Just a few hours ago, the news came, which has long been awaited in the scientific world. A group of scientists from several countries, working as part of the international project LIGO Scientific Collaboration, claim that with the help of several observatories-detectors they managed to record gravitational waves in laboratory conditions.
They are analyzing data from two Laser Interferometer Gravitational-Wave Observatories (LIGO) located in Louisiana and Washington, USA.
As mentioned at the press conference of the LIGO project, gravitational waves were recorded on September 14, 2015, first at one observatory, and then 7 milliseconds later at another.
Based on the analysis of the data obtained, which was carried out by scientists from many countries, including Russia, it was found that the gravitational wave was caused by the collision of two black holes with a mass of 29 and 36 times the mass of the Sun. After that, they merged into one large black hole.
This happened 1.3 billion years ago. The signal came to Earth from the direction of the Magellanic Cloud constellation.
Sergey Popov (astrophysicist at the Sternberg State Astronomical Institute, Moscow State University) explained what gravitational waves are and why it is so important to measure them.
Modern theories of gravity are geometric theories of gravity, more or less everything, starting with the theory of relativity. The geometric properties of space affect the movement of bodies or objects such as a light beam. And vice versa - the distribution of energy (this is the same as mass in space) affects the geometric properties of space. It's very cool, because it's easy to visualize - all this elastic plane lined in a cell has a certain physical meaning under it, although, of course, everything is not so literal.
Physicists use the word "metric". A metric is what describes the geometric properties of a space. And here we have bodies moving with acceleration. The simplest thing is that the cucumber rotates. It is important that it is, for example, not a ball or a flattened disc. It is easy to imagine that when such a cucumber is spinning on an elastic plane, ripples will run from it. Imagine that you are standing somewhere, and the cucumber will turn with one end to you, then the other. It affects space and time in different ways, a gravitational wave runs.
So, a gravitational wave is a ripple running along the space-time metric.
Beads in space
This is a fundamental property of our basic understanding of how gravity works, and people have wanted to test it for a hundred years. They want to make sure that there is an effect and that it is visible in the laboratory. In nature, this was seen already about three decades ago. How should gravitational waves manifest themselves in everyday life?
The easiest way to illustrate this is as follows: if you throw the beads in space so that they lie in a circle, and when the gravitational wave passes perpendicular to their plane, they will begin to turn into an ellipse, compressed in one direction, then in the other. The point is that the space around them will be outraged, and they will feel it.
"G" on Earth
This is about the kind of thing people do, only not in space, but on Earth.
At a distance of four kilometers from each other hang mirrors in the form of the letter "g" [referring to the American observatories LIGO].
Laser beams are running - this is an interferometer, a well-understood thing. Modern technology makes it possible to measure a fantastically small effect. I still do not really believe, I believe, but it just does not fit in my head - the displacement of the mirrors hanging at a distance of four kilometers from each other is less than the size of an atomic nucleus. This is small even compared to the wavelength of this laser. This was the catch: gravity is the weakest interaction, and therefore the displacements are very small.
It took a very long time, people have been trying to do this since the 1970s, they have spent their lives looking for gravitational waves. And now, only technical capabilities make it possible to obtain registration of a gravitational wave in laboratory conditions, that is, here it came, and the mirrors have shifted.
Direction
Within a year, if everything goes well, then three detectors will work in the world. Three detectors are very important, because these things are very bad at determining the direction of the signal. In about the same way as we do by ear, we poorly determine the direction of the source. “Sound from somewhere to the right” - these detectors feel something like that. But if there are three people at a distance from each other, and one hears a sound to the right, another to the left, and the third from behind, then we can very accurately determine the direction of the sound. The more detectors there are, the more they are scattered around the globe, the more accurately we can determine the direction to the source, and then astronomy will begin.
After all, the ultimate task is not only to confirm the general theory of relativity, but also to obtain new astronomical knowledge. Just imagine that there is a black hole weighing ten times the mass of the Sun. And it collides with another black hole weighing ten times the mass of the Sun. The collision occurs at the speed of light. Energy breakthrough. It's true. There is a fantastic amount of it. And its not in any way ... It's just ripples of space and time. I would say that detecting the merger of two black holes for a long time will be the most reliable confirmation that black holes are about the kind of black holes we think of.
Let's go over the issues and phenomena that she could reveal.
Do black holes really exist?
The signal expected from the LIGO announcement may have been produced by two merging black holes. Events like these are the most energetic known; the force of the gravitational waves emitted by them can briefly eclipse all the stars of the observed Universe in total. Merging black holes are also quite easy to interpret from very pure gravitational waves.
Merging black holes occurs when two black holes spiral around each other, emitting energy in the form of gravitational waves. These waves have a characteristic sound (chirp) that can be used to measure the mass of these two objects. After that, black holes usually merge.
“Imagine two soap bubbles that come close enough to form one bubble. The larger bubble deforms, ”says Tybalt Damour, a gravity theorist at the Institute of Advanced scientific research near Paris. The final black hole will be perfectly spherical, but must first emit predictable gravitational waves.
One of the most important scientific implications of discovering black hole mergers will be confirmation of the existence of black holes - at least perfectly circular objects made up of pure, empty, curved spacetime, as predicted by general relativity. Another consequence is that the merger proceeds as the scientists predicted. Astronomers have a lot of indirect confirmation of this phenomenon, but so far these have been observations of stars and superheated gas in the orbit of black holes, and not the black holes themselves.
“The scientific community, myself included, dislikes black holes. We take them for granted, ”says Frans Pretorius, a specialist in general relativity simulations at Princeton University in New Jersey. "But if you think about what an amazing prediction this is, we need some truly amazing proof."
Do gravitational waves move at the speed of light?
When scientists start comparing LIGO observations with those of other telescopes, the first thing they check is whether the signal arrived at the same time. Physicists believe that gravity is transmitted by graviton particles, the gravitational analogue of photons. If, like photons, these particles have no mass, then gravitational waves will move with the speed of light, corresponding to the prediction of the speed of gravitational waves in classical theory relativity. (Their speed can be influenced by the accelerating expansion of the Universe, but this should manifest itself at distances significantly exceeding those covered by LIGO).
It is quite possible, however, that gravitons have a small mass, which means that gravitational waves will move at a speed less than light. So, for example, if LIGO and Virgo detect gravitational waves and find out that the waves arrived on Earth later than associated with a cosmic event of gamma rays, this could have fatal consequences for fundamental physics.
Is spacetime made of cosmic strings?
An even stranger discovery could happen if bursts of gravitational waves are detected emanating from "cosmic strings." These hypothetical space-time curvature defects, which may or may not be related to string theories, should be infinitely thin but stretched to cosmic distances. Scientists predict that cosmic strings, if they exist, could bend accidentally; if the string bends, it will cause a gravitational surge that detectors like the LIGO or Virgo could measure.
Can neutron stars be jagged?
Neutron stars are the remnants of large stars that collapsed under their own weight and became so dense that electrons and protons began to melt into neutrons. Scientists have little understanding of the physics of neutron holes, but gravitational waves could tell a lot about them. For example, the intense gravity on their surface causes neutron stars to become almost perfectly spherical. But some scientists have suggested that they may also have "mountains" - a few millimeters high - that make these dense objects, no more than 10 kilometers in diameter, slightly asymmetrical. Neutron stars usually spin very quickly, so an asymmetric mass distribution will warp spacetime and produce a constant sinusoidal gravitational wave signal, slowing the star's rotation and emitting energy.
Pairs of neutron stars that revolve around each other also produce a constant signal. Like black holes, these stars spiral and ultimately merge with a distinctive sound. But its specificity differs from the specificity of the sound of black holes.
Why do stars explode?
Black holes and neutron stars form when massive stars stop shining and collapse into themselves. Astrophysicists think this process is at the heart of all common types of Type II supernova explosions. Simulations of such supernovae have not yet shown why they ignite, but listening to the gravitational wave bursts emitted by a real supernova is believed to provide an answer. Depending on what the burst waves look like, how loud they are, how often they occur, and how they correlate with supernovae tracked by electromagnetic telescopes, this data could help rule out a bunch of existing models.
How fast is the universe expanding?
The expansion of the universe means that distant objects that move away from our galaxy appear redder than they actually are, as the light they emit is stretched as they move. Cosmologists estimate the rate of expansion of the universe by comparing the redshift of galaxies to how far away they are from us. But this distance is usually estimated from the brightness of Type Ia supernovae, and this technique leaves a lot of uncertainties.
If several gravitational wave detectors around the world detect signals from the merger of the same neutron stars, together they can absolutely accurately estimate the signal's loudness, as well as the distance at which the merger took place. They will also be able to assess the direction, and with it, identify the galaxy in which the event occurred. By comparing the redshift of this galaxy with the distance to the merging stars, an independent rate of cosmic expansion can be obtained, possibly more accurate than current methods allow.
sources
http://www.bbc.com/russian/science/2016/02/160211_gravitational_waves
http://cont.ws/post/199519
Here we somehow found out, but what is and. See also how it looks The original article is on the site InfoGlaz.rf The link to the article this copy was made from isOn Thursday, February 11, a group of scientists from the international project LIGO Scientific Collaboration announced that they had succeeded, the existence of which Albert Einstein had predicted back in 1916. According to the researchers, on September 14, 2015, they recorded a gravitational wave, which was caused by the collision of two black holes with masses 29 and 36 times the mass of the Sun, after which they merged into one large black hole. According to them, this happened supposedly 1.3 billion years ago at a distance of 410 Megaparsecs from our galaxy.
LIGA.net told in detail about gravitational waves and the large-scale discovery Bogdan Hnatyk, Ukrainian scientist, astrophysicist, doctor of physical and mathematical sciences, leading researcher at the Astronomical Observatory of the Taras Shevchenko National University of Kiev, who headed the observatory from 2001 to 2004.
Theory simple language
Physics studies the interaction between bodies. It has been established that there are four types of interaction between bodies: electromagnetic, strong and weak nuclear interaction and gravitational interaction, which we all feel. Due to the gravitational interaction, the planets revolve around the Sun, the bodies have weight and fall to the ground. Man is constantly faced with gravitational interaction.
In 1916, 100 years ago, Albert Einstein built a theory of gravity, which improved the Newtonian theory of gravity, made it mathematically correct: it began to meet all the requirements of physics, began to take into account the fact that gravity propagates at a very high, but finite speed. This is rightfully one of the greatest achievements of Einstein, since he built a theory of gravity that corresponds to all the phenomena of physics that we observe today.
This theory also suggested the existence gravitational waves... The basis for this prediction was that gravitational waves exist as a result of the gravitational interaction that occurs due to the merger of two massive bodies.
What is gravitational wave
In a complicated language, this is the excitation of the space-time metric. “Let's say that space has a certain elasticity and waves can run through it. It’s like when we throw a pebble into the water and waves scatter from it,” Doctor of Physics and Mathematics told LIGA.net.
Scientists managed to experimentally prove that such an oscillation took place in the Universe and a gravitational wave ran in all directions. “Astrophysical method was the first to record the phenomenon of such a catastrophic evolution of a binary system, when two objects merge into one, and this merger leads to a very intense release of gravitational energy, which then propagates in space in the form of gravitational waves,” the scientist explained.
What it looks like (photo - EPA)
These gravitational waves are very weak and in order for them to shake space-time, the interaction of very large and massive bodies is necessary so that the strength of the gravitational field is large at the place of generation. But, despite their weakness, the observer will register this gravitational wave after a certain time (equal to the distance to the interaction divided by the signal propagation speed).
Let's give an example: if the Earth fell on the Sun, then gravitational interaction would occur: gravitational energy would be released, a gravitational spherically symmetric wave would be formed and the observer could register it. "A similar, but unique, from the point of view of astrophysics, phenomenon occurred here: two massive bodies collided - two black holes," said Gnatyk.
Back to theory
A black hole is another prediction of Einstein's general theory of relativity, which provides that a body that has a huge mass, but this mass is concentrated in a small volume, can significantly distort the space around itself, up to its closure. That is, it was assumed that when a critical concentration of the mass of this body is reached - such that the size of the body will be less than the so-called gravitational radius, then space will be closed around this body and its topology will be such that no signal from it will propagate beyond the confined space can not.
"That is, a black hole, in simple words, it is a massive object, which is so heavy that it closes space-time around itself, "says the scientist.
And we, according to him, can send any signals to this object, but he cannot send us any. That is, no signals can go beyond the black hole.
A black hole lives according to the usual physical laws, but as a result of strong gravity, not a single material body, not even a photon, is able to go beyond this critical surface. Black holes are formed during the evolution of ordinary stars, when the central core collapses and part of the star's matter, collapsing, turns into a black hole, and the other part of the star is ejected as a supernova envelope, turning into the so-called "flash" of the Supernova.
How we saw a gravitational wave
Let's give an example. When we have two floats on the surface of the water and the water is calm, the distance between them is constant. When a wave arrives, it displaces these floats and the distance between the floats will change. The wave has passed - and the floats return to their previous positions, and the distance between them is restored.
Similarly, a gravitational wave propagates in space-time: it compresses and stretches bodies and objects that meet in its path. "When an object is encountered on the path of the wave, it deforms along its axes, and after its passage, it returns to its previous shape. Under the action of a gravitational wave, all bodies are deformed, but these deformations are very insignificant," says Gnatyk.
When the wave passed, which was recorded by the scientists, the relative size of bodies in space changed by an amount of the order of 1 multiplied by 10 to the minus 21st power. For example, if you take a meter ruler, then it is compressed by such an amount that was its size, multiplied by 10 to the minus 21st power. This is a very meager amount. And the problem was that scientists needed to learn how to measure this distance. Conventional methods gave an accuracy of the order of 1 to 10 to the 9th power of a million, and here a much higher accuracy is needed. For this, the so-called gravitational antennas (gravitational wave detectors) were created.
LIGO Observatory (photo - EPA)
The antenna that recorded the gravitational waves is built in this way: there are two pipes, about 4 kilometers in length, located in the shape of the letter "L", but with the same shoulders and at right angles. When a gravitational wave hits the system, it deforms the antenna's wings, but depending on its orientation, it deforms one more and the other less. And then there is a path difference, the interference pattern of the signal changes - a total positive or negative amplitude appears.
"That is, the passage of a gravitational wave is similar to a wave on water passing between two floats: if we measured the distance between them during and after the passage of the wave, we would see that the distance would change, and then it became the same again," he said Gnatyk.
It also measures the relative change in distance between the two wings of the interferometer, each of which is about 4 kilometers long. And only very precision technology and the systems measure this microscopic displacement of the wings caused by the gravitational wave.
At the edge of the universe: where the wave came from
Scientists recorded the signal using two detectors located in the United States in two states: Louisiana and Washington at a distance of about 3 thousand kilometers. Scientists were able to estimate where this signal came from and from what distance. Estimates show that the signal came from a distance of 410 Megaparsecs. Megaparsec is the distance that light travels in three million years.
To make it easier to imagine: the closest active galaxy with a supermassive black hole in the center is Centaurus A, which is located at a distance of four Megaparsecs from ours, while the Andromeda Nebula is at a distance of 0.7 Megaparsecs. “That is, the distance from which the gravitational wave signal came is so great that the signal went to the Earth for about 1.3 billion years. These are cosmological distances that reach about 10% of the horizon of our Universe,” the scientist said.
At such a distance, in some distant galaxy, two black holes merged. These holes, on the one hand, were relatively small in size, and on the other hand, the high strength of the signal amplitude indicates that they were very heavy. It was found that their masses were 36 and 29 solar masses, respectively. The mass of the Sun, as you know, is a value that is 2 times 10 to the 30th power of a kilogram. After the merger, these two bodies merged and now one black hole has formed in their place, which has a mass equal to 62 times the mass of the Sun. At the same time, approximately three solar masses splashed out in the form of gravitational wave energy.
Who made the discovery and when
Scientists from the international LIGO project managed to detect a gravitational wave on September 14, 2015. LIGO (Laser Interferometry Gravitation Observatory) is an international project in which a number of states that have made a certain financial and scientific contribution, in particular the USA, Italy, Japan, are taking part, which are leading in the field of this research.
Professors Rainer Weiss and Kip Thorne (photo - EPA)
The following picture was recorded: there was a displacement of the wings of the gravitational detector, as a result of the real passage of a gravitational wave through our planet and through this installation. This was not reported then, because the signal had to be processed, "cleaned", found its amplitude and checked. This is a standard procedure: from the actual discovery to the announcement of the discovery - it takes several months in order to issue a reasoned statement. "Nobody wants to spoil their reputation. These are all secret data, before the publication of which nobody knew about them, there were only rumors," Hnatyk said.
History
Gravitational waves have been studied since the 70s of the last century. During this time, a number of detectors have been created and a number of fundamental studies have been carried out. In the 80s, the American scientist Joseph Weber built the first gravitational antenna in the form of an aluminum cylinder, which was about several meters in size, equipped with piezo sensors that were supposed to detect the passage of a gravitational wave.
The sensitivity of this device was a million times worse than today's detectors. And, of course, then he could not really fix the wave, although Weber said that he did it: the press wrote about it and there was a "gravitational boom" - the world immediately began to build gravitational antennas. Weber encouraged other scientists to study gravitational waves and continue experiments on this phenomenon, thanks to which it was possible to raise the sensitivity of the detectors a million times.
However, the very phenomenon of gravitational waves was recorded in the last century, when scientists discovered a double pulsar. This was an indirect registration of the fact that gravitational waves exist, proven through astronomical observations. The pulsar was discovered by Russell Hulse and Joseph Taylor in 1974 while observing the Arecibo Observatory's radio telescope. Scientists were awarded the Nobel Prize in 1993 "for the discovery of a new type of pulsar, which gave new possibilities in the study of gravity."
Research in the world and Ukraine
In Italy, a similar project called Virgo is close to completion. Japan also intends to launch a similar detector in a year, India is also preparing such an experiment. That is, in many parts of the world there are similar detectors, but they have not yet reached that sensitivity mode, so that we can talk about fixing gravitational waves.
"Officially, Ukraine is not a member of LIGO and also does not participate in the Italian and Japanese projects. Among such fundamental areas, Ukraine is now taking part in the LHC project (LHC - Large Hadron Collider) and in CERN" (we will officially become a member only after paying the entrance fee) "- Doctor of physical and mathematical sciences Bogdan Gnatyk told LIGA.net.
According to him, Ukraine has been a full-fledged member of the international collaboration CTA (MCHT-Cherenkov Telescope Array) since 2015, which is building a modern multi-telescope TeVgamma range (with photon energies up to 1014 eV). "The main sources of such photons are precisely the neighborhoods of supermassive black holes, the gravitational radiation of which was first detected by the LIGO detector. Therefore, the opening of new windows in astronomy - gravitational-wave and multi TeVthe new electromagnetic field promises us many more discoveries in the future, "the scientist adds.
What's next and how will new knowledge help people? Scientists disagree. Some say that this is just another step in understanding the mechanisms of the Universe. Others see this as the first steps towards new technologies for moving through time and space. One way or another - this discovery once again proved how little we understand and how much remains to be learned.
What are gravitational waves?
Gravitational waves - changes in the gravitational field, propagating like waves. They are emitted by moving masses, but after radiation they break away from them and exist independently of these masses. Mathematically related to the perturbation of the space-time metric and can be described as “ripples of space-time”.
In general relativity and most others modern theories gravity gravitational waves are generated by the motion of massive bodies with variable acceleration. Gravitational waves propagate freely in space at the speed of light. Due to relative weakness gravitational forces (in comparison with others) these waves have a very small value, which is difficult to register.
Gravitational waves are predicted by general relativity (GR). They were first detected directly in September 2015 by two LIGO twin detectors, which recorded gravitational waves, likely from the merger of two black holes and the formation of one more massive rotating black hole... Indirect evidence of their existence has been known since the 1970s - general relativity predicts the rate of convergence of close systems of binary stars coinciding with observations due to the loss of energy due to the radiation of gravitational waves. Direct registration of gravitational waves and their use to determine the parameters of astrophysical processes is an important task of modern physics and astronomy.
If we imagine our space-time as a network of coordinates, then gravitational waves are perturbations, ripples that will run along the grid when massive bodies (for example, black holes) distort the space around them.
It can be compared to an earthquake. Imagine that you live in a city. It has some kind of markers that create an urban space: houses, trees, and so on. They are motionless. When a major earthquake occurs somewhere near the city, the vibrations reach us - and even motionless houses and trees begin to vibrate. These vibrations are gravitational waves; and the objects that oscillate are space and time.
Why have scientists been unable to register gravitational waves for so long?
Concrete efforts to detect gravitational waves began in post-war period from several naive devices, the sensitivity of which, obviously, could not be sufficient to register such oscillations. Over time, it became clear that detectors for searching must be very large-scale - and they must use modern laser technology. It is with the development of modern laser technologies that it became possible to control the geometry, the disturbances of which are the gravitational wave. Powerful advances in technology played a key role in this discovery. No matter how ingenious the scientists were, 30-40 years ago it was technically impossible to do this.
Why is wave detection so important to physics?
Gravitational waves were predicted by Albert Einstein in general relativity about a hundred years ago. Throughout the 20th century, there were physicists who questioned this theory, although more and more confirmation appeared. And the presence of gravitational waves is such a critical confirmation of the theory.
In addition, before registering gravitational waves, we knew how gravity behaves only through the example of celestial mechanics, the interaction of celestial bodies. But it was clear that the gravitational field has waves and space-time can deform in a similar way. The fact that we had not seen gravitational waves before was a blank spot in modern physics... Now this white spot has been closed, another brick has been laid at the foundation of modern physical theory. This is a fundamental discovery. Nothing comparable for last years did not have.
"Waiting for waves and particles" - documentary about the search for gravitational waves(by Dmitry Zavilgelskiy)
There is also a practical point in the registration of gravitational waves. Probably, after further development of technology, it will be possible to talk about gravitational astronomy - about observing the traces of the most high-energy events in the Universe. But now it is too early to talk about this, we are talking only about the very fact of registration of waves, and not about clarifying the characteristics of the objects that generate these waves.
Wave your hand - and gravitational waves will run throughout the Universe.
S. Popov, M. Prokhorov. The ghostly waves of the universe
In astrophysics, an event has occurred that has been waiting for decades. After half a century of searching, gravitational waves have finally been discovered, the oscillations of space-time itself, predicted by Einstein a hundred years ago. On September 14, 2015, the updated LIGO observatory detected a gravitational wave burst generated by the merger of two black holes with masses of 29 and 36 solar masses in a distant galaxy at a distance of about 1.3 billion light years. Gravitational wave astronomy has become a full-fledged branch of physics; she told us new way observe the Universe and will allow you to study previously inaccessible effects of strong gravity.
Gravitational waves
Different theories of gravity can be invented. All of them will equally well describe our world, as long as we restrict ourselves to one single manifestation of it - Newton's law universal gravitation... But there are other, more subtle gravitational effects that have been experimentally tested on the scale of the solar system, and they point to one specific theory - general relativity (GR).
General relativity is not just a set of formulas, it is a fundamental view of the essence of gravity. If in ordinary physics space serves only as a background, a container for physical phenomena, in general relativity it itself becomes a phenomenon, a dynamic quantity that changes in accordance with the laws of general relativity. These distortions of space-time against an even background - or, in the language of geometry, distortions of the space-time metric - are felt as gravity. In short, general relativity reveals the geometric origin of gravity.
General relativity has an important prediction: gravitational waves. These are distortions of space-time, which are able to "break away from the source" and, self-sustaining, fly away. It’s gravity itself, nobody’s, its own. Albert Einstein finally formulated general relativity in 1915 and almost immediately realized that the equations he obtained admit the existence of such waves.
As with any honest theory, such a clear prediction of general relativity must be verified experimentally. Any moving body can emit gravitational waves: planets, a stone thrown up, and a wave of the hand. The problem, however, is that the gravitational interaction is so weak that no experimental setup is able to detect the radiation of gravitational waves from conventional "emitters".
To "drive" a powerful wave, you need to very much distort space-time. The ideal variant is two black holes revolving around each other in a close dance, at a distance of the order of their gravitational radius (Fig. 2). The distortions of the metric will be so strong that a noticeable part of the energy of this pair will be radiated into gravitational waves. Losing energy, the pair will approach each other, spinning faster and faster, distorting the metric more and more and generating even stronger gravitational waves - until, finally, a radical restructuring of the entire gravitational field of this pair occurs and two black holes merge into one.
Such a merger of black holes is an explosion of tremendous power, but all this radiated energy goes not into light, not into particles, but into vibrations of space. The radiated energy will make up a noticeable part of the original mass of black holes, and this radiation will splash out in a fraction of a second. Similar fluctuations will generate mergers of neutron stars. A slightly weaker gravitational-wave energy release accompanies other processes, for example, the collapse of a supernova core.
The gravitational wave burst from the merger of two compact objects has a very specific, well-calculated profile, shown in Fig. 3. The oscillation period is set orbital motion two objects around each other. Gravitational waves carry away energy; as a result, objects approach and spin faster - and this can be seen both in the acceleration of oscillations and in the amplification of the amplitude. At some point, a merger occurs, the last strong wave is emitted, and then a high-frequency "after-ring" follows ( ringdown) - the shaking of the formed black hole, which “throws off” all non-spherical distortions (this stage is not shown in the picture). Knowing this signature profile helps physicists look for a faint signal from such a fusion in highly noisy detector data.
Oscillations of the space-time metric - the gravitational-wave echo of a grandiose explosion - will scatter across the Universe in all directions from the source. Their amplitude weakens with distance, by analogy with how the brightness of a point source decreases with distance from it. When a burst from a distant galaxy reaches Earth, the metric fluctuations will be on the order of 10-22 or even less. In other words, the distance between objects that are not physically connected to each other will periodically increase and decrease by such a relative amount.
The order of magnitude of this number is easy to obtain from scale considerations (see the article by V.M. Lipunov). At the moment of the merger of neutron stars or black holes of stellar masses, the distortions of the metric right next to them are very large - on the order of 0.1, which is why there is strong gravity. Such severe distortion affects a region of the order of the sizes of these objects, that is, several kilometers. With distance from the source, the amplitude of the oscillation decreases in inverse proportion to the distance. This means that at a distance of 100 Mpc \u003d 3 × 10 21 km, the oscillation amplitude will drop by 21 orders of magnitude and become about 10 −22.
Of course, if the merger occurs in our home galaxy, the tremors of space-time that have reached Earth will be much stronger. But such events happen every few thousand years. Therefore, you should really count only on a detector that will be able to sense the merger of neutron stars or black holes at a distance of tens to hundreds of megaparsecs, which means it will cover many thousands and millions of galaxies.
Here it should be added that an indirect indication of the existence of gravitational waves has already been discovered, and even awarded the Nobel Prize in Physics for 1993. Long-term observations of the pulsar in the PSR B1913 + 16 binary system have shown that the orbital period decreases at exactly the rate predicted by general relativity, taking into account the energy losses due to gravitational radiation. For this reason, practically no scientist doubts the reality of gravitational waves; the only question is how to catch them.
Search history
The search for gravitational waves started about half a century ago - and almost immediately turned into a sensation. Joseph Weber of the University of Maryland designed the first resonant detector: a one-piece two-meter aluminum cylinder with sensitive piezoelectric sensors on the sides and good vibration isolation from extraneous vibrations (Fig. 4). When a gravitational wave passes, the cylinder resonates in time with the distortions of space-time, which should be recorded by the sensors. Weber built several such detectors, and in 1969, after analyzing their readings during one of the sessions, he stated in plain text that he had registered the "sounding of gravitational waves" in several detectors spaced two kilometers apart from each other (J. Weber, 1969 . Evidence for Discovery of Gravitational Radiation). The amplitude of oscillations declared by him turned out to be incredibly large, of the order of 10-16, that is, a million times greater than the typical expected value. Weber's message was met with great skepticism in the scientific community; in addition, other experimental groups, armed with similar detectors, were unable to catch a single similar signal in the future.
However, Weber's efforts gave impetus to this entire area of \u200b\u200bresearch and launched the hunt for waves. Since the 1970s, thanks to the efforts of Vladimir Braginsky and his colleagues from Moscow State University, the USSR also entered this race (see about the absence of gravitational wave signals). Interesting story about those times is in the essay If a girl falls into a hole .... Braginsky, by the way, is one of the classics of the whole theory of quantum optical measurements; he first came to the notion of the standard quantum measurement limit - a key limitation in optical measurements - and showed how, in principle, they could be overcome. Weber's resonant circuit was improved, and thanks to deep cooling of the installation, the noise was dramatically reduced (see the list and history of these projects). However, the accuracy of such all-metal detectors was still insufficient to reliably detect expected events, and besides, they are tuned to resonate only in a very narrow frequency range near the kilohertz.
Detectors seemed to be much more promising, in which not one resonating object is used, but the distance between two not connected with each other, independently suspended bodies, for example, two mirrors, is tracked. Due to the fluctuation of space caused by the gravitational wave, the distance between the mirrors will be slightly larger, sometimes slightly smaller. In this case, the longer the arm length, the greater the absolute displacement will be caused by a gravitational wave of a given amplitude. These vibrations can be felt by the laser beam running between the mirrors. Such a scheme is capable of recording oscillations in a wide frequency range, from 10 hertz to 10 kilohertz, and this is exactly the interval in which merging pairs of neutron stars or black holes of stellar masses will emit.
The modern implementation of this idea based on the Michelson interferometer is as follows (Fig. 5). Mirrors are suspended in two long, several kilometers long, perpendicular to each other vacuum chambers. At the entrance to the installation, the laser beam is split, goes through both chambers, is reflected from the mirrors, comes back and re-connects in a semitransparent mirror. The quality factor of the optical system is extremely high, so the laser beam does not just go back and forth once, but is delayed in this optical cavity for a long time. In the “quiet” state, the lengths are selected so that the two beams after reunification extinguish each other in the direction of the sensor, and then the photodetector is in full shadow. But as soon as the mirrors move a microscopic distance under the action of gravitational waves, the compensation of the two beams becomes incomplete and the photodetector will catch the light. And the stronger the displacement, the brighter the light will be seen by the photosensor.
The words "microscopic displacement" do not even come close to conveying the subtlety of the effect. The displacement of the mirrors by the wavelength of light, that is, microns, is easy to notice, even without any tweaks. But with an arm length of 4 km, this corresponds to space-time oscillations with an amplitude of 10 −10. It is also not a problem to notice the displacement of the mirrors by the diameter of the atom - it is enough to launch a laser beam that will run back and forth thousands of times and get the required phase incursion. But even this gives on the strength of 10 −14. And we need to go down the scale of displacements millions of times more, that is, learn how to register a mirror shift not even by one atom, but by thousandths of an atomic nucleus!
Physicists had to overcome many difficulties on the way to this truly amazing technology. Some of them are purely mechanical: you need to hang massive mirrors on a suspension that hangs on another suspension, that on a third suspension, and so on - and all in order to get rid of extraneous vibration as much as possible. Other problems are also instrumental, but optical. For example, the more powerful the beam circulating in the optical system, the weaker the displacement of the mirrors can be seen by the photosensor. But a too powerful beam will unevenly heat the optical elements, which will adversely affect the properties of the beam itself. This effect must be somehow compensated, and for this in the 2000s, a whole research program was launched on this matter (for a story about this study, see the news Overcoming the obstacle on the way to a highly sensitive gravitational wave detector, "Elements", 06/27/2006 ). Finally, there are purely fundamental physical limitations associated with the quantum behavior of photons in the resonator and the uncertainty principle. They limit the sensor's sensitivity to a value called the standard quantum limit. However, physicists have already learned to overcome it with the help of a cleverly prepared quantum state of laser light (J. Aasi et al., 2013. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light).
There is a whole list of countries in the race for gravitational waves; Russia also has its own installation, in the Baksan Observatory, and, by the way, is described in the documentary popular science film by Dmitry Zavilgelsky "Waiting for waves and particles" ... The leaders of this race are now two laboratories - the American project LIGO and the Italian Virgo detector. LIGO includes two identical detectors located in Hanford, Washington, and Livingston, Louisiana, 3,000 km apart. Having two installations is important for two reasons at once. Firstly, the signal will be considered registered only if both detectors see it simultaneously. And secondly, by the difference in the arrival of a gravitational-wave burst at two installations - and it can reach 10 milliseconds - one can roughly determine from which part of the sky this signal came. True, with two detectors the error will be very large, but when Virgo comes into operation, the accuracy will noticeably increase.
Strictly speaking, the idea of \u200b\u200binterferometric detection of gravitational waves was first proposed by Soviet physicists M. E. Hertsenstein and V. I. Pustovoit back in 1962. Then the laser was just invented, and Weber began to create his resonance detectors. However, this article was not noticed in the West and, to tell the truth, did not influence the development of real projects (see the historical review Physics of gravitational wave detection: resonant and interferometric detectors).
The creation of the LIGO gravity observatory was an initiative of three scientists from Massachusetts institute of Technology (MIT) and from the California Institute of Technology (Caltech). These are Rainer Weiss, who implemented the idea of \u200b\u200ban interferometric gravitational wave detector, Ronald Drever, who achieved sufficient laser light stability to register, and Kip Thorne, the theorist-master of the project, now well known to the general public as a scientific adviser. film "Interstellar". The early history of LIGO can be read in a recent interview with Rainer Weiss and in the flashbacks of John Preskill.
Activities related to the project of interferometric detection of gravitational waves began in the late 1970s, and at first the reality of this undertaking was also questioned by many. However, after demonstrating a number of prototypes, the current LIGO project was written and approved. It was built during the entire last decade of the 20th century.
Although the initial impetus for the project was given by the United States, the LIGO observatory is a truly international project. Fifteen countries have invested in it, financially and intellectually, and over a thousand people are members of the collaboration. Soviet and Russian physicists played an important role in the implementation of the project. From the very beginning, the already mentioned group of Vladimir Braginsky from Moscow State University took an active part in the implementation of the LIGO project, and later the Institute of Applied Physics from Nizhny Novgorod joined the collaboration.
The LIGO observatory was launched in 2002 and up to 2010 six scientific observation sessions took place on it. No gravitational wave bursts were reliably detected, and physicists could only set upper limits on the frequency of such events. This, however, did not surprise them too much: estimates showed that in that part of the Universe, which was then "listened to" by the detector, the probability of a sufficiently powerful cataclysm was small: about once every several decades.
Home stretch
From 2010 to 2015, the LIGO and Virgo collaborations radically modernized the equipment (Virgo, however, is still in the process of preparation). And now the long-awaited target was in line of sight. LIGO - or rather, aLIGO ( Advanced LIGO) - was now ready to catch bursts generated by neutron stars at a distance of 60 megaparsecs, and black holes - hundreds of megaparsecs. The volume of the Universe, open for gravitational-wave listening, has grown tenfold compared to previous sessions.
Of course, it is impossible to predict when and where the next gravitational wave "bang" will be. But the sensitivity of the updated detectors made it possible to count on several mergers of neutron stars per year, so that the first burst could be expected already during the first four-month observation session. If we talk about the whole project aLIGO lasting several years, then the verdict was very clear: either bursts will fall down one after another, or something in general relativity does not work in principle. Both will be a great discovery.
From September 18, 2015 to January 12, 2016, the first aLIGO observation session took place. During all this time, rumors about the registration of gravitational waves were circulating on the Internet, but the collaboration remained silent: "we are collecting and analyzing data and are not yet ready to report the results." An additional intrigue was created by the fact that in the process of analysis, the members of the collaboration themselves cannot be completely sure that they are seeing a real gravitational wave burst. The fact is that in LIGO, a splash generated on a computer is sometimes artificially introduced into the stream of real data. It is called "blind injection", and only three people from the whole group (!) Have access to the system, which carries out it at an arbitrary point in time. The team must track this surge, analyze it responsibly, and only at the very last stages of the analysis "cards open up" and the members of the collaboration will find out whether this was a real event or a test of vigilance. By the way, in one such case, in 2010, it even came to writing an article, but the signal detected then turned out to be just a "blind stuffing".
Lyrical digression
To once again feel the solemnity of the moment, I propose to look at this history from the other side, from the inside of science. When a difficult, unapproachable scientific task does not give in for several years, this is a normal working moment. When it does not give in for more than a generation, it is perceived in a completely different way.
As a schoolboy, you read popular science books and learn about this difficult to solve, but terribly interesting scientific riddle. As a student, you study physics, make reports, and sometimes, to the point or not, people around you remind of its existence. Then you do science yourself, work in another area of \u200b\u200bphysics, but you regularly hear about unsuccessful attempts to solve it. You, of course, understand that somewhere active work is underway to solve it, but the final result for you as a person from the outside remains unchanged. The problem is perceived as a static background, as a decoration, as eternal and almost unchanging on the scale of your scientific life element of physics. As a task that has always been and will be.
And then they solve it. And abruptly, on a scale of several days, you feel that the physical picture of the world has changed and that now it needs to be formulated in different expressions and asked other questions.
For people who are directly working on the search for gravitational waves, this task, of course, did not remain unchanged. They see the goal, they know what needs to be achieved. They, of course, hope that nature will also meet them halfway and throw a powerful outburst in some nearby galaxy, but at the same time they understand that even if nature is not so supportive, it will no longer hide from scientists. The only question is when exactly they will be able to achieve their technical goals. The story of this feeling from a man who has been searching for gravitational waves for several decades can be heard in the film already mentioned "Waiting for waves and particles".
Opening
In fig. 7 shows the main result: the profile of the signal recorded by both detectors. It can be seen that, against the background of the noise, at first, an oscillation of the desired shape appears weakly, and then increases in amplitude and frequency. Comparison with the results of numerical simulations allowed us to find out which objects we observed merging: these were black holes with masses of about 36 and 29 solar masses, which merged into one black hole with a mass of 62 solar masses (the error of all these numbers, corresponding to a 90% confidence interval, is 4 solar masses). The authors note in passing that the resulting black hole is the heaviest stellar mass black hole ever observed. The difference between the total mass of the two original objects and the final black hole is 3 ± 0.5 solar masses. This gravitational mass defect was completely transformed into the energy of the emitted gravitational waves in about 20 milliseconds. Calculations showed that the peak gravitational wave power reached 3.6 · 10 56 erg / s, or, in terms of mass, about 200 solar masses per second.
The statistical significance of the detected signal is 5.1σ. In other words, if we assume that these statistical fluctuations were superimposed on each other and purely accidentally produced such a burst, such an event would have to wait 200 thousand years. This allows us to state with confidence that the detected signal is not a fluctuation.
The time delay between the two detectors was approximately 7 milliseconds. This made it possible to estimate the direction of arrival of the signal (Fig. 9). Since there are only two detectors, the localization turned out to be very approximate: the region of the celestial sphere suitable in terms of parameters is 600 square degrees.
The LIGO collaboration did not confine itself to just stating the fact of registration of gravitational waves, but also carried out the first analysis of what this observation has for astrophysics. In the article Astrophysical implications of the binary black hole merger GW150914, published in the journal the same day The Astrophysical Journal Letters, the authors estimated the frequency with which such black hole mergers occur. There was at least one merge in a cubic gigaparsec per year, which is in line with the predictions of the most optimistic models in this regard.
What gravitational waves tell you about
The discovery of a new phenomenon after decades of searching is not the end, but only the beginning of a new branch of physics. Of course, the registration of gravitational waves from the merger of the black two is important in itself. This is a direct proof of the existence of black holes, and the existence of double black holes, and the reality of gravitational waves, and, generally speaking, proof of the correctness of the geometric approach to gravity, on which general relativity is based. But for physicists, no less valuable is the fact that gravitational-wave astronomy is becoming a new research tool, making it possible to study what was previously inaccessible.
First, it is a new way to view the Universe and study cosmic cataclysms. There are no obstacles for gravitational waves, they pass through everything in the Universe without any problems. They are self-sufficient: their profile carries information about the process that gave rise to them. Finally, if one grandiose explosion generates both an optical, neutrino, and gravitational burst, then you can try to catch all of them, compare with each other, and figure out in previously inaccessible details what happened there. To be able to catch and compare such different signals from one event is the main goal of all-signal astronomy.
When gravitational wave detectors become even more sensitive, they will be able to register the tremor of space-time not at the very moment of merging, but a few seconds before it. They will automatically send their warning signal to the general network of observing stations, and astrophysical satellites-telescopes, having calculated the coordinates of the proposed merger, will have time to turn in the right direction in these seconds and start shooting the sky before the optical burst begins.
Secondly, the gravitational wave burst will allow you to learn new things about neutron stars,. The merger of neutron stars is, in fact, the latest and most extreme experiment on neutron stars that nature can perform for us, and we, as spectators, will only have to observe the results. The observational consequences of such a merger can be varied (Fig. 10), and by typing their statistics, we can better understand the behavior of neutron stars in such exotic conditions. Overview current state cases in this direction can be found in the recent publication S. Rosswog, 2015. Multi-messenger picture of compact binary mergers.
Third, the registration of a burst that came from a supernova and its comparison with optical observations will allow us to finally understand the details of what is happening there inside, at the very beginning of the collapse. Now physicists still have difficulties with numerical modeling of this process.
Fourth, physicists dealing with the theory of gravity have a coveted "laboratory" for studying the effects of strong gravity. Until now, all the effects of general relativity that we could directly observe were related to gravity in weak fields. What happens in conditions of strong gravity, when space-time distortions begin to interact strongly with themselves, we could only guess from indirect manifestations, through the optical echo of cosmic catastrophes.
Fifth, it appears new opportunity to test exotic theories of gravity. There are already many such theories in modern physics, see, for example, the chapter dedicated to them from the popular book by AN Petrov "Gravity". Some of these theories resemble ordinary general relativity in the weak-field limit, but they can be very different when gravity becomes very strong. Others admit the existence of a new type of polarization for gravitational waves and predict a speed slightly different from the speed of light. Finally, there are theories that include additional spatial dimensions. What can be said about them on the basis of gravitational waves is an open question, but it is clear that some information can be profited here. We also recommend reading the opinion of the astrophysicists themselves about what will change with the discovery of gravitational waves in the selection on Post Science.
Plans for the future
The prospects for gravitational-wave astronomy are most encouraging. Now only the first, the shortest observation session of the aLIGO detector has ended - and already in this short time a clear signal was caught. It would be more accurate to say this: the first signal was caught even before the official start, and the collaboration has not yet reported on all four months of work. Who knows, maybe there are already some additional bursts in there? One way or another, but further, as the sensitivity of the detectors increases and the part of the Universe available for gravitational-wave observations expands, the number of registered events will grow like an avalanche.
The expected schedule of LIGO-Virgo sessions is shown in Fig. 11. The second, six-month session will begin at the end of this year, the third session will take almost the entire 2018, and at each stage the detector's sensitivity will increase. In the region of 2020, aLIGO should reach the planned sensitivity, which will allow the detector to probe the Universe for a merger of neutron stars located at a distance of up to 200 Mpc. For even more energetic events of the merger of black holes, the sensitivity can hit almost a gigaparsec. One way or another, the volume of the Universe available for observation will increase in comparison with the first session tenfold more.
The revamped Italian Virgo laboratory will also come into play later this year. Its sensitivity is slightly lower than that of LIGO, but also quite decent. Due to the triangulation method, a triple of spaced-apart detectors will allow much better reconstruction of the position of sources on celestial sphere... If now, with two detectors, the localization area reaches hundreds of square degrees, then three detectors will reduce it to tens. In addition, a similar KAGRA gravitational-wave antenna is currently under construction in Japan, which will begin operation in two to three years, and in India, in the region of 2022, it is planned to launch the LIGO-India detector. As a result, several years later, a whole network of gravitational-wave detectors will operate and regularly register signals (Fig. 13).
Finally, there are plans to launch gravitational wave instruments into space, in particular the eLISA project. Two months ago, the first test satellite was launched into orbit, the task of which will be to test technologies. The real detection of gravitational waves is still far from here. But when this group of satellites begins to collect data, it will open another window into the universe - through low-frequency gravitational waves. This all-wave approach to gravitational waves is the main long-term goal of this area.
Parallels
The discovery of gravitational waves has become the third case in recent years when physicists have finally broken through all the obstacles and got to the previously unknown subtleties of the structure of our world. In 2012, the Higgs boson was discovered - a particle predicted almost half a century away. In 2013, the IceCube neutrino detector proved the reality of astrophysical neutrinos and began to "look at the universe" in a completely new, previously inaccessible way - through high-energy neutrinos. And now nature has succumbed to man once again: a gravitational-wave "window" has opened for observing the universe and, at the same time, the effects of strong gravity have become available for direct study.
I must say, nowhere here there was any "freebie" from nature. The search was carried out for a very long time, but it did not give in because then, decades ago, the equipment did not reach the result in terms of energy, scale, or sensitivity. It was the steady, purposeful development of technologies that led to the goal, a development that was not stopped by either technical difficulties or the negative results of past years.
And in all three cases, the very fact of the discovery was not the end, but, on the contrary, the beginning of a new direction of research, became a new instrument for probing our world. The properties of the Higgs boson have become measurable - and in these data physicists are trying to discern the effects of New Physics. Thanks to the increased statistics of high-energy neutrinos, neutrino astrophysics is taking its first steps. At least the same is now expected from gravitational wave astronomy, and there is every reason for optimism.
Sources:
1) LIGO Scientific Coll. and Virgo Coll. Observation of Gravitational Waves from a Binary Black Hole Merger // Phys. Rev. Lett. Published 11 February 2016.
2) Detection Papers - A list of technical articles accompanying the main discovery article.
3) E. Berti. Viewpoint: The First Sounds of Merging Black Holes // Physics. 2016. V. 9.N. 17.
Review materials:
1) David Blair et al. Gravitational wave astronomy: the current status // arXiv: 1602.02872.
2) Benjamin P. Abbott and LIGO Scientific Collaboration and Virgo Collaboration. Prospects for Observing and Localizing Gravitational-Wave Transients with Advanced LIGO and Advanced Virgo // Living Rev. Relativity... 2016. V. 19.N. 1.
3) O. D. Aguiar. The Past, Present and Future of the Resonant-Mass Gravitational Wave Detectors // Res. Astron. Astrophys. 2011. V. 11.N. 1.
4) The search for gravitational waves - a selection of materials on the journal's website Science on the search for gravitational waves.
5) Matthew Pitkin, Stuart Reid, Sheila Rowan, Jim Hough. Gravitational Wave Detection by Interferometry (Ground and Space) // arXiv: 1102.3355.
6) V. B. Braginsky. Gravitational wave astronomy: new measurement methods // UFN... 2000.Vol. 170, pp. 743–752.
7) Peter R. Saulson.
The free surface of a liquid in equilibrium in the gravity field is flat. If, under the influence of some external influence, the surface of the liquid in some place is removed from its equilibrium position, then motion arises in the liquid. This movement will propagate along the entire surface of the liquid in the form of waves, called gravitational waves, since they are caused by the action of the gravity field. Gravitational waves occur mainly on the surface of the liquid, capturing its inner layers, the less the deeper these layers are located.
We will consider here such gravitational waves, in which the speed of the moving particles of the liquid is so small that in the Euler equation one can neglect the term in comparison with It is easy to find out what this condition means physically. During a time interval of the order of the period of oscillations made by liquid particles in a wave, these particles travel a distance of the order of the amplitude a of the wave; therefore, their velocity of movement is of the order of the Velocity v noticeably changes over time intervals of the order of and over distances of the order of waves). Therefore, the derivative of the velocity with respect to time is of the order and with respect to coordinates is of the order. Thus, the condition is equivalent to the requirement
that is, the amplitude of the oscillations in the wave must be small compared to the wavelength. In § 9 we saw that if the term in the equation of motion can be neglected, then the motion of the fluid is potential. Assuming the fluid is incompressible, we can therefore use equations (10.6) and (10.7). In equation (10.7) we can now neglect the term containing the square of the velocity; putting and entering the term in the gravity field, we get:
(12,2)
We choose the axis, as usual, vertically upward, and as the plane x, y we choose the equilibrium flat surface of the liquid.
We will denote - the coordinate of points of the liquid surface by; is a function of coordinates x, y and time t. In equilibrium, so there is a vertical displacement of the liquid surface during its oscillations.
Let constant pressure act on the surface of the liquid.Then we have on the surface according to (12.2)
The constant can be eliminated by redefining the potential (adding to it a coordinate-independent quantity.Then the condition on the liquid surface takes the form
The smallness of the amplitude of oscillations in the wave means that the displacement is small. Therefore, we can assume, in the same approximation, that the vertical component of the speed of movement of points on the surface coincides with the time derivative of the displacement But so we have:
Due to the smallness of oscillations, in this condition, we can take the values \u200b\u200bof the derivatives at instead of Thus, we finally obtain the following system of equations that determine the motion in a gravitational wave:
We will consider waves on the surface of a liquid, assuming this surface is unlimited. We will also assume that the wavelength is small compared to the depth of the liquid; then the liquid can be considered as infinitely deep. Therefore, we do not write boundary conditions at the lateral boundaries and at the bottom of the liquid.
Consider a gravitational wave propagating along the axis and uniform along the axis in such a wave, all quantities do not depend on the coordinate y. We will look for a solution that is a simple periodic function of time and coordinates x:
where (is the cyclic frequency (we will talk about it simply as a frequency), k is the wave vector of the wave, is the wavelength. Substituting this expression into the equation, we obtain the equation for the function
Its solution, which decays deep into the liquid (i.e., at):
We must still satisfy boundary condition (12.5) Substituting (12.5) into it, we find the relationship between the frequency b by the wave vector (or, as they say, the wave dispersion law):
The distribution of velocities in the fluid is obtained by differentiating the potential along the coordinates:
We see that the speed falls off exponentially in the direction of the liquid. At each given point in space (that is, for given x, z) the velocity vector rotates uniformly in the x-plane, remaining constant in magnitude.
Let us also define the trajectory of the liquid particles in the wave. Let us temporarily denote by x, z the coordinates of a moving fluid particle (and not the coordinates of a fixed point in space), but by - x values \u200b\u200bfor the equilibrium position of the particle. Then a on the right-hand side of (12.8) can be approximately written instead, taking advantage of the smallness of the oscillations. Integration over time then gives:
Thus, liquid particles describe circles around points with a radius exponentially decreasing in the direction into the depth of the liquid.
The velocity U of wave propagation is, as will be shown in § 67, Substituting here we find that the velocity of propagation of gravitational waves on an unbounded surface of an infinitely deep fluid is
It grows with increasing wavelength.
Long gravitational waves
Having considered gravitational waves, the length of which is small in comparison with the depth of the liquid, let us now dwell on the opposite limiting case of waves, the length of which is large in comparison with the depth of the liquid.
Such waves are called long waves.
Let us first consider the propagation of long waves in a channel. The length of the channel (directed along the x axis) will be assumed to be unlimited. The channel section can have an arbitrary shape and can vary along its length. The cross-sectional area of \u200b\u200bthe fluid in the channel is denoted by The depth and width of the channel are assumed to be small compared to the wavelength.
We will consider here longitudinal long waves in which the liquid moves along the channel. In such waves, the velocity component along the channel length is large in comparison with the components
By simply denoting v and omitting the small terms, we can write the -component of the Euler equation in the form
and -component - in the form
(we omit the terms quadratic in the velocity, since the wave amplitude is still considered small). From the second equation, we have, noting that on free surface ) should be
Substituting this expression into the first equation, we get:
The second equation for determining the two unknowns can be derived by a method similar to the derivation of the continuity equation. This equation is essentially the equation of continuity as applied to the case under consideration. Let us consider the volume of liquid enclosed between two planes of the channel cross-section located at a distance from each other. In a unit of time, a volume of liquid will enter through one plane, and a volume will exit through the other plane.Therefore, the volume of liquid between both planes will change by