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Best Ten Movies about Pregnancy to Binge-Watch
Congratulations, you're almost a mom! You must have slept, eaten and read books about pregnancy all day.
Yet, wouldn't you say you need a break from this:
It's hard to see the relationship between the main character of a romantic movie and involved in love, but everything can be imagined as a child.
So you have to watch films that will enable you to appreciate pregnancy. As always, we are here to guide you in the right direction.
From pre-craftsmanship Inappro Yu pregnancy to supplanting spontaneous high school pregnancies, look at their beneath 10 motion pictures that will keep you engaged while we hang tight for the most youthful individuals from your family.
Movie About Pregnancy N° 1
What To Expect When You're Expecting:
What to expect is that including romantic Comedy and acting star Jennifer Lopez and Cameron diplos all revolve around groups of women, fully covered by the pregnancy hormone.
From casual pregnancy to adoption, from morning sickness live to smooth pregnancy, the movie is a great way to get pregnant and have paternity issues, but in an entertaining way.
What you can expect when you come to a good pregnancy. But it's one of the best pregnancy movies that solve the emotional problems women face at this stage.
You will laugh out loud and during this movie, you will shed tears sometimes.
Nine Months:
Nine months starring Hugh Grant and Julian Moore is a movie you shouldn't miss at all when she's waiting.
ROM1995-com about Simon (chase award), the kind of duty fear and his better half, Rebecca(gene Moore). The couple had an about 5-year association with the ship until they found that Rebecca was pregnant.
While Rebecca is a future-obsessed mom, Simon struggles awkwardly and gracefully with the upcoming parenthood.
But the film ends happily, like a real blue movie of the 90s. the Couple at long last understands that the kid isn't the end, yet the start of another life.
Julianne Moore makes this incredible film with a heart.
You'll laugh out loud even in your old-fashioned ' 90s jokes. But the beloved riot hero is in Russia, and eventually unleashes his acting Robin Williams.
The delivery scene of this motion picture is funny. He has a lot of screams and even fainting. Luck.
That doesn't mean they can work for them. He tries to take the plane home to be there for delivery but loses it to Ethan Tremblay, who seeks to be an eccentric actor.
After that, he would be forced to join Ethan as much as Peter's sanity test.
At first hitchhiker felt a little uncomfortable, but in the end, his journey was a journey of pleasure.
The most effective method to watch movies about pregnancy.
Junior:
If you are looking for a thematic film about pregnancy with a plot of extravagant movies, the "Junior" can be your choice. The "younger" of the medical researchers,Dr. Hesse (Arnold Schwarzenegger), mother tried fertility meds for transportation under the threat of disappearance of children, the series of comic incidents unfolded, and, finally, Arnold was ready to undergo a cesarean section. Watch this movie about pregnancy with your mate for quite a long time of fun.
Away We Go:
Far away we travel across the country to find the best place to raise Bart's pregnancy and his girlfriend's baby, the first 5 months of pregnancy in Verona were mild, but when the sixth month began, Burt's mother, Gloria, and Jerry's mother gave birth to the baby.
When the couple announced that they had emigrated from Colorado, their travels to different cities with friends, i.e. Madison, Montreal, Phoenix. On the way, they discover that there is a house where the heart is.
The dramatic Comedy is full of interesting characters. You'll see all the mothers in action, from Lily, whose tongues are sour, reproached her child, smart, beautiful, but not so much.
Baby Mama:
If you want to watch movies about pregnancy to make you happy during pregnancy, go get baby mom.
"Baby mama" is the account of Kate, a vocation disapproved of a single lady who can't have an infant in view of her t-molded uterus.
She hires Angie, a broken, freedom-loving surrogate mother, to take her son.
Angie eventually moves in with Kate, who tells the couple everything about the pregnancy.
However, a carefully drawn up plan collapses when Kate discovers that Angie is pregnant, but her ex-husband was with her child from Carl.
Look Kate will have on a good Angie, but in the future, an ideal mother will hysterical and, too, urgent.
But Angie and the audacity of chutzpa delivered most of the laughter in the movie.
You'll adore the scene where she passes on the picture of a squirrel as an ultrasound, she additionally has a great deal of trust in Kate.
Expert Tips To Announce Your Pregnancy To Your Boss And Co-Workers.
The Business Of Being Born:
Just a few movies that you can choose from this list and you will be amazed at what you can do with them. The birth of a child is not an ordinary Omicnovel, but a brilliant film that every future mother sees.
This documentary by Ricky lake investigates the present understanding of pregnancy and labor in the United States.
These are cesarean section and natural childbirth, home delivery, induction, doula, epidural and everything else.
The film inspects the present reliance on mediations and drugs during labor.
It considers not the occasions of nature, however, the effect of how labor is seen as a medicinal crisis.
Fundamentally, the film readies your labor by giving all of you the data about labor.
Father Of The Bride 2:
For those who have planned their wedding while watching the bride's father repeat, this is the best pregnancy movie for you.
All your preferred characters will come back to this fun spin-off of "father of the bride."
The film pursues George's voyage as he gets ready to be a dad for the third time, just as to respect his first grandson.
The film is dedicated to the UPS and downs of pregnancy.
The main message of the film-to reunite with his family in a difficult moment, but it affects the late pregnancy.
Babies:
Baby is a long film about pregnancy 20 minutes of baby time.
This documentary recounts the account of a tyke's adventure from four unique nations to Japan, Mongolia, Namibia, and California.
Adorable babies made of Christmas will experience the life of a child in a timely manner and in detail.
Of these, despite the fact that there are written, geographical and cultural differences, the child gives birth to a child.
There is no dialogue in the film, but the picture is a highlight.
Movie About Pregnancy N° 10
Labor Pains:
The arrogant reckless Secretary of the publisher Thea pretends to be pregnant to save herself from being fired.
The arrangement works, and she lands the position.
She takes the child's shot from the mannequins and keeps on faking her pregnancy, choosing her best course of action (Movies About Pregnancy.
Thea starts to make the most of her pregnancy and gets exceptional treatment to the degree that pregnancy turns into a reality for her!
Did you enjoy our pregnancy-friendly movies list?
Let us know in the comments section below!
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WOMEN STICK TOGETHER, SHARING IS CARING!
Kingdomofbaby


In this Turkey Day edition of Podcast Unlocked, the crew discusses the aftermath of the Xbox One launch: what went well (Xbox Live), what didn't go so well (some people's disc drives), and what's annoying the heck out of us (crashes!). Plus, why one programmer's departure from 343 Industries is extremely bad news for Halo fans, why you should keep it clean on your Xbox One Upload Studio clips, EA's NBA Live apology, and the end of a legendary first-person shooter era. Oh, and we pick our favorite holidays in Get to Know Your Locksmiths!
The Xbox One is finally here, and we're talking about EVERYTHING: all the games (including the big wild-card, Ryse: Son of Rome), the Kinect, the Dashboard...all of it. A special occasion demands an extra-special crew, so we're bringing five Locksmiths to the table this week: Ryan, Mitch, Destin, Marty, and Naomi -- with a bonus appearance by Andrew Goldfarb. Plus: can Xbox One's Kinect see your junk?
Ryan has started playing the Xbox One version of Call of Duty: Ghosts...and hates the campaign. Mitch and Destin share their thoughts, while Marty (wisely) abstains. Meanwhile, you get another chance to acquire the rare white Xbox One and help a charity in the process, we see the Xbox One Dashboard in action, Xbox One reviews are coming in, an Xbox One launch game gets delayed, and a new Middle-earth game gets announced.
The Internet's #1 Xbox podcast tackles the entire Xbox One launch lineup this week after playing most of the games. Plus: Call of Duty: Ghosts' sales numbers, a special-edition Xbox One is going to raise a lot of money for charity, Peggle 2's good and bad news, Metal Gear Solid's strange move, the South Park RPG gets delayed yet again, and more.


Our mission is to enhance awareness of the value of carnivore presence on our planet by providing a wide range of creative, educational and research experiences for the people of New England, focusing on carnivores, their habitat, and their relationship to us.
We want to bring back a sense of community and empowerment to our people. That is why we wish to build this Place where we can be in each other's presence, and share. And we plan to build this Place in the Midcoast.
In our Native Peoples oral traditions, their Coyote Stories brought their community together to learn how to live their lives….and coyote was the Teacher. So to honor both our Native Wild Dog, Coyote and the traditions of our Native Peoples….we are Coyote Center.
For Educators: We wish to be of service to our educators on all levels here in Maine. So we will be offering workshops where teachers can learn about the complexity of carnivore ecology, and the many ways they can present it in classes they teach.
For Farmers: We are so blessed to have so many farmers who feed us with fresh Maine food. So we wish to give back to them by having gatherings where our farmers get to learn about carnivore ecology, learn effective animal husbandry practices and learn from each other. We will also offer opportunities for farmers to gather on each other's farms: a splendid experience for farmers and community members alike!
For our Children: The children are ready! We will engage our children in creative, experiential opportunities that are both fun and transforming. How do they learn about wild carnivores, especially Coyote…they become Coyotes. Our children have already let us know, that they love "pretending." But while they are pretending they are drinking in the realities of a Coyote family's life. And there will be so much more. This is just a peak! Watch your children light up with what is coming!
For Police and Animal Control Officers: In urban and suburban areas, the police or animal control officers are often the first persons to be contacted by a resident who observes a coyote on their property, on hiking trails or other public areas, and who express fear of them. These public servants, who have little if any knowledge of coyotes, are being asked to respond to a citizen concern, and the result is often tragic for the coyote. In addition, they are unable to give the resident any information or support that would help them to coexist with this resilient carnivore. To assist our public servants we will have workshops in which they will learn about coyote ecology and the skills needed when called upon by community members.
We have so many talented artists and writers in our communities who wish to use their talents on behalf of wildlife and our environment. So we will have many opportunities to engage them. We inspire one another by sharing what we love to create.
Here in the East, Our Land Trusts are protecting thousands of precious landscapes from development. But they also engage our communities in all forms of education about our environment. We wish to support them in their education efforts by holding workshops on Coyote and carnivore ecology so they can take it back to their communities.
Renowned ecologist, E.O. Wilson wrote that Citizen Science is a most powerful way to educate. And Yes! We shall be participating in Citizen Science on diverse levels. We plan to engage our young students in school as well as our adults in this very meaningful experience for each one of them.
We will collaborate with fellow scientists and organization, who will come and share their experiences, knowledge and expertise. So get ready!
Our First Step: It is our plan to purchase land in Midcoast, Maine. We wish for the Coyote Center to be easily accessible to many of our community members and visitors to Maine, but at the same time "a place apart." We are seeking generous benefactors that wish to leave a legacy of their own by donating funds that will enable us to purchase this land soon.
Photo below is a glimpse of this land.
Our Second Step: Building the Center: We have already chosen the architect who will plan our structure: Rick Renner Architects in Portland, Maine, architects who practice the highest standards of Green building practices. Under their professional guidance, we plan to build a structure that is fitting and respectful to this landscape.
Our Third Step: Opening the Center. What an exciting day that will be! We plan to begin with a celebration that will include an expression of our gratitude to all those who have helped to make the Vision become a Reality. Then onward to our Mission!
Help us arrive at our Final Goal: Leave Your Legacy!


FundToken volume of trade today and historical data of FundToken capitalization since start date of trade on the cryptocurrency exchange market.
Today's market capitalization for FundToken is 0 US dollars.
Today's FundToken volume of trade is 48 165 US dollars.
Capitalization of FundToken - total sum of all mined FundToken cryptocoins.
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Heads Up: As you may have guessed from the title, this post contains a bit of profanity — and it's the big one. I said it — a few times…in front of kids. These definitely were not the finest moments in my career as a teacher. I'm not really proud of what I did, but it definitely makes for a Travel Oops story.
Here I am…calm and collected.
Many teachers, at one time or another, have considered the potentially heavy consequences of dropping the "F-bomb" in class. On the other hand, you also consider the liberation of uttering the ultimate.


danglingonahook - Did You Get My Message?
Have you ever been lying in bed, almost asleep, floating in that distant in between place and heard a bump or a thump or a crash, or a far off rumble way in the distance?
Did your body pull together, tightened up from the inside out, go on alert?
You might have continued to lie there for a few seconds attentive, listening, waiting to see if there was more, or you might have gotten up, prowled through your home, restless and unsettled and finally, finding nothing, crawled back into bed, rolled over a few times and then drifted off to sleep.
Fred leaped out of bed, fully alert, me close behind him, clutching for our housecoats, scrambling through the boat.
The float home down the dock was in conflagration. Flames reached high into the sky and the heat was felt, hot and burning on our faces and legs, even where we were, several fingers away.
He talked to me two days later the young man who, with no thought for himself, took his tiny tug boat and pulled the float home away from the dock. Past the other float homes, past "Serendipity" and "Reality" and "Peggy Sue", past our small family, tied in a knot together on the dock watching, past our boat, our home and then out into the harbour.
I was folding clothes in the laundry and he came in, opened the dryer doors and looked inside and then closed them again. I was aware that there were no clothes in those dryers. "How ya doing?" I asked.
He put his hand on the counter where I was folding clothes and leaned on his arm and paused. He looked tired.
"How ya doing? You're a hero you know ", I advised him and his face crumpled.
"I keep hearing her screaming", he told me. "I wake up in the middle of the night and hear her screaming."
"She was tired", I told the hero. "Tired. She's not hurting anymore. Give yourself time. You did the best you could. It was good enough!"
He was a young man and angry, and he was hurting, and that day he was tired. "The newspapers got it all wrong", he told me pure pain on his face.
The night of the fire we stood on the dock, my companion, my daughter and I in a tight little knot, and watched two men, two more heroes from the Marina wrench the four foot propane tanks off the back of the float home, flames and heat and hell and salvation swirling in the air around them. We watched as the hero brought his tiny little tug boat around and watch still while yet another hero, garden hose in hand, tied the lines to the burning pyre. We watched other distant figures, distant heroes in the background spraying water and directing traffic and then we watched as the bonfire that once was a home was towed around the dock, past our own boat, our home, and then out into the bay.
"I keep hearing her screaming."
The next day and for weeks afterward we watched bouquets of flowers come fresh and beautiful, bought and hand picked, weeds and roses, laid on the burnt dock, silent homage, slowly wither in the spring air. Then the flowers would be pushed into the water and drift away with the tides. The next morning there'd be more.
Later on we watched the hero pull up the burnt planks and replace them one by one and bore witness while our neighbours hammered new tiles onto their float home roofs and struggled bringing awkward new windows in to replace those melted in the overwhelming heat.
Later still we watched the hero sitting on the dock across the bay, dipping his feet into the water and staring into the sky.
Have you ever been lying in bed, almost asleep, floating in that distant in between place and heard a bump or a thump or a crash or a rumble far off in the distance and your body pulled together, tightened from the inside out, went on alert?
or maybe it was a thump or a bump or a backfire far off in the distance.


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Master's Programs in New York
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Now accepting applications for Master's programs starting Fall 2019. Apply Now.
Founded by Sotheby's Auction House in 1969, Sotheby's Institute of Art is the first and foremost graduate school for the study of art and its markets. Our programs offer an important combination of art history, connoisseurship, and business training to prepare you for an enriching career in a competitive art market. The Institute has more than 8,000 alumni worldwide who hold top positions in all facets of the international art market.
Click here to receive a copy of the Sotheby's Institute Viewbook.
Graduate Certificate Option
A two-semester Graduate Certificate is available to students who complete the required coursework for an MA Degree program but who do not wish to complete a Master's Project or Thesis.
Sotheby's Institute of Art-New York offers three distinct Master's Degrees in Art Business, Contemporary Art, and Fine and Decorative Arts and Design. Our faculty of distinguished scholars and art professionals provide intensive training in specific subject areas of art history, connoisseurship, and art markets. In addition, courses and field study give students in-depth, insider access to the industry.
New York City is one of the largest art capitals in the world, making possible privileged access to pre-sale auction viewings, seminars with world-class experts and handling sessions in which important objects are examined in detail. Field Study takes students to the homes of legendary private collectors, artists' studios, international art fairs, biennials and triennials, and the world's finest museums.
Sotheby's Institute of Art-New York holds degree-granting authority from the Board of Regents of the University of the State of New York and is accredited by the National Association of Schools of Art and Design (NASAD).
Access a copy of the 2019-20 New York Master's Degree Curriculum and course descriptions.
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Presenting my recipe of Masala Butter Chicken, a treat for our Non-Vegetarian readers. I cook Non-Veg recipes when I have guests at home, this is the reason we do not have a lot of non-veg posts.
But the wait is worth for non-veg lovers as you cannot go wrong with any non-veg recipe on Evergreen Recipes. Butter chicken is usually a little sweeter because of the Kaaju paste that is used to enhance the flavour but I've put some little extra spices to make it spicier than the Butter Chicken.
My Husband, who is a non-vegetarian (and very tough to please with my culinary skills :p) has approved this recipe as a must have non-veg item whenever there are special guests at home.
Follow the easy step by step method to prepare this super delicious Butter Masala Chicken at home. I hope you will love it.
1. Clean and wash the chicken well. Marinate it with all the ingredients given in the table of "Ingredients for Marinate the Chicken". Keep it aside for 3-4 hours in a fridge. Take out the marinated chicken from the fridge, heat the wok with butter and add marinated chicken in the wok. Fry it on a medium flame for 4-5 minutes or till butter shows separately. Keep it aside.
2. Heat another wok with 2 tablespoon butter and add onion. Fry it till onions are translucent. Add in cumin seeds and green chilly.
3. Then add chopped tomato, coriander powder, ginger-garlic paste, kasoori methi, red chilly powder, cashew nuts, almonds, bay leaf and salt.
4. Cook the spices on medium flame for 5-6 minutes. Now switch off the flame and let it cool at the room temperature. Take the cooked spices in a grinder and grind it to make a fine paste.
5. Heat the wok with 6 tablespoon butter and add ground paste. Cook the spice on low flame till butter shows separately.
6. Add Kashmiri red chilly powder and cook for another 2-3 minutes. Add fried chicken in the masala/ wok and mix it well.
8. Switch off the flame, add cream to the gravy and mix it well. Delicious Butter Masala Chicken is ready to serve, top with some more butter, Serve hot with Naan, Chapatti or Pulav.


Ryan has a band named To The Waves which just released their first album. I just bought the last copy that was in stock on Amazon.com so hopefully the album is selling well.
Everyone should go order their own copy to help support Ryan's rock star lifestyle .
Gavin has learned to turn on his side recently. He usually does this while on his play mat. Before we know it he'll be rolling over. His other new trick is gurgling while trying to talk.


A short note on ransomware.
And one way these criminals will attack your enterprise is through your employees.
As usual, the best tool is prevention: patched software, up-to-date firewalls, top-flight anti-virus software and, of course, excellent employee policies and education to prevent the introduction of malware onto your system and to mitigate its effects once it is there.
Another key tool in the fight is business continuity planning. Without having built an adequately cyber-resilient organization, you may have to pay your attacker. This includes understanding what your most sensitive data is, where it is stored and ensuring that you have the ability to do business using back ups or via alternative means.
Next Post See you at Hofstra Law!


Miami Township's Community Development Department has staff dedicated to economic development activities. Let us know what we can do to help you to make the choice to locate within Miami Township!
Miami Township has a list of hotels, restaurants, and retail establishments within the community. Please look at our Business Directory for businesses within the community. If you are not on the directory and would like to be, please fill out this form.
Please review our demographic data. Miami Township is a community of contrasts with both a rural and urban mix. The township is located in the extreme south-central portion of Montgomery County, Ohio. The City of Miamisburg is located within Miami Township and occupies approximately 11 square miles. The unincorporated portion of Miami Township contains approximately 21 square miles of land. Miami Township is located just 11 miles from downtown Dayton and 41 miles from downtown Cincinnati.
Miami Township has numerous locations that are available for development. Please look at the developments tab to the left.


For the sake of supporting as many platforms as possible, Godot only accepts UNIX-style path separators (/). These work on all platforms, including Windows.
A path like C:\Projects will become C:/Projects.
As mentioned before, Godot considers that a project exists in any given folder that contains a project.godot text file, even if such file is empty.
Accessing project files can be done by opening any path with res:// as a base. For example, a texture located in the root of the project folder may be opened from the following path: res://some_texture.png.
While the project is running, it is a common scenario that the resource path will be read-only, due to it being inside a package, self-contained executable, or system-wide install location.
Storing persistent files in such scenarios should be done by using the user:// prefix, for example: user://game_save.txt.
On some devices (for example, mobile and consoles), this path is unique to the project. On desktop operating systems, the engine uses the typical ~/.local/share/godot/app_userdata/Name on macOS and Linux, and %APPDATA%/Name on Windows. Name is taken from the application name defined in the Project Settings, but it can be overridden on a per-platform basis using feature tags.


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arXiv Vanity renders academic papers from arXiv as responsive web pages so you don't have to squint at a PDF
View this paper on arXiv
Superradiant instabilities of rotating black holes in the time domain
Sam R. Dolan Consortium for Fundamental Physics, School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom
Bosonic fields on rotating black hole spacetimes are subject to amplification by superradiance, which induces exponentially-growing instabilities (the 'black hole bomb') in two scenarios: if the black hole is enclosed by a mirror, or if the bosonic field has rest mass. Here we present a time-domain study of the scalar field on Kerr spacetime which probes ultra-long timescales up to t≲5×106M, to reveal the growth of the instability. We describe an highly-efficient method for evolving the field, based on a spectral decomposition into a coupled set of 1+1D equations, and an absorbing boundary condition inspired by the 'perfectly-matched layers' paradigm. First, we examine the mirror case to study how the instability timescale and mode structure depend on mirror radius. Next, we examine the massive-field, whose rich spectrum (revealed through Fourier analysis) generates 'beating' effects which disguise the instability. We show that the instability is clearly revealed by tracking the stress-energy of the field in the exterior spacetime. We calculate the growth rate for a range of mass couplings, by applying a frequency-filter to isolate individual modal contributions to the time-domain signal. Our results are in accord with previous frequency-domain studies which put the maximum growth rate at τ−1≈1.72×10−7(GM/c3)−1 for the massive scalar field on Kerr spacetime.
pacs:
††preprint:
An exciting era is underway in astrophysics, in which precision measurements of black hole masses and spins are becoming possible Shafee ; Miller ; Brenneman ; Walton . Naturally, there is renewed interest in the range of physical processes that may occur in the vicinity of black holes. One of the most intriguing processes is the so-called 'black hole bomb' effect, first proposed by Press and Teukolsky in 1972 Press:1972 , in which the rotational energy of a Kerr black hole (BH) Kerr:1963 generates runaway amplification in a bosonic (i.e. integer-spin) field. The driving mechanism is superradiance Starobinskii:1973 : the stimulated emission of low-frequency radiation in co-rotating modes. A perturbation of frequency ω and azimuthal number m, incident upon a black hole of mass M and angular momentum J=aM has a reflection coefficient of greater than unity if the superradiance condition is met:
ω~ω<0, (1)
where ~ω≡ω−mΩ and Ω=a/(2Mr+) is the angular frequency of the event horizon, with r+ its radius. Via superradiance, a black hole may shed mass and angular momentum, but in such a way that the horizon area (A=8πMr+) actually increases. Since horizon area is associated with entropy Bardeen , superradiance may be thought of as a thermodynamically-favoured relaxation process.
The Kerr spacetime in vacuum is (apparently) stable under perturbation by massless fields (see e.g. Regge:1957 ; Press:1973 ; Whiting ; Dafermos1 ). For an instability to develop, some additional mechanism is needed to confine the field in the vicinity of the black hole. If the BH is surrounded by a reflecting boundary (i.e. a 'mirror'), then superradiant flux is reflected back on to the hole, generating an exponentially-growing instability Press:1972 ; Cardoso:2004-bomb . A boundary to the spacetime, as in Anti-de Sitter scenarios, may play a similar role Cardoso:2004-AdS . A more 'natural' astrophysical scenario arises if the bosonic field has a non-zero rest mass μ, in which case quasi-bound states Deruelle:1974 ; Damour:1976 ; Zouros:1979 ; Detweiler:1980 ; Gaina:1992 ; Gaina:1993 ; Cardoso:Yoshida:2005 ; Lasenby:2005-bs ; Dolan:2007 ; Grain:2007 ; Barranco may form around the BH, with Re(ω)≲μ. Quasi-bound states are localized modes which are associated with a minimum in the effective potential, induced by the gravitational attraction between the field and BH mass Damour:1976 . The spectrum of quasi-bound states on the Kerr spacetime depends on just two dimensionless quantities: the BH spin a/M and the mass coupling,
GMμℏc≡Mμ, (2)
where in the following we adopt units G=c=ℏ=1.
There has recently been fresh interest in a fundamental question: is the superradiant instability in the massive bosonic field of any relevance to astrophysics or particle theory? Here, we must be mindful of a strong constraint: the instability only applies if Mμ≲mMΩ, with MΩ≤1/2, and furthermore the growth rate is exponentially suppressed at large m Zouros:1979 . In other words, the instability is only physically relevant if there is a bosonic field with a Compton wavelength comparable to (or larger than) the horizon scale. Writing Mμ=7.5×109(M/M⊙)(μ/[1eV]), where M⊙ is the solar mass, makes it clear that this condition is not satisfied by any known massive bosons in astrophysical black hole scenarios. However, there are two more exotic scenarios which may be relevant: standard-model massive bosons interacting with 'primordial' black holes, or as-yet-undiscovered ultra-light bosons interacting with solar-mass or supermassive black holes. The latter scenario is motivated by compactifications of string theory, which suggest the existence of axions: a family of ultra-light bosons with masses spanning across many decades, possibly all the way down to the Hubble scale μ∼10−33eV. If extant, axions could ignite the superradiant instability in stellar-mass or supermassive black holes, generating 'gaps' in the black hole mass-spin spectrum Arvanitaki1 ; Arvanitaki2 . It has been suggested that precision measurement of the masses and spins of supermassive black holes can be used to put new constraints on the existence of ultra-light bosons Arvanitaki1 ; Arvanitaki2 ; Kodama:Yoshino ; Pani:PRL ; Pani:PRD . It is argued that unexpected gaps in the BH spectrum would be long-sought-after observational evidence in favour of string theory (or at least, in favour of a beyond-Standard-Model theory requiring compactification of higher dimensions which leads to a family of ultra-light bosons).
The majority of the literature to date has focussed on the instability caused by the scalar (i.e. spinless) field Zouros:1979 ; Detweiler:1980 ; Furuhashi:2004 ; Cardoso:Yoshida:2005 ; Dolan:2007 ; Rosa:2010 ; Lee:2011 ; Li:2012 ; Arvanitaki1 ; Arvanitaki2 ; Kodama:Yoshino ; Yoshino:Kodama . The massive scalar-field equations on Kerr spacetime are separable in the frequency domain Brill:1972 , hence a rather complete picture of the instability in the linear regime may be obtained via straightforward analysis of ordinary differential equations (ODEs). In Sec. II.2 we review the frequency-domain formalism and the key results in some detail.
The aim of this work is to study the instability in the time domain. Time domain methods offer a general toolkit for studying systems of partial differential equations of hyperbolic character, which are well-suited to cases where governing equations are non-separable and/or non-linear.
An interesting example of an apparently non-separable but linear system is the massive vector (Proca) field on Kerr spacetime. In the Proca-Schwarzschild case, the equations may be separated, by decomposing the field in vector spherical harmonics, leading to a single ODE describing the odd-parity part and a pair of coupled ODEs for the even-parity part Rosa:Dolan ; Herdeiro . In the Proca-Kerr case, impressive recent progress Pani:PRL ; Pani:PRD has been made by expanding the equations in powers of a/M, applying a method originally developed to analyze radiation reaction in rotating neutron stars Kojima:1992 . In a technical tour-de-force Pani:PRL ; Pani:PRD , it was shown that, at first and second orders in a/M, the equations may once more be separated with vector harmonics. This has opened a window on to Proca-Kerr phenomenonology, which has led to a new constraint on the mass of the photon Pani:PRL from measurements of supermassive BHs. New time-domain methods Witek offer a way to move beyond the slow-rotation regime, and to test the applicability of the approximation.
Non-linear effects will become important as the instability develops. An interesting question is: does the instability end in an explosive manner (with a 'bosenova' Arvanitaki1 ), or does the system relax towards a quasi-stable state? A recent time-domain study Yoshino:Kodama examined the evolution of an axion field with a non-linear self-interaction, and concluded that the system is prone to a sudden collapse of the axion cloud with a rapid release of energy.
However, even within the linear regime, studying the onset of the scalar-field instability in the time domain is a difficult challenge, because (as anticipated from frequency-domain analyses Furuhashi:2004 ; Cardoso:Yoshida:2005 ; Dolan:2007 ; Rosa:2010 ) the growth rate of the instability is rather slow, with a minimum e-folding time of ∼5.8×106M (see Sec. II.2 and Fig. 1). To date, this timescale has proven to be beyond the reach of multi-dimensional simulations Strafuss:2005 ; Witek ; Yoshino:Kodama , which (so far) have evolved the field up to t≲104M or less. Despite this limitation, a very recent time domain study Witek seems to have revealed the onset of the Proca-Kerr instability, confirming that it develops much more rapidly than the scalar-field instability. A key aim of this work is to construct a time-domain scheme capable of probing ultra-long timescales of the order of 106M or more. We will show that ultra-long evolutions, in combination with Fourier analysis, allow us to determine the physically-relevant part of the quasi-bound state spectrum to high accuracy. We will argue that the methods developed here are ripe for application in the Proca-Kerr case (amongst others).
Superradiant instabilities appear in a range of guises, for example, for BHs immersed in a magnetic field Konoplya:2009 , for higher-dimensional black holes, branes and strings Cardoso:2005-new ; Cardoso:Yoshida:2005 ; Lee:2011 ; Rosa:2012 , black holes in Anti-de Sitter (AdS) spacetimes Cardoso:2004-AdS ; Aliev:Delice:2008 ; Li:2012 ; Witek:box and alternative spacetimes Konoplya:Zhidenko:2011 and alternative gravitational theories Myung:2011 . The repulsive effect of a superradiant instability on orbits around a black hole has also been considered Press:1972 ; Cardoso:Chakrabarti:2011 . Charged BHs generate superradiance in charged bosonic fields which leads to an enhanced superradiant instability on charged rotating spacetimes Damour:1976 ; Furuhashi:2004 . On the other hand, non-rotating charged black holes appear to be stable under this effect Hod:2012 . Given the range of possible scenarios, it is possible that the time-domain method developed here could have many applications.
In Sec. II we describe the basic setup and review the properties of quasi-bound states as revealed by frequency domain analyses. In Sec. III we develop a time-domain method for evolving the massive scalar field on Kerr. We give a selection of numerical results in Sec. IV and conclude with a discussion in Sec. V.
Ii Foundations
In this section we recap the essential features of the scalar wave equation on the Kerr spacetime (II.1), and review the properties of the quasi-bound state spectrum in the frequency domain (II.2).
ii.1 Spacetime, wave equation and separability
The Kerr spacetime may be described via the Boyer-Lindquist coordinate system {t,r,θ,ϕ}, in which the line element ds2=gαβdxαdxβ is
ds2=−(1−2Mrρ2)dt2−4aMrsin2θρ2dtdϕ+ρ2Δdr2
+ρ2dθ2+(r2+a2+2Mra2sin2θρ2)sin2θdϕ2,
where ρ2=r2+a2cos2θ and Δ=r2−2Mr+a2. The Klein-Gordon equation governing a scalar field Φ of mass μ is
□Φ−μ2Φ=0, (3)
□Φ≡\tensorΦ;α;α=(−g)−1/2((−g)1/2gαβΦ,α),β, (4)
and the subscripts ';' and ',' denote covariant and partial differentiation, respectively. Here g=−ρ4sin2θ is the metric determinant, and gαβ is the contravariant version of the metric.
In the frequency domain, Eq. (3) admits separable solutions Brill:1972 ; Teukolsky:1972 of the form
Φ(t,r,θ,ϕ)=e−iωteimϕR(r)S(θ), (5)
where the radial and angular functions satisfy
ddr(ΔdRdr)+(K2/Δ−λ−r2μ2)R=0, (6)
1sinθddθ[sinθdSdθ]+
[Λ+a2(ω2−μ2)cos2θ−m2sin2θ]S=0, (7)
with K=(r2+a2)ω−am and λ=Λ−2amω+a2ω2. The solutions of the angular equation, S=Slm(θ), are spheroidal harmonics, and the angular eigenvalues can be found from a series expansion in the small-aω, aμ regime Seidel:1989 (with Λ→l(l+1) as aω,aμ→0), or by solving a continued-fraction equation Leaver:1985 .
The radial equation can be written in standard 2nd-order form, by applying a simple transformation Teukolsky:1973 ; Sasaki:Nakamura , R=X/√r2+a2, to obtain
d2Xdx2+U(x)X=0, (8)
U(x)=(r2+a2)−2[K2−Δ(λ+μ2r2)]+β2−∂xβ, (9)
where β=rΔ/(r2+a2)2. Here, the tortoise coordinate x (often denoted r∗) is defined via
dxdr=r2+a2Δ, (10)
or, after fixing the constant of integration,
x=r+2Mr+−r−(r+ln∣∣∣r−r+2M∣∣∣−r−ln∣∣∣r−r−2M∣∣∣). (11)
In the near-horizon limit, r→r+, the physical solution satisfies the ingoing boundary condition,
X∼exp(−i~ωx), (12)
where ~ω was defined below Eq. (1). In the far-field, the solution may be written as a linear superposition
X∼A+exp(qx)+A−exp(−qx), (13)
where q=√μ2−ω2 and we define the root so that Re(q)≥0.
ii.2 Quasi-bound state spectrum
The quasi-bound states are defined as the convergent solutions: those for which A+lm=0. The black hole is an open system as flux may pass into the horizon; hence it is natural that the quasi-bound states (like the quasi-normal modes Leaver:1985 ) have complex frequencies,
ω=^ω+iν. (14)
Here, the real part ^ω sets the oscillation frequency of the mode, and the imaginary part ν sets the growth (ν>0) or decay (ν<0) rate. The modes in the spectrum are labelled by integers for orbital angular momentum l, azimuthal angular momentum m, overtone number n and, in the non-zero spin case, polarization S. If the oscillation frequency ^ω lies inside the superradiant regime (^ω^~ω<0) then the bound state will grow (ν>0); if ^ω=mΩ then the mode is stationary (ν=0); otherwise the mode will decay (ν<0). All quasi-bound states on the non-rotating (Schwarzschild) spacetime are found to be decaying. We will confirm this picture in later sections.
The physically-relevant states grow (ν>0) or decay (ν<0) only slowly in comparison to the Compton and BH light-crossing times, i.e. M|ν|,|ν|/μ≪1. Foundational work by Detweiler Detweiler:1980 and Zouros & Eardley Zouros:1979 led to simple approximations for the growth rate in the low- and high-Mμ regimes. These studies imply a maximum growth rate νmax of approximately
Mνmax∼{ζ(Mμ)9(a/M−2μr+),Mμ≪1,10−7e−1.84Mμ,Mμ≫1. (15)
where ζ is a numerical constant 111Ref. Detweiler:1980 obtained ζ=1/24 (Eq. (28), whereas other analyses Rosa:Dolan ; Pani:PRD suggest ζ=1/48.. The maximum growth rate occurs in the intermediate regime Mμ∼1/2, which has been explored with numerical analyses. It turns out that, in close analogue to the quasinormal modes Leaver:1985 ; Simone:1992 ; Konoplya:2006 , the quasi-bound states are minimal solutions of a three-term recurrence relation Dolan:2007 . Finding minimal solutions is equivalent to finding the roots of a continued fraction equation; hence finding the spectrum numerically is 'easy' Dolan:2007 . A range of studies, by Furuhashi & Nambu Furuhashi:2004 , Cardoso & Yoshida Cardoso:Yoshida:2005 , Dolan Dolan:2007 , Rosa Rosa:2010 and Kodama & Yoshino Kodama:Yoshino , have presented numerical results which are in accord: the maximum growth rate νmax occurs in the fundamental mode n=0 of the co-rotating dipole mode l=m=1 for μ near to |Ω|. The rate νmax is strongly dependent on the black hole spin a/M.
Plots of the dependence of the growth rate ν on a/M and Mμ were previously been presented in (e.g.) Fig. 6 in Ref. Dolan:2007 , Fig. 2 in Ref. Cardoso:Yoshida:2005 , Fig. 6 in Ref. Furuhashi:2004 and Fig. 14 in Ref. Kodama:Yoshino . Typical values of the maximum growth rate for a range of spin rates a<M were given in Table I of Ref. Dolan:2007 . For example, the maximum rate at (e.g.) a=0.99M is Mν≈1.5×10−7, occurring at Mμ≈0.42. The extremal case a=M was considered in Rosa:2010 .
Figure 1 shows the growth rate as a function of Mμ, for the dominant mode (the l=m=1, n=0 quasi-bound state) of rapidly-rotating black holes. The plot shows that the maximum possible growth rate is Mνmax≈1.72×10−7M, corresponding to an e-folding time of 5.81×106M. Perhaps surprisingly, this maximum does not arise at the extremal limit (a=M), but rather at around a≈0.997M for Mμ≈0.45.
Figure 1: The growth rate Mν of the superradiant instability in the dominant mode of the scalar field (l=m=1, n=0), for rapidly-rotating black holes 0.99≤a/M≤1, as a function of mass coupling Mμ. The growth rates for a<M were found by computing the quasi-bound state spectrum with the continued-fraction method of Ref. Dolan:2007 . The growth rates for the extremal case a=M were found by direct integration of the radial equations. The plot shows that a maximal growth rate of Mν≈1.72×10−7 occurs for a≈0.997M and Mμ≈0.45.
Under the instability, the black hole will shed mass and angular momentum into the dominant bound state, in the ratio dJ/dM=m/^ω. Figure 2 illustrates the evolution of the black hole in the Mμ–(J/M2) plane. We make the simplifying assumptions that the process remains within the linear regime, and that the mass and angular momentum in the field are not re-absorbed by the BH as E and J shift. The plot illustrates that, once the instability is in progress, the BH will migrate from left to right along the coloured curves, spinning-down more rapidly than it loses mass, until it reaches the superradiant cut-off where ^ω=Ω (≈μ). The endpoint of the instability in the linear regime (i.e. the modes with ν=0) was the focus of a recent work on the extremal (a=M) case Hod:clouds . More realistic scenarios for the termination of the instability, taking into account non-linear effects, are the subject of recent studies Arvanitaki1 ; Arvanitaki2 ; Yoshino:Kodama .
Figure 2: Evolution of black hole parameters under superradiant instability induced by a scalar field of mass μ, in the linear approximation. In the superradiant regime, the black hole loses mass M and angular momentum J into the scalar field (predominantly emitting into the m=l=1, n=0 quasi-bound state). The BH parameters evolve along the coloured curves, from left to right (whereas the field mass μ is assumed to be constant). The instability ends at the marked points, which lie just above the (dotted) line μ=Ω. The inset, which shows μ2A/(2π) as a function of J/M2, illustrates that the horizon area A increases in this process, up to a maximum of A≈4πaΩ/μ2 (dotted line). [N.B. Units G=c=ℏ=1 are used].
It should be noted that the concordant view of the instability, as developed in Furuhashi:2004 ; Cardoso:Yoshida:2005 ; Dolan:2007 ; Rosa:2010 , is challenged by two outlier studies. First, a study of the extremal Kerr BH Hod:Hod:2009a predicted a maximum growth rate some four orders of magnitude faster than the consensus. However, it was shown in Rosa:2010 that the anomalous growth rate was likely due to a flaw in the matching method, because the functional forms used in the matching procedure were not valid across the full range of Mμ (the functional forms exhibited spurious poles which led to incorrect estimates). Second, a study of the massive scalar field in the time-domain in 2005 Strafuss:2005 reported evidence for a fast growth rate, Mν∼2×10−5 (see Fig. 6 in Ref. Strafuss:2005 ). It was recently argued in Ref. Witek that this finding was in fact due to a misinterpretation of the 'beating' effect, caused by interference between quasi-bound modes of similar frequencies, which leads to a slow modulation of the envelope of the field.
ii.3 Mirror states
The relevant states in the case of a black hole surrounded by a mirror are defined by Eq. (12) and
X(rm)=0 (16)
where rm is the mirror radius. The mirrored case was considered in detail in Ref. Cardoso:2004-bomb .
Once more, the states are labelled by angular momentum l, m and overtone n. But, in contrast to the massive-field case, the number of overtones in the spectrum depends on the mirror radius rm. The number of overtones nmax increases with rm, as more overtones 'fit into' the cavity between horizon and mirror.
The modes are approximately evenly-spaced in frequency, with a separation that decreases as the mirror radius increases. For moderate-to-large rm, a simple approximation for the oscillation frequency ^ωn of the nth overtone is given by Cardoso:2004-bomb ,
^ωn≈jl+1/2,nrm≈(n+1+l/2)πrm, (17)
where jl+1/2,n is the (n+1)th root of the Bessel function of the first kind Jl+1/2(⋅). Modes within the superradiant regime, ^ωn<mΩ, will grow with time, ν>0. An estimate of the growth rate is given in Eq. (35)–(36) of Ref. Cardoso:2004-bomb . The growth rate of the mirrored system can be much larger than the massive-field case without mirror, as shown in Fig. 10. This makes the mirrored case rather easier to observe in time-domain simulations.
Iii Time domain methods
Below we develop methods for solving the scalar-field wave equation in the time domain.
iii.1 2+1D equation
We begin by separating out the azimuthal dependence of the field,
Φ=1rΨ(t,r,θ)eimφ (18)
after introducing an alternative azimuthal coordinate φ via
dφ=dϕ+aΔdr. (19)
The new coordinate is introduced to handle the problem that the Boyer-Lindquist coordinate ϕ is undefined as the horizon is approached (equivalently, to remove a term in the potential, a2m2, which does not vanish as r→r+). This ansatz leads to a 2+1D equation,
{−Σ2∂tt+(r2+a2)2∂xx−4iamMr∂t
+[2iam(r2+a2)−2a2Δ/r]∂x−^V}Ψ = 0, (20)
where Σ2=[(r2+a2)2−a2Δ]+a2Δcos2θ, and ∂x, ∂xx denote 1st and 2nd partial derivatives with respect to the tortoise coordinate, and
^V = Δ[−{∂θθ+cotθ∂θ−m2sin2θ}+2Mr(1−a2Mr) (21)
+2iamr+(r2+a2cos2θ)μ2].
Eq. (20) can be solved with standard finite-difference methods, as in Refs. Strafuss:2005 ; DWB . Indeed, in Sec. III.7 we make use of a well-tested 2+1D code to check our implementation.
iii.2 Coupled 1+1D equations
Let us now make the further decomposition,
Ψ=∞∑j=|m|ψj(r)Yjm(θ), (22)
where Yjm(θ) denotes a standard spherical harmonic without the azimuthal part. Now we substitute (22) into Eq. (20) and, by projecting onto the basis of harmonics, we recover a set of coupled 1+1D equations. The cos2θ terms in Eq. (20) lead to some coupling between l modes; however, the coupling turns out to be rather straightforward. We make use of the following result Hughes:2000 ,
⟨lm|cos2θ|jm⟩ = 13δjl+23√2j+12l+1⟨j,2,m,0|l,m⟩× (23)
⟨j,2,0,0|l,0⟩≡cmjl
⟨lm|X(θ)|jm⟩=2π∫Y∗lm(θ)X(θ)Yjm(θ)d(cosθ). (24)
The numbers ⟨j,i,m,0|l,n⟩ are Clebsch-Gordan coefficients. The coupling coefficient cmjl is zero unless j=l−2, l or l+2. Hence, the even-l and odd-l sectors are completely decoupled (as expected from symmetry considerations), and, within each sector, the coupling between l-modes of the same parity is nearest-neighbour only.
Let us now write the equations in first-order-in-time form by introducing πl=∂tψl, so that
[Σ2(0)+a2Δcmll]˙πl+a2Δ(cml,l+2˙πl+2+cml,l−2˙πl−2)=
(r2+a2)2ψ′′l+(2iam(r2+a2)−2a2Δ/r)ψ′l
−4iamMrπl−Vl−a2μ2Δ(cl,l+2ψl+2+cl,l−2ψl−2)
where Σ2(0)=(r2+a2)2−a2Δ and
Vl = Δ[l(l+1)+2Mr(1−a2Mr)+2iamr+ (26)
μ2(r2+a2cmll)]ψl.
iii.3 Stress-energy
The field equation (3) is obtained from an action principle with a Lagrangian L, where
L=−12gαβ∂(αΦ∗∂β)Φ−12μ2Φ∗Φ. (27)
The associated stress-energy tensor \tensorTαβ is
\tensorTαβ=−gαγ∂(βΦ∗∂γ)Φ−δαβL, (28)
where X(αYβ)≡12(XαYβ+YαXβ). The stress-energy is symmetric in its indices, Tαβ=Tβα, and locally conserved, \tensorTαβ;β=0. By contracting the stress-energy with a Killing vector of the spacetime we may form a conserved vector field. There are two linearly-independent Killing vectors on Kerr spacetime, and we may form vectors associated with energy and azimuthal angular momentum in this way. By constructing a four-volume and applying Gauss' theorem (as in e.g. Sec. IIB of Ref. Dolan:2007 ), we obtain two conservation laws in the following form,
dEdt=−F(E),dJzdt=−F(J). (29)
The 'energy' E and 'azimuthal angular momentum' Jz are found from integrals over a timelike hypersurface. If we choose an 'exterior' region x≥x+ of a constant-t 3-surface, we may write
Here E and Jz are the energy and angular momentum densities, given by
E=∑lEl,Jz=∑lJl, (31)
2(r2+a2)El=Δ[l(l+1)+μ2(r2+a2cmll)]|Φl|2
+2Δa2μ2cml,l+2Re(Φ∗lΦl+2)
+(Σ2(0)+Δa2cmll)∣∣˙Φl∣∣2+2Δa2cml,l+2Re(˙Φ∗l˙Φl+2)
+(r2+a2)2∣∣Φ′l∣∣2+2am(r2+a2)Im(Φ∗lΦ′l), (32)
−(r2+a2)Jl=(Σ2(0)+a2Δcmll)mIm(˙ΦlΦ∗l)
+ma2Δcml,l+2Im(˙Φl+2Φ∗l+˙ΦlΦ∗l+2)
+2Marm2|Φl|2, (33)
where Φl=rψl, ˙Φl=∂tΦl, Φ′l=∂xΦl, and ~Φ′l=Φ′l+iamΦl/(r2+a2).
The fluxes are found from taking surface integrals over the spacelike surface at x=x+, leading to
F(E) = (r2+a2)∑lRe(˙Φ∗l~Φ′l), (34)
F(J) = −m(r2+a2)∑lIm(Φ∗l~Φ′l). (35)
Let us now consider a wave of frequency ω which is 'ingoing' in the near-horizon region, i.e. Φl∼Alexp(−iω(t+x)). Such a wave generates the fluxes F(E)∼2Mr+|Al|2ω~ω and F(J)∼(m/ω)F(E). These fluxes become negative in the superradiant regime 0<ω<mΩ. Hence, though the wave is moving inwards, energy and angular momentum are passing outwards into the exterior region.
The choice of inner boundary x+ is somewhat arbitrary (typically, we take x+=−100M), and necessitated by the use of a finite grid. Nevertheless, the idea of 'energy in the exterior spacetime' is a useful physical concept; we will see that, by tracking the energy and AM in the exterior, we may track the development of the instability. We note that shifting the boundary x+ merely has the effect of shifting the time-origin in the plots of E and Jz, providing that x+ remains in the near-horizon region (x+/M≪0).
The conservation laws (29) also provide a useful code test, because the following quantities (representing total energy and angular momentum) should remain constant in our evolution scheme, up to numerical error:
C(E)=E+∫t0F(E)dt,C(J)=Jz+∫t0F(J)dt. (36)
iii.4 Finite-difference method
We evolve the equations on a uniform grid with grid spacing Δx=M/n for n=4 to 16. We use time steps Δt=κΔx that are sufficiently small to satisfy the Courant-Friedrichs-Lewy stability condition, i.e. with κ=0.8, typically. We use the Method of Lines, with a fourth-order Runge-Kutta time step, and fourth-order finite differencing on spatial slices, i.e.
ψ′(i) ≈ [−ψ(i+2)+8ψ(i+1)−8ψ(i−1)+ψ(i−2)]/(12Δx),
ψ′′(i) ≈ [−ψ(i+2)+16ψ(i+1)−30ψ(i) (37)
+16ψ(i−1)−ψ(i−2)]/(12Δx2)
where ψ(i) denotes ψj(x=x++iΔx) (and here i is an integer).
The equation set (LABEL:eq:1+1-coupled) resembles
A˙x=b (38)
where xT=[πk,πk+2,πk+4,…] (where k=|m| or |m|+1, depending on parity) and A is a real, symmetric, diagonally-dominant, tridiagonal matrix. At each grid point, we solve (38) to find ˙x. Fortunately, as A is tridiagonal, the solution can be found in a numerically-efficient way, using (e.g.) the Thomas algorithm NumericalRecipes .
iii.5 Perfectly matched layers
To achieve computational efficiency, it is necessary to truncate the computational domain. As all physical solutions are 'ingoing' in the near-horizon region, one possibility is to impose the ingoing condition (∂t−∂x)ψl=0 on finite-difference molecules at the left boundary of the grid. However, this approach – imposing a condition at an interface – runs the risk of generating high-frequency artifacts, which can excite numerical instabilities, and which may be mistaken for the superradiant instabilities themselves. In this work, we take an alternative approach: we truncate the computational domain by introducing an artificial absorbing region, which smoothly attentuates the solution without generating reflections. This is the concept of the 'perfectly matched layer' (PML) Berenger:1994 .
PMLs are widely used in computational electrodynamics Taflove:Hagness , but the underlying idea can be applied to a range of wave equations. For example, PMLs have previously been used in simulations of 'analogue' black holes in Bose-Einstein condensates Farrell:Leonhardt ; Giovanazzi .
We may illustrate the PML approach with the simple example of the 1D wave equation, ¨u=u′′. This may be written in fully first-order form as ˙u=v′, ˙v=u′. In the frequency domain, it has wave solutions exp(−iω(t±x)). To create a PML, we may analytically-continue the wave solutions onto a complex-x contour and then make a coordinate transformation to express x as a function of a real coordinate. These steps are equivalent to making the replacements
∂∂x→11+iγ(x)/ω∂∂x, (39)
in the frequency-domain equations. In the time-domain equations this leads to
˙u = v′−γ(x)u, (40)
˙v = u′−γ(x)v, (41)
the solutions of which are given by
u∼exp(−iω(t±x)±∫xyγ(x)dx). (42)
and v=±u. It follows that a right-going wavepacket moving into a PML from the left (i.e. with y<x) will be attentuated; and also that a left-going wave moving into a PML from the right (i.e. with y>x) will also be attentuated. Note that here γ, the amplitude of the attentuation, is a function of x that may be varied smoothly, over the scale of a few wavelengths, without generating reflections (at the continuum level). Thus we may 'turn on' the PML in a smooth manner, which greatly reduces the risk of numerically-generated high-frequency reflections.
How may this idea applied to the wave equation on Kerr? First, we note that in the near-horizon region x/M≪0, the 1+1D equations decouple, reducing to
¨ψl−ψ′′l+2imΩ˙ψl−2imΩψ′l=0. (43)
This equation may be turned into a simple 1D wave equation, ¨u−u′′=0 by making the replacement ψl=exp(−imΩ(t+x))u. The PML modification can then be applied, at the expense of introducing an auxiliary variable to play the role of v. After transforming back to the original variables we are led to
˙ψ = π,
˙π = ψ′′+2imΩ(ψ′−π)−χ′−imΩχ−γ(π+imΩψ),
˙χ = γ(ψ′−χ+imΩψ)−imΩχ, (44)
(where the subscript l has been dropped for clarity). To get the ˙π equation, we combine the time-derivative of (40) with the spatial-derivative of (41) and introduce the modified auxiliary variable via χ=γexp(−imΩ(t+x))v. The ˙χ equation follows from (41).
Note that these equations only apply in the near-horizon region where terms ∼Δ are small enough to be neglected; this is the region where we add the Perfectly Matched Layer. Note also that χ, the auxiliary variable, does not propagate outside the PML region.
iii.6 Grid boundaries
Even with the use of PML, it is still necessary to take steps to handle the boundaries of the grid. We make use of one-sided spatial derivatives in the finite-difference molecules near the left and right boundaries. On the left boundary, beyond the PML region, we imposed a simple zero condition ψ=0=π on the final grid point. Although this simple condition creates reflections, it merely reflects attenuated flux back into the PML region, where it is attenuated once more, to a numerically-negligible level. On the right boundary, we also impose a simple zero condition on the final grid point. In the mirror case, this is physically well-motivated (see Sec. IV.1). In the massive-field case, we place the right boundary far from the BH, typically at rmax∼500M. As long as the boundary is far from the black hole, we do not expect it to significantly influence the lowest quasi-bound states, or the superradiant instability, as Ref. Cardoso:2004-bomb suggests that Mν∼(M/rmax)4 (for dipole of the massless field), so Mν∼1.6×10−11 for rmax=500M. We have carefully checked that the observed instability growth rate is not sensitive to the position of the right boundary.
iii.7 Validation
We have made a number of checks of the correctness of our implementation. First, we tested the numerical convergence of the finite-different code via a ratio test, and found strong evidence for the expected rate of convergence. Second, we compared the results of the 1+1D code with those of the well-validated 2+1D code, finding agreement up to the anticipated level of numerical error. Third, we tested the implementation of the PML, by comparing two data sets arising from evolving the same initial data (i) on a domain in x without a PML and sufficiently large that boundaries are not encountered, and (ii) on a restricted domain, x≥−200M, with a PML of width 10M centred around x=−100M. Figure 3 shows typical results; data sets (i) and (ii) are found to be in excellent agreement on the right-hand side of the PML, as expected. Fourth, we have checked that the 'quasinormal ringing' has the expected frequency and decay time. Fifth, we have checked that stress-energy 'constants' C(E/J) [Eq. (36)] are preserved in our runs, up to numerical error (see Fig. 11). Sixth, we have checked that the results presented here are robust under adjustments of 'internal' parameters (such as the width and amplitude of the PML; the maximum number of l modes used; and the grid resolution). Finally, the next sections show excellent agreement with frequency-domain results, which we take to be further evidence in favour of a valid implementation.
Figure 3: Validation of the Perfectly-Matched Layer method. Plot (a) compares snapshots taken at t=140M, from two evolutions of the same initial data (a time-symmetric Gaussian of width σx=5M centred around x0=10M in mode l=m=1) for a BH of spin a=0.99M, using (i) [red, solid] no PML, and (ii) [blue, dashed] a PML centred around x=−100M of amplitude γ(x) (with γ/10 shown as dashed magenta line). Outside the PML region, (i) and (ii) are in good agreement. Inside the PML region, incident waves are smoothly attenuated without reflection. Plot (b) shows the same data on a logarithmic scale. Plot (c), showing data extracted at x=x1=0 as a function of time, illustrates that, to the right of the PML, data sets (i) and (ii) are virtually indistinguishable.
Iv Results
In this section we present a selection of numerical results, for two cases in which an instability is present: the massless scalar field with a mirror (Sec. IV.1), and the massive scalar field without mirror (Sec. IV.2). In each case, we seek to stimulate the system with some broadband initial data – typically a Gaussian in x – and study the features of the response. In particular, we attempt to compute the spectrum of superradiant quasi-bound states which dominate the response at late times. We will focus on the m=1 mode, on which superradiance is expected to have the strongest effect.
iv.1 Scalar field with mirror
Let us begin by considering a BH surrounded by a mirror at rm=20M, subject to some smooth initial perturbation. Figure 4 compares the radial profile of the response of a Schwarzschild BH (a=0) with that of a rapidly-rotating Kerr BH (a=0.99M), over several decades of simulation time t/M=10,100,…105. Here the initial data is a time-symmetric Gaussian,
ψl=1=exp(−(x−x0)/[2σ2x]),ψl>1=0=˙ψl, (45)
with x0=10M and σx=2M. At early times (e.g. t=10M), the response of the two BHs is very similar. At intermediate times, t=100M–1000M, some part of the initial perturbation has been absorbed through the horizon, and some remains in the exterior; the responses of Schwarzschild and Kerr are now quite different, but the field amplitudes remain comparable. At late times, t=104M and t=105M we see that the scalar field on the Schwarzschild BH has essentially disappeared through the horizon. On the other hand, the scalar field on Kerr has experienced significant amplification (note vertical scale).
Figure 4: Time evolution of field on black hole background with 'mirror' at rm=20M. These plots show snapshots of ψl [Eq. (22)] for the dipole mode l=m=1 of the scalar field, as a function of tortoise coordinate x/M, at t=0 (top left), 10M, 100M, 103M, 104M and 105M (bottom right), for the Schwarzschild (a=0, solid line) and fast-rotating Kerr (a=0.99M, broken lines) BHs. Here the initial data is a time-symmetric Gaussian (45) of width σx=2M centred about x0=10M. By late times, the field has been absorbed by the Schwarzschild BH, and amplified by the Kerr BH.
These conclusions are supported by Fig. 5, which shows the radial profile of the energy density E [Eq. (32)] computed from the field shown in Fig. 4. The late time plots suggest that there is more than one growing mode in the Kerr case, and that the overtone with a peak closest to the mirror surface is emerging to dominate the signal.
Figure 5: Time evolution of energy density E on black hole spacetime with 'mirror' at rm=20M. These plots show snapshots of the energy density obtained from the field shown in Fig. 4 (see caption) via Eq. (32). The last plot (note the scale) shows that the energy of the field on the rotating BH (broken line) has increased by a factor of ∼100 over a timescale of ∼105M via the superradiant mechanism; whereas the field on the non-rotating spacetime (a=0) has drained away through the horizon.
Let us now consider the field measured at a fixed point x=x1 as a function of time. Figures 6 and 7 (log scale) show typical time series data for Schwarzschild and Kerr BHs. In the Schwarzchild case, the field decays exponentially, as the mirror reflects flux back onto the absorbing horizon. In general, the decay rate increases as the mirror is moved closer to the BH. In the Kerr case, the field is long-lived. Figure 6 shows that there is typically a 'beating' effect in the signal, generated by interference between modes of different frequencies. This leads to a modulation of the signal which can disguise the underlying growth and even lead to incorrect conclusions (as argued in Ref. Witek ). On the other hand, Fig. 7 shows that the total energy in the exterior E (Sec. III.3) gives a clear view of the underlying growth as: (i) E does not oscillate sinusoidally, and (ii) the variation of E depends only on the energy flux through the inner boundary, F(E) [Eq. (34)], which is only marginally influenced by beating between modes.
Figure 6: Time series data for the evolution of the field on a BH with mirror. The plot shows the (real part of) the field shown in Fig. 4 and 5 extracted at r=10M, as a function of time t/M. On the non-rotating spacetime (a=0, blue) the field decays, whereas on the rotating spacetime (a=0.99M, red) it is long-lived. The latter case shows evidence of 'beating', due to interference between modes of different frequencies, and growth (see Fig. 7).
Figure 7: These plots show the time evolution of the field as in Fig. 6, on a logarithmic scale, up to t=105M. The upper plot shows that the envelope of the Kerr (Schwarzschild) field grows (decays) exponentially. The lower plot shows that the energy in the exterior (see Sec. III.3) also grows (decays) exponentially without beating effects, and with twice the exponent, as expected.
iv.1.1 Mirror spectrum
The spectrum of the response is revealed by taking a Fast Fourier Transform of the time series data for the field (extracted from the grid at fixed x=x1). Let us introduce the (complex) Fourier amplitude, defined by
fl(ωj)=1NN−1∑k=0ψl(tk)exp(−iωjtk), (46)
where tk=kΔt, ωj=jΔω, and N is the number of data points, Δt=tmax/(N−1) and Δω=2π/tmax, and the power spectrum given by
Pl(ω)=|fl(ω)|2. (47)
Note that the frequency resolution Δω is inversely proportional to the simulation time tmax; hence longer runs yield a more refined view of the spectrum.
Figure 8 shows typical power spectra, for mirror radii rm=15M and rm=25M. The sharp peaks indicate the presence of modes in the spectrum. The plot makes it clear that the modes within the superradiant regime 0<Mω<mΩ have the largest amplitude (highest peaks), as expected.
Figure 8: Power spectrum of the scalar field on BH with mirror. The plot shows the power spectrum (47) as a function of frequency Mω, computed via a Fourier transform of the time-domain data (Fig. 7), for two mirror radii, rm=15M and rm=25M. Modes inside the superradiant regime, 0<ω<mΩ, are subject to exponential growth. The modes are approximately evenly-spaced in frequency [Eq. 17]. The plot illustrates that the effect of increasing the mirror radius rm is to bring more overtones within the superradiant window.
The spectrum has all the features anticipated in the frequency-domain study of the mirrored case, Ref. Cardoso:2004-bomb . The peaks are approximately evenly-spaced in frequency, with a separation that decreases as the mirror radius increases, as predicted by Eq. (17).
The width of a peak in the power spectrum provides information on the decay/growth rate of the corresponding mode. Consider a monochromatic mode with time dependence exp(−i(^ωt+iν)t). Its contribution to the power spectrum is
P(ω)≈1/[(ω−^ω)2+ν2]. (48)
Hence, fitting a quadratic to 1/P(ω) in the vicinity of a minimum ^ω should give an estimate of the decay/growth rate ν, whose sign may be found by examining f(ω)). Despite its promise, we found that this method was not sufficiently accurate to numerically estimate the rates of long-lived modes, and so we pursue an alternative method below.
iv.1.2 Growth rates from frequency-filter
To study the individual modes in the time-series data, we devised a simple 'frequency-filter' method. First, we identified the modes in the power-spectrum (e.g. Fig. 8) using a peak-finding algorithm, and used interpolation to obtain a better estimate for the real frequencies ^ωn. Next, for each mode, we (i) multiplied the Fourier transform by a narrow filter function centred on the peak, i.e. f(ω)→f(ω)F(ω−^ωn) with
F(x)=exp(−x4/w4) (49)
and w=kΔω with n∼5 (typically) and Δω defined above Eq. (47); (ii) reconstructed the modal contribution by taking the inverse Fourier transform of the filtered data; (iii) computed the square magnitude of the reconstructed field as a function of time; and (iv) computed the growth index using linear regression on the logarithm of the field magnitude.
These steps are illustrated in Fig. 9, which shows the square magnitude of the reconstructed field modes for the first few overtones. It shows that the reconstructed field modes at early and late times depend somewhat on the shape of the filter; but that the field around tmax/2 is a faithful reconstruction of the modal contribution.
Figure 9: The frequency-filter method. Each solid line in the plot corresponds to the (normalised) square amplitude of a particular mode extracted from the time-domain data. Each mode is isolated by applying a narrow frequency filter to the Fourier transform of the time domain signal, and then applying the inverse transformation (see text). The dotted lines represent the best fits. The plot shows the overtones n=0 to 3 for rm=32M and a=0.99M, for which case n=3 is the most rapidly-growing.
With the frequency-filter method, we may obtain accurate estimates for the frequency and growth/decay rate of the subdominant modes that are 'hidden' in the time-domain data. Figure 10 shows estimates of the growth rate of the instability, as a function of mirror radius, for the first few overtones. These estimates are compared with the growth rates obtained in Ref. Cardoso:2004-bomb via a frequency-domain analysis. The agreement is found to be excellent. It is gratifying that many overtones can be extracted from the time domain data, even in cases when the late-time signal is dominated by the fastest-growing mode.
Figure 10: The growth rate of the scalar field for rotating BH with mirror. The plot shows the growth rate of the first few exponentially-growing modes (∝exp(νt)), as a function of mirror radius rm. The points show the growth rates calculated from time-domain data, using runs up to t=105M, by applying the frequency-filter method (Sec. IV.1.2). The lines show the growth rates found in the frequency-domain analysis in Ref. Cardoso:2004-bomb (with thanks to the authors).
iv.2 Massive scalar field
In this section, we consider the case of a massive scalar field in vacuum. Non-zero mass leads to a spectrum of quasi-bound states with Re(ω)≲μ, as summarized in Sec. II.2.
Figure 11 shows the evolution of the energy and angular momentum of the massive scalar field in the exterior spacetime, for a typical simulation of the m=1 mode of a scalar field with Mμ=0.42 and Gaussian initial data, on a Kerr BH with a=0.99M. At 'early' times t≲104M, which have been probed by other studies Strafuss:2005 ; Yoshino:Kodama ; Witek , we see that the energy in the exterior initially drops off, as non-superradiant flux is absorbed by the BH, before approaching an apparent 'stationary' equilibrium. However, 'late' time evolutions t≫104M, reveal that the solution is not truly stationary, as the slow exponential growth in the quasi-bound states begins to dominate.
Figure 11: Time-domain evolution of a massive scalar field (Mμ=0.42) on rapidly-rotating BH spacetime (a=0.99M) with time-symmetric Gaussian initial data, with x0=20M and σx=10M [see Eq. (45)]. The plots show the evolution of the energy E and azimuthal angular momentum (AAM) Jz in the exterior spacetime (30), and the 'conserved' quantities (36). In the initial phase, t≲104M shown in plot (a), E decreases and Jz increases as non-superradiant flux, particularly the counter-rotating part of the field with ω<0, is absorbed by the BH. The 'total' energy and AAM, C(E) and C(J) [Eq. (36)], are numerically conserved to a good approximation. In the 'late' phase, t≳×104M shown in plot (b), the exterior energy and AAM increase, as growth in the superradiant quasi-bound states begins to dominate. Plot (c) shows that there is some numerical violation of the constraints which diminishes as resolution improves, and which is consistent with the resolution-dependence of the curves at late times seen in plot (b).
To check the validity of our results at late times, we have computed the energy and AM 'constants', defined in Eq. (36), from the numerical data. Figure 11(c) shows that there is a slow 'drift' in the numerical values of the constants, but that the drift is strongly resolution-dependent, as expected. We also see that the growth rate is resolution-dependent, but that it converges to a physical value as n is increased.
iv.2.1 Quasi-bound state spectrum
By extracting the field at a fixed point x=x1, and taking the Fourier transform, we may obtain the power spectrum Pl(ω) of the massive field (see Sec. IV.1.1). Figure 12 shows a typical spectrum in the case a=0.99M, Mμ=0.42. The fundamental mode and the first four overtones, n=0…4, are apparent as sharp peaks in the spectrum. With ultra-long timescale simulations, we gain sufficient frequency resolution to determine the bound-state frequencies with high accuracy. Figure 12 shows a comparison between the frequencies found by Fourier analysis of the time domain signal, and the frequencies found from frequency-domain analyses (e.g. Sec. II.2 and Ref. Dolan:2007 ).
Figure 12: Power spectrum of massive scalar field (Mμ=0.42) on fast-rotating BH spacetime (a=0.99M) in the co-rotating dipole l=m=1. The spectrum is computed from time-domain response to Gaussian initial data, up to t=6×105M. The fundamental mode and the first four overtones are visible. The frequencies of the first quasi-bound states, n=0…4 for l=m=1, computed by frequency-domain analysis, are shown at ω/μ≈0.9733, 0.9883, 0.9936, 0.9960 and 0.9980.
By applying the frequency-filter method described in Sec. IV.1.1 we may estimate the growth rate of the superradiant quasi-bound states. A key advantage of the method is that it allows us to separately analyse the many overtones that are in the spectrum, even though the time-domain signal may be dominated by a single mode. Furthermore, we find that we may extract good estimates for the growth rate of lowest overtones n=0,1 from runs of tmax=105M which are relatively 'short' in comparison to the e-folding time (ν−1>5.8×106M) of the instability. Figure 13 shows some typical results, comparing the growth rate extracted from time-domain simulations, with the much more accurate values found by frequency-domain analysis Dolan:2007 . The agreement is good, reinforcing once more the concordant view of quasi-bound states described in Sec. II.2.
Figure 13: Instability growth rates of massive scalar field on rapidly-rotating Kerr spacetime (a=0.99M). The lines show the growth rates of the fundamental and first-overtone bound states of the co-rotating dipole l=m=1, established via the frequency-domain method Dolan:2007 . The points show estimates of the growth rates found by applying the frequency-filter method (Sec. IV.1.2) to data extracted from time-domain runs (with tmax=105M and resolution dx=M/6). The observed agreement implies that the quasi-bound state spectrum can be accurately determined from 'ultra-long' time-domain datasets in cases, such as the Proca field on Kerr spacetime, where a lack of separability impedes a frequency-domain analysis.
V Discussion and conclusion
In this paper we have described a 'coupled 1+1D' time-domain scheme for evolving the scalar field on Kerr spacetime. The scheme enabled us to study the evolution of the scalar field, within the linear regime, over ultra-long timescales t∼106M, which represents a factor of 100 to 1000 improvement over previous time-domain studies Strafuss:2005 ; Yoshino:Kodama ; Witek . The efficiency of the scheme allowed us to track the massive-field superradiant instability for a significant fraction of an e-folding time. We have shown that accurate values for the quasi-bound state frequencies and growth rates may be extracted directly from time-domain data via Fourier analysis. We found that these values were in excellent agreement with the results of frequency-domain analyses (see Fig. 10 and Fig. 13). Thus, our technique offers a promising way forward for tackling scenarios where a frequency-domain analysis, reliant on the separability of the underlying equations, is not feasible. The method would seem to be ripe for application to the Proca-Kerr system Rosa:Dolan ; Pani:PRD ; Pani:PRL , recently examined by Witek et al. Witek in the time domain for the first time.
The efficiency of the method is primarily due to the dimensional reduction procedure, i.e., working in a 1+1D rather than 3+1D domain. In other words, the key step in our method is the decomposition of the field into a superposition of harmonics. Even though this does not lead on to a full decoupling, it turns out that the coupling between l modes is very simple. In the scalar-field case, the coupling is only between the nearest-neighbour modes of the same parity. In the Teukolsky equation (s≠0), we anticipate the coupling would also be simple, but instead between nearest- and next-to-nearest neighbours Hughes:2000 . In the higher-spin massive-field case (e.g. Proca) the coupling may be substantially more complicated, although the slow-rotation expansion of Pani:PRD ; Pani:PRL suggests it may be tractable.
Our 1+1D scheme is similar in some regards to that developed in Ref. Racz:Toth ; Csizmadia , to look at the late-time tails of massless fields on Kerr spacetime. In Ref. Racz:Toth , the coefficient of ∂ttΨ, which depends on cos2θ, appears in the denominator on the right-hand side of the equation. The denominator is converted to an infinite series in the numerator, which leads to coupling between all l modes after harmonic decomposition (see Sec. 2.2 in Racz:Toth ). Fortunately the couplings die away rapidly with increasing l and the method is useful in practice. An advantage of the approach of Ref. Racz:Toth is that a matrix-inversion step (Sec. III.4) is not required at each time step. This may offer an alternative route to tackling the Proca-Kerr case, where coupling terms may be substantially more complicated.
Our method also makes use of a technique from computational electrodynamics: the Perfectly Matched Layer Berenger:1994 ; Taflove:Hagness . We have shown that the PML offers a numerically-robust way of implementing the absorbing boundary condition at the event horizon. It seems that this technique may be readily incorporated into a range of linear wave equations, and yet to date it not been used widely in black hole perturbation theory. The PML also offers distinct advantages in multi-dimensional scenarios, where stable boundary conditions are more difficult to implement. It would be interesting to compare the efficacy of the PML with alternative methods, such as horizon-penetrating coordinate systems with excision.
We have run our time-domain code for a range of mass couplings, 0.1≲Mμ≲1, and BH spin rates 0≤a/M≤0.99, for low azimuthal numbers m. In every case, we have found that the growth of the field (if present) is due to growth in a superposition of quasi-bound states which lie within the superradiant regime. Furthermore, these quasi-bound states are found to have the expected frequencies and growth rates from the 'concordant' frequency-domain analyses (see Sec. II.2). We have found no evidence in favour of the more rapid growth rates reported in Strafuss:2005 and Hod:Hod:2009a .
Unabated, an exponentially-growing instability will inflate the magnitude of the scalar field until non-linear effects can no longer be ignored. Non-linear effects arise from (at least) two sources: the self-interaction of the field, and the back-reaction of the field upon the spacetime. In the context of axions, the former effect may be modelled by replacing μ2Φ with μ2sinΦ in the scalar-field equation, introducing a non-linear term at leading order μ2Φ3. A recent pioneering time-domain study Yoshino:Kodama looked at the effect of non-linear coupling. It concluded that, whereas small self-couplings will increase the rate of energy extraction, large self-couplings will cause a collapse of the axion cloud. This suggests that the endpoint of the instability is an explosive phenomenon Arvanitaki1 . As yet, the transition from small to large couplings has not been tracked via a time-domain simulation, due to strong limitations on run-time imposed by the computational expense of 3+1D codes. It may be possible that a dimensionally-reduced code could allow longer run-times; however, the non-linear terms would introduce couplings between m and l modes, which would substantially increase the complexity of implementation. A perhaps more tractable problem would be to use a time-domain code to study the generation of gravitational waves (i.e. linearized metric perturbations on the Kerr background) induced by the stress-energy of the scalar field.
A further prospect is to apply dimensional reduction to study the Proca field over long timescales within the linear regime. Here a 2+1D code, exploiting a simple separation into m-modes, would seem suitable. A 2+1D code would allow one to probe beyond the timescales achieved in Ref. Witek (tmax=3×103M). The resolution of the power spectrum scales with run-time, and so, with long-timescale datasets, one would be able to isolate the modes by applying a narrow filter to peaks in the power spectrum (as in Sec. IV.1.2) to determine the details of the quasi-bound states (i.e. frequencies and growth rates) to high precision. This is surely an interesting challenge.
In summary, time-domain simulations offer a way to examine the onset, development and termination of black holes superradiant instabilities which may be ignited in nature by ultra-light bosonic fields. It is important to understand these instabilities fully, if we are to use new observations of astrophysical black hole parameters to constrain the boson mass spectrum Arvanitaki1 ; Arvanitaki2 ; Kodama:Yoshino ; Yoshino:Kodama ; Pani:PRL ; Pani:PRD . Furthermore, the study of field configurations which are long-lived or quasi-stable is of foundational interest, because such configurations would seem to challenge to the spirit of the 'no-hair' conjecture. We have shown here that the scalar-field instability has been well understood within the linear regime. There remains much scope for pursuing the instability into the non-linear regime, and in the context of massive vector fields.
I am grateful to Joao Rosa, Helvi Witek, Barry Wardell, Elizabeth Winstanley and Leor Barack for helpful discussions, and to Vitor Cardoso and Shijun Yoshida for providing data for validation and comparison. I acknowledge the use of the following HPC resources: Iridis at the University of Southampton, Tesla at University College Dublin, and Iceberg at the University of Sheffield. Financial support in the early stages of this work was provided by the Engineering and Physical Sciences Research Council (EPSRC) under Grant No. EP/G049092/1.
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'I'm Coming Too': Forbidden History of the Female Orgasm
Home/Entertainment/Today's Films and Programs Tuesday 30 August 2022
Today's Films and Programs Tuesday 30 August 2022
David Martin December 19, 2022 Entertainment Leave a comment 8 Views
Movies Tonight on TV: Forever My Girl, Fast and Furious, Captive State, Tonya, Happy Birthday Mr. Grape, Guess Who, The Divergent Series: Allegiant. Programs, Fiction and Series Tonight on TV: I would like you to be a single hour, La Mantide, Yoga Radio Bruno Estate, Lie To Me.
You were probably looking for: Today's Films and Programs Wednesday 31 August 2022
Movie Tonight on TV Of Today Tuesday 30 August 2022. Among those on air early evening today on TV channels Unencrypted: Forever My Girlfriend, Fast and Furious, Captive State, Tonya, Happy Birthday Mr. Grape, Guess Who, The Divergent Series: Allegiant, Ex, The Great Trail, The Scorpion King, VIP, Something Personal.
All Movies tonight on TV:
Forever my girlthe movie on air tonight on tv at 21.20 on Rai 1: 2018 musical drama, gutless, musical by Bethany Ashton Wolf, starring Alex Roe, Jessica Rothe, Abby Ryder Fortson, Peter Cambor, Gillian Vigman, Judith Hoag, John Benjamin Hickey, Morgan Alexandria, Terayle Hill and Whitney Goin.
Fast and Furiousthe movie on air tonight on tv at 21.20 on Italia 1: Rob Cohen's 2001 action thriller film starring Paul Walker, Vin Diesel, Michelle Rodriguez, Jordana Brewster, Rick Yune, Chad Lindberg, Johnny Strong, Matt Schulze, Ja Rule and Ted Levine.
Formerthe movie on air tonight on tv at 21.25 on Nove: 2009 comedy film by Fausto Brizzi, with Claudio Bisio, Silvio Orlando, Cristiana Capotondi, Cécile Cassel, Fabio De Luigi, Alessandro Gassmann, Claudia Gerini, Flavio Insinna, Nancy Brilli, Elena Sofia Ricci, Vincenzo Salemme, Gian Marco Tognazzi, Giorgia Wurth, Martina Pinto, Malik Zidi, Carla Signoris and Angelo Infanti.
captive statethe movie on air tonight on tv at 21.20 on Rai 4: 2019 science fiction thriller film by Rupert Wyatt starring John Goodman, Vera Farmiga, Ashton Sanders, Madeline Brewer, Machine Gun Kelly, Alan Ruck, James Ransone, KiKi Layne, Kevin J. O'Connor, Ben Daniels, DB Sweeney and Jonathan Majors.
Tonyathe movie on air tonight on tv at 21.15 on Rai 5: 2017 biographical drama film by Craig Gillespie starring Margot Robbie, Sebastian Stan, Allison Janney, Julianne Nicholson, Bobby Cannavale, Paul Walter Hauser, Caitlin Carver, Bojana Novakovic and Mckenna Grace.
Happy Birthday Mr Grapethe movie on air tonight on tv at 21.10 on Rai Movie: 1993 drama film by Lasse Hallström, starring Johnny Depp, Leonardo DiCaprio, Juliette Lewis, Mary Steenburgen, Darlene Cates, Laura Harrington, Mary Kate Schellhardt, Kevin Tighe, John C. Reilly, Crispin Glover, Penelope Branning, Tim Green and Susan Loughran.
The great paththe movie on air tonight on tv at 21 on Iris: 1964 western film by John Ford, starring Richard Widmark, Carroll Baker, Karl Malden, Sal Mineo, Dolores Del Rio, Ricardo Montalban, Gilbert Roland, Arthur Kennedy, Patrick Wayne, Elizabeth Allen, John Carradine, Victor Jory, Mike Mazurki, George O'Brien, Sean McClory, Judson Pratt, Carmen D'Antonio, Ken Curtis, James Stewart, Edward G. Robinson, Harry Carey Jr., Walter Reed, Denver Pyle, Walter Baldwin, Ben Johnson, Shug Fischer and Nancy Hsue.
The Scorpion Kingthe movie on air tonight on tv at 21.15 on Italia 2: 2001 action-adventure-fantasy film by Chuck Russell, starring Dwayne Johnson, Stephen Brand, Kelly Hu, Michael Clarke Duncan, Grant Heslov, Peter Facinelli, Ralf Moeller, Scott L. Schwartz, Sherri Howard, Andrei Sterling and Bernard Hill.
Guess whothe movie on air tonight on tv at 21.15 on Sky: 2005 comedy, sentimental film by Kevin Rodney Sullivan, starring Bernie Mac, Ashton Kutcher, Zoe Saldana, Hal Williams, Kellee Stewart, Robert Curtis Brown, Ronreaco Lee, Paula Newsome, Phil Reeves, Sherri Shepherd, Nicole Sullivan, Jessica Cauffiel, Jonell Kennedy, Niecy Nash, Kimberly Scott, Denise Dowse, Gus Lynch, Amanda Tosch, Andy Morrow and Chad Gabriel.
vipthe movie on air tonight on tv at 21 on Cine34: 2008 comedy film by Carlo Vanzina, with Enrico Brignano, Matteo Branciamore, Carlo Buccirosso, Martina Colombari, Maurizio Mattioli, Monica Scattini, Alena Seredova and Maria Grazia Cucinotta.
The Divergent Series: Allegiantthe movie on air tonight on tv at 9pm: 2016 sci-fi action adventure film by Robert Schwentke starring Shailene Woodley, Theo James, Miles Teller, Ansel Elgort, Naomi Watts, Jonny Weston, Jeff Daniels, Zoë Kravitz, Maggie Q, Octavia Spencer and Bill Skarsgård.
Something personalthe movie on air tonight on tv at 21.10 on TwentySeven: 1996 sentimental film by Jon Avnet, with Robert Redford, Scott Bryce, Stockard Channing, Raymond Cruz, Kate Nelligan, Glenn Plummer, Dedee Pfeiffer, Michelle Pfeiffer, James Rebhorn, Miguel Sandoval and Joe Mantegna.
Programs, Fiction and Series Tonight on TV
Among the main programs Tonight on TV we report:
I would like you just one hourthe comedy show by Henry Brignanoon air from 21.20 on Rai 2
White paper (talk show, politics, current affairs), on air from 21.20 on Rai 3
Out of the core (talk show, politics, current affairs), on air at 21.20 on Rete 4
The Mantis (TV miniseries), aired at 21.20 on Canale 5
On air (talk show, politics, current affairs), on air from 8.35 pm on La7
Crystal hell (action, adventure film), aired from 21.30 on TV8
Yoga Radio Bruno Summer (music show), aired from 9.10 pm on La5
Lie To Me (TV series), aired at 21.30 on La7D
The Devil's Watches (fiction tv series), aired at 21.20 on Rai Premium
The Brokenwood Mysteries (TV series), aired from 21.10 on Yellow
Harrow (TV series), aired from 21.10 on Top Crime
Cougar Wives: love has no age (reality), on air from 21.20 on Real Time
Naked and raw (docu-reality), on air from 21.25 on DMAX
Focus nature – A perfect world (Docu-series), aired from 21.15 on Focus.
History of the World Special Edition (Docu-series), aired from 21.10 on Rai Storia.
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Q: Give permission according to data in Django I'm working a project wherein there are multiple institutions. All the institutes have the same hierarchy of students, teachers and staff members. My problem is how do I keep the institutions apart? For example: the data profile of student from institute A should not be editable by a staff member in institute B. And vice versa. I'm able to use AbstractUser class to give my own roles but what about users (like Staff Members) who have the same permissions but shouldn't interfere with other institutions?

A: Django allows you to use multiple databases as illustrated in the docs. You can use a separate database for each intitue and set your database router to direct you to the right db based on some attribute in the user profile.
An example of this process is available here

redpajama_set_name: RedPajamaStackExchange


CAE today announced the appointment of Mr. Nick Leontidis as Group President, Civil Simulation Products, Training and Services, effective immediately. Mr. Leontidis replaces Mr. Jeff Roberts who is leaving the company.
"We thank Jeff for his dedicated service to CAE; he has helped CAE's civil training and services business become the world leader it is today," said Mr. Marc Parent, CAE President and CEO. "We wish him the best in his future endeavours."
"We welcome Nick to his new role. He has been with CAE for the past 25 years, most recently as Executive Vice President, Strategy and Business Development. Since joining CAE, Nick has played a key role in the success of our civil business and the expansion of our global footprint. He brings strong customer relationships with our partner airlines and OEMs and excellent leadership qualities. Nick has outstanding international experience, in particular in China where he was the executive responsible for developing CAE's sales and relationships. His strong credibility and deep understanding of the global aviation market are critical assets to our company's continued growth," Mr. Parent said. "His credibility with our employees coupled with his intimate knowledge of our technology and products will allow CAE to offer innovative solutions."
Mr. Leontidis joined CAE as a software engineer in 1988 and rose to the leadership of CAE's Visuals Systems group in 1999. Since then, he has held roles of increasing responsibility within the company's products and services segments. His positions included Vice President, Marketing and Sales for the Civil unit; Vice President, Simulation Products; Executive Vice President, Civil Aviation Training and Equipment; and most recently, Executive Vice President, Strategy and Business Development.
Mr. Leontidis holds Bachelor's and Master's degrees in Engineering from Concordia University.


Sorry for the recent lack of posts. Right now I'm prepping the five books I'm editing for our Summer 2007 (April-August) list, planning a wedding, working on book #2 for my publisher, chairing a conference, and departing tomorrow with my friend Ishmael on a massive hunt for a white whale.
But today, make sure to check out new flash fiction from J.T. Ellison at Killer Year. Tomorrow (or Saturday), based on a suggestion from Allison Brennan, I'm planning to write a post on the thrilling world of book royalties. If you have any specific questions, post a comment, and I'll address it.
And tonight I'm meeting my baby cousin Hannah Sylvie for the first time. i'm so excited, she's absolutely adorable.
And according to Borders, I've earned $19.29 from their Rewards program which I can use for holiday shopping. Only $2.73 more and I can buy a copy of THE ALL NEW ULTIMATE SOUTHERN LIVING COOKBOOK.
Are you a cookbook fan, too? My favorite is How to Eat by Nigella Lawson. It's like the yummiest food-obsessed novel with recipes ever.