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Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. It replaced Newtonian notions of space and time, and incorporated electromagnetism as represented by Maxwell's equations. The theory is called "special" because it is a special case of Einstein's principle of relativity where the effects of gravity can be ignored. Ten years later, Einstein published the theory of general relativity, which incorporates gravitation.
Motivation for the theory of special relativityThe principle of relativity was introduced by Galileo. Overturning the old absolutist views of Aristotle, it held that motion, or at least uniform motion in a straight line, only had meaning relative to something else, and that there was no absolute reference frame by which all things could be measured. Galileo also assumed a set of transformations called the Galilean transformations, which seem like common sense today. Galileo produced five laws of motion. Next came Newton who inferred from his rotating bucket experiment an "absolute space", that is, an absolute reference frame. Nevertheless he kept the principle of relativity for what can be observed— uniform motion could not detect his absolute space. That concept he used for constructing an improved set of equations containing only three laws of motion. While these seemed to work well for everyday phenomena involving solid objects, light was still problematic. Newton believed that light was "corpuscular," but later physicists found that a transverse wave model of light was more useful. Mechanical waves travel in a medium, and so it was assumed for light. This hypothetical medium was called the "luminiferous aether." It seemed to have some conflicting properties, such as being extremely stiff, to account for the high speed of light, while at the same time being insubstantial, so as not to slow down the Earth, which moves through it. The idea of an aether seemed to reintroduce the idea of a detectable absolute frame of reference, one that is stationary with respect to the aether. In the early 19th century, light, electricity, and magnetism began to be understood as aspects of the electromagnetic aether field. Maxwell's equations showed that accelerating a charge produced electromagnetic radiation which always traveled at the speed of light. The equations were based on the ether idea in which the speed of radiation does not change with the speed of the source. This is consistent with analogies to mechanical waves. Presumably, however, the speed of the radiation relative to the observer would change based on the speed of the observer. Physicists tried to use this idea to measure the speed of the Earth with respect to the aether. The most famous such attempt was the Michelson-Morley experiment. While these experiments were controversial for some time, a consensus emerged that the speed of light does not vary with the speed of the observer, and since—according to Maxwell's equations—it does not vary with the speed of the source, the speed of light must be invariant for all observers. Before special relativity, Hendrik Lorentz and others had already noted that electromagnetic forces differed depending on the position of the observer. For example, one observer might see no magnetic field in a particular area while another moving relative to the first does. Lorentz suggested an aether theory in which objects and observers travelling with respect to a stationary aether underwent a physical shortening (Lorentz-Fitzgerald contraction) and a change in temporal rate (time dilation). This allowed what appeared at the time to be a reconciliation of electromagnetics and Newtonian physics by replacing the Galilean transformations. When the velocities involved are much less than the speed of light, the resulting laws simplify to the Galilean transformations. He proposed it to be valid for all forces. However, at that point in time he didn't realise the full power of his theory. The theory, today called Lorentz Ether Theory (LET) was criticized, even by Lorentz himself, because of its apparently ad hoc nature. For all practical purposes it is the same theory as SRT, and he taught it as such. While Lorentz suggested the Lorentz transformation equations, Einstein's contribution was, inter alia, to derive these equations from a more fundamental principle without assuming the presence of an aether. Einstein wanted to know what was invariant (the same) for all observers. Under Special Relativity, the seemingly complex transformations of Lorentz and Fitzgerald derived cleanly from simple geometry and the Pythagorean theorem. The original title for his theory was (translated from German) "On the Electrodynamics of Moving Bodies". Max Planck first suggested the term "relativity" to highlight the notion of transforming the laws of physics between observers moving relative to one another. Special relativity is usually concerned with the behaviour of objects and "observers" (inertial reference systems) which remain at rest or are moving at a constant velocity. In this case, the observer is said to be in an inertial frame of reference. Comparison of the position and time of events as recorded by different inertial observers can be done by using the Lorentz transformation equations. A common misstatement about relativity is that SR cannot be used to handle the case of objects and observers who are undergoing acceleration (non-inertial reference frames), but this is incorrect. For an example, see the relativistic rocket problem. SR can correctly predict the behaviour of accelerating bodies in the presence of a constant or zero gravitational field, or those in a rotating reference frame. It is not capable of accurately describing motion in varying gravitational fields. Postulates of special relativity1. First postulate (principle of relativity)
2. Second postulate (invariance of c)
Mathematical formulation of the postulatesIn the rigorous mathematical formulation of special relativity, we suppose that the universe exists on a four-dimensional spacetime M. Individual points in spacetime are known as events; physical objects in spacetime are described by worldlines (if the object is a point particle) or worldsheets (if the object is larger than a point). The worldline or worldsheet only describes the motion of the object; the object may also have several other physical characteristics such as energy, momentum, mass, charge, etc. In addition to events and physical objects, there are a class of inertial observers (which may or may not correspond to an actual physical object). Each inertial observer has associated to it an inertial frame of reference. This frame of reference provides a co-ordinate system <math>(x_1,x_2,x_3,t)<math> for events in the spacetime M. Furthermore, this frame of reference also gives co-ordinates to all other physical characteristics of objects in the spacetime, for instance it will provide co-ordinates <math>(p_1,p_2,p_3,E)<math> for the momentum and energy of an object, co-ordinates <math>(E_1,E_2,E_3,B_1,B_2,B_3)<math> for an electromagnetic field, and so forth. We assume that given any two inertial observers, there exists a coordinate transformation that converts the co-ordinates from one frame of reference to the co-ordinates in another frame of reference. This transformation not only provides a conversion for spacetime co-ordinates <math>(x_1,x_2,x_3,t)<math>, but will also provide a conversion for all other physical co-ordinates, such as a conversion law for momentum and energy <math>(p_1,p_2,p_3,E)<math>, etc. (In practice, these conversion laws can be efficiently handled using the mathematics of tensors). We also assume that the universe obeys a number of physical laws. Mathematically, each physical law can be expressed with respect to the co-ordinates given by an inertial frame of reference by a mathematical equation (for instance, a differential equation) which relates the various co-ordinates of the various objects in the spacetime. A typical example is Maxwell's equations. Another is Newton's first law. 1. First Postulate (Principle of relativity)
2. Second Postulate (Invariance of c)
Informally, the Second Postulate asserts that objects travelling at speed c in one reference frame will necessarily travel at speed c in all reference frames. It turns out that the Second Postulate can be mathematically deduced from the First Postulate and Maxwell's equations, in which case c is given by <math>c = 1/\sqrt{\mu_0 \epsilon_0}<math>, where <math>\mu_0<math> and <math>\epsilon_0<math> are the permeability and permittivity of vacuum respectively. Since Maxwell's equations govern the propagation of electromagnetic radiation such as light, it is thus common practice to refer to c as the speed of light, and one can interpret the Second Postulate as nothing more than an assertion that electrodynamics as described by Maxwell's equations is indeed correct, in contrast with the earlier theory of Galilean relativity which was in contradiction to Maxwell's equations (unless one postulated an aether). However, it is worth noting that the formulation of the Second Postulate as given above does not actually require the existence of electromagnetic radiation or Maxwell's equations. The second postulate can be used to imply a stronger version of itself, namely that the spacetime interval is invariant under changes of inertial reference frame. In the above notation, this means that
for any two events A, B. This can in turn be used to deduce the transformation laws between reference frames; see Lorentz transformation. The postulates of special relativity can be expressed very succinctly using the mathematical language of pseudo-Riemannian manifolds. The second postulate is then an assertion that the four-dimensional spacetime M is a pseudo-Riemannian manifold equipped with a Lorentzian metric g of signature (3,1), which is given by the flat Minkowski metric when measured in each inertial reference frame. This metric is viewed as one of the physical quantities of the theory, thus it transforms in a certain manner when the frame of reference is changed, and it can be legitimately used in describing the laws of physics. The first postulate is an assertion that the laws of physics are invariant when represented in any frame of reference for which g is given by the Minkowski metric. One advantage of this formulation is that it is now easy to compare special relativity with general relativity, in which the same two postulates hold but the assumption that the metric is required to be Minkowski is dropped. The theory of Galilean relativity is the limiting case of special relativity in the non-relativistic limit <math>c \to \infty<math>. In this theory, the first postulate remains unchanged, but the second postulate is modified to:
The physical theory given by classical mechanics, and Newtonian gravity is consistent with Galilean relativity, but not special relativity. Conversely, Maxwell's equations are not consistent with Galilean relativity unless one postulates the existence of a physical aether. In a surprising number of cases, the laws of physics in special relativity (such as the famous equation <math>E=mc^2<math>) can be deduced by combining the postulates of special relativity with the hypothesis that the laws of special relativity approach the laws of classical mechanics in the non-relativistic limit. Status of special relativitySpecial relativity is only accurate when gravitational effects are negligible or very weak, otherwise it must be replaced by general relativity. At very small scales, such as at the Planck length and below, it is also possible that special relativity breaks down, due to the effects of quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is now universally accepted by the physics community and experimental results which appear to contradict it are widely believed to be due to unreproducible experimental error. General relativity is still insufficiently confirmed by experiment to exclude certain alternative theories of gravitation such as the Brans-Dicke theory. Because of the freedom one has to select how one defines units of length and time in physics, it is possible to make one of the two postulates of relativity a tautological consequence of the definitions, but one cannot do this for both postulates simultaneously, as when combined they have consequences which are independent of one's choice of definition of length and time. For instance, if one defines units of length and time in terms of a physical object (e.g. by defining units of time in terms of transitions of a cesium atom, or length in terms of wavelengths of a krypton atom) then it becomes tautological that the law determining that unit of length or time will be the same in all reference frames, but then the invariance of c is non-trivial. Conversely, if one defines units of length and time in such a way that c is defined to be constant, then the second postulate becomes tautological, but the first one does not; for instance, if the length unit is defined in terms of the time unit and a predetermined fixed value of c, then there is no a priori reason why the number of wavelengths of krypton in a unit of length will be the same in all reference frames (or even in all orientations). Compatibility of special relativity with other physical theoriesAt a purely mathematical level, special relativity is internally consistent, being nothing more than the geometry of Minkowski space, together with a requirement that all laws of nature be Lorentz-invariant. Thus it is not possible to create a thought-experiment within special relativity that creates a genuine logical paradox, unless one assumes the existence of objects which do not seem to exist in special relativity (see below). However it is certainly possible to create thought-experiments which have unintuitive consequences, and which contradict some other theories. More specifically, special relativity is not compatible with the physical existence of the following objects, forces, or laws (except in the nonrelativistic limit in which all speeds are much less than c):
Special relativity is compatible with
As these examples show, special relativity affects all aspects of physics, and is not purely concerned with light. Indeed, in a very literal sense, light is merely the most visible phenomenon in physics which involves the constant c. Special relativity is partially consistent with Lorentz ether theory, in the sense that these theories give the same predictions in any experimental setting, however they are ontologically and philosophically very different, and they suggest different ways to extend or modify the theory, for instance to understand gravity. Consequences of special relativitySpecial relativity has several consequences that struck many people as bizarre, among which are:
See Consequences of Special Relativity. Lack of an absolute reference frameSpecial Relativity rejects the idea of any absolute ('unique' or 'special') frame of reference; rather physics must look the same to all observers travelling at a constant velocity (inertial frame). This 'principle of relativity' dates back to Galileo, and is incorporated into Newtonian Physics. In the late 19th Century, some physicists suggested that the universe was filled with a substance known as "aether" which transmited Electromagnetic waves. Aether constituted an absolute reference frame against which speeds could be measured. Aether had some wonderful properties: it was sufficiently elastic that it could support electromagnetic waves, those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, including the Michelson-Morley experiment, suggested that the Earth was always 'stationary' relative to the Aether - something that is difficult to explain. The most elegant solution was to discard the notion of Aether and an absolute frame, and to adopt Einstein's postulates. Mass, momentum, and energyIn addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated. There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR. Given an object of mass m traveling at velocity v the energy and momentum are given by
where γ (the Lorentz factor) is given by
and c is the speed of light. The term γ occurs frequently in relativity, and comes from the Lorentz transformation equations. The energy and momentum can be related through the formula
which is referred to as the relativistic energy-momentum equation. For velocities much smaller than those of light γ can be approximated using a Taylor series expansion and one finds that
Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities. Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and γ = 1) there is a non-zero energy remaining:
This energy is referred to as rest energy. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which matter. Taking this formula at face value, we see that in relativity, mass is simply another form of energy. This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by nuclear reactions, providing important information which was useful in the development of the nuclear bomb. The implications of this formula on 20th century life has made it one of the most famous equations in all of science. On massIt is often stated that in special relativity the mass of a body increases as its velocity increases, notably in older textbooks and in some introductory physics courses. However, this statement depends on one's definition of mass, and in SR there are actually two different notions of mass. The equations above use what is called the invariant mass or rest mass. This mass is an invariant quantity, meaning that it is the same for all inertial observers. In particular, the invariant mass does not increase with velocity. Another definition of mass is the relativistic mass which is given by
Since γ increases with velocity so does the relativistic mass. This definition is more consistent with (relativistic) length and time and convenient for some purposes. In particular, one can write the equations for energy and momentum as
which are valid in all reference frames. If the velocity is zero the relativistic mass and the invariant mass become equal. Neither definition is right or wrong. However, many physicists dislike the concept of relativistic mass because it changes under a Lorentz transformation; they prefer to formulate the special theory of relativity in terms of invariant quantities. The invariant mass is an important quantity in general relativity and quantum field theory. Thus many physicists simply refer to the mass when they actually mean the invariant mass, while they refer to relativistic energy instead of relativistic mass. Simultaneity and causalitySpecial relativity holds that events that are simultaneous in one frame of reference need not be simultaneous in another frame of reference. (See simultaneity for details.)  The interval AB in the diagram to the right is 'time-like'. I.e. there is a frame of reference in which event A and event B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect). The interval AC in the diagram is 'space-like'. I.e. there is a frame of reference in which event A and event C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. Barring some way of traveling faster than light, it is not possible for any matter (or information) to travel from A to C or from C to A. Thus there is no causal connection between A and C. The geometry of space-time in special relativitySR uses a 'flat' 4 dimensional Minkowski space, usually referred to as space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with. The differential of distance(ds) in cartesian 3D space is defined as:
where <math>(dx_1,dx_2,dx_3)<math> are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, is added, with units of c, so that the equation for the differential of distance becomes:
In many situations it may be convenient to treat time as imaginary (e.g. it may simplify equations), in which case <math>t<math> in the above equation is replaced by <math>i.t'<math>, and the metric becomes
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space,
We see that the null geodesics lie along a dual-cone:
defined by the equation
, or
Which is the equation of a circle with r=c*dt. If we extend this to three spatial dimensions, the null geodesics are continuous concentric spheres, with radius = distance = c*(+ or -)time.
This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old.", we are looking down this line of sight: a null geodesic. We are looking at an event <math>d = \sqrt{x_1^2+x_2^2+x_3^2} <math> meters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)
The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'. Tests of postulates of special relativity
Related topics
External links
References
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