BLACK HOLES: the OTHER SIDE of INFINITY General Information
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A Mathematical Derivation of the General Relativistic Schwarzschild
A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components ................. -
Astro 210 Lecture 37 April 23, 2018 Announcements
Astro 210 Lecture 37 April 23, 2018 Announcements: • HW 11: The Final Frontier posted, due 5:00pm Friday • Grades: we are catching up! keep checking Moodle 1 Last Time: Searching for Black Holes Black holes themselves are invisible∗ can can detect them via their strong gravitational effects on their close surroundings example: binary stars X-rays emitted from unseen massive companion ∗this ignores Hawking radiation–see below 2 Our Own Galactic Center central ∼ 30 pc of Galaxy: can’t see optically (Q: why?), but can in other wavelengths: extended (non-point) radio emission (Sagittarius A) from high-energy electrons radio source at center: Sgr A∗ size 2.4 AU(!), variable emission in radio, X-ray www: X-ray Sgr A∗ in infrared wavelengths: can see stars near Sgr A∗ and they move! www: Sgr A∗ movie elliptical paths! closest: period P = 15.2 yr semi-major axis: a = 4.64 × 10−3 pc 3 6 → enclosed mass (3.7 ± 1.5) × 10 M⊙ Q: and so? the center of our Galaxy contains a black hole! Sgr A∗ Schwarzschild radius 7 −7 rSch = 1.1 × 10 km=0.74 AU = 3.6 × 10 pc (1) → not resolved (yet) but: Event Horizon Telescope has data and right now is processing possible first images! Galactic black hole raises many questions: • how did it get there? • Sgr A∗ low luminosity, “quiet” compared to more “active” galactic nuclei www: AGN: M87 why? open question.... • in last few months: discovery of high-energy “bubbles” 4 above & below Galactic center www: gamma-ray images → remains of the most recent Sgr A∗ belch? Galaxies and Black Holes The Milky Way is not the only -
Arxiv:Gr-Qc/0612030 V1 5 Dec 2006
December 6, 2006 1:49 WSPC - Proceedings Trim Size: 9.75in x 6.5in main 1 STABLE DARK ENERGY STARS: AN ALTERNATIVE TO BLACK HOLES? FRANCISCO S. N. LOBO Centro de Astronomia e Astrof´ısica da Universidade de Lisboa, Campo Grande, Ed. C8 1749-016 Lisboa, Portugal flobo@cosmo.fis.fc.ul.pt In this work, a generalization of the Mazur-Mottola gravastar model is explored, by considering a matching of an interior solution governed by the dark energy equation of state, ω ≡ p/ρ < −1/3, to an exterior Schwarzschild vacuum solution at a junction interface, situated near to where the event horizon is expected to form. The motivation for implementing this generalization arises from the fact that recent observations have confirmed an accelerated cosmic expansion, for which dark energy is a possible candidate. Keywords: Gravastars; dark energy. Although evidence for the existence of black holes is very convincing, a certain amount of scepticism regarding the physical reality of singularities and event hori- zons is still encountered. In part, due to this scepticism, an alternative picture for the final state of gravitational collapse has emerged, where an interior compact ob- ject is matched to an exterior Schwarzschild vacuum spacetime, at or near where the event horizon is expected to form. Therefore, these alternative models do not possess a singularity at the origin and have no event horizon, as its rigid surface is arXiv:gr-qc/0612030 v1 5 Dec 2006 located at a radius slightly greater than the Schwarzschild radius. In particular, the gravastar (gravitational vacuum star) picture, proposed by Mazur and Mottola,1 has an effective phase transition at/near where the event horizon is expected to form, and the interior is replaced by a de Sitter condensate. -
Active Galactic Nuclei: a Brief Introduction
Active Galactic Nuclei: a brief introduction Manel Errando Washington University in St. Louis The discovery of quasars 3C 273: The first AGN z=0.158 2 <latexit sha1_base64="4D0JDPO4VKf1BWj0/SwyHGTHSAM=">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</latexit> <latexit sha1_base64="H7Rv+ZHksM7/70841dw/vasasCQ=">AAACQHicbVDLSgMxFM34tr6qLt0Ei+BCy0TEx0IoPsBlBWuFTlsyaVqDSWZI7ghlmE9z4ye4c+3GhSJuXZmxFXxdCDk599zk5ISxFBZ8/8EbGR0bn5icmi7MzM7NLxQXly5slBjGayySkbkMqeVSaF4DAZJfxoZTFUpeD6+P8n79hhsrIn0O/Zg3Fe1p0RWMgqPaxXpwzCVQfNIOFIUro1KdMJnhA+yXfX832FCsteVOOzgAobjFxG+lhGTBxpe+HrBOBNjiwd5rpZsky9rFUn5BXvgvIENQQsOqtov3QSdiieIamKTWNogfQzOlBgSTPCsEieUxZde0xxsOaurMNNPPADK85pgO7kbGLQ34k/0+kVJlbV+FTpm7tr97Oflfr5FAd6+ZCh0nwDUbPNRNJIYI52nijjCcgew7QJkRzitmV9RQBi7zgguB/P7yX3CxVSbb5f2z7VLlcBjHFFpBq2gdEbSLKugUVVENMXSLHtEzevHuvCfv1XsbSEe84cwy+lHe+wdR361Q</latexit> The power source of quasars • The luminosity (L) of quasars, i.e. how bright they are, can be as high as Lquasar ~ 1012 Lsun ~ 1040 W. • The energy source of quasars is accretion power: - Nuclear fusion: 2 11 1 ∆E =0.007 mc =6 10 W s g− -
Stephen Hawking: 'There Are No Black Holes' Notion of an 'Event Horizon', from Which Nothing Can Escape, Is Incompatible with Quantum Theory, Physicist Claims
NATURE | NEWS Stephen Hawking: 'There are no black holes' Notion of an 'event horizon', from which nothing can escape, is incompatible with quantum theory, physicist claims. Zeeya Merali 24 January 2014 Artist's impression VICTOR HABBICK VISIONS/SPL/Getty The defining characteristic of a black hole may have to give, if the two pillars of modern physics — general relativity and quantum theory — are both correct. Most physicists foolhardy enough to write a paper claiming that “there are no black holes” — at least not in the sense we usually imagine — would probably be dismissed as cranks. But when the call to redefine these cosmic crunchers comes from Stephen Hawking, it’s worth taking notice. In a paper posted online, the physicist, based at the University of Cambridge, UK, and one of the creators of modern black-hole theory, does away with the notion of an event horizon, the invisible boundary thought to shroud every black hole, beyond which nothing, not even light, can escape. In its stead, Hawking’s radical proposal is a much more benign “apparent horizon”, “There is no escape from which only temporarily holds matter and energy prisoner before eventually a black hole in classical releasing them, albeit in a more garbled form. theory, but quantum theory enables energy “There is no escape from a black hole in classical theory,” Hawking told Nature. Peter van den Berg/Photoshot and information to Quantum theory, however, “enables energy and information to escape from a escape.” black hole”. A full explanation of the process, the physicist admits, would require a theory that successfully merges gravity with the other fundamental forces of nature. -
Black Holes. the Universe. Today’S Lecture
Physics 311 General Relativity Lecture 18: Black holes. The Universe. Today’s lecture: • Schwarzschild metric: discontinuity and singularity • Discontinuity: the event horizon • Singularity: where all matter falls • Spinning black holes •The Universe – its origin, history and fate Schwarzschild metric – a vacuum solution • Recall that we got Schwarzschild metric as a solution of Einstein field equation in vacuum – outside a spherically-symmetric, non-rotating massive body. This metric does not apply inside the mass. • Take the case of the Sun: radius = 695980 km. Thus, Schwarzschild metric will describe spacetime from r = 695980 km outwards. The whole region inside the Sun is unreachable. • Matter can take more compact forms: - white dwarf of the same mass as Sun would have r = 5000 km - neutron star of the same mass as Sun would be only r = 10km • We can explore more spacetime with such compact objects! White dwarf Black hole – the limit of Schwarzschild metric • As the massive object keeps getting more and more compact, it collapses into a black hole. It is not just a denser star, it is something completely different! • In a black hole, Schwarzschild metric applies all the way to r = 0, the black hole is vacuum all the way through! • The entire mass of a black hole is concentrated in the center, in the place called the singularity. Event horizon • Let’s look at the functional form of Schwarzschild metric again: ds2 = [1-(2m/r)]dt2 – [1-(2m/r)]-1dr2 - r2dθ2 -r2sin2θdφ2 • We want to study the radial dependence only, and at fixed time, i.e. we set dφ = dθ = dt = 0. -
Undergraduate Thesis on Supermassive Black Holes
Into the Void: Mass Function of Supermassive Black Holes in the local universe A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of Arts Farhanul Hasan May 2018 Approved for the Division (Physics) Alison Crocker Acknowledgements Writing a thesis is a long and arduous process. There were times when it seemed further from my reach than the galaxies I studied. It’s with great relief and pride that I realize I made it this far and didn’t let it overpower me at the end. I have so many people to thank in very little space, and so much to be grateful for. Alison, you were more than a phenomenal thesis adviser, you inspired me to believe that Astro is cool. Your calm helped me stop freaking out at the end of February, when I had virtually no work to show for, and a back that ached with every step I took. Thank you for pushing me forward. Working with you in two different research projects were very enriching experiences, and I appreciate you not giving up on me, even after all the times I blanked on how to proceed forward. Thank you Johnny, for being so appreciative of my work, despite me bringing in a thesis that I myself barely understood when I brought it to your table. I was overjoyed when I heard you wanted to be on my thesis board! Thanks to Reed, for being the quirky, intellectual community that it prides itself on being. -
Review Study on “The Black Hole”
IJIRST –International Journal for Innovative Research in Science & Technology| Volume 2 | Issue 10 | March 2016 ISSN (online): 2349-6010 Review Study on “The Black Hole” Syed G. Ibrahim Department of Engineering Physics (Nanostructured Thin Film Materials Laboratory) Prof. Ram Meghe College of Engineering and Management, Badnera 444701, Maharashtra, India Abstract As a star grows old, swells, then collapses on itself, often you will hear the word “black hole” thrown around. The black hole is a gravitationally collapsed mass, from which no light, matter, or signal of any kind can escape. These exotic objects have captured our imagination ever since they were predicted by Einstein's Theory of General Relativity in 1915. So what exactly is a black hole? A black hole is what remains when a massive star dies. Not every star will become a black hole, only a select few with extremely large masses. In order to have the ability to become a black hole, a star will have to have about 20 times the mass of our Sun. No known process currently active in the universe can form black holes of less than stellar mass. This is because all present black hole formation is through gravitational collapse, and the smallest mass which can collapse to form a black hole produces a hole approximately 1.5-3.0 times the mass of the sun .Smaller masses collapse to form white dwarf stars or neutron stars. Keywords: Escape Velocity, Horizon, Schwarzschild Radius, Black Hole _______________________________________________________________________________________________________ I. INTRODUCTION Soon after Albert Einstein formulated theory of relativity, it was realized that his equations have solutions in closed form. -
Chapter 22 Neutron Stars and Black Holes Units of Chapter 22 22.1 Neutron Stars 22.2 Pulsars 22.3 Xxneutron-Star Binaries: X-Ray Bursters
Chapter 22 Neutron Stars and Black Holes Units of Chapter 22 22.1 Neutron Stars 22.2 Pulsars 22.3 XXNeutron-Star Binaries: X-ray bursters [Look at the slides and the pictures in your book, but I won’t test you on this in detail, and we may skip altogether in class.] 22.4 Gamma-Ray Bursts 22.5 Black Holes 22.6 XXEinstein’s Theories of Relativity Special Relativity 22.7 Space Travel Near Black Holes 22.8 Observational Evidence for Black Holes Tests of General Relativity Gravity Waves: A New Window on the Universe Neutron Stars and Pulsars (sec. 22.1, 2 in textbook) 22.1 Neutron Stars According to models for stellar explosions: After a carbon detonation supernova (white dwarf in binary), little or nothing remains of the original star. After a core collapse supernova, part of the core may survive. It is very dense—as dense as an atomic nucleus—and is called a neutron star. [Recall that during core collapse the iron core (ashes of previous fusion reactions) is disintegrated into protons and neutrons, the protons combine with the surrounding electrons to make more neutrons, so the core becomes pure neutron matter. Because of this, core collapse can be halted if the core’s mass is between 1.4 (the Chandrasekhar limit) and about 3-4 solar masses, by neutron degeneracy.] What do you get if the core mass is less than 1.4 solar masses? Greater than 3-4 solar masses? 22.1 Neutron Stars Neutron stars, although they have 1–3 solar masses, are so dense that they are very small. -
Radio Observations of the Supermassive Black Hole at the Galactic Center and Its Orbiting Magnetar
Radio Observations of the Supermassive Black Hole at the Galactic Center and its Orbiting Magnetar Rebecca Rimai Diesing Honors Thesis Department of Physics and Astronomy Northwestern University Spring 2017 Honors Thesis Advisor: Farhad Zadeh ! Radio Observations of the Supermassive Black Hole at the Galactic Center and its Orbiting Magnetar Rebecca Rimai Diesing Department of Physics and Astronomy Northwestern University Honors Thesis Advisor: Farhad Zadeh Department of Physics and Astronomy Northwestern University At the center of our galaxy a bright radio source, Sgr A*, coincides with a black hole four million times the mass of our sun. Orbiting Sgr A* at a distance of 3 arc seconds (an estimated 0.1 pc) and rotating with a period of 3.76 s is a magnetar, or pulsar⇠ with an extremely strong magnetic field. This magnetar exhibited an X-ray outburst in April 2013, with enhanced, highly variable radio emission detected 10 months later. In order to better understand the behavior of Sgr A* and the magnetar, we study their intensity variability as a function of both time and frequency. More specifically, we present the results of short (8 minute) and long (7 hour) radio continuum observations, taken using the Jansky Very Large Array (VLA) over multiple epochs during the summer of 2016. We find that Sgr A*’s flux density (a proxy for intensity) is highly variable on an hourly timescale, with a frequency dependence that di↵ers at low (34 GHz) and high (44 GHz) frequencies. We also find that the magnetar remains highly variable on both short (8 min) and long (monthly) timescales, in agreement with observations from 2014. -
Evolution of the Cosmological Horizons in a Concordance Universe
Evolution of the Cosmological Horizons in a Concordance Universe Berta Margalef–Bentabol 1 Juan Margalef–Bentabol 2;3 Jordi Cepa 1;4 [email protected] [email protected] [email protected] 1Departamento de Astrofísica, Universidad de la Laguna, E-38205 La Laguna, Tenerife, Spain: 2Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. 3Facultad de Ciencias Físicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. 4Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain. Abstract The particle and event horizons are widely known and studied concepts, but the study of their properties, in particular their evolution, have only been done so far considering a single state equation in a deceler- ating universe. This paper is the first of two where we study this problem from a general point of view. Specifically, this paper is devoted to the study of the evolution of these cosmological horizons in an accel- erated universe with two state equations, cosmological constant and dust. We have obtained closed-form expressions for the horizons, which have allowed us to compute their velocities in terms of their respective recession velocities that generalize the previous results for one state equation only. With the equations of state considered, it is proved that both velocities remain always positive. Keywords: Physics of the early universe – Dark energy theory – Cosmological simulations This is an author-created, un-copyedited version of an article accepted for publication in Journal of Cosmology and Astroparticle Physics. IOP Publishing Ltd/SISSA Medialab srl is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. -
BLACK HOLES: the OTHER SIDE of INFINITY General Information
BLACK HOLES: THE OTHER SIDE OF INFINITY General Information Deep in the middle of our Milky Way galaxy lies an object made famous by science fiction—a supermassive black hole. Scientists have long speculated about the existence of black holes. German astronomer Karl Schwarzschild theorized that black holes form when massive stars collapse. The resulting gravity from this collapse would be so strong that the matter would become more and more dense. The gravity would eventually become so strong that nothing, not even radiation moving at the speed of light, could escape. Schwarzschild’s theories were predicted by Einstein and then borne out mathematically in 1939 by American astrophysicists Robert Oppenheimer and Hartland Snyder. WHAT EXACTLY IS A BLACK HOLE? First, it’s not really a hole! A black hole is an extremely massive concentration of matter, created when the largest stars collapse at the end of their lives. Astronomers theorize that a point with infinite density—called a singularity—lies at the center of black holes. SO WHY IS IT CALLED A HOLE? Albert Einstein’s 1915 General Theory of Relativity deals largely with the effects of gravity, and in essence predicts the existence of black holes and singularities. Einstein hypothesized that gravity is a direct result of mass distorting space. He argued that space behaves like an invisible fabric with an elastic quality. Celestial bodies interact with this “fabric” of space-time, appearing to create depressions termed “gravity wells” and drawing nearby objects into orbit around them. Based on this principle, the more massive a body is in space, the deeper the gravity well it will create.