Bruker:Phidus/sandkasse-23

• Tidsdilatasjon
• Ø. Grøn, Relativitetsteorien og GPS-systemet, FFV 3, 69 (2006).
• Ø. Grøn, om Tesla, FFV 3 (2020)
• FFV, Utgaver 2023

• Skriv om norsk WP differensialgeometri, med moderne innhold ytre deriverte etc
• E. Cartan, Moving frames
• SNL, Riemannsk geometri
• SNL, Differensialgeometri
• Nynorsk WP, differensialgeometri, kort, men mye bedre en no-versjon
• Engelsk WP, Riemannian Geometry
• Fransk WP, Géométrie Riemannienne
• J.M. Lee, Introduction to Riemannian Manifolds, Springer Graduate Texts in Mathematics, New York (1997). ISBN 978-3-319-91754-2. Stored on my iPad

Darboux frame

• Fransk WP, Repère de Darboux, her skilles klart ut at disse bare finnes på flater, og ikke for kurver som kan man få inntrykk av på andre wikis. Etter å ha skrevet om krumning i Riemann rom med ortonormerte former.

• H.W. Guggenheimer, Differential Geometry], book in Oslo with Lie derivatives and Darboux frames on surfaces like Frenet frames along curves and Cartan frames in Riemann spaces.
• Tysk WP, Lie-Ableitung wo

${\displaystyle {\mathcal {L}}_{X}Y=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}(F_{t}^{*}Y)}$ ,

using pullback.

• NN, Pullback and Lie derivatives bases on Brian Greene lectures. Saved in 2021. Can from same address also find four similar lectures on Lie groups and fiber bundles.
• B. Owren NTNU, A crash course on manifolds, tangentvektorer og diff. forms.
• Stackexchange, Lie derivatives, found by different calculations
• UiO MAT4270, Lie groups and algebras, lectures (2012). See also extra notes by Geir Ellingsrud.
• UK, Lie derivatives from active coordinate transformations via pushforwards. Most useful.
• M.E. Fels, Introduction to Differential Geometry, in great detail with pullbacks, pushforwards og Lie derivative p. 174. Stored in 2021.
• Fionn, Line congruences and Lie derivatives. Also instructive.
• J.L. Coolidge, A History of Geometrical Methods, much about Lie sphere geometry and also Lie groups in good presentation in this Google Book.

• Juan Gerardo Alcazara, Heidi E.I. Dahl and Georg Muntingh, Symmetries of Canal Surfaces and Dupin Cyclides, much up-to-date background

• Tysk WP, Dupinsche Zyklide, good and gives Dupin's definition based on the three fixed spheres.

• Student, Dupin Cyclides, BS student thesis stored in 2021.
• Mathcurve, Dupin Cyclides with math and many illustrations
• U Oxford, Dupin Cyclides intro with figures
• U Wien, Dupin Cyclides with more math.
• Encyclopedia of math, Dupin Cyclide, concise summary.
• Chandru, Dutta and Hoffmann, On the Geometry of Dupin Cyclides, with historical, clear presentation and stored in 2021. Text without figures can be found here. Published by Springer here.
• Emanuel Huhnen-Venedey, Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides, diploma thesis using Lie sphere geometry.

• Engelsk WP, Laplace equation
• Italiensk WP, Funzione armonica med eksempler i 2-dim. Tysk WP har også 2-dim løsning i polarkoordinater
• Italiensk WP, Equazione di Laplace, med radiell løsning i n-dim
• Finnes allerede på norsk harmonisk funksjon
• Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, ISBN 0-8053-7002-1.
• NN, Separation of variables with simple examples
• Oxford, 2-dim Laplace equation, separation variables, also in polar coordinates.
• Fransk WP har 3-dim solutions inkludert Bessel-funksjoner i 3-dim for polarkoordinater
• Fitzpatrick, U Texas, Laplace in 3-dim cylindrical coordinates
• NN, Laplace eq with cylindrical coordinates
• Tysk WP om Laplace ligning
• UC Davis, Laplace equation and mean value theorem
• J. Oprea, The Mathematics of Soap Films: Explorations with Maple, American Mathematical Society, Providence RI (2000). ISBN 978-0-821-82118-3.
• J. Mathews and R.L. Walker, Mathematical Methods of Physics, W.A. Benjamin, New York (1970). ISBN 0-8053-7002-1.

Sfærisk harmoniske funksjoner som ofte omtales som kuleflatefunksjoner, er i stor grad bygget opp av assosierte Legendre-polynom. Den vanligste definisjonen som blir brukt i kvantemekanikken, er

${\displaystyle Y_{\ell ,m}(\theta ,\phi )={\sqrt {\frac {(2\ell +1)(\ell -m)!}{4\pi (\ell +m)!}}}\ P_{\ell }^{m}(\cos \theta )\ e^{im\phi }\qquad -\ell \leq m\leq \ell }$

Disse funksjonene er komplekse. Med de konvensjonene som er benyttet her, er

${\displaystyle Y_{\ell ,m}^{*}(\theta ,\phi )=(-1)^{m}Y_{\ell ,-m}(\theta ,\phi )}$

Som funksjoner av kuleflatekoordinatene (θ,φ) gir de direkte den romlige fordeling av elektronene i et atom som beskrevet av Schrödinger-ligningen. Det fører til å innordne elektronene i forskjellige elektronskall som i stor grad bestemmer deres kjemiske egenskaper.

• NN, Kuleflatefunksjoner og homogene polynom
• UPenn, Kuleflatefunksjoner og homogene polynom, også 2-dim i begynnelsen

• B. Gjevik, Lectures on Tides, UNIS, Longyearbyen (2011).

• Butikov, Theory of tides
• Fransk WP, Simplest Legendre expansion
• NN, Tidal potential with P₂(cosθ)

• Tidevannskrefter finnes allerede. Legg merke til også tidevannskraft som er noe gansk annet.
• Så må derfor omgjøre siden tidevannskrefter til tidekraft som får en underseksjon som heter tidevann?
• B. Gjevik, Lectures on Tides, UNIS, Longyearbyen (2011).
• Ref book, David Ross, Introduction to Oceanography
• Caltech, Tidal tensor

• Se tilsvarende seksjon i elektrostatikk
• Gravitasjonspotensial i kvanteteori.

• Fransk WP inneholder litt ganske mye multipolutvikling
• Kanskje ta med her potensialet fra en axisymmetrisk massefordeling og Legendre-polynom som finnes på engelsk WP. Også ta med historien om Laplace som innførte det, og Laplace-ligningen.

• Multipole expansion and MacCullach's formula also presented in fransk WP with Legendre expansion

• Fitzpatrick, U Texas, Fitzpatrick.
• Fitzpatrick, U Texas, Earth flattening
• Allerede finnes på norsk WP flattrykning og World Geodetic System som bruker sfæroider.
• Moments of inertia of spheroid given in English WP.
• Engelsk WP, Gravity acceleration formulas as function of latitude φ and standard gravity value here.
• IUPAC Gold Book, Standard acceleration of free fall

• MacTutor, Biography
• Walter William Rouse Ball, A Short Account of the History of Mathematics. 1908,
• Oxford Project, All about Wallis, also cryptography, many webpages

For bevis ved direkte integrasjon, se Pollack & Stump book on EM, p.53. Purcell book on EM in Berkeley series has on p. 27, saying that the proof he finally published in 1686 had delayed his theory of gravitation with almost 20 years.

• S. Chandrasekhar, Newton's Principia for the Common Reader, with Newton's original proofs and discussions.
• C. Schmid, Newton's superb theorem: An elementary geometric proof, American Journal of Physics, 79 (5), 536-539 (2011). Kopie auf Berliner Schreibtisch. With Newton's historical comments to its discovery.

Newton's "superb theorem" for the gravitational inverse-square-law force states that a spherically symmetric mass distribution attracts a body outside as if the entire mass were concentrated at the center. This theorem is crucial for Newton's comparison of the Moon's orbit with terrestrial gravity (the fall of an apple), which is evidence for the inverse-square-law. Newton's geometric proof in the Principia "must have left its readers in helpless wonder" according to S. Chandrasekhar and J.E. Littlewood. In this paper we give an elementary geometric proof, which is much simpler than Newton's geometric proof and more elementary than proofs using calculus.

Newton's superb theorem: An elementary geometric proof (PDF Download Available). Available from: https://www.researchgate.net/publication/221660870_Newton%27s_superb_theorem_An_elementary_geometric_proof [accessed Sep 29, 2017].

• English WP, Newton's Shell Theorem w/derivations. Even better on Italian and Dutsch versions!! French version is extremely short, pure geometry?
• NN, Derivation of Shell Theorem
• R. Borghi, On Newton's shell theorem, European Journal of Physics 35, 028003 (2014). With ref. to book by Chandrasekhar about Newton's Principe for laymen.

• I GR tilsvarer dette Jebsen-Birkhoffs teorem

• C.M. Hirata, Caltech, Reissner-Nordstrøm BH, lecture 25 in long series of lectures at Caltech.
• Chris Hirata, Caltech, Ph 236: General Relativity, Caltech lecture series following MTW
• P. Jerstad og B. Sletbak, Rom Stoff Tid, 3FY, J.W. Cappelens Forlag, Oslo (1998). ISBN 82-02-17155-5.
• D. Isaachsen, Lærebok i Fysikk for Realgymnaset, H. Aschehoug & Co, Oslo (1958).

• Engelsk WP, Latidude som inneholder referanse til Newton som viste i Principia at roterende Jord bestående av væske tok spheroide form.
• Fitzpatrick, U Texas, Rotating hydrodynamics and spheroids
• Clint Conrad, UiO, Centrifugal potential
• Clint Conrad, UiO, Earth shape and J_2
• Stargaze, Rotating Earth, lots of web pages
• Caltech, Shape of Earth with convincing calculations.
• Caltech, Earth gravity field lectures
• Purdue, Gravitational potential of Earth
• Bolvan, U Texas, Multipole expansion in general, excellent
• Fitzpatrick, U Texas, MacCullagh's formula, most direct derivation
• NN, MacCullagh's formula, similar derivation
• Yale U, History of gravity formula
• Fransk WP, Axial multipole expansion, with charges along z-axis. Griffith problem 3.38 with charged line segment along z-axis solved here.
• MIT, Legendre polynomials and multipole expansion
• Errede, UIUC, Multipole expansions, starts with discrete charges
• NN, Axial multipoles for rotating neutron star
• U Chicago, Multipole expansion, shape of Earth and moments of inertia and J_2.
• MIT, Gravitational field of Earth and multipole expansion. Using λ for latitude and θ for co-latitude.
• Turcotte and Shubert, U Chicago, Geodynamics, Chapter 5 can be found in Books on iPad. Includes connection to reference ellipsoid.
• Engelsk WP, Somigliana formula for normal gravity
• Engelsk WP, Legendre polynomials, example with point charge outside origin
• Oxford, Calculating shape of Earth using Legendre polynomials

• C.B. Boyer and U.C. Merzbach, A History of Mathematics, John Wiley & Sons, New Jersey (2010). ISBN 0-470-52548-7.
• Todhunter, Short Legendre bio
• Full vitenskapelig bio: Encyclopedia.com
• Morris Kline, Mathematical Thought From Ancient to Modern Times, Volume 3, contains in great detail Legendre derivation of his polynomials.
• Britannica, Adrien Marie Legendre, bio
• Encyclopedia.com, Legendre bio
• Legendre polynomials
• Runeberg, Nordisk Familjebok, andra upplagan, 1912.
• History of quadratic residues
• Essentials about famous mathematicians
• Boyer, History of Mathematics
• Wikipedia, Spherical harmonics with some history
• Wikipedia, Laplace expansion
• Wikipedia, Gravitational potential
• Google Book, History, p.404
• MIT, Earth Gravitational Potential, with derivation of Legendre polynomials from binomial formula.

• Sfærisk harmoniske funksjoner skulle heller bli kalt kuleflatefunksjoner som på tysk?
• Multipole expansion is very compactly done on English WP Multipole expansion by expanding Coulomb potentials in terms of higher derivatives. Faster method than using Legendre expansion.
• Multipole expansion and MacCullach's formula also presented in fransk WP with Legendre expansion
• Mathpages, Why solution Laplace equation is harmonic, and why solutions satisfy mean value theorem.
• NN 1938, History of Legendre and Laplace and their polynomials
• U Penn, Spherical harmonics and homogeneous polynomials
• Poul Olesen, Comments to Mathews and Walker, with short derivation of Rodrigues formula

Brukbare figurer

Masse ligger i ro i system som akselereres oppover. alakakak gfddyyu

• Tyngdekraft

• Newtons gravitasjonslov. Lag ny side Newtons skallteorem a la engelsk WP Shell theorem.
• Må også flytte på Newtons lov om universell gravitasjon som allerede er omdirigert til tyngdekraft.
• Kilogram er velskrevet og inneholder ny def basert på Plancks konstant. Nevner også Kibble-vekt
• Tyngde, tyngdekraft og fiktiv kraft. Kanskje i stedet skrive om treghetskraft.
• SNL, gravitasjon hvor det starter ut med å skille mellom tyngdekraft og gravitasjonskraft. Her er også fornuftig beskrivelse av tyngdefelt.
• Gravitasjonsfelt og tyngdeakselerasjon kan evt. slås sammen. Måling av g i gravimetri.
• Gravitasjonspotensial og gravitasjonsfelt
• Tyngdekraft som er omdirigert fra gravitasjon. Dette må omgjøres. Se også tyngde og tyngdeakselerasjon. Skal tyngdeakselerasjon g settes lik tyngdefeltet = gravitasjonsfelt som på tysk WP hvor det heter Schwerefeld? Bedre kanskje å innføre i Newtonsk teori gravitasjonsfelt som gradient av gravitasjonspotensialet, igjen som på tysk WP? Tyngdefeltet g får da generelt også bidrag fra akselerasjon. Også tenk på vekt og på engelsk WP weight som er viktig. Verdien for g på Jorden beregnet fra dens masse M og radius R, er gjort på italiensk WP. I artikkelen om tyngdekraft kan ta med Newtons argument med eple og måne a la boken til Cohen.
• Engelsk WP, Accelerated reference frame.
• Roterende referansesystem som gir utledning av sentrifugalkraften. I denne sammenhengen opptrer Euler-kraft som er omtalt på tysk WP. Ble innført av Lanczos i 1949 i sin bok om variasjonsregning: The variational principles of mechanics, University of Toronto Press 1949, p. 103: This third apparent force has no universally accepted name. The author likes to call it the Euler force in view of the outstanding investigations of Euler in this subject.
• Sentripetalakselerasjon er en vektor som sier hvordan hastigheten v forandrer seg. I to dimensjoner kan akselerasjonen a dekomponeres på to, gjensidig ortogonale vektorer. For vilkårlig bevegelse mest generelt å benytte tangentvector t og normalvektor n. Da er sentripetal akselerasjon gitt ved v²/ρ hvor ρ er krumningsradius til kurven, som vist av fisicalab. Eller hvis man bruker 2-dim polarkoordinater r og θ som gjort i MIT forelesning, så er en komponent av a langs e_r og en langs e_θ. Den første av disse inneholder v²/r som er den vanlige definisjonen. Grunnen er at den er forårsaket av en kraft som virker langs r, det vil si en sentralkraft som gravitasjon. Og det heter jo sentripetalakselerasjon, det vil si peker mot sentrum.
• Fitzpatrick, U Texas, Rotating reference frames, excellent

• Konvensjoner kan finnes på tysk WP

• Newtons gravitasjonslov. Lag ny side Newtons skallteorem a la engelsk WP Shell theorem.
• Må også flytte på Newtons lov om universell gravitasjon som allerede er omdirigert til tyngdekraft.
• C. Speake and T. Quinn, The search for Newton’s constant, Physics Today 67 (7) 7, 27-33 (2014).
• Kilogram er velskrevet og inneholder ny def basert på Plancks konstant. Nevner også Kibble-vekt
• Tyngde og tyngdekraft
• SNL, gravitasjon hvor det starter ut med å skille mellom tyngdekraft og gravitasjonskraft
• Gravitasjonsfelt og tyngdeakselerasjon kan evt. slås sammen. Måling av g i gravimetri.
• Gravitasjonspotensial og gravitasjonsfelt
• Tyngdekraft som er omdirigert fra gravitasjon. Dette må omgjøres. Se også tyngde og tyngdeakselerasjon. Skal tyngdeakselerasjon g settes lik tyngdefeltet = gravitasjonsfelt som på tysk WP hvor det heter Schwerefeld? Bedre kanskje å innføre i Newtonsk teori gravitasjonsfelt som gradient av gravitasjonspotensialet, igjen som på tysk WP? Tyngdefeltet g får da generelt også bidrag fra akselerasjon. Også tenk på vekt og på engelsk WP weight som er viktig. Verdien for g på Jorden beregnet fra dens masse M og radius R, er gjort på italiensk WP. I artikkelen om tyngdekraft kan ta med Newtons argument med eple og måne a la boken til Cohen.

• D. Overbye, LIGO Detects Fierce Collision of Neutron Stars for the First Time, New York Times, 16. oktober, 2017.
• Science 2017
• Gravitasjonskonstanten. Se også på masse som må forbedres.
• F.K. Hansen, Celestial Mechanics, forelesninger AST1100 i astrofysikk, UiO 2008.
• Geoide og WGS84, i.e. referanseellipsoide.
• München, Geoid tutorial]
• Tidrom, inneholder ref til bok fra 1974 hvor tidrom er benyttet, Ytrehus, Ottar (1974). Tidrommet : En innføring i det matematiske grunnlaget for den spesielle relativitetsteori. Oslo: Cappelen. ISBN 8202005078. 

• NN, Shape of Earth for beginners
• Stackexchange, Effects of rotation with a little calculation
• Caltech, Rotating Earth with calculation of bulge. Stored in Berlin WikiWorks 2018-3
• Norsk ref, Lien og Løvhøiden, Generell fysikk for universiteter og høgskoler, Bind 1 med innholdsfortegnelse som omtaler Newton gravitasjon, tyngdekraft etc.

• C. O’Raifeartaigh et al, One Hundred Years of the Cosmological Constant:, discusses original sign of cosmological constant etc.
• Sutton, Grav. waves with line element. In Oslo WikiWorks

• Engelsk WP, Introduction to GR, noe å ta hensyn til? Også den franske versjon er god, med flere figurer.
• Galina Einstein, Einstein, Schwarzschild and Perihelion Motion of Mercury and the rotating disk.
• Med omdirigering hit av Einsteins feltligninger. Ta her med alt om Riemann, Ricci og Einstein tensorer.
• Ta med Ricci-tensor her samt gjør bruk av Bianchi-identitet fra Riemanns differensialgeometri
• Riemanns differensialgeometri må inneholde utledning av geodetisk kurve fra ekstremalisering av integralet av ds.
• Engelsk WP, Rindler coordinates, with many historical references and comments. For Rindler-koordinater skriv også om Unruh effect.
• Engelsk WP, Ehrenfest paradox, with many historical references and comments. Kanskje jet skal kalle dette Generell relativitet på roterende skive?

• Physics Today 67 (2014), History of Expanding Universe, with links to papers
• SNL, Kosmologiens historie
• NN, Cosmological Principle and Symmetric Spaces, in Berlin WikiWorks
• Barbara Ryden, Introduction to Cosmology, whole book. Stored as Cosmology Barbara Ryden.
• NL, Cosmological Principle and RW metric
• FNAL, Intro Cosmology with clear discussion of Copernican and cosmological principle
• Kosmologi, det observerbare universet og Hubbles lov.
• F. Ravndal, Cosmolological Physics, FYS-5130 UiO
• Det kosmologiske prinsipp også godt fortalt på fransk WP
• U Cambridge, Newtonian cosmology
• J. Schombert, U Oregon, History cosmology for everyone
• J. Schombert, U Oregon, Copernican and Cosmological Principles, with clear discussion of implications of isotropy and homogeneity.
• J. Schombert, U Oregon, Lectures on Cosmology, wunderbar schön.

Bør ta med før kosmologi en oppsummering av tester av GR, inkludert Shapiros tidsforsinkelse eller Shapiro-forsinkelse

Utvid kosmologi, så skriv mer teknisk om Fysikalsk kosmologi eller kanskje bedre Kosmologisk fysikk a la Physical Cosmology på engelsk WP og mange andre språk. Likedan skriv ny side Relativistisk kosmologi som en parallell til Friedmann-Lemaître–Robertson–Walker metric på engelsk WP og andre språk. Møllers bok inneholder mye bra om Einsteins Univers som kan være både elliptisk og sfærisk.

• Engelsk WP, Friedmann equations
• Galina Weinstein, History Einstein two-body problem and Einstein-Rosen bridge
• Galina Weinstein, Einstein discovery of grav. waves

Relativistisk kosmologi

• Engelsk WP, Friedmann-Lemaître–Robertson–Walker metric som eksempel.
• Engelsk WP, Friedmann equations

Kosmologisk fysikk

• Her nuclear, photons, particle physics in the Universe.
• Svensk WP, Kosmo konst har god beskrivelse med numerics for Λ
• Italiensk WP har mange gode kosmologiske artikler, f.ex. om kvintessens
• A. Einstein 1917, Kosmologische Betracthungen zur allgemeinen Relativitätstheorie, original
• O’Raifeartaigh, Physics Today, Einstein's static universe
• O’Raifeartaigh, Einstein's cosmological model, great detail
• U. München, Einstein's static universe
• NN, Discussions around Λ and good comments about static and expanding de Sitter universe. The first had clocks that gave redshift in static field, while the expanding had cosmic time and grav. redshift.
• APS, Einstein and the static universe
• Carroll, Einstein's static universe

• Sutton, Grav. waves with line element. In Oslo WikiWorks
• Robert Gilmore, Lie Groups, Algebras and Applications
• Robert Gilmore, Lie Groups, Physics and Geometry, excellent Chapter 12 nice presentation of Riemann Symmetric Spaces.

• GENI, Sven Jesperson Ravndal
• S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, New York (1972). ISBN 0-471-92567-5.
• C. Møller, The Theory of Relativity, Oxford University Press, England (1960).

• Legg merke til at geodetiske koordinater er omdirigert til geografiske koordinater
• Ifølge MTW p. 286 er normalkoordinater alltid slike hvor Christoffel-symbolene er null i et punkt. Men dette kan gi opphav til mange forskjellige koordinatsystem. Det som er spesielt med riemannske normalkoordinater (som også kalles geodetiske koordinater) er at neste, høyere ordens ledd kan beregnes og er gitt ved Riemanns krumningstensor.
• Blau, p. 67 takes as simplest example polar coordinates in 2-dim. Geodesics through given point are radial lines, two of which can be taken as riemannske normalkoordinater når de står normal på hverandre.
• Weinberg in his Cosmology book writes on p. 340 that metrics of the form ${\displaystyle ds^{2}=dy^{2}-g_{ij}(x,y)dx^{i}dx^{j}}$ are in Gaussiske normalkoordinater. Det vil si også free-falling Robertson-Walker koordinater omtalt på side 413.
• As discussed in Blau p.71, for the metric ${\displaystyle ds^{2}=dy^{2}-g_{ij}(x,y)dx^{i}dx^{j}}$ the curves ${\displaystyle x^{i}=const}$ are geodesics, i.e. also for Robertson-Walker.
• Blau p. 94 viser hvordan man kan lage riemannske normalkoordinater på nordpolen til en kuleflate.
• L. Brewin, Monash, Riemann Normal Coordinates, excellent with detailed derivation of Riemann correction from his Habil. Vorlesung 1854. Stored in Oslo WikiWorks.
• Svante Jansson, Uppsala, Riemann Geometry and Maps, lots of applications. Stored in Oslo WikiWorks as RiemannGeometry-maps.
• NN, Normal Coordinates, very nice as mathematicians do it! Stored in Oslo WikiWorks as Normal Coordinates. This is part of Lectures on Riemannian Geometry which are very nice, similar to MTW.
• Student NTNU, Geodesics on Surfaces, detailed calculations, also normal coordinates.
• Mathoverflow, Riemann's formula for the metric in a normal neighbourhood, historical background and sensible discussion.
• L.P. Eisenhardt (1926), Riemannian Geometry, original book! Riemannian, normal and geodesic coordinates p. 53.
• Hvis hvordan kan lage generell, kovariant fysikk fra lokalt Minkowski-rom. Erstatte partialderiverte med kovariant deriverte.
• Skill mellom lokalt Minkowski-rom med normale koordinater i et lite område og ortogonal basis som bare gjelder i et punkt og er karakterisert med latinske indekser. Godt fremstilt hos Carroll.
• Mathpages, Riemann coordinates where Riemann normal coordinates are calculated for a 2-dim example. Understanding this, everything should be clear! This is part of very interesting book on Relativity
• Mathpages, Discussion of equivalence principle and Riemann normal coordinates
• Blau, GR lectures, contains everything. Redshift good discussed pp. 85, Riemann normal coordinates in Sections 2.11 and 7.9. Examples are θ and φ on spherical surface, Robertson-Walker coordinates where each observer follows geodesic. Also maximal symmetric spaces, Schwarzschild metric and Christoffel symbols calculated pp 400 etc, also done in Weinberg...... Stored in WikiWorks as GR-Blau and Dropbox..
• Engelsk WP, Normal coordinates, also discusses generalized spherical coordinates in given point.
• Tysk WP, Riemannsche Normalkoordinaten shows that metric in local region takes form (Riemann 1854)

${\displaystyle g_{\mu \nu }(q)=\eta _{\mu \nu }-{\frac {1}{3}}R_{\mu \lambda \nu \rho }q^{\lambda }q^{\rho }+...}$

• NN, Riemann normal coordinates
• Kachelriess NTNU, Lecture Notes Gravitation and Cosmology

Kilder generell relativitetsteori # # #

Med overgang far lokal, flat elevator til generelt koordinatsystem. Også geodet fra kovariant Lagrange-funksjon.

• Tysk WP, Hva er en observatør
• Engelsk WP, Shapiro time delay kan bli Shapiros tidsforsinkelse eller Shapiro-forsinkelse. God fremstilling i tysk WP.
• Einstein Online, Bending of light, explains why space is curved so to double final result. Nordstrom theory satisfies also equivalence principle, but space is curved in opposite direction so that final bending of light is zero!
• Minkowski Institute, Minkowski papers on relativity
• A. Einstein, Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen, Jahrbuch der Radioaktivität und Elektronik, 4, 411 - 462 (1907).
• H.M. Schwartz, Einsteins Jahrbuch articles in English, Am. J. Phys. 45 in 3 separate numbers. In WikiWorks
• MIT, Her is also same convention for Ricci tensor by contraction of Riemann
• Poul Olesen, GR lectures with negative metric and Ricci . In Oslo WikiWorks. Oversiktlig diskusjon om ekvivalensprinsippet.

• U. Washington, Grav time dilation up Mt. Rainier and publications in Physics Today. In Oslo WikiWorks as Mt. Rainier time delay
• NN, Grav time dilation with atomic clocks, experiment in Arizona.
• Hyperphysics, Grav time dilation
• Hafele-Keating experiment, Atomic Clock predictions, clearly written and in WikiWorks
• Ø. Grøn, Twin paradox and acceleration, in WikiWorks and on arXiv:1002.4154
• M. Gasperini, Twin paradox in gravitational field arXiv:1409.1818.
• Popular, Grav time dilation
• Sussex, Time dilation from tipping light cones
• Einstein online, Light deflection. For derivation see my Notes and also Grøn lectures.
• APS News, This Month in Physics, Einstein and GR.
• NN, History bending light and derivation
• FNAL, Simple derivation of bending for students
• J.D. Norton, Einstein's path to GR, comprehensive history
• J.D. Norton, Einstein’s Conflicting Heuristics: The Discovery of General Relativity, discuss how uniformly accelerated system played important role in developing theory
• J.D. Norton, How Einstein found his field equations: 1912-1915
• G. Weinstein, Einstein's Uniformly Rotating Disk and the Hole Argument, gives history of importance of rotating disk and Ehrenfest Paradox
• J. Stachel, The Hole Argument and Some Physical and Philosophical Implications, why it took three years for Einstein from metric to November 1915.
• C.H. Brans, Gravity and scalar fields
• P. Coles, CERN, Light bending - what actually happens?, with discussion of elevator derivation. This gives wrong result because 3d space with gravitation is actually curved!
• Augsburg, Einstein publications in Annalen der Physik
• NN, What Einstein did in the November revolution 1915
• Galina Weinstein, Einstein-Nordström theory, with detailed history. arXiv:1205.5966. Detailed history of scalar theory
• Einstein-Fokker, 1914, Improvement of Nordstrøm gravitation, original paper
• Engelsk WP, Nordström theory of gravitation, very good!
• J. Renn, The Genesis of General Relativity: Sources and Interpretations, with detailed discussions of Entwurf paper
• A. Einstein 25 november 1915, Die Feldgleichungen der Gravitation,
• M. Jansen, The Cambridge Companion to Einstein, Volume 1, with good discussions of Nordström etc.
• P. Halpern, Nordström, Ehrenfest and higher dimensions.
• P.C. Hemmer, Adriaan Daniel Fokker, bio pluss mye mer.
• A. Einstein, Einstein publikasjoner ved Prøyssisk Vitenskapsakademi
• K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 7, 189–196 (1916).
• S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, New York (1972). ISBN 0-471-92567-5.
• C. Møller, The Theory of Relativity, Oxford University Press, England (1960).

Utvid rødforskyvning. Engelsk WP er meget detaljert. Forbind med tidsdilatasjon hvor kan ta med GPS-effekter som på engelsk WP Relativistic corrections

• Hafele-Keating (original), Around-the-World Atomic Clocks, Science 177 (1972).

• Mørk energi

Entropy of Universe

• David Wallace, Gravity, entropy, and cosmology: in search of clarity
• P.C.W. Davies, The Physics of Time Asymmetry, whole book?
• Lineweaver, Entropy of U introduction
• Sabine Hossenfelder, 10 things to know
• Ø. Grøn, Entropy and Gravity, long and detailed.
• S. Frautschi, Entropy in an Expanding Universe, Science 217 (4560), 593–599 (1982).
• Morad og Ø. Grøn, Entropy of gravitationally collapsing matter in FRW universe models, Phys. Rev.

Må etterhvert skrive om gravitasjonell tidsdilatasjon, gravitasjonell rødforskyvning ( i samme artikkel?), gravitasjonell lysavbøyning

• God beskrivelse i Carroll book, som starter ut med maksimalt symmetriske rom Også Weinberg book som benytter stereografiske projeksjoner.
• Engelsk WP, De Sitter space
• Ta med de Sitter effect and redshift
• Engelsk WP, Willem de Sitter, with references to original papers in MNAS on his metric.
• McCrea, De Sitter bio
• Italian PhD thesis, De Sitter universes and de Sitter effect with simplest form of metric on stereographic form.
• Engelsk WP, De Sitter space and coordinate systems from different slicings.
• NN, France, De Sitter and AdS spaces, readable!
• U. Rochester, De Sitter and AdS spaces, one out of full lecture series on GR
• Chris Ripken, PhD thesis about coordinate systems for de Sitter metric.

Postive or Feynman metric with ${\displaystyle g_{00}=+(1+2\Phi )}$ as in Robertson-Noonan book, but even more identical to mine in Bergström-Goobar book. From the NR metric on p.231 Robertson calculates Christoffel-symbols and Riemann tensor. This is defined as in my old Notes and in tensor article. His results is given by

${\displaystyle R_{\;ij0}^{0}=R_{0ij0}={\partial ^{2}\Phi \over \partial x^{i}\partial x^{j}}}$

Then for leading component of Ricci tensor defines by ${\displaystyle R_{\mu \nu }=R_{\;\mu \alpha \nu }^{\alpha }}$ , i.e. with penultimate contraction as in old Notes, I get

${\displaystyle R_{00}=R_{\;0\mu 0}^{\mu }=R_{\;0i0}^{i}=-R_{i0i0}=+R_{0ii0}=\nabla ^{2}\Phi }$

exactly as found there! Thus Einstein eq. becomes

${\displaystyle E_{\mu \nu }=8\pi GT_{\mu \nu }}$

just as I wanted. Robertson's values for the components of his Einstein tensor ${\displaystyle G_{\mu \nu }}$ have therefore opposite signs from mine.

Bertschinger MIT calculates Ricci tensor for metric in weak-field limit on form

${\displaystyle ds^{2}=(1+2\Phi )dt^{2}-(1-2\Psi )d\mathbf {x} ^{2}}$

and even with non-diagonal terms. Dropping these and assuming no time dependence, then he finds

${\displaystyle R_{00}=\nabla ^{2}\Phi }$

${\displaystyle R_{ij}=-\partial _{i}\partial _{j}(\Phi -\Psi )+\delta _{ij}\nabla ^{2}\Psi }$

So when Φ = Ψ, then R_ij = δ_ij ∇² Φ

Bertschinger writes on pp 7-8 that although the motion of non-relativistic particles depend only on h₀₀, i.e. Φ, the non-relativistic equation for gravitational field (Poisson equation) depends on h_ij, i.e. Ψ. Interesting.

Jepsen and GR sources

• Leiden, Litt om Droste paper
• Mathpages, Schwarzschild-Droste coordinates
• Engelsk WP, Birkhoff theorem
• PhysicsForum, Simple proof Birkhoff, with missing 1/r? But general proof of Jebsen seems ok.
• U Oregon, Simple proof Birkhoff], which also is wrong
• Stanford U, Schwarzschild solution, correct and everything in detail with my metric conventions.

Ricci tensor can have different signs depending on how it is defined. In my old notes I defined it as contraction of upper index with penultimate lower index of Riemann tensor as MTW. This will also give different signs in Einstein equation E = 8πGT. I should use plus on RHS. While Robertson in his book defines Ricci with ultimate contraction, he gets a minus in the Einstein equation. But Poul Olesen which uses same metric, but penultimate contraction for Ricci, also gets a minus sign. Here must be a problem? No, his Riemann tensor is defined by opposite sign from all others!

Norsk Fysisk Selskap dannet i 1953 basert på Fysikkforeningen som igjen var dannet i 1938 av professorer og studenter ved UiO.Jørg Tofte Jebsen, Harald Schjelderup og Ole Colbjørnsen

• FFV, Jørg T. Jepsen og Birkhoffs teorem, FFV 4, 96-103 (2004).
• Voje Johansen Ravndal On Jørg Tofte Jebsen discovery of theorem, arXiv 0508163v2
• Niels Voje Johansen, Agder Vitenskapsakademi (2005), Einstein i Norge, pluss mye mer, også eksplisitt om at stedatter Ilse Einstein var med til Oslo som sekretær.
• Portogisisk WP, Jørg Tofte Jebsen, eneste på WP?
• Thormod Henriksen, Fysisk Selskaps Historie frem til 1923.
• A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 49 (7), 769 - 822 (1916).
• Astromaria 2017, På norsk heter det tidrom og ikke romtid!

János Bolyai (født 15. desember 1802, død 27. januar 1860) var en ungarsk matematiker, best kjent for sine arbeider innenfor ikke-euklidsk geometri. I ikke-euklidske geometrier forkaster man eller forsøker å finne andre alternative formuleringer av Euklids parallellaksiom.

Bolyai ble født i Kolozsvár i Transilvania (i dag Cluj-Napoca, Romania), og han var sønn av matematikeren Farkas Bolyai.

• MacTutor, Janos Bolyai bio
• George Bruce Halsted: Bolyai Farkas. Wolfang Bolyai., The American Mathematical Monthly 3, Januar 1896, S. 1–5 (englisch; mit Bild)

Farkas Bolyai (født 9. februar 1775, død 20. november 1856) var en ungarsk matematiker. På tysk er han kjent som Wolgang Bolyai. Han var studiekamerat og barndomsvenn av Carl Friedrich Gauss, og han er mest kjent for sin brevveksling med Gauss. Bolyai egnet store deler av livet til undersøkelser av geometriens grunnlag. Han var far til János Bolyai.

• B. O'Neill, Elementary Differential Geometry, Academic Press, New York (1966). ISBN 0-12-088735-5.
• E. Kreyzig, Differential Geometry, Dover Publications, New York (1991). ISBN 0-486-66721-9.

• G. Berge, Vektor og tensoranalyse, forelesninger ved Universitetet i Bergen (2004).
• N.M. Patrikalakis, Differential Geometry of Surfaces, forelesninger ved MIT (2009).
• Zürich, GR lecture notes

• Wikisource as external link, August .... Kundt

• Utvid gnomonisk kartprojeksjon med ny seksjon Matematisk beskrivelse og sjekk Stereografisk projeksjon, Konform avbildning samt Merkatorprojeksjon. Mye relevant stoff i boken til Kreyzig on Differential Geometry samt Felix Klein, Elementarmathematik vom höheren Standpunkte aus, Band II: Geometrie.
• J.P. Snyder, Flattening the Earth: Two Thousand Years of Map Projections, looks very good!
• Svante Jansson, Uppsala, Riemann Geometry and Maps, lots of applications. Stored in Oslo WikiWorks as RiemannGeometry-maps.
• A. Papadopoulos, History of Conformal Mappings
• A. Treibergs, Mapping the Earth, differential geometry for mappings with stereographic projection.
• NN, Map Projections and history
• NN, Lagrange map projection
• History of conformal map projections
• NN, Map projections
• Revolvy, Maps in different projections
• General map projections
• Conformal maps
• Maps and differential geometry
• Applications of differential geometry to cartography
• Gnomonic projection basics

• Geodesics and curvature
• Covariant derivative and tangent vectors

• Engelsk WP, Geodesics in general relativity for standard utledning.
• India, Variational calculations with examples, in Physics folder Berlin as Geodesics - examples.
• Pokorno, Geodesics, more thorough discussion with example of geodesics on sphere, torus and Fermat principle = geodesic on Poincare disk. In Physics folder Berlin as Geodesics - Pokorno.

• Berkeley, Interesting about hyperbolic plane

Beltrami-Klein-modellen

Poincaré-modellen

Hyperboloidemodellen

• Engelsk Wikipedia, Coordinate systems for the hyperbolic plane

1. # # # # # #

• AMS, Review of book about Janos Bolyai
• S.S. Costa, All coordinate systems for Hyperbolic Geometry, just what I need!
• Stillwell, Mathematics and its History, contains some hyperbolic geometry.
• Quantum Immortal, Horocircle coordinates with applications.
• Quantum Immortal, Introduction to Hyperbolic Geometry
• A. Ramsay and R.D. Richtmeyer, Introduction to Hyperbolic Geometry, excellent with emphasis on different coordinate systems.
• Blog, Calculating hyperbolic distance
• Master thesis, Hyperboloid models with distance functions.
• Bjørn Jahren, Hyperbolic Geometry, stored in iCloud.
• Royster, Beltrami coordinates, webpages.
• Royster, Beltrami coordinates, Chapter 6. Complete version in iCloud. Also in Appendix about circle inversions. Introductory us of complex mappings for Poincare model.
• French, extracts from book with simple use of horocircular coordinates. Also short biographies of B and L. Can view all chapters from 01 to 23. Quaternions and Octonions in 20.
• Stackexchange, Metric on one-sheet hyperboloid
• Stackexchange, Hyperbolic geometry models
• Stackexchange, Beltrami-Klein and Poincare models as gnomic and stereographic projection of hyperboloid model. Refers to book by P. Ryan, Euclidean and Non-Euclidean Geometry.
• Marywood, Hyperbolsk plan ved hekling
• NN, Litt om Gauss rundt 1800 i Braunschweig
• NN, Hyperbolic geometry in different models and relation to Relativistic Mechanics. Stored in iCloud as Pseudospheres and Hyperbolic Geometry
• Hitchin, Geometry of Surfaces, very nice and detailed!
• Cornell, Hyperbolic geometry from hyperboloid model.
• R. Hayter, Hyperbolic geometry and mappings between models, also from hyperboloid. Derives also distance formula. In Dropbox.
• E. Beltrami, Saggio di interpetrazione della geometria non-euclidea, original på italiensk!!
• Stackexchange, Discussion of Beltrami's models
• S. Coen, Mathematicians in Bologna, with details on Beltrami's Saggio.
• Ingemar Bengtsson, Hyperbolic and AdS geometries, very good!!
• N. Arcozzi, Beltrami and hyperbolic geometry
• Stackexchange, Metric on sphere from stereographic projection.
• Physicspages, Spherical metric from stereographic projection
• Manchester, Calculating metrics in spherical and hyperbolic geometry with stereographic projection.
• Stillwell, Geometry of Surfaces, consider planes in H³ and get H².
• Stillwell et al, Mathematical Evolutions with some details about Beltrami and the Hemisphere model.
• A. Treibergs, Mapping the Earth, differential geometry for mappings with stereographic projection.
• Stackexchange, Discussion about Beltrami and his hyperbolic works
• Fairfield, Saccheri and what he did, from jesuit point of view?
• H.S.M. Coxeter, Non-Euclidean Geometry, writes in beginning that Lambert found that for acute angle the angular defect is additive and can be use to define area in hyperbolic geometry. Comparing this with spherical geometry, found that hyperbolic case corresponds to imaginary radius.
• H.S.M. Coxeter, Angels and Arcs in the Hyperbolic Plane, short and essential! In folder Hyperbolic geometry in Oslo. Writes in his book Introduction to Geometry pp 297-299 that Gauss rejected hyperbolic geometry in a letter to father Bolyai in 1832 because he found that a triangle where all sides were infinitely long, had finite area. Det geniale med Lobachevsky 1826 derivation was that he used spherical geometry where the parallel postulate is not used.
• Heinrich Liebmann, Nichteuklidische Geometrie, Leipzig (1923).
• Arlan Ramsay, Robert D. Richtmyer, Introduction to Hyperbolic Geometry, looks to be good introduction!
• B.A. Rosenfeld, A History of Non-Euclidean Geometry, Springer-Verlag, New York (1988). ISBN 978-1-4612-6449-1. Starter ut med sfærisk geometri fra Menelaus! Plus mye mer av arabiske bidrag. Så om parelleller og så SLUTT!
• Coolidge, History of Geometrical Methods
• Engelsk Wikipedia, Giovanni Girolamo Saccheri
• Evelyn Lamb, Non-Euclidean geometry: a re-interpretation. Historia-Mathematica, stored in Dropbox as Hyperbolic geometry before Saccheri.
• Saccheri biography, very complete!
• János Bolyai biography
• NN, Litt om Saccheris argument som kan brukes.
• R.L. Cooke, The History of Mathematics: A Brief Course med noen brukbare kommentarer om hva Saccheri gjorde.
• G. Berge, Vektor og tensoranalyse, forelesninger ved Universitetet i Bergen (2004). Fine forelesninger i min stil. Første del om romkurver og partielle diff. ligninger. Finnes i folder Gerhard Berge i Mathematics, Berlin.
• NN, Map Projections and history
• M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, W.H. Freeman, New York (2008). ISBN 978-0-71679948-1. In Oslo.
• R. Bonola, Non-Euclidean Geometry: A Critical and Historical Study of Its Development, contains everything! Also shows how Taurinus found the trigonometry on hyperbolic plane from spherical geometry my letting curvature radius become imaginary, as Lambert had previously proposed. Original published in Bologna (1906). Dover Book from 1955.
• F. Engel und P. Stöckel, Die Theorie der Parallellinien von Euclid bis auf Gauss, Druck und Verlag B.G. Teubner, Leipzig (1895). Stor detalj om alt som Saccheri, Taurinus etc gjorde. Lagret som pdf-fil i Oslo.
• S. Braver, Lobachevski Explained, all is here!. Horocycles and angle of paralellism. PhD thesis. Stored in iCloud.
• S. Braver, Lobachevsky Illuminated, great details about horocycles p.91. Textbook version.
• M. Longair, Theoretical Concepts in Physics, got it! Excellent!
• S. Stahl, The Poincaré Half-plane: A Gateway to Modern Geometry, looks very good!
• J.W. Dauben and C.J. Scriba (eds), Writing the History of Mathematics: Its Historical Development, writes on p.79 that Beltrami published Saccheri"s Euclid ab omni naevo.. in 1889 with his discussion of angle of parallelism and more. Also possible that Bolyai and Lobachevsky had knowledge of Saccheri. Birkhäuser Verlag, Basel (2002). ISBN 3-7643-6167-0.
• G.E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, also intersting Google Book.
• B. Jahren, Hyperbolic Plane, UiO lectures. Derives in beginning metric for Poincare model from general considerations based on Möbius transformations. In Dropbox.
• NN MasterSc, Horospheres from a more modern point of view. Discusses relation between different disk models, projective model etc.
• W. Fenchel, Elementary Geometry in Hyperbolic space, based on quaternions, defined on p.1 from pairs of complex numbers (similar to how complex numbers can be defines by pairs of real numbers with certain combination rules)
• J. Wilson, Non-Euclidean Companion, derives angular defect from Saccheri and Lambert quadrangel på en systematisk og brukbar måte. Også for elliptisk geometri. Stored in iCloud som Non-Euclidean Companion.
• C. Walkden, Hyperbolic geometry, very good, with complex variables, Möbius-transformations and Fuchsian groups all explained. Stored in Dropbox as Hyperbolic geometry - Walkden.
• J. Parkkonen, Hyperbolic geometry, more mathematical, but very clear! Derives everything from hyperboloid model. Derives hyperbolic law of cosines same way as spherical law of cosines by cutting hyperboloid with different planes through origin.
• Finn, Spherical trigonometry. Stored in Dropbox as Spherical trigonometry - Finn.
• S. Costa, Coordinates in hyperbolic space, derives Lobachevsky = hyperbolic coordinates in addition to usual polar coordinates.
• P.J. Ryan, Euclidean and Non-Euclidean Geometry, used at Harvard.
• Miami, Different coordinates for Poincare disc, with transformation eqs.
• N.I. Lobachevsky, Pangeometry Lambert qudrilaterals
• N.I. Lobachevsky, Pangeometry, med originale arbeid og kommentarer. Also details about what F. Minding did and his biography. Also details about Lobachevsky coordinates,
• N. Hitchin, Hope page with many beautiful lectures!
• N. Hitchin, Geometry of surfaces with nice part about hyperbolic geometry.
• D. Hestenes, Notes on hyperbolic geometry using hyperboloid model. Is this due to Beltrami or Killing 20 years later?
• Engelsk Wikipedia, Hyperboloid model says that this was made by Killing, not Beltrami.
• J. Stilwell, Sources of Hyperbolic Geometry, contains detailed discussion of Beltrami with his first article Saggio and working out angle of parallelism in this particular metric.
• Mathematical Intelligencer, Hyperbolic trigonometry
• Webpages, Law of hyperbolic cosines, with many other clickable pages about hyperbolic geometry.
• NN. Hyperbolic trigonometry
• NN, More hyperbolic trigonometry, very useful!
• NN, Hyperbolic plane using complex variables.
• Cornell, Laws of sines and cosines
• Cornell, Stereographic projections relating Klein and Poincare models
• Cannon et al, Models of hyperbolic geometry. All derived from hyperboloid model. In iCloud as Cannon Hyperbolic geometry.
• Sketchpad, Constructing horocycles etc
• A. Papadopoulos, Strasbourg Master Class on Geometry, contains very detailed calculations of hyperbolic geometry. And compact derivation of spherical trigonometry.
• A. Papadopoulos, History hyperbolic geometry with trigonometric formulas. Based on Lobachevsky's way of derivation from his Pangeometry.
• A. Papadopoulos and W. Su, More derivations of hyperbolic trigonometry
• Norbert A'Campo et Athanase Papadopoulos, Notes on hyperbolic geometry, in: Strasbourg Master class on Geometry, p. 1-182, IRMA Lectures in Mathematics and Theoretical Physics, vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, 2012 (ISBN 978-3-03719-105-7),
• N. A'Campo and A. Papadopoulos, Klein's derivation of hyperbolic geometry with emphasis on projective geometry and contribution by Cayley and cross-ratio. Also contains bio of Beltrami.
• R. Hayter, Durham, The Hyperbolic Plane. Very gentle presentation. Discusses metrics, stereographic projections and clear derivation of Lorentz-transformations. Gets Klein-metric from hyperboloid by stereographic projection from origin (0,0,0) while others (Cannon et al) do it from (-1,0,0). This explains factor of 4 in some metrics. Stored in Dropbox as Hyperbolic plane - Durham. Contains simple proof on p. 27 for angular deficit of hyperbolic triangle, which I really should like to understand...
• L. Valaas, Hyperbolic triangles and trigonometry, deriving hyperbolic law of cosinus based on circle inversions as in Pedoe geometry book (in Berlin). In Dropbox as Hyperbolic trigonometry.
• Blog, Hyperbolic geometry for everyman
• Harvard, Notes on hyperbolic geometry
• NN, Hypercircles and horocycles
• Brouty, Lobachevsky geometry på fransk med forklaring av horocycles og utledning av paralellismevinkel.
• Fransk, Mer generelt om ikke-euklidsk geometri
• Hitchin, Oxford, Hyperbolic geometry and complex transformations. Relates distances in disk and upper-halfplane models.
• Caroline, Hyperbolic Geometry, alltid godt skrevet! Distances from cross-ratios with complex variables. God beskrivelse av horocykler. Symmetries and Fuchsian Groups. Stored in Dropbox as Hyperbolic Geometry - Caroline.
• Khudaverdian, Manchester, Home page
• Manchester, Hyperbolic geometry basics with Möbius etc.
• Manchester, Hyperbolic problems med noen gode observasjoner.
• Manchester, Some hyperbolic solutions. Very useful!
• Manchester, Geometry lectures. Very good and clear! Calculating hyperbolic metric and much more!
• Manchester, Galois theory.
• UCSD, Hyperbolic trigonometry
• Stackexchange, Laws of sines and cosines also in spherical geometry.
• R. Torretti, Philosophy of Geometry from Riemann to Poincaré, explains detailed what exactly Saccheri argued when he threw away abtuse angle.
• Denver, Non-euclidean geometry, talk.
• Tysk Wikipedia, Stereografische Projektion inneholder fine figurer og sammenheng med Astrolabium. Meget god figur som forklarer dette med stereografisk projeksjon, finnes på engelsk versjon Astrolabe.
• N. Arcozzi, Beltrami's models of non-euclidean geometry
• NN, Lambert and hyperbolic geometry
• Stackexchange, Metric for sphere in stereographic projection.
• Physics Pages, Metric for sphere in stereographic projection. Also calculated here:
• Manchester, Metric for sphere in stereographic projection.
• Stackexchange, Metrics for Beltrami-Klein model
• Engelsk Wikipedia, Angle of parallelism, med kompakt beskrivelse. Kanskje enda bedre på italiensk side!
• Engelsk Wikipedia, Beltrami-Klein model, with exemplary derivation of metric from Beltrami's hyperboloid. See also Discussion here. In this model geodesic lines are straight lines within disc. Is equivalent to gnomonisk kartprojeksjon av storsirkler i sfærisk geometri til rette linjer.
• Engelsk Wikipedia, Angle of parallelism, most compact derivation in upper halfplane model.
• Engelsk Wikipedia, Horocycle
• Engelsk Wikipedia, Ferdinand Minding
• Springer, Encyclopedia of Math, Non-euclidean geometries, very nice and short summary with connection to projective geometry.
• Springer, Encyclopedia of Math, Lobachevsky geometryy, also very clear!
• Engelsk Wikipedia, Horosphere med noe historie om Herr Wachter etc.
• NN, Surfaces in general and use of cross-ratio at the end about hyperbolic plane
• T. Traver, Trigonometry in the Hyperbolic Plane, excellent!. Stored in Dropbox as Hyperbolic Geometry - Excellent. Calculates hyperbolic angle. Hyperbolic distance based in cross-ratio in Poincare model which also leads to angle of parallelism.
• D. Royster, Hyperbolic geometry, excellent. Stored in iCloud.
• D. Royster, Hyperbolic geometry with geometric proof of angular defect.
• D. Royster, [1] Horocycles and circle inversions.
• D. Royster, Different coordinatsystems, explains Lobachevsky coordinates. Calculates sides in Saccheri quadrangle. Stored in Dropbox as Hyperbolic geometry - Royster.
• D. Royster, Hyperbolic plane lectures 2002, Excellent! Stored in
• D. Royster, Saccheri quadrangles, plus three more notes where in
• D. Royster, Poincare disc model angle of parallelism is calculated.
• D. Royster, Geometric webpages with all kinds of stuff.
• F. Rothe, Poincare disc model where 'poles and polars play important roles!. Also done in the Cederberg book. Stored in Dropbox.
• F. Rothe, Hyperbolic geometry with Beltrami-Klein model. Stored in iCloud.
• F. Rothe, Lectures on Geometry, the whole thing! 867p. Stored in folder Geometry, Berlin.
• S. Coen, Mathematicians in Bologna 1861–1960, much detail about Beltrami and different hyperbolic geometry metrics.
• J. Milnor, On hyperbolic geometry and its history.
• Papadopoulos, Developments of non-euclidean geometries and what F. Klein did.
• NN, More about F. Klein and non-euclidean geometries.
• Hagen, Elliptic and hyperbolic geometries
• A. Papadopoulos, Lobachevsky trigonometry in hyperbolic plane
• Plato, Stanford, Nineteenth Century Geometry, historisk oversikt med klar fremstilling av projectiv geometri som bakgrunn for all tre geometrier, elliptisk, parabolsk = euklidsk eller hyperbolsk avhengig av hvilket kjeglesnitt som velges å være invariant under collineations = projective transformations.
• A. Ramsay and R.D. Richtmyer, Introduction to Hyperbolic Geometry
• J. Ratcliffe, Foundations of Hyperbolic Manifolds, contains also good historical record of contributions. Says on p.33 that first surface with negative curvature was found by Minding in 1839, the tractroid (= pseudosphere?). Discusses also Beltrami's different contributions. Also p. 42 defines elliptic n-space as projective space RPⁿ with metric given by distance along great circles in spherical model. Each line (geodesic) in this elliptic space is doubly covered by great circle in spherical model. Also p.34 about Grassmann's extension of Euclidean geometry to n dimensions.
• J. Stillwell, The Four Pillars of Geometry, p. 205 writes that German mathematician Ferdinand Minding worked out trigonometry for negative curvature surface. Printed in same issue as paper by Lobachevsky, but no one saw the connection! This book by Stilwell contains good description of Beltrami's works. In Berlin! Contains also easy introduction to cross-ratio, projective line, involution, Möbius-transformations, circle inversions, hyperbolic geometry.
• Chinese, Hyperbolic trigonometry
• B.H. Lavanda, A New Perspective on Relativity: An Odyssey in Non-Euclidean Geometries, many interesting comments, but crazy ideas about classical electron models!
• S. Thorgeirsson, Uppsala, Hyperbolic geometry: history, models, and axioms, very useful. Stored in Dropbox as Hyperbolic geometry Uppsala.
• Cornell, M.C. Escher and hyperbolic geometry
• Singapore, Lectures on geometry, at the end interesting stuff about hyperbolic geometry, Saccheri and Lambert results in compact way! Stored in Geometry folder, Berlin.
• Martin Gardner, Odd One Out puzzle.
• A. Katok, Everything about surfaces
• NN, Hyperbolic plane with complex variables
• TU, Berlin, Projective geometries
• Jyvaskyla, Hyperbolic space and isometries, clear and based on hyperboloid model
• W.P. Thurston, Three-dimensional Geometry and Topology, Volume 1, excellent introduction! On p.100 relation between hyperbolic space H³ and Lie group SL(2, C). On p.32 describes what person A on North Pole of S² as another person crawls south versus South Pole. First B will for A look getting smaller and smaller until he reaches equator, i.e. increased θ by π/2. But after having passed Equator, B will start to look bigger and bigger until he reaches the South Pole and B fills the whole field of vision for A in every direction. In elliptic geometry B would also look smaller and smaller as he travels southwards. But he can only travel a distance π/2. But your whole field of view will also be filled up of the back of yourself, turned up-side down....

• Engelsk Wikipedia, Hyperbolic angle, include Liebmann derivation.

Finnes allrede på nn. Se Alfred Gray lectures no. 12 (Desktop i Oslo) for differensialgeometri på dem. Katenoide hovedkrumninger er regnet ut i Alfred Gray lecture 15.

The presentation below largely follows Gauss, but with important later contributions from other geometers. For a time Gauss was Cartographer to George III of Great Britain and Hanover; this royal patronage could explain why these papers contain practical calculations of the curvature of the earth based purely on measurements on the surface of the planet.

• Italiensk Wikipedia, Isometria, kan benyttes til å utvide isometri. Kan også være grunn til å lage ny side metrisk tensor a la italienske tensore metrico.

• Rothe, Metrics of pseudosphere and hyperbolic planes.
• Rothe, Metrics of pseudosphere and more about hyperbolic planes. Med Gauss formal for K uttrykt bed E, G og F.
• Rothe, Hyperbolic plane basics
• Rothe, Hyperbolic geometry from projective geometry
• Dartmouth, Hyperbolic Geometry, good read!
• Wolfram, Pseudosphere in different parametrizations
• S. Stahl, The Poincaré Half-plane: A Gateway to Modern Geometry
• Beltrami, Calculation of seudosphere metric
• Lavanda, ore about Beltrami and hyperbolic geometry
• Italian, Beltrami and the pseudosphere metric
• Australia, Pseudosphere metric in different coordinates
• Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume 3). Publish or Perish. ISBN 0-914098-72-1.
• Manfredo Perdigão do Carmo: Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs, New Jersey 1976, ISBN 0-13-212589-7.
• MIT, Differential geometry hyperbook, following book by Struik:
• D.J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley, Cambridge, Massachusetts (1950).
• B. O'Neill, Elementary Differential Geometry, Academic Press, New York (1997). ISBN 0-12-526750-9. A most beautiful book! Simple explanations of vectors as differential operators, covariant derivatives on curves, basic idea of moving frames (Darboux and E. Cartan) as on curves where derivatives of this basis is expressed on basis itself. Plus much more. On Desktop in Oslo.

Omtrent som det ble gjort historisk og fremstilt av Einstein 1916, Eddington, Weyl og mange andre. Riemannsk geometri kan gjøres ved å legge mangfoldigheten inn i høyere dimensjonalt euklidsk rom. Se engelsk Wikipedia Engelsk Wikipedia, Covariant derivative. Hvem gjorde dette først?

• Christian Gottlieb Kratzenstein, The Sound Pattern of Russian: A Linguistic and Acoustical Investigation by Morris Halle.
• E. Portnoy, Riemann's Contribution to Differential Geometry, Historia Mathematica 9, 1-18 (1982).

• Encyclopedia.com, Riemann biography, very good with mathematics
• St. Andrews, Riemann biography
• S. Coen, Mathematicians in Bologna 1861–1960, much detail about Beltrami and different hyperbolic geometry metrics.
• Bishop UIUC, Riemannian Geometry
• Engelsk Wikipedia, Luigi Bianchi
• Engelsk Wikipedia, Ricci Calculus
• Pietro Giuseppe Frè, Gravity, a Geometrical Course: Volume 1: Development of the Theory and Basic ... says that Bianchi discovered his identity for the Riemann-tensor i 1902. Men dette var allerede oppdaget av Ricci i 1880, men han hadde glemt å fått det publisert. Contains very good history of differential geometry with bio of Ricci etc.

Bianchi var professor i Scuola Superore i Pisa hele livet. 1886 publiserte han den viktige boken Lezioni di Geometria Differenziale hvor ordet differesiell geometri ble brukt for første gang. I boken til Frè er også bidrag fra fransk matematikk og spesielt Cartans formalisme med moving frames. Også en del historie om Riemann i Stillwell History of Mathematics hvor det fortelles at Riemann var mye i Italia pga. sin sykdom. I Pisa møtte han Betti og hans elever, inkludert Bianchi, Ricci og Beltrami.

Må først skrive om ytreprodukt (eller kileprodukt) og ytrederivasjon.

Mer moderne fremstilling med nablas, tangentvektorer som operatorer etc etc a la MTW. Også føger artikkel om differensielle former, med vekt på 3-dim, og kanskje også anvendt i 4-dim i forrige arikkel om differensialgeometri. Men kan kanskje også gjenta flategeometri med differensiell former som i Lecture 4 under Differensielle former i folder Curves and Surfaces på Desktop i Oslo. Også Lecture 2 i samme folder meget nytting om moving frames repere mobile på flate.

Skriv om geometri a la MTW: dP = dx^μe_μ på mangfoldigheter.Very nice summary of Riemannian geometry in modern notation to be found in

• Encyclopedia of Math, Riemannian Geoemrty
• Ray Weiss, Break-through 1972 paper gravitational radiation.
• Hannu Kurki-Suonio, Cosmology lectures and CMB
• Rindler, Hyperbolic motion
• NN, GR Problems-Solutions in iCloud.
• LIGO, Caltech Homepage
• V. Braginsky, Oral History, Caltech.
• Preskill, On LIGO and grav. rad. detection, 2016
• Kokkotas, Tübingen, Gravitational waves
• Braginsky, Thorne et al, Quantum Measurement, Google Book.
• J.D. Norton, Einstein, Nordström and the early demise of scalar, Lorentz-covariant theories of gravitation, Archive for History of Exact Sciences, '45 (1), 17-94 (1992). In iCloud as Einstein-Nordström.
• C. O’Raifeartaigh, Einstein’s 1917 Static Model of the Universe: A Centennial Review, arXiv-1701.07261.
• Einsteinpapers Princeton, Volume 6-7 with papers from GR and onwards.
• Øystein Elgarøy, 6 forelesninger om GR for kosmiske perturbasjoner, folder lagret i Dropbox.

• M.L. Boas, Mathematical Methods in the Physical Sciences, John Wiley & Sons, New York (1983). ISBN 0-471-04409-1.

• E. Weisstein, Spherical Coordinates, Wolfram MathWorld.
• Thesis, History diff. geometry and connections.

Fint gjort i O'Neill med bruk av mobile aksekors. Skriv om geometri a la MTW: dP = dx^μe_μ på mangfoldigheter.Very nice summary of Riemannian geometry in modern notation to be found in

• Encyclopedia of Math, Riemannian Geoemrty
• V.J. Katz, History of differential forms from Clairault to Poincare, Historia Mathematica 8, 161-188 (1981).
• Ray Weiss, Break-through 1972 paper gravitational radiation.
• LIGO, Caltech Homepage
• V. Braginsky, Oral History, Caltech.
• Preskill, On LIGO and grav. rad. detection, 2016
• Braginsky, Thorne et al, Quantum Measurement, Google Book.
• Berlin, Einstein arbeid publisert i Preussische Wissenschaftsakademie, også hans GR arbeid desember 1915.
• MIT, GR for weak fields

Georg Friedrich Bernhard Riemann (født 17. november 1826, død 20. juli 1866) var en tysk matematiker som leverte viktige bidrag til matematisk analyse og differensialgeometri. Noen av hans oppdagelser banet veien for den senere utviklingen av generell relativitet. Riemann var en av de mest innflytelsesrike matematikerne på midten av 1800-tallet, og selv om han publiserte lite åpnet han veien til nye områder av matematikken ved å kombinere analyse og geometri.

• J. Stillwell, Mathematics and its History has useful Riemann biography.
• Morris Kline, volume III has very good description of Riemann and his works and thinking about Riemannian Geometry.

• Riemanns zeta-funksjon
• Riemannsk sfære

• Sean Carroll, Lectures on GR
• Sean Carroll, Web-based lectures on GR, Christoffel symbol calculated in polar coordinates.
• Yvonne Choquet-Bruhat, Introduction to General Relativity, Black Holes, and Cosmology
• Warwick College, GR, elementary about tensors.
• Warwick College, GR, covariant derivative and Christoffel symbol calculated in polar coordinates. In lecture 10 derivation of els of motion for particle in central potential from Christoffel symbols.
• ZA, More covariant derivatives
• Lerner, More mathematical GR, but a bit clarification about covariant derivative notation.
• Heidelberg, Elementary lectures about tensors.
• Short Riemann biography with a little math. Riemann presented calculations of his geometry talk in a later paper on heat conduction written in Latin and never published:
• Ruth Farwell and C. Knee, Riemanns Commentatio
• Felix Klein: Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Springer-Verlag 1926, 1979. Also about Franz Neumann, Helmholtz and many others
• Felix Klein, Riemann und seine Bedeutung für die Entwickelung der modernen Mathematik, 1894.
• Deutsche Biographie, Bernhard Riemann
• Tazzioli, Riemann bio with a little math.
• R. Dedekind, Riemanns Lebenslauf
• Collected papers of Bernhard Riemann.
• B. Riemann, Paper on number of prime numbers

• Griffiths^[1]
• Rindler ^[2]
• Jackson ^[3]
• Møller ^[4]

Geodetisk kurve kalles den korteste linjen som forbinder to punkt. Navnet oppsto i geodesien som omhandler målinger av avstander på Jordens overflate. Her er en geodetisk kurve en del av en storsirkel. I et euklidsk rom eller flate er de geodetiske kurvene rette linjestykker.

Beregning av geodetiske kurver er mulig når metrikken eller avstanden mellom forskjellige punkt er kjent. De kan derfor finnes i metriske rom av hvilke det euklidske rom er det mest vanlige eksempel. Av stor betydning har også geodetiske kurver i rom med Riemannsk geometri. Her er metrikken gitt ved en tensor som gir avstanden mellom nærliggende punkt. I slike rom spiller geodetiske kurver den samme, sentrale rolle som rette linjer gjør i euklidsk geometri.

Einstein viste med sin generelle relativitetsteori at det firedimensjonale tidrommet omkring oss samt hele Universet er krummet på grunn av masse og energi på en slik måte at det må beskrives ved Riemannsk geometri. Lys og frie partikler vil da bevege seg langs geodetiske kurver som før Einstein ble forklart ved at de ble påvirket av gravitasjonskrefter.

Når flaten eller rommet under betraktning har en metrisk tensor g_μν i et visst krumlinjet koordinatsystem, er avstanden ds mellom to nærliggende punkter x^μ og x^μ + dx^μ  gitt ved

${\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }}$

Her benyttes Einsteins summekonvensjon hvor man for hver indeks som oppter dobbelt, summerer over alle rommets dimensjoner.

Utvide koordinatsystem med krumlinjete koordinater i euklidsk rom.

Se den finske versjonen Geodetics som også har interessante figurer. Oversett!! Men også den italienske er nyttig og forklarer hvorfor kan bruke variasjon av kvadrert action. Also very useful in en Wikipedia Geodesics in GR. Simple derivation can be found in Derivation of geodesic equation.

En termes mathématiques, ceci s'exprime de la manière suivante, avec ${\displaystyle \gamma (\lambda )}$ la courbe paramétrée représentant la géodésique et en notant par

${\displaystyle {\frac {{\rm {d}}\gamma (\lambda )}{{\rm {d}}\lambda }}=V=V^{\mu }{\frac {\partial }{\partial x^{\mu }}}}$

le vecteur tangent à la courbe (le vecteur vitesse si on identifie ${\displaystyle \lambda }$ avec le temps dans le référentiel du voyageur) dans le référentiel correspondant aux coordonnées ${\displaystyle x^{\mu }}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}\lambda }}V=\nabla _{V}V=V^{\mu }\nabla _{\mu }V=0}$

où ∇ est la connexion de Levi-Civita sur ${\displaystyle M}$ (équivalente à la dérivée covariante).

À partir de cette définition et de l'expression des composantes de la connexion de Levi-Civita, on obtient l'équation des géodésiques :

${\displaystyle {\frac {\mathrm {d} ^{2}x^{\alpha }}{\mathrm {d} \lambda ^{2}}}+{\Gamma ^{\alpha }}_{\gamma \beta }{\frac {\mathrm {d} x^{\gamma }}{\mathrm {d} \lambda }}{\frac {\mathrm {d} x^{\beta }}{\mathrm {d} \lambda }}=0}$

• T.J. Willmore, An Introduction to Differential Geometry, Clarendon Press, Oxford (1959). ISBN 0-486-48618-4.
• E. Kreyzig, Differential Geometry, Dover Publications, New York (1991). ISBN 0-486-66721-9.

• Blog med enkel diskusjon av kovariant derivert. Se utledning på siden om absolutt derivert.
• Metric of Mercator projection
• Italiensk Wikipedia, Conessione di Levi Civita
• Thesis, History diff. geometry and connections.

Morris Kline, Volume III gives inversion is a circle as simplest example of birational transformation. This is again a special case of a Cremona transformation which corresponds to Möbius transformation of complex plane.