Gravitational Lensing with a Large Deflection Angle as a Probe of General Relativity and the

202 Pages·2016·1.19 MB·English

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Publicly Accessible Penn Dissertations
Fall 2010
GGrraavviittaattiioonnaall LLeennssiinngg wwiitthh aa LLaarrggee DDeeflfleeccttiioonn AAnnggllee aass aa PPrroobbee ooff
GGeenneerraall RReellaattiivviittyy aanndd tthhee GGaallaaccttiicc CCeenntteerr
Amitai Yisrael Bin-Nun
University of Pennsylvania, [email protected]
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Part of the Cosmology, Relativity, and Gravity Commons, and the Elementary Particles and Fields and
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RReeccoommmmeennddeedd CCiittaattiioonn
Bin-Nun, Amitai Yisrael, "Gravitational Lensing with a Large Deflection Angle as a Probe of General
Relativity and the Galactic Center" (2010). Publicly Accessible Penn Dissertations. 252.
https://repository.upenn.edu/edissertations/252
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/252
For more information, please contact [email protected].
GGrraavviittaattiioonnaall LLeennssiinngg wwiitthh aa LLaarrggee DDeeflfleeccttiioonn AAnnggllee aass aa PPrroobbee ooff GGeenneerraall
RReellaattiivviittyy aanndd tthhee GGaallaaccttiicc CCeenntteerr
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Gravitational lensing is an important tool for the study of gravity. In this thesis, we use gravitational
lensing in the strong field to study a variety of phenomena. We begin with an overview of gravitational
lensing in the weak and strong deflection limits, including a formalism for the study of light that passes
close enough to a black hole to loop around it several times before reaching the observer. We move on to
discuss recent developments in the study of ``braneworld" models which present an interesting
framework for the effect of extra dimensions on gravity. We also discuss several potential black hole
metrics in the Randall-Sundrum II braneworld model. We then numerically study a variety of lensing
scenarios involving braneworld black holes, including a new form of the ``tidal Reissner-Nordstrom"
metric and find that a braneworld metric will produce results theoretically differentiable from a
Schwarzschild black hole. The analytical formalism we review is found to be an accurate reproduction of
the numerical results. We outline a test for the application of this analytical formalism to an arbitrary
static, spherically symmetric spacetime.
We then study the effects of gravitational lensing on the S stars orbiting Sgr A* in the galactic center. We
show that modifying the metric for the black hole at Sgr A* will produce different image properties for the
lensed S stars. We catalogue these image properties as a function of the metric and comment on the
observational prospects for these images and the specifics of their properties. Finally, we suppose that
the dark mass at the galactic center is a boson star and offer evidence that this will create observationally
significant lensing events for nearby stars.
DDeeggrreeee TTyyppee
Dissertation
DDeeggrreeee NNaammee
Doctor of Philosophy (PhD)
GGrraadduuaattee GGrroouupp
Physics & Astronomy
FFiirrsstt AAddvviissoorr
Justin Khoury
KKeeyywwoorrddss
Gravitational Lensing, Cosmology, Black Hole, Galactic Center, Branes, Boson Star
SSuubbjjeecctt CCaatteeggoorriieess
Cosmology, Relativity, and Gravity | Elementary Particles and Fields and String Theory
This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/252
Gravitational Lensing with a Large Deflection Angle as a Probe of
General Relativity and the Galactic Center
Amitai Yisrael Bin-Nun
A DISSERTATION
in
Physics and Astronomy
Presented to the Faculties of the University of Pennsylvania in Partial
Fulfillmentof theRequirementsforthe Degree of Doctor of Philosophy
2010
Justin Khoury Ravi Sheth
Supervisor of Dissertation Graduate Group Chairman
Dissertation Committee
Burt Ovrut, Professor of Physics
Gary Bernstein, Professor of Physics
Bhuvnesh Jain, Professor of Physics
Dedication
To my parents, who have always expected nothing less than my best, and serve as
an inspiration to keep trying.
To my Chanie, who expects me to keep trying, and is an inspiration to give my best.
To you, I dedicate my thesis.
ii
Acknowledgments
There is no way I could have completed this thesis and a PhD without the flexi-
bility, assistance, and understanding of the Associate Chair for Graduate Affairs,
Ravi Sheth. He has helped smooth the way and made done so much to ensure
my success. I would like to thank my advisor, Justin Khoury, for his invaluable
assistance and direction he has given my work and doing his best to keep me to
high standards. I would like to thank Burt Ovrut, Bhuvnesh Jain, and Gary Bern-
stein, my committee members, for reading my thesis as well as for offering helpful
advice, both scientific and professional. I would like to recognize Kumar Shwetketu
Virbhadra, who ignited my interest in lensing with large angles, introduced me to
the field, took a great deal of time to get me started on this path, and gave many
helpful suggestions in terms of research direction. Most of all, I would like to thank
him for his passionate and inspirational vision of the physicist’s dedication to his
craft.
I would like to thank Adam Lidz, Masao Sako, Mark Devlin, and Mark Trodden for
their helpful conversations, teaching me, and for making the fourth floor of DRL a
iii
pleasant place. I would like to thank my teachers Andrea Liu, Phil Nelson, Charles
Kane, Mark Goulian, Randy Kamien, Masao Sako, and Bhuvnesh Jain for sharing
their knowledge in the courses they taught. I would have never made it to Penn in
the first place without my mentors from my undergraduate days: Gabriel Cwilich,
Fredy Zypman, Lenny Tevlin, and Igor Kaganovich.
Life would not be the same without the companionship of my fellow members of
the windowless office of 4N5A, so I salute Joey Hyde, Michelle Caler, Jorge Moreno,
Andr´es Alejandro Plazas Malag´on, Alex Borisov, Matt Martino, Melinda Andrews,
Joseph Clampitt, and Alan Meert. I would also like to thank my fellow graduate
students Chris D’Andrea, Elizabeth Libby, Mike Ambroso, and Jennifer Mosher for
their friendship. Thank you also to Jacek Guzik, a former postdoc at Penn and a
good friend.
In my life outside of physics, I’d like to thank my roommates Josh Meisner, Nick
Jackson, Alex Phillips, DavidWise, Andew Lippa, and Marc Herman for being a di-
version from the constant demands of graduate study. I would also like to recognize
a few long time friends for their advice and patience. Thank you Burry Green-
baum, Yair Hindin, Ely Perlmutter, Simon Basalely, Didi Saiman, Ari Levine, Ariel
Fisher, ZachBodnar, EliSprecher, ErezHarrari,Tzachi Fried, YiskaDanziger, Ariel
Bayewitz, Ari Derdik, Brian Zawiski, and Shmuel Kadosh (the last is especially rec-
ognized for providing hosting during my frequent weekend “scholarly” visits to New
York City).
iv
I could not do justice to the support I receive from my family. Their passionate
commitment to scholarship and truthis the background from which Icome. I would
like to thank my siblings, Yair, Rafi, Chananya, Avrumie, Racheli, Yedidya, and
Ruchama for their support throughout the years. I also need to thank all of my
nieces and nephews, especially Avi, Dani, Zevi, Azaria, JoJo, Chana, and Yehuda
Mordechai, for being such fun playmates and asking important questions about sci-
ence. I would be completely remiss if I did not thank my parents, Rabbi Dov and
Tziona Bin-Nun, who serve as role models of the highest order, always giving their
best and working hard to succeed no matter the odds. What I have accomplished
rests on their sacrifice and support.
Finally, I would like to thank my dear wife Chanie, who has kindly put up with the
many nights in the office that this thesis entailed. Your love, patience, understand-
ing, and giving let me achieve.
v
ABSTRACT
Gravitational Lensing with a Large Deflection Angle as a Probe of General
Relativity and the Galactic Center
Amitai Yisrael Bin-Nun
Justin Khoury
Gravitational lensing is an important tool for the study of gravity. In this thesis,
we use gravitational lensing in the strong field to study a variety of phenomena. We
begin with an overview of gravitational lensing in the weak and strong deflection
limits, includingaformalismforthestudyoflightthatpassescloseenoughtoablack
hole to loop around it several times before reaching the observer. We move on to
discuss recent developments in the study of “braneworld” models which present an
interesting framework for the effect of extra dimensions on gravity. We also discuss
several potential black hole metrics in the Randall-Sundrum II braneworld model.
We then numerically study a variety of lensing scenarios involving braneworld black
holes, including a new form of the “tidal Reissner-Nordstrom” metric and find
that a braneworld metric will produce results theoretically differentiable from a
Schwarzschild black hole. The analytical formalism we review is found to be an
accurate reproduction of the numerical results. We outline a test for the application
of this analytical formalism to an arbitrary static, spherically symmetric spacetime.
We then study the effects of gravitational lensing on the S stars orbiting Sgr A* in
vi
the galactic center. We show that modifying the metric for the black hole at Sgr A*
will produce different image properties for the lensed S stars. We catalogue these
image properties as a function of the metric and comment on the observational
prospects for these images and the specifics of their properties. Finally, we suppose
that the dark mass at the galactic center is a boson star and offer evidence that
this will create observationally significant lensing events for nearby stars.
vii
Contents
1 Introduction 1
1.1 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 The Weak Field Limit . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Strong Deflection Limit . . . . . . . . . . . . . . . . . . . . . 16
1.2 Extra Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2.1 Universal Extra Dimensions . . . . . . . . . . . . . . . . . . 28
1.2.2 Braneworld . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3 Randall-Sundrum Braneworld . . . . . . . . . . . . . . . . . . . . . 35
1.3.1 Randall-Sundrum I . . . . . . . . . . . . . . . . . . . . . . . 36
1.3.2 Randall-Sundrum II . . . . . . . . . . . . . . . . . . . . . . 41
1.3.3 Black Hole Metrics in the Randall-Sundrum Braneworld . . 43
1.4 The Black Hole at the Center of the Galaxy . . . . . . . . . . . . . 52
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 Relativistic Images as a Probe of Modified Gravity 56
2.1 Gravitational Lensing in the Strong Deflection Limit . . . . . . . . 56
2.1.1 Numerical Approach to SDL Lensing . . . . . . . . . . . . . 62
2.1.2 Black Hole Lensing in the Braneworld . . . . . . . . . . . . 69
2.1.3 Primordial Black Holes . . . . . . . . . . . . . . . . . . . . . 79
2.1.4 Magnification in the Braneworld . . . . . . . . . . . . . . . . 83
2.2 Testing the SDL Approximation . . . . . . . . . . . . . . . . . . . 88
2.3 Comparison of SDL Bending Angles and Image Magnitudes . . . . 91
2.3.1 The Black Hole at Sgr A* . . . . . . . . . . . . . . . . . . . 93
2.3.2 Primordial Black Hole . . . . . . . . . . . . . . . . . . . . . 95
2.3.3 Miniature Black Hole . . . . . . . . . . . . . . . . . . . . . . 99
3 Secondary Images of S Stars as a Probe of Modified Gravity in the
Galactic Center 103
3.1 Lensing of S Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2.1 S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2.2 S14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
viii