# Special Relativity Einstein’s Special Relativity is built on two postulates: 1. The laws of physics are the same in all inertial frames. 2. The speed of light in vacuum is the same for all inertial observers, regardless of their motion or the motion of the source. The second one is the revolutionary bit. Before Einstein, people thought that light would behave like any other wave (like sound), and have its speed measured _relative to some medium_. But experiments (like Michelson–Morley) showed that **no such medium exists** — and light’s speed seems weirdly invariant. Einstein took that fact seriously and asked: _what kind of geometry makes this work?_ Answer: a geometry where space and time can mix, and where all observers agree on the speed of light—Minkowski spacetime. ### Minkowski Distance Metric The Minkowski Distance metric is: $s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$ This will yield a result in with units of distance (spatial dimension) squared. We could have written it to end up with time squared: $s^2 = -\Delta t^2 + \frac{\Delta x^2 + \Delta y^2 + \Delta z^2}{c^2}$ The speed of light is an [Invariant](Invariant.md). what does it mean to "define" a geometry? What does geometry refer to at it's core in this case? Is the real issue here that visualize light cone ### Overview Special relativity is a different version of classical mechanics than the newtonian version. It is a framework in which you can build *many theories*, such as: maxwells electromagnetism, relativistic quantum field theory, etc. Special relativity is the theory of fixed, flat spacetime, without gravity. Fill out: [Physical Effects of Approaching the Speed of Light](Physical%20Effects%20of%20Approaching%20the%20Speed%20of%20Light.md) ### History Chapter 6 of *Einstein* gives a great overview of the history, but we can list the main points here. * In 1632 Galileo articulated the principle that the laws of motion and mechanics (electromagnetism had not yet been discovered) were the same in all constant-velocity frames of reference. * Everyday experience tells us that relative velocities add up. For instance, if we were to stand on the shore and look at waves traveling 10 mph, we would observe them traveling at 10 mph, *relative* to ourselves. If we them hopped on a jet ski and headed against the waves at 40mph, we would observe the waves passing us at 50 mph, again *relative* to ourselves. * Einstein (as well as many others) asked the question: Does light behave the same way? * **Two Big Questions** * By the time of Einstein most physicists accepted that light should be treated as a wave. However, waves tend to propagate in a medium (for instance, a wave in the ocean is propagating via the medium *water*. A sound wave propagating via *air.*) * Additionally, we need to ask: What is light propagating *relative* to? * Originally, the answer to these questions was that light was propagated via an unseen medium called the *ether*. In other words, the ether was for light waves something akin to what air was for sound waves. * Physicists looked all over for the ether, but were never able to find it (pg. 112 *Einstein*). See [Michelson & Morley Experiment](https://www.youtube.com/watch?v=3G_Q6AggQF8). No matter how they measured the speed of light it was always the same. * **Einsteins Paradox** * Einstein had a though experiment about the state of affairs that troubled him: * "If I ride along side a beam of light with velocity $c$, I should observe such a beam of light as at rest, though oscillating in space. However, there seems to be no such thing, neither based on experiment or on Maxwell's equations. From the very beginning it appeared to me that everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. For how should the first observed know or be able to determine that he is in a state of fast uniform motion?" * This thought experiment violated Einsteins intuition that the laws of optics should obey the principle of relativity. In other words, Maxwell's equations which specify the speed of light, should be the same for all observers in constant-velocity motion. * **Einsteins Approach - Two Postulates** * Einstein decided to try and reconcile this paradox by creating a new theory from the top down: by deriving grand postulates and deducing the consequences. What postulates-basic assumptions of a general principle-would he start with? * **Postulate 1**: The first postulate was the [principle of relativity](Principle%20of%20Relativity.md). This asserts that the fundamental laws of physics, even Maxwell's equations governing electromagnetic waves, are the same for all observers moving at constant velocity relative to each other. * He then needed to decide upon a companion postulate. He had two choices: * 1) Emission theory: Light would shoot from it's source like particles from a gun. If the source was moving towards you, the light particle would come at your faster than if it was racing away. No ether is required. * 2) Constant Speed: The speed of light is constant, regardless of the motion that emitted it. This is consistent with wave theory. * Einstein explored 1, but eventually went with 2. He chose two because scientists always found the speed of light to be constant, and choosing 1 would require him to abandon Maxwell's equations. The more he thought about 1, the more problems he encountered. So he went with 2. * **Postulate 2**: The speed of light is constant. I.e., light always propagates in empty space with a definite velocity $V$ that is independent of the state of motion of the emitting body. * **Incompatibility of Postulates** * Now these postulates seemed to be incompatible! This is really worth exploring because it is crucial to understand the final special theory of relativity. * He used the following thought experiment to demonstrate their incompatibility: "Imagine a beam of light is sent along an embankment of a railway track. A man standing on the embankment would measure it's speed to be 186,000 mps, as it zipped past him. But now imagine a woman who is riding in a very fast carriage train that is racing away from the light source at 2,000 mps. We would assume that she would observe the beam to be zipping past her at only 184,000 mps. Hence, the velocity of light is smaller when observed from the carriage, compared to when it is observed from the embankment." * But this directly comes into conflict with postulate 1, the principle of relativity! This is because, like every other general law of nature, the law of the transmission of light must be the same when observed from the embankment or from the train (based on the principle of relativity). In other words, Maxwell's equations, which determine the speed at which light propagates, should operate the same way in the moving carriage as on the embankment. there should be no experiment that you could do, including measuring the speed of light, the distinguish which intertial frame of reference is at rest, and which is moving at constant velocity. * This made it seems as though Einstein's two postulate were incompatible! "The constant velocity of light is not consistent with the law of the addition of velocities". Einstein was considering abandoning this path, or at least one of the postulates, when he made of the most imaginative leaps in the history of physics. * **Time cannot be defined absolutely** * Einstein's leap was in realizing the *time cannot be absolutely defined*. Specifically, two events that appear simultaneous to one observer will not appear simultaneous to another observer who is moving rapidly. And there is no way to declare that one of the observes is really correct. In other words, there is no way to declare that the two events are truly simultaneous. * We can use a thought experiment to demonstrate. Consider the image below:  Suppose that two lightning bolts strike the trains tracks at two distance places, A and B. If we declare they struck simultaneously, what does that mean? Einstein stated that we would declare the two strikes simultaneous if we were standing exactly halfway between them and the light from each reached us at exactly the same time. So, if we were at location $M$ in the diagram above. Now let us consider being at location $M^t$ on the train. Suppose that at the exact instance (from the pov of the person on the embankment) when lightning strikes at points A and B, there is a passenger at the midpoint of the train, $M^t$. If the train was motionless relative to the embankment, the passenger inside would see the lightning flashes simultaneously, just as the observer on the embankment would. But if the train was moving to the right relative to the embankment, the observer inside will be rushing closer toward place B while the light signals are traveling. Thus he will be positioned slightly to the right by the time the light arrives. As a result he will see the lightning strike at B before he sees it strike at A. So he will assert that the lightning struck at B before A, and hence the events are not simultaneous. * We arrive at a very important result! Events that are simultaneous with respect to the embankment are not simultaneous with respect to the train! The principle of relativity says that their is no way to decree that the embankment is "at rest" and the train "in motion". So there is no "real" or "right answer". There is no way to say that any two events are "absolutely" or "really simultaneous". * This is a simple insight, but a very radical one! It means that *there is no absolute time*! Instead, all moving reference frames have their own relative time. ### Links * [Time Dilation](https://www.youtube.com/watch?v=iIEeSiT3SI4) * [Length Contraction](https://www.youtube.com/watch?v=FPzGAksFCbs) * [Absolute vs. Relative Space and Time](https://www.youtube.com/watch?v=Y5mzvHrvMwg&feature=youtu.be) * [Non Euclidean Geometry](https://www.youtube.com/watch?v=coPD3h8Iidc&feature=youtu.be) --- tags: #physics links: [Einstein](Einstein.md) [Physics](notes/Physics.md) [Principle of Relativity](Principle%20of%20Relativity.md) created: 2020-11-25 modified: 2020-11-25 References: Einstein, pgs (107)