Why is Newtonian mechanics incompatible with electromagnetism? How does special relativity fix that incompatibility? Solution: Maxwell's equations of electromagnetism predict that light moves at the same speed for all observers. But in Newtonian mechanics, different observers will see light move at different speeds. Special relativity fixes that by changing the way different observers see relative speeds. At slow speeds, special relativity is compatible with Newtonian mechanics (with only negligible differences), but at speeds close to light, and the speed of light itself, it works differently, so that all observers see light move at the exact same speed. What are the two fundamental postulates of special relativity? What are their origins? Solution: 1. The principle of relativity: the laws of physics are the same in all inertial frames of reference. This principle was already formulated by Galileo. 2. The principle of invariant speed of light: the speed of light in vacuum is the same in all inertial frames, regardless of the motion of the light source. This principle is new, and comes from Maxwell's equations of electromagnetism. What is a Lorentz transformation? How does it relate to the concept of spacetime? Solution: Spacetime is a 4-dimensional space which includes 3 spatial dimensions and 1 time dimension. In the spatial dimensions, we can perform a rotation in order to see things from the point of view of an observer looking in a different relative direction. Similarly, in spacetime, we can perform a Lorentz transformation, a "rotation" of a spatial dimension into the time dimension, to see things from the point of view of an observer moving at a different relative speed. Challenge question: A vertical line on a spacetime diagram represents an object with zero speed, since it stays at the same position at all times. What would a horizontal line represent? Solution: A horizontal line would represent the exact opposite: an object that exists at all positions at a single moment in time. This means that the object moves at "infinite" speed. This is not thought to be physically possible; indeed, just moving at the speed of light (45 degree angle) is already impossible for any massive object. Generally, we would not expect any physical objects to be represented by a line with more than a 45 degree angle relative to the vertical axis. What are three counterintuitive consequences of special relativity? Solution: 1. Relativity of simultaneity: observers moving at different relative speeds do not agree on whether two events are simultaneous. 2. Time dilation: observers moving at different relative speeds do not agree on the duration of events. 3. Length contraction: observers moving at different relative speeds do not agree on the length of objects. Alice is moving at 10 m/s relative to Bob. Which one has the larger Lorentz factor? (Think carefully about your answer!) Solution: Neither. There is no such thing as "absolute speed", so also no "absolute Lorentz factor". Alice has the same Lorentz factor relative to Bob as he has relative to her. Explain how the twin paradox is resolved. Solution: The paradox comes from the fact that we are assuming there is a perfect symmetry between the twins, so each twin expects to be older than the other twin when they meet again. However, this symmetry is broken because Bob is accelerating and Alice is not. Speed is relative, but acceleration is absolute, so Bob is physically doing something Alice is not, and therefore the two twins cannot be treated as physically equivalent. Alice is in an inertial frame, so she is correct when she predicts that she will be older than Bob. However, since Bob is accelerating he is not in an inertial frame, even if it's just for a short time when he turns around, so his calculation is wrong. What is the difference between special and general relativity? Solution: General relativity generalizes special relativity to include gravity. Special relativity is a special case of general relativity which only applies when there is no gravity. What is the weak equivalence principle? Solution: According to the weak equivalence principle, inertial mass is the same as gravitational mass. This is also known as the universality of free fall: objects will different masses still fall at the same rate, as was demonstrated by Galileo. What is the Einstein equivalence principle? Solution: According to the Einstein equivalence principle, acceleration is locally indistinguishable from gravity. Consider for example waking up in a box that is completely sealed, so you can't see or hear what's going on outside the box. You drop a ball and it falls to the floor. Does that mean you're on a planet and gravity is pulling the ball toward the floor? Not necessarily. You could also be in outer space with no gravity, but on a rocket that is accelerating upwards, so that the floor is moving toward the ball. There's no way to tell the difference. A similar thing happens if you drop the ball but it just stays floating in the air. This could be because the box is in outer space with no gravity, but it could also be because you're under the influence of a planet's gravity but in free fall, so both the ball and the box are moving at the same rate (until the box hits the ground). Again, there's no way to tell the difference. Challenge question: The equivalence principle predicts that light rays will curve in the presence of gravity. Can you figure out how? Solution: Consider the second case in the previous question. If the box is in outer space with no gravity, then if you shoot a ray of light (e.g. from a laser), it will move in a straight line. However, if you're actually free-falling in gravity, then the light must also move in a straight line, otherwise it would violate the equivalence principle - since you could know whether you're in gravity or not by whether the light moves in a curved path or not! But consider that if the box is falling, then the point on the wall where the light should have hit will be higher than the point where the light actually hits, because the box moved down while the light was moving to the side. Therefore the light must move in a curved path when it's in a gravitational field. You can see an animation illustrating this here. Right now, as you are reading this question, are you in an inertial frame? Solution: Not unless you're reading this while skydiving without a parachute! Is gravity a force? Why or why not? Solution: No, gravity is not a force. What we interpret as the "force of gravity" is actually the result of objects moving along geodesics due to the curvature of spacetime, with no forces acting on them. In Newtonian mechanics, gravity is a force, but today we know that Newtonian mechanics is incorrect. What are the differences between Newtonian gravity and general relativity in terms of where they can be applied? Solution: General relativity is much more precise than Newtonian gravity. It also has a more precise notion of time, because it takes into account that time flows differently in different places, which is a concept that does not exist in Newtonian gravity. Still, Newtonian gravity is good enough for most real-life tasks, such as building a house, where you don't need extreme precision. However, in some cases we do require the higher precision of general relativity. One example is the GPS system, where satellites must make very precise calculations in order to pinpoint your location to within a few meters, and must even take into account the subtle differences in the rate of time between you and the satellites. This would be impossible to do in Newtonian gravity. General relativity also allows us to perform precise astronomical calculations that would be impossible using Newtonian gravity alone, such as predicting the orbits of planets (e.g. the perihelion precession of Mercury) or setting the trajectory of spacecraft we send to other places in the solar system. What is gravitational redshift? Solution: Recall that "redshift" means the wavelength shifts "towards red", meaning that the wavelength increases (since red has the longest wavelength among the visible wavelengths). According to general relativity, the wavelength of a light ray increases when it goes from a region of strong gravity to a region with weaker gravity; this is called gravitational redshift. Conversely, the wavelength of a light decreases when it goes from a region of weak gravity to a region with stronger gravity; this is called gravitational blueshift. Challenge question: You can use a black hole (or another extremely dense and massive body) to "travel" to the future, by making time flow slower for you compared to Earth, due to gravitational time dilation. But can you use a black hole to travel to the past? Solution: No, because no matter how strong gravity is, it only slows down time - it doesn't make it flow backward. Describe two of the tests that first confirmed general relativity. Solution: The first test was the perihelion precession of mercury. The Newtonian prediction did not match the observed amount of precession. In 1915 Einstein showed that the predictions of general relativity, which also took into account the curvature of spacetime, did match the observations. The second test was the prediction that the stars will appear to be in different places when they're behind the Sun compared to when they're not behind the Sun. In general relativity, when the light from a star passes near the Sun, the large curvature of spacetime due to the Sun's mass causes the path of the light to deflect. This prediction was first confirmed in the Eddington experiment, during a solar eclipse in 1919. What is an Einstein ring? Solution: When a massive object curves spacetime, it bends the light from objects behind it. The massive object that curves spacetime is called a gravitational lens in this context, and the bending of light is called gravitational lensing. If the observer, the lens, and the light source are all perfectly aligned, and the lens is perfectly circular, then the light source will deform into the shape of a ring around the lens. This ideal case almost never happens, so usually we will see just a partial ring (or arc segment). Describe the first indirect and the first direct evidence for gravitational waves. Solution: The first indirect evidence was in 1974 from a binary system of a pulsar and a neutron star. This system was generating gravitational waves, causing it to lose energy over time. The loss of energy caused the stars to gradually get closer together. According to Kepler's laws, this means the orbital period must decrease. This was indeed detected, and interpreted as evidence for gravitational waves, since nothing else could explain this observation. The first direct evidence came much later, in 2015. A merger of two black holes generated gravitational waves, which, after traveling many light years, eventually reached Earth. Since gravitational waves are waves in the curvature of spacetime, they were detected by their effects on space itself - distances in space actually oscillated several times between longer and shorter. Since the same signal was detected in two different detectors, with the correct delay due to travel time between the detectors, we know that this was indeed a passing gravitational wave and not some local effect such as a tiny earthquake. What is multi-messenger astronomy? Give an example. Solution: Multi-messenger astronomy is the use of different types of signals to observe the same event. One example was the event GW170817, when two neutron stars merged, emitting signals both as gravitational waves and as electromagnetic waves. Both signals were successfully observed from Earth. Could interstellar travel be feasible? List all arguments for and against. Solution: With current technology, it is surely impossible, because a trip to even the closest star would take thousands of years. However, it may be possible with future technology. Given enough energy, we could accelerate close to the speed of light, making the trip much shorter due to time dilation. However, the energy requirements for this are immense and much beyond our current capabilities. Another option is to make the very long trip duration possible using a generation ship or a sleeper ship, or by extending human lifetimes. A much more hypothetical option is faster-than-light (FTL) travel; while this form of travel is, in principle, mathematically consistent with general relativity, we do not yet know if it is actually possible in our universe, even with arbitrarily advanced technology, since it may simply be physically impossible. What is a closed timelike curve? Solution: Object in spacetime move along geodesics. If the object is massive (i.e. it's not a photon or some other kind of massless particle), its geodesic is called a "timelike curve". If this curve is "closed", it forms a loop, so if you follow that curve, even though time has passed for you, you end up at the same point in time that you started at. Therefore, if you could somehow curve spacetime in a way that makes a geodesic into a closed timelike curve, you could hypothetically travel back in time. What is dark matter, and why do we think it might exist? Solution: Dark matter is a hypothetical type of matter that does not interact with the electromagnetic field, so it does not produce or reflect light. Photons of light just pass right through dark matter without any interaction. However, it has mass, and interacts with the gravitational field, so its gravitational influence can be detected even if it cannot be seen. The reason we think it might exist is that galaxies don't seem to rotate in the speed we expect them to. We would expect matter that is farther away from the center of the galaxy to rotate slower; but in reality, many galaxies, including our own, do not exhibit this behavior. This could be explained if there was additional mass in these galaxies that we cannot see but still influences the speed of the matter within the galaxy - we call that hypothetical mass dark matter. We've been trying to detect dark matter in various experiments for decades, but so far we haven't been able to do so, because it's obviously very hard to detect something that doesn't interact with light! However, some physicists think dark matter doesn't actually exist, and what really happens is that gravity works differently than we think. This is called "modified gravity", but most physicists don't like this approach, since general relativity is such a successful theory, validated by many experiments, and it's hard to modify it in any way without contradicting the results of at least some of these experiments. Galaxy A is located 1 billion light-years from the Milky Way and is moving away from us at 21,000 km/s. Galaxy B is located 2 billion light-years from the Milky Way. According to Hubble's law, at what speed is Galaxy B moving? Solution: According to Hubble's law, a galaxy's recessional velocity is proportional to its distance. Since Galaxy B is twice as far away from us compared to Galaxy A, it moves twice as fast, or 42,000 km/s. Galaxy C is moving away from us at 200,000 km/s. Galaxy D is moving away from us at 50,000 km/s. What can you say about the distances to both galaxies, based on Hubble's law? Solution: Since Galaxy C is moving 4 times faster than Galaxy D, it must also be 4 times farther away. According to Hubble's law, all galaxies are receding away from us at a velocity proportional to their distance. Does that mean that we are special? That we are the center of the universe? Solution: No, we are not special. Since the universe expands everywhere at the same rate, aliens on any other galaxy will also see all the galaxies recede away from them according to Hubble's law (although they will probably call it by some other name, like Xys7dflt3g's law). To understand why this is, take a straight rubber band and draw 3 points on it: A, B, and C. As you stretch the band, the 3 points will move away from each other. From the point of view of A, both B and C are moving away from it in the same direction. From the point of view of B, both A and C are moving away from it in opposite directions. From the point of view of C, both A and B are moving away from it in the same direction. So every galaxy in this example (A, B, or C) sees both the other galaxies move away from it due to the expansion of the universe (the rubber band). There is no one galaxy that is more "special" than the others. Challenge question: If the universe is expanding, does that mean that the Milky Way Galaxy is also expanding? What about the Earth? What about your body? Solution: The answer is no for all 3 cases, because the galaxy, the Earth, and the human body are all held together by forces - gravity in the first two cases, and the electromagnetic force in the latter case. It is only at distances of millions of light-years that the expansion of space is significant enough to overcome these binding forces. Is the expansion of the universe decelerating, accelerating, or unchanging? Why? Solution: The expansion of the universe is accelerating, most likely because the vacuum of space has energy, called "dark energy", which is fueling this acceleration. Without dark energy, the expansion of the universe would have continued at a constant rate, just like an object will keep moving at a constant speed unless we give it more energy. What is recombination? Solution: The moment, ~380,000 years after the Big Bang, when photons were first able to travel freely in space; before that they could not travel far, since they kept bumping into the very dense matter that was everywhere. The cosmic microwave background radiation was emitted at that time, and it is the earliest thing in the universe that we can actually see. |