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A reference frame is simply a selection of what constitutes a stationary object. Once the velocity of a certain object is arbitrarily defined to be zero, the velocity of everything else in the universe can be measured relative to that object.[Note 1] One oft-used example is the difference in measurements of objects on a train as made by an observer on the train compared to those made by one standing on a nearby platform as it passes. Consider the seats on the train car in which the passenger observer is sitting. The distances between these objects and the passenger observer do not change. Therefore, this observer measures all of the seats to be at rest, since he is stationary from his own perspective. The observer standing on the platform would see exactly the same objects but interpret them very differently. The distances between themself and the seats on the train car are changing, and so they conclude that they are moving forward, as is the whole train. Thus for one observer the seats are at rest, while for the other the seats are moving, and both are correct, since they are using different definitions of "at rest" and "moving". Each observer has a distinct "frame of reference" in which velocities are measured, the rest frame of the platform and the rest frame of the train – or simply the platform frame and the train frame. Why can't we select one of these frames to be the "correct" one? Or more generally, why is there not a frame we can select to be the basis for all measurements, an "absolutely stationary" frame? Aristotle imagined the Earth lying at the centre of the universe (the geocentric model), unmoving as other objects moved about it. In this worldview, one could select the surface of the Earth as the absolute frame. However, as the geocentric model was challenged and finally fell in the 1500s, it was realised that the Earth was not stationary at all, but both rotating on its axes as well as orbiting the Sun. In this case the Earth is clearly not the absolute frame. But perhaps there is some other frame one could select, perhaps the Sun's? Galileo challenged this idea and argued that the concept of an absolute frame, and thus absolute velocity, was unreal; all motion was relative. Galileo gave the common-sense "formula" for adding velocities: if particle P is moving at velocity v with respect to reference frame A and reference frame A is moving at velocity u with respect to reference frame B, then the velocity of P with respect to B is given by v + u. In modern terms, we expand the application of this concept from velocity to all physical measurements – according to what we now call the Galilean transformation, there is no absolute frame of reference. An observer on the train has no measurement that distinguishes whether the train is moving forward at a constant speed, or the platform is moving backwards at that same speed. The only meaningful statement is that the train and platform are moving relative to each other, and any observer can choose to define what constitutes a speed equal to zero. When considering trains moving by platforms it is generally convenient to select the frame of reference of the platform, but such a selection would not be convenient when considering planetary motion and is not intrinsically more valid. One can use this formula to explore whether or not any possible measurement would remain the same in different reference frames. For instance, if the passenger on the train threw a ball forward, he would measure one velocity for the ball, and the observer on the platform another. After applying the formula above, though, both would agree that the velocity of the ball is the same once corrected for a different choice of what speed is considered zero. This means that motion is "invariant". Laws of classical mechanics, like Newton's second law of motion, all obey this principle because they have the same form after applying the transformation. As Newton's law involves the derivative of velocity, any constant velocity added in a Galilean transformation to a different reference frame contributes nothing (the derivative of a constant is zero). This means that the Galilean transformation and the addition of velocities only apply to frames that are moving at a constant (relative) velocity. Since objects tend to retain their current velocity due to a property we call inertia, frames that refer to objects with constant speed are known as inertial reference frames. The Galilean transformation, then, does not apply to accelerations, only velocities, and classical mechanics is not invariant under acceleration. This mirrors the real world, where acceleration is easily distinguishable from smooth motion in any number of ways. For example, if an observer on a train saw a ball roll backward off a table, he would be able to infer that the train was accelerating forward, since the ball remains at rest unless acted upon by an external force. Therefore, the only explanation is that the train has moved underneath the ball, resulting in an apparent motion of the ball. Addition of a time-varying velocity, corresponding to an accelerated reference frame, changed the formula (see pseudo-force). Both the Aristotelian and Galilean views of motion contain an important assumption. Motion is defined as the change of position over time, but both of these quantities, position and time, are not defined within the system. It is assumed, explicitly in the Greek worldview, that space and time lie outside physical existence and are absolute even if the objects within them are measured relative to each other. The Galilean transformations can only be applied because both observers are assumed to be able to measure the same time and space, regardless of their frames' relative motions. So in spite of there being no absolute motion, it is assumed there is some, perhaps unknowable, absolute space and time. Classical physics and electromagnetism Through the era between Newton and around the start of the 20th century, the development of classical physics had made great strides. Newton's application of the inverse square law to gravity was the key to unlocking a wide variety of physical events, from heat to light, and calculus made the direct calculation of these effects tractable. Over time, new mathematical techniques, notably the Lagrangian, greatly simplified the application of these physical laws to more complex problems. As electricity and magnetism were better explored, it became clear that the two concepts were related. Over time, this work culminated in Maxwell's equations, a set of four equations that could be used to calculate the entirety of electromagnetism. One of the most interesting results of the application of these equations was that it was possible to construct a self-sustaining wave of electrical and magnetic fields that could propagate through space. When reduced, the math demonstrated that the speed of propagation was dependent on two universal constants, and their ratio was the speed of light. Light was an electromagnetic wave. Under the classic model, waves are displacements within a medium. In the case of light, the waves were thought to be displacements of a special medium known as the luminiferous aether, which extended through all space. This being the case, light travels in its own frame of reference, the frame of the aether. According to the Galilean transform, we should be able to measure the difference in velocities between the aether's frame and any other – a universal frame at last. Designing an experiment to actually carry out this measurement proved very difficult, however, as the speeds and timing involved made accurate measurement difficult. The measurement problem was eventually solved with the Michelson–Morley experiment. To everyone's surprise, no relative motion was seen. Either the aether was travelling at the same velocity as the Earth, difficult to imagine given the Earth's complex motion, or there was no aether. Follow-up experiments tested various possibilities, and by the start of the 20th century it was becoming increasingly difficult to escape the conclusion that the aether did not exist. These experiments all showed that light simply did not follow the Galilean transformation. And yet it was clear that physical objects emitted light, which led to unsolved problems. If one were to carry out the experiment on the train by "throwing light" instead of balls, if light does not follow the Galilean transformation then the observers should not agree on the results. Yet it was apparent that the universe disagreed; physical systems known to be at great speeds, like distant stars, had physics that were as similar to our own as measurements allowed. Some sort of transformation had to be acting on light, or better, a single transformation for both light and matter. The development of a suitable transformation to replace the Galilean transformation is the basis of special relativity. Invariance of length: the Euclidean picture Pythagoras' theorem In special relativity, space and time are joined into a unified four-dimensional continuum called spacetime. To gain a sense of what spacetime is like, we must first look at the Euclidean space of classical Newtonian physics. This approach to explaining the theory of special relativity begins with the concept of "length". In everyday experience, it seems that the length of objects remains the same no matter how they are rotated or moved from place to place; as a result the simple length of an object doesn't appear to change or is invariant. However, as is shown in the illustrations below, what is actually being suggested is that length seems to be invariant in a three-dimensional coordinate system. The length of a line in a two-dimensional Cartesian coordinate system is given by Pythagoras' theorem: h^2 = x^2 + y^2. \, One of the basic theorems of vector algebra is that the length of a vector does not change when it is rotated. However, a closer inspection tells us that this is only true if we consider rotations confined to the plane. If we introduce rotation in the third dimension, then we can tilt the line out of the plane. In this case the projection of the line on the plane will get shorter. Does this mean the line's length changes? – obviously not. The world is three-dimensional and in a 3D Cartesian coordinate system the length is given by the three-dimensional version of Pythagoras's theorem: k^2 = x^2 + y^2 + z^2. \, This is invariant under all rotations. The apparent violation of invariance of length only happened because we were "missing" a dimension. It seems that, provided all the directions in which an object can be tilted or arranged are represented within a coordinate system, the length of an object does not change under rotations. With time and space considered to be outside the realm of physics itself, under classical mechanics a 3-dimensional coordinate system is enough to describe the world. Note that invariance of length is not ordinarily considered a principle or law, not even a theorem. It is simply a statement about the fundamental nature of space itself. Space as we ordinarily conceive it is called a three-dimensional Euclidean space, because its geometrical structure is described by the principles of Euclidean geometry. The formula for distance between two points is a fundamental property of a Euclidean space, it is called the Euclidean metric tensor (or simply the Euclidean metric). In general, distance formulas are called metric tensors. Note that rotations are fundamentally related to the concept of length. In fact, one may define length or distance to be that which stays the same (is invariant) under rotations, or define rotations to be that which keep the length invariant. Given any one, it is possible to find the other. If we know the distance formula, we can find out the formula for transforming coordinates in a rotation. If, on the other hand, we have the formula for rotations then we can find out the distance formula. The Minkowski formulation: introduction of spacetime Main article: Spacetime After Einstein derived special relativity formally from the (at first sight counter-intuitive) assumption that the speed of light is the same to all observers, Hermann Minkowski built on mathematical approaches used in non-euclidean geometry[6] and on the mathematical work of Lorentz and Poincaré. Minkowski showed in 1908 that Einstein's new theory could also be explained by replacing the concept of a separate space and time with a four-dimensional continuum called spacetime. This was a groundbreaking concept, and Roger Penrose has said that relativity was not truly complete until Minkowski reformulated Einstein's work.[7] The concept of a four-dimensional space is hard to visualise. It may help at the beginning to think simply in terms of coordinates. In three-dimensional space, one needs three real numbers to refer to a point. In the Minkowski space, one needs four real numbers (three space coordinates and one time coordinate) to refer to a point at a particular instant of time. This point, specified by the four coordinates, is called an event. The distance between two different events is called the spacetime interval. A path through the four-dimensional spacetime (usually known as Minkowski space) is called a world line. Since it specifies both position and time, a particle having a known world line has a completely determined trajectory and velocity. This is just like graphing the displacement of a particle moving in a straight line against the time elapsed. The curve contains the complete motional information of the particle. In the same way as the measurement of distance in 3D space needed all three coordinates, we must include time as well as the three space coordinates when calculating the distance in Minkowski space (henceforth called M). In a sense, the spacetime interval provides a combined estimate of how far apart two events occur in space as well as the time that elapses between their occurrence. But there is a problem; time is related to the space coordinates, but they are not equivalent. Pythagoras' theorem treats all coordinates on an equal footing (see Euclidean space for more details). We can exchange two space coordinates without changing the length, but we can not simply exchange a space coordinate with time – they are fundamentally different. It is an entirely different thing for two events to be separated in space and to be separated in time. Minkowski proposed that the formula for distance needed a change. He found that the correct formula was actually quite simple, differing only by a sign from Pythagoras' theorem: s^2 = x^2 + y^2 + z^2 - (ct)^2 \, where c is a constant and t is the time coordinate.[Note 2] Multiplication by c, which has the dimensions L T −1, converts the time to units of length and this constant has the same value as the speed of light. So the spacetime interval between two distinct events is given by s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 - c^2 (t_2 - t_1)^2. \, There are two major points to be noted. Firstly, time is being measured in the same units as length by multiplying it by a constant conversion factor. Secondly, and more importantly, the time-coordinate has a different sign than the space coordinates. This means that in the four-dimensional spacetime, one coordinate is different from the others and influences the distance differently. This new "distance" may be zero or even negative. This new distance formula, called the metric of the spacetime, is at the heart of relativity. This distance formula is called the metric tensor of M. This minus sign means that a lot of our intuition about distances can not be directly carried over into spacetime intervals. For example, the spacetime interval between two events separated both in time and space may be zero (see below). From now on, the terms distance formula and metric tensor will be used interchangeably, as will be the terms Minkowski metric and spacetime interval. In Minkowski spacetime the spacetime interval is the invariant length, the ordinary 3D length is not required to be invariant. The spacetime interval must stay the same under rotations, but ordinary lengths can change. Just like before, we were missing a dimension. Note that everything thus far is merely definitions. We define a four-dimensional mathematical construct which has a special formula for distance, where distance means that which stays the same under rotations (alternatively, one may define a rotation to be that which keeps the distance unchanged). Now comes the physical part. Rotations in Minkowski space have a different interpretation than ordinary rotations. These rotations correspond to transformations of reference frames. Passing from one reference frame to another corresponds to rotating the Minkowski space. An intuitive justification for this is given below, but mathematically this is a dynamical postulate just like assuming that physical laws must stay the same under Galilean transformations (which seems so intuitive that we don't usually recognise it to be a postulate). Since by definition rotations must keep the distance same, passing to a different reference frame must keep the spacetime interval between two events unchanged. This requirement can be used to derive an explicit mathematical form for the transformation that must be applied to the laws of physics (compare with the application of Galilean transformations to classical laws) when shifting reference frames. These transformations are called the Lorentz transformations. Just like the Galilean transformations are the mathematical statement of the principle of Galilean relativity in classical mechanics, the Lorentz transformations are the mathematical form of Einstein's principle of relativity. Laws of physics must stay the same under Lorentz transformations. Maxwell's equations and Dirac's equation satisfy this property, and hence they are relativistically correct laws (but classically incorrect, since they don't transform correctly under Galilean transformations). With the statement of the Minkowski metric, the common name for the distance formula given above, the theoretical foundation of special relativity is complete. The entire basis for special relativity can be summed up by the geometric statement "changes of reference frame correspond to rotations in the 4D Minkowski spacetime, which is defined to have the distance formula given above". The unique dynamical predictions of SR stem from this geometrical property of spacetime. Special relativity may be said to be the physics of Minkowski spacetime.[8][9][10][11] In this case of spacetime, there are six independent rotations to be considered. Three of them are the standard rotations on a plane in two directions of space. The other three are rotations in a plane of both space and time: These rotations correspond to a change of velocity, and the Minkowski diagrams devised by him describe such rotations. As has been mentioned before, one can replace distance formulas with rotation formulas. Instead of starting with the invariance of the Minkowski metric as the fundamental property of spacetime, one may state (as was done in classical physics with Galilean relativity) the mathematical form of the Lorentz transformations and require that physical laws be invariant under these transformations. This makes no reference to the geometry of spacetime, but will produce the same result. This was in fact the traditional approach to SR, used originally by Einstein himself. However, this approach is often considered to offer less insight and be more cumbersome than the more natural Minkowski formalism. Reference frames and Lorentz transformations: relativity revisited Question book-new.svg This section does not cite any references or sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2013) Changes in reference frame, represented by velocity transformations in classical mechanics, are represented by rotations in Minkowski space. These rotations are called Lorentz transformations. They are different from the Galilean transformations because of the unique form of the Minkowski metric. The Lorentz transformations are the relativistic equivalent of Galilean transformations. Laws of physics, in order to be relativistically correct, must stay the same under Lorentz transformations. The physical statement that they must be the same in all inertial reference frames remains unchanged, but the mathematical transformation between different reference frames changes. Newton's laws of motion are invariant under Galilean rather than Lorentz transformations, so they are immediately recognisable as non-relativistic laws and must be discarded in relativistic physics. The Schrödinger equation is also non-relativistic. Maxwell's equations are written using vectors and at first glance appear to transform correctly under Galilean transformations. But on closer inspection, several questions are apparent that can not be satisfactorily resolved within classical mechanics (see History of special relativity). They are indeed invariant under Lorentz transformations and are relativistic, even though they were formulated before the discovery of special relativity. Classical electrodynamics can be said to be the first relativistic theory in physics. To make the relativistic character of equations apparent, they are written using four-component vector-like quantities called four-vectors. Four-vectors transform correctly under Lorentz transformations, so equations written using four-vectors are inherently relativistic. This is called the manifestly covariant form of equations. Four-vectors form a very important part of the formalism of special relativity. In physics, special relativity is a fundamental theory concerning space and time, developed by Albert Einstein in 1905[1] as a modification of Galilean relativity. (See "History of special relativity" for a detailed account and the contributions of Hendrik Lorentz and Henri Poincaré.) The theory was able to explain some pressing theoretical and experimental issues in the physics of the time involving light and electrodynamics, such as the failure of the 1887 Michelson–Morley experiment, which aimed to measure differences in the relative speed of light due to the Earth's motion through the hypothetical, and now discredited, luminiferous aether. The aether was then considered to be the medium of propagation of electromagnetic waves such as light. Einstein postulated that the speed of light in free space is the same for all observers, regardless of their motion relative to the light source, where we may think of an observer as an imaginary entity with a sophisticated set of measurement devices, at rest with respect to itself, that perfectly records the positions and times of all events in space and time. This postulate stemmed from the assumption that Maxwell's equations of electromagnetism, which predict a specific speed of light in a vacuum, hold in any inertial frame of reference[2] rather than, as was previously believed, just in the frame of the aether. This prediction contradicted the laws of classical mechanics, which had been accepted for centuries, by arguing that time and space are not fixed and in fact change to maintain a constant speed of light regardless of the relative motions of sources and observers. Einstein's approach was based on thought experiments, calculations, and the principle of relativity, which is the notion that all physical laws should appear the same (that is, take the same basic form) to all inertial observers. Today, the result is that the speed of light defines the metre as "the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second."[3] This relates that the speed of light is by convention 299 792 458 m/s (approximately 1.079 billion kilometres per hour, or 671 million miles per hour). The predictions of special relativity are almost identical to those of Galilean relativity for most everyday phenomena, in which speeds are much lower than the speed of light, but it makes different, non-obvious predictions for objects moving at very high speeds. These predictions have been experimentally tested on numerous occasions since the theory's inception and were confirmed by those experiments.[4] The major predictions of special relativity are: Relativity of simultaneity: Observers who are in motion with respect to each other may disagree on whether two events occurred at the same time or one occurred before the other. Time dilation (An observer watching two identical clocks, one moving and one at rest, will measure the moving clock to tick more slowly) Relativistic mass Length contraction (Along the direction of motion, a rod moving with respect to an observer will be measured to be shorter than an identical rod at rest), and The equivalence of mass and energy (written as E = mc2). Special relativity predicts a non-linear velocity addition formula which prevents speeds greater than that of light from being observed. In 1908, Hermann Minkowski reformulated the theory based on different postulates of a more geometrical nature.[5] This approach considers space and time as being different components of a single entity, the spacetime, which is "divided" in different ways by observers in relative motion. Likewise, energy and momentum are the components of the four-momentum, and the electric and magnetic field are the components of the electromagnetic tensor. As Galilean relativity is now considered an approximation of special relativity valid for low speeds, special relativity is considered an approximation of the theory of general relativity valid for weak gravitational fields. General relativity postulates that physical laws should appear the same to all observers (an accelerating frame of reference being equivalent to one in which a gravitational field acts), and that gravitation is the effect of the curvature of spacetime caused by energy (including mass). |
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