Physics Past Paper PDF

**SECTION- 1** **Newton's laws of Motion-** - **Newton\'s First Law (Law of Inertia):** 1. An object at rest will remain at rest, and an object in motion will remain in motion with a constant velocity unless acted upon by a net external force. 2. In simpler terms, obj...

**SECTION- 1** **Newton's laws of Motion-** - **Newton\'s First Law (Law of Inertia):** 1. An object at rest will remain at rest, and an object in motion will remain in motion with a constant velocity unless acted upon by a net external force. 2. In simpler terms, objects tend to maintain their state of motion (either at rest or moving in a straight line) unless an external force changes that state. - **Newton\'s Second Law:** 3. The acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the object\'s mass. 4. Mathematically, *F*=*ma*, where *F* is the force applied to the object, *m* is its mass, and *a* is the resulting acceleration. This law describes the relationship between force, mass, and acceleration. - **Newton\'s Third Law:** 5. For every action, there is an equal and opposite reaction. 6. This law states that if object A exerts a force on object B, then object B simultaneously exerts a force of equal magnitude in the opposite direction on object A. In other words, forces always occur in pairs. These laws provide a fundamental framework for understanding the motion of objects and the effects of forces in classical mechanics. They are essential principles in the study of physics and have broad applications in various fields. - **Other points -** 1. Every body attracts every other body with a force that is proportional to the mass of each body. 2. The farther apart the bodies, the smaller the force. 3. Newton believed in absolute time. The fact that light travels at a finite, but very high speed was first discovered in 1676 by the Danish astronomer Ole Christensen Roemer. He observed that the times at which the moons of Jupiter appeared to pass behind Jupiter were not evenly spaced, as one would expect if the moons went round Jupiter at a constant rate. As the earth and Jupiter orbit around the sun, the distance between them varies. Roemer noticed that eclipses of Jupiter's moons appeared later the farther we were from Jupiter. He argued that this was because the light from the moons took longer to reach us when we were farther away. A proper theory of the propagation of light didn't come until 1865, when the British physicist James Clerk Maxwell succeeded in unifying the partial theories that up to then had been used to describe the forces of electricity and magnetism. He did: - **Maxwell\'s Unification of Electricity and Magnetism:** 7. James Clerk Maxwell successfully unified the partial theories of electricity and magnetism into a single framework, known as Maxwell\'s equations. These equations predicted the existence of electromagnetic waves---wavelike disturbances in the combined electromagnetic field. - **Nature of Electromagnetic Waves:** 8. Maxwell\'s theory suggested that these electromagnetic waves could travel through space at a fixed speed, similar to ripples on a pond. The wavelength of these waves could vary, leading to different categories of waves such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. - **The Concept of Ether:** 9. To explain the fixed speed of light, scientists proposed the existence of a substance called the \"ether\" that permeated all of space, even in seemingly empty regions. According to this idea, light waves would travel through the ether in a manner similar to how sound waves travel through air. - **Relative Motion and the Ether:** 10. Since the concept of absolute rest had been discarded by Newton\'s theory, it was suggested that the speed of light should be measured relative to the ether. Different observers moving relative to the ether would see light coming toward them at different speeds, but the speed of light relative to the ether would remain constant. - **Michelson-Morley Experiment (1887):** 11. Albert Michelson and Edward Morley conducted a groundbreaking experiment to measure the speed of light in different directions, taking into account the Earth\'s motion through the supposed ether. According to the ether theory, the speed of light should vary depending on whether the Earth was moving toward or away from the light source. - **Surprising Result:** 12. The surprising outcome of the Michelson-Morley experiment was that they found the speed of light to be the same in all directions, regardless of the Earth\'s motion through the supposed ether. This result was inconsistent with the predictions of the ether theory. - **Significance:** 13. The Michelson-Morley experiment played a pivotal role in the development of physics. The unexpected result eventually led to the abandonment of the ether theory and paved the way for Albert Einstein\'s Special Theory of Relativity, which introduced a new understanding of space, time, and the constancy of the speed of light for all observers. Between 1887 and 1905 there were several attempts, most notably by the Dutch physicist Hendrik Lorentz, to explain the result of the Michelson-Morley experiment in terms of objects contracting and clocks slowing down when they moved through the ether. However, in a famous paper in 1905, a hitherto unknown clerk in the Swiss patent office, Albert Einstein, pointed out that the whole idea of an ether was unnecessary, providing one was willing to abandon the idea of absolute time. - **Key Points -** - **Michelson-Morley Experiment\'s Challenge:** 14. Following the Michelson-Morley experiment in 1887, there were attempts, notably by Hendrik Lorentz, to explain the results within the framework of an ether by proposing that objects would contract and clocks would slow down as they moved through the ether. - **Einstein\'s Insight (1905):** 15. In 1905, Albert Einstein, working as a clerk in the Swiss patent office, challenged the prevailing notion of the ether. He proposed that the entire concept of an ether was unnecessary, provided one was willing to give up the idea of absolute time. - **Abandoning Absolute Time:** 16. Einstein\'s key insight was that one could achieve a consistent explanation of the Michelson-Morley result by abandoning the concept of absolute time. Instead, he introduced the idea of relativity of simultaneity, suggesting that time could be relative and could elapse differently for observers in motion relative to each other. - **Henri Poincare\'s Similar Contribution:** 17. A few weeks later, Henri Poincare, a prominent French mathematician, independently made a similar point about the unnecessary nature of the ether. However, Einstein\'s arguments were regarded as more physical in nature, while Poincare approached the problem from a mathematical perspective. - **Postulate of the Theory of Relativity:** 18. The fundamental postulate of the theory, known as the theory of relativity, asserted that the laws of science should be the same for all freely moving observers, regardless of their speed. This extended the principle of relativity beyond Newton\'s laws of motion to include Maxwell\'s theory of electromagnetism and the speed of light. - **Consequences of the Postulate:** 19. The postulate had profound consequences, including the idea that all observers should measure the same speed of light, irrespective of their motion. This led to the concept of time dilation, length contraction, and the equivalence of mass and energy, famously expressed in Einstein\'s equation *E*=*mc*2. - **Einstein\'s Contribution and Poincare\'s Recognition:** 20. Einstein is usually credited with the development of the new theory, but Poincare\'s contribution is acknowledged, with his name attached to an important part of it. **Einstein's Theories of relativity -** ### **Special Theory of Relativity (1905):** \*\*1. **Principle of Relativity:** - The laws of physics are the same for all observers in unaccelerated motion (inertial frames). There is no privileged reference frame. \*\*2. **Invariance of the Speed of Light:** - The speed of light (*c*) is constant for all observers, regardless of their motion. This is a fundamental constant and is approximately *3×108*3×108 meters (about the height of the Statue of Liberty) per second. \*\*3. **Time Dilation:** - Time is relative and can dilate (slow down) or contract (speed up) depending on the relative motion of observers. Moving clocks tick more slowly than stationary clocks. \*\*4. **Length Contraction:** - Objects in motion appear shorter in the direction of their motion when observed by a stationary observer. \*\*5. **Mass-Energy Equivalence (E=mc²):** - Energy and mass are interchangeable. Mass can be converted into energy, and vice versa, according to the famous equation *E*=*mc*2. \*\*6. **No Simultaneity:** - Simultaneity is relative. Events that are simultaneous for one observer may not be simultaneous for another moving observer. ### **General Theory of Relativity (1915):** \*\*1. **Gravity as Curvature of Spacetime:** - Rather than a force, gravity is described as the curvature of spacetime caused by the presence of mass and energy. Massive objects warp the fabric of spacetime, and objects move along curved paths in response to this curvature. \*\*2. **Equivalence Principle:** - Acceleration due to gravity is equivalent to acceleration in a uniformly accelerating reference frame. This principle underlies the idea that gravity and acceleration are indistinguishable locally. \*\*3. **Geodesics:** - Objects in free fall follow paths known as geodesics in curved spacetime. These paths represent the natural motion of objects under the influence of gravity. \*\*4. **Time Dilation in Gravitational Fields:** - Clocks in stronger gravitational fields tick more slowly than clocks in weaker fields. This phenomenon, known as gravitational time dilation, has been experimentally confirmed. \*\*5. **Gravitational Waves:** - General Relativity predicts the existence of gravitational waves---ripples in spacetime caused by the acceleration of massive objects. These waves were detected in 2015, almost a century after Einstein\'s prediction. \*\*6. **Cosmological Implications:** - General Relativity has profound implications for the large-scale structure and evolution of the universe. It forms the basis for our understanding of cosmology and the dynamics of the cosmos. **How Einstein's theories described the so-called Space-Time -** ### **Special Relativity:** - **Relative Nature of Space and Time:** 21. Special Relativity challenges the classical Newtonian view of space and time as absolute and separate entities. Instead, it proposes that space and time are intertwined into a single, four-dimensional entity known as spacetime. - **Invariance of the Speed of Light:** 22. The theory introduces the concept that the speed of light (*c*) is constant for all observers, regardless of their motion. This constant speed of light is a fundamental feature of spacetime. - **Time Dilation:** 23. Special Relativity predicts that time is relative and can flow at different rates for observers moving relative to each other. The faster an object moves, the slower time appears to pass for it compared to a stationary observer. - **Length Contraction:** 24. Objects in motion appear shorter in the direction of their motion, according to the observer at rest. This effect is known as length contraction. ### **General Relativity:** - **Gravity as Curvature of Spacetime:** 25. General Relativity extends these ideas to include gravity. Instead of gravity being a force between masses, it describes gravity as the curvature of spacetime caused by the presence of mass and energy. - **Warped Spacetime:** 26. Massive objects like stars and planets warp the fabric of spacetime around them. Objects move along curves in this curved spacetime, which we perceive as the force of gravity. - **Time Dilation in Gravitational Fields:** 27. Clocks in stronger gravitational fields (e.g., closer to massive objects) tick more slowly than clocks in weaker fields. This effect, known as gravitational time dilation, has been experimentally confirmed. **Future and Past Light cone theory -** - **Future and Past Light Cones:** 28. Imagine an event, let\'s call it \"P.\" The light emitted from this event spreads out in all directions, forming a cone shape in the four-dimensional space-time (three dimensions of space and one of time). 29. There are two cones associated with this event: the \"future light cone\" and the \"past light cone.\" - **Future Light Cone:** 30. The future light cone includes all events that can be reached from the event \"P\" by anything traveling at or below the speed of light. 31. Events within or on this cone are in the future of \"P\" because they can be influenced by what happens at \"P.\" - **Past Light Cone:** 32. The past light cone includes all events from which a pulse of light could reach the event \"P\" while traveling at or below the speed of light. 33. Events within or on this cone are in the past of \"P\" and can potentially influence what happens at \"P.\" - **Three Classes of Events:** 34. All events in the universe can be categorized into three classes with respect to the event \"P.\" - **Future of P:** Events that can be influenced by what happens at \"P.\" They lie within or on the expanding sphere of light emitted from \"P.\" - **Past of P:** Events that can influence what happens at \"P.\" They are part of the set of events from which light can reach \"P.\" - **Elsewhere of P:** Events that neither influence nor are influenced by what happens at \"P.\" - **Example with the Sun:** 35. If we consider the sun as the event \"P,\" events on Earth at the present time are in the future of \"P\" because they can be influenced by the sunlight. On the other hand, events outside this future cone (like distant stars) are in the \"elsewhere\" category because they are not influenced by the sun at the present moment. - **Edwin Hubble\'s discovery -** Our modern picture of the universe dates back to only 1924, when the American astronomer Edwin Hubble demonstrated that ours was not the only galaxy. There were in fact many others, with vast tracts of empty space between them. In order to prove this, he needed to determine the distances to these other galaxies, which are so far away that, unlike nearby stars, they really do appear fixed. Hubble was forced, therefore, to use indirect methods to measure the distances. **The indirect method was -** Now, the apparent brightness of a star depends on two factors: how much light it radiates (its luminosity), and how far it is from us. For nearby stars, we can measure their apparent brightness and their distance, and so we can work out their luminosity. Conversely, if we knew the luminosity of stars in other galaxies, we could work out their distance by measuring their apparent brightness. Hubble noted that certain types of stars always have the same luminosity when they are near enough for us to measure; therefore, he argued, if we found such stars in another galaxy, we could assume that they had the same luminosity -- and so calculate the distance to that galaxy. If we could do this for a number of stars in the same galaxy, and our calculations always gave the same distance, we could be fairly confident of our estimate. - **Key Points-** Newton\'s discovery that sunlight, when passed through a prism, breaks up into its component colors (spectrum) led to the realization that different stars exhibit unique spectra. By observing these spectra, astronomers can determine a star\'s temperature and identify the elements present in its atmosphere. In the 1920s, when astronomers examined the spectra of stars in other galaxies, they observed a peculiar phenomenon: the characteristic missing colors in the spectra were consistently shifted toward the red end of the spectrum. This shift is explained by the Doppler effect, which affects the wavelengths of light emitted by objects in motion. Stars moving away from Earth exhibit a redshift in their spectra, while those moving toward Earth show a blueshift. The redshift observed in distant galaxies\' spectra led astronomers to a groundbreaking realization: the universe is expanding. This revelation laid the foundation for our understanding of the Big Bang theory, suggesting that the universe began as a hot, dense state and has been expanding ever since. - **Friedmann's assumptions -** - **Uniformity in All Directions:** 36. Friedmann assumed that, no matter which way we look in space, the universe looks the same. It\'s like saying that, on a large scale, the universe is uniform and consistent in every direction. - **Same View from Anywhere:** 37. He also assumed that if we were observing the universe from any other point in space, we would still see the same uniformity. In other words, the universe doesn\'t have a special center or direction; it looks the same from any vantage point. In 1965 two American physicists at the Bell Telephone Laboratories in New Jersey, Arno Penzias and Robert Wilson, were testing a very sensitive microwave detector. Here's what they found - - **Setting and Experiment:** 38. In 1965, two physicists, Arno Penzias and Robert Wilson, were working at the Bell Telephone Laboratories in New Jersey. They were testing a very sensitive microwave detector, which is a device that can pick up microwaves---similar to light waves but with a longer wavelength, around a centimeter. - **Unexpected Noise:** 39. While conducting their experiment, Penzias and Wilson noticed that their detector was picking up more noise than expected. This noise didn\'t seem to be coming from any specific direction. - **Investigation:** 40. They initially thought the excess noise might be due to issues like bird droppings in their detector. After ruling out these possibilities, they concluded that the noise was not originating from within the Earth\'s atmosphere. - **Consistent Noise:** 41. The unexpected noise remained the same regardless of the detector\'s direction, day or night, and throughout the year. This was surprising because, if the noise came from sources within our solar system or galaxy, it should have varied as the Earth moved and pointed the detector in different directions. - **Beyond the Solar System:** 42. Because the noise remained constant, Penzias and Wilson deduced that it must come from beyond the Solar System and even beyond our galaxy. This realization indicated that the source of the noise was not something local but had a cosmic origin. - **Cosmic Microwave Background Radiation:** 43. The consistent noise turned out to be the cosmic microwave background radiation, a faint glow of radiation filling the universe that is a remnant from the early stages of the Big Bang. This accidental discovery provided strong evidence for the Big Bang theory and significantly contributed to our understanding of the origin and evolution of the universe. - **Friedmann's models -** **The Laws of Thermodynamics -** ### **1. The Zeroth Law of Thermodynamics:** - **Concept:** If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other. - **Explanation:** This law helps define temperature. If object A is in thermal equilibrium with object B, and object B is in thermal equilibrium with object C, then A and C must be at the same temperature. ### **2. The First Law of Thermodynamics (Law of Energy Conservation):** - **Concept:** Energy cannot be created or destroyed, only transferred or transformed. - **Explanation:** The total energy in a closed system remains constant. For example, in an engine, the energy from fuel is converted into work and heat, but the total energy stays the same. ### **3. The Second Law of Thermodynamics (Entropy Law):** - **Concept:** The total entropy (disorder) of an isolated system always increases over time. - **Explanation:** Energy tends to spread out and become more disordered. For example, when you spill hot coffee, it cools down and the heat spreads out, increasing disorder. ### **4. The Third Law of Thermodynamics:** - **Concept:** As the temperature of a system approaches absolute zero (0 Kelvin or -273.15°C), the entropy of a perfect crystal approaches zero. - **Explanation:** At absolute zero, the particles of a perfectly ordered system would theoretically be at rest, so there would be no disorder or entropy. - **BLACK HOLES -** The term black hole is of very recent origin. It was coined in 1969 by the American scientist John Wheeler as a graphic description of an idea that goes back at least two hundred years, to a time when there were two theories about light.\ \ To understand how a black hole might be formed, we first need an understanding of the life cycle of a star. A star is formed when a large amount of gas (mostly hydrogen) starts to collapse in on itself due to its gravitational attraction. As it contracts, the atoms of the gas collide with each other more and more frequently and at greater and greater speeds -- the gas heats up. Eventually, the gas will be so hot that when the hydrogen atoms collide they no longer bounce off each other, but instead coalesce to form helium. The heat released in this reaction, which is like a controlled hydrogen bomb explosion, is what makes the star shine. This additional heat also increases the pressure of the gas until it is sufficient to balance the gravitational attraction, and the gas stops contracting. It is a bit like a balloon -- there is a balance between the pressure of the air inside, which is trying to make the balloon expand, and the tension in the rubber, which is trying to make the balloon smaller. Stars will remain stable like this for a long time, with heat from the nuclear reactions balancing the gravitational attraction. Eventually, however, the star will run out of its hydrogen and other nuclear fuels. Paradoxically, the more fuel a star starts off with, the sooner it runs out. This is because the more massive the star is, the hotter it needs to be to balance its gravitational attraction. And the hotter it is, the faster it will use up its fuel. In 1928 an Indian graduate student, Subrahmanyan Chandrasekhar, set sail for England to study at Cambridge with the British astronomer Sir Arthur Eddington, an expert on general relativity. During his voyage from India, Chandrasekhar worked out how big a star could be and still support itself against its own gravity after it had used up all its fuel. The idea was this: when the star becomes small, the matter particles get very near each other, and so according to the Pauli exclusion principle, they must have very different velocities. This makes them move away from each other and so tends to make the star expand. A star can therefore maintain itself at a constant radius by a balance between the attraction of gravity and the repulsion that arises from the exclusion principle, just as earlier in its life. Chandrasekhar realized, however, that there is a limit to the repulsion that the exclusion principle can provide. The theory of relativity limits the maximum difference in the velocities of the matter particles in the star to the speed of light. This means that when the star got sufficiently dense, the repulsion caused by the exclusion principle would be less than the attraction of gravity. Chandrasekhar calculated that a cold star of more than about one and a half times the mass of the sun would not be able to support itself against its own gravity. (This mass is now known as the Chandrasekhar limit.)\ If a star's mass is less than the Chandrasekhar limit, it can eventually stop contracting and settle down to a possible final state as a "white dwarf" with a radius of a few thousand miles and a density of hundreds of tons per cubic inch. A white dwarf is supported by the exclusion principle repulsion between the electrons in its matter. We observe a large number of these white dwarf stars. One of the first to be discovered is a star that is orbiting around Sirius, the brightest star in the night sky. - Russian scientist Lev Davidovich Landau pointed out that there was another possible final state for a star, also with a limiting mass of about one or two times the mass of the sun but much smaller even than a white dwarf. These stars would be supported by the exclusion principle repulsion between neutrons and protons, rather than between electrons. They were therefore called neutron stars. They would have a radius of only ten miles or so and a density of hundreds of millions of tons per cubic inch. At the time they were first predicted, there was no way that neutron stars could be observed.\ Stars with masses above the Chandrasekhar limit, on the other hand, have a big problem when they come to the end of their fuel. In some cases they may explode or manage to throw off enough matter to reduce their mass below the limit and so avoid catastrophic gravitational collapse, but it was difficult to believe that this always happened, no matter how big the star. How would it know that it had to lose weight? And even if every star managed to lose enough mass to avoid collapse, what would happen if you added more mass to a white dwarf \'or neutron star to take it over the limit? Would it collapse to infinite density? Eddington was shocked by that implication, and he refused to believe Chandrasekhar's result. Eddington thought it was simply not possible that a star could collapse to a point. This was the view of most scientists: Einstein himself wrote a paper in which he claimed that stars would not shrink to zero size. The hostility of other scientists, particularly Eddington, his former teacher and the leading authority on the structure of stars, persuaded Chandrasekhar to abandon this line of work and turn instead to other problems in astronomy.\ Chandrasekhar had shown that the exclusion principle could not halt the collapse of a star more massive than the Chandrasekhar limit, but the problem of understanding what would happen to such a star, according to general relativity, was first solved by a young American, Robert Oppenheimer, in 1939. His result, however, suggested that there would be no observational consequences that could be detected by the telescopes of the day.\ \ The picture that we now have from Oppenheimer's work is as follows. The gravitational field of the star changes the paths of light rays in space-time from what they would have been had the star not been present. The light cones, which indicate the paths followed in space and time by flashes of light emitted from their tips, are bent slightly inward near the surface of the star. This can be seen in the bending of light from distant stars observed during an eclipse of the sun. As the star contracts, the gravitational field at its surface gets stronger and the light cones get bent inward more. This makes it more difficult for light from the star to escape, and the light appears dimmer and redder to an observer at a distance. Eventually, when the star has shrunk to a certain critical radius, the gravitational field at the surface becomes so strong that the light cones are bent inward so much that light can no longer escape. According to the theory of relativity, nothing can travel faster than light. Thus if light cannot escape, neither can anything else; everything is dragged back by the gravitational field. So one has a set of events, a region of space-time, from which it is not possible to escape to reach a distant observer. This region is what we now call a black hole. Its boundary is called the event horizon and it coincides with the paths of light rays that just fail to escape from the black hole. In order to understand what you would see if you were watching a star collapse to form a black hole, one has to remember that in the theory of relativity there is no absolute time. Each observer has his own measure of time. The time for someone on a star will be different from that for someone at a distance, because of the gravitational field of the star.\ In 1967, the study of black holes was revolutionized by Werner Israel, a Canadian scientist (who was born in Berlin, brought up in South Africa, and took his doctoral degree in Ireland). Israel showed that, according to general relativity, non-rotating black holes must be very simple; they were perfectly spherical, their size depended only on their mass, and any two such black holes with the same mass were identical. They could, in fact, be described by a particular solution of Einstein's equations that had been known since 1917, found by Karl Schwarzschild shortly after the discovery of general relativity. At first many people, including Israel himself, argued that since black holes had to be perfectly spherical, a black hole could only form from the collapse of a perfectly spherical object. Any real star -- which would never be perfectly spherical -- could therefore only collapse to form a naked singularity. But Israel's result dealt with the case of black holes formed from non-rotating bodies only. In 1963, Roy Kerr, a New Zealander, found a set of solutions of the equations of general relativity that described rotating black holes. These "Kerr" black holes rotate at a constant rate, their size and shape depending only on their mass and rate of rotation. If the rotation is zero, the black hole is perfectly round and the solution is identical to the Schwarzschild solution. If the rotation is non-zero, the black hole bulges outward near its equator (just as the earth or the sun bulge due to their rotation), and the faster it rotates, the more it bulges. So, to extend Israel's result to include rotating bodies, it was conjectured that any rotating body that collapsed to form a black hole would eventually settle down to a stationary state described by the Kerr solution. In 1970 a colleague and fellow research student of mine at Cambridge, Brandon Carter, took the first step toward proving this conjecture. He showed that, provided a stationary rotating black hole had an axis of symmetry, like a spinning top, its size and shape would depend only on its mass and rate of rotation. Then, in 1971, I proved that any stationary rotating black hole would indeed have such an axis of symmetry. Finally, in 1973, David Robinson at Kings College, London, used Carter's and my results to show that the conjecture had been correct: such a black hole had indeed to be the Kerr solution. So after gravitational collapse a black hole must settle down into a state in which it could be rotating, but not pulsating. Moreover, its size and shape would depend only on its mass and rate of rotation, and not on the nature of the body that had collapsed to form it.\ \ **Black Hole Types -**\ \ **1. By Mass** : - Stellar-mass black holes: These are the most common type, formed from the collapse of massive stars (around 10 to 50 times the mass of our Sun). They can range in mass from about 3 to 15 solar masses. - Intermediate-mass black holes: These are less common and fall between stellar-mass and supermassive black holes, with masses roughly in the range of 100 to 100,000 solar masses. Their origins are still debated, but they might be formed from the mergers of smaller black holes or the collapse of dense star clusters. - Supermassive black holes: These behemoths reside at the hearts of most galaxies and can be millions or even billions of times more massive than the Sun. The Milky Way galaxy harbors a supermassive black hole called Sagittarius A\* with a mass of around 4 million solar masses. **2. By Spin and Charge:** - Schwarzschild black holes: These are the simplest type of black holes, with no electric charge and no spin (rotation). They are purely described by their mass. - Kerr black holes: These rotate and have angular momentum, making them more complex than Schwarzschild black holes. Their properties depend on both their mass and spin. - Reissner-Nordström black holes: These have electric charge but no spin. Their existence is purely theoretical, as charged black holes would quickly lose their charge through interactions with surrounding matter. - Kerr-Newman black holes: These combine the properties of Kerr and Reissner-Nordström, possessing both spin and electric charge. Like their charged counterparts, their existence is still hypothetical. **Hawking Radiation -**\ \ Because energy cannot be created out of nothing, one of the partners in a particle/antiparticle pair will have positive energy, and the other partner negative energy. The one with negative energy is condemned to be a short-lived virtual particle because real particles always have positive energy in normal situations. It must therefore seek out its partner and annihilate with it. However, a real particle close to a massive body has less energy than if it were far away, because it would take energy to lift it far away against the gravitational attraction of the body. Normally, the energy of the particle is still positive, but the gravitational field inside a black hole is so strong that even a real particle can have negative energy there. It is therefore possible, if a black hole is present, for the virtual particle with negative energy to fall into the black hole and become a real particle or antiparticle. In this case it no longer has to annihilate with its partner. Its forsaken partner may fall into the black hole as well. Or, having positive energy, it might also escape from the vicinity of the black hole as a real particle or antiparticle. To an observer at a distance, it will appear to have been emitted from the black hole. The smaller the black hole, the shorter the distance the particle with negative energy will have to go before it becomes a real particle, and thus the greater the rate of emission, and the apparent temperature, of the black hole. - Detailed:\ \ **Theory Origins:** 44. Proposed by physicist Stephen Hawking in 1974, Hawking radiation is a theoretical prediction based on the principles of quantum mechanics and general relativity. - **Black Hole Vacuum Fluctuations:** 45. According to quantum field theory, space is filled with virtual particles that spontaneously pop in and out of existence. Near the event horizon of a black hole, these particles can be influenced by the intense gravitational field. - **Pair Production:** 46. Virtual particle-antiparticle pairs continuously form and annihilate. Near a black hole, one of these particles may fall into the black hole, while its counterpart escapes into space. - **Energy Conservation:** 47. Hawking radiation is a consequence of energy conservation. To conserve energy, the particle falling into the black hole must have negative energy, effectively reducing the mass of the black hole. - **Effect on Black Hole Mass:** 48. Over time, the continuous process of particle-antiparticle creation near the event horizon leads to a gradual loss of mass for the black hole. This is a departure from classical physics, where nothing was expected to escape a black hole. - **Temperature of Black Holes:** 49. The process of Hawking radiation assigns a temperature to a black hole, known as the Hawking temperature. The temperature is inversely proportional to the mass of the black hole; smaller black holes have higher temperatures. - **Black Hole Evaporation:** 50. As a black hole radiates Hawking radiation and loses mass, its temperature increases, leading to a runaway process known as black hole evaporation. In the final stages, a black hole can theoretically vanish completely. - **Observational Challenges:** 51. Hawking radiation has not been directly observed, as it is extremely faint for stellar-mass black holes. For smaller black holes, such as primordial black holes, the radiation is more significant but still challenging to detect. - **Significance for Theoretical Physics:** 52. Hawking radiation represents a crucial bridge between quantum mechanics and general relativity. It implies that the vacuum of space is not truly empty, and even black holes, once thought to be perfect absorbers, emit radiation. - **Information Paradox:** 53. Hawking radiation has implications for the long-standing black hole information paradox. The information carried by particles falling into a black hole seems to be lost, contradicting the principles of quantum mechanics. Resolving this paradox is an ongoing challenge in theoretical physics. - **The Origin and Fate of the Universe -**\ \ The origin of the universe that we see traces back to the event known as "The Big Bang"**.** At the big bang itself the universe is thought to have had zero size, and so to have been infinitely hot. But as the universe expanded, the temperature of the radiation decreased. One second after the big bang, it would have fallen to about ten thousand million degrees. This is about a thousand times the temperature at the center of the sun, but temperatures as high as this are reached in H-bomb explosions. At this time the universe would have contained mostly photons, electrons, and neutrinos (extremely light particles that are affected only by the weak force and gravity) and their antiparticles, together with some protons and neutrons. As the universe continued to expand and the temperature to drop, the rate at which electron/antielectron pairs were being produced in collisions would have fallen below the rate at which they were being destroyed by annihilation. So most of the electrons and antielectrons would have annihilated with each other to produce more photons, leaving only a few electrons left over. The neutrinos and antineutrinos, however, would not have annihilated with each other, because these particles interact with themselves and with other particles only very weakly. So they should still be around today. About one hundred seconds after the big bang, the temperature would have fallen to one thousand million degrees, the temperature inside the hottest stars. At this temperature protons and neutrons would no longer have sufficient energy to escape the attraction of the strong nuclear force, and would have started to combine together to produce the nuclei of atoms of deuterium (heavy hydrogen), which contain one proton and one neutron. The deuterium nuclei would then have combined with more protons and neutrons to make helium nuclei, which contain two protons and two neutrons, and also small amounts of a couple of heavier elements, lithium and beryllium. One can calculate that in the hot big bang model about a quarter of the protons and neutrons would have been converted into helium nuclei, along with a small amount of heavy hydrogen and other elements. The remaining neutrons would have decayed into protons, which are the nuclei of ordinary hydrogen atoms. Within only a few hours of the big bang, the production of helium and other elements would have stopped. And after that, for the next million years or so, the universe would have just continued expanding, without anything much happening. Eventually, once the temperature had dropped to a few thousand degrees, and electrons and nuclei no longer had enough energy to overcome the electromagnetic attraction between them, they would have started combining to form atoms. The universe as a whole would have continued expanding and cooling, but in regions that were slightly denser than average, the expansion would have been slowed down by the extra gravitational attraction. This would eventually stop expansion in some regions and cause them to start to recollapse. As they were collapsing, the gravitational pull of matter outside these regions might start them rotating slightly. As the collapsing region got smaller, it would spin faster -- just as skaters spinning on ice spin faster as they draw in their arms. Eventually, when the region got small enough, it would be spinning fast enough to balance the attraction of gravity, and in this way disklike rotating galaxies were born. Other regions, which did not happen to pick up a rotation, would become oval-shaped objects called elliptical galaxies. In these, the region would stop collapsing because individual parts of the galaxy would be orbiting stably round its center, but the galaxy would have no overall rotation.\ \ As time went on, the hydrogen and helium gas in the galaxies would break up into smaller clouds that would collapse under their own gravity. As these contracted, and the atoms within them collided with one another, the temperature of the gas would increase, until eventually it became hot enough to start nuclear fusion reactions. These would convert the hydrogen into more helium, and the heat given off would raise the pressure, and so stop the clouds from contracting any further. They would remain stable in this state for a long time as stars like our sun, burning hydrogen into helium and radiating the resulting energy as heat and light. More massive stars would need to be hotter to balance their stronger gravitational attraction, making the nuclear fusion reactions proceed so much more rapidly that they would use up their hydrogen in as little as a hundred million years. They would then contract slightly, and as they heated up further, would start to convert helium into heavier elements like carbon or oxygen. This, however, would not release much more energy, so a crisis would occur, as was described in the chapter on black holes. What happens next is not completely clear, but it seems likely that the central regions of the star would collapse to a very dense state, such as a neutron star or black hole. The outer regions of the star may sometimes get blown off in a tremendous explosion called a supernova, which would outshine all the other stars in its galaxy. Some of the heavier elements produced near the end of the star's life would be flung back into the gas in the galaxy, and would provide some of the raw material for the next generation of stars. Our own sun contains about 2 percent of these heavier elements, because it is a second- or third-generation star, formed some five thousand million years ago out of a cloud of rotating gas containing the debris of earlier supernovas. Most of the gas in that cloud went to form the sun or got blown away, but a small amount of the heavier elements collected together to form the bodies that now orbit the sun as planets like the earth. **SECTION- 2** - **Newton's Flaws:** The Earth moves through absolute space, if for no other reason than its motion around the Sun; it moves in one direction in January, then in the opposite direction six months later, in June. Correspondingly, we on Earth should measure the speed of light to be different in different directions, and the differences should change with the seasons.\ \ To verify this prediction was a fascinating challenge for experimental physicists. Albert Michelson, a twenty-eight-year-old American, took up the challenge in 1881, using an exquisitely accurate experiential technique (now called \"Michelson interferometry\"11) that he had invented. But try as he might, Michelson could find no evidence whatsoever for any variation of light speed with direction. The speed turned out to be the same in all directions and at all seasons in his.ini6al 1881 experiments, and the same to much higher precision in later 1887 experiments that Michelson performed ill Cleveland, Ohio, jointly with a chemist, Edward Morley. - **Einstein's Discovery:**\ \ Einstein's solution to Newton's Flaws were: There is no such thing as absolute space. There is no such thing as absolute time. Newton \'s foundation for all of physics was flawed And as for the Aether: It does not exist.\ \ **Einstein\'s new foundation consisted of two new fundamental principles:** - **The principle of the absoluteness of the speed of light: Whatever might be their nature, space and time must be so constituted as to make the speed of light absolutely the same in all directions, and** **absolutely independent of the motion of the person who measures it.**\ \ This principle is a resounding affirmation that the Michelson\--Morley experiment was correct, and that regardless of how accurate light measuring devices may become in the future, they must always continue to give the same result: a universal speed of light. - **The principle of relativity: Whatever might be their nature, the laws of physics must treat all states or\" motion on an equal footing.**\ \ This principle is a resounding rejection of absolute space: If the laws of physics did not treat all states of motion (for example, that of the Sun and that of the Earth) on an equal footing, then using the laws of physics, physicists would be able to pick out some \"preferred\" state of motion (for example, the Sun\'s) and define it as the state of \"absolute rest.\" Absolute space would then have crept back into physics. - **Relativity of Simultaneity:**\ \ The relativity of simultaneity is a concept from Albert Einstein\'s theory of special relativity. It states that whether two spatially separated events occur at the same time is relative to the observer\'s frame of reference. In other words, simultaneity is not absolute but depends on the observer\'s state of motion. Here's a detailed explanation. **Einstein\'s Postulates:** - The principle of relativity: The laws of physics are the same in all inertial frames of reference. - The constancy of the speed of light: The speed of light in a vacuum is the same for all observers, regardless of their relative motion or the motion of the light source. 1. **Implications for Simultaneity:** - Because the speed of light is constant for all observers, observers moving relative to each other will measure different times for the same pair of events. **Thought Experiment:** - Imagine a train moving at a constant speed. A person standing in the middle of the train flashes a light in both directions, towards the front and the back of the train. - For an observer on the train, the light reaches both ends at the same time because the distances are equal and the speed of light is constant. - For an observer standing on the platform as the train moves, the light heading towards the back of the train travels a shorter distance than the light heading towards the front, because the train is moving forward. Thus, the observer on the platform concludes that the light reaches the back of the train first. **Relativity of Simultaneity:** - The two observers (one on the train and one on the platform) disagree on the simultaneity of the two events (the light reaching the ends of the train). This disagreement is due to their different states of motion. - This means that two events that are simultaneous in one frame of reference may not be simultaneous in another frame of reference moving relative to the first. ### **Mathematical Representation** - In special relativity, the transformation between different frames of reference is given by the Lorentz transformation. If two events occur at the same time in one frame of reference, their times in another frame moving at velocity *vv*v relative to the first are given by: *t′=γ(t−vxc2)t\' = \\gamma (t - \\frac{vx}{c\^2})*t′=γ(t−c2vx ) where: - *tt*t is the time in the original frame. - *t′t\'*t′ is the time in the moving frame. - *xx*x is the position of the event in the original frame. - *cc*c is the speed of light. - *γ\\gamma*γ is the Lorentz factor, *γ=11−v2c2\\gamma = \\frac{1}{\\sqrt{1 - \\frac{v\^2}{c\^2}}}*γ=1−c2v2 1. - This equation shows that the time *t′t\'*t′ depends on both the time *tt*t and the position *xx*x, highlighting that time and space are intertwined in relativity. - **Reference Frame:**\ \ A reference frame is a laboratory that contains all the measuring apparatus one might need for whatever measurements one wishes to make. The laboratory and all its apparatus must move through the Universe together; they must all undergo the same motion.\ \ Einstein expressed his principle of relativity not in terms of arbitrary reference frames, but. in terms of rather special ones: frames that move freely under their own inertia, neither pushed nor pulled by any forces, and that therefore continue always onward in the same state of unifom1 motion as they began. Einstein called such frames inertial because their motion is governed solely by their own inertia. - **The Warping of Space and Time:**\ \ The Idea of the Warpage of space and time began with Hermann Minkowski. Who stated that events in spacetime are analogous to points in 5pace, and there is an absolute interval between any two events in spacetime completely analogous to the straight\--line distance between any two points on a flat sheet of paper. Tbe absoluteness of this interval demonstrates that spacetime has an absolute reality; it is a four-dimensional fabric with properties that are independent of one\'s motion. - **Einstein's Thought Experiment:**\ \ In any small freely falling reference frame anywhere in our real gravity-endowed Universe, the laws of\' physics must. be the same as they are in an inertial reference frame in an idealized, gravity-free universe. Einstein called this the principle of equivalence, because it asserts that small, freely falling frames in the presence of gravity are equivalent to inertial frames in the absence of gravity.\ \ Wthin days after formulating his equivalence principle, Einstein used it to make an amazing prediction, called gravitational time dilation: if one is at rest relative to a gravitating body, then the nearer one is to the body. the more slowly one\'s time must flow.\ \ \ **Gravitational Time Dilation:**\ \ **Setup: Two Identical Clocks at Different Heights**: - One clock is placed on the floor, while another is attached to the ceiling. - The clock on the floor ticks according to the flow of time near the floor, and the ceiling clock ticks according to the flow of time near the ceiling. - **Clocks Falling Freely**: - The ceiling clock emits light pulses each time it ticks. Right before it emits its first pulse, the string holding it is cut, so it begins to fall freely. - Similarly, the floor clock is dropped just before the first pulse reaches it, so it also starts falling freely into a hole. - **Comparing Time Flows with Falling Clocks**: - Since the clocks are falling freely, they are both nearly stationary relative to their initial positions during the short time interval between pulses. - This allows us to use their ticking rates (regulated by the time flow at their respective heights) to compare how time flows at different heights. - **Einstein\'s Equivalence Principle**: - Einstein's principle states that the effects of gravity can be locally \"cancelled out\" in free-fall, allowing the laws of special relativity to apply. - According to special relativity, when an object moves toward you, you perceive the light it emits as Doppler-shifted---its waves get bunched up, appearing more frequent (a higher pitch or shorter wavelength). - **The Key Insight---Doppler Shift and Time Dilation**: - Since the ceiling clock is dropped first, it falls faster than the floor clock. - As the ceiling clock moves toward the floor clock, the light pulses it emits arrive more frequently (closer together in time) at the floor clock due to the Doppler effect. - This means that time is flowing faster at the ceiling (where the ceiling clock ticks) than at the floor (where the floor clock ticks), showing that gravity slows down time as you move closer to a massive object.