"Pace to break. The record for the most measles cases in a single year since the disease was eliminated two decades ago. Health officials are reporting five hundred fifty five cases so far this year and officials have identified a patient zero for one of the major outbreaks last November a man flew from Israel to Brooklyn, which was in the midst of a measles epidemic from there. He traveled to Michigan visiting ultra-orthodox Jewish communities unknowingly bringing the virus with him he stayed in private homes. She stopped by multiple synagogues, and ultimately infected thirty nine people in the Detroit area. Well here to talk about this and the nationwide outbreak is Lena sun health reporter for the Washington Post and welcome to the show. Thank you for having me. So could you I give us a big picture of measles nationwide right now. Well for the third week in a row US health officials have added dozens of new reports to the. The years list of confirmed measles cases, they are in twenty states. But the six current outbreaks are in California, New jersey, New York and Washington states and the biggest one is in the New York area concentrated in New York City's ultra-orthodox Jewish community. And of course, one of the real issues with measles is just how infectious it is. So if a an infected person walks through a room, how likely are you to contract measles? If you're not vaccinated, if you're not vaccinated, and you're in that same room up to two hours after the infected person has walked through you have a ninety percent chance of getting measles. Think about it. It is an exquisitely contagious disease, and what's the center for disease control telling people to do right now. You've been given the speed of of these cases the rate, I think they're urging everyone to get vaccinated. And especially if you're. Talking about travel during spring break or summer vacation to go to Europe or many of these other places where there are major outbreaks. It's important to get vaccinated usually children are vaccinated starting at twelve to fifteen months, but they say that if you're going to be traveling you can get the baby in vaccinated as early as six months. And now international travel. It's always a prime vector of measles in that that figures into this patient zero story to that. We're about to talk about remind us why that is that a lot of these start with travel travelers visiting the United States or US residents returning from abroad because right now around the world. There are measles. Outbreaks in many, many parts of the world, the W H O has just come out with a report that says measles cases increased three hundred percent worldwide in the first three months of twenty nineteen compared to the first three months in twenty eighteen many countries in Europe, big parts of Africa, south East, Asia, Japan. All have measles cases. Africa had the biggest rise in measles cases. And in Madagascar, for example, where infrastructure is a big challenge. They twelve hundred people have died from measles. And that's because they're fewer people who are back stated in a bright. And and. I understand. It's very easy to drop below the level of immunity if more and more people don't vaccinate, right. So because measles is so infectious in order for a community to pay protected you need to have ninety six percent of that community vaccinated or higher. So that's to protect against kids who are too little to be vaccinated or people who are immune compromised and can't get the vaccination that's something called her immunity. But it's a better way to think about it is community immunity, and every disease is different, but measles requires a really really high level. So you just have to slip below ninety something percent. And you lose that protection. And let's talk now about the Michigan's patient zero. And you've tracked the story. So this man, flew to the United States from Israel. And he was. Raising money for charity in different orthodox communities starting in a York, tell us what happened next he stayed in Brooklyn for about two months, and he was fundraising and he in early March decided to travel to the Detroit area, he drove overnight through the night. And he was beginning to feel sick on route and a doctor when he arrived in the Detroit area, but the doctor like many doctors these days had never seen measles. He did misdiagnose the man's fever and cough as bronchitis and gave the man and antibiotic and next day the man complained about having a rash. And the doctor thought always having an allergic reaction to the antibiotic. When in fact, the man was having the measles. Telltale rash? So you are infectious with measles four days before the rash and four days afterward and all that time when the man. Within the Detroit area. He had been living in people's homes going to synagogue, sometimes three times a day shopping. Kosher markets, and in very close proximity to hundreds and hundreds of adult is it relevant that this is all happening within an orthodox community. How does that factor into patients zero story? I think what's important to remember is that many communities whether orthodox Jewish or faith Christian or some Molly or eastern European when you live in proximity to one another and spend a lot of time interacting working and socializing. It doesn't really matter. What your cultural heritage is? Or your ethnicity is the fact that you're all tightly together, and that is how disease travels measles doesn't care. They just want to infect as many people as possible. So once diagnosed how did how? Officials react and snap into you know, epidemic mode. So when the health department got a call from the doctor saying that they suspect a case of measles with the man's contact information, they tried to reach the man the man cell phone was had a corrupted sim card. So it wasn't working."


One Stop Mobility offers children's stroller rentals for Disneyland, California in both single and double seats. Both versions are considered lightweight and can fold to fit in a car for easy travel and storage. Please note: These strollers are not intended for jogging.
A single stroller is the perfect choice if you would like to take the weight off you shoulders (literally). For trips and events where you will be walking all day such as at Disneyland, stroller rentals are a lifesaver. Traveling with your baby has never been easier. They can enjoy all the outdoor scenery without having to get tired from all the walking and vice versa. The single stroller comes with a single pouch storage area underneath and collapses for quick storage in your car.
Pave the way with your children using a double stroller. The double stroller features all-terrain wheels, a handlebar that adjusts, as well as a hand-operated brake. With the auto-lock mechanism device on all our stroller rentals, transportation and storage is also easier than ever.


text:
Freezing on the technological edge
My grandfather's brother, shortly after he went blind, used to tell me of how he would sleep whenever possible so that he could see the sun in his dreams...
AS a 10-year-old, I didn't really understand what he meant. Now, as I sit typing these words, the thermometer on the sauna wall shows 80°C, and I am drenched in perspiration, but I still close my eyes to try to imagine the sun.
This is night time in Helsinki, Finland. I spent most of today at a city called Oulu, some 230 km south of the perpetual winter darkness of the Arctic circle.
Dawn broke at about 10.30 this morning and a brief spattering of yellow light slipped between the clouds to reveal a chillingly beautiful landscape shrouded in a blanket of snow and ice. Then it was gone.
Dusk came rapidly at about 3.30 this afternoon. I blinked, and it was dark. My psyche, which has been pulled out of the warm Durban summer, has been kicking and screaming in protest
It's easy to understand why people in these parts suffer winter depression, frequently leading to suicide. And why the buildings are shimmering masterpieces in chrome and glass that snatch and enhance every glimmer of natural light there is to be had from the unyielding sky.
But there are very good reasons to be here at this time. Oulu Technopolis is the world's northernmost science park and home to some of the world's best telecommunications and electronics technology.
A steady stream of Japanese and American technology industry executives make their way to Oulu to tap that technological expertise. Most of that business is conducted right here in the saunas.
There are important lessons for South Africa to be found in Oulu, especially in the light of Pallo Jordan's decision to not allow Telkom - our shining example of technical expertise and customer satisfaction - to be opened up to competition.
By the end of the last century, Russian tsars controlled much of Finnish territory and the Finns developed many local phone companies to guard against Russian influence.
In 1994, 47 of these companies banded together to challenge the government owned Telecom Finland, which had a monopoly on trunk calls. Four days after they were given permission to enter the market, the coalition snapped up 60% of the business. Today, Finland's phone rates are among the lowest in the world.
This is particularly interesting in the light of the ANC's belief that a break-up of Telkom SA's monopoly would not address the issue of providing telephone services to the masses.
Finland now has more Internet users per capita than the USA, which is not surprising given that commercial multimedia network rates are as low as a tenth of those in the US, Britain, and France.
Nokia, the cellular phone giant which has seen a 2000% growth over the past four years, is one of the better known examples of Finnish technology.
But there are over 100 other companies in Oulu filling technological niches that have provided jobs and generate close to R4 billion annually.
Some of these technologies are good examples of the ability to create new markets. Such as a personal pulse monitoring device that is targeted at professional athletes which can then be computer processed to produce personalised readouts to enable them to improve performance. The system is expensive, but the target market can afford it.
There is no reason why South Africa should not be in a similar position.
The frequency-hopping signalling technology used in those ubiquitous M-Net decoders could be used for encryption of regular voice and fax communication.
The helmet-mounted gunsight technology developed for the Alpha HX-1 helicopter gunship could be reverse engineered for virtual reality headsets.
In a nutshell, there are many technologies that were funded by apartheid and which could now be put to good use in establishing South Africa as a world technological presence.
We have thousands of bright matriculants who are unemployed. Perhaps it's worth investing the money now to make those come together.
We need innovation. That comes from competition. Breaking up the Telkom monopoly would be a good start.
And I'd much rather talk business under the South African sun, on Durban's beaches, than in the sauna.
‹ Eurotrash is better than eco-slime
London calling... and they're talking about us ›

redpajama_set_name: RedPajamaCommonCrawl


Kyle Tuller, 28, of Alvine Road in Pittsgrove, was arrested at 2:31 a.m. on New Year's Day and charged with driving while intoxicated after a single vehicle crash at Dahliadel Drive and Carpy Avenue. He also was cited for reckless driving.
Clinton Hollister, 23, of West Main Street in Millville, was arrested for driving while intoxicated after a single-vehicle crash into a wood fence in the area of Delsea Drive and Elm Road at 11:36 p.m. on New Year's Day. He also was ticketed for driving on a suspended license.
Angel M. Rivera, 42, of North Delsea Drive, was charged with driving while intoxicated after a reported hit-and-run collision on North Delsea Drive on Dec. 25. He was issued summonses for reckless driving, failure to report an accident, and leaving the scene of an accident.
Jose A. Mendez, 47, of North East Avenue, was arrested Dec. 24 and charged with driving while intoxicated after a minor collision in the Wendy's Restaurant parking lot on South Delsea Drive. He was cited for reckless driving and refusal to submit a breath test.


text:
Q: For and Foreach Loop Container in SSIS What are differences and similarities between FOR and FOR EACH Loop containers in SQL Server Integration Services?

A: As stated in the documentation.
For Loop Container

The For Loop container defines a repeating control flow in a package.
  The loop implementation is similar to the For looping structure in
  programming languages. In each repeat of the loop, the For Loop
  container evaluates an expression and repeats its workflow until the
  expression evaluates to False.

Foreach Loop Container

The Foreach Loop container defines a repeating control flow in a
  package. The loop implementation is similar to Foreach looping
  structure in programming languages. In a package, looping is enabled
  by using a Foreach enumerator. The Foreach Loop container repeats the
  control flow for each member of a specified enumerator.

In short, a For Loop Container uses a loop condition that is defined by expressions whereas Foreach Loop Container iterates through a collection of objects.
With a Foreach Loop Container, you can perform variable mapping as well, which would be useful for tasks within the loop.

redpajama_set_name: RedPajamaStackExchange


Soon to have a soul brother in the White House? It's hard to say what will change in US relations with Israel. Obama generally said that he opposed what Avigdor Lieberman stood for, but never withheld any financial or political support for it. But right after Trump's election the Israeli government took the opportunity to "legalize" all of the informal seizures of Arab-owned land on the West Bank and in East Jerusalem to build segregated Jews-only housing projects.
These days are long past. We now used the term "United States" in the singular. The US does. The US thinks. The US votes.
In Israel, the very opposite prevails. Years ago, new judges were practically appointed by the old judges, "a friend brings a friend," as popular humor had it.
Later this system was changed a bit — Supreme Court judges are now chosen by a committee of nine, three of which are sitting judges, two others are politicians from the Knesset (one each from the government coalition and from the opposition), two are government ministers and two represent the bar association.
But the decisive point of the law is that any appointment must be made by a majority of seven members –- seven of nine. This means in practice that the three sitting judges on the committee have a veto power on any appointment. So have the politicians. A judge can only be appointed by compromise.
But it worked, right? It got him where he wanted to be, didn't it? It does not compel him to go on in the same vein, now that he has reached his goal. So, forget it.
WELL, I don't know. Nobody does. I tend to believe that he himself does not either.
Apart from their German-Austrian descent, they have nothing in common. Hitler was no billionaire. He was a real man from the people –- an unemployed nobody, who lived for some time in a public shelter.
And that may be bad enough. Maybe.


To register today - click on the following link www.lavote.net.
Nomination papers may be obtained from the City Clerk by appointment during normal business hours from 8:00 a.m. to 5:00 p.m., Monday through Friday. Nomination papers will be issued from July 16, 2018 through August 10, 2018. The last day for a candidate to withdraw nomination papers is August 10, 2018. However, if an incumbent does not file in this period (between July 16, 2018 and August 10, 2018), the filing period is extended to 5:00 p.m., August 15, 2018. Call City Clerk Beth Schott to schedule your appointment.
To hold any elective office within the City of Westlake Village, a person must be a United States citizen, 18 years or older, and must be a registered voter of the City of Westlake Village on the day you request nomination papers. Prior to issuing nomination papers, the registration of all prospective candidates will be verified with the Los Angeles County Registrar of Voters.
The City Clerk will not issue nomination papers if you are not a registered voter.


text:
<!DOCTYPE html>

  <meta charset="utf-8">
  <title>The Engineer's Lab</title>
  <meta name="author" content="Bereket Abraham">
  <meta name="keywords" content="JavaScript, fabricjs" />
  <meta name="description" content="FabricJS codelab" />
  <meta name="viewport" content="width=device-width,initial-scale=1,minimum-scale=1,maximum-scale=1">

  <link rel="icon" href="/static/favicon.ico" type="image/x-icon" >
  <link rel="shortcut icon" href="/static/favicon.ico" type="image/x-icon" >

  <link rel="stylesheet" href="//netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css">
  <link rel="stylesheet" href="//netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap-theme.min.css">
  <link rel="stylesheet" href="/static/personal.css">

Intuitive Color Changing

  <canvas id="table1" width="750" height="400">
    This browser does not support canvas elements.
  </canvas>

Depress the first box for a variable amount of time. The second box should change colors on its own.

redpajama_set_name: RedPajamaGithub


text:
Bankers Can Be Nice Too
Erasmus University's Jose argues that being nice has helped him professionally
By Maria Ahmed
Sat Aug 1 2009
Watch Video Interview
{{video:embed url="http://vimeo.com/5820504"}}
We caught up with Erasmus University's Jose when he was in London en route to the British Open golf in Scotland.
Jose, a native of Bogota, Columbia, just completed a one-year masters degree in Financial Economics.
With several years of commercial banking behind him, Jose is looking for a banking or insurance role in Europe, though he knows the odds are against him in the current climate. He's not averse to returning to Columbia, where he has a job offer, but he'll really miss working with people from around the world.
Impressively, ten hours aboard a coach on British motorways hasn't diminished his ability to field dumb questions.
How Can Sustainable Finance Combat Climate Change?
Master In Data Science Vs Business Analytics: Which Should You Choose?
Space, Sustainability & Future Energy | 5 Exciting New Business Masters For 2022


text:
Brace Hill is a mountain located in the Catskill Mountains of New York east-southeast of Walton, New York. Money Point is located southeast of Brace Hill and Colchester Mountain is located northeast.

References

Mountains of Delaware County, New York
Mountains of New York (state)

redpajama_set_name: RedPajamaWikipedia


text:
2 CalFire Employees Fired, 1 Resigns In Wake Of Homicide Investigation
Filed Under:CAL FIRE, California Department of Forestry and Fire Protection, Homicide, Orville Fleming, Sarah June Douglas
Credit: CBS13
SACRAMENTO, Calif. (AP) – Two state firefighters have been fired and another has resigned after an investigation prompted by the slaying of a CalFire battalion chief's girlfriend.
The California Department of Forestry and Fire Protection began serving employees with disciplinary papers on Wednesday. Spokeswoman Janet Upton tells The Associated Press that 13 other employees face discipline including suspensions, demotions or reductions in pay.
Fifteen of the 16 were instructors at the department's fire academy.
The $2 million investigation began in May after academy instructor Orville Fleming was charged with murder in the death of 26-year-old Sarah June Douglas, a former escort who became his girlfriend.
Fleming's estranged wife told Sacramento County sheriff's investigators and reporters that she had watched a video showing the victim having sex with her husband and other firefighters on fire trucks. Investigators never found a video and discredited the report.
But the resulting California Highway Patrol investigation uncovered problems including unauthorized use of a state vehicle; dishonesty; cheating while an employee was competing for a promotion; and improper conduct while instructors were on state time but after they were done teaching for the day.
"It's kind of ancillary to the original scope of the investigation, but it's things we couldn't overlook," Upton said. "We weren't going to sweep them under the rug. We felt they warranted the action we have taken."
All 16 were put on paid administrative leave last month, just as a new academy class began training for the coming fire season. One has since resigned and the others have been reassigned and no longer are at the academy.
The employees can appeal their discipline within the department and to the State Personnel Board. Upton said she could not disclose their names, specific offenses or other details without jeopardizing their due process rights.
"The CHP investigation brought to light the actions of a few individuals that violate the public trust. We are pursuing the disciplinary process to the furthest extent allowed. This type of behavior will not be tolerated," CalFire director Ken Pimlott said in a statement.
Although the highway patrol conducted the investigation, CalFire must pay for it from its annual budget.
Upton said the original sexual misconduct allegations were so serious that the department had no choice but to seek an outside investigation.
Fleming was arrested following a two-week manhunt after Douglas' body was discovered in the home they shared. Fleming has pleaded not guilty to a murder charge and is scheduled to appear in court in March.
Copyright 2015 The Associated Press.


Dogs are part of the family too, so why not treat them to these special homemade Chicken & Wild Rice Dog Biscuits.
Today is Take Your Dog to Work Day and since I've never experimented with making my own dog biscuits before, today seemed like as good a day as any to try my hand at these Chicken & Wild Rice Biscuits; especially since my two dogs come to work with me every day.
We have two Shih Tzu's, Tebow (age 3 1/2) and Koda (8 months). Tebow loves when I experiment with new recipes during the week and follows me everywhere. The kitchen; where all of the cooking takes place, the living room; where I photograph my food in the large bay window as it has the best natural light in the house.
He likes to just sit and watch and then into the dining room; where I have my sudo-office. He likes to hop up on the chair next to me to keep me company while I'm editing photos and writing posts like these.
Koda is still too little yet, so he just follows Tebow around everywhere he goes and is still trying to learn the ropes. I almost trip over him 10 times a day as he likes to be in the middle of all of the action. Growing up I only had one dog when I was about 4 or 5 years old; then my next dog wasn't until we had Casey; a Llaso Apso who passed away last Fall at the age of 12.
One thing I've learned since then is that dogs are truly part of your family and since I like to create family friendly meals, I thought it was about time I made something special for our four legged "children". I decided to go with these Chicken & Wild Rice Biscuits since the food they eat is also chicken and wild rice. I didn't want to give them ingredients they were unfamiliar with; especially the puppy.
These Chicken & Wild Rice Dog Biscuits were so easy to make and are prepared with natural ingredients like chicken, wild rice, flour, salt, chicken broth and eggs.
Start off by boiling a piece of boneless chicken breast and making wild rice; then allowed them to cool. You can do this a day ahead and keep refrigerated until ready to use.
Add flour, salt, egg and pureed chicken and rice to a bowl and mix well; then roll out the dough on a lightly floured board until 1/2-inch thick.
I purchased these dog bone cookie cutters from Amazon for about $5 (set of 3), but you can use any cookie cutter or glass to cut out the biscuits.
Once you cut out the shapes, bake for about 30 minutes; then allow to harden. You can store these in an airtight container in the freezer for up to 6 months (allow 10-20 minutes for them to defrost before serving).
My dogs L.O.V.E.D. these dog biscuits and I like how easy they were to make at home; not to mention they contain ingredients that you can pronounce. This batch made 32 bones and I usedthe 3.5″ bone cutter, but will probably use a smaller size next time as they were a little big for my two little "boys".
I hope your furry friends like these Chicken & Wild Rice Dog Biscuits as much as ours did. ~Enjoy!
Combine chicken, rice and chicken stock in a mini food processor or blender and pulse until a paste forms.
Add the flour, salt, egg and chicken paste into a bowl and mix well.
Sprinkle a little flour on a flat surface and knead dough until it's no longer sticky; then roll it out with a rolling pin about 1/2-inch thick.
Cut out shapes and place them on a parchment paper lined baking sheet. Mine yielded 32 biscuits using a 3.5" cookie cutter.
Bake for 25-30 minutes until light brown. Remove from oven and allow to cool completely before serving.
Keep these biscuits in an airtight container for up to one week. These biscuits can be stored in an airtight container in the freezer for up to 6 months. Allow to defrost completely (10-20 minutes) before serving.
As we all know, our pets like to get into anything and everything we could possibly drop on the floor.
Chocolate and Cocoa-The darker the chocolate, the more dangerous.
Xylitol (sugar free gum, sugar substitutes)-Can cause liver damage and drop in blood pressure.
There are many foods that are acceptable for dogs to consume. Here are some healthy foods to feed your pets. Tebow is a popcorn fiend. I usually pop mine on top of the stove and as soon as he hears the jar of kernels, he comes running!
For more details, visit PetMedicine.com.
*This post includes links to my affiliate account at Amazon.com where Carrie's Experimental Kitchen earns a few cents on the dollar when readers like yourself purchase the items I recommend. Thank you for supporting Carrie's Experimental Kitchen when you shop at Amazon!
I wood like to try it for my grandson dog and See if she likes the or not if you have more ill like to try them to thank you…???
These sound great but after they are baked do they need to be refrigerated?
I used coconut flour and added peanut butter (since the coconut flour made it a bit too dry) and they're cooking now! I let my pups have some of the dough and they went CRAZY. Thanks for the great recipe!
When I put the recipe into My Fitness Pal, it gives me this information: Total Fat 0g, Saturated Fat 0g, Polyunsaturated Fat 0g, Transfat 0g. I hope this helps!
are these ok for a 6 mos. puppy?
There is nothing in it that should harm the dog as it's all natural ingredients, but I would double check with your veterinarian first to make sure since puppy diets can be more limiting depending on the breed.
I will certainly try this recipe for my shih tzu Mopsey. She is 8 yrs.old and looks just like your puppy.
Awe, I hope Mopsey enjoys the treats!
What temp do you bake the treats?
Would you have a good recommendation besides chicken in this recipe? My little gal is allergic to chicken.
I'm sure you could use any other protein in it's place (beef, seafood, etc) though the consistency may change. I haven't tried it personally with anything other than chicken.


Nobody thinks about trucks in relation to the Olympics. But finally somebody thought about organizing a truck olympics. If you finish the race successfully you will be the one to win the prize! Use the arrows on the keyboard to continue!


text:
Q: How to declare the default indexed property in C++/CLI interface how is it possible to declare the default indexed property in an C++/CLI - Interface.
(Please excuse the repeating, full qualified notation with namespaces because I'm just learning C++/CLI and want to be sure that no acciential mixup of language primitives between C++ and C# happens)
Code is
public interface class ITestWithIndexer
{
    property System::String ^ default[System::Int32];
}

Compiler always throws "error C3289: 'default' a trivial property cannot be indexed".
Where is my error?
PS: in C#, it would simply be
public interface ITestWithIndexer
{
    System.String this[System.Int32] { get; set; }
}

How to translate this into C++/CLI?
Thanks!!

A: In C++/CLI, a 'trivial' property is one where the getter & setter is not declared. With a non-trivial property, the getter & setter are declared explicitly, with syntax that's more like a normal method declaration than C#'s property syntax.
public interface class IThisIsWhatANonIndexedNonTrivialPropertyLooksLike
{
    property String^ MyProperty { String^ get(); void set(String^ value); }
};

Since the trivial syntax isn't allowed for indexed properties, we need to do that for your indexed property.
public interface class ITestWithIndexer
{
    property String^ default[int]
    {
        String^ get(int index);
        void set(int index, String^ value);
    }
};

Here's my test code:
public ref class TestWithIndexer : public ITestWithIndexer
{
public:
    property String^ default[int]
    {
        virtual String^ get(int index)
        {
            Debug::WriteLine("TestWithIndexer::default::get({0}) called", index);
            return index.ToString();
        }
        virtual void set(int index, String^ value)
        {
            Debug::WriteLine("TestWithIndexer::default::set({0}) = {1}", index, value);
        }
    }
};

int main(array<System::String ^> ^args)
{
    ITestWithIndexer^ test = gcnew TestWithIndexer();
    Debug::WriteLine("The indexer returned '" + test[4] + "'");
    test[5] = "foo";
}

Output:
TestWithIndexer::default::get(4) called
The indexer returned '4'
TestWithIndexer::default::set(5) = foo

A: A "trivial property" is one where the compiler can auto-generate a getter and setter, deducing them from the property declaration.  That cannot work for an indexed property, the compiler has no idea what it is supposed to do with the index variable.  You must therefore declare the getter and setter explicitly.  Not unlike the C# declaration, minus the syntax sugar.  Ecma-372, chapter 25.2.2 has an example.  Adapted to your case:
public interface class ITestWithIndexer {
    property String ^ default[int] {
        String^ get(int);
        void set(int, String^);
    }
};


text:
G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}
Mathematical formulation
Fundamental concepts
Equivalence principle
Kepler problem
Gravitational lensing
Frame-dragging
Geodetic effect
Spacetime diagrams
Minkowski spacetime
Formalisms
Linearized gravity
Einstein field equations
Friedmann
Geodesics
Mathisson–Papapetrou–Dixon
Hamilton–Jacobi–Einstein
BSSN
Post-Newtonian
Advanced theory
Kaluza–Klein theory
Reissner–Nordström
Gödel
Kerr–Newman
Kasner
Lemaître–Tolman
Taub-NUT
Milne
Robertson–Walker
pp-wave
van Stockum dust
Poincaré
de Sitter
Reissner
Nordström
Weyl
Zwicky
Lemaître
Bardeen
Raychaudhuri
Hulse
van Stockum
Yau
In physics, gravitational waves are ripples in the curvature of spacetime that propagate as a wave, travelling outward from the source. Predicted in 1916[1][2] by Albert Einstein to exist on the basis of his theory of general relativity,[3][4] gravitational waves theoretically transport energy as gravitational radiation. Sources of detectable gravitational waves could possibly include binary star systems composed of white dwarfs, neutron stars, or black holes. The existence of gravitational waves is a possible consequence of the Lorentz invariance of general relativity since it brings the concept of a limiting speed of propagation of the physical interactions with it. Gravitational waves cannot exist in the Newtonian theory of gravitation, in which physical interactions propagate at infinite speed.
Although gravitational radiation has not been directly detected, there is indirect evidence for its existence. For example, the 1993 Nobel Prize in Physics was awarded for measurements of the Hulse–Taylor binary system that suggests gravitational waves are more than mathematical anomalies. Various gravitational wave detectors exist and on 17 March 2014, astronomers at the Harvard–Smithsonian Center for Astrophysics claimed that they had detected and produced "the first direct image of gravitational waves across the primordial sky" within the cosmic microwave background, providing strong evidence for inflation and the Big Bang.[5][6][7][8] Peer review will be needed before there can be any scientific consensus about these new findings.[9][10] On 19 June 2014, lowered confidence in confirming the cosmic inflation findings was reported;[11][12][13] and on 19 September 2014, a further reduction in confidence was reported.[14][15]
Effects of a passing gravitational wave 2
Sources of gravitational waves 3
Power radiated by orbiting bodies 3.1
Orbital decay from gravitational radiation 3.2
Orbital lifetime limits from gravitational radiation 3.3
Wave amplitudes from the Earth–Sun system 3.4
Radiation from other sources 3.5
Astrophysics and gravitational waves 4
Energy, momentum, and angular momentum carried by gravitational waves 4.1
Detecting gravitational waves 5
Difficulties in detection 5.1
Ground-based interferometers 5.2
Using pulsar timing arrays 5.3
Einstein@Home 5.4
Primordial gravitational waves 5.5
Linear approximation 6.1
Relation to the source 6.2
History of the Universe - gravitational waves are hypothesized to arise from cosmic inflation, a faster-than-light expansion just after the Big Bang (17 March 2014).[5][7][8]
In Einstein's theory of general relativity, gravity is treated as a phenomenon resulting from the curvature of spacetime. This curvature is caused by the presence of mass. Generally, the more mass that is contained within a given volume of space, the greater the curvature of spacetime will be at the boundary of this volume. As objects with mass move around in spacetime, the curvature changes to reflect the changed locations of those objects. In certain circumstances, accelerating objects generate changes in this curvature, which propagate outwards at the speed of light in a wave-like manner. These propagating phenomena are known as gravitational waves.
As a gravitational wave passes a distant observer, that observer will find spacetime distorted by the effects of strain. Distances between free objects increase and decrease rhythmically as the wave passes, at a frequency corresponding to that of the wave. This occurs despite such free objects never being subjected to an unbalanced force. The magnitude of this effect decreases inversely with distance from the source. Inspiralling binary neutron stars are predicted to be a powerful source of gravitational waves as they coalesce, due to the very large acceleration of their masses as they orbit close to one another. However, due to the astronomical distances to these sources the effects when measured on Earth are predicted to be very small, having strains of less than 1 part in 1020. Scientists are attempting to demonstrate the existence of these waves with ever more sensitive detectors. The current most sensitive measurement is about one part in 5×1022 (as of 2012) provided by the LIGO and VIRGO observatories.[16] The lack of detection in these observatories provides an upper limit on the frequency of such powerful sources.[17][18] A space based observatory, the Laser Interferometer Space Antenna, is currently under development by ESA.
Linearly polarised gravitational wave
Gravitational waves should penetrate regions of space that electromagnetic waves cannot. It is hypothesized that they will be able to provide observers on Earth with information about black holes and other exotic objects in the distant Universe. Such systems cannot be observed with more traditional means such as optical telescopes and radio telescopes. In particular, gravitational waves could be of interest to cosmologists as they offer a possible way of observing the very early universe. This is not possible with conventional astronomy, since before recombination the universe was opaque to electromagnetic radiation.[19] Precise measurements of gravitational waves will also allow scientists to test the general theory of relativity more thoroughly.
In principle, gravitational waves could exist at any frequency. However, very low frequency waves would be impossible to detect and there is no credible source for detectable waves of very high frequency. Stephen W. Hawking and Werner Israel list different frequency bands for gravitational waves that could be plausibly detected, ranging from 10−7 Hz up to 1011 Hz.[20]
Effects of a passing gravitational wave
The effect of a plus-polarized gravitational wave on a ring of particles.
The effect of a cross-polarized gravitational wave on a ring of particles.
The effects of a passing gravitational wave can be visualized by imagining a perfectly flat region of spacetime with a group of motionless test particles lying in a plane (the surface of your screen). As a gravitational wave passes through the particles along a line perpendicular to the plane of the particles (i.e. following your line of vision into the screen), the particles will follow the distortion in spacetime, oscillating in a "cruciform" manner, as shown in the animations. The area enclosed by the test particles does not change and there is no motion along the direction of propagation.
The oscillations depicted here in the animation are exaggerated for the purpose of discussion—in reality a gravitational wave has a very small amplitude (as formulated in linearized gravity). However they enable us to visualize the kind of oscillations associated with gravitational waves as produced for example by a pair of masses in a circular orbit. In this case the amplitude of the gravitational wave is a constant, but its plane of polarization changes or rotates at twice the orbital rate and so the time-varying gravitational wave size (or 'periodic spacetime strain') exhibits a variation as shown in the animation.[21] If the orbit is elliptical then the gravitational wave's amplitude also varies with time according Einstein's quadrupole formula.[22]
Like other waves, there are a few useful characteristics describing a gravitational wave:
Amplitude: Usually denoted h, this is the size of the wave — the fraction of stretching or squeezing in the animation. The amplitude shown here is roughly h=0.5 (or 50%). Gravitational waves passing through the Earth are many billions times weaker than this — h \approx 10^{-20}. Note that this is not the quantity that would be analogous to what is usually called the amplitude of an electromagnetic wave, which would be \frac{\mathrm{d}h}{\mathrm{d}t}.
Frequency: Usually denoted f, this is the frequency with which the wave oscillates (1 divided by the amount of time between two successive maximum stretches or squeezes)
Wavelength: Usually denoted \lambda, this is the distance along the wave between points of maximum stretch or squeeze.
Speed: This is the speed at which a point on the wave (for example, a point of maximum stretch or squeeze) travels. For gravitational waves with small amplitudes, this is equal to the speed of light, c.
The speed, wavelength, and frequency of a gravitational wave are related by the equation c = λ f, just like the equation for a light wave. For example, the animations shown here oscillate roughly once every two seconds. This would correspond to a frequency of 0.5 Hz, and a wavelength of about 600,000 km, or 47 times the diameter of the Earth.
In the example just discussed, we actually assume something special about the wave. We have assumed that the wave is linearly polarized, with a "plus" polarization, written h_{\,+}. Polarization of a gravitational wave is just like polarization of a light wave except that the polarizations of a gravitational wave are at 45 degrees, as opposed to 90 degrees. In particular, if we had a "cross"-polarized gravitational wave, h_{\,\times}, the effect on the test particles would be basically the same, but rotated by 45 degrees, as shown in the second animation. Just as with light polarization, the polarizations of gravitational waves may also be expressed in terms of circularly polarized waves. Gravitational waves are polarized because of the nature of their sources. The polarization of a wave depends on the angle from the source, as we will see in the next section.
Sources of gravitational waves
In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like an expanding or contracting sphere) or cylindrically symmetric (like a spinning disk or sphere). A simple example of this principle is provided by the spinning dumbbell. If the dumbbell spins like wheels on an axle, it will not radiate gravitational waves; if it tumbles end over end like two planets orbiting each other, it will radiate gravitational waves. The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. If we imagine an extreme case in which the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off.
Some more detailed examples:
Two objects orbiting each other in a quasi-Keplerian planar orbit (basically, as a planet would orbit the Sun) will radiate.
A spinning non-axisymmetric planetoid — say with a large bump or dimple on the equator — will radiate.
A supernova will radiate except in the unlikely event that the explosion is perfectly symmetric.
An isolated non-spinning solid object moving at a constant velocity will not radiate. This can be regarded as a consequence of the principle of conservation of linear momentum.
A spinning disk will not radiate. This can be regarded as a consequence of the principle of conservation of angular momentum. However, it will show gravitomagnetic effects.
A spherically pulsating spherical star (non-zero monopole moment or mass, but zero quadrupole moment) will not radiate, in agreement with Birkhoff's theorem.
More technically, the third time derivative of the quadrupole moment (or the l-th time derivative of the l-th multipole moment) of an isolated system's stress–energy tensor must be nonzero in order for it to emit gravitational radiation. This is analogous to the changing dipole moment of charge or current necessary for electromagnetic radiation.
Power radiated by orbiting bodies
Two stars of dissimilar mass are in circular orbits. Each revolves about their common center of mass (denoted by the small red cross) in a circle with the larger mass having the smaller orbit.
Two stars of similar mass are in circular orbits about their center of mass
Two stars of similar mass are in highly elliptical orbits about their center of mass
Gravitational waves carry energy away from their sources and, in the case of orbiting bodies, this is associated with an inspiral or decrease in orbit. Imagine for example a simple system of two masses — such as the Earth-Sun system — moving slowly compared to the speed of light in circular orbits. Assume that these two masses orbit each other in a circular orbit in the x-y plane. To a good approximation, the masses follow simple Keplerian orbits. However, such an orbit represents a changing quadrupole moment. That is, the system will give off gravitational waves.
Suppose that the two masses are m_1 and m_2, and they are separated by a distance r. The power given off (radiated) by this system is:
P = \frac{\mathrm{d}E}{\mathrm{d}t} = - \frac{32}{5}\, \frac{G^4}{c^5}\, \frac{(m_1m_2)^2 (m_1+m_2)}{r^5} ,[23]
where G is the gravitational constant, c is the speed of light in vacuum and where the negative sign means that power is being given off by the system, rather than received. For a system like the Sun and Earth, r is about 1.5×1011 m and m_1 and m_2 are about 2×1030 and 6×1024 kg respectively. In this case, the power is about 200 watts. This is truly tiny compared to the total electromagnetic radiation given off by the Sun (roughly 3.86×1026 watts).
In theory, the loss of energy through gravitational radiation could eventually drop the Earth into the Sun. However, the total energy of the Earth orbiting the Sun (kinetic energy + gravitational potential energy) is about 1.14×1036 joules of which only 200 joules per second is lost through gravitational radiation, leading to a decay in the orbit by about 1×10−15 meters per day or roughly the diameter of a proton. At this rate, it would take the Earth approximately 1×1013 times more than the current age of the Universe to spiral onto the Sun. This estimate overlooks the decrease in r over time, but the majority of the time the bodies are far apart and only radiating slowly, so the difference is unimportant in this example.
A more dramatic example of radiated gravitational energy is represented by two solar mass (M☉) neutron stars orbiting at a distance from each other of 1.89×108 m (only 0.63 light-seconds apart). [The Sun is 8 light minutes from the Earth.] Plugging their masses into the above equation shows that the gravitational radiation from them would be 1.38×1028 watts, which is about 100 times more than the Sun's electromagnetic radiation.
Orbital decay from gravitational radiation
Gravitational radiation robs the orbiting bodies of energy. It first circularizes their orbits and then gradually shrinks their radius. As the energy of the orbit is reduced, the distance between the bodies decreases, and they rotate more rapidly. The overall angular momentum is reduced however. This reduction corresponds to the angular momentum carried off by gravitational radiation. The rate of decrease of distance between the bodies versus time is given by:[23]
\frac{\mathrm{d}r}{\mathrm{d}t} = - \frac{64}{5}\, \frac{G^3}{c^5}\, \frac{(m_1m_2)(m_1+m_2)}{r^3}\ ,
where the variables are the same as in the previous equation.
The orbit decays at a rate proportional to the inverse third power of the radius. When the radius has shrunk to half its initial value, it is shrinking eight times faster than before. By Kepler's Third Law, the new rotation rate at this point will be faster by \sqrt{8}=2.828, or nearly three times the previous orbital frequency. As the radius decreases, the power lost to gravitational radiation increases even more. As can be seen from the previous equation, power radiated varies as the inverse fifth power of the radius, or 32 times more in this case.
If we use the previous values for the Sun and the Earth, we find that the Earth's orbit shrinks by 1.1×10−20 meter per second. This is 3.5×10−13 m per year, which is about 1/300 the diameter of a hydrogen atom. The effect of gravitational radiation on the size of the Earth's orbit is negligible over the age of the universe. This is not true for closer orbits.
A more practical example is the orbit of a Sun-like star around a heavy black hole. Our Milky Way has a, potential, 4 million M☉ black hole at its center in Sagittarius A. Such supermassive black holes are being found in the center of almost all galaxies. For this example take a 2 million M☉ black hole with a solar-mass star orbiting it at a radius of 1.89×1010 m (63 light-seconds). The mass of the black hole will be 4×1036 kg and its gravitational radius will be 6×109 m. The orbital period will be 1,000 seconds, or a little under 17 minutes. The solar-mass star will draw closer to the black hole by 7.4 meters per second or 7.4 km per orbit. A collision will not be long in coming.
Assume that a pair of 1 M☉ neutron stars are in circular orbits at a distance of 1.89×108 m (189,000 km). This is a little less than 1/7 the diameter of the Sun or 0.63 light-seconds. Their orbital period would be 1,000 seconds. Substituting the new mass and radius in the above formula gives a rate of orbit decrease of 3.7×10−6 m/s or 3.7 mm per orbit. This is 116 meters per year and is not negligible over cosmic time scales.
Suppose instead that these two neutron stars were orbiting at a distance of 1.89×106 m (1890 km). Their period would be 1 second and their orbital velocity would be about 1/50 of the speed of light. Their orbit would now shrink by 3.7 meters per orbit. A collision is imminent. A runaway loss of energy from the orbit results in an ever more rapid decrease in the distance between the stars. They will eventually merge to form a black hole and cease to radiate gravitational waves. This is referred to as the inspiral.
The above equation can not be applied directly for calculating the lifetime of the orbit, because the rate of change in radius depends on the radius itself, and is thus non-constant with time. The lifetime can be computed by integration of this equation (see next section).
Orbital lifetime limits from gravitational radiation
Orbital lifetime is one of the most important properties of gravitational radiation sources. It determines the average number of binary stars in the universe that are close enough to be detected. Short lifetime binaries are strong sources of gravitational radiation but are few in number. Long lifetime binaries are more plentiful but they are weak sources of gravitational waves. LIGO is most sensitive in the frequency band where two neutron stars are about to merge. This time frame is only a few seconds. It takes luck for the detector to see this blink in time out of a million year orbital lifetime. It is predicted that such a merger will only be seen once per decade or so.
The lifetime of an orbit is given by:[23]
t= \frac{5}{256}\, \frac{c^5}{G^3}\, \frac{r^4}{(m_1m_2)(m_1+m_2)}\ ,
where r is the initial distance between the orbiting bodies. This equation can be derived by integrating the previous equation for the rate of radius decrease. It predicts the time for the radius of the orbit to shrink to zero. As the orbital speed becomes a significant fraction of the speed of light, this equation becomes inaccurate. It is useful for inspirals until the last few milliseconds before the merger of the objects.
Substituting the values for the mass of the Sun and Earth as well as the orbital radius gives a very large lifetime of 3.44×1030 seconds or 1.09×1023 years (that is approximately 1013 times larger than the age of the universe). The actual figure would be slightly less than that. The Earth will break apart from tidal forces if it orbits closer than a few radii from the Sun. This would form a ring around the Sun and instantly stop the emission of gravitational waves.
If we use a 2 million M☉ black hole with a solar mass star orbiting it at 1.89×1010 meters, we get a lifetime of 6.50×108 seconds or 20.7 years.
Assume that a pair of solar mass neutron stars with a diameter of 10 kilometers are in circular orbits at a distance of 1.89×108 m (189,000 km). Their lifetime is 1.30×1013 seconds or about 414,000 years. Their orbital period will be 1,000 seconds and it could be observed by LISA if they were not too far away. A far greater number of white dwarf binaries exist with orbital periods in this range. White dwarf binaries have masses on the order of our Sun and diameters on the order of our Earth. They cannot get much closer together than 10,000 km before they will merge and cease to radiate gravitational waves. This results in the creation of either a neutron star or a black hole. Until then, their gravitational radiation will be comparable to that of a neutron star binary. LISA is the only gravitational wave experiment that is likely to succeed in detecting such types of binaries.
If the orbit of a neutron star binary has decayed to 1.89×106m (1890 km), its remaining lifetime is 130,000 seconds or about 36 hours. The orbital frequency will vary from 1 revolution per second at the start and 918 revolutions per second when the orbit has shrunk to 20 km at merger. The gravitational radiation emitted will be at twice the orbital frequency. Just before merger, the inspiral can be observed by LIGO if the binary is close enough. LIGO has only a few minutes to observe this merger out of a total orbital lifetime that may have been billions of years. The chance of success with LIGO as initially constructed is quite low despite the large number of such mergers occurring in the universe, because the sensitivity of the instrument does not 'reach' out to enough systems to see events frequently. No mergers have been seen in the few years that initial LIGO has been in operation, and it is thought that a merger should be seen about once per several tens of years of observing time with initial LIGO.[24] The upgraded Advanced LIGO detector, with a ten times greater sensitivity, 'reaches' out 10 times further—encompassing a volume 1000 times greater, and seeing 1000 times as many candidate sources. Thus, the expectation is that detections will be made at the rate of tens per year.
Wave amplitudes from the Earth–Sun system
We can also think in terms of the amplitude of the wave from a system in circular orbits. Let \theta be the angle between the perpendicular to the plane of the orbit and the line of sight of the observer. Suppose that an observer is outside the system at a distance R from its center of mass. If R is much greater than a wavelength, the two polarizations of the wave will be
h_{+} = -\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{2 m_1 m_2}{r} (1+\cos^2\theta) \cos\left[2\omega(t - R)\right],
h_{\times} = -\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{4 m_1 m_2}{r}\, (\cos{\theta})\sin \left[2\omega(t-R)\right].
Here, we use the constant angular velocity of a circular orbit in Newtonian physics:
\omega=\sqrt{G(m_1+m_2)/r^3}.
For example, if the observer is in the x-y plane then \theta=\pi/2, and \cos (\theta) = 0, so the h_\times polarization is always zero. We also see that the frequency of the wave given off is twice the rotation frequency. If we put in numbers for the Earth-Sun system, we find:
h_{+} =-\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{4m_1 m_2}{r} = -\frac{1}{R}\, 1.7\times 10^{-10}\, \mathrm{m}.
In this case, the minimum distance to find waves is R ≈ 1 light-year, so typical amplitudes will be h ≈ 10−26. That is, a ring of particles would stretch or squeeze by just one part in 1026. This is well under the detectability limit of all conceivable detectors.
Radiation from other sources
Although the waves from the Earth-Sun system are minuscule, astronomers can point to other sources for which the radiation should be substantial. One important example is the Hulse-Taylor binary — a pair of stars, one of which is a pulsar.[25] The characteristics of their orbit can be deduced from the Doppler shifting of radio signals given off by the pulsar. Each of the stars are about 1.4 M☉ and the size of their orbit is about 1/75 of the Earth-Sun orbit. This means the distance between the two stars is just a few times larger than the diameter of our own Sun. The combination of greater masses and smaller separation means that the energy given off by the Hulse-Taylor binary will be far greater than the energy given off by the Earth-Sun system — roughly 1022 times as much.
The information about the orbit can be used to predict just how much energy (and angular momentum) should be given off in the form of gravitational waves. As the energy is carried off, the stars should draw closer to each other. This effect is called an inspiral, and it can be observed in the pulsar's signals. The measurements on the Hulse-Taylor system have been carried out over more than 30 years. It has been shown that the gravitational radiation predicted by general relativity allows these observations to be matched within 0.2 percent. In 1993, Russell Hulse and Joe Taylor were awarded the Nobel Prize in Physics for this work, which was the first indirect evidence for gravitational waves. Unfortunately, the orbital lifetime of this binary system before merger is about 1.84 billion years. This is a substantial fraction of the age of the universe.
Inspirals are very important sources of gravitational waves. Any time two compact objects (white dwarfs, neutron stars, or black holes) are in close orbits, they send out intense gravitational waves. As they spiral closer to each other, these waves become more intense. At some point they should become so intense that direct detection by their effect on objects on Earth or in space is possible. This direct detection is the goal of several large scale experiments.[26]
The only difficulty is that most systems like the Hulse-Taylor binary are so far away. The amplitude of waves given off by the Hulse-Taylor binary as seen on Earth would be roughly h ≈ 10−26. There are some sources, however, that astrophysicists expect to find with much larger amplitudes of h ≈ 10−20. At least eight other binary pulsars have been discovered.[27]
Astrophysics and gravitational waves
List of unsolved problems in physics
Can gravitational waves be detected experimentally?
Two-dimensional representation of gravitational waves generated by two neutron stars orbiting each other.
During the past century, astronomy has been revolutionized by the use of new methods for observing the universe. Astronomical observations were originally made using visible light. Galileo Galilei pioneered the use of telescopes to enhance these observations. However, visible light is only a small portion of the electromagnetic spectrum, and not all objects in the distant universe shine strongly in this particular band. More useful information may be found, for example, in radio wavelengths. Using radio telescopes, astronomers have found pulsars, quasars, and other extreme objects that push the limits of our understanding of physics. Observations in the microwave band have opened our eyes to the faint imprints of the Big Bang, a discovery Stephen Hawking called the "greatest discovery of the century, if not all time". Similar advances in observations using gamma rays, x-rays, ultraviolet light, and infrared light have also brought new insights to astronomy. As each of these regions of the spectrum has opened, new discoveries have been made that could not have been made otherwise. Astronomers hope that the same holds true of gravitational waves.
Gravitational waves have two important and unique properties. First, there is no need for any type of matter to be present nearby in order for the waves to be generated by a binary system of uncharged black holes, which would emit no electromagnetic radiation. Second, gravitational waves can pass through any intervening matter without being scattered significantly. Whereas light from distant stars may be blocked out by interstellar dust, for example, gravitational waves will pass through essentially unimpeded. These two features allow gravitational waves to carry information about astronomical phenomena never before observed by humans.
The sources of gravitational waves described above are in the low-frequency end of the gravitational-wave spectrum (10−7 to 105 Hz). An astrophysical source at the high-frequency end of the gravitational-wave spectrum (above 105 Hz and probably 1010 Hz) generates relic gravitational waves that are theorized to be faint imprints of the Big Bang like the cosmic microwave background (see gravitational wave background).[28] At these high frequencies it is potentially possible that the sources may be "man made"[20] that is, gravitational waves generated and detected in the laboratory.[29][30]
Energy, momentum, and angular momentum carried by gravitational waves
Waves familiar from other areas of physics such as water waves, sound waves, and electromagnetic waves are able to carry energy, momentum, and angular momentum. By carrying these away from a source, waves are able to rob that source of its energy as well as its linear and angular momentum. Gravitational waves perform the same function. Thus, for example, a binary system loses angular momentum as the two orbiting objects spiral towards each other—the angular momentum is radiated away by gravitational waves.
The waves can also carry off linear momentum, a possibility that has some interesting implications for astrophysics.[31] After two supermassive black holes coalesce, emission of linear momentum can produce a "kick" with amplitude as large as 4000 km/s. This is fast enough to eject the coalesced black hole completely from its host galaxy. Even if the kick is too small to eject the black hole completely, it can remove it temporarily from the nucleus of the galaxy, after which it will oscillate about the center, eventually coming to rest.[32] A kicked black hole can also carry a star cluster with it, forming a hyper-compact stellar system.[33] Or it may carry gas, allowing the recoiling black hole to appear temporarily as a "naked quasar". The quasar SDSS J092712.65+294344.0 is believed to contain a recoiling supermassive black hole.[34]
Detecting gravitational waves
Difficulties in detection
Evidence of gravitational waves in the infant universe may have been uncovered by the BICEP2 radio telescope. The microscopic examination of the focal plane of the BICEP2 detector is shown here.[5][6][7][8][10]
Gravitational waves are not easily detectable. This knowledge gap is primarily due to the massive presence of noise in the low frequencies where antennas currently operate. Gravitational waves are expected to have frequencies 10^{-16}\, \mathrm{Hz}<10^4\, \mathrm{Hz}.[35]
Ground-based interferometers
Though the Hulse-Taylor observations were very important, they give only indirect evidence for gravitational waves. A more conclusive observation would be a direct measurement of the effect of a passing gravitational wave, which could also provide more information about the system that generated it. Any such direct detection is complicated by the extraordinarily small effect the waves would produce on a detector. The amplitude of a spherical wave will fall off as the inverse of the distance from the source (the 1/R term in the formulas for h above). Thus, even waves from extreme systems like merging binary black holes die out to very small amplitude by the time they reach the Earth. Astrophysicists expect that some gravitational waves passing the Earth may be as large as h ≈ 10−20, but generally no bigger.
A simple device theorised to detect the expected wave motion is called a Weber bar — a large, solid bar of metal isolated from outside vibrations. This type of instrument was the first type of gravitational wave detector. Strains in space due to an incident gravitational wave excite the bar's resonant frequency and could thus be amplified to detectable levels. Conceivably, a nearby supernova might be strong enough to be seen without resonant amplification. With this instrument, Joseph Weber claimed to have detected daily signals of gravitational waves. His results, however, were contested in 1974 by physicists Richard Garwin and David Douglass. Modern forms of the Weber bar are still operated, cryogenically cooled, with superconducting quantum interference devices to detect vibration. Weber bars are not sensitive enough to detect anything but extremely powerful gravitational waves.[36]
MiniGRAIL is a spherical gravitational wave antenna using this principle. It is based at Leiden University, consisting of an exactingly machined 1150 kg sphere cryogenically cooled to 20 mK.[37] The spherical configuration allows for equal sensitivity in all directions, and is somewhat experimentally simpler than larger linear devices requiring high vacuum. Events are detected by measuring deformation of the detector sphere. MiniGRAIL is highly sensitive in the 2–4 kHz range, suitable for detecting gravitational waves from rotating neutron star instabilities or small black hole mergers.[38]
A schematic diagram of a laser interferometer
A more sensitive class of detector uses laser interferometry to measure gravitational-wave induced motion between separated 'free' masses.[39] This allows the masses to be separated by large distances (increasing the signal size); a further advantage is that it is sensitive to a wide range of frequencies (not just those near a resonance as is the case for Weber bars). Ground-based interferometers are now operational. Currently, the most sensitive is LIGO — the Laser Interferometer Gravitational Wave Observatory. LIGO has three detectors: one in Livingston, Louisiana; the other two (in the same vacuum tubes) at the Hanford site in Richland, Washington. Each consists of two light storage arms that are 2 to 4 kilometers in length. These are at 90 degree angles to each other, with the light passing through 1m diameter vacuum tubes running the entire 4 kilometers. A passing gravitational wave will slightly stretch one arm as it shortens the other. This is precisely the motion to which an interferometer is most sensitive.
Even with such long arms, the strongest gravitational waves will only change the distance between the ends of the arms by at most roughly 10−18 meters. LIGO should be able to detect gravitational waves as small as h \sim 5\times 10^{-20}. Upgrades to LIGO and other detectors such as Virgo, GEO 600, and TAMA 300 should increase the sensitivity still further; the next generation of instruments (Advanced LIGO and Advanced Virgo) will be more than ten times more sensitive. Another highly sensitive interferometer (LCGT) is currently in the design phase. A key point is that a tenfold increase in sensitivity (radius of 'reach') increases the volume of space accessible to the instrument by one thousand times. This increases the rate at which detectable signals should be seen from one per tens of years of observation, to tens per year.[24]
Interferometric detectors are limited at high frequencies by shot noise, which occurs because the lasers produce photons randomly; one analogy is to rainfall—the rate of rainfall, like the laser intensity, is measurable, but the raindrops, like photons, fall at random times, causing fluctuations around the average value. This leads to noise at the output of the detector, much like radio static. In addition, for sufficiently high laser power, the random momentum transferred to the test masses by the laser photons shakes the mirrors, masking signals at low frequencies. Thermal noise (e.g., Brownian motion) is another limit to sensitivity. In addition to these 'stationary' (constant) noise sources, all ground-based detectors are also limited at low frequencies by seismic noise and other forms of environmental vibration, and other 'non-stationary' noise sources; creaks in mechanical structures, lightning or other large electrical disturbances, etc. may also create noise masking an event or may even imitate an event. All these must be taken into account and excluded by analysis before a detection may be considered a true gravitational wave event.
Space-based interferometers, such as LISA and DECIGO, are also being developed. LISA's design calls for three test masses forming an equilateral triangle, with lasers from each spacecraft to each other spacecraft forming two independent interferometers. LISA is planned to occupy a solar orbit trailing the Earth, with each arm of the triangle being five million kilometers. This puts the detector in an excellent vacuum far from Earth-based sources of noise, though it will still be susceptible to shot noise, as well as artifacts caused by cosmic rays and solar wind.
There are currently two detectors focusing on detection at the higher end of the gravitational wave spectrum (10−7 to 105 Hz): one at University of Birmingham, England, and the other at INFN Genoa, Italy. A third is under development at Chongqing University, China. The Birmingham detector measures changes in the polarization state of a microwave beam circulating in a closed loop about one meter across. Two have been fabricated and they are currently expected to be sensitive to periodic spacetime strains of h\sim{2 \times 10^{-13}/\sqrt{\mathrm{Hz}}} , given as an amplitude spectral density. The INFN Genoa detector is a resonant antenna consisting of two coupled spherical superconducting harmonic oscillators a few centimeters in diameter. The oscillators are designed to have (when uncoupled) almost equal resonant frequencies. The system is currently expected to have a sensitivity to periodic spacetime strains of h\sim{2 \times 10^{-17}/\sqrt{\mathrm{Hz}}} , with an expectation to reach a sensitivity of h\sim{2 \times 10^{-20}/\sqrt{\mathrm{Hz}}} . The Chongqing University detector is planned to detect relic high-frequency gravitational waves with the predicted typical parameters ?g ~1010 Hz (10 GHz) and h ~10−30-10−31.
Using pulsar timing arrays
Pulsars are rapidly rotating stars. A pulsar emits beams of radio waves that, like lighthouse beams, sweep through the sky as the pulsar rotates. The signal from a pulsar can be detected by radio telescopes as a series of regularly spaced pulses, essentially like the ticks of a clock. Gravitational waves affect the time it takes the pulses to travel from the pulsar to a telescope on Earth. A pulsar timing array uses millisecond pulsars to seek out perturbations due to gravitational waves in measurements of pulse arrival times at a telescope, in other words, to look for deviations in the clock ticks. In particular, pulsar timing arrays can search for a distinct pattern of correlation and anti-correlation between the signals over an array of different pulsars (resulting in the name "pulsar timing array"). Although pulsar pulses travel through space for hundreds or thousands of years to reach us, pulsar timing arrays are sensitive to perturbations in their travel time of much less than a millionth of a second. Globally there are three active pulsar timing array projects. The North American Nanohertz Gravitational Wave Observatory uses data collected by the Arecibo Radio Telescope and Green Bank Telescope. The Parkes Pulsar Timing Array at the Parkes radio-telescope has been collecting data since March 2005. The European Pulsar Timing Array uses data from the four largest telescopes in Europe: the Lovell Telescope, the Westerbork Synthesis Radio Telescope, the Effelsberg Telescope and the Nancay Radio Telescope. (Upon completion the Sardinia Radio Telescope will be added to the EPTA also.) These three projects have begun collaborating under the title of the International Pulsar Timing Array project.
In some sense, the easiest signals to detect should be constant sources. Supernovae and neutron star or black hole mergers should have larger amplitudes and be more interesting, but the waves generated will be more complicated. The waves given off by a spinning, aspherical neutron star would be "monochromatic"—like a pure tone in acoustics. It would not change very much in amplitude or frequency.
The Einstein@Home project is a distributed computing project similar to SETI@home intended to detect this type of simple gravitational wave. By taking data from LIGO and GEO, and sending it out in little pieces to thousands of volunteers for parallel analysis on their home computers, Einstein@Home can sift through the data far more quickly than would be possible otherwise.[40]
Primordial gravitational waves
Primordial gravitational waves are gravitational waves observed in the cosmic microwave background. They were allegedly detected by the BICEP2 instrument, an announcement being made on 17 March 2014.
Einstein's equations form the fundamental law of general relativity. The curvature of spacetime can be expressed mathematically using the metric tensor — denoted g_{\mu \nu} \,. The metric holds information regarding how distances are measured in the space under consideration. Because the propagation of gravitational waves through space and time change distances, we will need to use this to find the solution to the wave equation.
Spacetime curvature is also expressed with respect to a covariant derivative, \nabla \,, in the form of the Einstein tensor, G_{\mu \nu}. This curvature is related to the stress–energy tensor, T_{\mu\nu}, by the key equation
G_{\mu \nu} = \frac{8\pi G_N}{c^4} T_{\mu \nu},
where G_N is Newton's gravitational constant, and c is the speed of light. We assume geometrized units, so G_N = 1 = c.
With some simple assumptions, Einstein's equations can be rewritten to show explicitly that they are wave equations. To begin with, we adopt some coordinate system, like (t,r,\theta,\phi) \,. We define the "flat-space metric" \eta_{\mu\nu} \, to be the quantity that — in this coordinate system — has the components we would expect for the flat space metric. For example, in these spherical coordinates, we have
\eta_{\mu \nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2\theta \end{bmatrix}.
This flat-space metric has no physical significance; it is a purely mathematical device necessary for the analysis. Tensor indices are raised and lowered using this "flat-space metric".
Now, we can also think of the physical metric g_{\mu \nu} as a matrix, and find its determinant, \det g. Finally, we define a quantity
\bar{h}^{\alpha \beta} \equiv \eta^{\alpha \beta} - \sqrt{|\det g|} g^{\alpha \beta} \, .
This is the crucial field, which will represent the radiation. It is possible (at least in an asymptotically flat spacetime) to choose the coordinates in such a way that this quantity satisfies the "de Donder" gauge conditions (conditions on the coordinates):
\nabla_\beta\, \bar{h}^{\alpha \beta} = 0,
where \nabla represents the flat-space derivative operator. These equations say that the divergence of the field is zero. The linear Einstein equations can now be written[41] as
\Box \bar{h}^{\alpha \beta} = -16\pi \tau^{\alpha \beta} \, ,
where \Box = -\partial_t^2 + \Delta \, represents the flat-space d'Alembertian operator, and \tau^{\alpha \beta} \, represents the stress–energy tensor plus quadratic terms involving \bar{h}^{\alpha \beta} \,. This is just a wave equation for the field with a source, despite the fact that the source involves terms quadratic in the field itself. That is, it can be shown that solutions to this equation are waves traveling with velocity 1 in these coordinates.
Linear approximation
The equations above are valid everywhere — near a black hole, for instance. However, because of the complicated source term, the solution is generally too difficult to find analytically. We can often assume that space is nearly flat, so the metric is nearly equal to the \eta^{\alpha \beta} \, tensor. In this case, we can neglect terms quadratic in \bar{h}^{\alpha \beta} \,, which means that the \tau^{\alpha \beta} \, field reduces to the usual stress–energy tensor T^{\alpha \beta} \,. That is, Einstein's equations become
\Box \bar{h}^{\alpha \beta} = -16\pi T^{\alpha \beta} \, .
If we are interested in the field far from a source, however, we can treat the source as a point source; everywhere else, the stress–energy tensor would be zero, so
\Box \bar{h}^{\alpha \beta} = 0 \, .
Now, this is the usual homogeneous wave equation — one for each component of \bar{h}^{\alpha \beta} \,. Solutions to this equation are well known. For a wave moving away from a point source, the radiated part (meaning the part that dies off as 1/r \, far from the source) can always be written in the form A(t-r,\theta,\phi)/r \,, where A \, is just some function. It can be shown[42] that — to a linear approximation — it is always possible to make the field traceless. Now, if we further assume that the source is positioned at r=0, the general solution to the wave equation in spherical coordinates is
\begin{array}{lcl} \bar{h}^{\alpha \beta} & = & \frac{1}{r}\, \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & A_{+}(t-r,\theta,\phi) & A_{\times}(t-r,\theta,\phi) \\ 0 & 0 & A_{\times}(t-r,\theta,\phi) & -A_{+}(t-r,\theta,\phi) \end{bmatrix} \\ \\ & \equiv & \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & h_{+}(t-r,r,\theta,\phi) & h_{\times}(t-r,r,\theta,\phi) \\ 0 & 0 & h_{\times}(t-r,r,\theta,\phi) & -h_{+}(t-r,r,\theta,\phi) \end{bmatrix} \end{array} \,
where we now see the origin of the two polarizations.
Relation to the source
If we know the details of a source — for instance, the parameters of the orbit of a binary — we can relate the source's motion to the gravitational radiation observed far away. With the relation
we can write the solution in terms of the tensorial Green's function for the d'Alembertian operator:[41]
\bar{h}^{\alpha \beta} (t,\vec{x}) = -16\pi \int\, G^{\alpha \beta}_{\gamma \delta} (t,\vec{x};t',\vec{x}')\, \tau^{\gamma \delta}(t',\vec{x}')\, \mathrm{d}t'\, \mathrm{d}^3x' .
Though it is possible to expand the Green's function in tensor spherical harmonics, it is easier to simply use the form
G^{\alpha \beta}_{\gamma \delta} (t,\vec{x};t',\vec{x}') = \frac{1}{4\pi} \delta_{\gamma}^\alpha\, \delta_{\delta}^\beta\, \frac{\delta(t\pm|\vec{x}-\vec{x}'|-t')} {|\vec{x}-\vec{x}'|} ,
where the positive and negative signs correspond to ingoing and outgoing solutions, respectively. Generally, we are interested in the outgoing solutions, so
\bar{h}^{\alpha \beta} (t,\vec{x}) = -4 \int\, \frac{\tau^{\alpha \beta}(t-|\vec{x}-\vec{x}'|,\vec{x}')}{|\vec{x}-\vec{x}'|}\, \mathrm{d}^3x' .
If the source is confined to a small region very far away, to an excellent approximation we have:
\bar{h}^{\alpha \beta} (t,\vec{x}) \approx -\frac{4}{r}\, \int\, \tau^{\alpha \beta}(t-r,\vec{x}')\, \mathrm{d}^3x' ,
where r=|\vec{x}| .
Now, because we will eventually only be interested in the spatial components of this equation (time components can be set to zero with a coordinate transformation), and we are integrating this quantity — presumably over a region of which there is no boundary — we can put this in a different form. Ignoring divergences with the help of Stokes' theorem and an empty boundary, we can see that
\int\, \tau^{i j}(t-r,\vec{x}')\, \mathrm{d}^3x' = \int\, x'^i x'^j \nabla_k \nabla_l \tau^{k l} (t-r,\vec{x}')\, \mathrm{d}^3x' ,
Inserting this into the above equation, we arrive at
\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \int\, x'^i x'^j \nabla_k \nabla_l \tau^{k l} (t-r,\vec{x}')\, \mathrm{d}^3x' ,
Finally, because we have chosen to work in coordinates for which \nabla_\beta\, \bar{h}^{\alpha \beta} = 0, we know that \nabla_\beta\, \tau^{\alpha \beta} = 0. With a few simple manipulations, we can use this to prove that
\nabla_0 \nabla_0 \tau^{00} = \nabla_j \nabla_k \tau^{jk} .
With this relation, the expression for the radiated field is
\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \frac{\mathrm{d}^2}{\mathrm{d}t^2}\, \int\, x'^i x'^j \tau^{0 0} (t-r,\vec{x}')\, \mathrm{d}^3x' .
In the linear case, \tau^{00} = \rho, the density of mass-energy.
To a very good approximation, the density of a simple binary can be described by a pair of delta-functions, which eliminates the integral. Explicitly, if the masses of the two objects are M_1 and M_2, and the positions are \vec{x}_1 and \vec{x}_2, then
\rho(t-r,\vec{x}') = M_1 \delta^3(\vec{x}'-\vec{x}_1(t-r)) + M_2 \delta^3(\vec{x}'-\vec{x}_2(t-r)) .
We can use this expression to do the integral above:
\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \frac{\mathrm{d}^2}{\mathrm{d}t^2}\, \left\{ M_1 x_1^i(t-r) x_1^j(t-r) + M_2 x_2^i(t-r) x_2^j(t-r) \right\} .
Using mass-centered coordinates, and assuming a circular binary, this is
\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \frac{M_1 M_2}{R}\, n^i(t-r) n^j(t-r) ,
where \vec{n} = \vec{x}_1 / |\vec{x}_1|. Plugging in the known values of \vec{x}_1(t-r), we obtain the expressions given above for the radiation from a simple binary.
Big Bang Observer (BBO), proposed successor to LISA
DECIGO "Deci-hertz Interferometer Gravitational wave Observatory", the planned laser interferometric detector in space
Gravitational field
Gravitomagnetism
Gravitational wave astronomy
Hawking radiation, for gravitationally induced electromagnetic radiation from black holes
HM Cancri
LIGO, VIRGO, GEO 600, and TAMA 300 — Gravitational wave detectors
Linearised Einstein field equations
LISA the proposed Laser Interferometer Space Antenna
Peres metric
pp-wave spacetime, for an important class of exact solutions modelling gravitational radiation
Spin-flip, a consequence of gravitational wave emission from binary supermassive black holes
Sticky bead argument, for a physical way to see that gravitational radiation should carry energy
Tidal force
^ Einstein, A (June 1916). "Näherungsweise Integration der Feldgleichungen der Gravitation".
^ Einstein, A (1918). "Über Gravitationswellen". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 154-167.
^ Finley, Dave. "Einstein's gravity theory passes toughest test yet: Bizarre binary star system pushes study of relativity to new limits.". Phys.Org.
^ The Detection of Gravitational Waves using LIGO, B. Barish
^ a b c Staff (17 March 2014). "BICEP2 2014 Results Release".
^ a b ""First Direct Evidence of Cosmic Inflation"". http://www.cfa.harvard.edu. Harvard-Smithsonian Center for Astrophysics. 17 March 2014. Retrieved 17 March 2014.
^ a b c Clavin, Whitney (17 March 2014). "NASA Technology Views Birth of the Universe".
^ a b c
^ Cosmic inflation: 'Spectacular' discovery hailedAstronomers discover echoes from expansion after Big BangGravitational Waves: The Big Bang's Smoking GunGravitational Waves from Big Bang Detected
^ Amos, Jonathan (19 June 2014). "Cosmic inflation: Confidence lowered for Big Bang signal".
^ Ade, P.A.R. et al (BICEP2 Collaboration) (19 June 2014). "Detection of B-Mode Polarization at Degree Angular Scales by BICEP2" (PDF).
^ Planck Collaboration Team (19 September 2014). "Planck intermediate results. XXX. The angular power spectrum of polarized dust emission at intermediate and high Galactic latitudes".
^ LIGO Scientific Collaboration; Virgo Collaboration (2012). "Search for Gravitational Waves from Low Mass Compact Binary Coalescence in LIGO's Sixth Science Run and Virgo's Science Runs 2 and 3". Physical Review D 85: 082002.
^ LIGO Scientific Collaboration; Virgo Collaboration (2012). "All-sky search for gravitational-wave bursts in the second joint LIGO-Virgo run". Physical Review D 85: 122007.
^ LIGO Scientific Collaboration; Virgo Collaboration (2013). "Search for gravitational waves from binary black hole inspiral, merger, and ringdown in LIGO-Virgo data from 2009-2010". Physical Review D 87: 022002.
^ Krauss, LM; Dodelson, S; Meyer, S (2010). "Primordial Gravitational Waves and Cosmology". Science 328 (5981): 989–992.
^ a b Hawking, S. W. and Israel, W., General Relativity: An Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979, 98.
^ Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields. Fourth Revised English Edition, Pergamon Press., 1975, 356–357.
^ Einstein, A., "Über Gravitationswellen." Sitzungsberichte, Preussische Akademie der Wissenschaften, 154, (1918).
^ a b c Gravitational Radiation
^ a b LIGO Scientific Collaboration; Virgo Collaboration (2010). "Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectors". Classical and Quantum Gravity 27: 17300.
^ Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis
^ Crashing Black Holes
^ Binary and Millisecond Pulsars
^ L. P. Grishchuk (1976), "Primordial Gravitons and the Possibility of Their Observation", Sov. Phys. JETP Lett. 23, p. 293.
^ Braginsky, V. B., Rudenko and Valentin, N. Section 7: "Generation of gravitational waves in the laboratory", Physics Report (Review section of Physics Letters), 46, No. 5. 165–200, (1978).
^ Li, Fangyu, Baker, R. M L, Jr., and Woods, R. C., "Piezoelectric-Crystal-Resonator High-Frequency Gravitational Wave Generation and Synchro-Resonance Detection", in the proceedings of Space Technology and Applications International Forum (STAIF-2006), edited by M.S. El-Genk, American Institute of Physics Conference Proceedings, Melville NY 813: 2006.
^ Gualandris, A.;
^ Komossa, S.; Zhou, H.; Lu, H. (May 2008). "A Recoiling Supermassive Black Hole in the Quasar SDSS J092712.65+294344.0?". The Astrophysical Journal 678 (2): L81–L84.
^ Thorne, Kip S. (1995). "Gravitational Waves". Cornell University Library.
^ For a review of early experiments using Weber bars, see Levine, J. (April 2004). "Early Gravity-Wave Detection Experiments, 1960–1975". Physics in Perspective (Birkhäuser Basel) 6 (1): 42–75.
^ Gravitational Radiation Antenna In Leiden
^ de Waard, Arlette; Luciano Gottardi; Giorgio Frossati (July 2000). "Spherical Gravitational Wave Detectors: cooling and quality factor of a small CuAl6% sphere" (PDF). Marcel Grossmann meeting on General Relativity. Rome, Italy: World Scientific Publishing Co. Pte. Ltd. (published December 2002). pp. 1899–1901.
^ The idea of using laser interferometry for gravitational wave detection was first mentioned by Gerstenstein and Pustovoit 1963 Sov. Phys.–JETP 16 433. Weber mentioned it in an unpublished laboratory notebook. Rainer Weiss first described in detail a practical solution with an analysis of realistic limitations to the technique in R. Weiss (1972). "Electromagetically Coupled Broadband Gravitational Antenna". Quarterly Progress Report, Research Laboratory of Electronics, MIT 105: 54.
^ Einstein@Home
^ a b Thorne, Kip (April 1980). "Multipole expansions of gravitational radiation". Reviews of Modern Physics 52 (2): 299.
^ C. W. Misner, K. S. Thorne, and J. A. Wheeler (1973). Gravitation. W. H. Freeman and Co.
Chakrabarty, Indrajit, "Gravitational Waves: An Introduction". arXiv:physics/9908041 v1, Aug 21, 1999.
Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields (Pergamon Press),(1987).
Will, Clifford M., The Confrontation between General Relativity and Experiment. Living Rev. Relativity 9 (2006) 3.
Peter Saulson, "Fundamentals of Interferometric Gravitational Wave Detectors", World Scientific, 1994.
J. Bicak, W.N. Rudienko, "Gravitacionnyje wolny w OTO i probliema ich obnarużenija", Izdatielstwo Moskovskovo Universitieta, 1987.
A. Kułak, "Electromagnetic Detectors of Gravitational Radiation", PhD Thesis, Cracow 1980 (In Polish).
P. Tatrocki, "On intuitive description of graviton detector", www.philica.com .
P. Tatrocki, "Can the LIGO, VIRGO, GEO600, AIGO, TAMA, LISA detectors really detect?", www.philica.com .
Berry, Michael, Principles of cosmology and gravitation (Adam Hilger, Philadelphia, 1989). ISBN 0-85274-037-9
Collins, Harry, Gravity's Shadow: the search for gravitational waves, University of Chicago Press, 2004.
P. J. E. Peebles, Principles of Physical Cosmology (Princeton University Press, Princeton, 1993). ISBN 0-691-01933-9.
Wheeler, John Archibald and Ciufolini, Ignazio, Gravitation and Inertia (Princeton University Press, Princeton, 1995). ISBN 0-691-03323-4.
Woolf, Harry, ed., Some Strangeness in the Proportion (Addison–Wesley, Reading, Massachusetts, 1980). ISBN 0-201-09924-1.
Gravitational waves at Encyclopædia Britannica
Gravitational Waves on In Our Time at the BBC. (listen now)
The LISA Brownbag – Selection of the most significant e-prints related to LISA science
Astroparticle.org. To know everything about astroparticle physics, including gravitational waves
Caltech's Physics 237-2002 Gravitational Waves by Kip Thorne Video plus notes: Graduate level but does not assume knowledge of General Relativity, Tensor Analysis, or Differential Geometry; Part 1: Theory (10 lectures), Part 2: Detection (9 lectures)
www.astronomycast.com January 14, 2008 Episode 71: Gravitational Waves
Laser Interferometer Gravitational Wave Observatory. LIGO Laboratory, operated by the California Institute of Technology and the Massachusetts Institute of Technology
The LIGO Scientific Collaboration
Einstein's Messengers – The LIGO Movie by NSF
Home page for Einstein@Home project, a distributed computing project processing raw data from LIGO Laboratory, searching for gravitational waves
The National Center for Supercomputing Applications – a numerical relativity group
Caltech Relativity Tutorial – A basic introduction to gravitational waves, and astrophysical systems giving off gravitational waves
Resource Letter GrW-1: Gravitational waves – a list of books, journals and web resources compiled by Joan Centrella for research into gravitational waves
Mathematical and Physical Perspectives on Gravitational Radiation – written by B F Schutz of the Max Planck Institute explaining the significance and background of some key concepts in gravitational radiation
Binary BH Merger – estimating the radiated power and merger time of a BH binary using dimensional analysis
Gravitational wave detectors
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Mario Schenberg
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Principle of relativity
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WorldHeritage articles needing clarification from May 2009
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