibrium configuration. The Schwarzschild geometry must therefore represent the gravitational field of such an object up to r = 0. This being said, the Schwarzschild metric in its original form is singular, not only at r = 0 where the curvature diverges, but also at r = 2m, a surface which is crossed by geodesics.

Galileo Unbound

2018 ◽  
Author(s):  
David D. Nolte
Keyword(s):  
Free Fall ◽  
The Sun ◽  
Quantum Field ◽  
Many Worlds ◽  
Quantum Particles ◽  
Light Rays ◽  
History Of ◽  
And Control ◽  
The Universe ◽  
Insight Into

Galileo Unbound: A Path Across Life, The Universe and Everything traces the journey that brought us from Galileo’s law of free fall to today’s geneticists measuring evolutionary drift, entangled quantum particles moving among many worlds, and our lives as trajectories traversing a health space with thousands of dimensions. Remarkably, common themes persist that predict the evolution of species as readily as the orbits of planets or the collapse of stars into black holes. This book tells the history of spaces of expanding dimension and increasing abstraction and how they continue today to give new insight into the physics of complex systems. Galileo published the first modern law of motion, the Law of Fall, that was ideal and simple, laying the foundation upon which Newton built the first theory of dynamics. Early in the twentieth century, geometry became the cause of motion rather than the result when Einstein envisioned the fabric of space-time warped by mass and energy, forcing light rays to bend past the Sun. Possibly more radical was Feynman’s dilemma of quantum particles taking all paths at once—setting the stage for the modern fields of quantum field theory and quantum computing. Yet as concepts of motion have evolved, one thing has remained constant, the need to track ever more complex changes and to capture their essence, to find patterns in the chaos as we try to predict and control our world.

Traversing a black-and-white hole in free fall and rise

2020 ◽  
Vol 33 (4) ◽  
pp. 460-465
Author(s):  
Andreas Trupp
Keyword(s):  
Black Holes ◽  
Gravity Field ◽  
Free Fall ◽  
Spherical Body ◽  
Relativity Principle ◽  
White Hole ◽  
Schwarzschild Metric ◽  
Temporal Interval ◽  
Black And White ◽  
Schwarzschild Radius

It is shown that a traverse of a Black-and-White Hole (through a shaft in the interior of the central, spherical body) in free radial fall and rise is described by the Schwarzschild metric without any ambiguity. In other words, all Black Holes can also be White Holes. The relativity principle, according to which both the freely falling/rising observer Alice and a second observer Bob (sitting outside of the gravity field) have to measure the same temporal interval for the complete trip, is observed [(Δt)/(Δτ) = 1]. In the interior of the Schwarzschild radius, Alice's time τ is reversed. Kruskal charts do not present an obstacle to this result, since quadrant II can be used for ingoing traffic only, but not for outgoing traffic.

scholarly journals Thermodynamic insight into the growth of nanoscale inclusion of Al-deoxidation in Fe–O–Al melt

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Hong Lei ◽  
Yuanyou Xiao ◽  
Guocheng Wang ◽  
Hongwei Zhang ◽  
Wei Jin ◽  
...  
Keyword(s):  
Experimental Data ◽  
Free Energy ◽  
Gibbs Free Energy ◽  
Liquid Iron ◽  
Reaction Products ◽  
Straight Line ◽  
Nano Alumina ◽  
Increasing Temperature ◽  
The Relationship ◽  
Insight Into

Abstract Products of Al-deoxidation reaction in iron melt are the most common inclusions and play an important effect on steel performance. Understanding the thermodynamics on nano-alumina (or nano-hercynite) is very critical to explore the relationship between Al-deoxidation reaction and products growth in iron melt. In present study, a thermodynamic modeling of nano-alumina inclusions in Fe–O–Al melt has been developed. The thermodynamic results show that the Gibbs free energy changes for the formation of nano-Al2O3 and nano-FeAl2O4 decrease with the increasing size and increase with the increasing temperature. The Gibbs free energy changes for transformation of nano-Al2O3 into bulk-Al2O3 increase with the increasing size and temperature. The thermodynamic curve of nano-alumina (or nano-hercynite) and the equilibrium curve of bulk-alumina (or bulk-hercynite) obtained in this work are agree with the published experimental data of Al-deoxidation equilibria in liquid iron. In addition, the thermodynamic coexisting points about Al2O3 and FeAl2O4 in liquid iron are in a straight line and coincide with the various previous data. It suggested that these scattered experimental data maybe in the different thermodynamic state of Al-deoxidized liquid iron and the reaction products for most of the previous Al-deoxidation experiments are nano-alumina (or nano-hercynite).

scholarly journals A trajectory equation for walking droplets: hydrodynamic pilot-wave theory

2013 ◽  
Vol 737 ◽  
pp. 552-570 ◽  
Author(s):  
Anand U. Oza ◽  
Rodolfo R. Rosales ◽  
John W. M. Bush
Keyword(s):  
Chaotic Dynamics ◽  
Wave Theory ◽  
Pitchfork Bifurcation ◽  
Horizontal Motion ◽  
Straight Line ◽  
Pilot Wave ◽  
The Stability ◽  
Trajectory Equation ◽  
Insight Into ◽  
Pilot Wave Theory

AbstractWe present the results of a theoretical investigation of droplets bouncing on a vertically vibrating fluid bath. An integro-differential equation describing the horizontal motion of the drop is developed by approximating the drop as a continuous moving source of standing waves. Our model indicates that, as the forcing acceleration is increased, the bouncing state destabilizes into steady horizontal motion along a straight line, a walking state, via a supercritical pitchfork bifurcation. Predictions for the dependence of the walking threshold and drop speed on the system parameters compare favourably with experimental data. By considering the stability of the walking state, we show that the drop is stable to perturbations in the direction of motion and neutrally stable to lateral perturbations. This result lends insight into the possibility of chaotic dynamics emerging when droplets walk in complex geometries.

Stability Analysis Hinged Vertical Flat Plate Rotation in a Uniform Flow

2012 ◽  
Author(s):  
Sina Mirzaei Sefat ◽  
Antonio Carlos Fernandes
Keyword(s):  
Stability Analysis ◽  
Dynamic Models ◽  
Fundamental Problem ◽  
Chaotic Oscillation ◽  
Free Fall ◽  
Vertical Axis ◽  
Sea Bed ◽  
Uniform Current ◽  
Free Falling ◽  
Insight Into

The free falling of objects is a phenomenon that has been observed in the nature. The Pendulous Installation Method (PIM) of heavy devices is an example of free falling which occurs in the installation of heavy offshore devices on sea bed. Hence, the motivation of the present research is to study the fluttering and tumbling (autorotation) motions that may occur during the free fall of object. The fluttering is a periodic or chaotic oscillation of body about a vertical axis. On the other hand, the tumbling is end-over-end rotation of body. In order to access the main physical aspects, the present work decided to attack a more fundamental problem and describes the investigations on fluttering and autorotation motions of the interaction of uniform current and freely rotating plate about a fixed vertical axis. A quasi-steady model is suggested to model the trajectories of flow induced rotation phenomenon and a stability analysis performed to gain insight into the nature of the bifurcation from fluttering to autorotation. At first, the fixed points for different models of motion is obtained and each point analyzed by using the linearized equation. Secondly, the phase diagrams as a function of angular velocity and angle of rotation have been presented for different dynamic models.

scholarly journals Islam and Art

1986 ◽  
Vol 3 (2) ◽  
pp. 327-330
Author(s):  
Mohamed Zakariya
Keyword(s):  
University Of Pennsylvania ◽  
Islamic Art ◽  
Complex Field ◽  
Central Question ◽  
The Arts ◽  
Straight Line ◽  
True Answer ◽  
The University ◽  
Arabic Calligraphy ◽  
Insight Into

I met Dr. Lois Lamya al Faruqi last March, while I was giving a lectureon Arabic calligraphy at the University of Pennsylvania in Philadelphia. I hadcorresponded with her previously and had been impressed by her graciousnessand her insight into the complex field of Islamic art. Now, here she and herhusband were, and I was pleased to have these bright lights of the Muslimcommunity in my audience. At the end of my remarks, someone asked theinevitable question, "What is the significance, the symbolism of the designsused to highlight a calligraphic piece?" I replied that, while some Islamicdesigns have their origins in the material world, they become abstracted andstylized but do not take on additional symbolic meaning. In other words, Islamicart, at its best, does not depend on visual symbols as clues to its meaning.A flower Wing remains a flower drawing, no matter how abstract it becomes.Dr. Faruqi's was interested in my interpretation and, I think somewhat amused.As she left that evening, I saw a definite twinkle in her eye, and I feltI had found a congenial colleague. Two weeks later, I received an autographedcopy of Islam and Art from her. I never saw her again.I am pleased to have this opportunity to review this volume, Dr. Faruqi'slast published work. Let me make my own position clear: I am neither anacademician nor a genuine scholar of Islamic art. Rather, I approach the subjectas a practitioner; therefore, my interest in Islamic art-and in this book- isat once personal, practical, and professional.Why do the arts of the Muslim peoples-and by arts, I include graphicarts, architecture, crafts, and music -develop with such obvious consistencyfrom people to people, and in such a straight line from their inception to thepresent? This question has baffled scholars for at least a century. Dr. Faruqi'sinsight guides the reader in the direction of a true answer, yet it is an answereach of us must experience for ourselves, through study and contemplation.In short, Dr. Faruqi's answer to this central question is that the arts ofthe Muslim peoples did not develop by chance, but rather, as an attempt toexpress by various media the Quranic doctrine of tawhid, the immense ...

scholarly journals A drop does not fall in a straight line: a rationale for the width of stalagmites

2019 ◽  
Vol 475 (2231) ◽  
pp. 20190556 ◽  
Author(s):  
J. Parmentier ◽  
S. Lejeune ◽  
M. Maréchal ◽  
F. Bourges ◽  
D. Genty ◽  
...  
Keyword(s):  
Calcium Ions ◽  
High Speed ◽  
Free Fall ◽  
Impact Point ◽  
Calcite Precipitation ◽  
High Speed Imaging ◽  
Point Position ◽  
Straight Line ◽  
The Impact

Drops loaded in calcium ions detach from stalactites and impact the underlying stalagmites, thereby allowing these latter to grow through calcite precipitation. Nevertheless, little is known about the influence of the drop free fall and splash dynamics on stalagmite shape and width. Through high-speed imaging of impacting drops on stalagmites from several caves, we observed that the impact point position of the drops is scattered, sometimes over several centimetres. We show that this dispersal has no external cause and must, therefore, be self-induced. Using a Langevin-like equation, we then propose a prediction of the impact point dispersal as a function of the falling height travelled by the drops. We finally show that measured stalagmite widths are correlated to the dispersal in the impact point position of the drop.

scholarly journals A foundation for reference models for drug combinations with an application to Loewe’s reference model

2020 ◽  
Vol 21 (1) ◽  
Author(s):  
Wim De Mulder ◽  
Martin Kuiper
Keyword(s):  
Dose Response ◽  
Reference Model ◽  
Synergistic Effects ◽  
Null Model ◽  
Drug Combinations ◽  
Response Curves ◽  
Reference Models ◽  
Straight Line ◽  
Long Time ◽  
Insight Into

Abstract Background Treating patients with combinations of drugs that have synergistic effects has become widespread practice in the clinic. Drugs work synergistically when the observed effect of a drug combination is larger than the effect predicted by the reference model. The reference model is a theoretical null model that returns the combined effect of given doses of drugs under the assumption that these drugs do not interact. There is ongoing debate on what it means for drugs to not interact. The controversy transcends mathematical punctuality, as different non-interaction principles result in different reference models. A famous reference model that has been in existence for already a long time is Loewe’s reference model. Loewe’s vision on non-interaction was purely intuitive: two drugs do not interact if all combinations of doses that result in a certain given effect lie on a straight line. Results We show that Loewe’s reference model can be obtained from much more fundamental principles. First, we introduce the new notion of complementary dose. Secondly, we reformulate the existing concept of equivalent dose, whereby our formulation is more general than existing ones. Finally, a very general non-interaction principle is put forward. The proposed non-interaction principle represents a certain interplay between complementary and equivalent doses: drugs are non-interacting if complementarity is preserved under equivalence. It is then shown that Loewe’s reference model naturally follows from these principles by an appropriate choice of complementarity. Conclusions The presented work increases insight into Loewe’s reference model for drug combinations, which is realized by the introduction of a very general non-interaction principle that does not refer to any specific dose-response curve, nor to any property of applicable dose-response curves.

Calibration problems in hydrogen luminosities

1966 ◽  
Vol 24 ◽  
pp. 322-330
Author(s):  
A. Beer
Keyword(s):  
Galactic Structure ◽  
B Stars ◽  
Insight Into

The investigations which I should like to summarize in this paper concern recent photo-electric luminosity determinations of O and B stars. Their final aim has been the derivation of new stellar distances, and some insight into certain patterns of galactic structure.

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