text:
Prose stubs, Target novelisations, 1987 novels,
Second Doctor novelisations
Stories set outside time and space
Novels set in the Land of Fiction
The Mind Robber (novelisation)
The Mind Robber
Master of the Land, Master Brain, White Robots, Clockwork soldiers
Land of Fiction
Peter Ling
David McAllister
Hardcover and paperback editions; 10 Chapters, 144 Pages
The Ark The Faceless Ones
TV series order
The Dominators The Invasion
The Mind Robber was a novelisation based on the 1968 television serial The Mind Robber.
1987 Target Books edition Edit
To escape a catastrophic volcanic eruption the Doctor takes the TARDIS out of space and time - and into a void he can only describe as 'nowhere'.
But the crisis is far from over and when the time-machine's circuits overload, the TARDIS explodes.
The Doctor, Jamie and Zoe come to in a dark unearthly forest. There they encounter a host of characters who seem somehow familiar: a beautiful princess with long flaxen hair, a sea traveler dressed in eighteenth-century clothes, and a white rabbit frantically consulting his pocket watch...
What is happening to the three time-travellers? What strange power guides their actions? In the Land of Fiction who can really tell?
'BUT WE CAN'T BE NOWHERE - THAT'S IMPOSSIBLE.'
Zoe, the Doctor's astrophysicist companion, can't understand how the TARDIS can be beyond both time and space; but the Doctor's time-travelling craft has taken him, Zoe and Jamie to a place where fiction is reality.
Beguiled and threatened on all sides by characters from myth and literature, the Doctor must discover who or what controls the all-too-lifelike apparitions.
The Mind Robber, with Patrick Troughton as the Doctor, was first broadcast in 1968. This novelisation is by Peter Ling, who wrote the scripts of four of the five television episodes.
Doctor Who - The Mind Robber is available as a BBC video, and will be broadcast on BSB television during 1990.
Chapter titles Edit
The Doctor Abhors a Vacuum
Boys and Girls Come Out to Play
Into the Labyrinth
The Facts of Fiction
'I Am the Karkus'
A Meeting of Masters
The Doctor Has the Last Word
Deviations from televised story Edit
The novel opens in the Land of Fiction with the first episode of the story depicted as a flashback.
Rather than escaping Dulkis following the events of TV: The Dominators, the TARDIS instead materialises on the slopes of Vesuvius moments before it erupts.
The story contains many scenes written specially for the novel, including a firing squad test before the charging unicorn and several references to Lewis Carroll's Alice in Wonderland, among many others.
The labyrinth is presented as a hacienda from an unnamed Spanish crime novel, containing Miss Haversham's wedding cake. The cave is from C.S. Lewis's Narnia.
The Karkus speaks using speech bubbles.
The Master of the Land of Fiction leaves with the Doctor, Zoe and Jamie in the TARDIS.
Writing and publishing notes Edit
Several amendments were made to Peter Ling's original submission. Of particular note was his mistaken belief that Zoe had long blonde hair. The error was not realised until a viewing of the BBC video. Incidentally, the character of Zoe was at one point considered to be a male, with blond hair.
Dedication: For Dave Baldock with love and gratitude.
The 1990 Target/Virgin edition utilised the same new cover artwork as used on the BBC video release.
Additional cover images Edit
1986 Hardback edition
1990 edition.
Cover by Alister Pearson
British publication history Edit
Hardback (November 1986)
W.H.Allen & Co. Ltd. UK ISBN:0491036825, copies priced £7.25 (UK))
Paperback (April 1987)
Target / W.H. Allen & Co. Ltd. Single paperback edition, estimated print run: 32,500, priced £1.75 (UK).
Paperback (August 1990)
Target / Virgin Publishing New cover artwork by Alister Pearson, priced £2.50 (UK).
Audiobook release Edit
This Target Book was released on 6 August 2009 complete and unabridged by BBC Audio and read by Derek Jacobi.
The cover blurb and thumbnail illustrations were retained in the accompanying booklet with sleevenotes by David J. Howe. Music and sound effects by Simon Power.
Audiobook cover
Doctor Who novelisations
Susan, Barbara, Ian
Doctor Who and an Unearthly Child • Doctor Who in an Exciting Adventure with the Daleks • The Edge of Destruction • Marco Polo • Doctor Who and the Keys of Marinus • The Aztecs • The Sensorites • The Reign of Terror • Planet of Giants • Doctor Who and the Dalek Invasion of Earth
Barbara, Ian, Vicki
The Rescue • The Romans • Doctor Who and the Zarbi • Doctor Who and the Crusaders • The Space Museum
Barbara, Ian, Vicki, Steven
The Time Meddler • Galaxy Four
Vicki, Steven, Katarina
The Myth Makers
Steven, Katarina, Sara
Mission to the Unknown
Steven, Sara
The Mutation of Time
The Massacre
Steven, Dodo
The Ark • The Celestial Toymaker • The Gunfighters • The Savages
Dodo, Ben, Polly
The War Machines
The Smugglers • Doctor Who and the Tenth Planet
Ben, Polly, Jamie
The Highlanders • The Underwater Menace • Doctor Who and the Cybermen • The Macra Terror • The Faceless Ones
The Evil of the Daleks • Doctor Who and the Tomb of the Cybermen • Doctor Who and the Abominable Snowmen • Doctor Who and the Ice Warriors • Doctor Who and the Enemy of the World • Doctor Who and the Web of Fear • Fury from the Deep
The Wheel in Space • The Dominators • The Mind Robber • The Invasion • The Krotons • The Seeds of Death • The Space Pirates • Doctor Who and the War Games
Doctor Who and the Auton Invasion • Doctor Who and the Cave-Monsters • The Ambassadors of Death • Inferno
Doctor Who and the Terror of the Autons • The Mind of Evil • Doctor Who and the Claws of Axos • Doctor Who and the Doomsday Weapon • Doctor Who and the Dæmons • Doctor Who and the Day of the Daleks • Doctor Who and the Curse of Peladon • Doctor Who and the Sea-Devils • Doctor Who and the Mutants • The Time Monster • The Three Doctors • Doctor Who and the Carnival of Monsters • Doctor Who and the Space War • Doctor Who and the Planet of the Daleks • Doctor Who and the Green Death
Doctor Who and the Time Warrior • The Paradise of Death • Doctor Who and the Dinosaur Invasion • Death to the Daleks • Doctor Who and the Monster of Peladon • Doctor Who and the Planet of the Spiders
Doctor Who and the Giant Robot • Doctor Who and the Ark in Space • Doctor Who and the Sontaran Experiment • Doctor Who and the Genesis of the Daleks • The Revenge of the Cybermen • Scratchman • Doctor Who and the Loch Ness Monster
The Planet of Evil • Doctor Who and the Pyramids of Mars • Doctor Who and the Android Invasion • Doctor Who and the Brain of Morbius • Doctor Who and the Seeds of Doom • Doctor Who and the Masque of Mandragora • The Pescatons • Doctor Who and the Hand of Fear
Doctor Who and the Deadly Assassin
Doctor Who and the Face of Evil • Doctor Who and the Robots of Death • Doctor Who and the Talons of Weng-Chiang • Doctor Who and the Horror of Fang Rock
Leela, K9 Mark I
Doctor Who and the Invisible Enemy • Doctor Who and the Image of the Fendahl • Doctor Who and the Sunmakers • Doctor Who and the Underworld • Doctor Who and the Invasion of Time
Romana I, K9 Mark II
Doctor Who and the Ribos Operation • The Pirate Planet • Doctor Who and the Stones of Blood • Doctor Who and the Androids of Tara • Doctor Who and the Power of Kroll • Doctor Who and the Armageddon Factor
Romana II
Doctor Who and the Destiny of the Daleks • City of Death
Romana II, K9 Mark II
Doctor Who and the Creature from the Pit • Doctor Who and the Nightmare of Eden • Doctor Who and the Horns of Nimon • Shada • Doctor Who and the Krikkitmen • Doctor Who and the Leisure Hive • Meglos
Romana II, K9 Mark II, Adric
Full Circle • Doctor Who and the State of Decay • Doctor Who and Warriors' Gate
Doctor Who and the Keeper of Traken
Logopolis
Castrovalva • Four to Doomsday • Kinda • Doctor Who and the Visitation • Black Orchid • Earthshock
Time-Flight • Arc of Infinity • Snakedance
Nyssa, Tegan, Turlough
Mawdryn Undead • Terminus
Tegan, Turlough
Enlightenment • The King's Demons • The Five Doctors • Warriors of the Deep • The Awakening • Frontios • Resurrection of the Daleks
Turlough, Peri
Planet of Fire
The Caves of Androzani
The Twin Dilemma • Attack of the Cybermen • Vengeance on Varos • The Mark of the Rani • The Two Doctors • Timelash • Revelation of the Daleks • Slipback • The Mysterious Planet • Mindwarp
Terror of the Vervoids • The Ultimate Foe
Time and the Rani • Paradise Towers • Delta and the Bannermen
Mel, Ace
Remembrance of the Daleks • The Happiness Patrol • Silver Nemesis • The Greatest Show in the Galaxy • Battlefield • Ghost Light • The Curse of Fenric • Survival
The Novel of the Film
The Christmas Invasion
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Amy, River
The Time of Angels
Amy, Craig
Robot of Sherwood • Mummy on the Orient Express • Flatline • The Girl Who Died • Face the Raven
The Woman Who Lived • Twice Upon a Time
Retrieved from "https://tardis.fandom.com/wiki/The_Mind_Robber_(novelisation)?oldid=2794962"
Target novelisations


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redpajama_set_name: RedPajamaStackExchange


By putting the lives of Thelonius Monk veteran Ahmed Abdul-Malik, Transatlantic fusion band leader Randy Weston, Ghanian African music visionary Guy Warren and South African composer Sathima Bea Benjamin under the microscope — Africa Speaks, America Answers brings overdue insights into a musical cultural exchange that took seed in the 1950s, bloomed in the 60s and continues today.
Many European, Africans and Asians without background or experience in the English speaking regions of North America, find it difficult to understand the passion and energy some Americans place in the process of discovering or re-discovering their roots.
In the case of African-Americans well into the 20th century, other than the memes transmitted, and cultural traits transmuted in families and communities, information about (and communication with) Africa was far from common place. Compared to other diasporas and immigrant groups (ie. English, Irish, Italian, German, Slavic, Jewish), African traditional literature and central texts were not readily available. As anthropological/ethnography texts emerged in the late 19th century by Europeans, its remarkable that the majority of these were not written in English but instead appeared in French, German and Dutch.
This all adds drama and poignancy to the events Kelly documents; when during the 1950s and 1960s a powerful and heady confluence of the US Civil rights movement and the liberation of African colonial states provided increased opportunities for communication and collaboration.
Such points of view confounded artists and made for less than an evolutionary statement in the mainstream marketplace. This is especially evident in the first chapter which looks at the problematic career of Ghanaian composer Guy Warren [AKA Kofi Ghanaba].
Warren/Ghanaba's progressive vision of African music fused jazz with a West-African folklore sources that reached far beyond polyrhythmic drums. His work included complex juxtapositions of processional chants, rhythms with melodic interludes of piano and saxophone, that sought graceful currency not just as "African music" but as contemporary art.
It is with no shortage of chagrin and disappointment that Warren returned to Ghana after a short and less than successful recording career in North America. All the while watching artists he considered amateur at best such as the Nigerian Babatunde Olatunji better fit the market places expectations of African tradition as an upbeat drum dominated stereotype.
While Warren's story may be the most symbolic, Kelly's chapters on Randy Weston and Ahmed Adbul-Malik demonstrate 2 New York artists ultimately finding their musical path bounce off of West Africa only to be better fulfilled in North Africa.
While Randy Weston would embrace a set of inspirations spread between Nigerian High Life and Moroccan Gnawa, Ahmed Adbul-Malik focused on a Jazz expression to match his liberal Moorish-American inflected Islamic cosmology. As Kelly quotes from a 1963 interview in Downbeat.
Kelly completes the book with a chapter on the South African composer singer Sathima Bea Benjamin. As Kelly chronicles, Benjamin offered to the world a voice that gained her critical acclaim on both sides of the ocean but little popularity during her prime. For instance her 1963 recording with Duke Ellington 'A Morning in Paris' was not placed on the market until 1997, as well Kelly sadly points out Benjamin produced sessions that could have been the first Jazz LPs produced in South Africa – but they were never released.
Kelly's chapters are packed with well researched and documented facts shedding light on the colonial, marketing and political forces that shaped both helped and hindered the careers of these 4 exceptional artists. Their rich legacies of these creative personalities serve well to illustrate and illuminate current and recent events — while leaving the musically curious reader with a variety of musical impulses to further explore. Jazz aficionados, who over the years have been enjoying works like John Coltrane's Africa Brass, should run not walk to seek out Ahmed Abdul-Malik's Jazz Sounds of Africa. World-Music listeners who over the years have regarded Olatunji has the premier champion of African music in the 20th century North America, should give Guy Warren a chance to re-write narrower definitions of contemporary African music.
The paths to discovery in Kelly's Africa Speaks, America Answers are diverse and the context of the book is not limited to the effective careers of the inspiring artists mentioned in this review. Kelly provides detail of many of the other formidable players in the movements and scenes. A scene populated and informed by; pioneers of the early 20th century Harlem scene, Art Blakely, the NYC Caribbean music artists, the early African field recordings of Harold Courlander, and the philosophical dialogues between writers such as James Baldwin and Aimé Césaire.


Aaron's Aura (Japanese: アーロンの波導 Arlon's Wave) is a Supporter card. It is part of the Movie Commemoration VS Pack: Aura's Lucario.
Search your deck for Aura's Lucario and a Metal Energy card, show them to your opponent, and put them into your hand. Shuffle your deck afterward.
This card depicts Sir Aaron, a character with the ability to manipulate Aura in the movie Lucario and the Mystery of Mew.
This page was last edited on 20 April 2013, at 12:49.


Deep vein thrombosis is a blood clot typically in the leg of the patient. Hospital staff are constantly on high alert for this as it can be a result of the lack of activity, that is common during hospital visits.
The main symptoms are leg pain and inflammation. However, some patients may not feel any symptoms and the longer the clot is there the more potential it has to damage vascular structures. Therefore, it's important to keep circulation flowing by exercising and wearing compression socks. If deep vein thrombosis is found, there are a number of ways that it can be resolved.
Depending on the size of the clot, your physician may determine that physically removing it will be necessary. One way that they can do this is to utilize EKOS catheter thrombolysis. To clarify, the term catheter here is not used to aid with urinary matters, but simply refers to a hollow tube utilized in surgery. Additionally, thrombosis is another term for a blood clot.
The process is fairly quick and has good results when considering that it removes the clot, reduces the amount of medication needed, and aids in blood thinning. The entry site will vary depending on the location of your clot, but as the instruments used are small, the entry site will not be large. First, the catheter will be inserted and pushed through the entire length of the clot using a guiding wire. Then, the guiding wire will be removed, and an ultrasonic core will be inserted through the catheter.
This ultrasonic core will emit ultrasonic pulses which will help to break up the clot by aiding in the dispersion of the fibrinolytic drug, or drug that breaks up the fibers that are holding the clot together. The acoustics of the ultrasonic core also assist with the breakup of these fibers. From there, the tools will be removed.
In most cases, your physician will prescribe you a blood thinner to take on a regular basis. There are several anticoagulants that have been around for decades so their risks are better understood. Your physician will consider the drug's effects as well as your personal history. This includes things like your family history, other medications, and co-morbid conditions.
It is important to note that these will not be selected as a way to get rid of existing blood clots. Instead, these medications will be prescribed to prevent further clots. This is especially true if it has been determined that you have a clotting factor, or other genetic vulnerability, which lead to the first one.
Another option that your healthcare provider may consider is a filter. If a filter is chosen, it will be placed in one of the veins leading to your lungs. Its function will be to break up any potential clots before they get to your lungs. This is usually selected if other conditions prevent you from being able to take blood thinners. However, it may also be chosen based on the location of any prior clots.


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Mr Wong's Emporium
[Episode 11 | 1908 : Evelyn]
Evelyn, Edward and Freddie try to replace the fireworks. They visit Mr Wong's emporium to buy a replacement box but find they can't afford it. They then try to obtain some gunpowder in an attempt to make their own fireworks.
The Australian curriculum: History
Show curriculum details
The Australian Curriculum: History aims to ensure that students develop:
interest in, and enjoyment of, historical study for lifelong learning and work, including their capacity and willingness to be informed and active citizens
knowledge, understanding and appreciation of the past and the forces that shape societies, including Australian society
understanding and use of historical concepts, such as evidence, continuity and change, cause and effect, perspectives, empathy, significance and contestability
capacity to undertake historical inquiry, including skills in the analysis and use of sources, and in explanation and communication.
This resource contains extracts from the Australian Curriculum and is current as at 25 May 2011. © Australian Curriculum, Assessment and Reporting Authority 2010.
ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. You can find the unaltered and most up to date version of this material at http://www.australiancurriculum.edu.au/Home
This material is reproduced with the permission of ACARA.
History activities [5]
Activity 1: Multicultural Australia
Subtheme(s): Multiculturalism; Politics; Social order and education
In the early years of the twentieth century, at the time of the Federation of the colonial states and territories, many important events occurred in Australia. These include:
the implementation of laws to improve working conditions and devise fairer wages
the extension of the vote to non-Indigenous women
the introduction of the White Australia policy, which restricted immigration.
The new Australian Constitution discriminated against Indigenous Australians by excluding them from the national census and preventing parliament from making laws about them.
At the time of Federation in 1901, three quarters of the population were Australian born, the majority of English, Scottish and Irish descent. The Chinese, who had represented 3.3% of the population in 1861, represented 0.8% of the population at the time of Federation.
Students can research why and when Chinese people came to Australia. Mr Wong is a significant character in this episode and his story could be a good starting point for discussing the reasons Chinese people came to Australia at different times. Why might Mr Wong have come to live in Australia?
As a class, look at the laws and policies of Australia that have had an impact on Chinese Australians. How might Mr Wong have been affected by Australian laws and policies?
Students could develop a timeline of Chinese migration to Australia. The timeline should use dates, images and reasons for coming to Australia.
Student Activity Sheet H11.4: Timeline of Chinese immigration
Activity 2: Mr Wong
Subtheme(s): Multiculturalism
Mr Wong's Emporium seems to be a hub for the local people. He sells everything from fireworks to paper products. He knows the local children and is friendly and cheerful. We know that Mr Wong's Emporium is quite established, as it appeared in Episode 12: 1898: Rowley.
Students can research the Chinese community in Australia in the early 1900s. A good place to start might be National Archives of Australia, 'Chinese Australians', http://www.naa.gov.au/whats-on/online/showcases/chinese-australians/index.aspx
Who is Mr Wong? Students can create an imaginary profile for Mr Wong using the 'historyface' template in Student Activity Sheet H11.5, with information and old photos from their research on the Chinese community in Australia in the early 1900s. Students need to include his date of birth, background information, family information, status updates and anything else of interest.
Ask students to design a postcard that Mr Wong might have sent back to China, informing his relatives about his life in Australia. The postcard should have an image on one side and text on the other.
Student Activity Sheet H11.5: A profile of Mr Wong
Activity 3: Currency
Subtheme(s): Currency
Loose change doesn't buy much these days, and it seems to be the same for Evelyn and her companions. Loose change couldn't buy them the fireworks they wanted from Mr Wong's Emporium.
Coins and notes from the early 1900s were different to the coins and notes of today, and so was the amount families needed to spend on everyday items like milk and bread.
Research the currency of the 1900s using the following websites:
Museum of Australian Currency Notes, http://www.rba.gov.au/Museum/Displays
Printing, 'A New Currency: 1900–1910', http://printingcolor.blogspot.com/2009/09/new-currency-1900-1920.html
The reference book, Australian decimal currency: an introduction for teachers by the Australian Decimal Currency Board (1965) has some interesting information.
Ask students to bring current Australian coins of each denomination to class. Photocopy or download pictures of coins from the early 1900s. Students can compare the coins they use today to the coins of the 1900s by using the table in Student Activity Sheet H11.6. They can trace or rub over the coins they have brought into class, as well as cutting and pasting the pictures of the old coins.
Student Activity Sheet H11.6: Comparing coins
Activity 4: The price of bread
Subtheme(s): Currency; Food
What did a loaf of bread cost back then? In pairs, students research the cost of everyday essential items like bread and milk in the early 1900s, and compare it to the cost of the same items today. Ask students to conduct an online search and utilise reference books in the library.
Students can develop a cost-comparison chart by listing the items they have researched and converting the pre-decimal amounts into decimal currency. They can use the list on the template provided in Student Activity Sheet H11.7 and add other items if they wish. On the chart, include images of actual advertisements for these commodities.
Student Activity Sheet H11.7: Comparing prices
Activity 5: Gunpowder
Subtheme(s): Celebrations; Inventions and electronic media
In this clip, Evelyn, Edward and Freddie try to create fireworks by obtaining gunpowder. Why does this plan fail? What are the ingredients of gunpowder; where and when was it invented, and by whom? A useful website to start your research with is Wikipedia, 'Gunpowder', http://cunnan.sca.org.au/wiki/Gunpowder
As a class, discuss the benefits and risks associated with gunpowder.
Ask students to research the origins of gunpowder and list some technological advances in its use across time. They could develop a timeline of the developments associated with gunpowder.
Student Activity Sheet H11.8: Timeline on gunpowder
Episode clips
Games from India
Chores and punishment
Cracker nights
Evelyn holding pennies and halfpennies
Buying Mr Wong's 'No 5' assortment box
Evelyn, Freddie and Eddie with Mr Wong's shop sign
Decade timeline – 1900s
Aligned resources
TLF digital curriculum resources
L2707 This house: settling in Broome
R967 Flash Jack from Gundagai
R4210 Melbourne street decorations, 1901
R8056 Children taking part in Federation celebrations, 1901
R9106 Bryce Higgins - Family and Holiday Scenes, 1909: Mount Wellington
Teaching activities Episode 11 | 1908 : Evelyn
| Pdf 40kb
Word 64kb
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Word 687kb

redpajama_set_name: RedPajamaCommonCrawl


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\section{Introduction}
The RT/HRT/EW formula \cite{Ryu:2006bv, Hubeny:2007xt, Engelhardt:2014gca} for computing entanglement entropies is a remarkable entry in the holographic dictionary.
We are instructed to find a codimension-two surface in the bulk that minimizes the generalized entropy functional.\footnote{The generalized entropy functional \cite{Bekenstein:1972tm, Bekenstein:1973ur} depends on a codimension-two surface and equals the area of this surface, plus the entropy of matter fields on the outer half of a Cauchy slice passing through this surface.}
This codimension-two surface is called the quantum extremal surface (QES) and the value of the generalized entropy functional on the QES gives the entanglement entropy.
Furthermore, the bulk region between the QES and the boundary, the entanglement wedge, can be reconstructed just using the knowledge of the corresponding boundary subregion \cite{Almheiri:2014lwa, Dong:2016eik, Almheiri:2017fbd, Cotler:2017erl, Jafferis:2014lza, Jafferis:2015del}.

The papers \cite{Penington:2019npb, Almheiri:2019psf} considered the coupling of a large AdS black hole to a flat space bath region, allowing the black hole to evaporate.
The entanglement entropy of the black hole was seen to undergo a first order phase transition following the appearance of a new nontrivial quantum extremal surface at late times.

Following this idea, \cite{Almheiri:2019hni} considered a two-dimensional gravity+matter theory, where the matter sector has a three-dimensional holographic dual.
The main result of \cite{Almheiri:2019hni} is that the entanglement wedge of Hawking radiation at late times contains an ``island" that lies in the interior of the black hole.
This was also suggested in \cite{Penington:2019npb}.
From a 2d viewpoint, this island is completely disconnected and spacelike separated from the naive domain of dependence of the region where the Hawking radiation lives.
The 3d geometry connects these two pieces of the entanglement wedge.

The general lesson is that one should include contributions from islands in order to compute entanglement wedges and entropies of quantum systems coupled to gravity.
The role of islands becomes crucial if there is a lot of entanglement between the bulk fields in the naive region and the island, for then, it can be beneficial to pay a cost proportional to the area of the island while incurring lots of savings in the bulk entropy.
A prototypical case is to compute the entanglement entropy of the Hawking radiation that lies in the asymptotically-flat, weak-gravity region.

The state considered in \cite{Almheiri:2019hni} was time-dependent since the black hole is evaporating.
In \cite{Almheiri:2019yqk}, the situation was simplified and it was demonstrated that islands exist even in large AdS black holes that are in equilibrium with a flat space bath region (and hence the geometry is static).
All explicit computations in \cite{Almheiri:2019yqk} were also for a two-dimensional gravity+matter theory, since this allows for some simple analytic expressions.

The goal of this note is to demonstrate that islands also exist in higher dimensions.
For that purpose, we consider the equilibrium setup of \cite{Almheiri:2019yqk}, but in \emph{four}-dimensional gravity+matter theories.
To facilitate the computation of quantum extremal surfaces, we use the trick from \cite{Almheiri:2019hni} of taking the matter CFT$_4$ to have a five-dimensional holographic dual.
In other words, we consider a Randall-Sundrum type setup with a 4d brane in a 5d ambient spacetime \cite{Randall:1999vf, Karch:2000ct, Dvali:2000hr}.
In this setup, quantum extremal surfaces in 4d become ordinary RT surfaces in 5d, and thus it becomes a tractable problem to compute them.

In particular, we will focus on the version of the information paradox described in section 4 of \cite{Almheiri:2019yqk}, see also \cite{Mathur:2014dia}.\footnote{This is different than the paradox discussed in \cite{Maldacena:2001kr}. See also \cite{Papadodimas:2015xma} for a discussion of local operators behind the horizon in the eternal black hole.}
This involves the thermofield double of a black hole coupled to, and in equilibrium with, a bath at some temperature.
That is, there are two black holes, both coupled to their own baths.\footnote{Note that unlike \cite{Almheiri:2019hni}, the paper \cite{Almheiri:2019yqk} did not assume that the matter sector has a three-dimensional dual. In other words the computations of \cite{Almheiri:2019yqk} were all done in two dimensions.}
One starts with a Cauchy slice through the middle of the Penrose diagram, and moves it forward in time on both sides.
See figure \ref{fig:cartoon}.
The question is what is the entanglement entropy of the union of the two baths as a function of this time?
Naively, this entropy increases linearly in time, forever.
This happens because the bath is exchanging particles with the black hole: Hawking particles enter the bath and their entangled partners fall into the black hole.
The mass of the black hole is not changing because we are in the Hartle-Hawking state, but the underlying exchange of quanta goes on.
This is the analog of Hawking's calculation.

At late times, however, the entanglement wedge of the union of the bath regions contains an island that extends \emph{outside} the horizon \cite{Almheiri:2019yqk}.
The generalized entropy of this QES saturates at late time, and is approximately equal to twice the Bekenstein-Hawking of a single black hole.
This happens because the island contains the Hawking partners, and thus by including the island we save on the $S_\text{bulk}$ term in the generalized entropy.
Thus, overall, the entropy grows linearly in time for a while before saturating.

In this note, we demonstrate that the same resolution works even in higher dimensions.
The problem of setting up the above paradox in the 4d eternal black hole with a matter sector that has a 5d holographic dual reduces to finding a static 5d geometry with the correct boundary conditions.
We construct this geometry numerically using the DeTurck trick \cite{Headrick:2009pv,Wiseman:2011by,Dias:2015nua}.
This involves solving coupled PDEs for five functions of two variables each, see (\ref{eq:ansatz}) for the ansatz for the line element.
For a picture of the integration domain and the behavior of the space near the conformal boundaries and the Planck brane, see figure \ref{fig:intdom}.

As already noted, quantum extremal surfaces in 4d become ordinary extremal surfaces in 5d.
In the numerically constructed 5d geometry, we numerically find the extremal surfaces that are relevant for computing the entropy of the union of the two baths.
There are two qualitatively different types of extremal surfaces, see figure \ref{fig-minsurfaces}.
The extremal surface that dominates at early times goes through the horizon, and the entropy computed using this surface increases linearly in time.
This is because of the stretching of space inside the horizon, as described in \cite{Hartman:2013qma}.
However, there is another extremal surface that dominates at late times.
This extremal surfaces always stays outside the horizon and ends on the Planck brane.
The entropy computed using this surface saturates at late times, essentially because, being completely outside the horizon, it does not get affected by the stretching of space inside the horizon.

Thus, our results provide a highly nontrivial check that the results of \cite{Penington:2019npb, Almheiri:2019psf, Almheiri:2019hni, Almheiri:2019yqk} are unchanged upon increasing the spacetime dimension: The information paradox is averted by the emergence of an island in the relevant entanglement wedge at late times.

Increasing the dimensionality of the setup of section 4 of \cite{Almheiri:2019yqk} is a significant step forward because a possible criticism of \cite{Almheiri:2019hni, Almheiri:2019yqk, Almheiri:2019psf} is that the explicit computations were only done in 2d AdS-JT gravity \cite{Almheiri:2014cka, Engelsoy:2016xyb, Maldacena:2016upp, Jensen:2016pah}, which is known to be dual to an ensemble of Hamiltonians, rather than a single fixed Hamiltonian \cite{Saad:2019lba, Saad:2019pqd}.
So one might wonder if 2d AdS-JT gravity is somehow not representative of a typical gravity theory, and that islands might not exist in higher dimensions.
Even from the perspective of the gravity equations of motion, it is not completely obvious that the gravity computations of \cite{Almheiri:2019hni, Almheiri:2019yqk, Almheiri:2019psf} generalize to higher dimensions.
By our numerical construction, we explicitly provide this generalization.

The organization of this paper is as follows.
In section \ref{sec-background}, we discuss the action for the 5d gravity theory and the boundary conditions.
In section \ref{sec-numerics}, we describe the technique for numerically finding the static geometry.
In section \ref{sec-surfaces}, we describe the relevant extremal surfaces, including the one that corresponds to having an island, and discuss how it avoids the information paradox in this setting.
We conclude in section \ref{sec-discussion} with some discussion and future directions.
Appendix \ref{app:convergence} contains some details about the convergence of the numerical methods used.

\section{Setup of the problem}
\label{sec-background}

\begin{figure}[t!]
    \centering
    \includegraphics[scale=0.4]{graphics/zwplane.pdf}
    \caption{A simple geometry with a RS or a Planck brane, discussed in \cite{Takayanagi:2011zk}.
    The RS or the Planck brane along lies along the locus $z=-w \tan \theta$.
    The induced geometry on the brane is AdS$_4$ with length scale (\ref{eq:L4}).
    The angle $\theta$ is fixed by the tension parameter $\alpha$ in the action (\ref{eq:action}) via the relationship (\ref{eq:alphathetarelation}).}
    \label{fig:zwplane}
\end{figure}

As mentioned in the introduction, following \cite{Almheiri:2019hni}, we want to consider a ``doubly-holographic" setup, but in higher dimensions.
We take a 4d AdS gravity theory coupled to a matter CFT$_4$ that has a 5d holographic dual.
We wish to consider a large black hole in this theory that is in equilibrium with a flat space bath region containing the same matter CFT$_4$.
Thus, we are led to consider the following action:
\begin{equation}
I = \frac{1}{16\pi G_5}\int_{\mathcal{M}}\mathrm{d}^5x \sqrt{-g} \left(R+\frac{12}{L^2}\right)+\frac{1}{8\pi G_5}\int_{\mathcal{B}} \mathrm{d}^{4} x \sqrt{-h} \left(K - \alpha\right)\, .
\label{eq:action}
\end{equation}
Here $\mathcal{B}$ denotes the Planck or the RS brane \cite{Randall:1999vf} and it should be seen as one of the boundary components of the bulk spacetime.
The quantity $L$ is the AdS$_5$ length scale, and $\alpha$ is proportional to the tension of the brane, see (\ref{eq:tension}) below.
The Gibbons-Hawking term at the UV boundary has been omitted to avoid clutter.

As discussed in detail in \cite{Almheiri:2019hni}, the fundamental description of such a system should be taken to be a (2+1)-d holographic theory, coupled to a (3+1)-d bath system.
In the first step, one replaces the (2+1)-d holographic theory with an AdS$_4$ spacetime.
This AdS$_4$ spacetime is coupled to a flat space reservoir, and we have (3+1)-d matter fields propagating on this hybrid spacetime.
In the second step, one assumes that the (3+1)-d matter fields are also holographic and replaces them with the (4+1)-d geometry.
The (3+1)-d gravitational fields are represented by a Randall-Sundrum brane embedded inside this (4+1)-d geometry.

Varying the action (\ref{eq:action}) with respect to the metric gives us the Einstein equations
\begin{equation}
R_{AB}-\frac{R}{2}g_{AB}-\frac{6}{L^2} g_{AB}=0\qquad \text{on}\qquad \mathcal{M}\, .
\label{eq:einstein}
\end{equation}
Henceforth, upper case Latin indices will refer to the five-dimensional indices and lower case Latin indices will refer to coordinates along the brane.
The boundary term in (\ref{eq:action}) allows for the usual Dirichlet boundary conditions, but we will infact impose the other possible alternative
\begin{equation}
K_{ab}-K h_{ab}+\alpha h_{ab}=0\qquad
\text{on}\qquad \mathcal{B}\,.
\label{eq:bc}
\end{equation}
Recall from (\ref{eq:action}) that $h_{ab}$ is the induced metric on $\mathcal{B}$.
This boundary condition is what one would call a Neumann boundary condition.
In fact, imposing this Neumann boundary condition rather than the Dirichlet one is what allows a given boundary component of $\mathcal{M}$ to be called a Planck brane.

Before proceeding, let us review a simple Randall-Sundrum geometry \cite{Randall:1999vf, Karch:2000ct}, which has a Planck brane ending on the conformal boundary of AdS.
This configuration was also considered in \cite{Takayanagi:2011zk, Fujita:2011fp} in the context of AdS/BCFT.
We start with pure AdS written in Poincar\'e coordinates
\begin{equation}
\mathrm{d}s^2 = \frac{L^2}{z^2}\left(-\mathrm{d}t^2+\mathrm{d}z^2+\mathrm{d}w^2+\mathrm{d}w_1^2+\mathrm{d}w_2^2\right)\,.
\label{eq:poincare}
\end{equation}
Now consider the surface $z = - w\tan \theta$, where $\theta$ is some angle between $0$ and $\pi/2$.
We only keep the region $z> - w \tan \theta$ for $w<0$.
Of course, we always restrict to $z>0$ even for $w>0$.
See figure \ref{fig:zwplane}.

We now want to implement the boundary condition (\ref{eq:bc}), which is a form of the Israel junction conditions \cite{Israel:1966rt}.
Computing the extrinsic curvature on the surface $w + z \tan \theta=0$, we get
\begin{equation}
K_{ab} = \frac{\cos \theta}{L}\, h_{ab}\,.
\end{equation}
Plugging this into (\ref{eq:bc}), we get that the parameter $\alpha$ in the action (\ref{eq:action}) determines the angle $\theta$ via the relationship
\begin{align}
    \alpha = \frac{3 \cos \theta}{L}\, .
\label{eq:alphathetarelation}
\end{align}
From the Israel junction condition we know that the quantity
\begin{equation}
-\frac{1}{8\pi G_5}(K_{ab}-h_{ab} K)=
\frac{1}{8\pi G_5} \, \frac{3\cos \theta}{L}\, h_{ab}\, .
\end{equation}
can be interpreted as the stress tensor of a codimension one object.
This stress-tensor can be interpreted as arising from $3$-brane with tension
\begin{equation}
T_3= \frac{\alpha}{8\pi G_5}\, .
\label{eq:tension}
\end{equation}
This value of the brane tension is consistent with the fact that the last term in the action (\ref{eq:action}) is equal to $\frac{\alpha}{8\pi G_5}$ times the worldvolume of the brane.
Finally, note that by substituting $w = - z \cot \theta$ in the AdS$_5$ line element (\ref{eq:poincare}), and rescaling $z$, we see that the induced metric on the brane is nothing but AdS$_4$ with a length scale
\begin{equation}
L_4= \frac{L}{\sin \theta}\,.
\label{eq:L4}
\end{equation}
Let us now turn to the description of the actual numerical solution that we seek.

\section{The numerical solution}
\label{sec-numerics}

\subsection{The DeTurck trick}
To find solutions, we will use the so-called DeTurck trick, which was first proposed in \cite{Headrick:2009pv}, and reviewed extensively in \cite{Wiseman:2011by,Dias:2015nua}.

Let us first write the Einstein equation (\ref{eq:einstein}) in trace reversed form
\begin{equation}
R_{AB}+\frac{4}{L^2} g_{AB}=0\,.
\label{eq:einsteinr}
\end{equation}
The idea is that, instead of directly solving (\ref{eq:einsteinr}), one considers the modified equation
\begin{equation}
R_{AB}+\frac{4}{L^2} g_{AB} -\nabla_{(A}\xi_{B)} =0\,,
\label{eq:deturck}
\end{equation}
where $\xi^A := \left[\Gamma^A_{BC}(g)-\Gamma^A_{BC}(\bar{g})\right]g^{BC}$ is the so-called DeTurck vector, and $\bar{g}$ is a reference metric.\footnote{Also, $\Gamma^A_{BC}(\mathfrak{g})$ is the Christoffel symbol associated with a metric $\mathfrak{g}$.}
The reference metric $\bar{g}$ is required only to be regular and satisfy the same boundary conditions as $g$ on Dirichlet boundaries, but is otherwise arbitrary.
In particular, if there are Neumann boundaries, the reference metric $\bar{g}$ is not required to satisfy the Neumann boundary condition there.
The equation (\ref{eq:deturck}) is nice because the choice of gauge needed to solve Einstein's equations now appears as a choice of $\bar{g}$.
Further, if we are looking for static solutions, then (\ref{eq:deturck}) together with either Dirichlet or Neumann boundary conditions is an elliptic problem, and is thus locally well-posed.
(For Neumann boundaries, the DeTurck vector is also required to satisfy $\xi \cdot n = 0$, where $n$ is the normal vector to the boundary.)
This is a major advantage over the original Einstein equation (\ref{eq:einsteinr}), whose character depends on the gauge choice even when seeking static solutions.

Solutions of (\ref{eq:deturck}) are not necessarily solutions of (\ref{eq:einsteinr}), because of the new added term $\nabla_{(A}\xi_{B)}$.
Possible solutions with $\xi\neq0$ are called DeTurck solitons.
It can be shown that DeTurck solitons do \emph{not} exist for static and certain stationary solutions of (\ref{eq:deturck}) with purely Dirichlet boundaries \cite{Figueras:2011va,Figueras:2016nmo}.
In this case there is a complete equivalence between solutions of (\ref{eq:deturck}) and (\ref{eq:einsteinr}).
On solutions with $\xi=0$, the gauge choice is a generalisation of harmonic coordinates, given by $\bigtriangleup x^A = \Gamma^A_{BC}(\bar{g})g^{BC}$, where $\bigtriangleup$ stands for the scalar Laplacian in the metric $g$.

However, for Neumann boundary conditions on the metric, of the form (\ref{eq:bc}), this has never been proved.
Although in this case one cannot prove $\xi=0$ on solutions of (\ref{eq:deturck}), one can still make progress because solutions of elliptic equations are locally unique.
Hence, an Einstein solution cannot be arbitrarily close to a DeTurck soliton, and one should be able to distinguish the Einstein solutions of interest from DeTurck solitons by monitoring the quantity $\xi_A \xi^A$ appropriately.

\subsection{The metric ansatz}

\begin{figure}[t!]
    \centering
    \includegraphics[scale=0.5]{graphics/integration-domain.pdf}
    \caption{
    On the left, we show the Penrose diagram of the 4d geometry. We have a two-sided AdS black hole, with each side coupled to a bath.
    On the right, we show the integration domain used in the numerics $x\in (0,1)$ and $y \in (-1,1)$.
    The objective is to solve for five metric functions $Q_1, \ldots, Q_5$ of two variables each (\ref{eq:ansatz}), in this domain.
    We numerically solve only in the region $y>0$, the rest is obtained simply by symmetry.
    On the left edge of this diagram, at $x=0$, we have the RS or the Planck brane where the 4d gravity region lives and the boundary condition (\ref{eq:bc}) is imposed. On the top and bottom edges we have the two baths. As $x \to 1$, the metric approaches that of a 5d planar AdS-Schwarzschild black hole.
    The reader might find it useful to note the points $ABHCD$ on both diagrams.
    The precise induced geometry on the segment $BC$ is determined by the numerical solution, and the left picture is just a cartoon.
    }
    \label{fig:intdom}
\end{figure}

Let us define $\widetilde{w} := w + z \cot \theta$.
For numerical purposes, we take the domain of integration to be $\widetilde{w}>0$ and impose (\ref{eq:bc}) together with $\xi^A n_A=0$ on the edge of the computation domain.
Since we are interested in the Hartle-Hawking state, we want to have a bulk horizon that intersects the brane.
Furthermore, the geometry should be such that at large $\widetilde{w}$ it should approach a five-dimensional planar black hole, whose line element reads
\begin{equation}
\mathrm{d}s^2_{\mathbb{P}} = \frac{L^2}{z^2}\left[-\left(1-\frac{z^4}{z_+^4}\right)\mathrm{d}t^2+
\left(1-\frac{z^4}{z_+^4}\right)^{-1} \mathrm{d}z^2
+\mathrm{d}\widetilde{w}^2+\mathrm{d}w_1^2+\mathrm{d}w_2^2\right]\,.
\end{equation}
For numerical convenience we want to work with compact coordinates only, so we define a new coordinate $x \in (0,1)$ via
\begin{equation}
\frac{x}{1-x} := \widetilde{w} = w + z \cot \theta \,.
\end{equation}
Note that $x=1$ is the asymptotic region $\widetilde{w}\to+\infty$.
Finally, we change from $z$ to a coordinate $y$ where constant $t$ slices are manifestly regular at the event horizon $z=z_+$.
One such choice is given by
\begin{equation}
y := \sqrt{1 - \frac{z}{z_+}}\, .
\end{equation}
In terms of $(t,x,y,w_1,w_2)$ coordinates, the planar black hole reduces to
\begin{equation}
\mathrm{d}s^2_{\mathbb{P}} = \frac{L^2}{(1-y^2)^2}\left\{-y^2\,G(y)\,y_+^2\,\mathrm{d}t^2+\frac{4\,\mathrm{d}y^2}{G(y)}+y_+^2\left[\frac{\mathrm{d}x^2}{(1-x)^4}+\mathrm{d}w_1^2+\mathrm{d}w_2^2\right]\right\}\,,
\end{equation}
where
\begin{equation}
y_+ := z_+^{-1}\,,\quad\text{and}\quad G(y) := \left(2-y^2\right) \left(2-2 y^2+y^4\right)\,.
\end{equation}

We are finally ready to present our metric ansatz:
\begin{multline}
\mathrm{d}s^2 = \frac{L^2}{(1-y^2)^2}\Bigg\{-y^2\,G(y)\,y_+^2\,Q_1\,\mathrm{d}t^2+\frac{4\,Q_2\,\mathrm{d}y^2}{G(y)}+
\\
\frac{Q_4}{(1-x)^4}\left[y_+\mathrm{d}x+2(1-x)^2\,y\,Q_3\,\mathrm{d}y\right]^2+y_+^2\,Q_5\,\left(\mathrm{d}w_1^2+\mathrm{d}w_2^2\right)
\Bigg\}\, .
\label{eq:ansatz}
\end{multline}
Here $Q_I$, with $I\in\{1,2,3,4,5\}$, are functions of $(x,y)\in(0,1)^2$ to be determined by solving (\ref{eq:deturck}).
For the reference metric we take the line element (\ref{eq:ansatz}) with $Q_1=Q_2=Q_4=Q_5=1$ and $Q_3=\cot \theta$.

Let us now discuss the boundary conditions.
At the horizon, located at $y=0$, we impose Neumann boundary conditions for all variables, \emph{i.e.} $ \left.\partial_y Q_I\right|_{y=0} =0$ together with $Q_1(x,0)=Q_2(x,0)$, which in turn enforces the Hawking temperature to be
\begin{equation}
T_H=\frac{y_+}{\pi}\,.
\label{eq:hawking5D}
\end{equation}
At the conformal boundary, located at $y=1$, and also at $x=1$ we demand $g$ to approach the reference metric $\bar{g}$, that is to say $Q_1=Q_2=Q_4=Q_5=1$ and $Q_3=\cot \theta$. Finally, at the brane location, that is $x=0$, we demand the boundary condition (\ref{eq:bc}) together with $Q_3(0,y)=\cot \theta$ and $\xi^a n_a=-\xi^x=0$.
See figure \ref{fig:intdom} for a cartoon depiction of the integration domain.

These boundary conditions yield Robin-type boundary conditions on $Q_1$, $Q_2$, $Q_4$ and $Q_5$ at $x=0$.
It is then a simple exercise to show that (\ref{eq:deturck}) with such boundary conditions, gives rise to an elliptic problem \cite{Figueras:2011va}.

To solve the resulting system partial differential equations, we used a standard pseudospectral collocation approximation on Chebyshev-Gauss-Lobatto points and solved the resulting non-linear algebraic equations using a damped Newton-Raphson method.
The resulting method does \emph{not} exhibit exponential convergence in the continuum limit due to the existence of non-analytic behaviour close the conformal boundary\cite{Donos:2014yya,Marolf:2019wkz}.\footnote{This non-analytic behaviour is a consequence of the DeTurck trick.}
Instead, we will find a power law convergence as we approach the continuum limit.

\subsection{Induced geometry on the brane}

\begin{figure}[t!]
    \begin{center}
    \includegraphics[width=0.9\textwidth]{graphics/induced_geometry.pdf}
    \end{center}
    \caption{Plots of $-g_{tt}/L_4^2$ and $g_{w_1\,w_1}/L_4^2$ on the RS brane (located at $x=0$) as a function of the proper distance from the horizon $\mathcal{P}_{4D}$. In the top row, $\theta \approx 1.47113$, and in the bottom row, $\theta \approx 0.343024$.
    The blue disks correspond to the numerical data, and the solid blue lines are obtained from the 4d planar AdS black hole geometry.
    It is clear that as $\theta$ becomes smaller, the induced geometry on the brane gets closer to that of a 4d planar AdS black hole}
\label{fig:comp}
\end{figure}

Recall that the boundary condition for the metric on the brane is Neumann rather than Dirichlet.
Hence, the actual induced metric on the brane is determined numerically, and does not have a simple analytic expression.
All we know is that there is a horizon at $y=0$.
In this subsection, we characterize the behavior of the induced geometry as $\theta$ becomes small.
The upshot is that, in the limit $\theta\ll 1$, the induced geometry on the brane is close to that of a 4d planar AdS black hole.

In order to see this, consider the auxiliary line element of a four-dimensional planar black hole with horizon located at $Z_+$ and AdS$_4$ length scale $L_4$:
\begin{equation}
\mathrm{d}s^2_{4D} = \frac{L_4^2}{Z^2}\left[-\left(1-\frac{Z^3}{Z_+^3}\right)\mathrm{d}t^2
+ \left( 1-\frac{Z^3}{Z_+^3}\right)^{-1} \mathrm{d}Z^2
+\mathrm{d}w_1^2+\mathrm{d}w_2^2\right]\,.
\label{eq:planar4D}
\end{equation}
Its associated Hawking temperature is given by
$T_{4D} = \frac{3}{4\pi Z_+}$.
If we want to match the temperature of our numerical solution reported in (\ref{eq:hawking5D}), we should impose $Z_+ = \frac{3}{4\,y_+}$.

To compare the line element (\ref{eq:planar4D}) with the induced metric on the brane, we change to a new set of coordinates $\{t,\mathcal{P}_{4D},w_1,w_2\}$, where $\mathcal{P}_{4D}$ is the proper distance from the horizon:
\begin{equation}
\frac{\mathcal{P}_{4D}(Z)}{L_4} = \int_{Z}^{Z_+}\frac{\mathrm{d}\tilde{Z}}{\tilde{Z}}\,
\left(1-\frac{\tilde{Z}^3}{Z_+^3} \right)^{-\frac{1}{2}}
=\frac{2}{3} \log \left(\sqrt{1-\frac{Z^3}{Z_+^3}}+1\right)-\log \left(\frac{Z}{Z_+}\right)\,.
\label{eq:planarproper4D}
\end{equation}
We then look at $-g_{tt}/L_4^2$ and $g_{w_1\,w_1}/L_4^2$ as functions of $(\mathcal{P}_{4D})$, and compare with the results obtained from computing the same quantities using the induced metric on the brane.

More explicitly, the induced metric on the brane can be read off from (\ref{eq:ansatz}) and is given by
\begin{multline}
\mathrm{d}s_{\mathcal{B}}^2 = \frac{L^2}{(1-y^2)^2}\Bigg\{-y^2\,G(y)\,y_+^2\,Q_1(0,y)\,\mathrm{d}t^2+4\left[\frac{Q_2(0,y)}{G(y)}+y^2 \cot^2\theta\,Q_4(0,y)\right]\mathrm{d}y^2
\\
+y_+^2\,Q_5(0,y)\,\left(\mathrm{d}w_1^2+\mathrm{d}w_2^2\right)\Bigg\}\,.
\end{multline}
Again, we can change to proper distance coordinates $\{t,\mathcal{P}_{4D},w_1,w_2\}$ by defining
\begin{equation}
\frac{\mathcal{P}_{4D}(y)}{L_4}=\frac{2\,L}{L_4}\int_0^y \frac{\mathrm{d} \tilde{y}}{(1-\tilde{y}^2)}\sqrt{\frac{Q_2(0,\tilde{y})}{G(\tilde{y})}+\tilde{y}^2 \cot^2\theta\,Q_4(0,\tilde{y})}\,,
\end{equation}
where $L_4$ was is given by (\ref{eq:L4}).
Numerically, computing $\mathcal{P}_{4D}(y)$ can be tricky, because of the divergence of the integrand in the limit $\tilde{y}\to1^-$. To bypass this difficulty, we consider instead
\begin{multline}
\frac{\mathcal{P}_{4D}(y)}{L_4}=\frac{2\,L}{L_4}\int_0^y \frac{\mathrm{d} \tilde{y}}{(1-\tilde{y}^2)}\left[\sqrt{\frac{Q_2(0,\tilde{y})}{G(\tilde{y})}+\tilde{y}^2 \cot^2\theta\,Q_4(0,\tilde{y})}-\sqrt{1+\tilde{y}^2 \cot^2\theta}\right]
\\
+\frac{2\,L}{L_4}\int_0^y \frac{\mathrm{d} \tilde{y}}{(1-\tilde{y}^2)}\sqrt{1+\tilde{y}^2 \cot^2\theta}\,.
\end{multline}
One can show that the integrand in the first line is finite\footnote{Explicitly, we have $\displaystyle \lim_{\tilde{y}\to1^-}\frac{2\,L}{(1-\tilde{y}^2)}\left[\left(\frac{Q_2(0,\tilde{y})}{G(\tilde{y})}+\tilde{y}^2 \cot^2\theta\,Q_4(0,\tilde{y})\right)^\frac{1}{2}
-\left(1+\tilde{y}^2 \cot^2\theta\right)^\frac{1}{2}\right]
=-L \sin \theta$.} as $\tilde{y}\to1^-$, while the integral in the second line can be readily done analytically, and carries all the divergences:
\begin{equation}
\frac{L}{L_4}\int_0^y \frac{\mathrm{d} \tilde{y}}{(1-\tilde{y}^2)}\sqrt{1+\tilde{y}^2 \cot^2\theta}=\mathrm{arctanh}\left( \frac{y}{\sin \theta\sqrt{1+y^2 \cot ^2\theta}}\right)-\cos \theta\, \mathrm{arcsinh}(y \cot \theta )\,.
\end{equation}

We plot our comparisons in figure \ref{fig:comp}.
The top row corresponds to $\theta \approx 1.47113$ and the bottom row corresponds to $\theta \approx 0.343024$.
In all plots in figure \ref{fig:comp}, we have taken $y_+=1$.
The blue disks correspond to the numerical data, and the solid blue lines are obtained from the 4d planar black hole geometry, as detailed above.
The trend is clear: As $\theta$ becomes smaller, the induced geometry gets closer to that of a 4d planar AdS black hole.

\section{Extremal surfaces and the island}
\label{sec-surfaces}

Recall that we want to consider a version of the information paradox in 4d gravity theory coupled to a 4d matter sector.
In this theory, we are considering a black hole coupled to, and in equilibrium with, a bath at nonzero temperature.
We are also working in the two-sided purification, or the thermofield double, of the coupled system.
So there are two black holes and two baths.
We would like to compute the von Neumann entropy of the union of the left and the right bath regions as a function of time, where the time dependence is introduced by moving time forwards on both sides.
See figure \ref{fig:cartoon}.
The two-dimensional version of this problem was considered in section 4 of \cite{Almheiri:2019yqk}.
See also \cite{Mathur:2014dia} and the recent paper \cite{Rozali:2019day}.

We would like to compute 4d quantum extremal surfaces \cite{Engelhardt:2014gca} for the union of the blue regions in figure \ref{fig:cartoon}.
Since this is a very hard problem, we have made the simplification that the matter CFT$_4$ has a 5d holographic dual, as in \cite{Almheiri:2019hni}, and so the 4d quantum extremal surfaces become ordinary 5d RT surfaces.
Note that we are imagining toroidally compactifying the transverse directions to get IR-finite entropies.
\begin{figure}[t!]
    \centering
    \includegraphics[scale=0.4]{graphics/hh_paradox.pdf}
    \caption{Shown here is a two-sided 4d black hole (with two of the spatial dimensions suppressed) coupled to two baths. See also figure \ref{fig:intdom}. We want to compute the entanglement entropy of the union of the two blue regions shown. This diagram lives on the boundary of a static 5d spacetime whose exterior region was computed numerically in section \ref{sec-numerics}.}
    \label{fig:cartoon}
\end{figure}

\subsection{Extremal surfaces at $t=0$}

\begin{figure}[t!]
    \begin{center}
    \includegraphics[width=.45\textwidth]{graphics/example1.pdf}
    \end{center}
    \caption{The two types of extremal surfaces, computed numerically at $t=0$ in the background geometry found numerically in section \ref{sec-numerics}.
    In this figure, we have taken $\theta = \pi/4$ and $x_{\partial}=1/2$.
    The horizontal black dotted lines $\partial$ at the top and bottom are the left and right baths.
    The dashed black line $\mathcal{B}$ along the left edge is the location of the brane, which contains the 4d black hole.
    The horizontal red dashed-dotted line in the middle is the 5d bifurcate horizon, which meets the brane at the 4d horizon.
    Compare with figure \ref{fig:intdom}.
    The orange curve corresponds to an extremal surface ending on the brane with $y_{\mathcal{B}}\approx0.31602(1)$, while the blue curve correspond to an extremal surface that penetrates the bifurcating Killing surface smoothly.
    There is, in fact, a continuous family of orange extremal surfaces and there is a unique one amongst them with the smallest area, see figure \ref{fig:mar}.}
\label{fig-minsurfaces}
\end{figure}

We would like to compute these 5d RT surfaces \cite{Ryu:2006bv}.
More precisely, we would like to extract the extremal surfaces that anchor at the boundary at a given location $x=x_\partial>0$.
We will numerically compute extremal surfaces  on the $t=0$ slice of the line element (\ref{eq:ansatz}).

As emphasized in \cite{Almheiri:2019hni}, there are two extremal surfaces of interest emanating from $x_{\partial}$: the ones that penetrate the horizon, and the ones that end up anchoring on the brane (recall that the brane is located at $x=0$), see figure \ref{fig-minsurfaces}.
We will denote the area of the surface that penetrates the horizon by $\mathcal{A}_{\mathcal{H}}(x_\partial)$ and the area of the surface that ends on the Planck brane by $\mathcal{A}_{\partial \mathcal{M}}(x_\partial,y_{\mathcal{B}})$, where $y_{\mathcal{B}}$ is the value of $y$ at which this surface intersects the brane.
Formally, both these areas are infinite, because of the divergence at the conformal boundary.
However, the difference between these two is well defined.

Let us define the area difference
\begin{equation}
\Delta \mathcal{A}(x_{\partial},y_{\mathcal{B}}):=\mathcal{A}_{\partial \mathcal{M}}(x_{\partial},y_{\mathcal{B}})-\mathcal{A}_{\mathcal{H}}(x_{\partial})\,,
\label{eq:diff}
\end{equation}
which is finite for any pair $(x_\partial,y_{\mathcal{B}})$.
We should also minimize this with respect to $y_\mathcal{B}$ and define
\begin{equation}
\Delta \mathcal{A}(x_{\partial}) := \min_{y_{\mathcal{B}}\in(0,1)}\Delta \mathcal{A}(x_{\partial},y_{\mathcal{B}})\,.
\end{equation}
We will simply compute $\Delta \mathcal{A}(x_{\partial},y_{\mathcal{B}})$ for several values of $y_{\mathcal{B}}$ and look for a minimum.
We will see that there is unique value of $y_\mathcal{B}$ that minimizes $\Delta \mathcal{A}(x_{\partial},y_{\mathcal{B}})$.

Our extremal surfaces are parametrized by coordinates $\sigma^{\hat{\mu}}$, with $\hat{\mu}=1,2,3$.
For the surfaces that penetrate the horizon, we choose $\sigma^1=y$, $\sigma^{2}=w_1$ and $\sigma^{3}=w_2$, so that the extremal surfaces can be parametrized by $x=F(y)$ in the $(x,y)$ plane.
To compute such curves we look at the Euler-Lagrange equations derived from
\begin{align}
S &= \int \mathrm{d}^3\sigma \sqrt{\mathrm{det}\left[ (\partial_{\sigma^{\hat{\mu}}} x^{\dot a})\,(\partial_{\sigma^{\hat{\nu}}} x^{\dot b})\,g_{\dot a\dot b}\right]} \nonumber
\\
&=\Delta w_1\Delta w_2 \int_{0}^{1}\mathrm{d}y\, \frac{L^3 y_+^2 Q_5(F(y),y)}{\left(1-y^2\right)^3}\times \nonumber
\\
&\left(\frac{4 Q_2(F(y),y)}{G}+\frac{Q_4(F(y),y)}{[1-F(y)]^4} \left[y_+ \frac{\mathrm{d}F(y)}{\mathrm{d} y}+2 [1-F(y)]^2 y Q_3(F(y),y)\right]^2\right)^\frac{1}{2}\, ,
\label{eq:minimal}
\end{align}
where dotted indices run over the spatial coordinates $x,y,w_1,w_2$, but not over time.
We will not present the explicit equations of motion following from (\ref{eq:minimal}) because they are not illuminating.
Suffice it to say that they are second order ODEs in $F(y)$, and thus can only be solved once two boundary conditions are supplied.
One of these boundary conditions is imposed at the conformal boundary, where we demand $F(1)=x_\partial$, while at the horizon we demand $F^\prime(0)=0$.

For the surfaces that end on the Planck brane, one has to proceed with more care, because if we try to think of these surfaces as a function $y(x)$ or $x(y)$, these functions will be multi-valued, see the orange curve in figure \ref{fig-minsurfaces}.
To bypass this, we introduce two parametrizations in two different parts of the surface.
For a range $y\in(1,y_c)$ we take $x=F(y)$, \emph{i.e.} we choose $\sigma^1=y$, $\sigma^{2}=w_1$ and $\sigma^{3}=w_2$.
As boundary conditions we demand $F(1)=x_{\partial}$ and $F(y_c)=x_c>x_{\partial}$, which yields a unique solution in this interval for given values of $x_{\partial}$, $x_c$ and $y_c$.
For $x\in(0,x_c)$ we choose $\sigma^1=x$, $\sigma^{2}=w_1$ and $\sigma^{3}=w_2$ with $y=P(x)$.
We view the resulting second order ordinary differential equation as an initial value problem, where we demand $P(x_c)=y_c$ and $P^\prime(x_c)=F^\prime(y_c)^{-1}$.
Finally, we read off $y_{\mathcal{B}}=P(0)$ from the integration procedure.

For numerical stability, we found that it was crucial to use the same parametrization for both surfaces near the boundary, as the leading divergences in (\ref{eq:diff}) were easier to cancel.

The results are shown in figure \ref{fig-minsurfaces}, where we plot an example of the two types of curves in the $(x,y)$ plane.
In this figure we used $\theta = \pi/4$ and $x_{\partial}=1/2$. Also, the value of $y_\mathcal{B} \approx  0.31602(1)$ for this specific plot.
For the surface that ends on the brane, we vary $x_c$ and $y_c$, which in turn varies $y_\mathcal{B}$.
As we do so, we compute $\Delta \mathcal{A}(x_{\partial},y_{\mathcal{B}})$ as in the left panel of figure \ref{fig:mar}.
In the right panel of figure \ref{fig:mar}, we zoom in close to the point where the horizon intersects the brane and find that $\Delta \mathcal{A}(x_{\partial},y_{\mathcal{B}})$ is minimized for some value $y_{\mathcal{B}}=y_{\mathcal{B}}^\star>0$.
For the particular run shown in figure \ref{fig:mar}, we find that this occurs for $y_{\mathcal{B}}^\star\approx 0.067224(5)$.
Since the minimum is very shallow, one might wonder whether this is a numerical artefact.
To show that this is not the case, we also plot error bars in figure \ref{fig:mar}, which are estimated via the numerical convergence studies performed in appendix \ref{app:convergence}.
\begin{figure}[t!]
    \begin{center}
    \includegraphics[width=.9\textwidth]{graphics/example2.pdf}
    \end{center}
    \caption{This figure depicts $\Delta \mathcal{A}(x_{\partial},y_{\mathcal{B}})$ as a function of $y_{\mathcal{B}}$, computed for $x_{\partial}=1/2$ and $\theta = \pi/4$. In the left panel we have $y_{\mathcal{B}}\in[0.0028(7),0.35379(9)]$, whereas on the right we zoom in close to the point where the horizon intersects the brane. The surface corresponding to the minimum in this figure is the correct RT surface at late times.}
\label{fig:mar}
\end{figure}

\subsection{Time dependence of the entropy and the island}
As shown in figure \ref{fig:mar}, the surface that penetrates the horizon (blue in figure \ref{fig-minsurfaces}) has smaller area at $t=0$ and thus is the correct RT surface to use.
The time dependence we are considering involves moving the two sides forwards in time, see figure \ref{fig:cartoon}.

As in \cite{Almheiri:2019yqk, Hartman:2013qma}, the area of of this surface increases (linearly after a few thermal times) as we perform the time evolution.
As explained nicely in \cite{Hartman:2013qma}, the intuitive reason behind this is the stretching of space behind a black hole horizon \cite{Mathur:2009hf, Polchinski:1995ta}.
We would have an information paradox if this entropy increase continued forever, because the von Neumann entropy of the union of the two baths should saturate close to $2 S_\text{BH}$.
(Note that we are imagining the transverse directions to be toroidally compactified.)

The resolution is that the area of the surface that ends on the Planck brane (orange in figure \ref{fig-minsurfaces}) approaches approximately $2 S_\text{BH}$ at late times, as in \cite{Almheiri:2019yqk}.
Again, the intuitive reason is that since this surface does not penetrate the horizon, it does not get affected by the stretching of space inside the horizon.
Thus, the surface that ends on the Planck brane will win at late times.

The exchange of dominance between these two surfaces leads to an entropy that increases linearly for a while before saturating.
This is the resolution of the information paradox in this setting.

Note that at late times, the entanglement wedge of the union of the left and the right baths contains an island.
The island is the region on the left vertical line in figure \ref{fig-minsurfaces} in between the two points where the orange curves intersect it.

\subsection{Relation to Penington's work and comments on greybody factors}

Higher-dimensional evaporating black holes were also discussed in section 2.4 of \cite{Penington:2019npb}, where an argument using the intermediate value theorem was presented for the existence of a quantum extremal surface behind the horizon.
The quantitative location of the QES in higher dimensions was left undetermined.
While we believe that a more accurate computation will not qualitatively change the results in \cite{Penington:2019npb}, at the same time it is desirable to obtain quantitatively the location of the QES.

Our work sidesteps the complications of an evaporating black hole and direct computations of $S_\text{bulk}$ by considering an eternal black hole in the doubly holographic setup.
In this toy setup, we are able to make quantitative predictions because $S_\text{bulk}$ gets geometrized as the area of the five-dimensional RT surface.

We would also like to make some comments on greybody factors which are present in higher dimensions.\footnote{We thank the referees of SciPost for emphasizing the issue of greybody factors.}
The direct computation of greybody factors is hard.
In our doubly holographic setup, the effect of 4d greybody factors is packaged into the numerical 5d geometry.
Thus, even though we have not been explicit about greybody factors, they are taken into account in our computations.

However, note that we have not explicitly computed the time dependence of the initial RT surface.
In particular, we do not know the precise coefficient of the linear growth of entropy.
It would be desirable to do so, but the time dependent calculation is numerically more involved and is beyond the scope of this work.
In principle, one could compute that coefficient and compare it to the same coefficient in the Hartman-Maldacena setup \cite{Hartman:2013qma}, which does not have the RS brane.
One could also compare the times at which the entropy saturates.
We expect that the coefficient of linear growth in our setup with the RS brane should be smaller, and the time to saturation should be larger than the corresponding quantities in the Hartman-Maldacena setup because of greybody effects.

\section{Discussion}
\label{sec-discussion}

Following section 4 of \cite{Almheiri:2019yqk}, we discussed a version of the information paradox in a four-dimensional black hole coupled to a bath in the Hartle-Hawking state.
Time dependence is introduced by moving time forwards on both sides.
We ask for the von Neumann entropy of the union of the left and right baths as a function of time, as depicted in figure \ref{fig:cartoon}.
To facilitate computations of quantum extremal surfaces, the matter is described by a CFT$_4$ that has a five-dimensional holographic dual \cite{Almheiri:2019hni}.

We numerically solved Einstein's equations using the DeTurck trick and found a static five-dimensional geometry having two flat UV boundaries and a Planck brane, see figure \ref{fig:intdom}.
This geometry has a bifurcate horizon that intersects the Planck brane.
In this setup, quantum extremal surfaces in 4d become usual RT surfaces in 5d.
We have computed the extremal surfaces that correspond to computing the entropy of the union of the left and the right baths.
There are two types of surfaces, as shown in figure \ref{fig-minsurfaces}.
One type of extremal surface (blue in figure \ref{fig-minsurfaces}) penetrates the horizon and is the dominant one at early times.
However, its area increases as a function of time because of the stretching of space inside the horizon \cite{Hartman:2013qma}.
If there was no competing extremal surface, this would lead to an indefinite growth of entropy.
However, we know that the entropy of the union of the two baths, being equal to the entropy of the two black holes, should saturate close to $2 S_{\text{BH}}$.
The resolution is that, in fact there is a second type of surface (orange in figure \ref{fig-minsurfaces}) that ends on the Planck brane.
Its area saturates at $2 S_\text{BH}$, and thus, it wins at late times.
Overall, we get an entropy that grows linearly and then saturates.

This also means that the entanglement wedge of the union of the two baths contains an island at late times \cite{Almheiri:2019yqk}.
In figure \ref{fig-minsurfaces}, this island is the region on the left vertical edge between the two points where the orange curves intersect the vertical line.

The results of this paper unambiguously show that at least some of the gravity computations of \cite{Almheiri:2019psf, Almheiri:2019yqk, Almheiri:2019hni} done in AdS-JT gravity generalize to higher dimensions.
In particular, the microscopic fact that 2d AdS-JT gravity is dual to an ensemble of Hamiltonians, rather than a single one, plays no crucial role as far as gravity computations are concerned.
One can speculate about the possibility that quantities computed using path integrals over metrics should always be interpreted as suitably-ensemble-averaged quantities, and that to reproduce \emph{all} the features present in observables of normal unitarily evolving quantum systems, one perhaps needs stringy physics in the bulk and all sorts of additional effects.
See \cite{Bousso:2019ykv} for a recent discussion of ensemble averages vs. unitary evolution in this context.

In conclusion, this paper provides the first setup where quantitatively precise entanglement islands \cite{Penington:2019npb, Almheiri:2019psf, Almheiri:2019hni, Almheiri:2019yqk} have been computed in higher dimensions.
The conclusion is the same: Islands appear in entanglement wedge of the Hawking radiation at late times and this stops the indefinite growth of von Neumann entropy, giving an answer consistent with unitarity and a finite density of states.

There are quite a few natural extensions of our work.
We found the static geometry and the two types of extremal surfaces numerically at $t=0$, and then used general reasoning to deduce the time dependence of the areas.
It would be interesting to explicitly compute the time dependence of the extremal area surfaces.
It would also be interesting to see if one can make any analytic statements in the limit $\theta \to 0$.
This is the limit where the length scale of AdS$_4$ (\ref{eq:L4}) goes to infinity.
Finally, it would be interesting to see if the scenario of ``uberholography" \cite{Pastawski:2016qrs}, found to hold in the 2d/3d setup of \cite{Almheiri:2019hni} in the recent paper \cite{Chen:2019uhq}, persists in higher dimensions.

\subsection*{Acknowledgments}
We thank Juan Maldacena and Ying Zhao for many discussions and collaboration in the initial stages of this work.
A.~A. is supported by funds from the Ministry of Presidential Affairs, UAE.
R.~M. is supported by US Department of Energy grant No.\ DE-SC0016244.
J.~E.~S. is supported in part by STFC grants PHY-1504541 and ST/P000681/1.
J.~E.~S. also acknowledges support from a J.~Robert~Oppenheimer Visiting~Professorship. This work used the DIRAC Shared Memory Processing system at the University of Cambridge, operated by the COSMOS Project at the Department of Applied Mathematics and Theoretical Physics on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E- infrastructure capital grant ST/J005673/1, STFC capital grant ST/H008586/1, and STFC DiRAC Operations grant ST/K00333X/1. DiRAC is part of the National e-Infrastructure.

redpajama_set_name: RedPajamaArXiv


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World >
Diplomacy has been 'weaponized' with memes, trolling, online polls
By Alastair Jamieson with NBC News World News • Updated: 09/04/2018
A protest sign outside of Facebook headquarters in Menlo Park, California, on April 5. - Copyright Justin Sullivan
LONDON — A growing army of diplomats is spreading targeted propaganda online.
Techniques for promoting news articles and memes on social media are being used by governments around the world to influence opinion about global events and to undermine public confidence in rival countries.
The spy poisoning case and Russian interference in the 2016 U.S. presidential election have proved to be fertile ground for their work.
Russia has taken to trolling adversaries using official diplomatic Twitter accounts — creating tongue-in-cheek online polls and even snarky memes.
Coming in the wake of fake news factories and troll farms, experts warn such digital diplomacy is the latest way in which public trust in official information is being eroded.
Researchers say that while some governments are using social media transparently to streamline their message, others are engaged in covert disinformation campaigns that disrupt traditional foreign relations.
"There is a now a sense that public diplomacy has become weaponized," said Jan Melissen, a professor of diplomacy at the University of Antwerp and a senior research fellow at the Netherlands Institute of International Relations Clingendael. "These technologies are being used in ways that we didn't anticipate."
It is a far cry from the Arab Spring uprisings of 2011 when campaigners used Facebook to mobilize grassroots pro-democracy protests in repressive Middle Eastern countries.
Corneliu Bjola, an associate professor of diplomatic studies at the University of Oxford and head of the Oxford Digital Diplomacy Research Group, said the poisoning of Russian former double agent Sergei Skripal highlighted the extent of online statecraft.
According to Bjola, methods of online diplomacy include:
using Facebook's publicly-available option to pay for posts to be targeted at particular interest groups;
"gaming" Facebook's constantly changing news feed filter in order to boost page views;
building communities of sympathetic users more likely to reshare links or retweet and approaching other so-called influencers to do the same. "People are more likely to trust links that come from their friends, their groups or pages they have liked, and they are also more likely to reshare them," Bjola said. "Something that resonates has a much better chance of being passed on, regardless of how accurate it is.
In extreme examples, it can involve the dissemination of misleading or fake information.
Such techniques are "getting much greater reach than through traditional media," Melissen said. "In 2012 after the Arab Spring, Facebook was seen as a democratizing medium where sharing everything meant people reaching out to each other. That's still taking place but we've the same medium being used in a more intrusive way, including using algorithms not to reach the networks of your friends, like in normal public diplomacy, but to enter the networks of your opponents."
Wael Ghonim, a former Google executive who spearheaded the Arab Spring in Egypt with his online activism, lamented how authoritarian governments had responded by drowning out dissent with propaganda. "I thought the Internet was a power that was granted to the people and that would never be weakened," he said in an interview with MIT Technology Review. "But I was wrong."
'A tool for contesting reality'
Russia's London Embassy boasts a Digital Club that offers Twitter users "regular competitions and prize draws" and even invitations to the ambassador's residence in exchange for allowing their account to be used for automatic retweets of official posts.
Most official diplomacy remains benign, even in areas of tension. The U.S. operates a "virtual embassy" for Iranians, in the absence of formal relations. The Palestinian Authority, which has no presence in Israel, runs a Facebook page in Hebrew aimed at communicating directly with ordinary Israelis.
There have been some successes; the U.S. Embassy in Beijing's decision to publish air quality data on its Twitter feed is credited with pressuring Chinese authorities into tackling the city's chronic pollution.
And President Donald Trump's Twitter pressure on Kim Jong Un — including his "Little Rocket Man" jibe — was last month praised by South Korea for nudging Pyongyang toward talks.
However, Russian meddling in Ukraine in 2014 — which ended with the annexation of Crimea — saw digital diplomacy used as "a tool for contesting reality," according to Bjola's fellow Oxford researcher, Ilan Manor.
"During the first two months of the Crimean crisis … Russian digital diplomacy stated that there were no Russian troops in Ukraine while the U.S. argued the opposite," Manor wrote. "Once Russia admitted it had forces in the area, it portrayed its actions as an attempt to safeguard the lives of Russian minorities coming under threat from Ukrainian neo-Nazis. The U.S., and its allies, alleged that no such neo-Nazis existed and portrayed Russia's actions as the invasion of another sovereign state. Social media users were thus faced with two conflicting realities."
Western nations are learning to play Russia at its own game. Moscow traded barbs with the U.S. and Britain over the Skripal case, posting snarky memes ridiculing the U.K. government, the State Department and even American weather.
"It seems all very entertaining but the Russians are systematically trying undermine confidence in the U.K. and U.S. government position," said James Pamment, assistant professor at Lund University in Sweden and an expert on digital diplomacy. "They got caught out this time because the British Foreign Office quickly used social media quickly to frame the issue not as a spy being injured but about the use of chemical weapons in peacetime. By doing so, Britain was able to build an international consensus behind diplomatic expulsions and I don't think Russia was expecting that."
It is not just Russia involved in the darker side of digital diplomacy.
"There are many other players — more than we know — and some of them, such as Iran and Sudan, you wouldn't immediately expect in this field," Melissen said.
In particular, Israel has embraced online communication as a way of getting its message across and to underscore its reputation as one of one the world's leading technology hubs. The Israeli Defense Forces (IDF) has a social media unit that provides fast information on clashes and stabbings, helping it shape the media narrative on events such as the recent killings of Palestinian protesters in Gaza.
Elad Ratson, one of a number of Israeli diplomats to regularly approach reporters on social media, describes himself in his Twitter bio as "pioneering digital & algorithmic diplomacy."
His apparently automated direct messages invite journalists to retweet positive news about Israel or to join the Israel Twitter Hive, "a community of few thousands of people from around the world who enable their Twitter account to automatically amplify important factual information about Israel, thus contributing to its large exposure."
Ratson agreed to discuss his work with NBC News but the request was overruled, without explanation, by a spokeswoman.
Lund University's Pamment predicted that digital diplomacy would never replace traditional relationship-building between nations.
"The real battles are still taking place in traditional diplomacy, such as trying to agree the terms for talks or find common ground," he said. "Digital diplomacy is just an extension of what happens in the real world."
Can a social media platform really eradicate anything negative? MyAngels.com says it can
Russian woman faces 6 years in jail for reposting memes
WORLD NEWS NEWS


text:
Q: How to recive the output from cUrl i've been tasked converting a payment gateway for magento from asp to php, the asp one is outdated for an old site and what's required is a new one, looking at all of the asp code i can easily do it but there is one thing that gets me.
Magento uses this Varien_Http_Adapter_Curl to send xml to the payment gateway and while i understand how the .asp file get the xml data (using Request.InputStream) i can't seem to duplicate that in a php file. this is as close as i can get
function Page_Load()
{
    $cUrl = curl_init();

    $dump = curl_exec($cUrl);

    $file = fopen("Gateway.txt","w");
    echo fwrite($file,var_export($dump,true));
    fclose($file);

    var_dump($dump);
}

Page_Load();

i changed the payment gateway URL in magento backend to my php file and went though the checkout, it creates the .txt file but all it contains is false
so how do i receive the output from magento in my Page_Load function, at the moment i'm just outputting it to a file just to confirm that i am getting a response
UPDATE: i have chnaged the code to this
function Page_Load()
{
    $cUrl = curl_init();

    curl_setopt($cUrl,CURLOPT_URL,"http://my.site.local/Current-Build/gateway/Gateway.php");
    curl_setopt($cUrl,CURLOPT_RETURNTRANSFER,true);

    $dump = curl_exec($cUrl);

    if(!$dump)
    {
        $dump = curl_error($cUrl);
    }

    $file = fopen("Gateway.txt","a");
    echo fwrite($file,var_export($dump,true));
    fclose($file);

    curl_close($cUrl);

    var_dump($dump);
}

Page_Load();

the site is on a wamp served that's been put online so anyone on in my development team can access it and it is also the same path that is specified in magento in the payment methods for the payment gateway url which was set to the old aspx file
when i then go though the checkout, while magento gives me the normal error cause it's not getting a response my output file isn't even being created, when i try to go back to magento it just sits there loading and i have to restart WAMP Server a couple of times cause it sits on the orange logo for ages


In 1998, Jasper, Tex., was the location of the horrific murder of an African-American man named James Byrd Jr. (Roy T. Anderson), who was brutally dragged to his death at the hands of three white men who tied him to a pickup truck. With media swarming around, white sheriff Billy Rowles (Jon Voight) and black mayor R.C. Horn (Louis Gossett Jr.) try to come to terms with the ghastly crime while the town threatens to erupt from racial friction.


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Eden Autism Welcomes Two New Members to its Board of Trustees
Eden Autism, a leading autism service provider since 1975, is pleased to announce two new members to its Board of Trustees, Scott D. Kent and Jeffrey Vamos, DDiv. These individuals were nominated and elected based on their extraordinary commitment to advancing Eden's mission.
"We are thrilled to have these talented, driven individuals join our Board of Trustees," said Eden Autism President and CEO Michael Decker. "Their expertise, passion for philanthropy, and community knowledge will be an incredible asset to Eden Autism as we further our mission to improve the lives of individuals with autism."
Scott Kent
Scott Kent is an executive with Wawa, Inc., serving as a multi-unit operations leader who oversees 22 general managers and over 600 associates in the Central/Northern New Jersey market. Kent has a long history of community involvement, having worked with numerous organizations to better the lives of individuals with disabilities, including Special Olympics, Eden Autism, IGA Nephrophathy Foundation, and KidsBridge.
Kent, a Hamilton resident, has been involved with Eden for more than 30 years. "I feel Eden was one of the first in the nation to address the needs of individuals who were autistic when there were few players in this field," said Kent. "I know where Eden came from, and I feel I can assist from a strategic viewpoint."
Kent holds an undergraduate degree in Organizational Dynamics from Immaculata College and a master's degree in Food Marketing from Saint Joseph's University.
The Reverend Doctor Jeffrey A. Vamos is Pastor and Head of Staff of The Presbyterian Church of Lawrenceville, a position he's held since 2005. He is leading the church through a capital campaign and new innovative forms of ministry, including a community center and job training program for formerly incarcerated persons.
Dr. Vamos is active in the community and currently serves as board chair of Every Child Valued, an afterschool program in Eggert's Crossing
Jeff Vamos
Village in Lawrenceville. He served on the Committee on Ministry for the New Brunswick Presbytery, which focused on forming a collaboration among Trenton-area congregations to explore the future of Presbyterian ministry in the city. He also served on the executive committee of Planned Parenthood of the Mercer Area.
About his desire to serve on the Eden board, Dr. Vamos said, "Our congregation is involved with Eden via employing a person through Eden's employment program. We are also at the beginning stages of developing a community center, called The Community Well, which would have some synergies with Eden's mission and programming."
Dr. Vamos, a Lawrence resident, holds a D.Min., from Wesley Theological Seminary and M.Div. from Union Theological Seminary. He received a bachelor's degree from Duke University in Political Science and Public Policy Studies.
Eden's Board of Trustees plays a crucial role in determining the strategic course of Eden Autism, ensuring that the organization continues to operate in the best interest of past, present, and future members. To learn more about Eden Autism and its Board of Trustees, please visit edenautism.org/about-us.
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Pazos MR, Cinquina V, Gómez A, Layunta R, Santos M, Fernández-Ruiz J, Martínez-Orgado J.
score, 0-5), magnetic resonance spectroscopy and Western blotting.
novel object recognition[NOR]) were carried out 30 days after HI (P37).
neuroprotective agent for neonatal HI.


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Insurance SMI Insurance is available at special SMAR rates. They will evaluate your needs and make recommendations to close gaps, enhance coverage, and reduce costs. They offer business, personal, life & health, retirement and estate planning services. Contact Kenneth Wyvill at 301-390-0099.
The Maryland REALTORS® Forms Committee works to provide our members with standardized forms to conduct day-to-day real estate business. All forms published have been reviewed by their Legal Department.
Legal Hotline: Free, authoritative legal information is available. Questions on all aspects of real estate law are answered, including license law, agency, contracts, deposits and disclosures. An appropriate written response may be provided to the caller and the identified broker or office manager.
Annual Conference & Expo - Held in Ocean City each September, this conference is an excellent way to complete continuing education requirements, attend professional development courses and the expo.
Committee Service MAR maintains several committees which assist in establishing policy, promoting educational training, forms, events, member services, and much more.
Legislative Monitoring/Government Affairs: Maryland REALTORS® makes your voice heard in Congress and at the State Legislature. The Legislative and Public Policy Committees develop a straightforward political agenda to: support and develop legislation that helps our industry and oppose legislation that hurts the free transfer of real estate.
Below are just a few highlights - click here to learn more!
Center for Financial Wellness: The National Association of REALTORS® proudly introduces the Center for REALTOR® Financial Wellness, a new resource designed exclusively to meet the specific financial planning needs of REALTORS®. This new comprehensive program includes educational materials and resources for wealth building, business planning, and investing in real estate.
Conferences: NAR organizes two conferences each year - REALTORS® Legislative Meetings & Trade Expo held in May in DC & the Annual Conference held in November at rotating cities across the US. Each are excellent networking opportunities where you'll learn from top industry professionals.
Commitment to Excellence: The National Association of REALTORS® proudly introduces Commitment to Excellence (C2EX), a program free to members that empowers REALTORS® to demonstrate their professionalism and commitment to conducting business at the highest standards.
REALTOR.COM: Are you harnessing the power of your Realtor.com profile? Every REALTOR® automatically has a profile on realtor.com - it's included in your REALTOR® membership. Claim your profile today!
REALTOR® Benefits Program: Designed with you in mind, the REALTOR Benefits® Program is your official member benefits resource, bringing you savings and unique offers on products and services just for REALTORS®.
REALTORS® Content Resource: The best FREE way to keep in touch with your clients. REALTORS® Content Resource: Your source for ready-to-use articles on all aspects of homeownership, including homebuying and selling, for your consumer communications.
REALTORS® Property Resource: A powerful platform designed for REALTORS® in need of comprehensive data, powerful analytics, and dynamic reports.
FREE Safety Resources: Take advantage of free webinars, blogs, tips and more to keep you safe on the job. There are even resources for brokers.
ZipForms: Includes zipTMS™ robust transaction management system, unlimited zipVault® document storage, an exclusive NAR library of REALTOR® Forms & Templates, and access to the industry-standard zipForm® Plus forms engine where you can add and manage our own custom documents.


text:
David Chia
Short stories and essays
David Chia has read his short stories and essays at WMOS since 2001. David currently leads the writers' group at the Medford Public Library. His work has been published in several editions of the Doc Kountze Annual. David has also received writing grants from the Medford Arts Council. David has taken a number of classes in fiction and creative writing at the Cambridge Center of Adult Education.
David is a member of the WMOS board of directors. He is also on the board of the Cambridge Chamber Singers and a trustee for the Medford Public Library. In 2015, he was assistant curator for "Lines Connecting Lines," Malden exhibit of poems paired with paintings. He is a lifetime member of the Girl Scouts.
David has written grants for the Pine Street Inn. David has a B.S. in Engineering and Applied Science from Yale University and a S.M. in Technology & Policy from the Massachusetts Institute of Technology. He was a reporter and senior editor for the Yale Daily News and an opinion writer for The Tech, MIT's student newspaper.
Words are small but powerful, entities. I try to make sure each of my words say what I want it to say. I am interested in the details of everyday life of ordinary and not so ordinary people. A listener has said that there is "always an enjoyable wistfulness" to my stories. I think that was a compliment.
Read: "Wait foo Me"
Read: "Me and the Girls"