In the last few years a revolutionary new technique has been developed to edit genes. In this VERITAS I will describe the technique and how it has the potential to completely transform the way we fight diseases and how it may help us come up with new disease resistant varieties of plants and animals.

The name of the technique is CRISPR-Cas9. It gives scientists a way to quickly and precisely cut a piece of DNA of any animal or plant and replace it with a different piece of DNA. Scientists have already done a lot of experiments with this technique and have achieved amazing results. Scientists in Japan have used this technique to create tomatoes that have a longer life by turning off genes that ripen them. Chinese scientists have created a variety of wheat that is extremely resistant to mildew. Scientists are proposing that people in haemophilia can be treated by editing their stem cells outside their body and then reinjecting these cells into the body. And as we will see in this article, scientists are trying to use this technique to conquer cancer, malaria and various other diseases.

A potential application of CRISPR-Cas9 is organ transplant. Every year millions of people around the world get into the waiting list to receive organ transplants. Most of them never get the organs and they die waiting. For many decades scientists have been proposing the idea of using animal organs to help people with organ transplant needs. The animal that has been identified is the pig and that is because the size of a pig’s organs is very close to that of humans. However, there is a problem: PERVs. The pig’s DNA is full of virus like structures called PERVs and these are able to infect humans. Till you solve the PERV problem, you cannot use pig organs in humans. Scientists at Harvard used CRISPR-Cas9 to edit the DNA in a pig cell to remove all 62 PERVs. And when these edited cells were mixed with human cells, none of the human cells got infected. Scientists are confident that in a few years people will be able to receive pig organs and not have to rely on human donors alone. This will solve the shortage of organs immediately and also put an end to the trafficking of human organs.

Now, lets look at how CRISPR works. The idea of genetic engineering is not new. More than 40 years ago scientists discovered how to take a piece of DNA of one organism and pasting it into the DNA of another organism to create a desired trait. This technique is called Recombinant DNA technology. But there lies a danger in this approach. One could also transfer viruses from one species to another. Also, recombinant genes are not “natural” in the sense that in nature the two species may never have mated. For example, people with diabetes are treated with insulin that is produced by inserting human insulin gene into E. Coli or Yeast. So, it works but it is unnatural and this is one of the reasons for the huge public furore over genetically modified(GM) crops. CRISPR is different. DNA sequences from other organisms are not pasted into one organism’s genome. CRISPR is a technique to edit the existing DNA using “molecular scissors”.

The fundamental idea behind this revolutionary technique was not invented by us. It was discovered by observing bacteria and how they fight against invading viruses. In 1980s and 1990s scientists discovered that the DNA of bacteria contained short, repeating DNA sequences separated by short, non-repeating, “spacer” DNA sequences. These sequences were called CRISPR(Clustered regularly interspaced short palindromic repeats ). At first scientists did not understand the reason of these patterns. But then they found that these patterns are an essential components of the immune system of these bacteria. This is how it works: When a previously unknown virus invades a bacteria, a particular enzyme called Cas cuts a small piece of the invading virus’ DNA and inserts it as a spacer in the bacteria’s CRISPR DNA. So the spacer sections of a bacteria’a DNA are small pieces of the DNA of viruses that invaded it in the past. Thus the bacteria is keeping a memory of the genetic structure of viruses that have attacked it in the past. If in the future the bacteria is attacked by a virus of the same kind, the bacteria’s immune system can react very quickly. This is because it has a copy of a part of the virus’ DNA from the previous attack. The bacteria’s CRISPR system would now make two small strands of RNA with exactly the same sequence as the Virus’s saved DNA – these two RNA strands are called Crispr RNA ( crRNA). A Cas9 molecule attaches to the CrRNA and together this system goes to attack the virus. The CrRNA is used to find the exact location of the virus’s DNA where the attack will happen. When the match between the CrRNA and the virus’s DNA occurs, the Cas9 cuts the virus’s DNA rendering its gene ineffective and making the virus unable to continue the attack. So, the CrRNA acts as a guide to where the cutting will take place and the Cas9 acts as a molecular knife which does the cutting. This is shown in the following image from Wikipedia.

Scientists thought of using this technique to edit human, animal and plant DNA for our benefit. We can create a RNA sequence of any base pairs. The first step is to create a RNA sequence of the DNA sequence that we are targeting. They called this guideRNA. If we combine this with a Cas9 molecule, we have a very precise DNA cutting tool. The guideRNA will go to the right place in the target organism’s DNA and then Cas9 will cut it off. So we can remove genes, or even modify them. Once we cut one sequence of the DNA, we can also replace it with a similar sequence with slight modifications. For this to happen, scientists also send a repair template. When the target DNA is cut, it tries to repair the cut. A DNA strand typically repairs the cut based on the sequence of the complementary strand. But if we use CRISPR-Cas9 to cut both DNA strands, then the DNA has no way of knowing what sequence to recover while repairing. If we do nothing, the DNA will simply bind the two cut portions together. But if we send a repair template, the DNA will think that it is the complementary DNA strand and creates a sequence just like it. So, it “looks” at the repair template and creates a sequence just like that. So we fool the target DNA to repair its cut portion but instead it making the same sequence as the original, we create a slightly different sequence- one that is beneficial to us. This technique of fooling the DNA to create a new sequence based on the template we send it is called homology directed repair.

Thus we see that we can use the CRISPR-Cas9 mechanism to either remove some harmful genes by cutting them off both strands of the DNA or we can modify the genes by cutting off the original gene and then fooling the DNA to repair it using the template that we give it. Thus using this technique we can achieve very precise gene editing results. And it can be done at a small fraction of the cost and time compared to traditional genetic engineering methods. It is estimated that CRISPR techniques cost about one hundredth of the cost of traditional genetic engineering methods. Also, the time taken to do one CRISPR experiment is about a week whereas traditional genetic engineering methods could take more than a year. And to top it all, CRISPR is a very precise technique- something that traditional techniques cannot even hope to match. So, we have a revolutionary new scientific tool in our hands. And the possibilities are endless.

Scientists have tried to use CRISPR to cure mice of HIV. And they have been successful. Scientists were able to cure live mice of HIV infection using this technique. Scientists also believe that CRISPR techniques will help us battle and possibly cure cancer. If we can use CRISPR to modify the DNA of the cells of the immune system of our bodies and command these cells to seek out and destroy the cancer cells, we will win the battle over this dreaded disease.

There is another way in which scientists are thinking of using CRISPR-Cas9 system: Fighting mosquito borne diseases like Malaria, Dengue and Zika. The mosquito has been the cause of billions of deaths in human history. Mosquito borne diseases have caused more deaths than all wars in history combined together. Every year nearly half a million people die of malaria. But mosquiotoes do not themselves cause malaria or any other disease that we associate with mosquitoes. Mosquitoes are carriers of the microbes that cause these diseases. So how about using CRISPR-Cas9 to change the DNA of mosquitoes to be resistant to these mosquitoes. If we do that, mosquitoes will not carry the microbes and will not spread diseases. But there is a problem: how many mosquitoes will scientists modify? If they modify just a few, they will not be able to make a big difference. The aim should be to create a DNA change that is carried by the offspring also. And the DNA change has to be dominant so that when a genetically modified mosquito mates with a wild one, the changed DNA is carried on to the next generation. So the gene modifications that we do should be dominant over the original gene. Scientists have devised such a mechanism- again using CRISPR-Cas9. It is called gene drives. We have already created genetically modified mosquitoes that are not capable of spreading malaria and with gene modifications that will always be dominant. If we use this technique the hope is that within a few generations most mosquitoes will be resistant to the microbes carrying deadly diseases.

So, we have a very powerful new tool in our hands. But we also need to be very careful. Whenever in history humans have had access to a powerful tool, we have also misused it and caused immense harm to ourselves. With CRISPR we need to be careful not to use it to satisfy the lust for power or wealth. For instance, a rogue country may use it to create an army of super-humans. Or someone could use it to create new kinds of diseases and use those during wars. We have a great responsibility to use this power carefully. The aim of science should be to elevate the human race and not to satisfy our lowest animal desires and needs.

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

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Emmy Noether was born to a Jewish family in the German town of Erlangen in 1882. Her father was a professor of mathematics at the University of Erlangen. At that time middle class European girls were expected to learn languages, arts, music, dancing and also household activities. When she passed out of high school she got a certificate which would allow her to become a language teacher. But Emmy had other plans. At the age of 18 she decided to study mathematics at her father’s university. At that time women were not allowed to take classes at the university. She was, however, given permission to “audit” classes. So, she sat in the classes but not as a regular student. But she did so well in her studies that she passed out 7 years later(in 1907) with a doctorate in mathematics. From 1908 to 1915 she worked without pay and without title at the Mathematical Institute of Erlangen. This was because the university had a policy that did not allow hiring women professors. In 1915 she was invited by David Hilbert and Felix Klein to join them at the university of Gottingen. Hilbert and Klein are very famous names in the history of mathematics and physics. Though she was allowed to lecture there, it was many years later that she was formally recognised as a teacher at the university. It was at the university of Gottingen that she formulated a very important theorem of physics that is now known as Noether’s theorem. We will learn more about this theorem after finishing our story about Emmy’s life.

In 1920s Emmy did a lot of work on abstract Algebra. This work is considered to be of fundamental importance by mathematicians. She worked on group theory, ring theory and number theory. During this time she also lectured throughout Europe and was recognized as a mathematician of the first rank. In 1932, The International Mathematical Congress in Zurich asked her to give a plenary lecture. This was the high point of her mathematical career. But, in 1933 the rise of Hitler changed all that. The Nazis removed Jews from all government jobs. Emmy lost her right to lecture at the university. Many Jewish scientists left Germany and migrated to other counties. Even Einstein had to leave Germany and he joined Princeton University in USA. Emmy too left Germany and joined Bryn Mawr College in USA. She died two years later from complications following a surgery. She was 53.

Now, lets look at Noether’s theorem and understand why it is of fundamental importance to Physics. When I first read this theorem I was amazed at what this theorem showed. This theorem showed that two of the deepest concepts in Physics and Mathematics are related at a very deep level. The two ideas are symmetry and conservation laws. Emmy Noether showed that every symmetry implies the presence of a conservation law and vice versa. It is a beautiful idea and a very surprizing one too. To understand Emmy’s wonderful theorem, lets first define what we mean by symmetry and conservation laws.

We all know about symmetry. However, the definition of symmetry in everyday life is different from the definition in mathematics. In everyday life we typically call an object symmetric if we can draw an imaginary line through it and the two parts on either sides of the line look exactly the same. For example a circle is symmetric because if we draw a line through the centre, the two parts look exactly the same. We humans are attracted to symmetry. We find it visually appealing and we try to create objects that exhibit some form of symmetry. We ourselves are symmetric to a large extent – if we draw a vertical line through the centre of our nose, the two sides look nearly the same. But this is one kind of symmetry- mirror symmetry. Mathematicians generalized it and formed more kinds of symmetry. For example, if you take and object and rotate it by an angle and if the rotated object looks the same as the original object, it is said to have rotational symmetry. Take a square, and rotate it about its centre by 90 degrees(clockwise or anticlockwise), the result is a square that looks the same as the original square. So a square possesses rotational symmetry( of course, it also possesses mirror symmetry). So the general mathematical definition of symmetry is this: take an object or system; do some transformation on it; if the transformed object or system is the same as the original one then we can say that the system is symmetric with respect to the transformation. I will give you some examples later when we discuss Noether’s theorem.

Now, lets briefly talk about Conservation laws. In Physics there are some quantities that never change if the system is closed. Such quantities are said to be conserved and the laws that “assure” this are called conservation laws for these quantities. Lets take an example: The law of conservation of energy says that the total energy of a closed system always remains the same. Energy can change form- Kinetic energy can change to potential energy. Energy can change to mass and mass can turn into energy. But the sum total of all the energy remains constant. When we use the word closed system, we mean that no energy comes in or flows out of the system. Like the conservation of energy, there are many other conservation laws- law of conservation of momentum, law of conservation of angular momentum, law of conservation of charge etc. Many years back( in the year 2000), I wrote a VERITAS series on the conservation laws. You can see the series here: https://unvarnishedveritas.wordpress.com/tag/conservationlawsweek/

Emmy noticed a very deep connection between conservation laws and symmetry and this is what Noether’s theorem is all about. Emmy showed that every symmetry in nature results in a conservation law of physics and vice versa.

But from a physics point of view, what does it mean when we say that a system is symmetric with respect to a transformation. So, in other words, when we compare a system with respect to the transformed system, how do we check if they are the “same”? From the point of view of Physics, the two systems are the same if their Lagrangian is equal. The Lagrangian is a quantity named after the great mathematician Joseph-Louis Lagrange. It is the difference between the kinetic and potential energy. So L= T-V where T is kinetic energy and V is potential energy. If a system’s Lagrangian stays the same when it is transformed then we can say that the system is symmetric with respect to the transformation.

Lets take some examples of conservation laws and find out which symmetry results from each of them. The simplest case to understand is that of translational symmetry. A system is said to have translational symmetry if we can change its coordinates by a fixed amount( by addition or substraction) without changing the equations of physics that govern it. Or, in other words, if we take a system and then add a fixed amount to its x, y and z coordinates and if the Lagrangian stays the same, then we sat that the system has translational symmetry. And Noether’s theorem tells us that translational symmetry implies law of conservation of linear momentum. Isn’t it remarkable? Lets look at it this way: The reason why linear momentum is conserved in our universe is because particles or systems of particles display translational symmetry.

Lets now look at the law of conservation of angular momentum. What symmetry is it based on? Noether was able to show that rotational symmetry implies the law of conservation of angular momentum. In more precise terms, if you take a system and then rotate it and the Lagrangian of the system does not change due to the rotation, then the system will exhibit the law of conservation of angular momentum.

There is one symmetry-conservation law relationship that I find particularly interesting. And very surprising. Lets take a system and calculate its Lagrangian. Let’s now wait for some time and calculate its Lagrangian again. When I say wait for some time, I mean any length of time. So if the Lagrangian is the same in both the cases then we can say that the system was symmetrical with respect to time change. In this case the quantity that is conserved is Energy. Isn’t that amazing! The very familiar conservation of Energy is due to symmetry with respect to time changes. This symmetry is different from the usual notion of symmetry which is spatial in nature. This particular symmetry has nothing to do with space and is based on time alone. So Noether was able to show a deep relationship between Energy and Time. And later when Quantum Mechanics was discovered( or rather formulated), we again found that Energy and Time are related through Heisenberg’s uncertainty relationship.

Another conservation law that we are familiar with is the law of conservation of electric charge. What symmetry is at the root of this? Quantum Field theory tells us that charge conservation is due to gauge invariance( or symmetry) of the electrostatic potential and vector potential. It is not easy to explain this in simple terms. But it is interesting to note that this relationship between conservation of charge and gauge symmetry was discovered several decades after Noether proposed her famous theorem.

There are other conservation laws too: CPT( charge, parity and time), colour charge etc and each of these conservation laws has been shown to be related to a symmetry of some transformation. Note that some of these do not follow directly from Noether’s theorem. There is also a Quantum version of Noether’s theorem. It is called Ward-Takahashi identity.

Noether did not just give us the relationship between a few conservation laws and the underlying symmetry of nature. She gave us a recipe to find the symmetry if the conservation law is known and vice versa. So, if you discover a previously unknown symmetry in nature, you have not just discovered a symmetry. You have also discovered a new law of conservation. You can use Noether’s theorem to find the conserved quantity.

So we see that Emmy Noether made tremendous contributions to Physics. We have just explored a simplified version of her famous theorem. We have not been able to even briefly look at her contributions to mathematics. The next time some kid mentions the names of Isaac Newton and Albert Einstein as great scientists, please do name Emmy Noether also. It is important that women mathematicians get recognized for their great contributions.

Many years back, in 2002 , I wrote an article about another great woman mathematician, Sophie Germain: https://unvarnishedveritas.wordpress.com/2002/11/27/women-of-mathematics-sophie-germain/

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

====== ======= =========== ============================== =============

This is the seventh and last part of the VERITAS series on General Relativity. Just to remind our readers: We are doing this series on Einstein’s Theory of Relativity to celebrate the 100th Anniversary of General Theory of Relativity. If you have missed any of the first six parts, you can read them here:

https://unvarnishedveritas.wordpress.com/tag/100-years-of-general-relativity/

In the previous parts we have studied the basic ideas behind special and general relativity. We discussed how gravity bends space time and slows down time. We have also explored one of the most interesting and astonishing results of relativity: black holes. And then we applied general relativity to the whole universe to understand how it evolved after Big Bang and how it may end up billions of years from now. If you have not read the previous parts, I suggest you read them before reading this one.

In this article we will discuss Gravitational waves which are a consequence of Einstein’s theory of General Relativity and were predicted by him in 1916. Gravitational waves have recently been detected for the first time- an event that made headlines in scientific journals and also in newspapers. We will also discuss the implications of this discovery and why this discovery is of immense importance to science and technology.

In the earlier episodes of this series we saw that according to General Relativity, Gravity is a distortion in space-time caused by objects with mass. What happens when this mass moves? The distortion moves too. Now imagine what will happen if the mass accelerated? According to General Relativity, a massive accelerating body would create ripples in space-time or gravitational waves that travel at the speed of light. In other words, waves of distorted space would move from the accelerating body outwards and travel in all directions at the speed of light. Though these were predicted in 1916, they had never been detected till September 2015.

Any accelerating body will produce gravitational waves. So, I, you, our cars, our pets, every object that we know produces gravitational waves when accelerated. However most of these gravitational waves are so weak that there is no hope of detecting them. Let’s look at objects/phenomenon that generate powerful gravitational waves. The most powerful gravitational waves would be generated by colliding black holes, supernovas( the collapse of stellar cores), coalescing neutron stars(when two neutron stars come together to form one), coalescing white dwarf stars, the wobbly rotation of imperfectly shaped neutron stars and the remnants of gravitational radiation created by Big Bang.

Here are some properties of gravitational waves:

1) Gravitational waves can theoretically have any frequency. The frequency depends on the kind of acceleration that the cause of these waves undergoes. Astrophysical sources are likely to produce waves that have a frequency ranging from 10^-16Hz to 10^4 Hz. Higher frequencies would come from waves that were created by Big Bang.

2) When gravitational waves pass an observer, he will notice that the distance between objects increase and decrease at the frequency of the wave. This happens because gravitational waves are waves of distortion of space. We will see how this effect was used to detect these waves.

3) Unlike electromagnetic radiation, gravitational waves interact very weakly with matter. So they travel through the universe virtually unchanged. This is a very important property and we will discuss this later in this article.

Now lets understand how gravitational waves can be detected and the technique used for the first detection that happened in sept 2015. The amplitude of the waves that reach earth are so small that detecting and measuring them is a very complex engineering task. Let’s take an example to show you how tiny are the wave amplitudes that reach earth. Let’s take a binary star system- a binary star system is two stars that revolve around each other. So these stars are being accelerated as they revolve around each other. To make the acceleration larger, let’s make one of these stars a Pulsar. So we have a “normal” star revolving around a Pulsar. This system would produce gravitational waves of an amplitude of 10^-26 m on Earth. Detecting a wave of such a small amplitude would take an engineering marvel.

So we need to do two things to detect gravitational waves: we need to find a way to detect extremely small amplitudes and we need to find stronger sources of gravitational waves. Merging black holes may produce gravitational waves that may have an amplitude of 10^-20 m on Earth. Since these may the most powerful sources of gravitational waves, we need to be able to detect amplitudes of this order of magnitude to claim that we have found gravitational waves.

The property of gravitational waves that we can use for their detection is that when gravitational waves pass an observer, he will notice that the distance between objects increase and decrease at the frequency of the wave. See the following animation from Wikipedia:

( if you see a static image and not an animation, please click this link to see it: https://en.wikipedia.org/wiki/Gravitational_wave#/media/File:GravitationalWave_PlusPolarization.gif)

Imagine a set of particles set into a circular ring. When there are no gravitational waves passing the ring, it appears perfectly circular. However, when gravitational waves pass through the ring, the space in which the ring resides gets distorted and exhibits an “oscillation” like the one depicted in the animation. Now, what you see in this animation is hugely exaggerated. The effect is so small that it is extremely difficult to observe. And that brings us to the marvellous experiment that actually observed these waves.

Laser Interferometer Gravitational-Wave Observatory (LIGO) is one of the grandest and most ambitious experiments ever performed. There are two LIGO interferometers, one in Louisiana state in USA and the other in Washington state in USA. Each of these two observatories is shaped like a L with each arm of the L 4 km long. See the Wikipedia image below:

In the above figure we see that light from a laser source hits a beam splitter and is split into two beams. Each of these beams travels along one of the arms of the L. As discussed earlier, each of these arms is 4 Km long. From the diagram it seems that the two light beams will reflect back after travelling the 4 km arms and meet again. But that is not the case. It is not shown in the diagram, but to improve the accuracy of the experiment, each beam is reflected 280 times before it is allowed to meet the other beam. The setting of the mirrors is such that when the two beams meet after having travelled a distance of 280X8 km, they will be completely out of phase ie, when the meet each other they neutralize each other( this is called destructive interference) and the photo detector is not “fired”. Now, lets see what happens if a gravitational wave passes through the observatory. Depending on the direction of propogation of the wave, one of the arms will become slightly shorter or longer than the other one. This is because the wave will cause a distortion of space and the length will change. This causes the two beams to loose their out of phase setting. They become slightly in-phase and thus do not cancel each other completely. And therefore the photodetector fires. Note that I have simplified the method of functioning of the observatory but the basic principle of the observatory is as described. The amazing thing is the accuracy with which it can measure the difference in length between the two arms and the amplitude of gravitational waves that can be detected. The LIGO can detect gravitational waves of amplitude 5 X 10^-22 m! That, I would say, is an astounding level of accuracy! But what happens if a LIGO detector experiences a vibration due to a seismic event on earth? That may also disturb the legs of the L and cause a wrong detection. And that is the reason why we have two LIGO detectors thousands of miles apart. Any gravitational wave reading at one LIGO must be followed by another reading at the second LIGO detector and the difference in time between these two detections must match the time required by light to travel the distance( gravitational waves travel at the speed of light).

An interesting thing about LIGO is that it is connected to Einstein and his ideas in many ways: LIGO is based on Michelson-Morley experiment performed in 1887. The results of this was used by Einstein to formulate Special Relativity. LIGO uses Lasers. The idea of lasers came from a 1917 paper by Einstein. LIGO uses a photo detector- and we know that Einstein explained photoelectric effect in 1905. And finally LIGO is used to detect Gravitational Waves which Einstein predicted in 1916.

On 14th Sept, 2015, LIGO detected the first gravitational wave. The characteristics of the waveform matched what Einstein had predicted exactly a 100 years earlier. Scientists analysed this signal and concluded that it was caused by the merger of two black holes. Scientists estimated that the black holes were about 29 and 36 times the mass of the sun and the event took place 1.3 billion years ago( light took 1.3 billion years to reach us). About 3 times the mass of the sun was converted into gravitational waves in a fraction of a second. There was a 7 ms difference between the detection at the two LIGO observatories. Scientists could use this information to determine the direction of the black holes relative to the earth. And then on 26th December 2015, another signal was detected at the LIGO detectors. This signal again came from the merger of two black holes, one 14 times the mass of the sun and the other 7 times the mass of the sun. The combined entity was a black hole of about 20 solar masses. Thus a mass of about 1 solar mass( mass of our sun) was shot outwards as gravitational waves. The event took place about 1.4 billion light years away.

Now, let’s understand why gravitational waves are important. First, gravitational waves was the only idea proposed by Einstein’s general theory of relativity that had still not been confirmed. And now we know that Einstein’s theory of general relativity was correct in all aspects. And the detection of gravitational waves exactly a 100 years after Einstein proposed his theory was a wonderful celebration of the 100th anniversary of the great theory. But there is a bigger significance of the detection. Gravitational waves opens up a new window to study the universe. Our current window to study the universe was opened when Galileo first looked at the night sky using a telescope. That changed the world! Since then scientists have relied on electromagnetic waves( light, radio waves, x rays etc) to study the universe. Electromagnetic waves have a problem: they interact with matter and this interaction can disrupt them or impede them. Gravitational waves pass through the universe without any obstacles. So, while light from distant stars may be blocked by interstellar dust or other objects, gravitational waves will pass through without problems. So we will be able to “see” more, much more. Also, not all objects and events emit electromagnetic radiation. For example, two colliding black holes do not emit electromagnetic radiation but they emit a lot of gravitational waves. So, we have found a new way to look at the Universe. And many people believe that this will make a huge impact on the evolution of science from now on.

Now that gravitational waves have been detected, and we know how to do it, I am certain that we will see many gravitational wave detectors come up all over the world. There is a proposal for two Gravitational wave detectors in Europe( Virgo and GEO600), Japan( KAGRA) and even one in India( INDIGO). And scientists will find ways to make the detectors more sensitive to be able to measure waves of even smaller amplitude. So, this is really the beginning of a grand new adventure to unravel the mysteries of the universe using gravitational waves. And this is a very exciting time to do physics!

And with this we end our celebration of the 100th anniversary of Einstein’s General Relativity. It really is an amazing theory and a tribute to human intellect. And people who think that science is just about cold hard logic and has nothing to do with beauty, should read this theory once because I believe that it is one of the most beautiful creations of the human mind.

“The most beautiful thing we can experience is the mysterious. It is the source of all art and science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead; his eyes are closed.” – Albert Einstein

kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

====== ======= =========== ============================== =============

https://unvarnishedveritas.wordpress.com/tag/100-years-of-general-relativity/

In the previous parts we have studied the basic ideas behind special and general relativity. We have discussed how gravity bends space time and slows down time. We have also explored one of the most interesting and astonishing results of relativity: black holes. If you have not read the previous parts, I suggest you read them before reading this one.

In this article we will apply General Relativity to the biggest possible problem that can be thought of: the whole universe. We will understand the current theory of how the universe originated and how it evolved after that. We will also see how the universe will evolve into the future. We will also learn about dark energy and dark matter. But before we move further, think about the magnitude of what modern physics allows us to do: we, tiny creatures living on a small planet revolving around a small star in one corner of a medium sized galaxy( which contains over 100 billion stars) have developed the ability to ask and try to answer questions about the whole universe which contains billions of galaxies like our own!

Our view of how big the universe is and what our place in the universe is has evolved over time. For most of history we have thought that the universe is small and we are its centre- a special world that God has created and continues to administer on a day to day basis J. In the 16th century Copernicus placed Sun at the center of the universe. This was a big change in the human view of the universe and our place in it. Of course, religious authorities were disgusted at this idea- you see Religion is not so much about the importance of God, it is more about the importance of man in a universe that God has specifically created for him. Copernicus’ model placed sun at the center of the universe but the actual extent of the universe was not known. It was only in the 1920s that the first galaxy outside our own( Andromeda) was identified- it was referred to as an “island universe”. This was another big change as now, the milky way was not the only galaxy. Suddenly the universe was much much bigger than what we had imagined. Since the time of Copernicus and Galileo, the size of the earth relative to the universe and its importance in the grand scheme of things has kept steadily decreasing. And then in the 1930s the human “ego” suffered another blow- not only is Earth an insignificant planet, the universe is expanding and thus increasing our insignificance enormously with the passage of every moment . The universe is getting bigger! Edwin Hubble had examined the relationship between the distance to galaxies beyond the milky way and “red shift” in the light received from them and had come to this conclusion.

The universe is expanding. In whatever direction you look, galaxies are receding away from us. Now this does not mean that we are the cente of the universe. Think of a cake with raisins in it. When the cake is being baked, it expands. The raisins seem to move away from each other. If you were on one of the raisins, you will see all raisins move away from you. It does not matter which raisin you are on, you will see the same thing. This is similar to our universe. No matter which galaxy you are on, all galaxies in all directions seem to move away from you. So, there is no center of the universe. Also, if the size of the cake doubles, the distance between the raisins doubles. So if at one particular time the distance between your raisin and the neighboring one is 10 cm, it will become 20cm when the cake doubles. Now imagine a raisin 20 cm away from you. When the cake doubles, the raisin’s distance from you becomes 40 cm. So we see that the raisins that are further away from us appear to move away faster than the ones that are closer. This is similar to the universe. The galaxies that are far away from us appear to move away at a faster rate. The relationship between distance of a galaxy and the velocity at which it appears to move away is called Hubble’s law. It is v = H X r. r is the distance, v is the velocity of the galaxy going away from you. H is the Hubble constant. The Hubble’s law can be derived from the General Theory of Relativity.

In 1927, Georges Lemaître proposed the idea that if all galaxies are moving away from each other, there would a time in the past when they would all be at the same place. So, at some distant time in the past, the entire mass of the universe was concentrated at a single point. And then an event known as the Big Bang occurred and that caused the universe to come into existence. The universe has been expanding ever since. The Big Bang is a consequence of Einstein’s theory of relativity applied to the entire universe. How does one apply a theory like relativity to the whole universe? Physicists first make a model of the universe and then apply the equations of relativity to it to see how the universe would have been earlier and what may happen to it as time passes. The simplest possible model of the universe is that of a fluid with a positive mass density but no pressure. Such models are called dust models( see Wikipedia: https://en.wikipedia.org/wiki/Dust_solution). One of the most interesting and important examples of the dust model is the Friedmann–Lemaître–Robertson–Walker model. This describes a universe that is isotropic and homogenous. By isotropic we mean that no matter where you go in the universe, you will see similar stuff around you. So on a large scale the universe in terms of matter density and radiation is the same everywhere. By homogenous we mean that in whichever direction you look you will see similar stuff. So if I look towards west, the galaxies are moving away from me according to Hubble’s law. The same will be true even if I look toward the right or towards any direction. So by isotropy and homegeniety we mean that the universe is the same( on a large scale) at every place and thus has no preferred place( no centre, of course) and no preferred direction. Once we have a model of the universe we can then apply Relativity to the model of the universe and come up with solutions as to its past and its future.

The following Wikipedia image shows how the universe evolved according to our current understanding:

So, the universe began about 13.7 billion years ago. The most interesting part of this evolution is just after the big bang. The universe expanded 10^26 times in a fraction of a second! This period is known as cosmic inflation. After that initial period of superexpansion, the universe settled into a slow regular expansion that has continued to this time. The first stars were formed about 400 million years after the big bang. The development of galaxies happened much later. Our own milky way is 9 billion years old and the sun is about 5 billion years old. Note that during the time of inflation, the universe expanded at a much faster rate than the speed of light. This does not violate Einstein’s relativity. Nothing in the universe can travel faster than light but the universe itself can expand faster than the speed of light. If you look at the far right of the above figure, you can see that the universe seems to be expanding at an accelerated rate. Not only is the universe expanding, the expansion is accelerating. This fact was discovered very recently – in 1998. We will have more to say about this later.

Now, the above image shows what has happened till now. How will the universe evolve in the future? In the earlier versions of this series we learnt that gravity is actually a distortion in space-time. It would be natural to think that the universe may also have a shape that is determined by the sum total of its gravity. The shape of universe thus determines the total amount of gravitational energy in it and that ultimately determines its fate. See the following image from Wikipedia:

The omega in the above image is known as the density parameter. If it is greater than 1, the universe is said to be closed. In that case, the universe may not expand forever. It will stop expanding and then start contracting into a tiny lump of matter with “infinite” density. This scenario is called the big crunch. If the universe’s density parameter is > 1 the universe will continue to expand forever at an accelerated rate. In that case the universe will be “torn apart” due to huge expansion called the big rip. The third scenario is known as the big freeze- the universe will keep expanding at a slower rate and ultimately each particle would be far away from any other particle in the universe- it would be a uninteresting universe without any stars or galaxies: just a cold, dark, lifeless universe.

Can we determine the density parameter of the universe and determine which of the three possibilities, big rip, big crunch and big freeze will happen? We cannot determine the exact value of density parameter because of two big unknowns- dark energy and dark matter. When scientists compared the estimated mass of galaxies by summing the masses of stars to how the gravitational field changes as we move outwards from the centre of the galaxy, they observed something strange. There was more mass in the galaxy than could be accounted for by the sum of the masses of all the stars. They called this unknown and unseen matter, dark matter. Astonishingly, they calculated that there is more dark matter than “normal” matter in the galaxy. Now lets talk about Dark Energy. I have stated earlier that the universe’s expansion is accelerating. Why should the universe accelerate while exapanding? All the gravity in the universe should pull it inside. What is the force that is making the universe accelerate by countering gravity that is trying to stop the expansion? The force is still a mystery. Scientists have theorized that there may be energy of a different kind that is causing this accelerated expansion. They call it Dark Energy. Note that according to Quantum Mechanics, even vacuum has some energy called zero point energy. Some scientists think that zero point enegy may be dark energy. However the exact nature of dark energy is not known.

The ultimate fate of the universe depends on the amount of dark energy and dark matter in the universe and the physics of these mysterious forms of matter/energy. The current estimate is that 68% of the universe is dark energy, 27% is dark matter and only 5 % is “normal” matter that we see around us( normal matter is the matter that makes up the visible stars, planets, galaxies etc). So it is safe to say that we have no idea about 95% of the matter/energy in the universe. Dark energy and Dark matter are some of the biggest mysteries in physics.

Another question that gets asked is: what happened before Big Bang? In the previous VERITAS article we studied that every Black Hole has at its centre a place that we call the singularity. A singularity is a place with infinite density and it is a place where the laws of physics break down. The Big Bang was also a singularity. So we cannot know about the Big Bang or what happened before it using our current understanding of physics. We can only use physics to understand what happened after the Big Bang.

Isn’t it amazing that Einstein’s theory of relativity, which he composed while sitting on a desk can be used to understand the whole universe. As Einstein said “The most incomprehensible thing about the universe is that it is comprehensible”

Regards

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

====== ======= =========== ============================== =============

]]>This is the fifth part of the VERITAS series on General Relativity. Just to remind our readers: We are doing this series on Einstein’s Theory of Relativity to celebrate the 100th Anniversary of General Theory of Relativity. If you have missed any of the first four parts, you can read them here:

https://unvarnishedveritas.wordpress.com/tag/100-years-of-general-relativity/

In the previous parts we discussed the fundamental idea behind General Relativity- The Equivalence Principle and its consequences. We used the equivalence principle to show that light is bent by gravity. And then we reasoned that gravity bends space-time and that objects moving along this distorted space-time seem to be moving under the influence of a force. We also saw that gravity slows down time. If you have not read the previous parts, I suggest you read them before reading this one.

We have all heard or read about black holes in popular science books/articles. The general idea is that a black hole is a region from which nothing, not even light can escape. And many of us know that black holes are formed when large stars reach the end of their lives.

The physics of Black Holes is very complex and vast. But the basic idea is that a black hole is such a deep depression in space-time that even light cannot escape. Note that anything can become a black hole if it is compressed to a high density. Every object in the universe has what is known as Schwarzschild radius and if you compress the mass of that object into a sphere of that radius you will get a black hole. For Earth the Schwarzschild radius is about 9mm( the size of a peanut). So if you want to make a black hole using Earth you will have to compress the whole mass of Earth into the size of a peanut! The Schwarzschild radius for the sun is about 3 Km.

There are three kinds of black holes:

1) Stellar black holes: When most us talk about black holes, this is the kind we are referring to. These are formed when large stars collapse under the influence of gravity. The mass of these black holes ranges from 3 solar masses( ie 3 times the mass of our sun) to several tens of solar masses. To understand how these are formed we will have to first understand how a star is formed. A star starts its life with the gravitational collapse of a clould of interstellar gas consisting mostly of hydrogen. Compressional heating raises the core temperature to such a high level that thermonuclear reactions are ignited- hydrogen is fused to form helium and this process releases energy. Lots of it! The star reaches a steady state in which the energy lost to radiation is balanced by the energy produced by thermonuclear reactions. At this time our Sun is in a steady state. But after billions of years the star may run out of hydrogen to burn. And then gravitational collapse starts again. Smaller stars end up as white dwarfs or neutron stars. But if the original star was larger than about 3 times the mass of our sun, the gravitational collapse continues forever and what we get is a black hole. Nothing, not even light can escape its pull. How are such black holes detected? By their effect on nearby stars, gravitational lensing and Hawking radiation. These effects are very interesting but I cannot talk about them in this short article. There are many objects which scientists suspect are black holes. The nearest one to Earth is known as A0620-00 and is about 3000 light years away and has a mass of about 12 times that of our sun.

2) Supermassive black holes: A supermassive black hole, as the name suggests is huge! Much bigger than stellar black holes. The mass of a supermassive black hole ranges from millions to billions of solar masses. These black holes are typically found in the centres of galaxies. Scientists believe that our own galaxy, the milky way also has a supermassive black hole in its center. These black holes are formed when the centre of a galaxy collapses under extreme gravity. However, this is not the only way: Sometimes gravitational collapse occurs when galaxies collide or merge and sometimes a galaxy forms around an preexisting supermassive black hole. The nearest supermassive blackhole to Earth is the one at the center of our galaxy. It is about 26000 light years away and has the mass of about 4 million suns!

3) Primordial Black Holes: Primordial black holes are very different from stellar and supermassive black holes. Whereas stellar and supermassive black holes can form at any time due to the collapse of stars or centres of galaxies, primordial black holes could only have been formed during the early stages of the universe. Immediately after the Big Bang the universe became a place with enormous temperature and pressure. Today we know( from the study of cosmic background radiation) that the universe at that time was very smooth but had some tiny fluctuations in the density- in other words, the matter density all over the universe was the same but there were some places where the density was different. Some of these places with density fluctuations may have undergone a gravitational collapse to create small black holes. The mass of these black holes could be as small as 10^-8 kg ( The mass of a flea’s egg)! Of course, they could also be bigger. No primordial black hole has yet been detected but some scientists think that primordial black holes may be the prime candidate for dark matter in our universe.

Now lets compare these three kind of black holes using the Hawking radiation coming out from them. In 1974 Stephen Hawking applied quantum mechanics to the study of black holes and found that they must radiate and the radiation is inversely proportional to their mass. Hawking also showed that black holes decrease in mass due to this radiation. For stellar and supermassive black holes the radiation is very small and does not cause much mass loss. In fact, for stellar and supermassive black holes the mass gain due to absorption of nearby matter completely overwhelms any mass loss to Hawking radiation. But even if the black hole did not absorb anything the rate of mass loss due to Hawking radiation for a stellar or supermassive black hole is very small over the age of the universe. But for a primordial black hole this is very different. Since the mass is very small the Hawking radiation is much more.

As a primordial black hole radiates, it decreases in mass and therefore radiates more. This results in runaway evaporation resulting in a massive explosion just before the black hole completely vanishes. Many small primordial black holes would have already exploded because of Hawking radiation. Scientists have calculated that primordial black holes of the mass of about 10^11 Kg( the mass of a mountain on earth) would be exploding now. So to detect primordial black holes, scientists are looking for explosions which are caused by Hawking radiation. These explosions would be coming from a very small area- the Schwarzschild radius for a black hole of this mass is less than a nanometer! If a primordial black hole is detected we will have very strong evidence of Big Bang, Einstein’s general relativity, the theory of how the universe evolved after Big Bang and also Hawking radiation.

The physics of black holes is very interesting. Einstein’s General theory of Relativity can tell us how black holes are formed and some of their properties but there is a lot of stuff that we still do not know. And the biggest mystery is at the centre of a black hole- a place known as the singularity. At this place the known laws of physics completely break down. This is the place of infinite gravity and curvature. This is the place where both quantum mechanics and general relativity completely fail! But we absolutely do want to understand singularities! This is because the universe at the time of the Big Bang was a singularity. So to understand the beginning of our universe we must understand singularities.

But the most interesting thing( and also the most frustrating thing!) about a singularity is that it may never ever be observed. And this is because of something known as the Cosmic Censorship Hypothesis. Basically what it says is this: all singularities are covered by black holes. So there is no “naked” singularity. In other words: An observer can NEVER observe a singularity. What happens inside a black hole stays within the black hole!

Apart from the singularity, there is another interesting place- it is known as the Event Horizon. The event horizon is the point of no escape. You can go close to a black hole and return, if you do not cross its event horizon. If anything crosses the event horizon( even light), there is no hope for it. It will be pulled into the black hole and will reach the singularity at the center. So the structure of a black hole is surprisingly simple- there is a singularity at the center and an event horizon some distance around that.

Now, let’s imagine what will happen if you cross the event horizon of a black hole. Suppose you and your friend decide to do an experiment- your friend will stay outside the black hole and you are to enter the black hole( cross the event horizon). You tell your friend that you will send him a message every 5 minutes informing him about what you see. You know that you will ultimately die but you still want to enter the black hole for the sake of scientific research . Now let’s see how things will progress. You keep moving towards the black hole and keep sending messages every 5 minutes. However, as you go closer to the black hole, your friend will see that the messages are delayed. They may come after 10 and then 20 minutes. This is because Gravity slows down time( see the last article in this series for more). When you enter the event horizon, you will see see strange optical distortions of the sky around you from the bending of light around the black hole. After you enter the black hole, your friend will not be able to get any messages- this is because nothing can escape from the event horizon. Soon you will be spaghettified – there will be enormous tidal forces—forces due to the curvature of space-time—which will squash you and you spaceship in some directions and stretch them in another until you look like a piece of spaghetti. Your spaghetti form will move to the singularity and what happens after that is a great scientific mystery. See a very interesting video about this here: https://www.youtube.com/watch?v=OGn_w-3pjMc

So black holes and their singularities are the ultimate mysteries of Physics. Nothing else even comes close.

Black Holes were not predicted by Einstein. A few months after Einstein wrote his equations of General Relativity in 1915, Karl Schwarzschild solved these equations for the simplest possible case- that of a non-rotating spherical star. Schwarzschild’s solution would later lead to the prediction of black holes. Schwarzschild’s solution contain the singularity at a particular point( now called Schwarzschild radius) but he did not realize the amazing consequences of the idea. When Schwarzschild solved Einstein’s equations, he was serving as a German soldier stationed on the Russian Front. That was the time of the first world war. Schwarzschild sent his solution to Einstein with a note that read “As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas”. He died a few months later in 1916.

Regards

Kanwar

============= ============ =================== ============== ========= =====

Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

====== ======= =========== ============================== =============

Friends,

This is the fourth part of the VERITAS series on General Relativity. Just to remind our readers: We are doing this series on Einstein’s Theory of Relativity to celebrate the 100th Anniversary of General Theory of Relativity. If you have missed any of the first three parts, you can read them here:

https://unvarnishedveritas.wordpress.com/tag/100-years-of-general-relativity/

In the previous part we discussed the fundamental idea behind General Relativity- The Equivalence Principle. We used the equivalence principle to show that light is bent by gravity. And then we reasoned that gravity bends space-time and that objects moving along this distorted space-time seem to be moving under the influence of a force. If you have not read that part( or the previous ones), I suggest you read them before reading this one.

In this episode of this VERITAS series we will show how gravity causes time to slow down. We will also consider a practical application of this phenomenon.

If you have seen the movie Interstellar, you would know that time flows differently on Miller’s planet. Miller’s planet orbits the supermassive Black Hole Gargantua. Because Miller’s planet is under the influence of Gargantua’s gigantic gravitational forces, time flows very slowly there compared to Earth. We saw in the movie that one hour on Miller’s planet was equal to 7 years on Earth.! We also saw that tidal waves of more than 4000 feet raged across the planet. These massive tidal waves were also caused by Gargantua’s gravitational pull. If you have not seen Interstellar, I strongly recommend that you see it. It is a rare science-fiction movie in which science is not sacrificed. Almost every amazing thing shown in the movie is scientifically possible. This level of scientific accuracy was possible because Kip Thorne, one of the leading experts in General Relativity was the scientific consultant for the film. He solved Einstein’s equations for various situations described in the movie and then computer simulations were created for the solutions which were used in the movie. Kip Thorne has written a book, “The Science of Interstellar” which he describes more scientific aspects of the movie.

Now, let’s use equivalence principle to understand the effect of gravity on the flow of time.

See the above images that describe two situations. In the first case we have a rocket moving in empty space far away from any source of gravity. This rocket is accelerating at the rate of 9.8 m/s^2. In this rocket we have two clocks, one at the floor and one on the ceiling. In the middle we have an observer. The clocks are attached to a mechanism that sends a pulse of light at regular intervals( say 5 seconds). So the top clock and the bottom clock send a beam of light every 5 seconds. The beams reach the observer who records the times. Since the rocket in case 1 is accelerating at the rate of g( 9.8 m/s^2), the observer is moving towards the top clock and away from the lower clock. When the top clock’s light beam reaches the observer, the lower clock’s light beam has still not reached- it has a longer distance to cover because the observer has moved away from it. Thus the observer sees that the beams from the top clock come at a faster rate than the beams of the lower clock. The observer has the right to think that the bottom clock is running slower than the top clock. Now, according to equivalence principle( which we discussed in the last VERITAS of this series), an accelerating lift is equivalent to what happens in a gravitational field with the same g. So case 1 and case 2 are equivalent. What happens in case 1 also happens in case 2. Case 2 is the same arrangement but it is stationary on the surface of the earth. So even in case 2 the observer in the middle of the rocket will find that the beams of the upper clock come at a faster rate than the beams from the lower clock. So the observer will think that the clock at the floor of the rocket is slower than the clock at the top. Therefore, gravity slows down time! An clock in a stronger gravitational field slows down more compared to a clock in a weaker field. Note: as I said when we talked about time dilation in special relativity( episode 2 of this series), this is just one mechanism that shows that gravity slows down time. No matter what arrangement you come up with, you will find the exact same result- gravity does slow down time. This is called Gravitational time dilation.

The effect is so small that we do not notice it in our daily lives. Here is an example that would give you an idea of the magnitude of time dilation. Let’s take a 30 m high building( about 10 storeys). The clock on the ground floor would appear slower than the clock on the top floor by about a few microseconds in a 100 year period. So a person that resides on the top story( and never leaves home) would age faster than a person on the ground floor by a very small amount. This is because even the human heart is a kind of clock and this too should follow the principles of physics. Many experiments have experimentally verified Einstein’s calculations on gravitational time dilation. Physicists have observed the difference in time on an aeroplane vs the time on earth using atomic clocks.

Though the effect of gravitational time dilation is small, it is not without technological importance. The Global Positioning System which helps us navigate would not work properly if it did not take into account the time difference caused by gravitational time dilation. The GPS is a collection of 24 satellites with each satellite carrying an atomic clock. Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/hour. The satellites are so positioned that at any point on the earth, at least 4 satellites are always visible. Each satellite reports its position and time( from its atomic clock) at regular intervals. These signals( travelling at the speed of light) are intercepted by your GPS receiver, which calculates how far away each satellite is based on how long it took for the messages to arrive. The GPS receivers( like your smartphone) compare the time of reception of signals from 4 satellites to accurately determine your position on earth to an accuracy of about 5 meters. But since GPS calculations involve accurate knowledge of time, we have to take into account the time dilation caused by Gravitation and also special relativity. If time dilation is not accounted for, the GPS receivers will have a location error of about 10 Km per day! So general relativity is not just of interest to theoretical physicists studying black holes, even your smartphone’s GPS will not work without accounting for it.

In the next episode of this series, we will explore one of the most exotic objects in the universe: Black Holes.

Before we end, here is an Einstein quote:

“Whoever undertakes to set himself up as a judge of Truth and Knowledge is shipwrecked by the laughter of the gods.”

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run,

correct old time, regulate the sun

====== ======= =========== ============================== =============

]]>This is the third part of the VERITAS series on General Relativity. Just to remind our readers: We are doing this series on Einstein’s Theory of Relativity to celebrate the 100th Anniversary of General Theory of Relativity. If you have missed any of the first two parts, you can read them here:

https://unvarnishedveritas.wordpress.com/tag/100-years-of-general-relativity/

In the previous part we reviewed the Special Theory of Relativity that Einstein proposed in 1905. That theory combined space and time into a single concept, space-time and showed us that time does not flow equally for every object in the universe- time is a relative concept and its flow can vary based on the observer’s frame of reference. We also learnt that if an observer finds that two observations are simultaneous, a different observer in a different reference frame may not find those events simultaneous. We also learnt about some amazing and fantastic consequences of Special Relativity- time dilation, length contraction, twin paradox etc. If you have not read the VERITAS on Special Relativity, I suggest you read it before you start reading this one.

Around 1907 Einstein started think about expanding Special Theory of Relativity to include non-inertial( accelerating) frames. As we have already discussed, Special Relativity only considered uniformly moving frames. In 1907 Einstein was still working as a clerk in the Swiss patent office. His 1905 papers had created quite a stir in the scientific community. But he was still unknown outside the scientific community.

One day Einstein thought of a situation which he would later describe as “the happiest thought of my life”. Einstein imagined a freely falling person and wondered if the freely falling person would feel his own weight. Then he imagined a person in an accelerating lift. Using these “thought experiments” or Gedankenexperiment( in German), he came up with a principle that he called the “Equivalence Principle”. And this principle is at the core of the General Theory of Relativity. To understand General Relativity we will first need to learn about The Equivalence Principle.

We have two concepts when it comes to the notion of mass: there is the gravitational mass which determines the force of attraction to another body. And then there is the inertial mass: this determines the force required to move the object or stop it from moving. The Equivalence principle states that gravitational mass and inertial mass are same( or equivalent). A consequence of this is that all bodies should fall towards the earth at the same rate. This was demonstrated by Galileo when ( according to some stories), he dropped two balls of different masses from the Leaning Tower of Pisa and they landed on the the earth at the same time. Let’s now trace Einstein’s thought experiments. Imagine a person in a windowless rocket that is accelerating in outer space at the rate of 9.8 m/s^2. Imagine another person sitting in a closed room on the surface of the earth. See the two scenarios in the following image from Wikipedia( Wikimedia commons):

When the person on earth drops a ball( 2nd image), the ball falls towards the earth( or floor of the room) at the rate of 9.8 m/s^2. Now, what happens if the person travelling in the rocket drops a ball? Since the rocket it accelerating, the floor of the rocket moves towards the ball at 9.8 m/s^2. From the person’s point of view, the ball is “falling” towards the floor at 9.8 m/s^2. So, both observers see the same thing. If the windows of the rocket and room are closed, these two observers have no idea whether they are in a gravitational field or in an accelerating rocket. So, according to Einstein, both these situations are completely equivalent.

Now let’s consider another situation: let’s have one person in a stationary( or moving with constant speed) spaceship in outer space far away from any source of gravity. You would agree that the person would experience weightlessness. Now imagine another person in a freely falling lift on earth. This person too would experience weightlessness- this is because he, the lift and any other object in the lift are falling at the same rate. So they are not falling with respect to each other- from the point of view of the observer inside the lift, the objects in the lift are all floating around. So a freely falling lift on earth is equivalent to a non-accelerating spaceship in outer space.

So Einstein observed that an accelerating lift is equivalent to a gravitational field and a freely falling lift is equivalent to zero gravity. Now this is a very powerful idea. To understand Gravity, Einstein did not need to study the forces created by large masses. To understand gravity we just need to study accelerating lifts! It is such a simple but at the same time a beautiful and immensely powerful idea. And using this one simple idea Einstein was able to formulate the Theory of General Relativity.

Now lets apply the equivalence principle to light in a Gravitational field. As discussed earlier, to understand how light behaves in a gravitational field we need to just consider how light behaves in an accelerating lift. See the following figure:

In this diagram we see a lift and an observer inside it. At point A on one of the sides of the lift, there is a mechanism to send a beam of light. If the lift is stationary, the person inside the lift would see the beam travel straight from A to a point exactly at the same height on the opposite wall- this is marked as the dotted line. However, if the lift is accelerating, the light beam that starts from A will hit the opposite wall at a lower point. This is because by the time the light beam reaches the opposite side, the lift has already moved forward and thus light hits it at a lower point. So we can say that light bends in an accelerating lift. But we know that an accelerating lift is equivalent to a gravitational field. So gravity must bend light!

How about another way of looking at this? How about saying that light travels in straight line but Gravity bends space-time and thus light travelling through bent space time appears to be bent. So we come to the amazing and beautiful conclusion – Gravity bends space-time. Note: I have presented the simplest argument to claim that Gravity bends space-time. There are other arguments also which one can use to conclude that gravity bends light. But we will not go into them in this article.

If Gravity bends space-time, then we come to a radically different way of looking at Gravitational fields. Gravity is not a force as Newton suggested. It is the bending of space-time. So the physics of Gravitational fields is a problem of geometry. If we know the bending caused by an object, we can calculate the paths of objects near it. For example, lets consider how earth moves around the sun. The sun bends space-time i.e. causes a distortion in space time. When earth moves through that distortion, it has to move in an elliptical path. You can imagine putting a large weight on a matress. The weight will cause a deformation in the matress. Now if you roll a marble on the matress, its path will be deflected towards the weight that is causing the distortion. This is how gravitational attraction works. Here is an excellent youtube video that can help you visualize this: https://www.youtube.com/watch?v=MTY1Kje0yLg

Lets now talk about the experimental verification of this theory. Einstein was a theoretical physicist. He worked on the ideas and mathematics of this theory from 1907 to 1915. In 1915, he published his results which included the new equations of gravity called Einstein’s field equations. Einstein never did any experiments. And this is typical in Physics: the theorists work on ideas, mathematics and theories; the experimental physicists test these theories and try to find their limits. The first major test for General Relativity came in 1919 when Sir Arthur Eddington measured the bending of light caused by the sun. He did that during an expedition to South America during a total solar eclipse. The deflection was exactly the same as what Einstein had predicted. And immediately after this experiment, Einstein attained immediate unparalleled world fame. He became the superstar of science and people started associating the word genius with him. His long unkempt hair, his violin, his quotes about just every subject, absentmindedness – everything became famous. Most of these people did not understand relativity but that did not stop them from idolizing Einstein.

Another major confirmation of the theory was that Einstein was able to predict the anamolous precession of Mercury’s orbit. As we all know, Newton’s theory of Gravity was very successful in predicting the paths of almost all planets. Using Newton’s theory scientists were able to even discover new planets. But the orbit of Mercury behaved in a rather strange manner and Newton’s theory could not account for it. Einstein’s theory solved the problem of Mercuty’s orbit and explained its strange precession. Einstein’s relativity has been tested by hundreds of experiments since then. The equivalence principle that is the basis of this theory has been tested to more than 1 part in 10^12! For more details of the amazing experiments that test the Equivalence principle see the VERITAS article on Lunar Ranging experiment( Modern Version of Galileo’s Leaning Tower of Pisa experiment) : https://unvarnishedveritas.wordpress.com/2014/05/01/modern-version-of-galileos-leaning-tower-of-pisa-experiment/

In the next episode of this series we will discuss the time dilation caused by Gravity- you have seen this phenomenon in the movie Interstellar.

Before we end this episode, here is a quote by Einstein:

“The important thing is not to stop questioning. Curiosity has its own reason for existence. One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvelous structure of reality. It is enough if one tries merely to comprehend a little of this mystery each day. Never lose a holy curiosity”

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run,

correct old time, regulate the sun

====== ======= =========== ============================== =============

]]>As you know, in this series we are celebrating 100th anniversary of General Theory of Relativity. This is the second part of this series. You can read the first part at : https://unvarnishedveritas.wordpress.com/

Before we discuss General Theory of Relativity, we need to understand the concepts of Special Theory of Relativity. In this VERITAS article, I will give you an introduction to the Special Theory of Relativity.

1905 was a very important year for Einstein and his scientific pursuits. Einstein turned 26 years old in March of that year. He was employed as a clerk in the Swiss patent office in Bern, he was already married and had two sons. His work at the Swiss patent office was neither creative nor did it involve much science. Einstein balanced his work in office, his family life and his real interest, Physics. Like today’s age, the greatest scientific breakthroughs in those days came from universities and laboratories. Einstein was far away from these institutions and was working completely alone( he did discuss his ideas with his friends but none of them were professional scientists). And he had limited access to scientific journals. But he still published 4 papers that year and each one of those papers was a breakthrough in the world of physics. The first of these papers explained the photoelectric effect and laid the foundations of quantum physics, the second paper dealt with Brownian motion, the third paper proposed the Special Theory of Relativity and in the fourth paper Einstein derived the most famous equation of Physics: E= mc^2. It is amazing that a 26 year old clerk could come up with 4 amazing ideas that would change the course of physics in a single year. This year( 1905) is often referred to as Einstein’s Annus Mirabilis ( Miracle year). If you want to read Einstein’s biography, I would recommend “Einstein: The Life and Times” by Ronald Clark and “Einstein: His life and Universe” by Walter Isaacson.

The hallmark of Einstein’s work is that he starts with the simplest of ideas or axioms and builds a complex, beautiful theory using those ideas. The special theory of relativity is based on only two assumptions/axioms:

The laws of Physics are the same in all inertial frames of reference. An inertial frame of reference is one that is not accelerating. For example a car moving at a constant speed, or a rocket moving in space at a constant speed are examples of inertial frames of reference. So Einstein basically says that any person in any inertial reference frame will see the same results of any experiment performed in it. There is no inertial reference frame better than any other.

The speed of light in free space has the same value ( scientists write it as c) in all inertial reference frames. So a person moving in a rocket or moving in a bus or stationary on a chair will all get the same value if they measure the speed of light.

So just these two assumptions are needed to derive the entire theory of Special Relativity. These assumptions come from experimental results( like Michelson-Morley experiment). When Einstein explored the consequences of these two assumptions, he found some surprizing results. The first casualty was simultaneity. The concept that two events may occur at the same time is an essential part of how we perceive this world. Indeed, even our concept of time is based on the assumption that all clocks reach a particular time all at once( or simultaneously). But Einstein showed that events that are simultaneous in one reference frame may not be simultaneous in another reference frame.

See the figure above. We see a moving train and two observers. One observer is inside the train and the other is standing outside on the platform( but can see the events inside the train). At a certain time t there is a flash of light at the center of the train. For the observer inside the train the light from the flash reaches the front of the train and the back of the train at the same time. But the observer on the platform sees a different sequence of events: according to him the back of the train is moving towards the flash of the light and so light has to travel less distance to meet it. The front of the train is moving away from the flash of the light and light has to travel more distance to meet it. So he sees that the two events: light meeting the two sides of the train are not simultaneous . So we have shown that simultaneity is a relative concept: it depends on the frame of reference.

Now lets look at another situation illustrated in the following diagrams:

In the first diagram we see a moving train in which there are two mirrors: one on the floor and other on the ceiling. There is a person on the train who is trying to measure time by reflecting light from the mirrors. He sees a beam of light coming from the bottom mirror, strikes the top and returns to a detector on the bottom mirror. So he says that the time needed for light to come back to the original position is : 2d/c where d is the distance between the mirrors and c is the speed of light. Now, lets consider how a person standing on the platform will see the experiment. See the second diagram. This person sees that the light from the mirror at the bottom has to travel at an angle to meet the mirror on the top. This is because by the time the light travels from the bottom mirror to the top, the top mirror has already moved ahead because the train is in motion. Similarly when the light reflects from the top mirror it has to travel at an angle to meet the bottom mirror. The time spent in this round trip is not 2d/c. it is more because light has to travel a larger distance. So if the two observers are using this mirror-light mechanism to measure time then we have to say that the time recorded by the two observers is different! If you do the calculations carefully using basic trigonometry you will be able to derive Einstein’s formula for time dilation: Moving clocks run slower than stationary clocks. For the first observer the clock( light mirror mechanism ) is stationary, for the 2nd observer the clock is moving and we have seen that the time intervals in the moving clock( for observer 2) are slower than time intervals recorded for the stationary clock( observer 1). So time slows down when a body is in motion! You may say: wait a minute! You have done this experiment with light and mirrors but I don’t use that to measure time: I use a quartz watch. I will say this: This is only an illustration. You can use any mechanism to measure time and you will see that time observed for a moving clock will be slower than time observed for a stationary clock!

From these thought experiments( experiments imagined by a scientist), it became clear to Einstein that we cannot think of a universal time that flows equally for all observers. The duration of time elapsed and the events which occur at the same time are dependent on the frame of reference of the observer. So time is not an absolute quantity- Time is relative. This idea is of fundamental importance and created an enormous change in the course of physics. Before special relativity, scientists would be content with marking the coordinates of any object by three variables: x, y and z. Einstein told us that the three coordinates of space( x, y and z) are not enough to describe the physics of any situation: we need time too. This is how pure space and pure time died and a new concept was born: space-time. So modern physicists rely on 4 coordinates: x, y , z and t to describe any event or location of any object. Space-time diagrams help us understand how objects move through 4 dimensional space-time and how they interact with each other. For more information refer to space-time diagrams on the internet.

Einstein’s Special Relativity has some amazing and mindboggling consequences. Here are some of them:

Einstein was able to show that for moving objects the length shortens along the direction of motion. This is known as length contraction.

We have seen that moving clocks run slower. This leads us to a very interesting situation known as the twin paradox. Imagine two twins: one stays on earth and the other sits in a space ship and travels into space at a speed very close to the speed of light. After about 2 years she come back and finds that her sister on earth has aged 50 years. So the sister that travelled in the rocket grew only 2 years older but the one that stayed on earth was 50 years older. Sure, one can say that moving clocks run slower so the time elapsed on the spaceship would be less. But the situation is more complicated. The sister in spaceship could say: according to my point of view, I was stationary but my sister on Earth moved away from me at a speed close to light, so she should be younger. So we have a “paradox” because looking at the same situation from two different angles give us two different results. But there is really no paradox. We can solve this problem using space-time diagrams and that will tell us that the sister in the spacecraft would be younger. Essentially this is because the situation is asymmetrical. In order to meet, one of them has to turn around and it is that person who would appear younger. If the sister who goes into space turns around and comes back to earth to meet her twin, she would be the one who is younger. It is the turning around that causes the asymmetry. Note: this is just a brief explanation. You can read up on twin paradox on the internet for more details. A similar paradox for length contraction is known as the ladder paradox and is also very interesting.

Time dilation and length contraction depend on the speed of the moving body. At the speed of light, time completely stops and length becomes equal to 0.

Einstein was able to show that momentum and energy are related at a deeper level ( just as space and time are related). This resulted in the famous equation E= mc^2

Einstein was also able to show that Electricity and Momentum are just two views of the same phenomenon- so relativity can help explain how magnetism “arises” from electricity.

Now, let me discuss a common question that gets asked when someone learns that time slows down for an object in motion. This makes people think that this is an illusion and not a real phenomenon. People also argue that the twin paradox is not possible because people cannot age faster or slower. These questions are based on an incorrect premise: that biology and physics are different. The distinction between biology, physics and chemistry is artificial and was created to make the lives of school and university teachers more convenient. All chemistry and biology is ultimately based on atoms and molecules and therefore all theories of matter, energy, space and time must apply to biology also. So if Relativity says that time slows down in a moving rocket, it slows down no matter how you measure it- using a swiss watch, or a cellphone, or a clock based on atomic phenomenon, or a biological clock or any other type of clock that you can imagine. But note: if you move at close to the speed of light with respect to say an object A, you will not notice the slowdown in your clock- this is because every possible measure of time has also slowed down. Also, for you, the clock is stationary and you are stationary. Only another person who is stationary with respect to A can compare his clock with yours and will be able to see the slowdown in your time. And this is important: different people will measure time differently. Time is relative. There is no absolute time.

Special Relativity is a very successful theory and its predictions have been tested through thousands of experiments to a very high level of accuracy. Even its strangest consequence- time dilation has been measured in the decay of elementary particles. There are too many experiments and I cannot into the details of these in this short article. But I will mention that in 2010 an experiment showed that time dilation occurs exactly as predicted by Einstein’s equations in a body that was moving at just 10 m/sec which is the speed of a car on the crowded streets of Delhi( 36 km/hour).

If you want to study Special Relativity in detail, I would recommend the book “Introduction to Special relativity” by Robert Resnick. Note: this does not cover General relativity. I will tell you about a few good books for General Relativity when we discuss that theory.

Friends, in today’s short article we have discussed the basic ideas that Einstein proposed in 1905. These ideas form the basis of a even greater theory that was proposed by Einstein in 1915- The General Theory of Relativity. And that would be the topic of discussion in the subsequent articles in this series.

And before we end this article here is a quote by Einstein:

Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.

Regards,

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

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]]>Friends,

Exactly a hundred years back, in November 1915, Albert Einstein published a paper that would change the way we look at the universe. He called this theory, “The General Theory of Relativity”. Ten years earlier Einstein had published the Special Theory of Relativity which for the first time combined space and time into a single inseparable “fabric” known as space-time. That was a revolution too but at a slightly smaller scale. The Special Theory applied only to objects moving at a uniform rate. The General Theory of Relativity included accelerating objects and explored the relationship between acceleration and gravity.

After 1915 physics evolved along two different trajectories: 1) Quantum Physics explained the physics of the ultrasmall( atoms, electrons etc) 2) General Relativity explained the physics of the ultralarge ( stars, galaxies, the universe!). So the grand structure of physics was held stable by two pillars: Quantum Mechanics and General relativity.

It is very difficult to study physics in some detail and not be amazed and mesmerized by its beauty. So all of physics is beautiful. But if you ask physicists to name the theory that they consider the most beautiful, you will find that most will say “General Relativity”. I spent the last one year studying General Relativity and I absolutely agree that General Relativity is the most beautiful theory of science.

100 years after its birth, General Relativity continues to be one of the two pillars of modern Physics. Scientists around the world are celebrating the 100th anniversary of General Relativity. No lover of physics can miss the importance of this year. I have also participated in this celebration in my own little way. As I said earlier, I spent the whole of this year studying General Relativy. This summer we also travelled to Switzerland to visit Einstein’s university( ETH, Zurich) and his house in Bern. My son and daughter are giving small presentations on General Relativity in their schools. And this VERITAS article is the first of a series that will celebrate this beautiful theory. In the next few weeks I will post VERITAS articles which will explore the history, basics and consequences of this theory. This is what I plan to cover over the next few weeks ( or months depending on how much time I get to write):

1) Special Relativity: Combining space and time. Twin paradox, Time dilation, length contraction etc

2) What is General relativity: The core idea, the history, the thought experiments that led to it.

3) Space time curvature and gravity.

4) Black Holes

5) The Universe- the history and its future.

6) Other amazing consequences of General relativity

The history of General Relativity is also the story of Albert Einstein. Einstein is regarded as one of the greatest scientists in history. If you search for the word “genius” on the internet, you will see the search results dominated by links and pictures of Einstein. We must understand Special and General Relativity to explain why Einstein’s name has become synonymous with the word genius. In my opinion, the story of Einstein and how he formulated the theories of relativity shows the astounding power of the human mind and its ability to comprehend this incredible mystery that surrounds us. You will see that Einstein started with very simple ideas and developed them into the most profound theories by pure thought alone and while working in solitude.

General relativity is a theory of Gravity. When we studied the theory of gravity is school we studied the one proposed by Isaac Newton. For nearly 250 years that was THE theory Gravity. It enjoyed tremendous success as scientists and mathematicians used it to understand, predict and analyse the paths of planets and moons. Newton thought of Gravity as a force- an attractive force that extended to infinity and could act without any intermediate medium. There were very few phenomena involving gravity that could not be explained by Newton’s theory. The precession of Mercury’s orbit was one such irritating phenomenon. In 1915 Einstein’s theory not only explained everything that Newton’s theory did, it also explained Mercury’s orbit perfectly. And Einstein’s theory was not a small correction to Newton’s equation. It gave us a completely different perspective of gravity. So Newton’s 250 years old theory of Gravity was toppled exactly a 100 years back and what emerged was an even more beautiful, profound and amazing theory that we shall talk about in the subsequent articles in this VERITAS series.

Note: It is not possible to completely understand Special and General relativity in 5-6 short articles. That is not my aim. My aim is simple. I want to show you why Theory of Relativity is important, interesting and beautiful. And hopefully some of you would be inspired to learn more about it. If any of you is interested in understanding this theory in greater depth, I can tell you which books you can read to do that.

Let me end this VERITAS with the following Einstein quote:

There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle.

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

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]]>Friends,

A few weeks back I started teaching my 11 year old son Python programming language. It is turning out to be an amazing experience as I see the delight on his face when his program works and when I see the worry when he is debugging a program that he thought he wrote “perfectly”. One of the programs that he recently wrote was to find the first n primes where n is a user specified value. When his programmed worked( after some debug), he asked me if we can run it long enough to be able to find the biggest prime ever discovered. A few years ago I had told him that there are prizes for finding the largest known prime number. He thought that he could win the prize by running his program overnight! That was an excellent starting point to discuss algorithms and why some algorithms are better than others. I also told him about the limits of a personal computer. We then searched on the internet for the prime number world records and how the largest prime numbers are found. And that is the subject of this VERITAS.

When scientists search for the largest primes, they focus on a special kind of primes known as Mersenne primes. First let’s talk about Mersenne numbers. A Mersenne number M(n) is a number of the form 2^n -1. In other words, a Mersenne number is one less than a power of 2. Note that I am writing 2 raised to power n as 2^n. Let’s calculate M(4). It is 2^4 -1 = 2X2X2X2 -1 = 15. So 15 is a Mersenne number. If a Mersenne number is a prime then we call it Mersenne Prime. Let’s take an example. M(3) is 2^3 -1 ie 7. So M(3) is a Mersenne Prime. The next Mersenne prime is M(5) which is 2 ^ 5 -1 = 2X2X2X2X2 -1 = 31. This is followed by M(7) ie 2^7 -1 = 127. At the time of writing this article 48 Mersenne primes are known.

Mersenne primes are named after Marin Mersenne, a 17th century French theologian, philosopher and mathematician. He was also a music theorist and is considered by many as the “father of acoustics”. In 1637 he published a book in which he described the laws governing the frequency of a stretched string. These laws are now known as Mersenne’s Laws. So Mersenne’s laws govern the sounds produced by musical instruments such as pianos, harps, guitars etc.

Marin Mersenne also edited the works of Euclid, Archimedes and many other ancient Greek methematicians. But perhaps his greatest contributions to the science and mathematics of that age was that he was a great connector of scientists and mathematicians. At that time, there we no scientific journals that people could read to know about the latest advances. All information exchange happened through people meeting each other or exchanging letters. Mersenne was at the centre of the scientific world and knew who was working on which problem. So if you wanted to know about the latest advances in the science of optics, for instance, you could ask Mersenne and he would tell you about the scientists working on that problem and may even introduce you to them.

Mersenne studied the primes which were later named after him. He found all Mersenne primes till M(31) ( i.e 2 ^31 -1). He actually suggested that M(67) and M(257) are also primes but that was later found to be incorrect. So we can say that Mersenne found Mersenne Primes till M(31). After that his list became inaccurate. M(31) was verified to be a prime in 1772 by Euler. Mersenne had also suggested that M(127) is a prime and that was verified by Lucas in 1876.

Now the question is: why is the search for the largest primes focused on Mersenne primes? There are two reasons for this: 1) We do not need to check every odd number one by one. We can just focus on powers of 2 ( and subtract 1) and may find some really big prime numbers. 2) There exist some very efficient and fast methods to check if a Mersenne number is prime or not. The Lucas-Lehmer test for primality of Mersenne primes is the one that most people use to search for large prime numbers.

In the 1940s and 50s mathematicians started using computers to search for Mersenne primes. Alan Turing and others used Manchester Mark 1 computer to search for these primes in 1949. The computer ran for nine hours but could not discover a new Mersenne Prime. The first Mersenne prime found by a computer was M(521) found by the SWAC computer running at University of California, LA in 1952. Humans just cannot compete with computers when it comes to complex calculations like these. This statement is borne out by the fact that this was the first prime to be discovered in 38 years. And then 2 hours later the computer found a new Mersenne prime : M(607)!

Before we go into the latest world records in the search for Mersenne primes, let’s discuss two very interesting theorems about Mersenne Primes. We won’t prove them. We will just list them to show how beautiful some mathematical theorems can be and how seemingly different mathematical ideas may have a deep connection.

1) Mersenne Primes and Perfect numbers are related: A perfect number is one whose proper factors( all factors except the number itself) add to make that number. One example is 6. It has 3 proper factors: 1, 2 and 3. And the sum of the factors( 3 + 2 +1) is 6. The next perfect number is 28. This is because the sum of its factors( 1 + 2 + 4 + 7+ 14) is 28. The interesting theorem is that a number can be a perfect number only if it is of the form 2^(n-1) X (2^n -1) and (2^n -1 ) is a prime. And we know that Mersenne primes are of the form 2^n -1. The converse theorem is that if 2^n -1 is a prime then 2^(n-1) X (2^n -1 ) has to be a perfect number. This is known as the Euclid-Euler Theorem. Note that 2^(n-1) is a even number since it is a power of two. So we can say that the search for Mersenne primes is equivalent to the search for even Perfect numbers! Mathematicians still do not know if there are any odd perfect numbers.

Another way to look at the Euclid-Euler theorem is that every Mersenne Prime is uniquely associated with a Perfect number. Let’s take an example: M(7) is a Mersenne Prime ie 2^7 -1 = 127 is a prime. So we can immediately say that (2^7 -1)X( 2^(7-1) is a perfect number. This number is 127 X 2^6 ie 8128 which is a perfect number. So if we find a Mersenne prime we also find a perfect number and vice versa.

2) A Mersenne number M(n) = 2^ n -1 can be a prime only if n is a prime. So, 2^7 -1 is a Mersenne prime and we know that 7 is a prime. We only need to check 2 to the power of primes when we search for Mersenne primes. That further simplifies our search.

We know that there are an infinite number of prime numbers- Euclid proved that in 300 BC! The number of Mersenne primes may also be infinite.

Today mathematicians use GIMPS ( The Great Internet Mersenne Prime Number Search) to search for large primes. This program was launched in 1996 and it uses the power of hundreds of computers of participants to find large Mersenne prime numbers. GIMPS was one of the first large scale distributed computing projects. The first success came in 1996 itself when the GIMPS program found that M(1398269) is a Mersenne Prime! This was a huge milestone because at that time it was the largest prime ever discovered- it has 420, 921 digits! Since that date GIMPS has kept breaking its own prime number world records. The 10 largest known prime numbers have all been found by GIMPS! The last prime discovered by GIMPS was in 2013- It is M(57, 885, 161) and has 17, 425, 170 digits! We have seen that when we search for new Mersenne numbers we also find perfect numbers. So, by finding the largest prime, GIMPS is also helping us find the largest perfect numbers. The largest perfect number, the one associated with M(57, 885, 161) has 34, 850, 340 digits! GIMPS shows us the power of distributed computing. The 10 largest prime numbers were discovered on normal PCs like yours and mine but that is because the task was distributed on many computers. You can also help find the next greatest prime( and there is a huge $ reward for that) if you participate in the GIMPS program ( go to http://www.mersenne.org/ to find out how you can join.)

Prime numbers have always fascinated us. They are like sparkling gems in the beautiful jungle of mathematics- amazingly beautiful but appearing magically without any pattern. The human mind continues to try and find a pattern in the appearance of primes. But we have been unsuccessful. So we entertain ourselves by finding larger and larger primes.

Let me end with a quote by Euler:

“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.”

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

====== ======= =========== ============================== =============

( Image Source: Wikimedia Commons)

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