From ad7aef821b654e54347c4a574070b0a4d99fda84 Mon Sep 17 00:00:00 2001 From: Rahul Kashyap Date: Thu, 22 Feb 2024 15:22:35 -0500 Subject: [PATCH] updated redirect notice to new github.io --- index.html | 15 +++++++++------ 1 file changed, 9 insertions(+), 6 deletions(-) diff --git a/index.html b/index.html index 2eae056..796d943 100644 --- a/index.html +++ b/index.html @@ -472,6 +472,8 @@
  • rahulkashyap411_githubIO_published
    • +
  • Redirect
    • +
  • Rendall 2008
  • Research Highlights
    • @@ -601,7 +603,7 @@ {"title":"$:/core/ui/SideBar/Recent","text":"\u003C$macrocall $name=\"timeline\" format={{$:/language/RecentChanges/DateFormat}}/>\n","caption":"Recent","created":"20210721044754002","creator":"Rahul Kashyap","is-dropdown":"yes","modified":"20210727164130313","modifier":"rkashyap","tags":"$:/tags/SideBar"}, {"title":"$:/core/ui/SideBar/Tools","text":"\\define lingo-base() $:/language/ControlPanel/\n\\define config-title()\n$:/config/PageControlButtons/Visibility/$(listItem)$\n\\end\n\n\u003C\u003Clingo Basics/Version/Prompt>> \u003C\u003Cversion>>\n\n\u003C$set name=\"tv-config-toolbar-icons\" value=\"yes\">\n\n\u003C$set name=\"tv-config-toolbar-text\" value=\"yes\">\n\n\u003C$set name=\"tv-config-toolbar-class\" value=\"\">\n\n\u003C$list filter=\"[all[shadows+tiddlers]tag[$:/tags/PageControls]!has[draft.of]]\" variable=\"listItem\">\n\n\u003Cdiv style=\"position:relative;\" class={{{ [\u003ClistItem>encodeuricomponent[]addprefix[tc-btn-]] }}}>\n\n\u003C$checkbox tiddler=\u003C\u003Cconfig-title>> field=\"text\" checked=\"show\" unchecked=\"hide\" default=\"show\"/> \u003C$transclude tiddler=\u003C\u003ClistItem>>/> \u003Ci class=\"tc-muted\">\u003C$transclude tiddler=\u003C\u003ClistItem>> field=\"description\"/>\u003C/i>\n\n\u003C/div>\n\n\u003C/$list>\n\n\u003C/$set>\n\n\u003C/$set>\n\n\u003C/$set>\n","caption":"{{$:/language/SideBar/Tools/Caption}}","created":"20210721041246345","creator":"Rahul Kashyap","modified":"20210726023728655","modifier":"rkashyap","tags":"rahul_private $:/tags/SideBar"}, {"title":"$:/core/ui/ViewTemplate/tags","text":"\u003C$reveal type=\"nomatch\" stateTitle=\u003C\u003Cfolded-state>> text=\"hide\" tag=\"div\" retain=\"yes\" animate=\"yes\">\n\u003Cdiv class=\"tc-tags-wrapper\">\u003C$list filter=\"[all[current]tags[]sort[title]] -[enlist{$:/config/HideTags}]\" template=\"$:/core/ui/TagTemplate\" storyview=\"pop\"/>\u003C/div>\n\u003C/$reveal>\n","created":"20210726014052460","creator":"rkashyap","modified":"20210726014054933","modifier":"rkashyap","tags":"$:/tags/ViewTemplate"}, -{"title":"$:/DefaultTiddlers","text":"About","created":"20210106062508083","creator":"Rahul Kashyap","modified":"20210726024732310","modifier":"rkashyap"}, +{"created":"20210106062508083","creator":"Rahul Kashyap","title":"$:/DefaultTiddlers","text":"Redirect","modified":"20240222201828275","modifier":"rkashyap"}, {"title":"$:/favorites/favlist","text":"","created":"20210721040952164","creator":"Rahul Kashyap","list":"[[TOV and Tidal Deformability of Neutron Stars]]","modified":"20210728032212238","modifier":"rkashyap"}, {"title":"$:/favorites/folder-002","text":"","caption":"Physics Posts","created":"20210721163420789","creator":"rkashyap","list":"","modified":"20210725125555465","modifier":"rkashyap","tags":"$:/tags/Favorites/Folder"}, {"title":"$:/Import","text":"The following tiddlers were imported:\n\n# [[Tolman1939-jb]]","status":"complete"}, @@ -648,8 +650,8 @@ {"created":"20231025193323619","creator":"rkashyap","title":"$:/state/Link/--678957325/search","text":"","modified":"20231025193326600","modifier":"rkashyap"}, {"created":"20220609080115514","creator":"rkashyap","title":"$:/state/Link/--69918751/search","text":"","modified":"20220609080120084","modifier":"rkashyap"}, {"created":"20211006165350005","creator":"rkashyap","title":"$:/state/showeditpreview","text":"yes","modified":"20220921154218473","modifier":"rkashyap"}, -{"created":"20160725112947956","creator":"TonGerner","text":"no","title":"$:/state/sidebar","modifier":"rkashyap","modified":"20230123182901269"}, -{"created":"20211122152912534","creator":"rkashyap","title":"$:/state/tab-1749438307","text":"$:/core/ui/ControlPanel/Plugins","modified":"20211122152912534","modifier":"rkashyap"}, +{"created":"20160725112947956","creator":"TonGerner","text":"no","title":"$:/state/sidebar","modifier":"rkashyap","modified":"20240222201923140"}, +{"created":"20211122152912534","creator":"rkashyap","title":"$:/state/tab-1749438307","text":"$:/core/ui/ControlPanel/Info","modified":"20240222201816352","modifier":"rkashyap"}, {"created":"20211006165819856","creator":"rkashyap","title":"$:/state/tab/moresidebar-1718121225","text":"$:/core/ui/MoreSideBar/Plugins","modified":"20211006165819856","modifier":"rkashyap"}, {"created":"20211006165817333","creator":"rkashyap","title":"$:/state/tab/sidebar--1435920229","text":"$:/plugins/kookma/refnotes/ui/bibtexlibrary","modified":"20220921153656245","modifier":"rkashyap"}, {"created":"20211125021830964","creator":"rkashyap","title":"$:/state/toc/Lecture Notes-An Introduction to General Relativity-1605685012","text":"close","modified":"20220921233921708","modifier":"rkashyap"}, @@ -658,7 +660,7 @@ {"created":"20220921231354437","creator":"rkashyap","title":"$:/state/toc/TableOfContents-Lecture Notes-263827834","text":"close","modified":"20220922053804640","modifier":"rkashyap"}, {"title":"$:/status/RequireReloadDueToPluginChange","text":"no"}, {"title":"$:/status/UserName","text":"rkashyap","created":"20210106081059820","creator":"r","modified":"20210721162329576","modifier":"rakashyap"}, -{"title":"$:/StoryList","created":"20231025192205148","creator":"rkashyap","text":"","list":"About","modified":"20231025193427333","modifier":"rkashyap"}, +{"title":"$:/StoryList","text":"","list":"Redirect"}, {"title":"$:/tags/Favorites/Folder","text":"","created":"20210721163420792","creator":"rkashyap","list":"$:/favorites/folder-002","modified":"20210721163420792","modifier":"rkashyap"}, {"title":"$:/theme","text":"$:/themes/tiddlywiki/snowwhite","created":"20210106083031677","creator":"Rahul Kashyap","modified":"20210106083031677","modifier":"Rahul Kashyap"}, {"title":"$:/themes/tiddlywiki/snowwhite","name":"Snow White","author":"JeremyRuston","core-version":">=5.0.0","plugin-type":"theme","description":"Emphasises individual tiddlers","dependents":"$:/themes/tiddlywiki/vanilla","plugin-priority":"0","version":"5.2.1-prerelease","type":"application/json","text":"{\"tiddlers\":{\"$:/themes/tiddlywiki/snowwhite/base\":{\"title\":\"$:/themes/tiddlywiki/snowwhite/base\",\"tags\":\"[[$:/tags/Stylesheet]]\",\"text\":\"\\\\define sidebarbreakpoint-minus-one()\\n\u003C$text text={{{ [{$:/themes/tiddlywiki/vanilla/metrics/sidebarbreakpoint}removesuffix[px]subtract[1]addsuffix[px]] ~[{$:/themes/tiddlywiki/vanilla/metrics/sidebarbreakpoint}] }}}/>\\n\\\\end\\n\\n\\\\rules only filteredtranscludeinline transcludeinline macrodef macrocallinline\\n\\n.tc-sidebar-header {\\n\\ttext-shadow: 0 1px 0 \u003C\u003Ccolour sidebar-foreground-shadow>>;\\n}\\n\\n.tc-tiddler-info {\\n\\t\u003C\u003Cbox-shadow \\\"inset 1px 2px 3px rgba(0,0,0,0.1)\\\">>\\n}\\n\\n@media screen {\\n\\t.tc-tiddler-frame {\\n\\t\\t\u003C\u003Cbox-shadow \\\"1px 1px 5px rgba(0, 0, 0, 0.3)\\\">>\\n\\t}\\n}\\n\\n@media (max-width: \u003C\u003Csidebarbreakpoint-minus-one>>) {\\n\\t.tc-tiddler-frame {\\n\\t\\t\u003C\u003Cbox-shadow none>>\\n\\t}\\n}\\n\\n.tc-page-controls button svg, .tc-tiddler-controls button svg, .tc-topbar button svg {\\n\\t\u003C\u003Ctransition \\\"fill 150ms ease-in-out\\\">>\\n}\\n\\n.tc-tiddler-controls button.tc-selected,\\n.tc-page-controls button.tc-selected {\\n\\t\u003C\u003Cfilter \\\"drop-shadow(0px -1px 2px rgba(0,0,0,0.25))\\\">>\\n}\\n\\n.tc-tiddler-frame input.tc-edit-texteditor {\\n\\t\u003C\u003Cbox-shadow \\\"inset 0 1px 8px rgba(0, 0, 0, 0.15)\\\">>\\n}\\n\\n.tc-edit-tags {\\n\\t\u003C\u003Cbox-shadow \\\"inset 0 1px 8px rgba(0, 0, 0, 0.15)\\\">>\\n}\\n\\n.tc-tiddler-frame .tc-edit-tags input.tc-edit-texteditor {\\n\\t\u003C\u003Cbox-shadow \\\"none\\\">>\\n\\tborder: none;\\n\\toutline: none;\\n}\\n\\ntextarea.tc-edit-texteditor {\\n\\tfont-family: {{$:/themes/tiddlywiki/vanilla/settings/editorfontfamily}};\\n}\\n\\ncanvas.tc-edit-bitmapeditor {\\n\\t\u003C\u003Cbox-shadow \\\"2px 2px 5px rgba(0, 0, 0, 0.5)\\\">>\\n}\\n\\n.tc-drop-down {\\n\\tborder-radius: 4px;\\n\\t\u003C\u003Cbox-shadow \\\"2px 2px 10px rgba(0, 0, 0, 0.5)\\\">>\\n}\\n\\n.tc-block-dropdown {\\n\\tborder-radius: 4px;\\n\\t\u003C\u003Cbox-shadow \\\"2px 2px 10px rgba(0, 0, 0, 0.5)\\\">>\\n}\\n\\n.tc-modal {\\n\\tborder-radius: 6px;\\n\\t\u003C\u003Cbox-shadow \\\"0 3px 7px rgba(0,0,0,0.3)\\\">>\\n}\\n\\n.tc-modal-footer {\\n\\tborder-radius: 0 0 6px 6px;\\n\\t\u003C\u003Cbox-shadow \\\"inset 0 1px 0 #fff\\\">>;\\n}\\n\\n\\n.tc-alert {\\n\\tborder-radius: 6px;\\n\\t\u003C\u003Cbox-shadow \\\"0 3px 7px rgba(0,0,0,0.6)\\\">>\\n}\\n\\n.tc-notification {\\n\\tborder-radius: 6px;\\n\\t\u003C\u003Cbox-shadow \\\"0 3px 7px rgba(0,0,0,0.3)\\\">>\\n\\ttext-shadow: 0 1px 0 rgba(255,255,255, 0.8);\\n}\\n\\n.tc-sidebar-lists .tc-tab-set .tc-tab-divider {\\n\\tborder-top: none;\\n\\theight: 1px;\\n\\t\u003C\u003Cbackground-linear-gradient \\\"left, rgba(0,0,0,0.15) 0%, rgba(0,0,0,0.0) 100%\\\">>\\n}\\n\\n.tc-more-sidebar > .tc-tab-set > .tc-tab-buttons > button {\\n\\t\u003C\u003Cbackground-linear-gradient \\\"left, rgba(0,0,0,0.01) 0%, rgba(0,0,0,0.1) 100%\\\">>\\n}\\n\\n.tc-more-sidebar > .tc-tab-set > .tc-tab-buttons > button.tc-tab-selected {\\n\\t\u003C\u003Cbackground-linear-gradient \\\"left, rgba(0,0,0,0.05) 0%, rgba(255,255,255,0.05) 100%\\\">>\\n}\\n\\n.tc-message-box img {\\n\\t\u003C\u003Cbox-shadow \\\"1px 1px 3px rgba(0,0,0,0.5)\\\">>\\n}\\n\\n.tc-plugin-info {\\n\\t\u003C\u003Cbox-shadow \\\"1px 1px 3px rgba(0,0,0,0.5)\\\">>\\n}\\n\"}}}"}, @@ -671,7 +673,7 @@ {"title":"$:/themes/tiddlywiki/vanilla/options/sidebarlayout","text":"fixed-fluid","created":"20210718205058855","creator":"Rahul Kashyap","modified":"20210721042620603","modifier":"Rahul Kashyap"}, {"title":"$:/view","text":"zoomin","created":"20210106083015091","creator":"Rahul Kashyap","modified":"20210721043007207","modifier":"Rahul Kashyap"}, {"created":"20210726023957827","creator":"rkashyap","text":"Binary white dwarfs are the natural end state of binary stellar evolution for stars with initial masses between 0.4 to 8 times mass of our sun (M$$ _\\odot$$). With our collaborators, I have worked out a detailed investigation of the end state of such binary systems.\n\nThe initial state has been simulated by our collaborators, Pablo Loren Aguilar et al using SPH methods. Here is a video produced by them -- \n\n\u003Ciframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/bHfOWjf8z7Y\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen>\u003C/iframe>\n\n As shown here, in most of the simulations by other groups as well, the post-merger state has the similar outcomes - the more massive white dwarf surrounded by a thick disk formed from the disruption of the secondary white dwarf. The question of the fate of such post-merger state has been largely unexplored so far which we have followed using 3D hydrodynamical simulations for a range of binary white dwarf systems. We have found that spiral gravitational instability makes the disk unstable resulting in transport of angular momentum outward on dynamical timescales. This causes hot disk matter to mix with the degenerate material of the white dwarf. This mixing along with possible turbulent heating causes the degenerate carbon-oxygen mixture to detonate (\u003C\u003Cref [[Kashyap2015-zq]]>>). Below is a movie showing the mid-plane density of 3D simulation of 1.1 M$$ _\\odot$$ and 1 M$$ _\\odot$$ white dwarf merger and it's subsequent detonation to form an SN Ia.\n \n \u003Ciframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/-HgPvPckLEI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen>\u003C/iframe>\n\nFurther, radiative transfer calculations of the explosion model shows similarities of such exploding systems with SN 2002ay, a type Ia supernova (SN Ia) which earlier required exotic possibilities (\u003C\u003Cref [[Van_Rossum2016-oh]]>>). My analysis of a range of white dwarf masses in the binary shows similar physical effects.\n\nRelevant peer-reviewed publications are (Astrophysical Journal Letters and Astrophysical Journal) \u003C\u003Cref [[Kashyap2017-fh]]>>. \n\n*https://doi.org/10.1088/2041-8205/800/1/L7\n*https://doi.org/10.3847/0004-637X/827/2/128\n*https://doi.org/10.3847/1538-4357/aa6afb\n\nWe have also simulated a heavier binary WD system (with 1.2M$$ _\\odot$$ Oxygen-Neon (ONe) and 1.1M$$ _\\odot$$ Carbon-Oxygen (CO)) which does not lead to the complete disruption but, only a carbon flash. We believe this system might be a progenitor of a faint SN Iax. This system could produce electromagnetic transients if it goes another episode of carbon outburst. Since, it is quite difficult to detonate ONe WD, this system could collapse to form a neutron star and act as a source of also a of gravitational wave burst and an electromagnetic transient \u003C\u003Cref [[Kashyap2018-vi]]>>. The findings are accepted for publication in ApJ -- \nhttps://doi.org/10.3847/1538-4357/aaedb7 \n\nBinary white dwarfs with thin He layer on one or both have been proposed as a progenitor of SN Ia. We looked at such a system where we find a deflagration wave in He layer but, fails to detonate the carbon-oxygen core. Our findings are consistent with others in the literature. We are looking for other interesting physics active in this system.\n\n\u003C\u003Cshowrefs title:\"References:\">>","title":"A Stellar Graveyard: Binary White Dwarf Mergers","modified":"20220922143456349","modifier":"rkashyap","tags":"[[Research Highlights]] [[White Dwarfs]] [[Binary White Dwarf Mergers]]"}, -{"created":"20210718200500108","creator":"Rahul Kashyap","text":"!Hello and Welcome.\n\n\u003C\u003Cimage-pretty img:\"images/rahul_Dec2018.JPG\" width:\"200px\" align:\"right\" tooltip:\"demo of image-pretty macro\">>\n\nMy name is Rahul Kashyap. I study astrophysical objects using both, computational and analytical methods. I work on hydrodynamical simulations of binary white dwarf (BWD) and binary neutron star (BNS) mergers and transients obtained from them. I am also interested in combining information from gravitational wave data analysis and several physical constraints obtained from them. I am a member of LIGO-Virgo collaboration participating in enriching our astrophysical knowledge by performing hydrodynamical simulations as well as gravitational-wave data analysis. More specifically, I am working on hydrodynamical modeling of the kilonova ejecta and astrophysical implications from parametric Bayesian inferences using individual gravitational wave sources.\n\n Currently, I am a Postdoctoral Scholar, in the Eberly College of Science, Department of Physics, Pennsylvania State University, USA. I work with the PAX group led by by [[Prof. Bangalore Sathyaprakash|https://sites.psu.edu/infinity/]] and Numerical Relativity group (Please visit group site — [[Rahul Kashyap – NumRel@PSU|https://sites.psu.edu/numrel/members/rahul-kashyap/]] to know more about numerical relativity group at Penn State led by [[Prof. David Radice|https://sites.psu.edu/numrel/members/david-radice/]]). Prior to Penn State, I was working with [[Prof. Parameswaran Ajith|https://home.icts.res.in/~ajith/Home.html]] as Max Planck postdoctoral fellow at International Centre for Theoretical Sciences, Bengaluru, India.\n\n\n\nI have obtained my Ph. D. in Computational Astrophysics from UMASS Dartmouth with [[Prof. Robert Fisher|https://sites.google.com/site/fishercompgroup]] where I discovered a new possible mechanism for the formation of Type Ia supernovae from binary white dwarf mergers. You can see my Ph.D. thesis [[here|https://umassd.userservices.exlibrisgroup.com/discovery/delivery/01MA_DM_INST:umassd_library/1287406980001301?lang=en&viewerServiceCode=AlmaViewer]]. I am studying the evolution of post-merger binary WD merger evolution and their outcomes such as accretion-induced-collapse to form neutron stars.\n\nMy [[CV]] and [[publication list|Publication List]] are available if you are interested.\n\nI use this site to share my research work and some thoughts on topics of my other interests in the form of short infrequent posts. Send me an email at this address to talk about interesting natural philosophy i.e. modern science. My institute home page is here. ","title":"About","modified":"20220922053748726","modifier":"rkashyap","tags":"About [[Publications and CV]] rahulkashyap411_githubIO_published","tmap.id":"bb623368-a043-46a2-a415-711af26d98a4"}, +{"created":"20210718200500108","creator":"Rahul Kashyap","text":"> ''Please redirect yourself here -- '' https://rahulkashyap-phy.github.io/#About\n!Hello and Welcome.\n\n\u003C\u003Cimage-pretty img:\"images/rahul_Dec2018.JPG\" width:\"200px\" align:\"right\" tooltip:\"demo of image-pretty macro\">>\n\nMy name is Rahul Kashyap. I study astrophysical objects using both, computational and analytical methods. I work on hydrodynamical simulations of binary white dwarf (BWD) and binary neutron star (BNS) mergers and transients obtained from them. I am also interested in combining information from gravitational wave data analysis and several physical constraints obtained from them. I am a member of LIGO-Virgo collaboration participating in enriching our astrophysical knowledge by performing hydrodynamical simulations as well as gravitational-wave data analysis. More specifically, I am working on hydrodynamical modeling of the kilonova ejecta and astrophysical implications from parametric Bayesian inferences using individual gravitational wave sources.\n\n Currently, I am a Postdoctoral Scholar, in the Eberly College of Science, Department of Physics, Pennsylvania State University, USA. I work with the PAX group led by by [[Prof. Bangalore Sathyaprakash|https://sites.psu.edu/infinity/]] and Numerical Relativity group (Please visit group site — [[Rahul Kashyap – NumRel@PSU|https://sites.psu.edu/numrel/members/rahul-kashyap/]] to know more about numerical relativity group at Penn State led by [[Prof. David Radice|https://sites.psu.edu/numrel/members/david-radice/]]). Prior to Penn State, I was working with [[Prof. Parameswaran Ajith|https://home.icts.res.in/~ajith/Home.html]] as Max Planck postdoctoral fellow at International Centre for Theoretical Sciences, Bengaluru, India.\n\n\n\nI have obtained my Ph. D. in Computational Astrophysics from UMASS Dartmouth with [[Prof. Robert Fisher|https://sites.google.com/site/fishercompgroup]] where I discovered a new possible mechanism for the formation of Type Ia supernovae from binary white dwarf mergers. You can see my Ph.D. thesis [[here|https://umassd.userservices.exlibrisgroup.com/discovery/delivery/01MA_DM_INST:umassd_library/1287406980001301?lang=en&viewerServiceCode=AlmaViewer]]. I am studying the evolution of post-merger binary WD merger evolution and their outcomes such as accretion-induced-collapse to form neutron stars.\n\nMy [[CV]] and [[publication list|Publication List]] are available if you are interested.\n\nI use this site to share my research work and some thoughts on topics of my other interests in the form of short infrequent posts. Send me an email at this address to talk about interesting natural philosophy i.e. modern science. My institute home page is here. ","title":"About","modified":"20240222201757770","modifier":"rkashyap","tags":"About [[Publications and CV]] rahulkashyap411_githubIO_published","tmap.id":"bb623368-a043-46a2-a415-711af26d98a4"}, {"created":"20190506111639355","creator":"rkashyap","text":"\n> This is outline of a course on General Theory of Relativity organized by Institute of Advanced Research, Ahmedabad.\n\n\u003Cdiv class=\"nr\">\n\n! [[An Introduction to General Relativity]]\n\n!! Grades: \n\n\u003C\u003Calert info src:\"Tuesday and Friday morning, 8 AM\">>\n\n|Assignments | 50% |\n|Mid-sem exam | 20% |\n|End-sem exam | 30% |\n|Computational excercises | Python, Mathematica, SAGE, Jupyter notebook |\n\n\n!! Syllabus and references:\n\n!!! Week-1,2: ''Prerequisites''\n\n* Mathematical Methods: \n ** Kreyszig Ch-7,8,9,10; Appendix-A3.4\n# [[Newtonian Gravitation]]\n ## Gravity vs electrostatic force: Weak force but biggest player! \n ##* ''\"How heavy and how small\" is what matters!''\n ## Derivation and use of ''Poisson's equation''.\n\t## Two body problem in Newtonian gravity for the small body moving around massive object. Expressions for $$\\frac{dr}{dt}, \\frac{d\\phi}{dt}$$ (later it will be contrasted with results using relativistic gravity); Bound and unbound systems; Stability of orbits. \n ## Equations of hydrodynamics and Energy-momentum tensor in classical gravity: \n ##* Construction of a star under hydrostatic equillibrium. \n ## Virial theorem of collection of particles. \n\n# [[Special Relativity]]\n ## [Preclass]: Suggestion for scientific explanatory video: https://youtu.be/JmSV8mISWmU : World line of an object.\n ## Using Lorentz transformation to realize and calculate length contraction, time dilation and other special relativistic effects.\n ## Minkowski's idea of 4D spacetime geometry -- world line, events and line interval. Line elements in pseudo-Euclidean geometry.\n ## Spacelike, timelike and null intervals and its invariance as proper time-interval.\n ## Derivation of Lorentz boost in one direction and relation with the nature of hyperbolic geometry.\n ## Writing Lorentz transformation as matrix form.\n ## Principle of least action: writing Lagrangian for relativistic and non-relativistic systems.\n ##* importance of boundary conditions. \n ## Statements and examples of ''Lorentz Covariance''\n ## Please derive the classical equations of hydrodynamics starting from a more general conservation equation that is of ''energy-momentum'' tensor ($$T_{\\mu\\nu}$$) of relativistic flow with the definition of energy-momentum tensor as $$T_{\\mu\\nu} = (p+\\epsilon)u^\\mu u^\\nu + pg^{\\mu\\nu} $$ (''Why this form the most general for an ideal fluid?'') Show correspondence with classical notion of energy-momentum tensor i.e. mass density, internal energy density, momentum fluxes \u003C\u003Cref [[Poisson and Will 2014]]>>. \n\n> Classical Gravity and Special Relativity: Inconsistencies and resolution. \n\n!!! ''Week-3,4'': Origins of General Relativity\n\n #* [Preclass]: https://youtu.be/P11Txh5C1UI: lectures by Prof. G. Srinivasan.\n ## Using Galilean transformation to obtain transformation of elementary physical quantities such as velocity, acceleration, energy between two inertial reference frames. \n ##* ''How does equations change in non-inertial reference frames? Do they look same in two different accelerated reference frames?''\n \n ## ''Equivalence Principle:'' Equality between gravitational and inertial masses. Newton's argument and Einstein's extension as Weak Equivalence Principle. \n ##* Meaning of \"locally flat\". \n ## Writing line element (distance between two nearby points) in orthogonal systems such as Cartesian system: $$ds^2 = dx^2 + dy^2 + dz^2$$, cylindrical and spherical. Identifying all the components of metric tensor and variation of unit vector along a curve. \n ## Obtaining the components of metric tensor in cylindrical/spherical coordinates using metric tensor in Cartesian system and a given transformation law between these systems and vice-versa.\n ## Realizing the invariance of line elements in 3D.\n\n# General Covariance and Einstein's Equations\n ## Galiliean Relativity: Invariance of mass, charge and length and time intervals. Covariance of Physical laws (conservations laws) between inertial reference frames.\n ## [Revisit] Special Relativity: Invariance of speed of light, Covariant form of physical laws for inertial reference frames (''Lorentz Covariance'')\n ## Covariance in general: Need for tensor formulation of physical laws. \n ### Analysis on Manifolds \u003C\u003Cref [[Schutz 1985]]>>.\n ## Properties of Energy-momentum tensor, $$T_{\\mu\\nu}$$ from action.\n ##* Different types of idealized examples -- pressure-less dust, ideal fluid with isotropic pressure, radiation, electromagnetic field. \n\t##* Conservation law for energy-momentum tensor. \n ## \"Derivation\" of Einstein's Field equations: a ''historical note:'' https://arxiv.org/pdf/1608.05752.pdf \n## ''Hilbert-Einstein Action:'' Using Hamilton's action principle to derive Einstein's equation. \n\n ## [Additional Topics]''Setting up coordinate systems for an accelerated observers'' \n ## Derivation of Geodesic equation: \n \t##* [Preclass]: https://youtu.be/3NnZzRb7L58\n ## [Additional Topics] Uniqueness of Einstein tensor and $$f(R)$$ gravity.\n\n!!! Week-5,6,7,8:\n\n# Solutions of EE: \n #* [Preclass]: https://youtu.be/sEDFHMLPaW8 , https://youtu.be/HJlhBPci_Bg \n ## ; (see [Preclass] https://youtu.be/-UPSiKugRW0); given $$g_{\\mu\\nu}$$, find $$\\Gamma^{\\mu\\nu}_\\lambda, R_{\\mu \\nu \\sigma \\lambda}, R_{\\mu \\nu}, R$$. Complete few examples with pen-paper calculation and then using Mathematica notebooks. \n ### Invariant quantities constructed from the quantities and their significance in distinguishing physical and unphysical properties of spacetime\n\t### Calculate ''Kretschmann Scalar''.\n ## Starting with Einstein's equation with unknown constant, $$G^{\\mu \\nu}=\\kappa T^{\\mu \\nu}$$. In the weak field limit, $$g_{\\mu \\nu}=\\eta _{\\mu \\nu}+h_{\\mu \\nu}$$ with $$h_{\\mu \\nu} \u003C\u003C 1$$ and \n ### [Preclass]: https://youtu.be/JKQMre-bze4 \n ### $$T^{\\mu \\nu}=[\\rho c^2 ,0,0,0]$$ (slowly moving matter with no pressure) and compare with the Newtonian Gravity, $$\\nabla ^2 \\phi = 4 \\pi G \\rho$$ to find out the constant, $$\\kappa$$.\n\t### $$T^{\\mu \\nu}=[0,0,0,0]$$ (empty space solution `just like light!`): wave solutions. \n ## Start with the Schwarzschild metric in spherical coordinate system to show that it satisfies Einstein's equation in vaccum.\n ## Schwarszchild solution: \n ### ''methods of finding solutions'': \u003C\u003Cref [[Chandrasekhar 1983]]>>\n ### Physicality of a solution and Coordinate singularity. \n\t### Schwarzschild metric in different coordinates. \n\n ## Relativistic Stars \u003C\u003Cref [[Zeldovich and Novikov 1967]]>>\n ### Structure of a neutron star; ''TOV equation'' and Tidal Deformability.\n\t### Problem of equation of state. \n\t\n ## Kerr solution\n ### Slowly rotating Stars: approximate treatment in GR.\n \t### Ergoregion and dual-horizons. \n\t### Existence of maximum spin. \n## Evidence and realization of Schwarzschild and Kerr solution in nature. \n\n!!! ''Week- 9,10'':\n\n# Particle Path and Photon path in Schwarzchild and Kerr Geometry:\n ## [Pre-class]: Following \u003C\u003Cref [[Hartle 2003]]>>.\n ## Geodesic equation for particles.\n ## Bending of light \u003C\u003Cref [[Hartle 2003]]>>; Gravitational lensing and applications in astrophysics.\n\t## precession of perihelion \n\t## Shapiro delay\n\t## Tests of GR and future directions.\n\t\t### Gravitational Wave observations. \n \n!!! Overview of Relativistic Astrophysics \n\n# Blackholes, Neutron Stars and Cosmology\n ## accretion disks\n\t## Pulsars\n\t## Extreme mass-ratio binary black holes. \n\t## Binary compact mergers\n\t ### [Additional Topics] Wave solution produced by Post-Newtonian sources; solution as a Green's function given sources. ''Lecture by Prof. Bala Iyer, Prof. P. Ajith''.\n\t ### [Advanced] Analytical and numerical computation of waveforms: Hence, the need for Numerical Relativity which is possible only after we establish a well-posed initial value problem. (more on this later in [[Cauchy Problem in General Relativity]])\n\t\n!!! ''Week- 11,12'':\n\n# Relativistic Cosmology\n #* [Preclass]: https://youtu.be/HJlhBPci_Bg \n ## Homogeity and Isotropy of cosmos: an observational fact and Copernican principle. \n ## Form of metric for the universe; ''Friedman-Robertson-Walker'' Models. \n ### Constant curvature geoemtries: de Sitter model. \n ## Hubble Constant and its measurement. \n ## Cosmological Redshift and Thermal history; temperature of cosmic microwave background radiation. \n ## Homogeneous non-isotropic cosmologies: Bianchi models. \n \n# [Advanced] [[Cauchy Problem in General Relativity]]\n ## \u003C\u003Cref [[Wald 1974]]>> for an introduction. \n \n\n \n \n\u003C\u003Cshowrefs title:\"References\">>\n\n\u003C/div>","title":"An Introduction to General Relativity","list":"","modified":"20230809150337992","modifier":"rkashyap","tags":"TableOfContents [[Lecture Notes]] GeneralRelativity rahulkashyap411_githubIO_published","tmap.edges":"{\"a6a1c029-ceea-484a-9799-25f293366287\":{\"to\":\"f92d1372-abbc-4f1c-af60-4e68e6aece8f\",\"type\":\"rk_teachingModule\"},\"662312bd-55e8-40b7-aa9a-25335cc5ebec\":{\"to\":\"085c71ab-0ad5-4974-98be-7d366165e8f5\",\"type\":\"tmap:unknown\"},\"22b03737-9295-4a31-971c-bc8c04a6185d\":{\"to\":\"9933145f-8d88-44d8-a4cb-94e2dad4ad6b\",\"type\":\"tmap:unknown\"}}","tmap.id":"d1a3fac8-2097-4adf-b5d0-c7d7aed4befd","week":"0"}, {"created":"20211118133129210","creator":"Rahul Kashyap","title":"Bauswein:2013jpa","bibtex-entry-type":"ARTICLE","bibtex-title":"Prompt merger collapse and the maximum mass of neutron stars","bibtex-author":"Bauswein, A and Baumgarte, T W and Janka, H-T","bibtex-affiliation":"Max-Planck-Institut f{\\\"u}r Astrophysik,\n Karl-Schwarzschild-Strasse 1, D-85748 Garching, Germany and\n Department of Physics, Aristotle University of Thessaloniki,\n GR-54124 Thessaloniki, Greece.","bibtex-abstract":"We perform hydrodynamical simulations of neutron-star mergers\n for a large sample of temperature-dependent nuclear equations\n of state and determine the threshold mass above which the\n merger remnant promptly collapses to form a black hole. We\n find that, depending on the equation of state, the threshold\n mass is larger than the maximum mass of a nonrotating star in\n isolation by between 30 and 70 percent. Our simulations also\n show that the ratio between the threshold mass and maximum\n mass is tightly correlated with the compactness of the\n nonrotating maximum-mass configuration. We speculate on how\n this relation can be used to derive constraints on\n neutron-star properties from future observations.","bibtex-journal":"Phys. Rev. Lett.","bibtex-volume":"111","bibtex-number":"13","bibtex-pages":"131101","bibtex-month":"sep","bibtex-year":"2013","bibtex-url":"http://dx.doi.org/10.1103/PhysRevLett.111.131101","bibtex-file":"All_Papers/2013,Starred_Papers/Bauswein_et_al._2013_-_Prompt_merger_collapse_and_the_maximum_mass_of_neutron_stars.pdf","bibtex-keywords":"Binary Neutron\n Star;GravitationalCollapse;BNS\\_Prompt\\_BH\\_Collapse","bibtex-language":"en","bibtex-issn":"0031-9007, 1079-7114","bibtex-pmid":"24116763","bibtex-doi":"10.1103/PhysRevLett.111.131101","modified":"20211118133205275","modifier":"Rahul Kashyap","tags":"$:/tags/Commander/Working","import_date":"18.Nov.2021","import_from":"Paperpile","tmap.id":"964cdfc5-bb58-429f-bd76-9c52f206fcc3","text":""}, {"created":"20211118133128289","creator":"Rahul Kashyap","title":"Bauswein:2017vtn","bibtex-entry-type":"ARTICLE","bibtex-title":"Neutron-star Radius Constraints from {GW170817} and Future\n Detections","bibtex-author":"Bauswein, Andreas and Just, Oliver and Janka, Hans-Thomas and\n Stergioulas, Nikolaos","bibtex-abstract":"We introduce a new, powerful method to constrain properties of\n neutron stars (NSs). We show that the total mass of GW170817\n provides a reliable constraint on the stellar radius if the\n merger did not result in a prompt collapse as suggested by the\n interpretation of associated electromagnetic emission. The\n radius of nonrotating NSs with a mass of 1.6 can be constrained\n to be larger than km, and the radius Rmax of the nonrotating\n maximum-mass configuration must be larger than km. We point out\n that detections of future events will further improve these\n constraints. Moreover, we show that a future event with a\n signature of a prompt collapse of the merger remnant will\n establish even stronger constraints on the NS radius from above\n and the maximum mass Mmax of NSs from above. These constraints\n are particularly robust because they only require a measurement\n of the chirp mass and a distinction between prompt and delayed\n collapse of the merger remnant, which may be inferred from the\n electromagnetic signal or even from the presence/absence of a\n ringdown gravitational-wave (GW) signal. This prospect\n strengthens the case of our novel method of constraining NS\n properties, which is directly applicable to future GW events\n with accompanying electromagnetic counterpart observations. We\n emphasize that this procedure is a new way of constraining NS\n radii from GW detections independent of existing efforts to\n infer radius information from the late inspiral phase or\n post-merger oscillations, and it does not require particularly\n loud GW events.","bibtex-journal":"ApJL","bibtex-publisher":"IOP Publishing","bibtex-volume":"850","bibtex-number":"2","bibtex-pages":"L34","bibtex-month":"nov","bibtex-year":"2017","bibtex-url":"https://iopscience.iop.org/article/10.3847/2041-8213/aa9994/meta","bibtex-file":"All_Papers/2017/Bauswein_et_al._2017_-_Neutron-star_Radius_Constraints_from_GW170817_and_Future_Detections.pdf","bibtex-keywords":"BNS\\_EoS\\_3GRadErr;BNS\\_Prompt\\_BH\\_Collapse","bibtex-language":"en","bibtex-issn":"2041-8205","bibtex-doi":"10.3847/2041-8213/aa9994","modified":"20211118133204132","modifier":"Rahul Kashyap","tags":"$:/tags/Commander/Working","import_date":"18.Nov.2021","import_from":"Paperpile","tmap.id":"f9455965-83c7-4dc0-ae2e-d37647271edd","text":""}, @@ -717,6 +719,7 @@ {"title":"Publications and CV","text":"\u003Cdiv class=\"tc-table-of-contents\">\n\n\u003C\u003Ctoc-selective-expandable 'Publications and CV' \"sort{arbitrary_field}\">>\n\n\u003C/div>","caption":"Publications and CV","created":"20210726024542385","creator":"rkashyap","is-dropdown":"yes","modified":"20210727164839899","modifier":"rkashyap","tags":"$:/tags/MenuBar TableOfContents"}, {"title":"rahul_N-N_TW_settings","text":"/* multicol */\n\n.fourcolumns {\n display:block;\n -moz-column-count:4;\n -moz-column-gap:1em;\n -webkit-column-count: 4;\n -webkit-column-gap:1em;\n}\n.small { font-size:80%; line-height:1.3em; }\n.nowrap { white-space:wrap; }\n\n.columns5 { display:block; -moz-column-count:5; -moz-column-gap:1em; -webkit-column-count:5; -webkit-column-gap:1em; }\n.small { font-size:80%; line-height:1.3em; }\n.nowrap { white-space:nowrap; }\n\n.columns4 { display:block; -moz-column-count:4; -moz-column-gap:1em; -webkit-column-count:4; -webkit-column-gap:1em; }\n.small { font-size:80%; line-height:1.3em; }\n.nowrap { white-space:wrap; }\n\n/* end multicol */\n\n\n\nbody.tc-body {\n\tbackground: #f2f2f7;\n}\n\n\nbody {font-size:5.9;font-family: CMU Serif Roman, CMU Serif, Roman; margin:\n0; padding:0;color:#000000; color: #000000;\nbackground: yellow;\n}\n\n\n.tc-titlebar h2 {\nfont-size: 1em;\ndisplay: inline; color:green;\n}\n\nol {list-style-type:decimal;}\nol ol {list-style-type:lower-alpha;}\nol ol ol {list-style-type:lower-roman;}\nol ol ol ol {list-style-type:decimal;}\nol ol ol ol ol {list-style-type:lower-alpha;}\nol ol ol ol ol ol {list-style-type:lower-roman;}\nol ol ol ol ol ol ol {list-style-type:decimal;}\n\n.bimlas-highlight-searched-text.bimlas-sidebar .bimlas-next-match svg, .bimlas-highlight-searched-text.bimlas-sticky-find-in-page .bimlas-next-match svg {\n fill: green;\n}\n.bimlas-highlight-searched-text.bimlas-sidebar .bimlas-previous-match svg, .bimlas-highlight-searched-text.bimlas-sticky-find-in-page .bimlas-previous-match svg {\n fill: red;\n}\n\n\n.doc-icon svg {\n\twidth: 1em;\n\theight: 1em;\n vertical-align: middle;\n}\n\n\n.doc-icon-block {\n\tborder-left: 2px solid \u003C\u003Ccolour code-border>>;\n\tmargin-left: 3em;\n\tpadding-left: 0.6em;\n\tposition: relative;\n}\n\n\n.doc-block-icon {\n\tposition: absolute;\n\tleft: -3em;\n\ttop: 0.2em;\n}\n.doc-block-icon .tc-image-tip {\n\tfill: \u003C\u003Ccolour primary>>;\n}\n.doc-block-icon .tc-image-warning {\n\tfill: \u003C\u003Ccolour alert-highlight>>;\n}\n\n\n\n```\n######### INFORMATION ###################\n## Jan.2021/Rahul Kashyap (rahulkashyap411@gmail.com): This tiddlywiki is modified using personally defined macros as well as plugins from various developers whose details are unmodified in their respective plugin documentations. \nPlease use at your own risk and give credit to original creators. \n\n```\n\n","created":"20200406212253116","creator":"Rahul Kashyap","modified":"20210117103627969","modifier":"Rahul Kashyap","tags":"Tools rahul_N-N_TW_settings $:/tags/Stylesheet","tmap.id":"d965e852-32d5-4aca-aa23-63d68d6b730b"}, {"title":"rahulkashyap411_githubIO_published","text":"","created":"20210725222625901","creator":"Rahul Kashyap","modified":"20210726024846203","modifier":"rkashyap","tags":"rahulkashyap411_githubIO_published rahul_N-N_TW_settings","tmap.id":"156bac23-4080-4750-aa21-9107914a8e39"}, +{"created":"20240222201832922","creator":"rkashyap","text":"> ''Please redirect yourself here -- '' https://rahulkashyap-phy.github.io/#About","tags":"","title":"Redirect","modified":"20240222201843071","modifier":"rkashyap"}, {"title":"Rendall 2008","text":"","bibtex-author":"Rendall, Alan D","bibtex-doi":"10.1007/s10714-009-0792-z","bibtex-entry-type":"BOOK","bibtex-keywords":"Analytical Relativity","bibtex-pages":"1675--1676","bibtex-title":"Partial differential equations in General Relativity","bibtex-volume":"41","bibtex-year":"2008"}, {"title":"Research Highlights","text":"\u003Cdiv class=\"tc-table-of-contents\">\n\n\u003C\u003Ctoc-selective-expandable 'Research Highlights' \"sort{arbitrary_field}\">>\n\n\u003C/div>","caption":"Research Highlights","created":"20210726023741564","creator":"rkashyap","is-dropdown":"yes","modified":"20210727164825686","modifier":"rkashyap","tags":"$:/tags/MenuBar TableOfContents"}, {"title":"Resources","text":"!Resources ","created":"20210116162200078","creator":"Rahul Kashyap","modified":"20210116174715106","modifier":"Rahul Kashyap","tags":"CoursePlan"}, @@ -726,7 +729,7 @@ {"created":"20211118133128120","creator":"Rahul Kashyap","title":"Shibata:2005ss","bibtex-entry-type":"ARTICLE","bibtex-title":"Merger of binary neutron stars with realistic equations of state\n in full general relativity","bibtex-author":"Shibata, Masaru and Taniguchi, Keisuke and Ury{\\=u}, K{\\=o}ji","bibtex-journal":"Phys. Rev. D","bibtex-publisher":"American Physical Society","bibtex-volume":"71","bibtex-number":"8","bibtex-pages":"084021","bibtex-month":"apr","bibtex-year":"2005","bibtex-url":"https://link.aps.org/doi/10.1103/PhysRevD.71.084021","bibtex-file":"All_Papers/2005/Shibata_et_al._2005_-_Merger_of_binary_neutron_stars_with_realistic_equations_of_state_in_full_general_relativity.pdf","bibtex-keywords":"BNS\\_Prompt\\_BH\\_Collapse","bibtex-doi":"10.1103/PhysRevD.71.084021","modified":"20211118133203913","modifier":"Rahul Kashyap","tags":"$:/tags/Commander/Working","import_date":"18.Nov.2021","import_from":"Paperpile","tmap.id":"bef6bce0-2dec-452e-abee-414a31119719","text":""}, {"created":"20210727182858744","creator":"rkashyap","text":"In this post, I will summarise our discussion with Reza Katebi about my research on formation of supernovae, explosions of stars and use of computers to simulate them.\n\n>;What are stars? How do they shine?\n\nStars are big hot balls of gas. One may ask why doesn’t the gas making up the stars fly away or, flow away? Well, it’s the same reason as to why we are not flying away from the Earth into space — gravity. Gravity of the different parts of the gas pull each other. You may wonder then, why don’t they just keep falling in and stop at nothing, squeezing everything to a point? The answer to this question is related to another obvious question about stars — why do they shine?\n\nWe see shining stars every day and night in the sky. They shine because they produce energy; a large fraction of which comes out in the form of light. This energy is produced by fusing elements together to form new elements near core of stars dues to high pressure. Almost all of the energy comes from fusing 4 Hydrogen (H) nuclei to form one Helium (He) nucleus. Such reactions produces energy because the mass of one He nucleus is slightly less than the mass of four H nuclei. The difference in mass gets converted into energy in accordance with Einstein’s famous formula — $$E=mc^2$$.\n\nThe light coming out of the star’s center not only make it shine but also prevent it to collapse under its own gravity. The produced energy in form of light i.e. photons bump into the gas and keep them from falling. They maintain what is technically called hydrostatic equilibrium (hydro: water (because of fluid approximation in this case), static: at rest).\n\nBut, stars have only so much Hydrogen to spend! Usually, these reactions proceed at a slow and constant pace. That’s why they keep producing energy for millions and millions of years. If the mass of the star is more, the reactions happen at a faster rate and it consumes its fuel in a shorter time and reaches its end state sooner than than a low mass star.\n\n>;How do stars live their life?\n\nStars live on Hydrogen and they have only so much to spend and hence so much to live! Usually, the nuclear reactions proceed at a slow and constant pace. That’s why they keep producing energy for millions and millions of years. If the mass of the star is more, the reactions happen at faster rate and it consumes its fuel in shorter time and reaches its end state sooner than than a low mass star.\nThis is the reason for a famous fact that most of the shining stars — hot stars (high surface temperature) are blue, cold stars are red.\n\n>;What leads the star to explode and go supernova?\n\nSometime towards end of their life, stars reach a stage where the stellar fuel sits there like a cracker, large amount of highly compact fuel. Just like a bomb, it burns, but fails to expand in time because of the heat produced. High concentration of fuel means higher production of heat and higher heat means faster reactions. This positive feedback can go unstable and a runaway process may result, commonly known as detonation. These are so powerful that it produces as much energy as our Sun will produce in its entire lifetime!! We use these bright events to study our universe; just like sailors use light house to guide their ships in oceans.\n\nThe first recorded supernova in history is considered to be the one recorded by Chinese astronomers as “guest star” in 1054 AD. It is also believed to be painted in the caves by native Americans. With modern analysis, we came to know that the that neutron star at the center of Crab nebula was produced by the very same supernova explosions seen by our ancestors about thousand years ago.\n\n>;White Dwarfs, Subrahmanyann Chandrasekhar and Arthur Eddington\n\nWhite dwarfs are peculiar stars whose white color indicates very high surface temperature. On the other hand, observations shows they are also very small. They don’t follow the trend set by most of the stars. Painstaking observations and complex arguments by several scientists unveiled that they are actually the end state of low mass stars (with masses up to 8 solar mass).\n\nDuring the early twentieth century detailed theory of structure of matter were laid out by Planck, Einstein, Heisenberg, Schrödinger and others. Among them Enrico Fermi and Paul Dirac were thinking about collective properties of electrons as gas. This is how electrons behave inside metals.\nSubrahmanyan Chandrasekhar (people used to call him Chandra) used these recently discovered theories to find the structure of white dwarfs. He found that as the mass of a white dwarf increases, its radius decreases going to zero at a critical mass, called Chandrasekhar mass.\n\n\nFamous astrophysicist and Chandra’s advisor, Arthur Eddington was not ready to accept this conclusion. He suspected that “some” physical process has to intervene and stop the collapse when the mass of white dwarf increases. He was right too in some sense, however he failed to recognize the importance of Chandrasekhar’s result that the maximum mass limit is direct consequence of statistical properties of electrons. Neutrons have same statistical properties hence they will also have maximum mass. This general result can be taken to be inspired from Chandrasekhar’s work. In fact, later general relativity is used to establish a much general result about existence of maximum mass independent of statistical properties i.e. equation of state.\n\n>;How do we simulate star explosions on a computer?\n\n\nSimulations are an extension, though a very sophisticated one, of the calculation that we do using pen and paper. Simulations are a way to test our scientific understanding of nature and also predict new phenomena. Our concept of scientific understanding of the world is different for two distinct types of natural phenomena. In one case, we do experiments where we can manipulate the variables that control the behavior of output or, observables such as fluid flow, sound, light, heat etc. While we also have natural systems which do not allow us to manipulate their conditions (such as astrophysical and biological evolutionary phenomena). In the later case, we can only observe the multitude of conditions that are already present. We can imagine the varying conditions of the system akin to manipulating the controlling variables. In this sense, the universe itself is a laboratory presenting us myriad possibilities of the input variables. This picture also tells us that our understanding of nature in terms of science is probabilistic.\n\n>;How much computation power is needed to solve the problems of astrophysics?\n\n\nIn principle, one can put all the governing equations in the form of computer programs and solve it for any arbitrary initial and boundary conditions but, there are several conceptual and practical difficulties in such a scheme of answering astrophysical questions. Firstly, there will this objection — Are the equations used and implementations of the computer program valid in all possible domains pertaining to the problem at hand? The answer is yes, if we take the most fundamental theories of space, time and matter -- General Theory of Relativity and Quantum Field Theory. However, this is too big a task even to cast them into a computational problem. Even without such objections, such a simulation would be unfeasible because it would require ridiculously large amount of computational time. Hence, there is always a certain amount of “art” involved in performing the computational work. This artistic freedom here is not subjective but, an objective one. The demand of computational power depends on our question of what observable we want to compare with observation and what accuracy we need. The answer to this question then determines which pieces of physics must be included which are not important. Many of such answers can be obtained using a single laptop but, the questions at the fore front of knowledge demand large scale supercomputing facilities.\n\n>> \u003Ciframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/hGe8bHAUQlM\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen>\u003C/iframe>\n\n\n>''Acknowledgement:'' I would like to dedicate this post to my pre-college teacher ([[Abhay Kumar Singh|https://www.amazon.com/Abhay-Kumar-Singh/e/B001ICHAAW?ref=sr_ntt_srch_lnk_2&qid=1628084401&sr=1-2]]) who taught me most of what I know about thinking about the beauty of mathematics as applied to physical problems after I.Sc. IInd year.","title":"Shining stars, exploding stars","modified":"20231025193112932","modifier":"rkashyap","tags":"[[Science Outreach]] rahulkashyap411_githubIO_published","tmap.id":"fa5da22d-af6a-4ba4-9471-4abe3257fc33"}, {"created":"20220922053832827","creator":"rkashyap","text":"\nHere is my older wordpress blog/site which is retired now -- https://naturennumbers.wordpress.com/","tags":"TableOfContents","title":"Society and Life","modified":"20220922053854090","modifier":"rkashyap"}, -{"created":"20200321035908902","creator":"Rahul Kashyap","text":"\u003Cdiv class=\"nr\">\n\n\n!Special Relativity\n\nAny observer-independent physical phenomena is completely charaterised by 4 real numbers ($$t, x, y, z$$: known as an //event//) with suitably defined coordinate system. We stick to phenomena for which we can measure its position and time coordinates. \n\nThe path of a single particle (localized field or a wavefunction\u003C!--\n($$\\int_{-\\infty}^{+\\infty} |\\phi(x)|^2 dx \u003C\\infty $$ and $$\\int_{-L}^{+L} |\\phi(x)|^2 dx = \\mathcal{E} $$ \nfor a finite measurable size, $$L$$ and energy )-->) travelling through spacetime is described by four numbers\n\nA Lorentz invariant interval constructed from $$dx^i$$ (a 4-vector) in spacetime is \n$$\ndx^i dx_i=d\\tau ^2=(-c\\Delta t)^2 +(\\Delta x)^2+(\\Delta y)^2+(\\Delta z)^2\n$$\nwhere three different regions of spacetime is defines as $$dx^i dx_i\u003C0$$ (space like), $$dx^i dx_i>0$$ (time like) and $$dx^i dx_i=0$$ (light like).\n\n\u003C\u003Cquestion \"\n# We saw in the class that if $$ds^2=0$$ in one reference frame then it is zero in all reference frame due to contancy of speed of light. Prove that this is true for any value of $$ds^2$$. (Refer Landau-Lifshitz for proof). \n\n\">>\n\nThis can be used as an axiom replacing Einstein's second axiom of constancy of speed of light. \n\n!! General Lorentz transformation \n$$\n\\left[\\begin{array}{c}\nc t^{\\prime} \\\\\nx^{\\prime} \\\\\ny^{\\prime} \\\\\nz^{\\prime}\n\\end{array}\\right] = \\bf{L} \\left[\\begin{array}{c}\nc t \\\\\nx \\\\\ny \\\\\nz\n\\end{array}\\right]\n$$\n\nFor two frames travelling with respect to each other along $$x$$-axis -- \n$$\n \\bf{L} =\\left[\\begin{array}{cccc}\n\\gamma & -\\gamma \\beta & 0 & 0 \\\\\n-\\gamma \\beta & \\gamma & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{array}\\right]\n$$\n\nMore generally for arbitrary direction of travel, we get -- \n$$\n\\mathrm{L}=\\left[\\begin{array}{cccc}\n\\gamma & -\\gamma \\beta_{x} & -\\gamma \\beta_{y} & -\\gamma \\beta_{z} \\\\\n-\\gamma \\beta_{x} & 1+(\\gamma-1) \\frac{\\beta_{x}^{2}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{y}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{z}}{\\beta^{2}} \\\\\n-\\gamma \\beta_{y} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{y}}{\\beta^{2}} & 1+(\\gamma-1) \\frac{\\beta_{y}^{2}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{y} \\beta_{z}}{\\beta^{2}} \\\\\n-\\gamma \\beta_{z} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{z}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{z} \\beta_{y}}{\\beta^{2}} & 1+(\\gamma-1) \\frac{\\beta_{z}^{2}}{\\beta^{2}}\n\\end{array}\\right]\n$$\n\n\n!! 4-vectors and invariant quantities\n\nTo preserve the covariance of equations, it is useful to write equations as linear functional of vectors like $$(ct,x,y,z)$$ (known as ''4-vectors''). This makes sure that an equation like \n$$\nf^\\mu = \\alpha A^\\mu + \\beta B^\\mu \\\\\n\\quad \\\\\n\n\\textbf{L} (f^\\mu = \\alpha A^\\mu + \\beta B^\\mu) \\\\\n\\tilde{f}^\\mu = \\alpha \\tilde{A}^\\mu + \\beta \\tilde{B}^\\mu\n$$\nremains ''Covariant''. In other words, the relationship between the physical quantities, $$f^\\mu, A^\\mu, B^\\mu$$ remains the same as measured in both reference frames. \nMore often we need a relationship between first and/or second differential of a vector field. \n\n\u003C\u003Cquestion \"\n# What kind of equation(s) involving derivatives will remain covariant under Lorentz transformation ?\">>\n\nOther 4-vectors comes from demanding Lorentz Invariant kinematical and dynamical quantities such as velocity, acceleration and force. Now, differentiating $$dx^i$$ w.r.t. $$dt$$, we get $$u^i = \\frac{dx^i}{dt} \\neq \\frac{dx'^i}{dt'}$$ i.e. not Lorentz Invariant. So, we construct a 4-velocity as $$u^i = \\frac{dx^i}{d\\tau} = (c\\gamma,v^i \\gamma)$$ which transforms according to Lorentz transformation. Some properties --\n$$\n u^i u_i = c^2\\\\\n p^i = m_0 u^i \\\\\n p^i p_i = m_0 ^2 c^4 = E^2 -p^2 c^2\\\\\n \\textrm{4-acceleration, } a^i \\coloneq \\frac{du^i}{d\\tau}, u^i a_i = 0 \\textrm{ i.e. } u \\perp a \\textrm{ in 4-D }\n$$\n\n; 4-force and Mass-energy theorem\n$$\nG^i \\coloneq \\frac{dp^i}{d\\tau} = m_0 a^i \n$$\n\n\u003C\u003Cquestion \"\n# Derive the $$v/c \u003C\u003C 1$$ limit of these equations to get back Newtonian expressions for energy and momentum.\n# A particle of rest mass $$m_0$$ is subject to a constant force F (ordinary 3-force). If it starts from rest at the origin (at time, t=0) find its position $$x$$, as a function of time. Using your answer, argue that it is possible to outrun a light ray, if you're given a sufficient headstart, and your feet generate a constant force.\n\">>\n\n\n!! Electromagnetism \n\n\u003C\u003Cproblem \"\nPerform the Lorentz transformation to the wave equation from coordinates $$(ct,x,y,z) \\rightarrow (ct',x',y',z')$$, $$\n\\nabla^2 \\phi = \\frac{1}{c^2} \\frac{\\partial^2 \\phi}{\\partial t^2}\n$$ \nPerform the same transformation to Schrodinger equation, $$\n\\frac{- \\hbar}{2m} \\nabla^2 \\phi + V \\phi = i\\hbar \\frac{\\partial \\phi}{\\partial t}\n$$ where $$V(\\vec{r},t)$$ is scalar function. \n\">>\n\n\n!!! First and second set of Maxwell's eqns:-\n$$\nF_{ij}=\\left[ {\\begin{array}{cccc} 0 & \\frac{E_x}{c} & \\frac{E_y}{c} & \\frac{E_z}{c} \\\\ %EM tensor\n-\\frac{E_x}{c} & 0 & -B_z & B_y \\\\\n-\\frac{E_y}{c} & B_z & 0 & -B_x\\\\\n-\\frac{E_z}{c} & -B_y & B_x & 0 \\end{array} } \\right]\\\\\n\n$$\n\nFor current density, $$J^\\mu = (c\\rho,\\vec{j})$$\n$$\n\\partial _i F_{jk} + \\partial_jF_{ki}+\\partial_kF_{ij}=0 \\\\\n\\partial_\\nu F^{\\nu\\mu} = -\\mu_o J^\\mu\n$$\n\n>> Dot product of two 4-vectors are defined as \n$$\n\\mathbf{A \\cdot B} = A^\\mu B^\\nu \\eta_{\\mu\\nu}\n$$\n\n\u003C\u003Cexcercise \"\n# A particle of rest mass $$m_0$$ is subject to a constant force F (ordinary 3-force). If it starts from rest at the origin (at time, t=0) find its position $$x$$, as a function of time. Using your answer, argue that it is possible to outrun a light ray, if you're given a sufficient headstart, and your feet generate a constant force.\n#A particle with rest mass, $$m$$ and 4-momentum $$\\mathbf{p}$$ is examined by an observer with 4-velocity, $$\\mathbf{u}$$ \n ## show that the energy measured by the observer is $$E=-\\mathbf{p\\cdot u}$$.\n ## What is the 3-momentum and 3-velocity of particle measured by the observer.\n# Write down the equations of motion of a relativistic particle in electromagnetic field using 4-vector formalism. \n\n\">>\n\n\n\u003C\u003Cimage-pretty img:\"images/STR_details.png\" width:\"1524px\" align:\"center\" caption:\"([[img|./images/STR_details.png]])Descriptive Graphics -1: Length contraction and time dilation using only Pythagorus theorem and two basic postulates of Special Relativity as described by Einstein in his book //Relativity: The special and general theory//\" >> \n\n!!! 4-vector for a single photon and a collection of photons\n\n;Photon 4-momentum: \n\n> see Box 4.3 (\u003C\u003Cref [[Poisson and Will 2014]]>>) for proper justification of defining such 4-vectors for photons. \n$$\nM^\\alpha = \\frac{h\\nu}{c} (1,\\vec{n})\n$$\n\n;Photon propagation 4-vector \n$$\nK^\\alpha = \\frac{2\\pi\\nu}{c}(1,\\vec{n})\n$$\n\n\u003C\u003Cquestion \"\n\n;Assume that there are two reference frames, $$K, K'$$ traveling at constant velocity, $$ \\vec{v}=v_x \\mathbf{e_x} + v_y \\mathbf{e_y} + v_z \\mathbf{e_z}$$. Single or multiple events will be observed, analysed and compared from both reference frames. \n\n#Use Lorentz transformation on above 4-vectors for photons to derive relativistic doppler shift as well as relativistic beaming effect?\n ## Relativistic Doppler Shift.\n ## Relativistic Beaming Effect.\n ##*''In this problem we look at a single traveling photon from two different inertial reference frames.''\n \">>\n \n \n \n \u003C\u003Cexcercise \"\n# Box4.1, [[Hartle 2003]]: Michaelson-Morley Experiment and its improvement by Brillet and Hall in 1978. Explain how the observed relative error on $$\\Delta f/f=(1.5 \\pm 2.5) \\times 10^{-15}$$ against the classical prediction of $$(V_E / c)^2 \\approx 10^{-8}$$\n# sec-4.1.7: Relativistic mass-energy theorem as zeroth component of F=ma using 4-vectors. \n# Write down the equations of motion of a relativistic particle in electromagnetic field using 4-vector formalism. sec-3.1 MTW. \n\n\n\">> \n\n\n\n\n\n\n\n\n \u003C/div>\n","title":"Special Relativity","modified":"20220609080448619","modifier":"rkashyap","tags":"[[Special Relativity]] GeneralRelativity TableOfContents Prerequisites_GeneralRelativity rahulkashyap411_githubIO_published [[Lecture Notes]]","tmap.edges":"{\"9b095235-1086-407e-a80c-43df0f81bd8a\":{\"to\":\"f92d1372-abbc-4f1c-af60-4e68e6aece8f\",\"type\":\"tmap:unknown\"}}","tmap.id":"7437f34b-fd01-4943-afaa-6694d2a75d83","week":"1"}, +{"created":"20200321035908902","creator":"Rahul Kashyap","text":"\u003Cdiv class=\"nr\">\n\n\n!Special Relativity\n\nAny observer-independent physical phenomena is completely charaterised by 4 real numbers ($$t, x, y, z$$: known as an //event//) with suitably defined coordinate system. We stick to phenomena for which we can measure its position and time coordinates. \n\nA Lorentz invariant interval constructed from $$dx^i$$ (a 4-vector) in spacetime is \n$$\ndx^i dx_i=d\\tau ^2=(-c\\Delta t)^2 +(\\Delta x)^2+(\\Delta y)^2+(\\Delta z)^2\n$$\nwhere three different regions of spacetime is defines as $$dx^i dx_i\u003C0$$ (space like), $$dx^i dx_i>0$$ (time like) and $$dx^i dx_i=0$$ (light like).\n\n\u003C\u003Cquestion \"\n# We saw in the class that if $$ds^2=0$$ in one reference frame then it is zero in all reference frame due to contancy of speed of light. Prove that this is true for any value of $$ds^2$$. (Refer Landau-Lifshitz for proof). \n\n\">>\n\nThis can be used as an axiom replacing Einstein's second axiom of constancy of speed of light. \n\n!! General Lorentz transformation \n$$\n\\left[\\begin{array}{c}\nc t^{\\prime} \\\\\nx^{\\prime} \\\\\ny^{\\prime} \\\\\nz^{\\prime}\n\\end{array}\\right] = \\bf{L} \\left[\\begin{array}{c}\nc t \\\\\nx \\\\\ny \\\\\nz\n\\end{array}\\right]\n$$\n\nFor two frames travelling with respect to each other along $$x$$-axis -- \n$$\n \\bf{L} =\\left[\\begin{array}{cccc}\n\\gamma & -\\gamma \\beta & 0 & 0 \\\\\n-\\gamma \\beta & \\gamma & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{array}\\right]\n$$\n\nMore generally for arbitrary direction of travel, we get -- \n$$\n\\mathrm{L}=\\left[\\begin{array}{cccc}\n\\gamma & -\\gamma \\beta_{x} & -\\gamma \\beta_{y} & -\\gamma \\beta_{z} \\\\\n-\\gamma \\beta_{x} & 1+(\\gamma-1) \\frac{\\beta_{x}^{2}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{y}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{z}}{\\beta^{2}} \\\\\n-\\gamma \\beta_{y} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{y}}{\\beta^{2}} & 1+(\\gamma-1) \\frac{\\beta_{y}^{2}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{y} \\beta_{z}}{\\beta^{2}} \\\\\n-\\gamma \\beta_{z} & (\\gamma-1) \\frac{\\beta_{x} \\beta_{z}}{\\beta^{2}} & (\\gamma-1) \\frac{\\beta_{z} \\beta_{y}}{\\beta^{2}} & 1+(\\gamma-1) \\frac{\\beta_{z}^{2}}{\\beta^{2}}\n\\end{array}\\right]\n$$\n\n\n!! 4-vectors and invariant quantities\n\nTo preserve the covariance of equations, it is useful to write equations as linear functional of vectors like $$(ct,x,y,z)$$ (known as ''4-vectors''). This makes sure that an equation like \n$$\nf^\\mu = \\alpha A^\\mu + \\beta B^\\mu \\\\\n\\quad \\\\\n\n\\textbf{L} (f^\\mu = \\alpha A^\\mu + \\beta B^\\mu) \\\\\n\\tilde{f}^\\mu = \\alpha \\tilde{A}^\\mu + \\beta \\tilde{B}^\\mu\n$$\nremains ''Covariant''. In other words, the relationship between the physical quantities, $$f^\\mu, A^\\mu, B^\\mu$$ remains the same as measured in both reference frames. \nMore often we need a relationship between first and/or second differential of a vector field. \n\n\u003C\u003Cquestion \"\n# What kind of equation(s) involving derivatives will remain covariant under Lorentz transformation ?\">>\n\nOther 4-vectors comes from demanding Lorentz Invariant kinematical and dynamical quantities such as velocity, acceleration and force. Now, differentiating $$dx^i$$ w.r.t. $$dt$$, we get $$u^i = \\frac{dx^i}{dt} \\neq \\frac{dx'^i}{dt'}$$ i.e. not Lorentz Invariant. So, we construct a 4-velocity as $$u^i = \\frac{dx^i}{d\\tau} = (c\\gamma,v^i \\gamma)$$ which transforms according to Lorentz transformation. Some properties --\n$$\n u^i u_i = c^2\\\\\n p^i = m_0 u^i \\\\\n p^i p_i = m_0 ^2 c^4 = E^2 -p^2 c^2\\\\\n \\textrm{4-acceleration, } a^i \\coloneq \\frac{du^i}{d\\tau}, u^i a_i = 0 \\textrm{ i.e. } u \\perp a \\textrm{ in 4-D }\n$$\n\n; 4-force and Mass-energy theorem\n$$\nG^i \\coloneq \\frac{dp^i}{d\\tau} = m_0 a^i \n$$\n\n\u003C\u003Cquestion \"\n# Derive the $$v/c \u003C\u003C 1$$ limit of these equations to get back Newtonian expressions for energy and momentum.\n# A particle of rest mass $$m_0$$ is subject to a constant force F (ordinary 3-force). If it starts from rest at the origin (at time, t=0) find its position $$x$$, as a function of time. Using your answer, argue that it is possible to outrun a light ray, if you're given a sufficient headstart, and your feet generate a constant force.\n\">>\n\n\n!! Electromagnetism \n\n\u003C\u003Cproblem \"\nPerform the Lorentz transformation to the wave equation from coordinates $$(ct,x,y,z) \\rightarrow (ct',x',y',z')$$, $$\n\\nabla^2 \\phi = \\frac{1}{c^2} \\frac{\\partial^2 \\phi}{\\partial t^2}\n$$ \nPerform the same transformation to Schrodinger equation, $$\n\\frac{- \\hbar}{2m} \\nabla^2 \\phi + V \\phi = i\\hbar \\frac{\\partial \\phi}{\\partial t}\n$$ where $$V(\\vec{r},t)$$ is scalar function. \n\">>\n\n\n!!! First and second set of Maxwell's eqns:-\n$$\nF_{ij}=\\left[ {\\begin{array}{cccc} 0 & \\frac{E_x}{c} & \\frac{E_y}{c} & \\frac{E_z}{c} \\\\ %EM tensor\n-\\frac{E_x}{c} & 0 & -B_z & B_y \\\\\n-\\frac{E_y}{c} & B_z & 0 & -B_x\\\\\n-\\frac{E_z}{c} & -B_y & B_x & 0 \\end{array} } \\right]\\\\\n\n$$\n\nFor current density, $$J^\\mu = (c\\rho,\\vec{j})$$\n$$\n\\partial _i F_{jk} + \\partial_jF_{ki}+\\partial_kF_{ij}=0 \\\\\n\\partial_\\nu F^{\\nu\\mu} = -\\mu_o J^\\mu\n$$\n\n>> Dot product of two 4-vectors are defined as \n$$\n\\mathbf{A \\cdot B} = A^\\mu B^\\nu \\eta_{\\mu\\nu}\n$$\n\n\u003C\u003Cexcercise \"\n# A particle of rest mass $$m_0$$ is subject to a constant force F (ordinary 3-force). If it starts from rest at the origin (at time, t=0) find its position $$x$$, as a function of time. Using your answer, argue that it is possible to outrun a light ray, if you're given a sufficient headstart, and your feet generate a constant force.\n#A particle with rest mass, $$m$$ and 4-momentum $$\\mathbf{p}$$ is examined by an observer with 4-velocity, $$\\mathbf{u}$$ \n ## show that the energy measured by the observer is $$E=-\\mathbf{p\\cdot u}$$.\n ## What is the 3-momentum and 3-velocity of particle measured by the observer.\n# Write down the equations of motion of a relativistic particle in electromagnetic field using 4-vector formalism. \n\n\">>\n\n\n\u003C\u003Cimage-pretty img:\"images/STR_details.png\" width:\"1524px\" align:\"center\" caption:\"([[img|./images/STR_details.png]])Descriptive Graphics -1: Length contraction and time dilation using only Pythagorus theorem and two basic postulates of Special Relativity as described by Einstein in his book //Relativity: The special and general theory//\" >> \n\n!!! 4-vector for a single photon and a collection of photons\n\n;Photon 4-momentum: \n\n> see Box 4.3 (\u003C\u003Cref [[Poisson and Will 2014]]>>) for proper justification of defining such 4-vectors for photons. \n$$\nM^\\alpha = \\frac{h\\nu}{c} (1,\\vec{n})\n$$\n\n;Photon propagation 4-vector \n$$\nK^\\alpha = \\frac{2\\pi\\nu}{c}(1,\\vec{n})\n$$\n\n\u003C\u003Cquestion \"\n\n;Assume that there are two reference frames, $$K, K'$$ traveling at constant velocity, $$ \\vec{v}=v_x \\mathbf{e_x} + v_y \\mathbf{e_y} + v_z \\mathbf{e_z}$$. Single or multiple events will be observed, analysed and compared from both reference frames. \n\n#Use Lorentz transformation on above 4-vectors for photons to derive relativistic doppler shift as well as relativistic beaming effect?\n ## Relativistic Doppler Shift.\n ## Relativistic Beaming Effect.\n ##*''In this problem we look at a single traveling photon from two different inertial reference frames.''\n \">>\n \n \n \n \u003C\u003Cexcercise \"\n# Box4.1, [[Hartle 2003]]: ''Michaelson-Morley Experiment and its improvement by Brillet and Hall in 1978'' -- Explain how the observed relative error on $$\\Delta f/f=(1.5 \\pm 2.5) \\times 10^{-15}$$ acts as a falsifiable prediction of Newtonian transformation law for velocity of light with source velocity, $$V_E$$ of the order, $$(V_E / c)^2 \\approx 10^{-8}$$. \n# How the classical theory of ether and absolute space is ruled out from this experiment?\n# sec-4.1.7: Relativistic mass-energy theorem as zeroth component of F=ma using 4-vectors. \n# Write down the equations of motion of a relativistic particle in electromagnetic field using 4-vector formalism. sec-3.1 MTW. \n\n\n\">> \n\n\n\n\n\n\n\n\n \u003C/div>\n","title":"Special Relativity","modified":"20240129112609484","modifier":"rkashyap","tags":"[[Special Relativity]] GeneralRelativity TableOfContents Prerequisites_GeneralRelativity rahulkashyap411_githubIO_published [[Lecture Notes]]","tmap.edges":"{\"9b095235-1086-407e-a80c-43df0f81bd8a\":{\"to\":\"f92d1372-abbc-4f1c-af60-4e68e6aece8f\",\"type\":\"tmap:unknown\"}}","tmap.id":"7437f34b-fd01-4943-afaa-6694d2a75d83","week":"1"}, {"title":"Stephani et. al. 1980","text":"","bibtex-author":"Stephani, H and Kramer, D and MacCallum, M and Herlt, E","bibtex-entry-type":"ARTICLE","bibtex-file":"All_Papers/1980/Kramer_et_al._1980_-_Exact_solutions_of_Einstein's_field_equations.pdf","bibtex-journal":"Berlin","bibtex-keywords":"Analytical Relativity","bibtex-publisher":"s3.cern.ch","bibtex-title":"Exact solutions of Einstein's field equations","bibtex-url":"https://s3.cern.ch/inspire-prod-files-5/503faf611d92da5006b6f72f44150e43#page=75","bibtex-year":"1980","created":"20210107034132254","creator":"Rahul Kashyap","modified":"20210107034154577","modifier":"Rahul Kashyap"}, {"title":"TableOfContents","text":"\u003Cdiv class=\"tc-table-of-contents\">\n\n\u003C\u003Ctoc-selective-expandable 'TableOfContents' \"sort{arbitrary_field}\">>\n\n\u003C/div>","created":"20210116171418332","creator":"Rahul Kashyap","modified":"20210721044417995","modifier":"Rahul Kashyap","tags":"$:/tags/SideBar"}, {"created":"20210727163629891","creator":"rkashyap","text":"White dwarf in a binary system are birth place of several astrophysical transients, most notably Type Ia supernovae (SN Ia). Analytical and detailed computational modelling of turbulent flame interaction in the white dwarf core near Chandrasekhar mass has been going on for last 40 years.\n\nWe still have very few observational evidence to support single degenerate channel as a route to majority of SN Ia. We have analyzed a detailed recent observation of a remnant 3C397 against 2-dimensional simulations of various models. The initial observations indicated the remnant to come from the explosion of a WD with 5 times solar luminosity. We have shown that the same observation can be explained by a high central density deflagration model of near Chandrasekhar mass object.\n\nCheck out this paper for details — \u003C\u003Cref [[Dave2017-yh]]>>\n\n\u003C\u003Cshowrefs title:\"References\">>","title":"The Fate of a Chandrasekhar Mass White Dwarfs","modified":"20220922143811966","modifier":"rkashyap","tags":"[[Research Highlights]]"},