Physics Astrophysics for Dummies Cheat Sheet Cheat Sheet / Updated 03-14-2024 The path to understanding astrophysics is both thought-provoking and brain-stretching. How did the universe come into existence, when will it end, and what role do our familiar planets and stars play in the grand scheme of the cosmos? There are many more questions in astrophysics than there are answers. The goal of this book is to put you in a position where you’re able to better formulate those questions, and know where to go for answers. As you’re getting started on your journey, use this Cheat Sheet to answer some of the first questions that come to mind. View Cheat Sheet

Physics The Theory of Parallel Universes Article / Updated 09-14-2023 The multiverse is a theory that suggests our universe is not the only one, and that many universes exist parallel to each other. These distinct universes within the multiverse theory are called parallel universes. A variety of different theories lend themselves to a multiverse viewpoint. Not all physicists really believe that these universes exist. Even fewer believe that it would ever be possible to contact these parallel universes. Following, are descriptions of different levels, or types of parallel universes, scientists have discussed. Level 1: If you go far enough, you’ll get back home The idea of Level 1 parallel universes basically says that space is so big that the rules of probability imply that surely, somewhere else out there, are other planets exactly like Earth. In fact, an infinite universe would have infinitely many planets, and on some of them, the events that play out would be virtually identical to those on our own Earth. We don’t see these other universes because our cosmic vision is limited by the speed of light — the ultimate speed limit. Light started traveling at the moment of the big bang, about 14 billion years ago, and so we can’t see any further than about 14 billion light-years (a bit farther, since space is expanding). This volume of space is called the Hubble volume and represents our observable universe. The existence of Level 1 parallel universes depends on two assumptions: The universe is infinite (or virtually so). Within an infinite universe, every single possible configuration of particles in a Hubble volume takes place multiple times. If Level 1 parallel universes do exist, reaching one is virtually (but not entirely) impossible. For one thing, we wouldn’t know where to look for one because, by definition, a Level 1 parallel universe is so far away that no message can ever get from us to them, or them to us. (Remember, we can only get messages from within our own Hubble volume.) Level 2: If you go far enough, you’ll fall into wonderland In a Level 2 parallel universe, regions of space are continuing to undergo an inflation phase. Because of the continuing inflationary phase in these universes, space between us and the other universes is literally expanding faster than the speed of light — and they are, therefore, completely unreachable. Two possible theories present reasons to believe that Level 2 parallel universes may exist: eternal inflation and ekpyrotic theory. In eternal inflation, recall that the quantum fluctuations in the early universe’s vacuum energy caused bubble universes to be created all over the place, expanding through their inflation stages at different rates. The initial condition of these universes is assumed to be at a maximum energy level, although at least one variant, chaotic inflation, predicts that the initial condition can be chaotically chosen as any energy level, which may have no maximum, and the results will be the same. The findings of eternal inflation mean that when inflation starts, it produces not just one universe, but an infinite number of universes. Right now, the only noninflationary model that carries any kind of weight is the ekpyrotic model, which is so new that it’s still highly speculative. In the ekpyrotic theory picture, if the universe is the region that results when two branes collide, then the branes could actually collide in multiple locations. Consider flapping a sheet up and down rapidly onto the surface of a bed. The sheet doesn’t touch the bed only in one location, but rather touches it in multiple locations. If the sheet were a brane, then each point of collision would create its own universe with its own initial conditions. There’s no reason to expect that branes collide in only one place, so the ekpyrotic theory makes it very probable that there are other universes in other locations, expanding even as you consider this possibility. Level 3: If you stay where you are, you’ll run into yourself A Level 3 parallel universe is a consequence of the many worlds interpretation (MWI) from quantum physics in which every single quantum possibility inherent in the quantum wavefunction becomes a real possibility in some reality. When the average person (especially a science fiction fan) thinks of a “parallel universe,” he’s probably thinking of Level 3 parallel universes. Level 3 parallel universes are different from the others posed because they take place in the same space and time as our own universe, but you still have no way to access them. You have never had and will never have contact with any Level 1 or Level 2 universe (we assume), but you’re continually in contact with Level 3 universes — every moment of your life, every decision you make, is causing a split of your “now” self into an infinite number of future selves, all of which are unaware of each other. Though we talk of the universe “splitting,” this isn’t precisely true. From a mathematical standpoint, there’s only one wavefunction, and it evolves over time. The superpositions of different universes all coexist simultaneously in the same infinite-dimensional Hilbert space. These separate, coexisting universes interfere with each other, yielding the bizarre quantum behaviors. Of the four types of universes, Level 3 parallel universes have the least to do with string theory directly. Level 4: Somewhere over the rainbow, there’s a magical land A Level 4 parallel universe is the strangest place (and most controversial prediction) of all, because it would follow fundamentally different mathematical laws of nature than our universe. In short, any universe that physicists can get to work out on paper would exist, based on the mathematical democracy principle: Any universe that is mathematically possible has equal possibility of actually existing. View Article

Physics Conservative and Nonconservative Forces in Physics Article / Updated 07-31-2023 In physics, it’s important to know the difference between conservative and nonconservative forces. The work a conservative force does on an object is path-independent; the actual path taken by the object makes no difference. Fifty meters up in the air has the same gravitational potential energy whether you get there by taking the steps or by hopping on a Ferris wheel. That’s different from the force of friction, which dissipates kinetic energy as heat. When friction is involved, the path you take matters — a longer path will dissipate more kinetic energy than a short one. For that reason, friction is a nonconservative force. For example, suppose you and some buddies arrive at Mt. Newton, a majestic peak that rises h meters into the air. You can take two ways up — the quick way or the scenic route. Your friends drive up the quick route, and you drive up the scenic way, taking time out to have a picnic and to solve a few physics problems. They greet you at the top by saying, “Guess what — our potential energy compared to before is mgh greater.” “Mine, too,” you say, looking out over the view. You pull out this equation: ΔPE = mg(hf - hi) This equation basically states that the actual path you take when going vertically from hi to hf doesn’t matter. All that matters is your beginning height compared to your ending height. Because the path taken by the object against gravity doesn’t matter, gravity is a conservative force. Here’s another way of looking at conservative and nonconservative forces. Say you’re vacationing in the Alps and your hotel is at the top of Mt. Newton. You spend the whole day driving around — down to a lake one minute, to the top of a higher peak the next. At the end of the day, you end up back at the same location: your hotel on top of Mt. Newton. What’s the change in your gravitational potential energy? In other words, how much net work did gravity perform on you during the day? Gravity is a conservative force, so the change in your gravitational potential energy is 0. Because you’ve experienced no net change in your gravitational potential energy, gravity did no net work on you during the day. The road exerted a normal force on your car as you drove around, but that force was always perpendicular to the road (meaning no force parallel to your motion), so it didn’t do any work, either. Conservative forces are easier to work with in physics because they don’t “leak” energy as you move around a path — if you end up in the same place, you have the same amount of energy. If you have to deal with nonconservative forces such as friction, including air friction, the situation is different. If you’re dragging something over a field carpeted with sandpaper, for example, the force of friction does different amounts of work on you depending on your path. A path that’s twice as long will involve twice as much work to overcome friction. What’s really not being conserved around a track with friction is the total potential and kinetic energy, which taken together is mechanical energy. When friction is involved, the loss in mechanical energy goes into heat energy. You can say that the total amount of energy doesn’t change if you include that heat energy. However, the heat energy dissipates into the environment quickly, so it isn’t recoverable or convertible. For that and other reasons, physicists often work in terms of mechanical energy. View Article

Physics Physics: How to Find the Final Height of a Moving Object Article / Updated 06-28-2023 Thanks to the principle of conservation of mechanical energy, you can use physics to determine the final height of a moving object. At this very moment, for example, suppose Tarzan is swinging on a vine over a crocodile-infested river at a speed of 13.0 meters/second. He needs to reach the opposite river bank 9.0 meters above his present position in order to be safe. Can he swing it? The principle of conservation of mechanical energy gives you the answer: At Tarzan’s maximum height at the end of the swing, his speed, v2, will be 0 meters/second, and assuming h1 = 0 meters — meaning that he started swinging from the same height as the tree branch he's swinging to — you can relate h2 to v1 like this: Solving for h2, this means that Tarzan will come up 0.4 meters short of the 9.0 meters he needs to be safe, so he needs some help. View Article

Physics Calculating Torque Perpendicular to the Applied Force Article / Updated 06-06-2023 In physics, how much torque you exert on an object depends on two things: the force you exert, F; and the lever arm. Also called the moment
arm, the lever arm is the perpendicular distance from the pivot point to the point at which you exert your force and is related to the distance from the axis, r, by is the angle between the force and a line from the axis to the point where the force is applied. The torque you exert on a door depends on where you push it. Assume that you’re trying to open a door, as in the various scenarios in the figure. You know that if you push on the hinge, as in diagram A, the door won’t open; if you push the middle of the door, as in diagram B, the door will open; but if you push the edge of the door, as in diagram C, the door will open more easily. In the figure, the lever arm, l, is distance r from the hinge to the point at which you exert your force. The torque is the product of the magnitude of the perpendicular force multiplied by the lever arm. It has a special symbol, the Greek letter tau: The units of torque are force units multiplied by distance units, which are newton-meters in the MKS (meter-kilogram-second) system and foot-pounds in the foot-pound-second system. For example, the lever arm in the figure is distance r (because this lever arm is perpendicular to the force), so If you push with a force of 200 newtons and r is 0.5 meters, what’s the torque you see in the figure? In diagram A, you push on the hinge, so your distance from the pivot point is zero, which means the lever arm is zero. Therefore, the magnitude of the torque is zero. In diagram B, you exert the 200 newtons of force at a distance of 0.5 meters perpendicular to the hinge, so The magnitude of the torque here is 100 newton-meters. But now take a look at diagram C. You push with 200 newtons of force at a distance of 2r perpendicular to the hinge, which makes the lever arm 2r or 1.0 meter, so you get this torque: Now you have 200 newton-meters of torque, because you push at a point twice as far away from the pivot point. In other words, you double the magnitude of your torque. But what would happen if, say, the door were partially open when you exerted your force? Well, you would calculate the torque easily, if you have lever-arm mastery. View Article

Physics Calculating a Spring’s Potential & Kinetic Energy Article / Updated 05-03-2023 In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretching the spring must go into the energy stored in the spring. That energy is called elastic potential energy and is equal to the force, F, times the distance, s: W = Fs As you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement). The distance (or displacement), s, is just the difference in position, xf – xi, and the average force is (1/2)(Ff + Fi). Therefore, you can rewrite the equation as follows: Hooke’s law says that F = –kx. Therefore, you can substitute –kxf and –kxi for Ff and Fi: Distributing and simplifying the equation gives you the equation for work in terms of the spring constant and position: The work done on the spring changes the potential energy stored in the spring. Here’s how you give that potential energy, or the elastic potential energy: For example, suppose a spring is elastic and has a spring constant, k, of and you compress the spring by 10.0 centimeters. You store the following amount of energy in it: You can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved: PE1 + KE1 = PE2 + KE2 When you compress the spring 10.0 centimeters, you know that you have of energy stored up. When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is by the conservation of mechanical energy. View Article

Physics Acceleration & Einstein's Relativity Theory Article / Updated 04-27-2023 General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity. Einstein immediately realized that his theory of special relativity worked only when an object moved in a straight line at a constant speed. What about when one of the spaceships accelerated or traveled in a curve? Einstein came to realize the principle of equivalence, and it states that an accelerated system is completely physically equivalent to a system inside a gravitational field. As Einstein later related the discovery, he was sitting in a chair thinking about the problem when he realized that if someone fell from the roof of a house, he wouldn’t feel his own weight. This suddenly gave him an understanding of the equivalence principle. As with most of Einstein’s major insights, he introduced the idea as a thought experiment. If a group of scientists were in an accelerating spaceship and performed a series of experiments, they would get exactly the same results as if sitting still on a planet whose gravity provided that same acceleration, as shown in this figure. Einstein’s brilliance was that after he realized an idea applied to reality, he applied it uniformly to every physics situation he could think of. For example, if a beam of light entered an accelerating spaceship, then the beam would appear to curve slightly, as in the left picture of the following figure. The beam is trying to go straight, but the ship is accelerating, so the path, as viewed inside the ship, would be a curve. By the principle of equivalence, this meant that gravity should also bend light, as shown in the right picture of the figure above. When Einstein first realized this in 1907, he had no way to calculate the effect, other than to predict that it would probably be very small. Ultimately, though, this exact effect would be the one used to give general relativity its strongest support. View Article

Physics What Is Einstein's General Relativity Theory? Article / Updated 04-14-2023 General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity. The theory of the space-time continuum already existed, but under general relativity Einstein was able to describe gravity as the bending of space-time geometry. Einstein defined a set of field equations, which represented the way that gravity behaved in response to matter in space-time. These field equations could be used to represent the geometry of space-time that was at the heart of the theory of general relativity. As Einstein developed his general theory of relativity, he had to refine the accepted notion of the space-time continuum into a more precise mathematical framework. He also introduced another principle, the principle of covariance. This principle states that the laws of physics must take the same form in all coordinate systems. In other words, all space-time coordinates are treated the same by the laws of physics — in the form of Einstein’s field equations. This is similar to the relativity principle, which states that the laws of physics are the same for all observers moving at constant speeds. In fact, after general relativity was developed, it was clear that the principles of special relativity were a special case. Einstein’s basic principle was that no matter where you are — Toledo, Mount Everest, Jupiter, or the Andromeda galaxy — the same laws apply. This time, though, the laws were the field equations, and your motion could very definitely impact what solutions came out of the field equations. Applying the principle of covariance meant that the space-time coordinates in a gravitational field had to work exactly the same way as the space-time coordinates on a spaceship that was accelerating. If you’re accelerating through empty space (where the space-time field is flat, as in the left picture of this figure), the geometry of space-time would appear to curve. This meant that if there’s an object with mass generating a gravitational field, it had to curve the space-time field as well (as shown in the right picture of the figure). Without matter, space-time is flat (left), but it curves when matter is present (right). In other words, Einstein had succeeded in explaining the Newtonian mystery of where gravity came from! Gravity resulted from massive objects bending space-time geometry itself. Because space-time curved, the objects moving through space would follow the “straightest” path along the curve, which explains the motion of the planets. They follow a curved path around the sun because the sun bends space-time around it. Again, you can think of this by analogy. If you’re flying by plane on Earth, you follow a path that curves around the Earth. In fact, if you take a flat map and draw a straight line between the start and end points of a trip, that would not be the shortest path to follow. The shortest path is actually the one formed by a “great circle” that you’d get if you cut the Earth directly in half, with both points along the outside of the cut. Traveling from New York City to northern Australia involves flying up along southern Canada and Alaska — nowhere close to a straight line on the flat maps we’re used to. Similarly, the planets in the solar system follow the shortest paths — those that require the least amount of energy — and that results in the motion we observe. In 1911, Einstein had done enough work on general relativity to predict how much the light should curve in this situation, which should be visible to astronomers during an eclipse. When he published his complete theory of general relativity in 1915, Einstein had corrected a couple of errors and in 1919, an expedition set out to observe the deflection of light by the sun during an eclipse, in to the west African island of Principe. The expedition leader was British astronomer Arthur Eddington, a strong supporter of Einstein. Eddington returned to England with the pictures he needed, and his calculations showed that the deflection of light precisely matched Einstein’s predictions. General relativity had made a prediction that matched observation. Albert Einstein had successfully created a theory that explained the gravitational forces of the universe and had done so by applying a handful of basic principles. To the degree possible, the work had been confirmed, and most of the physics world agreed with it. Almost overnight, Einstein’s name became world famous. In 1921, Einstein traveled through the United States to a media circus that probably wasn’t matched until the Beatlemania of the 1960s. View Article

Physics String Theory and The Hierarchy Problem in Physics Article / Updated 02-07-2023 Many physicists feel that string theory will ultimately be successful at resolving the hierarchy problem of the Standard Model of particle physics. Although it is an astounding success, the Standard Model hasn’t answered every question that physics hands to it. One of the major questions that remains is the hierarchy problem, which seeks an explanation for the diverse values that the Standard Model lets physicists work with. For example, if you count the theoretical Higgs boson (and both types of W bosons), the Standard Model of particle physics has 18 elementary particles. The masses of these particles aren’t predicted by the Standard Model. Physicists had to find these by experiment and plug them into the equations to get everything to work out right. You notice three families of particles among the fermions, which seems like a lot of unnecessary duplication. If we already have an electron, why does nature need to have a muon that’s 200 times as heavy? Why do we have so many types of quarks? Beyond that, when you look at the energy scales associated with the quantum field theories of the Standard Model, as shown in this figure, even more questions may occur to you. Why is there a gap of 16 orders of magnitude (16 zeroes!) between the intensity of the Planck scale energy and the weak scale? At the bottom of this scale is the vacuum energy, which is the energy generated by all the strange quantum behavior in empty space — virtual particles exploding into existence and quantum fields fluctuating wildly due to the uncertainty principle. The hierarchy problem occurs because the fundamental parameters of the Standard Model don’t reveal anything about these scales of energy. Just as physicists have to put the particles and their masses into the theory by hand, so too have they had to construct the energy scales by hand. Fundamental principles of physics don’t tell scientists how to transition smoothly from talking about the weak scale to talking about the Planck scale. Trying to understand the “gap” between the weak scale and the Planck scale is one of the major motivating factors behind trying to search for a quantum gravity theory in general, and string theory in particular. Many physicists would like a single theory that could be applied at all scales, without the need for renormalization (the mathematical process of removing infinities), or at least to understand what properties of nature determine the rules that work for different scales. Others are perfectly happy with renormalization, which has been a major tool of physics for nearly 40 years and works in virtually every problem that physicists run into. View Article

Physics How to Calculate a Spring Constant Using Hooke's Law Article / Updated 12-23-2022 Any physicist knows that if an object applies a force to a spring, then the spring applies an equal and opposite force to the object. Hooke’s law gives the force a spring exerts on an object attached to it with the following equation: F = –kx The minus sign shows that this force is in the opposite direction of the force that’s stretching or compressing the spring. The variables of the equation are F, which represents force, k, which is called the spring constant and measures how stiff and strong the spring is, and x, the distance the spring is stretched or compressed away from its equilibrium or rest position. The force exerted by a spring is called a restoring force; it always acts to restore the spring toward equilibrium. In Hooke’s law, the negative sign on the spring’s force means that the force exerted by the spring opposes the spring’s displacement. Understanding springs and their direction of force The direction of force exerted by a spring The preceding figure shows a ball attached to a spring. You can see that if the spring isn’t stretched or compressed, it exerts no force on the ball. If you push the spring, however, it pushes back, and if you pull the spring, it pulls back. Hooke’s law is valid as long as the elastic material you’re dealing with stays elastic — that is, it stays within its elastic limit. If you pull a spring too far, it loses its stretchy ability. As long as a spring stays within its elastic limit, you can say that F = –kx. When a spring stays within its elastic limit and obeys Hooke’s law, the spring is called an ideal spring. How to find the spring constant (example problem) Suppose that a group of car designers knocks on your door and asks whether you can help design a suspension system. “Sure,” you say. They inform you that the car will have a mass of 1,000 kilograms, and you have four shock absorbers, each 0.5 meters long, to work with. How strong do the springs have to be? Assuming these shock absorbers use springs, each one has to support a mass of at least 250 kilograms, which weighs the following: F = mg = (250 kg)(9.8 m/s2) = 2,450 N where F equals force, m equals the mass of the object, and g equals the acceleration due to gravity, 9.8 meters per second2. The spring in the shock absorber will, at a minimum, have to give you 2,450 newtons of force at the maximum compression of 0.5 meters. What does this mean the spring constant should be? In order to figure out how to calculate the spring constant, we must remember what Hooke’s law says: F = –kx Now, we need to rework the equation so that we are calculating for the missing metric, which is the spring constant, or k. Looking only at the magnitudes and therefore omitting the negative sign, you get Time to plug in the numbers: The springs used in the shock absorbers must have spring constants of at least 4,900 newtons per meter. The car designers rush out, ecstatic, but you call after them, “Don’t forget, you need to at least double that if you actually want your car to be able to handle potholes.” View Article