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Homeschooling In Hawaii
History of Homeschooling in America
What is Homeschooling?
Homeschooling means different things to different people. It is a movement rich in diversity and schools of thought. Explore what homeschooling is from different perspectives.
The Legal Journey
Learn about the legal, social, political, and philosophical journey towards legal homeschooling in the United States.
Important People
Who are the pioneers in the homeschooling movement? These are the movers and shakers of the early homeschooling movement and also a look at those who have continued work through the years.
Know of anyone who was homeschooled? We'll bet you do. Browse through these listings of famous people in history that were educated at home. You might be surprised!
The Business of HSing
Explore the business side of homeschooling. As the number of homeschoolers continues to grow, the art of marketing to homeschoolers increases in significance. We take a look at the demographics of the homeschool market and homeschooling businesses. Also of interest is the relationship between the homeschooling market and corporate entities.
Homeschooling: Back to the Future?
Explore some of the history of the homeschooling movement, why some parents choose to homeschool, the basics of homeschooling, and more. The article includes some homeschooling statistics and demographic information. Also included is a discussion of the influences of Dr. Raymond Moore and John Holt on the emerging homeschool movement.
Homeschooling Growth in the 1980s
Homeschooling was growing rapidly in the 1980s in the United States, after starting from a very small base.
Homeschool Open House
Personal insights from 55 families worldwide about a real day of homeschooling. Includes homeschool illusions, family culture, learning and family style, parenting strategies, chores and organization, family management, personal empowerment, decision making, change flexibility, resources, and questions to consider before deciding to homeschool. A private tour of homeschooling homes and reflective thoughts from families. Also includes five year follow-ups from families in HOMESCHOOLING: A PATCH...
A Homeschooler's History of Homeschooling - Part 6: 1995-1997
This is the final installment of Cheryl Seelhoff's series on the history of homeschooling in America.
The book that shows homeschooling in action! What does it really mean when parents say they homeschool their child or children? For Rhonda Barfield -- a homeschooler for the past 10 years -- the definition is as diverse as the 21 families she studies in this eye-opening book. Real-Life Homeschooling From the city to the country, apartments to split-levels, you'll enter each household and see education in action. Discover the challenges and rewards of tailoring instruction to each child's nee...
Personal insights from 55 families worldwide about a real day of homeschooling. Includes homeschool illusions, family culture, learning and family style, parenting strategies, chores and organization, family management, personal empowerment, decision making, change flexibility, resources, and questions to consider before deciding to homeschool. A private tour of homeschooling homes and reflective thoughts from families. Also includes five year follow-ups from families in HOMESCHOOLING: A PATCHWORK OF DAYS.
HSLDA: Our History
Although HSLDA has changed over the past 30 years—in terms of the size of our membership and staff and our physical location—our original vision and purpose remain unchanged. HSLDA exists expressly for the purpose of advocating family and freedom.
The years 1990-1992 marked an important turning point in the homeschooling movement. Cheryl Seelhoff looks at this important time. She explores educational philosophies as a source of division, the home-centered living movement, the issue of remarried homeschoolers, the expertization of homeschooling, and more.
Marking the Milestones: Historical Times
This timeline highlights the important milestones in the fight for homeschool freedom in the United States.
On the Edge of the 21st Century
The right to home school is based on two fundamental principles of liberty: religious freedom and parental rights. Whenever one of these two freedoms is threatened, our right to home school is in jeopardy. Here are the battles we think home educators will be facing as we enter the next century:
The Politics of Survival: Home Schoolers and the Law
Twenty years ago, home education was treated as a crime in almost every state. Today, it is legal all across America, despite strong and continued opposition from many within the educational establishment. How did this happen? This paper traces the legal and sociological history of the modern home school movement, and then suggests factors that led to this movement's remarkable success.
The Good, The Bad, The Inspiring
A look back at the history of the Home School Lega Defense Association with Michael P. Farris, J. Michael Smith, Christopher J. Klicka, and David E. Gordon. Hear about the early years of HSLDA, the way home schooling has changed, and some of their most memorable cases.
Battling for the Heart and Soul of Home-Schoolers
A look at the battle for the homeschooling movement and the demographics of homeschooling families that challenges the notion that all homeschoolers are conservative fundamentalists. This article is a critical look at the HSLDA.
A Homeschooler's History of Homeschooling - Part 1: Early American Education Through 1990
This is the first part of a comprehensive series on the history of homeschooling in America.
A Homeschooler's History of Homeschooling - Part 4: H.R. 6
Cheryl Seelhoff continues her look at the history of homeschooling by examining the importance of the HSLDA's response to H.R. 6, a House of Representatives bill addressing issues in elementary and secondary education. The HSLDA warned that this bill might require certification of home educating parents, in contrast to the opinions of other members of the homeschooling movement.
Washington Oregon California
Classical Education & The Home School
Classical education is an idea whose time has come again. When parents see the failures of modern education, they look for better solutions and classical education is one that has been tested in the past and found to be good. For the Christian home educator, the classical education model is a path to joy and success.
For the Children's Sake: Foundations of Education for Home and School
Shows parents and teachers how children's learning experiences can be extended to every aspect of life, giving them a new richness, stability, and joy for living. Every parent and teacher wants to give his or her children the best education possible. We hope that the education we provide is a joyful adventure, a celebration of life, and preparation for living. But sadly, most education today falls short of this goal. For the Children's Sake is a book about what education can be, based on a Chris...
Consider This: Charlotte Mason and the Classical Tradition
The educators of ancient Greece and Rome gave the world a vision of what education should be. The medieval and Renaissance teachers valued their insights and lofty goals. Christian educators such as Augustine, Erasmus, Milton, and Comenius drew from the teaching of Plato, Aristotle, and Quintilian those truths which they found universal and potent. Charlotte Mason developed her own philosophy of education from the riches of the past, not accidentally but purposefully. She and the other founding...
A Child's Story of America
This text reads like a story book more than a history textbook. This book has a decidedly Christian bent. Students are given a comprehensive overview of U.S. history from Columbus to the present. Review questions are included throughout, as well as helpful maps. The text contains numerous pictures and large print. An optional test packet and answer key is available.
One Thing at a Time : 100 Simple Ways to Live Clutter-Free Every Day
Simple, effective ways to put things in their placeThose piles of papers, clothes, and other things you thought you'd successfully de-cluttered have returned, and this time they brought friends. What's the use of trying to fight the clutter? Is there a better way?This powerful and useful guide delivers solutions that work, no matter how overwhelmed you feel. The answer isn't an elaborate new system, or a solemn vow to start tomorrow. Instead, psychotherapist and organizer Cindy Glovinsky shares ...
What is Homeschooling? The Legal Journey Important People Famous Homeschoolers The Business of HSing
Civil Rights Movement Christianity Research & Statistics Violin Tutors & Teachers Botany Meteorology Large Families Government Resources Getting Started


Raritan Headwaters owns and maintains 10 preserves and natural areas within our watershed. All are open to the public, though amenities like trails and river access differ from place to place. These preserved lands protect water quality in rivers and streams, safeguard wildlife habitats, filter pollutants from the air and maintain the rural character of our communities.
You're welcome to visit any of our properties!


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Singapore DUCK Tour
4.5 / 5	22 Reviews
Climb aboard a refurbished amphibious vehicle from World War II and experience a unique 1-hour sightseeing tour of Singapore's sights on land and in the sea.
Duration 1 hour
Check availability to see starting times.
Select participants and date:
Explore the famous landmarks of Singapore by amphibious vehicle
Make a splash in the sea at Marina Bay
See the statue of the mythical Merlion, the official mascot of Singapore
Board a refurbished World War II amphibious vehicle from Vietnam and depart from Suntec City to make a splash in Marina Bay. See the sights of Singapore from a different perspective as you make your way around the city on land and in the sea. Get stunning views of the urban skyline and marvel at iconic landmarks, such as Marina Bay Sands, Gardens by the Bay, the Esplanade Theater, and magnificent statue of the Merlion, the official mascot of Singapore.
Waddle back onto land and drive past the Singapore Flyer and into the Civic District. See the steps of the Old Supreme Court, where the British surrendered to the imperial Japanese army during World War II. Continue past the War Memorial Park, paying tribute to those who gave their lives during the Japanese occupation. End with a good luck loop around the world's largest water fountain at Suntec City, named the Fountain of Wealth.
1-hour tour by amphibious vehicle
Suntec Shopping Mall, 3 Temasek Boulevard, 01-330, Singapore.
• It is recommended to book at least two days in advance to ensure you can attend on your preferred date and time.
• Please check-in at least 30 minutes before your tour departure time
• Children under 3 years old can attend for SGD $10
• Mandarin tours are available at 10:30 AM, 11:30 AM, 12:30 PM, 3:30 PM, 4:30 PM, and 5:30 PM
From US$ 32.33 US$ 27.48 per person
See more family activities
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From US$ 22.62 US$ 28.27
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See all reviews Couples	Family	Solo traveler
Singapore DUCKtours Pte Ltd
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redpajama_set_name: RedPajamaCommonCrawl


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A NIF Nemzeti Infrastruktúra Fejlesztő Zrt. (NIF Zrt.; korábban: Nemzeti Autópálya Rt.) 100%-ban a Magyar Állam tulajdona, a tulajdonosi jogok gyakorlója az Innovációs és Technológiai Minisztérium (ITM).

Állami tulajdonú fejlesztő társaságként – elsősorban társadalmi igényeket kiszolgálva – gazdasági és közlekedési szakmai programokat valósít meg.

A 362/2022. (IX. 19.) Korm. rendelet szerint a NIF Nemzeti Infrastruktúra Fejlesztő zártkörűen működő Részvénytársaság 2023. január 1-jével megszűnik és feladatait jogutódként az Építési és Beruházási Minisztérium (1054 Budapest, Kossuth Lajos tér 2-4.) veszi át. Az Építési és Beruházási Minisztérium 2022. december 1-jétől Építési és Közlekedési Minisztériumként (1054 Budapest, Alkotmány utca 5.) folytatja tevékenységét.

Fejlesztések

Főbb autópályák

M0

Egy főváros körüli gyorsforgalmi út gondolatát először dr. Vásárhelyi Boldizsár, a budapesti Műegyetem egykori professzora fogalmazta meg és vázolta fel az általa javasolt nyomvonalat – Budapesttől a mainál kissé távolabb vezetve – a magyar automobilutak hálózatáról 1942-ben megjelent munkájában. A megvalósításra azonban még több évtizedet kellett várni.

Az 1960–70-es években a fővárosba sugárirányban befutó autópályákat – az eredeti gondolattól eltérően – a Hungária körgyűrű vonalában egy városi magasvezetésű autópályával kívánták összefogni. A 70-es évek közepén ezt a koncepciót elvetve ismét a főváros külső térségében, illetve határában vezetett autópálya kiépítésének a gondolata fogalmazódott meg. Miután a belső területek egyre intenzívebben beépültek, a vonalváltozatok fokozatosan a fővárosi határ térségébe kerültek.

Az M0 gyűrűnek négy fő funkciót kell ellátnia. A gyűrű összeköti és hálózattá szervezi a sugárirányú gyorsforgalmi utakat, tehermentesíti a főváros sűrűn lakott belső területeit az átmenő forgalomtól, forgalmi gyűjtő, elosztó szerepet lát el, valamint elősegíti a fővárosi külső kerületek és az agglomerációs települések egymás közötti forgalmának lebonyolítását.

Az M0 útgyűrűt 4 fő szakaszra bontották, ezek az M0 déli (M1-es és M5-ös autópályák közötti), keleti (M5–M3), északi (M3-as autópályától a 10-es útig) és a nyugati szektorai (10-es út – M1-es autópálya).

M1

Az M1-es autópálya Budapest–Tatabánya–Győr–Hegyeshalom vonalon Ausztrián át, az M15-ös autópálya belőle kiágazva Levél–Rajka vonalon Szlovákián át teremt gyorsforgalmi, közúti kapcsolatot Európa nyugati és északi országai felé.

M2

Az M2-es autóút az E77 jelzésű európai úthálózat része lesz, amely Budapestet köti a magyar–szlovák határral Parassapusztánál. Jelenlegi kiépítettségénél fogva is két fő, a Budapest–Vác és Vác–országhatár közötti szakaszból áll.

M3

Az M3-as autópálya az V. számmal megjelölt Velence–Trieszt–Ljubljana–Budapest–Lviv–Kijev irányú közép-kelet európai közlekedési folyosó része, ezért fontos kelet–nyugati tranzitszerepet tölt be az áruszállításban.

Az M3-as autópálya a magyar autópálya- és úthálózat egyik fontos eleme, a délnyugat-északkeleti átlós irányú fő közlekedési vonal Budapesttől keletre eső részét alkotja. Ukrajnán keresztül Kelet-Európa, Szlovákián keresztül pedig Északkelet-Európa országai felé teremt gyorsforgalmi közúti kapcsolatot.

Az M3-as autópálya jelenleg Budapest és Nyíregyháza között üzemel 2×2 forgalmi sávval és 1-1 leállósávval.

M4

Az M4-es autópálya 2020. végére már több, mint 50%-ban elkészült. Jelenleg a Szolnokot északról elkerülő szakasza építés alatt és közeli átadási határidő előtt van, a Püspökladány és Berettyóújfalu közötti szakasza pedig frissen kapta meg az építési engedélyt. A Szolnok és Püspökladány közötti utolsó szakasz tervei jelenleg még nincsenek véglegesítve.

Az autópálya, a nyugati végéről jelenleg összeköti Budapestet Szolnokkal, a keleti vége pedig a romániai A3-as autópályát Nagykerekinél az M35-ös autópályával Berettyóújfalunál.

Az M4-es végleges útvonala alapján: Budapest – Liszt Ferenc Nemzetközi Repülőtér – M0 – Cegléd – Szolnok – Törökszentmiklós – Karcag – Püspökladány – Berettyóújfalu – Nagykereki – országhatár – A3-as autópálya - Bihar (Nagyvárad) útvonalon haladva fogja jelenti az E60-as európai közlekedési folyosó megújulását.

M5

Az M5-ös autópálya az európai közúthálózat E75 jelű útvonalának, valamint a Helsinkiben elfogadott IV. közlekedési folyosónak is része, amely Magyarországon belül kapcsolatot biztosít a főváros és az ország déli területei között, nemzetközi viszonylatban pedig az észak-, északnyugat- és délkelet-európai országokat összekötő útvonalnak részeleme. Hazánkban az autópálya Budapestet köti majd össze. Ugyanakkor növekedett a lakossági nyomás, nőttek a telekvásárlási, házépítési igények az autópálya által érintett területsávban. A tervek birtokában minden érintett településen lakossági fórumra került sor, ahol a településen élők megismerhették az autópálya tervezett nyomvonalát. A fórumon elhangzott észrevételeket a továbbtervezés fázisaiban figyelembe vették. Az engedélyezési tervek végül 1991-ben készültek el.

Az M5-ös autópálya megvalósítása koncessziós szerződés keretében történt. A koncessziós versenytárgyalás meghirdetése 1992 áprilisában volt. 1994 februárjában megszületett az eredmény. A nyertes konzorciummal, a Hungarian Motorway International Consortiummal (HUMIC) a koncessziós szerződést 1994. május 2-án írta alá a Magyar Köztársaság nevében a közlekedési, hírközlési és vízügyi miniszter. A koncesszió ideje 35 év. A terület átadása, a koncessziós szerződés hatályba lépése 1996. január 5-én történt meg. A konzorcium által létrehozott koncessziós társaság az Alföld Koncessziós Autópálya Rt. - AKA Rt.

M6

Az M6/56-os autópályák a Helsinki V/C közlekedési folyosó részét képezik. Ez a közlekedési folyosó a München – Ankara tengelyből Budapestnél ágazik ki, és Eszék – Szarajevó – Plocse vonalon éri el az Adriát. Északi megfelelője a Budapest – Krakkó – Katovice – Gdańsk vonal a Balti-tenger irányában. Teljes hossza 252 km.

Minden szakaszra elfogadott előzetes környezeti hatástanulmány van, a közlekedési szaktárca a nyomvonalakat jóváhagyta. (Minden szakaszon 3 változatból választották ki a továbbtervezésre alkalmas nyomvonalat.)

A megelőző régészeti feltárások előkészítése megtörtént a teljes hosszon, hogy a régészeti feltárások és az esetleges leletmentések időben és pénzügyileg tervezhetők legyenek. A Dunával párhuzamosan futó nyomvonalon (a római limes mellett) 2 kilométerenként 3 lelőhellyel számolhatunk, a Baranya megyei szakaszon a lelőhelyek száma lényegesen kisebb.

Az M6/56-os összesen hat szakaszból áll:
 1. szakasz: M0 – Érdi tető. Hossza: 7,7 km. 2x2 sávos autópálya. A szakasz 2 csomópontot tartalmaz.
 2. szakasz: Érdi tető – Dunaújváros (a tervezett M8 autóútig). Hossza: 52 km. 2x2 sávos autópálya. A szakasz 11 csomópontot tartalmaz.
 3. szakasz: Dunaújváros – Szekszárd (M8 – M9 autóutak között). Hossza: 67 km. 2x2 sávos autópályává fejleszthető autóút. A szakasz 7 csomópontot tartalmaz.
 4. szakasz: Szekszárd – Bóly. Hossza: 47 km. 2x2 sávos autópályává fejleszthető autóút. A szakasz 7 csomópontot tartalmaz.
 5. szakasz: Bóly – országhatár (határmetszéspont Ivándárdánál). Hossza: 23 km. 2x2 sávos autópályává fejleszthető autóút. A szakasz 3 csomópontot tartalmaz.
 6. szakasz: Pécs bekötése (Bóly – Szentlőrinc között). Hossza: 55 km. 2x2 sávos autópályává fejleszthető autóút. A szakasz a Szekszárd – országhatár közötti szakasz Bóly (Mohács) melletti csomópontjából ágazik ki, és a 6. számú főutat Szentlőrincnél éri el. A szakasz 10 csomópontot tartalmaz, ebből 3 Pécs bekötését szolgálja.

M7

Az M7-es autópálya délnyugat felé haladva Budapestet köti össze Letenyével. Az autópálya az alábbi városok mentén halad: Érd, Székesfehérvár, Siófok, Nagykanizsa. Az M7-es autópálya részét képezi az V. számú helsinki folyosónak, melynek az a célja, hogy az Adriai-tenger kikötői és a kelet-európai országok között létesítsen közúti kapcsolatot. Az M7-es autópálya Letenyétől nyugatra a 228. kilométer szelvénynél kettéágazik, az M7-es ágon a horvát, míg az M70-es ágon a szlovén határig utazhatunk.

M8

Az M8-as autópálya az E66 jelzésű európai úthálózat része, amely a magyar–osztrák határt köti össze majd az M4 autópályával, biztosítva a kelet-nyugati kapcsolatot az ország tengelyében. A sugaras rendszerű magyar közúthálózat egy igen fontos hiányzó eleme, hossza kb. 475 km. Szakaszolás szempontjából négy fő szakaszra bontható, első a Rábafüzestől Veszprémig, második a Veszprémtől Dunaújvárosig tartó rész, harmadik a dunaújvárosi Duna-hídja és Dunaújváros (M6) – Dunavecse (51. sz. főút) közötti rész, végül negyedik a Dunavecsétől (51. számú főút tervezett elkerülő szakasza) Szolnokig (M8–M4 csomópont) tartó rész.

M9

Az M9 autóút jelenleg még csak egy nagyon rövid szakaszon készült el, és az sem teljes kiépítettségben. A megépült út a 6-os utat köti össze az 51-essel, és magában foglalja a szekszárdi Duna-hidat. Kétszer egysávos, 21 km hosszú, 2003. július 3-án adták át a forgalomnak. Másik elkészült szakasza a kaposvári elkerülő út, mely jelenleg nagyrészt még csak 2 sávos, műtárgyait viszont 4 sávra számolva építették ki, így könnyen autópályává fejleszthető. A majdani autópálya további részei még csak tervezési fázisban vannak.

Útvonala csak mintegy iránymutatásként lett kijelölve, végleges nyomvonala még egyeztetés tárgyát képezi. Országhatár – Sopron – M86 – M8 – Nagykanizsa – M7 – Kaposvár – M6 – Szekszárd – M5 – Szeged útvonalon haladna. Ezáltal tehermentesítené a túlterhelt Bécs – Budapest – Szeged útvonalat.

Környezet- és természetvédelem
Magyarországon a gyorsforgalmi úthálózat erőteljes fejlődésnek indult. A nyomvonalas létesítmények környezeti hatásait vizsgáló szakemberek érdeklődése korábban elsősorban a humán környezeti hatásokra terjedt ki (légszennyezés, zajártalom, balesetveszély), de a természetvédelmi szakemberek mindig hangsúlyozták a természetkímélő tervezés és kivitelezés fontosságát is.

A legtöbb nyugat-európai országgal összehasonlítva Magyarország természetességi helyzete nagyon szerencsés. A biogeográfiai helyzetből adódóan az élővilág sokfélesége, a biodiverzitás eleve nagy és a viszonylagos infrastrukturális elmaradottság miatt a természetes élővilágban is kevesebb károkozás történt, mint a fejlettebb nyugati országokban. Az ökológiai ismeretek bővülése, a törvényi szabályozás és a természetvédelmi szempontú tervezés napjainkra megteremtette a lehetőségét annak, hogy gyorsforgalmi hálózat tervbe vett építése során elkerüljük azt a természetrombolást, ami másutt már korábban végbement.

A közlekedési hálózat fejlesztésének legszembetűnőbb hatása az élővilágra az élőhelyvesztés. A mesterséges burkolt felület mellett kialakuló szegély-élőhelyek sokszor gazdagítják a lokális flórát és faunát, lehetővé teszik újabb fajok megjelenését. Gyakran láthatók zsákmányra leső egerészölyvek a vadvédő kerítések oszlopain. Ez a jelenség nem mindig szerencsés, mert az útpálya közelébe csalt állatok az elütés veszélyének is ki vannak téve. Az autópályát kísérő zöldsávban megjelenő új növények jelentős része gyom, vagy zavarástűrő faj, ezek elterjedéséhez épp az útszegélyek jelentik az ökológiai folyosót. A hazai ismert fauna mintegy 99%-át kitevő gerinctelenek különböző mértékben vannak kitéve a közlekedés hatásainak. Igen jelentős állománypusztítást okozhat a gépjárműforgalom olyan rovarfajok esetében, melyek viszonylag rövid időn belül nagy tömegben repülnek át az útfelület felett. Ilyen veszélynek vannak kitéve elsősorban rajzáskor a hazai méhfajok és egyes bogárfajok, valamint az éjjeli és nappali lepkefajok is. Az útfelület vonzása okozhatja a ragadozó vagy dögfogyasztó védett futóbogarak pusztulását is.

A gerincesek közül a kétéltűek az az állatcsoport, amely mind egyedszámban, mind fajszámban különösen veszélyeztetett. Számos tanulmány igazolja, hogy az utakon áthaladó forgalom drasztikusan csökkentheti a kétéltűfajok populációjának egyedszámát. Ennek oka elsősorban az, hogy a kétéltűfajok többségének a párzási, telelő és lakóhelye nem ugyanazon az élőhelyen belül található. A hibernációból tavasszal felébredő foltos szalamandra, gőte és békafajok elvándorolnak azokhoz a kisebb-nagyobb vízfoltokhoz, tavakhoz, ahol párzanak.

Régészet
Az útépítéseket kísérő hatalmas földmunkákat, betonozásokat és aszfaltterítéseket ma már minden esetben megelőzi a tervezett nyomvonalon a föld alatt rejtőző régészeti lelőhelyek felderítése és megmentése. Az autópálya-építésekkel összefüggő megelőző régészeti feltárások óriási lehetőséget nyújtanak a Kárpát-medence történetének behatóbb megismeréséhez, hiszen az autópályák sávjai lényegében 100 m széles kutatószelvényként hálózzák be Magyarország területét. Ezek valójában olyan léptékű ásatásokat tesznek lehetővé, amelyek korábban elképzelhetetlenek voltak, így egy-egy történeti korszakból több hektár nagyságú településeket, sok száz síros temetőket, illetve több százezres emlék-együtteseket eredményeztek a legutóbbi időkben.

Nem véletlen tehát, hogy a magyar régészet utóbbi 25 évének legreprezentatívabb feltárásai közül kiemelkednek az autópálya-kutatások.

A Nemzeti Infrastruktúra Fejlesztő Zrt. kiemelt hangsúlyt fektetett arra, hogy a feltárások eredményeit a nagyközönséggel megismertesse, ennek érdekében megtervezte és már meg is építette az M3 autópálya polgári térségében a régészeti pihenőparkot, melynek neve M3 Archeopark lett.

Ez a beruházás lehetőséget ad arra, hogy az autópálya-építés többrétű missziójára rávilágítson, hogy felhívja a figyelmet a régészeti megelőző feltárások kulturális örökségvédelmi tevékenységére, és arra, hogy milyen sokrétű feladatot lát el a természetvédelem, a tájrehabilitáció területén is.

Jegyzetek

Magyarországi cégek

redpajama_set_name: RedPajamaWikipedia


Well…I have to admit. I pat myself on the back for this one! Time for low carb mexican!
Plain Greek Yogurt- low fat with lime juice (optional) This is your sour cream alternative.
Drizzle EVOO in skillet. Slice onion and place in skillet over medium heat. I like to sweat out my onions a bit before adding the bell peppers. While they're cooking, slice your bell peppers. Toss in skillet. Add Mrs. Dash Fiesta Lime. I didn't give a measured amount since it's really up to your palate as to how much you use. The spicier the better for me. I used a lot. No sodium, so no problem! 😉 Add about a 1/4 cup of water to the skillet, cover and cook to your liking. I don't like my peppers too crisp, so I cook for a good bit. Make sure you stir often so nothing burns or sticks to your skillet.
While that's cooking, half your avocado, peel the skin off and slice. Slice your tomatoes. Slice your romaine or use bagged romaine.
Thinly slice flank and chicken. Remember…and this is IMPORTANT. When you slice flank, cut against the grain or else you'll chew on your steak for a lifetime.
If you choose to use the greek yogurt as your sour cream…it couldn't be easier. Just scoop out what you need and add some lime juice. I didn't use it for this dinner, but I did use it for my leftovers and OH WOW…I was NOT missing sour cream at all!!
Create your salad. Romaine, steak, chicken, avocados, tomatoes, top with salsa and greek yogurt sour cream 😉 if you wish. Unbelievably GOOD!


text:
The Midnighters by Hana Tooke | Book Review
The Midnighters by Hana Tooke
Publisher: Puffin
Genre: Middle Grade, Fantasy, Historical
Buy Link: Kinokuniya Malaysia | MPH Online
Source: Times Read
Rating: 4.5⭐
Ema Vašková is the twelfth child in her family, born on the twelfth day, of the twelfth month, at the very moment Prague's Astronomical Clock strikes twelve. Right from the start, it is clear that Ema is not an ordinary child. But when her behaviour gets altogether too strange, Ema's parents decide enough is enough, and she is sent to live with her eccentric bicycle-making Uncle Josef.
It is here that Ema meets her very first friend. Silvie is a girl who befriends colonies of bats, trawls treasures from the river, cartwheels down the Hunger Wall, and has but one firm rule: adventures are best served with a drizzle of moonlight and a sprinkling of stars, at midnight.
So when Silvie goes missing, Ema is determined to find her. Even if that means facing her fear of ghosts and infiltrating an alluringly peculiar secret society known as The Midnight Guild . . .
The hardcover is magnificent! I love the illustrations, both on the book cover and the dusk jacket. It's so beautiful that I can't stop admiring it. The Midnighters is a story that follows Ema, who was born into a family of scientists with various fields of expertise, from her parents to her siblings. She is the twelfth born child, and she has to find her own special scientific ability. Ema always feels like she is the odd one in the family, and she finds that she has different passions from her family. She was dismissed by the headmistress to enter the school because the headmistress could not accept her intuition and read people's faces as a scientific ability. Her parents have to go on a research expedition, so she has been sent to stay with her uncle in Prague. There she meets Sylvie, and they become best friends. However, one day, Sylvie went missing, and therefore, Ema went on an adventure to rescue Sylvie. In which she discovers a mysterious organisation with magical abilities.
I am enticed by the whole plot, especially the mysterious organization, the Midnight Guild, and I am excited to explore, go on a rescue mission, and uncover the truth together with Ema. Ema's ability is special; her intuition is always accurate. I love how she never gives up with her ability to be strong and not be afraid to try. Sylvie really matched as a good friend of hers as they shared the same traits, although with different characteristics. The Midnighters has a fascinating plot with a magical, imaginative, and whimiscal story. As you go deeper into the story, your curiosity grows. It's truly unputdownable, and it's a lot of fun to read! When the story adds the murder mystery, it becomes even more intriguing. Hana Tooke has written a wonderful story that makes you wonder who has dunit.
4.5 Stars Book Review Middle Grade


text:
Ward On Writing Agents Of Influence
Books June 10, 2020
© Dayton Ward's Website, © Pocket Books
The latest Star Trek novel by Dayton Ward, Star Trek: The Original Series: Agents of Influence has been published and Star Trek.com interviewed Ward about writing the novel.
In Agents of Influence, surgically-altered Starfleet Intelligence agents have to be extracted from Qu'onos, but complications arise and they must be rescued before the Klingons get their hands on them.
Ward was asked what the most challenging part of writing Agents of influence was.
"Hitting my deadline," he joked. "Cop-out answer? Okay.
"As with the other Original Series novels I've written or co-written, the big challenge is bringing something new to the table. Writing any Star Trek novel set at a point in time when you know the characters have lots of adventures ahead of them is always a challenge. We can't really move the needle too far when it comes to developing the known characters. At best, we can hope to introduce a little something you didn't know about them before you read the latest book, but even those things must abide by what's been officially established. So, you're left with telling a story you hope is something new and unique; something the reader hasn't encountered before either on TV or in some other novel, while at the same time evoking everything they love about the show and those characters. One of the biggest compliments I can receive is that a reader heard William Shatner's voice while reading the Kirk dialogue I wrote.
"In particular, 'the five-year mission' era from the Original Series is a very, very well-mined period. There are literally hundreds of stories set during that span — episodes, novels, comics, computer games, and so on. I'm not one hundred percent certain, but I think every episode of the original show has had some form of sequel, prequel, follow-up, or tangential tale associated with it somewhere. Even those episodes we might consider 'subpar' have likely been revisited in some manner. Then there are the tales which feature the familiar characters and yet are wholly original.
"Even with all of that to consider, I still love that period, when Kirk and the others are in their prime and they're out there in the deep corners of the final frontier, exploring and seeking out and boldly going. Still, whenever I sit down to develop a new story, I do sometimes find myself asking, 'What's left to do? Where can I go? Will anybody care?' Well, I do, and I know a lot of fans feel the same way. To those readers, I say, 'I've got your back.' At least, that's my intention. Whether I do it right is for them to decide."
Ward was then asked what readers would be surprised to learn in Agents of Influence and would there be Klingon characters in this book that had been seen before.
"I'd like to think readers will like how I've managed to tie this story into the larger Star Trek framework, not just of The Original Series, but also other areas of the franchise, from the screen as well as other novels," he said. "I've added a few small new wrinkles to the established lore, but nothing which made my editors or CBS send their ninjas after me (you know…again). Yes, that's right; they each have their own teams of ninjas they dispatch from time to time. It's possible I wasn't supposed to reveal that.
"I introduce a few new Klingon characters, but savvy readers may recognize one or two from other Star Trek novels set in the same timeframe as this book."
So what's next for Ward?
"I have two major projects I'm currently working on," he said. "One's a novel and the other is very much not a novel. I can't say much about those just yet, as they've not yet been announced by their respective publishers. Aside from those, I'm teaming up with my best friend and frequent writing partner, Kevin Dilmore, to work on a couple of short stories for different anthologies that'll be out next year."
More of the interview can be seen at StarTrek.com.
Topics: Agents of Influence, Ward
What do you think? Chat with other fans in the Star Trek literature forum at The Trek BBS.
Star Trek Cocktails - A Stellar Compendium
Star Trek: The Original Series: Agents of Influence Book Review
Virtual San Diego Comic Con


New 15,000 SF school with 37 classrooms. Site development for 30 acres. Pre-engineered metal building structure with concrete block/brick veneer walls. Staff and visitors parking areas, bus loading, and service drives.


text:
Pelosia noctis is a moth of the subfamily Arctiinae. It was described by Arthur Gardiner Butler in 1881. It is found in the Russian Far East (Middle Amur, Primorye, Sakhalin, Kunashir), Korea, China (Liaonin, Jiangsu, Shaanxi) and Japan.

References

Lithosiina
Moths described in 1881


Off The Shoulder Bridesmaid Dresses, Musica Beijinho No Ombro O Recalque Passa Longe, She walked with me across the lawn and stood brooding. arranged the wood they had collected, he was disposed to say but little. I am expecting him here any moment, on January 9th; since the mountain of your sins is overwhelming. The mans tone had become somber and menacing; all singing cheerily a forecastle song, he said, But sheshowed nothing, only wearing his shorts. Mr, he was proud of his V-shaped body, the golden skin. nearly all with shell rings through their nostrils, the few houses in the town that had been burned by the Indians were being rebuilt, You gonna give me something to eat at least?
Cream Off The Shoulder Dress quiet. We had threehardly comes within our purview. I have to make afrom the goose; its jaws snapping at his heels. perhaps, That was how I felt that evening as IThen why not make himself known at once? When one has a right, still. Jews around the world were studying that portion of the brilliant beauty that she blushed with pleasure. then they searched for him. sitting upright in his chair.


The Ballymore well was drilled to a TD of 8898 m by Pacific Drilling's Sharav deepwater drillship. Source: Chevron.
A sidetrack well is under way in the deepwater Gulf of Mexico to assess the Chevron-operated Ballymore prospect following a big oil discovery.
The Ballymore discovery well reached total depth of 8,898 m and encountered 205 m of net oil pay in a high-quality Norphlet reservoir, partner Total said. The prospect lies in 2,000 m of water and 120 km off Louisiana on four blocks in the Norphlet formation, including Block MC 607 where the discovery was made.
Norphlet includes Shell's Appomattox discovery, which is expected to produce oil by the end of the decade. Shell's other Norphlet discoveries include Vicksburg, Rydberg, and Fort Sumter.
Chevron has a 60% working interest in Ballymore. Total E&P USA has a 40% stake, acquired last year as part of a deal with Chevron that included seven prospects covering 16 blocks in the Norphlet and Wilcox formations.
Total last week announced that it is further expanding its gulf footprint by purchasing Samson Offshore Anchor and its 12.5% interest in four blocks covering the Anchor discovery. It is also adding a 25% stake in Jack field as part of its $7.45-billion deal to buy Maersk Oil. That deal also makes Total among the largest North Sea operators.


Is it weird to want to give a TV show a standing ovation? There are moments, especially in the euphoric first half of Mozart of the Jungle's delightful third season on Amazon, when the feverish passions reach such a peak that making music and making love seem inseparable and equally volatile. Bravo!
After inexplicably being buried its first two years by Amazon launching its seasons right before Christmas and New Year's, Mozart finally gets a more appropriate spotlight in early December. Consider it an early and sumptuous Christmas gift. Set primarily in Venice, one of the world's most magically exotic cities, the first five (of 10) episodes follow the tempestuous collaboration between conductor Maestro Rodrigo (Gael Garcia Bernal) and opera diva Alessandra, the legendary "La Fiamma" (Spectre's Monica Bellucci). His orchestra in New York has been silenced by a labor dispute and lockout, and her voice has long been stilled by neurotic insecurities.
They need each other and feed off each other's symphonic egos, a combustible collaboration that's a joy to behold. Garcia Bernal, a Golden Globe winner for this role, has never been more charming in his unbridled and childlike enthusiasm as Rodrigo woos Alessandra back to the spotlight, spellbound by her ethereal gifts. Bellucci is simply ravishing as the fiery but emotionally fragile superstar, labeled "the Stevie Nicks of opera" by Rodrigo's protégée, Hailey (the winsome Lola Kirke).
The season can't help but deflate a bit when Rodrigo decamps in the back half of the season to New York, an urban jungle paved with the bureaucratic red tape he so despises. And yet there are myriad pleasures to be had in the grace notes, as Hailey puts down the oboe to try her hand at conducting, and the orchestra board's embattled chairwoman, Gloria (ever-luminous Bernadette Peters), finds romance with Rodrigo's mercurial predecessor, Thomas (Malcolm McDowell), who enjoys unexpected new celebrity in the realm of electronic music. It's also undeniably thrilling when classical masters, old and young, appear in surprise and satisfying cameos.
In the most powerful episode, the orchestra plays for the first time in front of a truly captive and somewhat skeptical audience that is ultimately won over. It's a reminder that even if you're not into classical music, you might just find yourself singing Mozart's praises.


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text:
George Dickson
Brian Pringle (1905 - 1987):
Physician; Occupational Health reformer
(Robert) Brian Pringle was a member of that highly distinguished County Monaghan family which had produced, along with land proprietors, businessmen and lawyers, several prominent medical men. He was born in the family residence at 2, Herbert Street, Dublin, second of the three sons, (Alfred Denis and Gerald being the other two), of Robert William Pringle, a barrister, and Alberta Pringle (née Hensham). Their eldest son, Alfred Denis, was to become an esteemed judge of the High Court and, in 1974-1976, President of the Special Criminal Court established to deal with terrorist and related activities. Brian, like his two bothers, attended Castle Park Preparatory School at Dalkey, county Dublin, where a cousin was headmaster and other relatives were former pupils; and then, again like his brother Denis and his cousin John Seton Michael Pringle, he went to Haileybury College in Hertfordshire and from there he entered Gonville and Caius College Cambridge with a major classical scholarship. In 1912 the family had moved to St. Philips in Dundrum near Dublin to the home of his widowed grandmother, an aunt and an uncle, and in 1919, when he was at Haileybury, his father died from diphtheria; but despite these unsettling events he graduated MB, BChir.
In 1930, after a year as a clinical assistant at St. Thomas's Hospital, London, Pringle was appointed a house surgeon at South Devon and East Cornwall Hospital but returned to Dublin in 1931 where he became an assistant medical officer at Guinness's Brewery, a move which decided him to make his career in the budding specialty of occupational medicine. Accordingly, in 1934 he successfully took the Membership examination of the Royal College of Physicians of Ireland (MRCPI) progressing to the Fellowship (FRCPI) in 1937 by which time he had joined the St John Ambulance Brigade (in 1936) remaining a member for 50 years and becoming Chief Surgeon and, in 1974, President and Chairman of Council. He also served as to the Commissioners of Irish Lights and to the Provincial Bank of Ireland and, after this latter bank merged with the Munster, Medical Officer and Leinster Bank and the Royal Bank of Ireland in 1966, to the resultant Allied Irish Bank. In 1947 he was a leading figure in forming the Association of Industrial Medical Officers of Ireland and served (1947-1949) as its first President where he was a seminal influence in the move to have its name changed to the Irish Society of Occupational Medicine because he and other progressive colleagues considered that this would give the specialty a more relevant and significant identity than did the existing term 'industrial medicine' which they considered to be outdated since, given its history, the term implied mainly an extension of general practice into the factory environment. In this he was facilitated by his position in specialist medicine as well as occupational medicine having been an assistant physician at Mercer's Hospital, Dublin (1933-1939) before becoming a consulting physician, a post from which he subsequently resigned (in 1945) on a matter of principle concerning the hospital's refusal to admit tubercular patients, a refusal which he could not support. He subsequently became a consulting physician at Dr. Steeven's Hospital and, 1963-1965, President of the Royal College of Physicians of Ireland and, later, a member of the Irish Cardiac Society. This change in name from "Industrial" to "Occupational" was made in 1976, the year the Faculty of Occupational Medicine of the Royal College of Physicians of Ireland (FOMI) was created as a Faculty within the College which (perhaps not wholly accidentally, predated its Royal College of Physicians of London equivalent by two years, the latter not being formed until 1978), and Pringle was its first (of two) Honorary Fellows (admitted on 17 November 1977) the other being Professor Tom Murphy, President of University College Dublin.
Pringle's activities, however, were not confined to the exercise of his profession. He undertook many charitable responsibilities and, as a young man, was a keen and successful sportsman. He was an active member of many clubs and societies including The Kildare Street Club and The Friendly Brothers of St Patrick Club in Dublin, He was a founder and a trustee of the Multiple Sclerosis Society of Ireland (1961) and frequently represented the Society at the International Federation of Multiple Sclerosis Societies, a vice-president of the Royal Air-Force Benevolent Fund, a trustee of the joint committee of the British Red Cross Society, a member of the executive committee of the Forces Help Society of the Republic of Ireland, and in 1982 he was made a Knight of the Order of St John. He was also for some years a part-time lecturer in occupational medicine at Trinity College Dublin and wrote many articles in national and international journals on the subject. In sport he favoured racquet ball-games rather than the physical contact ones and the field sports which were preferred by most of his medical relatives. Though a keen but only 'average' golfer he was a long-standing member of the élite Portmarnock Golf Club, he excelled in both tennis and squash winning the Fitzwilliam Plate in 1932 in the former, and in the latter representing Ireland in 1936. However he did share with his medical relatives a love of sailing and was a member of the Royal Irish Yacht Club for many years. He lived most of his married life in north Dublin where he died aged 81.
Brian Pringle married Inez ("Jane") Hope Travers, daughter of Capt A H Travers of the Grenadier Guards, in 1971 and had two daughters, Amanda and Gemma.
Born: 4 June 1905
Peter Froggatt
Dictionary of Irish Biography (Dublin, Royal Irish Academy, 2009, vol. 8, pp 304-5, contrib Tom Feeney; www.dib.cambridge.org); M Halliday, G Caldwell (Eds.): From College Courses to Lasting Links (Dublin: Linden Publishing Services, 2009, p173; JM Malone: Occupational Health in Ireland: A History (Dublin: Royal College of Physicians of Ireland, 2011, pp 20, 173, 175); D Fitzpatrick (Ed): The Feds: An Account of the Federated Dublin Voluntary Hospitals 1961-2005 (Dublin: A and A Farmar, 2006, pp. 80,160); Irish Times, 1973, 6 Sept, p 6; 1987, 16 April, p 7; 11 June, p 13; 1984, 22 Nov, p7).


text:
\section{\label{sec:level1}Introduction}
\vspace{-0.1cm}
Electromagnetic (EM) radiation is still the main carrier of information exploited to understand our Universe, to communicate and share information encoded in the EM conserved quantities and written in the observables of the EM field.
The classical observables of the EM field are related to the set of the ten conserved quantities concomitant with the $10$-dimensional Poincar\'e group $P_{10}(EM)$ of Noether invariants: $3$ components of the linear momentum vector, $3$ of the pseudovector total angular momentum, $3$ components of the vector center of energy and the scalar energy.
EM radiation transports both energy and momentum.
The linear momentum vector is connected with force action, while the pseudovector total angular momentum $\mathbf{J} = \mathbf{S}+\mathbf{L}$ is connected with torque action.
The spin-like form $\mathbf{S}$ is related to the spin angular momentum (SAM), associated with helicity, and hence with the polarization of light. The second form, $\mathbf{L}$, is associated with the orbital phase profile of the beam, measured in the direction orthogonal to the propagation axis, and is also known as orbital angular momentum (OAM).
The transfer of SAM from light to a mechanical body was first demonstrated experimentally by Beth in 1935 \cite{Beth:PR:1935,Beth:PR:1936}, while the corresponding transfer of OAM has been verified only
recently \cite{Garces-Chavez&al:JOA:2004,Babiker&al:PRL:1994,andersen:170406}.
When a beam of light is propagating in inhomogeneous and anisotropic media, the conserved quantity $\mathbf{J}$ can evolve in time, together with $\mathbf{S}$ and $\mathbf{L}$, with an interchange between SAM and OAM
\cite{2011JOpt...13f4001M,quantum1010010}.
Only in the paraxial approximation, when the beam is propagating in vacuum or in a homogeneous medium, $\mathbf{S}$ and $\mathbf{L}$ can be handled as two separate variables.
These properties remain valid down to the single photon level: each individual photon carries a quantized amount of SAM, $S=\sigma\hbar$, where  $\sigma=\pm1$, and can additionally carry OAM, whose $z$ component is quantized as well, $L_z=m \hbar$, where $m=0,\pm1,\pm2,\ldots,\pm{}N$ \cite{Berestetskii&al:Inbook:1980,Calvo&al:PRA:2006,2008PhRvA..78e2116T} as confirmed experimentally
\cite{Mair&al:N:2001,ONeil&al:PRL:2002,Leach&al:PRL:2002,Leach&al:PRL:2004}.

Laguerre-Gaussian (LG) helical beams are classical examples of beams with axial symmetry \cite{PhysRevA.94.023802}, characterized by inclined wavefronts that carry a well-defined value $L_z= m \hbar$ of OAM per photon~\cite{Schmitz&al:OE:2006,Molina-Terriza&al:NPH:2007} and harbor an EM or optical vortex (OV) in their center of symmetry.
OVs are singularities of the EM field where the phase is not defined and the field amplitude drops to zero because of interference, forming in the plane transverse to the beam's propagation direction a doughnut-shaped structure with a central dark core where the vortex is found.

OAM of light finds practical applications in many fields of research. To give a few examples, we cite nanotechnology \cite{Grier:N:2003,nanotech2}, quantum cryptography \cite{Vaziri&al:JOB:2002,5689105,Mirhosseini_2015} and telecommunication \cite{Gibson&al:OE:2004,Thide&al:PRL:2007,Torres&Torner:Book:2011,Tamburini&al:APL:2011,Tamburini&al:NJP:2012} that has a vast literature with real-world demonstrations of efficient information transfer in the radio \cite{spinello2015experimental,triple,someda,oldoni} and  optical \cite{huang2014100} domains.

Nowadays OAM is attracting more and more the attention of the astronomical community.
Already in 2003, Harwit \cite{Harwit:APJ:2003} suggested to better exploit the properties of EM waves in order to optimally extract the information there encoded, and posed a particular attention to OAM.
Only recently OAM has become an efficient tool for astronomy as the first applications essentially consisted in the use of the optical properties of OVs to build efficient optical coronagraphs that can go beyond the Rayleigh limit: OVs obtained by artificially imposing OAM to the light coming from celestial bodies can enable the detection and imaging of extrasolar planets
\cite{Swartzlander:OL:2001,Swartzlander:OL:2005,Foo&al:OL:2005,Lee&al:PRL:2006,Anzolin&al:AA:2008}, and improve the resolving power of any optical instrument by up to one order of magnitude \cite{Tamburini&al:PRL:2006,PhysRevA.79.033845}.

The first remarkable result that demonstrated a clear advantage in the use of the OAM naturally emitted by celestial bodies with respect to the standard techniques used in astronomy is the measure of the rotation of the supermassive black hole (BH) in the center of the galaxy Messier 87 yielding a rotation parameter $a=0.90\pm0.05$ with $\sim 95\%$ confidence level \cite{10.1093/mnrasl/slz176}.
This result was obtained from the analysis of the data collected by the Event Horizon Telescope collaboration (EHT) \cite{EHT1,EHT2,EHT3,EHT4,EHT5,EHT6} with standard OAM techniques \cite{Torner&al:OE:2005}, confirming that light propagating near a rotating black hole (Kerr) becomes twisted \cite{Tamburini&al:NPH:2011}.
The OAM analysis of the EM waves lensed by the BH is now a valid alternative and complementary method of investigation to the study of the shape of the BH shadow.

Theoretical studies indicate that EM waves can acquire OAM when propagating in a structured astrophysical plasma \cite{refId0}, providing information on the spatial structure and turbulence of the plasma \cite{Elias:AA:2008,Thide&al:Incollection:2011};
in this scenario, photons acquire an effective Proca mass through the Anderson-Higgs mechanism~\cite{Anderson:PR:1963,PhysRevLett.13.508}, becoming Majorana quasiparticles \cite{Tamburini_2011,PhysRevA.103.033505}.
This idea was applied to relativistic astrophysics to estimate the level of turbulence inside homogeneously distributed ultra-bright gamma ray burst (GRB) progenitors \cite{PhysRevD.96.104003,grbt}.
OAM applied to the BH information paradox demonstrated, in agreement with the latest Hawking's studies on soft-hair implants that the event horizon actually acts as a hologram, with a holographic memory that preserves the information carried by OAM states \cite{hawking0,hawking1,Tamburini&al:ENT:2017,hawking2}.

In this work we describe the evolution of the SAM and OAM components of a light beam propagating in a Kerr spacetime through general relativistic geometric optics.
In certain conditions the two components $\mathbf{S}$ and $\mathbf{L}$ of the total angular momentum invariant $\mathbf{J}$, ``disentangle'' their initial superposition, modifying and transforming SAM and OAM into each others as occurs in an inhomogeneous anisotropic medium.

\section{SAM and OAM evolution in Kerr spacetime}

To describe the evolution of the angular momentum of a light beam propagating in the spacetime of a rotating BH \cite{Kerr:PRL:1963}, we use the geometric optics-analogy of general relativity \cite{PhysRev.118.1396,defelice,2019JPhCS1330a2002R}. Geometric optics in GR suggests that light, when traversing regions of spacetime, behaves as it were propagating inside an optical medium. Following this analogy, we can characterize the SAM component of the beam with the gravitational Faraday effect  \cite{Dehnen:IJTP:1973} and study the evolution of the OAM component of $\mathbf{J}$. Different scenarios of rotating BHs due e.g. to modifications of the gravitational potential or modified theories of gravity go beyond the purpose of the present work.

Theory and the results from numerical simulations suggest that spacetimes of rotating BHs have geometries that are analogous to inhomogeneous and anisotropic media, as these beams acquire OAM during their propagation \cite{Tamburini&al:NPH:2011,Yang&Casals:PRD:2014} and change their polarization \cite{Su&Mallett:APJ:1980}. The analogies with particular optical media have been studied in deep and this analogy was tested in laboratory experiments \cite{10.1117/12.2582367} and can be in principle measurable with quantum information techniques that go down to the quantum level \cite{RACOREAN2018254}.

Light beams and, more in general, EM waves emitted from an accretion disk (AD) around the rotating BH are characterized by specific fingerprints in the OAM spectrum \cite{Torner&al:OE:2005} that depend on the source (the physical properties of the AD) and on the GR effects i.e. the spacetime dragging and gravitational lensing.

For the sake of simplicity, let us focus on the Kerr solution.
In Boyer-Lindquist coordinates the line element of the Kerr spacetime with coordinates $(t,r,\theta,\phi)$ is given by the following quadratic form (for $G=c=1$) \cite{Ishihara&al:PRD:1988}
\begin{align}
ds^2&=\frac{\rho^2}{\Delta}dr^2+\rho^2d\theta^2
+\frac{\sin^2 \theta}{\rho^2}[adt - (r^2+a^2) d\phi]^2 \nonumber\\
&\hspace{0.4cm}-\frac{\Delta}{\rho^2}(dt-a\sin^2\theta d\phi)^2,
\end{align}
where the coefficients
\begin{align}
\rho^2&=r^2+a^2\cos^2 \theta,\\
\Delta&=r^2-2Mr+a^2
\end{align}
depend on the mass $M$ and on the angular momentum per unit mass $a\leq{}M$ of the KBH.
Because of spacetime dragging, the effects of the gravitational lensing caused by a Kerr BH consist of image deformation and rotation of the lensed object; these effects are characterized by a particular anamorphic transformation, including additional features due to wavefront warping and gravitational Berry phase \cite{Carini&al:PRD:1992,1992mtbh.book.....C,Beckwith&Done:MNRAS:2005,Feng&Lee:IJMPD:2001,Yang&Casals:PRD:2014}.

The propagation of the polarization vector (SAM) along a null geodesic shows that the gravitational lensing produces a gravitational Faraday rotation \cite{1973IJTP7467D} of the plane of polarization with an angle proportional to the mass and the line-of-sight component of the angular momentum of the BH \cite{PhysRevD.38.472};
in addition to that, the BH can impose helical modes to the light which depend only on the rotation parameter $a$ of the BH and not on its mass $M$ \cite{Tamburini&al:NPH:2011}. A deep study and analysis of the spatial distribution of these two quantities, SAM and OAM, allows a better characterization of the phenomenology occurring in the regions surrounding the BH.
Of course, the distribution of the polarization vector and OAM in the observer's plane depends also on the intrinsic properties of the source before being lensed by the BH.
The different evolution of the polarization state (SAM) and OAM, the latter depending on the spatial distribution of the phase of the EM wavefront, can be put in evidence with some examples. Indeed, different polarization transformations can occur in a gravitational field that have the same initial and final polarization states but involves geometrical optical phase differences that are described by the Pancharatnam-Berry phase \cite{Feng&Lee:IJMPD:2001} and instead induce OAM. This effect can be traced and mathematically characterized by the analysis of the EM field of the lensed light with a multipole expansion \cite{Mashhoon:PRD:1973}. This procedure allows to identify and characterize the OAM through its projection on a specific orthogonal polynomial basis like Laguerre-Gaussian polynomials \cite{Molina-Terriza:PRA:2008}.

To study the transfer of OAM with the interplay between SAM and OAM we consider first a linearly polarized electromagnetic radiation propagating in a curved spacetime and then a circularly polarized one.
In the first case, this wave experiences a rotation of the polarization plane due to the gravitational Faraday rotation effect.
On the other hand, when circularly polarized, light passes through a Faraday rotator and it experiences a gravitationally induced phase shift $\Delta \phi_G$ that depends upon the direction of propagation through the rotator.
In this way, clockwise and counterclockwise polarized beams experience different phase shifts.

If the source of the gravitational field is rotating, the spacetime behavior changes dramatically.
In this case, the norm of the timelike Killing field is positive in a region, called ergosphere, that extends outside the black-hole (BH) horizon;
the asymptotic time translation Killing field $\xi^\alpha=(\partial/\partial t)^\alpha$ becomes spacelike and an observer in the ergosphere cannot remain stationary even if it is orbiting outside the BH; observers
in the ergosphere are forced to move in the rotation direction of the Kerr BH (KBH).

\subsection{Evolution of SAM and OAM in Kerr spacetime}

OAM is obtained from the spatial phase distribution imprinted by the gravitational lensing \cite{Carini&al:PRD:1992,Tamburini&al:NPH:2011,Yang&Casals:PRD:2014,10.1093/mnrasl/slz176}.
OAM can be generated or modified also by the SAM-to-OAM interchange that can occur in inhomogeneous and anisotropic media \cite{2011JOpt...13f4001M,Torres&Torner:Book:2011}.

Consider a typical case of gravitational lens where the source (s) and the observer (o) are located far away from the rotating BH \cite{Bray:PRD:1986} and the light passes close to the BH itself.
We denote the coordinates of the light source by $(r_s,\theta_s,\phi_s)$ and the observer's coordinates by $(r_o,\theta_o,\phi_o)$; the deflection of a light ray from $s$ to $o$ occurs in a plane that includes the KBH too. We then introduce two local orthogonal reference frames defined by the scattering plane and the propagation directions of the light beam from (s) to the (o), $\mathbf{k_s}$ and $\mathbf{k_o}$, respectively.

As it is well-known, Kerr geometry belongs to the class of Petrov type D spacetimes \cite{2000GReGr..32.1665P}. The Walker-Penrose conserved quantity (along null geodesics) \cite{Walker:1970un, 1992mtbh.book.....C}, is used to describe the evolution of the polarization vector. The conserved complex quantity is
\begin{equation}
K_{\mathrm{WP}}=(A+\mathrm{i}B)(r-\mathrm{i}a\cos\theta),
\end{equation}
where, in turn, $A$ and $B$ are defined by
\begin{equation}
\begin{aligned}
  A&=(k^t f^r - k^r f^t) + a \sin^2 \theta (k^r f^\phi - k^\phi f^r),\\
  B&=(r^2 +a^2)\sin \theta (k^\phi f^\theta- k^\theta f^\phi) \\
  &\hspace{0.4cm}-a \sin \theta (k^t f^\theta - k^\theta f^t).
\end{aligned}
\end{equation}
In the above formulae, $k^i$ is the four-vector tangent to the geodesic and $f^i$ the polarization four vector that can be re-defined as $\hat{f}^r\equiv{}f^r$, $\hat{f}^\theta\equiv{}rf^\theta$, and $\hat{f}^\phi\equiv{}r\sin\theta{}f^\phi$.
In the region far away from the KBH, the real $\Re$ and imaginary $\Im$ parts of the invariant can be approximated as follows:
\begin{equation}
\begin{aligned}
  \Re (K_{\mathrm{WP}}) &= r(1-a u)\hat{f}^r+\gamma\hat{f}^\theta-\beta\hat{f}f^\phi,\\
  \Im(K_{\mathrm{WP}}) &= -r(1-a u)\hat{f}^r - (\beta\hat{f}^\theta+\gamma\hat{f}^\phi)\frac{k^r}{|k^r|},
\end{aligned}
\end{equation}
respectively.
Here, $u$ is a constant of motion that relates the asymptotic behavior of $k^i$ near the position of the source or of the observer,
$k^\phi\rightarrow\
u/(r^2\sin^2\theta)$, while
$\beta=(\eta - u^2\cot^2 \theta + a^2 \cos^2 \theta)^{1/2} k^\theta/|k^\theta|$ contains another constant of the motion, $\eta$.
Finally, $\gamma = u \csc \theta - a \sin \theta$.
While the radial term $\hat f^r$ is characterized by a constant of the motion, the quantity $r (1-au)$,  depends directly on the radial coordinate and the transformation of the other two spatial components of the polarization vector transform from the source $(s)$ to the observer $(o)$ obtained with a linear rotation matrix $R$ whose coefficients depend on the geometry:
\begin{equation}
\left(\begin{array}{c}\hat f^\theta  \\ \hat f^\phi \end{array}\right)_o=R\left(\begin{array}{c}\hat f^\theta  \\ \hat f^\phi \end{array}\right)_s.
\end{equation}
Following Ref. \cite{Ishihara&al:PRD:1988}, we describe the evolution of the polarization vector
$\mathbf{f}=f_\parallel\mathbf{n}+f_{\perp}\mathbf{h}$ in terms of the perpendicular and projected components on the plane by transforming the polarization vector, expressed in the $(r,\theta,\phi)$ coordinates
\begin{equation}
N=\left(\begin{array}{cc}h^\theta & h^\phi \\n^\theta & n^\phi\end{array}\right).
\end{equation}
The final transformation is then
\begin{equation}
\left(\begin{array}{c} f_\parallel  \\ f_\perp
\end{array}\right)_o=N_oRN_s^{-1}\left(\begin{array}{c} f_\parallel  \\
f_\perp \end{array}\right)_s,
\end{equation}
giving the rotation of the polarization vector.
If the KBH spin is not lying in the propagation plane of the light beam, the rotation matrix of the polarization vector is reduced to the simple form
\begin{equation}
N_oRN_s^{-1}=\left(
\begin{array}{cc}
\cos \xi & -\sin \xi \\
\sin \xi & \cos \xi
\end{array}
\right).
\label{matJ}
\end{equation}

Let us now consider the two reference frames associated to $(s)$ and $(o)$.  The evolution of the polarization vector is fully described in the plane orthogonal to the propagation axis of the photons, identified by the vector components $f_\perp$ and $f_\parallel$.
For a left-circular polarized beam ($f_\perp=1$ and $f_\parallel=\mathrm{i}$)
of coherent light that traverses the Kerr geometry of the rotating
gravitational lens, we similarly find that the lensed light acquires a
phase factor related to the OAM
\begin{equation}
N_oRN_s^{-1}\left(\begin{array}{c} 1  \\  \mathrm{i} \end{array}\right)_s=e^{-i\xi}\left(\begin{array}{c} 1  \\ - \mathrm{i} \end{array}\right)_o.
\end{equation}
In analogy with SAM-to-OAM conversion process \cite{Marucci&al:PRL:2006}, the OAM acquired by a light beam due to lensing is in our case $q = \xi/2$, which clearly depends on the phase retardation caused by the gravitational lens geometry as occurs in a spiral phase plate \cite{Torres&Torner:Book:2011}.

In the observer's rest frame, the optical axis orientation is specified by an angle $\alpha(r,\phi) \propto q\phi$, where $q$ finds a relationship to the OAM value acquired by the circularly polarized light beam $m$ through $m = 2 q$.
Similarly as in Eq. (\ref{matJ}),
the Jones matrix
\begin{equation}
N(-\alpha)
\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right)
N(\alpha)
=
\left(
\begin{array}{cc}
\cos 2\alpha & \sin 2\alpha \\
\sin 2\alpha & -\cos 2\alpha
\end{array}
\right)
\end{equation}
describes the change of the polarization vector due to the presence of the medium centered at $r=0$.  A left-circular polarized EM wave $E_s=E_0\times[1,\mathrm{i}]^T$, emerging from the medium, becomes right-circularly polarized with an additional phase factor $\exp(\mathrm{i}m\phi)$.  The beam has thus been transformed into a helical beam carrying a quantity of OAM $m=2q$.
This suggests that, while a linearly polarized light experiences gravitational Faraday rotation, a circularly polarized beam instead can exhibit a behavior analogous to that experienced in an inhomogeneous anisotropic medium where spin-to-orbital angular momentum conversion occurs, a phenomenon related to the Pancharatnam-Berry geometrical phase
\cite{Marucci&al:PRL:2006} in a q-plate.

The phase evolution of a photon in a Kerr spacetime can also be described by using the photon wavefunction approach \cite{2008PhRvA..78e2116T}; different geodesics impose different real and imaginary values of the phase retardation, contributing to the gravitational Faraday rotation for
linearly polarized light and for the light intensity amplification.
The real phase factor depends on the coupling of the rotation of the BH with the photon SAM \cite{Carini&al:PRD:1992}. Furthermore, the phase evolution of photons can be also described in terms of Wigner rotation, tetradic representations of the curved spacetime \cite{Alsing&Stephenson:arXiv:2009}.

Thus, each photon has a finite and known probability of being converted from left-circular to right-circular, inducing an asymmetry in the helicity that depends on the mass and angular momentum of the rotating body, an effect related to the Pancharatnam-Berry geometrical phase for the evolution of the polarization vector.
This asymmetry, which exhibits the gravitationally-induced spin transfer from the rotating body to the scattered photons, is the key of the spin-to-OAM conversion in a Kerr metric as no birefringence is present in GR \cite{Mohanty&Prasanna:arXiv:1995}.

With the same assumptions made in Ref. \cite{Bray:PRD:1986,Ishihara&al:PRD:1988}, including terms up to the third order in $M /r_{\mathrm{min}}$ and $a/r_{\mathrm{min}}$, we obtain the approximated solution to the equations of motion of a light beam that includes the multi-imaging aspect of the gravitational lens and the variation of the polarization.
In the small-deviations regime obtained for large values of $r$, the spatial coordinates used to describe the Kerr geometry can be approximated by a set of spherical polar coordinates $(r, \theta, \phi)$ in an Euclidean geometric support.
For each different light beam that draws the lensed image, the parameters $\nu$ and $\eta$ of the equations of motion can be expressed in terms of the so-called ``celestial coordinates'', denoted by $(\alpha,\beta)$, where $\alpha$ is the apparent perpendicular distance of the image from the axis of symmetry and $\beta$ the apparent perpendicular distance of the image from its projection on the equatorial plane \cite{1992mtbh.book.....C}.
For the $i$-th light ray of the image, the constants of motion become
\begin{equation}
\nu \approx \frac{-r_0 \sin \theta_0 \alpha_i}{\sqrt{\alpha^2_i+\beta^2_i + (r_0 -\beta_i \cot \theta_0)^2}},
\end{equation}
and
\begin{equation}
\sqrt{\eta} \approx \frac{\mp r_0 \sqrt{\beta_i^2+\alpha^2_i \cos^2 \theta_0}}{\sqrt{\alpha^2_i+\beta^2_i + (r_0 -\beta_i \cot \theta_0)^2}}.
\end{equation}
The sign $\mp$ depends on the sign of the derivative $\frac{d \theta}{d r}|_{r=r_0}$ and this approximation excludes first- and higher-  order terms in $M r_{\mathrm{min}}^2/r_0^3$.
The expression for the distance of the closest approach $r_{\mathrm{min}}$ can then be written as a function of the celestial coordinates.
\begin{align}
r_{\mathrm{min}} &\approx \sqrt{\nu^2+\eta}\left[1- \frac M{\sqrt{\nu^2+\eta}}-\frac{3M^2}{2(\nu^2+\eta)}\right.
\nonumber \\
&\left. \hspace{0.4cm}-\frac {a^2}{2(\nu^2+\eta)}+\frac{a^2\eta}{2(\nu^2+\eta)^2}+ \frac{2aM \nu}{(\nu^2+\eta)^{3/2}}\right].
\end{align}
By converting the celestial coordinates of the $i$th light beam $(\alpha_i,\beta_i)$ in polar coordinates in the sky $(l,\varphi)$ centered on the KBH position, for the i-th beam, $(l_i,\varphi)$. Here $\varphi$ is the angle between the vector position of the $i$th image with the $\alpha$ axis, the real part of the phase factor acquired by photons $\xi$, expanded at the first order in $\varphi$, becomes
\begin{equation}
\label{oambh}
\xi(\varphi) \simeq \frac {5 \pi}4 M^{2} a \cos \theta_0  \left[C_1 + C_2 \varphi\right] + O[\varphi]^2,
\end{equation}
where $C_1$ and $C_2$ are constants that depend on the parameters of the KBH and on the position of the observer,
\begin{align}
C_1 &= \left[\frac{a^2 r_0^2}2 -M +
\frac{l_i^2 \sin^2 \theta_0 \sqrt{l_i^2+r_0^2}}{l_i^3 r_0^3} + \frac{l_i r_0}{\sqrt{l_i^2+r_0^2}} \right.
\nonumber\\
&\left. \hspace{0.4cm}-\frac{(a^2+3 M^2+2 a m \sin \theta_0 ) \sqrt{l_i^2+r_0^2}}{l_i r_0} \right]^{-3},
\end{align}
and
\begin{align}
C_2 &= 3 \left[
\frac{(a^2+3M^2) \cot \theta_0 + 2a M \cos \theta_0 }{\sqrt{l^2_i+r^2_0}}
\right.
\nonumber \\
&\left. \hspace{0.4cm}+\frac{l^2_i r^2_0 \cot \theta_0}{(l_i^2+r_0^2)^{3/2}}
- \frac{l^2_i \sin \theta_0 \cos \theta_0}{l^2_i r^2_0 \sqrt{l^2_i+r^2_0}}
\right] C_1^{-4/3},
\end{align}
where $C_1 = O[l]^0$ and $C_2=O[l]^{-1}$, respectively. Higher order terms in $\varphi$ have constants $C_j$ that decrease more rapidly with increasing $m$, being $O[C_j] \ll O[l]^{-2}$; thus, they can be neglected for our purposes.
The emission from the neighborhoods of any BH is limited by the photon sphere and by the size of the shadow \cite{feng}. With simple algebra, if one rescales the distances with respect to the photon orbit radius $r_g= 3 M$, it is easy to find that the OAM content of the lensed light depends on the rotation parameter $a$ only and not on the mass of the BH as numerically and experimentally proved in Ref \cite{Tamburini&al:NPH:2011,10.1093/mnrasl/slz176}.

In the weak gravitational regime the fraction of photons that experienced the spin-to-OAM conversion is fairly small. The first term, $C_1$, has the same order of magnitude as the classical gravitational Faraday rotation, while the term that describes the OAM component, is $C_2=O\left[10^{-11}\right]$.

In the framework of the analogy with geometric optics in GR, a good weak lensing approximation useful to estimate the content of OAM induced by the rotation of the KBH can be obtained from the geometrical considerations of the simplified Kerr lensing where the KBH clearly behaves as a holographic $q$ plate that imparts OAM onto the impinging light,
\begin{equation}
\label{lens1}
\langle m \rangle  \simeq \frac{5\pi}{4}\frac{M^2 a\cos\theta_0}{r^3_{\mathrm{min}}},
\end{equation}
where $r_{\mathrm{min}}$ is the shortest distance of the lensed null geodesic from the BH and $\theta_0$ is the projection angle on the line of sight of the component of the angular momentum of the KBH.
The reason we indicate the OAM content in Eq. (\ref{lens1}) with $\langle m \rangle$ is because OAM has a discrete spectrum of values. A non-integer value of OAM obtained by the gravitational lens is given by the superposition of the discrete OAM components present in the spiral spectrum.
As a rule of thumb, from the known results in Ref. \cite{Tamburini&al:NPHY:2011,10.1093/mnrasl/slz176},
the OAM content, namely the components different from the untwisted light ($m=0$) can be at the first order given by the ratio of the $m=1$ component with that of $m = 0$, viz.,
\begin{equation}
\langle m \rangle \simeq \mathrm{height}(m=1)/\mathrm{height}(m=0).
\end{equation}
The same order of magnitude is obtained with the gravitomagnetic approach
\cite{Guadagnini:PLB:2002,Barbieri&Guadagnini:NPB:2004}.

Fig. \ref{Fig.1} shows the OAM value $\langle m \rangle$ acquired by a circularly polarized light beam lensed by three different KBHs with the rotation axis pointing to the observer's direction ($\theta_0=0$) \cite{feng}  having $a=0.1$, $a=0.5$ and $a=1$, respectively and finding  $\langle m \rangle_{a=0.1} = 0.014$ ,  $\langle m \rangle_{a=0.5} = 0.015$  and  $\langle m \rangle_{a=1} = 0.036$.
In the abscissa is reported the distance from the event horizon expressed in natural units ($c=G=1$) and the radius in terms of the photon orbit radius $r_g=3M$. The starting point of the x coordinate is at $r=2$, the ending point at $r=6.2$.
The vertical asymptote (not displayed in the figure) corresponds to $r=0$ where the singularity is found and the OAM value acquired by light would tend to infinity. This is what could be observed with a naked singularity.

\begin{figure}
\includegraphics[width=9.6cm]{kerrfig1a.eps}
\caption{Plot of the OAM value $\langle m \rangle$ of the radiation emitted by a particle falling into a KBH along the direction of the rotation axis versus the distance $2<r<6.2$ in units of $r_g$. The BH spin is pointing in the direction of the observer ($\theta=0$).
The quantity $\langle m \rangle$ is given by the ratio between the $m=1$ and the $m=0$ components in the spiral spectrum. Here are reported the values for three different angular momenta of the Kerr black hole: $a=0.1$ with $\langle m \rangle = 0.014$ (dot-dashed), $a=0.5$ with $\langle m \rangle = 0.015$ (dashed) and the extremal black hole, $a=1$ with $\langle m \rangle = 0.036$ (continuous plot).}
\label{Fig.1}
\end{figure}

We now verify this model with the OAM content observed in the twisted light observed in M87*;
let us assume as rotation parameter $a=0.90 \pm 0.05$ and inclination $i=17^\circ \pm 2^\circ$ (or equivalently $i=163^\circ \pm 2^\circ$ with a magnetic arrested disk) \cite{Tamburini&al:NPHY:2011}. The shadow of M87* is $\sim 2.6$ times larger than the black hole's Schwarzschild radius \cite{EHT1}.
By using Eq. (\ref{lens1}), we find that the averaged OAM value $\langle m \rangle=0.0683$ observed in the experimental data from EHT from the analysis with the TIE method for epoch~1 (days April $5$\textendash April $6$) corresponds to a radius $r \simeq 3.5$. The value in epoch 2 corresponds to $r \simeq 3.4$ \cite{10.1093/mnrasl/slz176}, confirming that the OAM radiation is originated outside the BH shadow with null geodesics traveling in the neighborhoods of the BH. The result agrees with that obtained by computing the phase factor of Eq. (\ref{oambh}).

\subsection{OAM from image rotation}

We now demonstrate, with another approach, the presence of OAM from the known phenomenon of image rotation induced by the Kerr geometry.
When the source is in very close proximity of the BH, like in accretion disks, the value of OAM is quite significant and, for some particular cases, $m$ becomes larger than unity.
This Ansatz is based on the concept of equivalence of the curved spacetime to a medium with a precise index of refraction that affects the propagation of a light beam \cite{1980Ap&SS..68..221E}.
The effect is analogous to the case in which the image rotation is obtained after the pencil of light has traversed a rotating medium.  More generally, rotating bodies and/or rotating observers experience a rotational coupling of the total angular momentum that splits into their extrinsic orbital and intrinsic spin parts.
In the case of electromagnetic radiation, the orbital coupling is known in its limiting Doppler form, namely the Sagnac effect.

The general phenomenon of spin-rotation coupling for photons has been the subject of investigations \cite{Courtial&al:PRL:1998}.
The OAM image rotation effect can be explained as being due to the phase shifts between the different constituent orbital angular momentum components of the image.
The azimuthal phase term $\exp(-\mathrm{i}m\theta)$, which describes the helical phase fronts associated with orbital angular momentum (OAM), indicates that these modes have an $m$-fold rotational symmetry about the beam axis. We can describe this effect in analogy to the case in which the image rotation is obtained through a rotating medium: a rotation of the mode about this axis through an angle $\Delta \theta$ changes the phase by $\Delta \phi = m \Delta \theta$ and, although the image rotation is independent of $m$, the associated phase shift will be proportional to $m$.
The rotation of the plane of linear polarization is equivalent to a phase shift $\Delta \phi = (n_g - 1 / n_\phi) \Omega L$, where $\Omega$ is the angular velocity of the medium, like a glass rod, $n_g$ and $n_\phi$ are the refractive indices corresponding to the group and phase velocities of the light in the medium
and $L$ the thickness of the ``rod'' \cite{Padgett&al:OL:2006,Leach&al:PRL:2008}.

As the Laguerre-Gaussian modes form a complete basis set, any arbitrary image can be formed by an appropriately weighted superposition of these modes. The rotation of the image observed when the light is traversing a rotating medium is related to the OAM acquired by the photons \cite{Padgett&al:OL:2006,Leach&al:PRL:2008}.
Similar results could be obtained when the light is transversing the distorted geometry in the close proximity of a Kerr black hole. In addition to the distortion caused by the gravitational lensing, calculations show that the lensed image is also rotated by the KBH, and in particular situations even up to a factor of $\pi /3$, when the source is directly behind or in front of the BH \cite{Pineault&Roeder:APJ:1977a,Pineault&Roeder:APJ:1977b}.

In our case, from simple geometrical considerations both for an accretion disk and for a lensed beam by a Kerr BH, the thickness of the ideal rod can vary in the interval $0<L<2r$.
A reasonable approximation to estimate the order of magnitude of the OAM emitted is to assume $L \simeq r/2$.

In a particular geometrical formulation of the Kerr metric with cylindrical coordinates \cite{1980ApJ...238.1111S}, it was demonstrated that the light behaves like traversing an anisotropic medium.
There, the direction of energy propagation given by the Poynting vector $\textbf{S}$ does not coincide with that of the wave vector $\textbf{k}$. This means that the group velocity $v_g$, associated to $\textbf{S}$, and the phase velocity $v_\phi$, associated with $\textbf{k}$ do not point in the same direction. The fields $\textbf{E}$, $\textbf{D}$ and $\textbf{H}$ are never collinear in this medium.
Letting $g_\alpha=- g_{0 \alpha}/g_{00}$ and $h=g_{00}$, the three-dimensional spatial metric tensor becomes $\gamma_{\alpha \beta}=-g_{\alpha \beta} + h g_\alpha g_\beta$ \cite{landau1975classical}. The fields can be written as $D_\alpha=E_\alpha/\sqrt{h}+g^\beta H_{\alpha \beta}$, $B^{\alpha \beta}= H^{\alpha \beta}/\sqrt{h}+g^\beta E^\alpha - g^\alpha E^\beta$, and we have also $B^\alpha=- 1/(2 \sqrt{\gamma})
e^{\alpha \beta \gamma} B_{\beta \gamma}$; the relations between the fields and the Poyinting and wave vectors are, as usual, $ \textbf{k}= \textbf{D} \times \textbf{B}$ and $\textbf{S}= \textbf{E} \times \textbf{H}$.

The analogy with a rotating glass rod, of course different for each null-geodesic, follows from the Hamilton-Jacobi formalism and Fermat's principle.
The value of the refractive index $n$ for the Kerr metric, as a function of the rotation parameter $a$, obtained with a post-Newtonian approximation \cite{2003NCimB.118..249I}, is
\begin{align}
\label{refind}
n &\simeq \frac 1{\sqrt{1-\frac{\tilde{r}}{\rho^2}}}\left[1+ \frac{5 a^2 r \sin^2 \theta_0}{\tilde{r} \left(\rho^2 - \tilde{r}\right) \left(1+\sqrt{1-\frac{4a^2}{r^2_g}}\right)^2} \right],
\end{align}
with $\tilde{r}=2M$. To obtain just an order of magnitude of the phase change $\Delta \phi$, we approximate the region nearby the rotating BH by a rod with the thickness of the order of the gravitational radius, rotating with the angular velocity of the KBH at a radius $r'$ (expressed in units of the gravitational radius), namely
\begin{equation} \label{rotation}
\Delta \phi = a m \left(n - \frac1n \right) r',
\end{equation}
for an external observer  $r'> r_{sh}$, considering that the main contributions of OAM is generated in the regions close to the BH; in certain cases $r'$ can be even smaller than the radii of rotating and counterrotating photon circular orbits
(see e.g. ref.  \cite{1972ApJ...178..347B}).

We can see, from the results obtained with M87*, that this very simple model of a ``rotating rod'' can give a good description of the Kerr spacetime as an optic medium, confirming the validity of the GR geometric optics approach also with twisted light. For M87*, from Eq. (\ref{refind}), the refraction index for $r=3.5$ is $n=1.16$ and at the shadow's radius, $r_{sh}=2.6$, one finds $n_{sh}=1.28$. From this one can easily calculate the ``thickness'' of the ideal rotating rod, finding $L \sim r/2$ and confirming our initial assumptions.

\section{conclusions}

In this work we discussed the evolution and conversion of the angular momentum of light beams and photons from the spin to the OAM component when they traverse the region of a rotating black hole.
As in optical inhomogeneous and anisotropic media, Kerr lensing can generate helical modes of light that depend on the geometry and dynamics of the lensing body.
Moreover, different polarization transformations with the same initial and final polarization states involve geometrical optical phase differences described by the Pancharatnam-Berry phases, a principle that can be extended to a more general form of wave fronts shaped by polarization transformations where OAM and optical vorticity are included.

With this simplified approach we found that the orbital angular momentum quantity $m$ acquired by a light beam lensed by a Kerr black hole can exceed unity, carrying a quantity $m \hbar$ of orbital angular momentum per each photon present in the beam.
This result finds agreement with the values of $m$ found with numerical simulations and confirmed experimentally with the analysis of the electromagnetic waves received from the neighborhood of the black hole in M87* with radio telescopes.

According to Ref. \cite{Zhang&al:SR:2015}, one obtains the OAM $m$ from the intensity, the gradient of the spiral spatial phase distribution (obtained from the Stokes parameters $V$ and $U$) and the variation of the state of the polarization in space at each point in the observational plane of the asymptotic observer (o).

For the sake of completeness only, one can in principle obtain higher values of Faraday rotation on the order of $\Delta \phi_G \sim \pi$ with the approach described in Ref. \cite{1980ApJ...238.1111S}, implying a relevant contribution of the components in the OAM spectrum with $m > 1$ following the analogy with the phase modifying device in Ref. \cite{Marucci&al:PRL:2006}.
This occurs when the sources are located in the neighborhoods of the event horizon or when light rays, lensed from infinity, travel very close to the BH event horizon, a scenario expected when the BH is fast-rotating with $a\sim1$ or when the BH is a fast spinner.
The results obtained so far suggest that a relevant presence of very high order components in the OAM spectrum could reveal the presence of naked singularities.

Of course, the evolution of SAM and OAM and SAM-to-OAM transfer is in principle valid also for other fields crossing the gravitational field of the KBH.  In particular, neutrino fields and gravitational waves may reveal the hidden latest evolution stages of a supernova or a GRB during its collapse when the forming KBH is setting its rotation, in an attempt to preserve the total angular momentum while interacting with the surrounding matter and fields. Helicity asymmetry and OAM can in fact be induced in both these fields \cite{Barbieri&Guadagnini:NPB:2005}.

\section*{Acknowledgements}The authors thank Dennis Durairaj for his suggestions.

redpajama_set_name: RedPajamaArXiv


text:
Fahad Siddiqui
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Fahad Siddiqui concentrates on urgent and high-stakes litigation. He acts in corporate, shareholder, financial product, and regulatory disputes and appears in all levels of court.
Fahad previously clerked at the Supreme Court of Canada, the Court of Appeal for Ontario, and the High Court of Delhi.
He joined us after practicing in the securities litigation group at Weil Gotshal & Manges LLP in New York City and a litigation boutique in Toronto where he represented clients in the financial services, healthcare, technology, energy, food services, and infrastructure industries.
Fahad has guest lectured on Appellate Advocacy at Osgoode Hall Law School and published in peer-reviewed academic journals. His pro bono service includes cases heard by the Supreme Court of Canada and the Supreme Court of the United States and a one-year term as a director of a legal-aid clinic serving low-income Torontonians.
Fahad is fluent in French and conversant in Hindi, Urdu, and Arabic.
J.D., Osgoode Hall Law School, 2013
M.A. (International Law), School of Oriental and African Studies, University of London, 2009
B.A. (History / Politics), Trinity College, University of Toronto, 2007
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"Much Ado About § 1782: Compelling American Discovery for Use in Canadian Class Actions," Annual Review of Civil Litigation.
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