The famous physicist spent his whole life trying to “make friends” between the theory of gravity and quantum theory, dreamed of flying into space and reminded earthlings of the inevitable space emigration

Moscow. March 14th. website - On Wednesday, March 14, it became known that at the age of 76, one of the most famous theoretical physicists of our time and popularizer of science, Stephen Hawking, who spent his whole life trying to reconcile the theory of gravity and quantum theory.

The secret of Hawking's popularity is his intelligent eccentricity, his inability to isolate himself within any boundaries, and his openness to people with whom he tried to dialogue on equal terms, speaking in simple language about complex things.

His active lifestyle contributed to the popularization of science: the scientist traveled a lot, more than once became the hero of cartoons in “The Simpsons” and “Futurama”, in which he voiced his character, and even starred in films as himself - in one of the episodes of the TV series “Star Trek: The Next Generation" and in an episode of the comedy series "The Big Bang Theory", the scientist was a supporter of nuclear disarmament and fought against climate change.

The German popularizer of science Hubert Mania in his book “Stephen Hawking” describes the British physicist this way: “An almost perfect embodiment of a free spirit, a huge intellect, a man who courageously overcomes physical weakness, devoting all his strength to deciphering the “divine plan.”

At the age of 20, Hawking began to show signs of a chronic disease of the central nervous system, which later led to complete paralysis. However, a serious illness that confined the scientist to a wheelchair for almost 40 years did not prevent him from showing the world the diversity of the Universe. The scientist himself dreamed of going into space, and in the last years of his life he repeatedly warned that humanity was doomed, and the Earth would die from an asteroid strike, high temperatures or overpopulation, and that it was only a matter of time.

Hawking began his research activities while studying at Cambridge, taught the theory of gravity, gravitational physics, and worked at the Institute of Astronomy, at the Department of Applied Mathematics and Theoretical Physics at Cambridge. At the California Institute of Technology, where he was invited in 1974, he worked, in particular, on issues of general relativity. In 1979, the physicist received the position of Lucasian Professor at the University of Cambridge and held it until 2009.

For more than 20 years, Hawking led a group dealing with problems around the theory of relativity and questions of gravity. In 2007, he founded the Center for Theoretical Cosmology at the University of Cambridge.

"Hawking radiation"

Cambridge University professor Hawking is known, in particular, for his theoretical prediction of the radiation of black holes, due to which they gradually evaporate, losing mass, and therefore information about the objects that have fallen into it. The discovery was called "Hawking radiation". It has significantly changed modern cosmological ideas. According to generally accepted ideas, an external observer cannot look inside a black hole and obtain any information about objects located beyond the event horizon. However, theoretically, Hawking radiation allows us to look inside a black hole, that is, to determine its internal topology.

Hawking radiation does not result from the movement of charges. It occurs when the properties of the vacuum change as a result of the formation of a black hole. If charges and masses produce only electromagnetic and gravitational waves, then electrons, positrons, protons and other particles can appear as a result of quantum Hawking radiation.

In Hawking radiation, the black hole will emit like an ordinary source heated to a certain temperature. In this case, the temperature will be inversely proportional to its mass: the larger the hole, the “colder” it is. When a black hole radiates, its mass decreases and its temperature increases, this follows from the correspondence between energy and mass according to the formula E=mc2. In this case, all characteristics of particles, except mass and charge, are emitted with equal probability.

The paradox of information loss

This paradox is formulated at the interface between quantum field theory and general relativity, so its resolution can help in formulating the theory of quantum gravity.

One of the pressing problems in modern theoretical physics is the disappearance of information in a black hole. The physicist offered his explanation. In his opinion, the information does not disappear and is not recorded somewhere inside the black hole - instead, it is stored on the surface of the event horizon of a supermassive object in the form of a hologram. The event horizon is the surface of a black hole, from within which light cannot escape. If the radiation source is located directly on the horizon, then the field it creates is visible as not changing over time, that is, there is no radiation. According to the holographic principle, if everything about the dynamics at the horizon is known, then the dynamics inside the black hole can be reconstructed.

Hawking described in his paper how each act of radiation is reflected on the event horizon of a black hole. In his opinion, using the holographic principle, it is possible to describe the details of the process of formation of black hole radiation. Hawking believes that if something happened inside or outside the black hole, then some kind of event is happening on the horizon.

In September 2015, Hawking announced a new idea that he believed would help resolve the 40-year-old paradox of information loss in black holes. The scientist referred in his message to some special properties of space. If you use them correctly, you can indicate how and in what form information leaves the black hole. The paper argues that Hawking radiation will have an infinite number of characteristics, not just a temperature distribution depending on mass, charge and angular momentum, and using these characteristics it will be possible to fully characterize the state of the black hole.

Prophet of the end of the world

One of Hawking's most popular works is A Brief History of Time. Published in 1988 with the subtitle “From the Big Bang to Black Holes,” the book immediately became a bestseller. Its circulation amounted to 10 million copies, translated into 40 languages. Hawking later wrote two more books: Black Holes and Young Universes (1993) and The World in a Nutshell (2001). In 2005, A Brief History of Time was published, a new edition of the 1988 bestseller.

Hawking tried to refute the postulate about the immutability of the Universe in an accessible language. “The light from distant galaxies is shifted towards the red part of the spectrum. This means that they are moving away from us, that the Universe is expanding,” he wrote.

“A dying star, contracting under its own gravity, eventually collapses into a singularity—a point of infinite density and zero size. If we reverse the course of time so that contraction turns into expansion, it will be possible to prove that the Universe had a beginning. However, "The proof based on Einstein's theory of relativity also showed that it was impossible to understand how the universe came into being: it demonstrated that all theories did not apply at the moment the universe began," the scientist notes.

He wondered what would happen when the Universe stopped expanding and began to contract. “It seemed to me that when the compression began, the Universe would return to an orderly state. In this case, with the beginning of the compression, time should have turned back. People at this stage would live their lives backwards and get younger as the Universe contracts,” he said.

Later, he comes to the conclusion that time will not turn back when the Universe contracts. "In the real time in which we live, the Universe has two possible fates. It can continue to expand forever. Or it can begin to contract and cease to exist at the moment of the 'big flattening'. It will be like a big bang, but in reverse." , - the physicist believes.

Hawking believed in the existence of extraterrestrial life. "In a Universe with 100 billion galaxies, each containing hundreds of millions of stars, it is unlikely that Earth is the only place where life develops. From a purely mathematical point of view, the numbers alone make the idea of ​​the existence of alien life absolutely reasonable. A real problem "is what aliens might look like, whether earthlings would like their appearance. After all, they could be microbes or single-celled animals, or worms that have inhabited the Earth for millions of years," Hawking said.

According to Hawking, the Universe will still have a finale, and humanity will have no choice but to conquer space and explore new planets, and we should start with the Moon and Mars. “Space settlement will completely change the future of humanity. It will determine whether we will have any future at all,” the scientist said at a science festival in 2017. “It is clear that we are entering a new space age. We are on the threshold of a new era. Human colonization of other planets is no longer science fiction, it could become scientific fact."

He was one of the outstanding theoretical physicists of our time. Over his difficult 76-year history (1942-2018), Hawking managed to form a number of fundamental assumptions and theories, thanks to which we today know a little more about how space and the physics of the surrounding world works.

Let's look back at some of the most important achievements of the talented scientist:

Seminal work on the study of singularity

A gravitational singularity is considered to be a one-dimensional point that contains an infinitely large mass in an infinitely small space. At a singularity, gravity becomes infinite, the space-time continuum bends indefinitely, and the laws of physics as we know them cease to exist.

Together with the English mathematical physicist Roger Penrose, Stephen Hawking carried out innovative research work, which resulted in a well-reasoned assumption of the existence of a singularity, and the theory that the Universe began with it.

Working on the discovery of the four laws of black hole mechanics

Together with James Barnid and Brandon Carter, Stephen Hawking discovered the four laws of black hole mechanics. These laws are similar to the laws of thermodynamics, and describe the physical properties that are believed to be inherent to black holes. In January 1971, Hawking's essay on black holes won a prestigious award. Gravity Research Foundation Award.

The theory that black holes emit radiation

For many years, physicists believed that nothing could escape the confines of a black hole. In 1974, Stephen Hawking proposed a theoretical argument that black holes still emit radiation, which continues to move until it expends all the energy and dissolves. Despite the controversy surrounding Stephen's theory, a new term has appeared in physics - “Hawking radiation”, work on the study of which continues today as part of experimental research at .

Contribution to the theory of cosmic inflation

Introduced by Alan Guth in 1980, the theory of cosmic inflation suggests that after the Big Bang the universe expanded exponentially before slowing down. Stephen Hawking was one of the first scientists to calculate the quantum fluctuations that arose during cosmic inflation, and also offered an explanation for how they could affect the expansion of galaxies in the Universe. Today, the inflationary model of the Universe has many supporters among the scientific community, but it also has a number of critics, including the previously mentioned English physicist Roger Penrose.

Assumption of an important model of the initial state of the Universe

In 1983, together with James Hartle, Stephen Hawking published a theoretical model known today as the Hartle-Hawking state. The theory suggested that time did not exist before the Big Bang, so the concept of the beginning of the universe is meaningless. The universe in the Hartle-Hawking state has no beginning, since it has no initial boundaries in time and space. The model remains one of the best-supported theories about the original state of the universe.

Work on the theory of “descending cosmology”

In 2006, Stephen Hawking, together with Thomas Hertog, proposed the theory of "top-down cosmology", which was that the Universe did not have one unique initial state, but consisted of superpositions of many possible initial states. Thus, if we do not know the initial state at the beginning of the Universe, we cannot obtain a bottom-up model. This leaves the possibility only for a top-down approach, since we know the final state of the Universe - this is the state we are in now. The theory found many supporters, largely due to the fact that it was also combined with the well-known string theory.

Authorship of the book "A Brief History of Time"

Stephen Hawking's book was published in 1988. "A Brief History of Time". In it, the scientist covers a range of topics from cosmology, including the Big Bang, black holes and cones of light. The book is written in down-to-earth, non-technical language to convey complex ideas clearly to the average reader. After its release "A Brief History of Time" it immediately became a bestseller and over 20 years sold more than 10 million copies. The book also managed to remain on the newspaper's bestseller list for a record 237 weeks British Sunday Times. The publication of the work seriously strengthened Hawking's international reputation as a scientist, after which the media came up with a nickname for Stephen - “master of the Universe.”

Work on a number of other popular books

After "A Brief History of Time" Some of Hawking's other works were published and became quite popular, including Black Holes and Baby Universes and Other Essays (1993), The Universe in a Nutshell (2001), On The Shoulders of Giants (2002), and God Created the Integers: The Mathematical Breakthroughs That Changed History (2005). Stephen also co-authored a series of fiction novels with his daughter Lucy Hawking.

Scientific awards

In 1974, a few weeks after the announcement of Hawking's radiation theory, Stephen became one of the youngest members of the Royal Society in London. In 1982 he was appointed Commander of the Order of the British Empire. In 1985 he received the Gold Medal of the Royal Astronomical Society. In 1987 he was awarded the Dirac Medal, awarded by the Institute of Physics for his invaluable contribution to science. Literally every new year, Stephen Hawking's research will be recognized with various awards from scientific communities around the world.

Place in the ranking of 100 Great Britons

In a 2002 poll conducted by the BBC in Great Britain, the public determined that Stephen Hawking deserves 25th place on the list of the 100 greatest Britons in history for his achievements.

You can also learn about other interesting moments from the life of an outstanding scientist from the documentary about Stephen William Hawking.

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Stephen Hawking, who made many scientific discoveries and assumptions about the structure of the world, was one of the most famous and popular physicists of our time. Hawking's main area of ​​research is cosmology and quantum gravity.

For his achievements, Hawking became a member of the Royal Society of London in 1974, and in 1975, thanks to the development of the theory of the quantum process of “evaporation” of black holes, he became a world famous scientist. The discovery of this radiation, which leads to the elimination of black holes, is named after Stephen Hawking.
The essence of the Hawking Radiation phenomenon is the process of emission of various elementary particles by a black hole. According to this theory, a black hole not only absorbs everything around it, but also emits various particles itself, mainly photons. When a black hole has nothing left to absorb around itself, it must begin to shrink, that is, emit particles outward. This should ultimately lead to her disappearance through an explosion in the final stages. This emission is called “Hawking Radiation” or Black Hole Evaporation.

According to Einstein's general theory of relativity, primordial black holes could be born during the formation of the Universe, and some of them should theoretically finish evaporating in our time. Since the intensity of evaporation, on the contrary, increases as the size of the black hole decreases, the last stages should, in fact, be an explosion of the black hole. However, no such explosions have been recorded so far.
By the way, Hawking radiation turned out to be a serious argument for the assumption of the emergence of small black holes during experiments conducted at the Large Hadron Collider. However, a microscopic black hole formed in this way, according to the same assumption, should instantly evaporate due to Hawking radiation and cannot pose a threat to the life of earthlings...
…. “I think that our mind is a program, while our brain is analogous to a computer. It is theoretically possible to copy the contents of the brain onto a computer and thus create a form of eternal life. Today, however, this is not in our power,” Hawking concluded.
According to Hawking, the past is just a possibility. One of the implications of the theory of quantum mechanics is that events that happened in the past did not happen in any particular way. Instead, they happened in every possible way. This is difficult for us to understand, and is due to the probabilistic nature of matter and energy according to quantum mechanics: until there is an outside observer, everything will float in uncertainty.....
…..Based on his knowledge, Stephen Hawking warns that artificial intelligence may well put an end to the human race, and the first contact with an intelligent extraterrestrial civilization may be the last. According to him: if alien technology surpasses that of Earth, they will definitely form their own colony on Earth and enslave humanity.
But there is also good news. Recently, a famous English physicist said that modern science is on the eve of a revolution, when a unified theory will be created that explains all the fundamental principles of the physical world and existence. Moreover, according to Hawking, the discovery will be made within the framework of the M-theory, which assumes the presence of parallel worlds and numerous physical forces that are still unknown to modern science.
The so-called M-theory or “Theory of Everything,” proposed by Edward Witten in the 90s of the 20th century, was conceptualized and refined by Hawking and his colleague Leonard Mlodinow. M-theory is a branch of string theory and describes the entire Universe at once. According to it, at the smallest level, all particles consist of branes - multidimensional membranes, the properties of which can explain absolutely all processes occurring in our Universe. By the way, this theory also assumes the existence of a huge number of universes in which physical laws different from ours apply.
Hawking claims that the creation of the Last Theory will finally complete the orderly edifice of theoretical physics. "We will learn the fundamental laws that govern the Universe." Well, given the fact that over the last hundred years of its history, humanity has made a huge innovation leap, there is no reason not to trust the prediction of the supremely intelligent scientist - Stephen Hawking.

Stephen Hawking

Theory of everything

Translation of the original edition:

The Theory of Everything

Reprinted with permission Waterside Productions Inc and the literary agency "Synopsis".

© Phoenix Books and Audio, 2006

© AST Publishing House LLC, 2017 (translation into Russian)

Introduction

In this series of lectures, I will try to outline our understanding of the history of the Universe from the Big Bang to the formation of black holes. The first lecture is devoted to a brief overview of the ideas about the structure of the Universe that were held in the past, and a story about how the modern picture of the world was built. This part can be called the history of the development of ideas about the history of the Universe.

In the second lecture, I will describe how Newton's and Einstein's theories of gravity led to the understanding that the Universe cannot remain unchanged - it must either expand or contract. From this, in turn, it follows that at some time in the interval from 10 to 20 billion years ago the density of the Universe was infinite. This point on the time axis is called the Big Bang. Apparently, this moment was the beginning of the existence of the Universe.

In the third lecture I will talk about black holes. They are formed when a massive star or larger cosmic body collapses under its own gravity. According to Einstein's general theory of relativity, anyone foolish enough to fall into a black hole will remain there forever. No one will be able to get out of there. At the singularity, the history of the existence of any object comes to an end. However, the general theory of relativity is a classical theory, that is, it does not take into account the quantum mechanical principle of uncertainty.

In the fourth lecture, I will explain how quantum mechanics allows energy to escape from a black hole. Black holes are not as black as they are made out to be.

In the fifth lecture, I will talk about the application of the ideas of quantum mechanics to solving questions related to the Big Bang and the origin of the Universe. This will lead us to understand that spacetime can be finite, but have no boundary or edge. It resembles the surface of the Earth, but with two more dimensions added.

In the sixth lecture I will show how this new boundary assumption can explain why the past is so different from the future even though the laws of physics are time symmetrical.

Finally, in the seventh lecture, I will talk about attempts to formulate a unified theory that covers quantum mechanics, gravity, and all other physical interactions. If we succeed, we will truly be able to understand the Universe and our place in it.

Lecture one

Ideas about the Universe

Back in 340 BC. e. Aristotle, in his treatise On the Heavens, formulated two compelling arguments in favor of the fact that the Earth is spherical and not flat like a plate. First, he realized that lunar eclipses are caused by the Earth passing between the Sun and the Moon. The shadow of the Earth on the Moon is always round, and this is only possible if the Earth has a spherical shape. If the Earth were a flat disk, the shadow would be elongated and elliptical unless the Sun was directly above the center of the disk at the time of the eclipse.

Secondly, from the experience of their travels, the Greeks knew that in the southern regions the North Star is lower above the horizon than in the more northern regions. Based on the difference in the apparent positions of the North Star in Egypt and Greece, Aristotle even gives an estimate of the circumference of the Earth - 400 thousand stadia. It is not known exactly what one stage is equal to (perhaps about 180 meters). Aristotle's estimate is then almost twice the value currently accepted.

The ancient Greeks also had a third argument in favor of the fact that the Earth should be spherical: otherwise why do the sails of an approaching ship first appear on the horizon and only then does its hull become visible? Aristotle thought that the Earth was stationary, and the Sun, Moon, planets and stars moved in circular orbits around it. He thought so because, due to mystical considerations, he was convinced that the Earth is the center of the Universe, and circular motion is the most perfect.

Aristotle believed that the Earth is motionless, and the Sun, Moon, planets and stars move in circular orbits around it.

In the 1st century AD e. this idea was developed by Ptolemy into a holistic cosmological model. The Earth is located in the center, surrounded by eight spheres bearing the Moon, Sun, stars and the five planets known at that time: Mercury, Venus, Mars, Jupiter and Saturn. The planets move in circles of smaller radii, which are associated with the corresponding spheres. This was required to explain their rather complex observed trajectories of movement across the sky. On the outer sphere there are the so-called fixed stars, which maintain their positions relative to each other, but all together make a circular motion across the sky. What lies beyond the outer sphere remained unclear, but this part of the Universe was undoubtedly inaccessible to observation.

Ptolemy's model made it possible to quite accurately predict the positions of celestial bodies in the sky. But to do this, Ptolemy had to admit that sometimes the Moon comes twice closer to the Earth than at other moments of its movement along the predicted trajectory. This meant that periodically the Moon should appear twice its normal size. Ptolemy was aware of this shortcoming, but despite this, his model was accepted by most, although not all. It received the approval of the Christian Church as a picture of the world consistent with the Holy Scriptures. After all, this model had a huge advantage, since it left enough space for heaven and hell behind the sphere of the fixed stars.

An ancient drawing depicting various cosmological models that explained the movement of the planets. The central diagram shows a heliocentric (the Sun is in the center) model of the movement of the six planets known at that time, their satellites and other celestial bodies revolving around the Sun. From the second century, the geocentric (Earth in the center) Ptolemaic system (top left) became the dominant model. It was succeeded by Copernicus's heliocentric system, published in 1543 (bottom right). The Egyptian model (bottom left) and Tycho Brahe's model (top right) attempted to preserve the idea of ​​a stationary Earth as the center of the universe. Details of the planets' orbits are given on the left and right.

From the Illustrated Atlas by Johann Georg Heck, 1860.

However, in 1514, the Polish priest Nicolaus Copernicus proposed a much simpler model. At first, fearing accusations of heresy, he published his model anonymously. He believed that the stationary Sun was in the center, and the Earth and planets moved around it in circular orbits. Unfortunately for Copernicus, it was almost a hundred years before his ideas were taken seriously. Then two astronomers - the German Johannes Kepler and the Italian Galileo Galilei - publicly came out in support of the Copernican theory, despite the fact that the orbits predicted on the basis of this theory were somewhat different from those observed. The dominance of the Aristotle-Ptolemy theory came to an end in 1609, when Galileo Galilei began studying the night sky using the newly invented telescope.

In 1609, Galileo Galilei began studying the night sky using a newly invented telescope.

While observing Jupiter, Galileo noticed that the planet was accompanied by several small satellites (moons) that orbited around it. This meant that not all celestial bodies had to revolve around the Earth, as Aristotle and Ptolemy thought. Of course, it was still possible to assume that the Earth is motionless and located at the center of the Universe, and the satellites of Jupiter move along extremely complex trajectories around the Earth, so that the appearance of their revolution around Jupiter is created. However, Copernicus' theory was much simpler.

At the same time, Kepler developed the Copernican theory, suggesting that the planets move not in circular orbits, but in elliptical ones. Now the theory's predictions have finally coincided with observations. As far as Kepler was concerned, elliptical orbits were only an artificial hypothesis, and a very unfortunate one, since the ellipse was considered a less perfect figure than a circle. Having discovered (almost by accident) that elliptical orbits corresponded well with observations, he could not reconcile this with his idea that the planets revolve around the Sun under the influence of magnetic forces.

The explanation was found much later, in 1687, when Newton published his work "Mathematical principles of natural philosophy". This is perhaps the most important work on physics ever published. In it, Newton not only proposed a theory of the movement of bodies in space and time, but also developed a mathematical apparatus for analyzing this movement. In addition, he formulated the law of universal gravitation. This law states that all bodies in the Universe are attracted to each other with a force, which is greater, the greater the mass of the bodies and the closer to each other they are located. This is the same force that causes objects to fall to the ground. The story of the apple falling on Newton is almost certainly fictional. Newton himself only mentioned that the idea of ​​gravity came to him when he was in a contemplative mood and noticed an apple falling.

One of the most famous and popular physicists of our time, Stephen William Hawking, died at the age of 77. During his life, he made many discoveries and assumptions about the structure of the world. TengriMIX shares some of them with you.

The past is a possibility

Hawking believed that one of the implications of quantum mechanics was that events in the past did not happen in any particular way, but in all possible ways. This is due to the probabilistic nature of matter and energy. Until there is an outside observer, everything will float in uncertainty.

“No matter what memories you currently hold of the past, the past, like the future, is uncertain and exists as a spectrum of possibilities,” Hawking said.

Aquarium fish are depressed

Several years ago, the city council of Monza in Italy banned pet owners from keeping ornamental fish in round aquariums. The law was intended to protect poor fish from distorted perceptions of reality, since indirect light could give them a distorted picture of their habitat.

Hawking used this example to illustrate the fact that it is impossible to know the pure nature of reality. We think we have an accurate picture of what is happening around us. But what if we knew that we were living in a metaphorical giant fishbowl with bulging sides and no way of knowing what was actually going on around us to compare?

There is a "Theory of Everything"

If there is any "theory of everything" that can describe the entire universe, it is M-theory. It was first proposed by Edward Witten in the 1990s and was later conceptualized and refined by Hawking and his colleague Leonard Mlodinow. M-theory is a branch of string theory and describes the entire Universe at once. According to it, at the smallest level, all particles consist of branes - multidimensional membranes, the properties of which can explain absolutely all processes occurring in our Universe. By the way, this theory also assumes the existence of a huge number of universes in which physical laws different from ours apply.

Light power

Here's a fun fact: a one-watt night light emits billions and billions of photons per second. Photons are small packets in which light comes. They, like all particles, behave both as particles and as waves of particles.

Quarks are never alone

Quarks, the "building blocks" of protons and neutrons, exist only in groups and never alone. The force that binds quarks together increases with the distance between them, so if you try to pull one quark away from another, the harder you pull, the harder it will try to break free and come back. Free quarks do not occur in nature.

The universe gave birth to itself

Hawking was a convinced atheist. He devoted a lot of time to scientifically proving that God is not needed for life to exist. One of his famous sayings is: "Because there is such a force as gravity, the universe could and did create itself out of nothing. Spontaneous creation is the reason why the universe exists, why we exist. There is no need for God to "ignite" "fire and make the universe work."

General relativity has implications for navigation system errors

The general theory of relativity was formulated by Einstein in 1915. It postulates that gravitational effects are caused not by the force interaction of bodies and fields located in space-time, but by the deformation of space-time itself, which is associated, in particular, with the presence of mass-energy.

Hawking acted as a popularizer of this theory. He argued that if general relativity was not taken into account in GPS navigation satellite systems, errors in determining global positions would accumulate at a rate of about ten kilometers per day. It is important to understand that the closer an object is to Earth, the slower time passes. Thus, depending on how far the satellites are from Earth, their onboard clocks will operate at different speeds. We could compensate this difference automatically if this effect were taken into account."

Prepared by: Nurlyaiym Nursain