Schwarzschild’s Spacetime

2018 ◽  
Author(s):  
Prasenjit Saha ◽  
Paul A. Taylor
Keyword(s):  
Gravitational Lensing ◽  
Orbital Motion ◽  
Proper Time ◽  
Dynamical Problem ◽  
Light Deflection ◽  
Basic Form ◽  
Field Equations ◽  
Stable Orbit ◽  
Rotating Black Holes ◽  
Newtonian Systems

The concept of a metric is motivated and introduced, along with the introduction of relativistic quantities of spacetime, proper time, and Einstein’s field equations. Geodesics are cast in basic form as a Hamiltonian dynamical problem, which readers are guided towards exploring numerically themselves. The specific case of the Schwarzschild metric is presented, which is applicable to space around non-rotating black holes, and orbital motion around such objects is contrasted with that of Newtonian systems. Some well-known formulas for black hole phenomena are derived, such as those for light deflection (also known as gravitational lensing) and the innermost stable orbit, and their consequences discussed.

scholarly journals Null Geodesics and Strong Field Gravitational Lensing in a String Cloud Background

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
M. Sharif ◽  
Sehrish Iftikhar
Keyword(s):  
Black Hole ◽  
Gravitational Lensing ◽  
Galactic Center ◽  
Deflection Angle ◽  
Orbital Motion ◽  
Strong Field ◽  
Schwarzschild Black Hole ◽  
Field Limit ◽  
Null Geodesics ◽  
Supermassive Black Hole

This paper is devoted to studying two interesting issues of a black hole with string cloud background. Firstly, we investigate null geodesics and find unstable orbital motion of particles. Secondly, we calculate deflection angle in strong field limit. We then find positions, magnifications, and observables of relativistic images for supermassive black hole at the galactic center. We conclude that string parameter highly affects the lensing process and results turn out to be quite different from the Schwarzschild black hole.

Demonstrating Gravitational Lensing Using Solar Eclipses

2021 ◽  
Vol 03 (03) ◽  
pp. 2150009
Author(s):  
Gillian Foo ◽  
Jhoon Yong Tan ◽  
Edmund Yuen ◽  
Laurentcia Arlany ◽  
A. Yang ◽  
...  
Keyword(s):  
Gravitational Lensing ◽  
Angular Separation ◽  
Total Solar Eclipse ◽  
Light Deflection ◽  
Interesting Paper ◽  
National University ◽  
Solar Eclipses ◽  
The Individual ◽  
Pixel Scale ◽  
Plate Solution

As encouraged by the interesting paper “Solar eclipses as a teaching opportunity in relativity” by Overduin et al.,awe made measurements of the angular deflections of neighboring stars during the 9 March 2016 total solar eclipse as imaged by National University of Singapore (NUS) students, to verify a result of general relativity. In this project, we used these images and measured the stars’ pixel positions and transformed them to equatorial coordinates using a similar approach to Overduin et al., with a few modifications. Instead of solving to determine the pixel scale and rotation, we performed a plate solution using the software AstroImageJ which enables accounting for the image’s higher order distortion. This data is found in the image’s Flexible Image Transport System (FITS) header. Image star pair separations were then compared to their database separations after determining how the individual deflections affect angular separation. Our experimental results have large uncertainties and were deemed imprecise to confirm the effects of gravitational light deflection. We include a detailed analysis and discussion on this educational project.

General Relativity

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Field Equation ◽  
Einstein Field Equation ◽  
Kerr Metric ◽  
Gravitational Lenses ◽  
Field Equations ◽  
Spacetime Geometry ◽  
Tests Of General Relativity ◽  
Rotating Black Holes

This chapter presents the physical motivation for general relativity, derives the Einstein field equation and gives concise derivations of the main results of the theory. It begins with the equivalence principle, tidal forces in Newtonian gravity and their connection to curved spacetime geometry. This leads to a derivation of the field equation. Tests of general relativity are considered: Mercury’s perihelion advance, gravitational redshift, the deflection of starlight and gravitational lenses. The exterior and interior Schwarzschild solutions are discussed. Eddington–Finkelstein coordinates are used to describe objects falling into non-rotating black holes. The Kerr metric is used to describe rotating black holes and their astrophysical consequences. Gravitational waves are described and used to explain the orbital decay of binary neutron stars. Their recent detection by LIGO and the beginning of a new era of gravitational wave astronomy is discussed. Finally, the gravitational field equations are derived from the Einstein–Hilbert action.

Solar Eclipses as a Teaching Opportunity in Relativity

2020 ◽  
Vol 02 (02) ◽  
pp. 2050010
Author(s):  
James Overduin ◽  
Kelsey Glazer ◽  
Keri McClelland ◽  
Amelia Genus ◽  
Chris Miskiewicz
Keyword(s):  
General Relativity ◽  
Gravitational Lensing ◽  
Pivotal Role ◽  
Light Deflection ◽  
British Government ◽  
Composite Image ◽  
Theory Of Gravity ◽  
Solar Eclipses

Total solar eclipses represent a challenging but spectacular opportunity to introduce curious students to the wonders of general relativity through the phenomenon of light deflection (gravitational lensing). During the Great American Eclipse of 2017, we were among a small number of teams attempting to repeat Eddington’s iconic observations of 1919, which played a pivotal role in establishing Einstein’s theory as the governing theory of gravity. We were not quite successful on the observational front, but acquired an excellent composite image from a fellow astronomer. Analysis of this image allowed us to obtain a result consistent with Einstein’s theory. It is remarkable that such an experiment, which once required the resources of the British government, can now be attempted with reasonable hope of success by teachers and their students. We look forward to our next chance in 2024.

scholarly journals Axially Symmetric Null Dust Space-Time, Naked Singularity, and Cosmic Time Machine

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Faizuddin Ahmed
Keyword(s):  
Gravitational Lensing ◽  
Cosmic Time ◽  
Naked Singularity ◽  
Space Time ◽  
Axially Symmetric ◽  
Time Machine ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Closed Orbits ◽  
Group A

We present a gravitational collapse null dust solution of the Einstein field equations. The space-time is regular everywhere except on the symmetry axis where it possesses a naked curvature singularity and admits one parameter isometry group, a generator of axial symmetry along the cylinder which has closed orbits. The space-time admits closed timelike curves (CTCs) which develop at some particular moment in a causally well-behaved manner and may represent a Cosmic Time Machine. The radial geodesics near the singularity and the gravitational lensing (GL) will be discussed. The physical interpretation of this solution, based on the study of the equation of the geodesic deviation, will be presented. It was demonstrated that this solution depends on the local gravitational field consisting of two components with amplitudes Ψ2 and Ψ4.

scholarly journals Gravitational Lensing by Stars and Machos and the Orbital Motion of the Earth

1995 ◽  
Vol 166 ◽  
pp. 305-308
Author(s):  
M. Hosokawa ◽  
K. Ohnishi ◽  
T. Fukushima ◽  
M. Takeuti
Keyword(s):  
Light Curve ◽  
Gravitational Lensing ◽  
Orbital Motion ◽  
Single Star ◽  
The Earth ◽  
Primary Image ◽  
Gravitational Deflection

We showed that it is feasible to measure the mass of a single star by observing the variation of gravitational deflection caused by the orbital motion of the Earth. When the distance of a star is less than 60 pc and some appropriate sources are within 1 arcsec. in its background, not only the distance but also the mass of the star may be determined by measuring the deflection with an accuracy of 10 μ arcsec. In the case of photometric microlensing by a MACHO, the observation of astrometric gravitational deflection is also useful. By measuring the separation between the primary image and the secondary image, the ratio of mass to distance of the MACHO will be obtained. Further, the orbital motion of the Earth modifying the light curve of the source is discussed.

σ-MODELS AND STRING THEORIES

1987 ◽  
Vol 02 (03) ◽  
pp. 667-693 ◽  
Author(s):  
S. RANDJBAR-DAEMI ◽  
ABDUS SALAM ◽  
J. A. STRATHDEE
Keyword(s):  
Bosonic Strings ◽  
Proper Time ◽  
Background Field ◽  
Dimensional Manifold ◽  
Field Equations ◽  
World Sheet ◽  
Global Features ◽  
String Loop ◽  
Definition Of ◽  
Background Fields

The propagation of closed bosonic strings interacting with background gravitational and dilaton fields is reviewed. The string is treated as a quantum field theory on a compact 2-dimensional manifold. The question is posed as to how the conditions for the vanishing trace anomaly and the ensuing background field equations may depend on global features of the manifold. It is shown that to the leading order in σ-model perturbation theory the string loop effects do not modify the gravitational and the dilaton field equations. However for the purely bosonic strings new terms involving the modular parameter of the world sheet are induced by quantum effects which can be absorbed into a re-definition of the background fields. We also discuss some aspects of several regularization schemes such as dimensional, Pauli-Villars and the proper-time cut off in an appendix.

scholarly journals Twisted light, a new tool for general relativity and beyond — Revealing the properties of rotating black holes with the vorticity of light —

2021 ◽  
Author(s):  
F. Tamburini ◽  
F. Feleppa ◽  
B. Thidé
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Black Hole ◽  
Angular Momentum ◽  
Orbital Angular Momentum ◽  
Gravitational Lensing ◽  
Compact Object ◽  
Additional Tool ◽  
Physical Observable ◽  
Rotating Black Holes

We describe and present the first observational evidence that light propagating near a rotating black hole is twisted in phase and carries orbital angular momentum. The novel use of this physical observable as an additional tool for the previously known techniques of gravitational lensing allows us to directly measure, for the first time, the spin parameter of a black hole. With the additional information encoded in the orbital angular momentum, not only can we reveal the actual rotation of the compact object, but we can also use rotating black holes as probes to test general relativity.

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