Newman-Penrose calculations for SageMath
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itemprop=\"text\"\u003e\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eSageNP: Newman-Penrose calculations for SageMath.\u003c/h1\u003e\u003ca id=\"user-content-sagenp-newman-penrose-calculations-for-sagemath\" class=\"anchor\" aria-label=\"Permalink: SageNP: Newman-Penrose calculations for SageMath.\" href=\"#sagenp-newman-penrose-calculations-for-sagemath\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis work is funded by TUBITAK 1001 Program, Grant Number 123R114.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe class \u003cstrong\u003eSageNP\u003c/strong\u003e includes functions for some calculations defined in the Newman-Penrose formalism. The code is based on SageManifolds.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eCoded by:\u003c/h1\u003e\u003ca id=\"user-content-coded-by\" class=\"anchor\" aria-label=\"Permalink: Coded by:\" href=\"#coded-by\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://web.itu.edu.tr/birkandant/\" rel=\"nofollow\"\u003eTolga Birkandan\u003c/a\u003e (Corr.: \u003ca href=\"mailto:birkandant@itu.edu.tr\"\u003ebirkandant@itu.edu.tr\u003c/a\u003e)\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://www.linkedin.com/in/onur-arman-709478337/\" rel=\"nofollow\"\u003eOnur Arman\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://scholar.google.com/citations?user=kq9ia_oAAAAJ\u0026amp;hl=en\" rel=\"nofollow\"\u003eEmir Baysazan\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://scholar.google.com/citations?user=OjiOpogAAAAJ\u0026amp;hl=tr\" rel=\"nofollow\"\u003eSelinay Sude Binici\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://www.linkedin.com/in/pelin-%C3%B6zt%C3%BCrk-3904572b2/?utm_source=share\u0026amp;utm_campaign=share_via\u0026amp;utm_content=profile\u0026amp;utm_medium=ios_app\" rel=\"nofollow\"\u003ePelin Ozturk\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eEfe Özyürek\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eSpecial thanks to \u003ca href=\"https://luth.obspm.fr/~luthier/gourgoulhon/en/\" rel=\"nofollow\"\u003eEric Gourgoulhon\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eFILES:\u003c/h1\u003e\u003ca id=\"user-content-files\" class=\"anchor\" aria-label=\"Permalink: FILES:\" href=\"#files\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eSageNP.py\u003c/strong\u003e: Main file to import in SageMath.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003e\u003ca href=\"https://github.com/tbirkandan/SageNP/blob/main/Notebooks/SageNP_Tutorial.ipynb\"\u003eSageNP_Tutorial.ipynb\u003c/a\u003e\u003c/strong\u003e: Tutorial (ipynb file) - Definitions and calculations for the Schwarzschild (with covariant null-tetrad vectors) and Reissner-Nordstrom (with contravariant null-tetrad vectors) spacetimes.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003e\u003ca href=\"https://github.com/tbirkandan/SageNP/blob/main/Notebooks/SageNP_Tutorial.pdf\"\u003eSageNP_Tutorial.pdf\u003c/a\u003e\u003c/strong\u003e: Tutorial (PDF file)\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eREFERENCE:\u003c/h1\u003e\u003ca id=\"user-content-reference\" class=\"anchor\" aria-label=\"Permalink: REFERENCE:\" href=\"#reference\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eMain reference for all definitions and calculations:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eH. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, \"Exact Solutions of Einstein’s Field Equations\", 2nd ed. Cambridge: Cambridge University Press, 2003.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eBASIC DEFINITIONS AND NOTATION:\u003c/h1\u003e\u003ca id=\"user-content-basic-definitions-and-notation\" class=\"anchor\" aria-label=\"Permalink: BASIC DEFINITIONS AND NOTATION:\" href=\"#basic-definitions-and-notation\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eWe will use the Metric signature: (- + + +)\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eFor the null-tetrad vector names, the ref. book uses (k,l,m,mbar). However, in the code we will use (l,n,m,mbar) like the rest of the literature. Therefore, one should set k-\u0026gt;l, l-\u0026gt;n in the ref. book\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eProducts of the vectors are given by: l\u003cem\u003en = -1, m\u003c/em\u003embar = 1, all others zero.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eThe metric is found using the covariant null-tetrad vectors as:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eg = -2\u003cem\u003el\u003c/em\u003en + 2\u003cem\u003em\u003c/em\u003embar\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eand,\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eg = [[0 1 0 0], [1 0 0 0], [0 0 0 -1], [0 0 -1 0]]\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003ePlease check the reference book for the details and further definitions.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eINSTRUCTIONS WITH EXAMPLES:\u003c/h1\u003e\u003ca id=\"user-content-instructions-with-examples\" class=\"anchor\" aria-label=\"Permalink: INSTRUCTIONS WITH EXAMPLES:\" href=\"#instructions-with-examples\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eImport the class:\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003efrom SageNP import NewmanPenrose\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDefine your manifold:\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eMyManifold = Manifold(4 , 'MyManifold', r'\\mathcal{Man}')\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDefine your coordinates:\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eMyCoordinates.\u0026lt;t,r,th,ph\u0026gt; = MyManifold.chart(r't r th:\\theta ph:\\phi')\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDefine the metric functions (if needed):\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003evar('M')\nDelta=r^2-2\u003cem\u003eM\u003c/em\u003er\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eEnter null tetrad elements:\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003elvec=[1,-(r^2)/Delta,0,0]\u003c/p\u003e\n\u003cp dir=\"auto\"\u003envec=[Delta/(2*r^2),1/2,0,0]\u003c/p\u003e\n\u003cp dir=\"auto\"\u003emvec=[0,0,(-r/sqrt(2)),(-I*r/sqrt(2))*sin(th)]\u003c/p\u003e\n\u003cp dir=\"auto\"\u003embarvec=[0,0,(-r/sqrt(2)),(I*r/sqrt(2))*sin(th)]\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\u003cem\u003eHere, the element ordering is the same as the coordinate ordering. (The first element is the t element, the second is the r element, etc.)\u003c/em\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDefine an object of the class:\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eschw=SageNP(MyManifold,MyCoordinates,lvec,nvec,mvec,mbarvec,'covariant')\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cem\u003eHere, our null-tetrad vectors lvec, nvec, mvec and mbarvec are covariant. Thus we used the keyword 'covariant'.\u003c/em\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cem\u003eIf they were contravariant, then we should use the keyword 'contravariant'.\u003c/em\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eOnce the object is defined, the code calculates the metric and displays it on the screen. It is recommended that you check your metric.\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eFUNCTIONS:\u003c/h1\u003e\u003ca id=\"user-content-functions\" class=\"anchor\" aria-label=\"Permalink: FUNCTIONS:\" href=\"#functions\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cem\u003eAll page and equation numbers belong to the reference book.\u003c/em\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003etest_nulltetrad()\u003c/strong\u003e: Checks the products of the vectors l\u003cem\u003en = -1, m\u003c/em\u003embar = 1, all others zero.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eSpin coefficients (Page 75-76, Eq.(7.2))\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_spincoefficients()\u003c/strong\u003e: Calculates the spin coefficients.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_spincoefficients()\u003c/strong\u003e: Displays the spin coefficients\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eAll spin coefficients are available under their names:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ekappaNP, kappabarNP, tauNP, taubarNP, sigmaNP, sigmabarNP,\nrhoNP, rhobarNP, piNP, pibarNP, nuNP, nubarNP, muNP, mubarNP,\nlambdaNP, lambdabarNP, epsilonNP, epsilonbarNP, gammaNP, gammabarNP,\nbetaNP, betabarNP, alphaNP, alphabarNP\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDirectional derivatives (Page 43, Eq.(3.82))\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDlNP(X)\u003c/strong\u003e: Given X, calculates the D derivative (l direction).\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDeltanNP(X)\u003c/strong\u003e: Given X, calculates the Delta derivative (n direction)\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003edeltamNP(X)\u003c/strong\u003e: Given X, calculates the delta derivative (m direction)\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003edeltambarNP(X)\u003c/strong\u003e: Given X, calculates the deltabar derivative (mbar direction)\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eCommutators (Page 77, Eq.(7.6))\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cem\u003eThe right-hand sides of the commutation relations are calculated.\u003c/em\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDeltan_Dl_commNP(X)\u003c/strong\u003e: Given X, calculates the [Delta,D] commutator.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003edeltam_Dl_commNP(X)\u003c/strong\u003e: Given X, calculates the [delta,D] commutator.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003edeltam_Deltan_commNP(X)\u003c/strong\u003e: Given X, calculates the [delta,Delta] commutator.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003edeltambar_deltam_commNP(X)\u003c/strong\u003e: Given X, calculates the [deltabar,delta] commutator.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eWeyl tensor components (Page 38, Eq.(3.59))\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_Weyl()\u003c/strong\u003e: Calculates the Weyl tensor components.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_Weyl()\u003c/strong\u003e: Displays the Weyl tensor components.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eAll Weyl tensor components are available under their names:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePsi0NP, Psi1NP, Psi2NP, Psi3NP, Psi4NP\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eRicci components (Page 78, Eq.(7.10-7.15))\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_Ricci()\u003c/strong\u003e: Calculates the Ricci tensor components.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_Ricci()\u003c/strong\u003e: Displays the Ricci tensor components.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eAll Ricci tensor components are available under their names:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePhi00NP, Phi01NP, Phi10NP, Phi02NP, Phi20NP,\nPhi11NP, Phi12NP, Phi21NP, Phi22NP, LambdaNP\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eRicci (Newman-Penrose) equations (Page 79, Eq.(7.21))\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cem\u003eAll Newman-Penrose equations are defined as 0 = -(left hand side)+(right hand side) of the equations.\u003c/em\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_NPeq()\u003c/strong\u003e: Calculates the Newman-Penrose equations\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_NPeq()\u003c/strong\u003e: Displays the Newman-Penrose equations\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eAll Newman-Penrose equations are available under their names\nin the order they are given in the reference:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eNPeq1, NPeq2, NPeq3, NPeq4, NPeq5, NPeq6, NPeq7, NPeq8, NPeq9, NPeq10,\nNPeq11, NPeq12, NPeq13, NPeq14, NPeq15, NPeq16, NPeq17, NPeq18\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eBianchi identities (Page 81, Eq.(7.32))\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cem\u003eAll Bianchi identities are defined as 0 = -(left hand side)+(right hand side) of the equations.\u003c/em\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_Bianchi()\u003c/strong\u003e: Calculates the Bianchi identities\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_Bianchi()\u003c/strong\u003e: Displays the Bianchi identities\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eAll Bianchi identities are available under their names\nin the order they are given in the reference:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eBI1, BI2, BI3, BI4, BI5, BI6, BI7, BI8, BI9, BI10, BI11\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePetrov invariants I, J, K, L, N (Kramer p.121, 9.6; p.54, 4.19)\u003c/strong\u003e:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cem\u003e(also check diagram Fig. 9.1 on p. 122)\u003c/em\u003e\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_PetrovinvINP()\u003c/strong\u003e: Calculates the Petrov invariant I\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_PetrovinvJNP()\u003c/strong\u003e: Calculates the Petrov invariant J\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_PetrovinvKNP()\u003c/strong\u003e: Calculates the Petrov invariant K\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_PetrovinvLNP()\u003c/strong\u003e: Calculates the Petrov invariant L\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_PetrovinvNNP()\u003c/strong\u003e: Calculates the Petrov invariant N\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eAll Petrov invariants are available under their names:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePetrovinvINP, PetrovinvJNP, PetrovinvKNP, PetrovinvLNP, PetrovinvNNP\u003c/strong\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePetrov type of the spacetime\u003c/strong\u003e:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePetrov_frominvariants()\u003c/strong\u003e: Calculates the Petrov type using I, J, K, L, N.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePetrov_fromWeyl()\u003c/strong\u003e: Calculates the Petrov type using the Weyl components\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eMassive Klein-Gordon equation\u003c/strong\u003e:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ekleingordon(Phi,M2)\u003c/strong\u003e: Calculates the Klein-Gordon equation for a massive scalar field where Phi is a scalar field on the manifold and M2 is the mass of the scalar field. (Ref.: G. Silva-Ortigoza, Rev. Mex. Fis. 4, 543 (1996))\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003eThe result is available under the name: \u003cstrong\u003ekgNP\u003c/strong\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eMassive Dirac equation\u003c/strong\u003e:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003edirac(f1,f2,g1,g2,M2)\u003c/strong\u003e: Calculates the Dirac equation for a massive spinor field where f1,f2,g1,g2 are the components of the spinor field (defined as scalar fields on the manifold) and M2 is the mass of the spinor field. (Ref.: S. Chandrasekhar, \"Mathematical Theory of Black Holes\", Oxford Univ. Press, New York (1983), p.544.)\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003eThe result is available under the name: \u003cstrong\u003ediracNP\u003c/strong\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eSL(2,C) Transformations\u003c/strong\u003e: SL(2,C) transformations as defined in Carmeli and Kaye, Annals of Physics 99, 188 (1976).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003etype_A_transformation(z)\u003c/strong\u003e: Calculates the Type A transformations where z is a complex variable.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_type_A_transformation()\u003c/strong\u003e: Shows the results of the Type A transformations.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003etype_B_transformation(z)\u003c/strong\u003e: Calculates the Type B transformations where z is a complex variable.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_type_B_transformation()\u003c/strong\u003e: Shows the results of the Type B transformations.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003etype_C_transformation(z)\u003c/strong\u003e: Calculates the Type C transformations where z is a complex variable.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_type_C_transformation()\u003c/strong\u003e: Shows the results of the Type C transformations.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe results are available under their names:\n\u003cstrong\u003elNP_trA, nNP_trA, mNP_trA, mbarNP_trA, kappaNP_trA, rhoNP_trA, etc., Psi0NP_trA, Psi1NP_trA, etc., Phi00NP_trA, Phi01NP_trA, etc.\u003c/strong\u003e and the same notation for \u003cstrong\u003elNP_trB, nNP_trB, etc.\u003c/strong\u003e and \u003cstrong\u003elNP_trC, nNP_trC, etc.\u003c/strong\u003e for other types.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ecalculate_allNP()\u003c/strong\u003e: Runs the following functions:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003ecalculate_spincoefficients()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003ecalculate_Weyl()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003ecalculate_Ricci()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003ecalculate_NPeq()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003ecalculate_Bianchi()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003ePetrov_fromWeyl()\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eshow_allNP()\u003c/strong\u003e: Runs the following functions:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eshow_spincoefficients()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eshow_Weyl()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eshow_Ricci()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eshow_NPeq()\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp dir=\"auto\"\u003eshow_Bianchi()\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/article\u003e","renderedFileInfo":null,"shortPath":null,"symbolsEnabled":true,"tabSize":4,"topBannersInfo":{"overridingGlobalFundingFile":false,"globalPreferredFundingPath":null,"showInvalidCitationWarning":false,"citationHelpUrl":"https://docs.github.com/github/creating-cloning-and-archiving-repositories/creating-a-repository-on-github/about-citation-files","actionsOnboardingTip":null},"truncated":false,"viewable":true,"workflowRedirectUrl":null,"symbols":{"timed_out":false,"not_analyzed":false,"symbols":[{"name":"SageNP: Newman-Penrose calculations for SageMath.","kind":"section_1","ident_start":2,"ident_end":51,"extent_start":0,"extent_end":262,"fully_qualified_name":"SageNP: Newman-Penrose calculations for 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This work is funded by TUBITAK 1001 Program, Grant Number 123R114.
The class SageNP includes functions for some calculations defined in the Newman-Penrose formalism. The code is based on SageManifolds.
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Tolga Birkandan (Corr.: birkandant@itu.edu.tr)
-
Efe Özyürek
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Special thanks to Eric Gourgoulhon
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SageNP.py: Main file to import in SageMath.
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SageNP_Tutorial.ipynb: Tutorial (ipynb file) - Definitions and calculations for the Schwarzschild (with covariant null-tetrad vectors) and Reissner-Nordstrom (with contravariant null-tetrad vectors) spacetimes.
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SageNP_Tutorial.pdf: Tutorial (PDF file)
Main reference for all definitions and calculations:
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, "Exact Solutions of Einstein’s Field Equations", 2nd ed. Cambridge: Cambridge University Press, 2003.
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We will use the Metric signature: (- + + +)
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For the null-tetrad vector names, the ref. book uses (k,l,m,mbar). However, in the code we will use (l,n,m,mbar) like the rest of the literature. Therefore, one should set k->l, l->n in the ref. book
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Products of the vectors are given by: ln = -1, mmbar = 1, all others zero.
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The metric is found using the covariant null-tetrad vectors as:
g = -2ln + 2mmbar
and,
g = [[0 1 0 0], [1 0 0 0], [0 0 0 -1], [0 0 -1 0]]
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Please check the reference book for the details and further definitions.
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Import the class:
from SageNP import NewmanPenrose
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Define your manifold:
MyManifold = Manifold(4 , 'MyManifold', r'\mathcal{Man}')
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Define your coordinates:
MyCoordinates.<t,r,th,ph> = MyManifold.chart(r't r th:\theta ph:\phi')
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Define the metric functions (if needed):
var('M') Delta=r^2-2Mr
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Enter null tetrad elements:
lvec=[1,-(r^2)/Delta,0,0]
nvec=[Delta/(2*r^2),1/2,0,0]
mvec=[0,0,(-r/sqrt(2)),(-I*r/sqrt(2))*sin(th)]
mbarvec=[0,0,(-r/sqrt(2)),(I*r/sqrt(2))*sin(th)]
- Here, the element ordering is the same as the coordinate ordering. (The first element is the t element, the second is the r element, etc.)
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Define an object of the class:
schw=SageNP(MyManifold,MyCoordinates,lvec,nvec,mvec,mbarvec,'covariant')
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Here, our null-tetrad vectors lvec, nvec, mvec and mbarvec are covariant. Thus we used the keyword 'covariant'.
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If they were contravariant, then we should use the keyword 'contravariant'.
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Once the object is defined, the code calculates the metric and displays it on the screen. It is recommended that you check your metric.
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All page and equation numbers belong to the reference book.
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test_nulltetrad(): Checks the products of the vectors ln = -1, mmbar = 1, all others zero.
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Spin coefficients (Page 75-76, Eq.(7.2)):
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calculate_spincoefficients(): Calculates the spin coefficients.
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show_spincoefficients(): Displays the spin coefficients
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All spin coefficients are available under their names:
kappaNP, kappabarNP, tauNP, taubarNP, sigmaNP, sigmabarNP, rhoNP, rhobarNP, piNP, pibarNP, nuNP, nubarNP, muNP, mubarNP, lambdaNP, lambdabarNP, epsilonNP, epsilonbarNP, gammaNP, gammabarNP, betaNP, betabarNP, alphaNP, alphabarNP
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Directional derivatives (Page 43, Eq.(3.82)):
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DlNP(X): Given X, calculates the D derivative (l direction).
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DeltanNP(X): Given X, calculates the Delta derivative (n direction)
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deltamNP(X): Given X, calculates the delta derivative (m direction)
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deltambarNP(X): Given X, calculates the deltabar derivative (mbar direction)
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Commutators (Page 77, Eq.(7.6)):
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The right-hand sides of the commutation relations are calculated.
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Deltan_Dl_commNP(X): Given X, calculates the [Delta,D] commutator.
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deltam_Dl_commNP(X): Given X, calculates the [delta,D] commutator.
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deltam_Deltan_commNP(X): Given X, calculates the [delta,Delta] commutator.
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deltambar_deltam_commNP(X): Given X, calculates the [deltabar,delta] commutator.
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Weyl tensor components (Page 38, Eq.(3.59)):
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calculate_Weyl(): Calculates the Weyl tensor components.
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show_Weyl(): Displays the Weyl tensor components.
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All Weyl tensor components are available under their names:
Psi0NP, Psi1NP, Psi2NP, Psi3NP, Psi4NP
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Ricci components (Page 78, Eq.(7.10-7.15)):
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calculate_Ricci(): Calculates the Ricci tensor components.
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show_Ricci(): Displays the Ricci tensor components.
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All Ricci tensor components are available under their names:
Phi00NP, Phi01NP, Phi10NP, Phi02NP, Phi20NP, Phi11NP, Phi12NP, Phi21NP, Phi22NP, LambdaNP
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Ricci (Newman-Penrose) equations (Page 79, Eq.(7.21)):
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All Newman-Penrose equations are defined as 0 = -(left hand side)+(right hand side) of the equations.
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calculate_NPeq(): Calculates the Newman-Penrose equations
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show_NPeq(): Displays the Newman-Penrose equations
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All Newman-Penrose equations are available under their names in the order they are given in the reference:
NPeq1, NPeq2, NPeq3, NPeq4, NPeq5, NPeq6, NPeq7, NPeq8, NPeq9, NPeq10, NPeq11, NPeq12, NPeq13, NPeq14, NPeq15, NPeq16, NPeq17, NPeq18
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Bianchi identities (Page 81, Eq.(7.32)):
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All Bianchi identities are defined as 0 = -(left hand side)+(right hand side) of the equations.
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calculate_Bianchi(): Calculates the Bianchi identities
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show_Bianchi(): Displays the Bianchi identities
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All Bianchi identities are available under their names in the order they are given in the reference:
BI1, BI2, BI3, BI4, BI5, BI6, BI7, BI8, BI9, BI10, BI11
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Petrov invariants I, J, K, L, N (Kramer p.121, 9.6; p.54, 4.19):
(also check diagram Fig. 9.1 on p. 122)
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calculate_PetrovinvINP(): Calculates the Petrov invariant I
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calculate_PetrovinvJNP(): Calculates the Petrov invariant J
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calculate_PetrovinvKNP(): Calculates the Petrov invariant K
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calculate_PetrovinvLNP(): Calculates the Petrov invariant L
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calculate_PetrovinvNNP(): Calculates the Petrov invariant N
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All Petrov invariants are available under their names:
PetrovinvINP, PetrovinvJNP, PetrovinvKNP, PetrovinvLNP, PetrovinvNNP
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Petrov type of the spacetime:
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Petrov_frominvariants(): Calculates the Petrov type using I, J, K, L, N.
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Petrov_fromWeyl(): Calculates the Petrov type using the Weyl components
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Massive Klein-Gordon equation:
kleingordon(Phi,M2): Calculates the Klein-Gordon equation for a massive scalar field where Phi is a scalar field on the manifold and M2 is the mass of the scalar field. (Ref.: G. Silva-Ortigoza, Rev. Mex. Fis. 4, 543 (1996))
- The result is available under the name: kgNP
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Massive Dirac equation:
dirac(f1,f2,g1,g2,M2): Calculates the Dirac equation for a massive spinor field where f1,f2,g1,g2 are the components of the spinor field (defined as scalar fields on the manifold) and M2 is the mass of the spinor field. (Ref.: S. Chandrasekhar, "Mathematical Theory of Black Holes", Oxford Univ. Press, New York (1983), p.544.)
- The result is available under the name: diracNP
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SL(2,C) Transformations: SL(2,C) transformations as defined in Carmeli and Kaye, Annals of Physics 99, 188 (1976).
type_A_transformation(z): Calculates the Type A transformations where z is a complex variable.
show_type_A_transformation(): Shows the results of the Type A transformations.
type_B_transformation(z): Calculates the Type B transformations where z is a complex variable.
show_type_B_transformation(): Shows the results of the Type B transformations.
type_C_transformation(z): Calculates the Type C transformations where z is a complex variable.
show_type_C_transformation(): Shows the results of the Type C transformations.
The results are available under their names: lNP_trA, nNP_trA, mNP_trA, mbarNP_trA, kappaNP_trA, rhoNP_trA, etc., Psi0NP_trA, Psi1NP_trA, etc., Phi00NP_trA, Phi01NP_trA, etc. and the same notation for lNP_trB, nNP_trB, etc. and lNP_trC, nNP_trC, etc. for other types.
-
calculate_allNP(): Runs the following functions:
-
calculate_spincoefficients()
-
calculate_Weyl()
-
calculate_Ricci()
-
calculate_NPeq()
-
calculate_Bianchi()
-
Petrov_fromWeyl()
-
-
show_allNP(): Runs the following functions:
-
show_spincoefficients()
-
show_Weyl()
-
show_Ricci()
-
show_NPeq()
-
show_Bianchi()
-
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