Special relativity

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 relativity

The principle of relativity was introduced by Galileo. Overturning the old absolute 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. Newton accepted the principle of relativity when constructing an improved set, 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 as it passes through. The idea of an aether seemed to reintroduce the idea of an 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 field. Maxwell's equations showed that accelerating a charge produced electromagnetic radiation which always traveled at the speed of light. The equations showed that the speed of the radiation did not change based upon the speed of the source. This is consistent with analogies to mechanical waves. Presumably, however, the speed of the radiation 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 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 speed of light, the resulting laws simplify to the Galilean transformations. The theory, known as Lorentz Ether Theory (LET) was criticized, even by Lorentz himself, because of its ad hoc nature.

While Lorentz suggested the Lorentz transformation equations, Einstein's contribution was, inter alia, to derive these equations from a more fundamental theory, a theory which did not require 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) "Theory of Invariants". It was Max Planck who 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 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 relativity

1. First postulate (principle of relativity)

Observation of physical phenomena by more than one inertial observer must result in agreement between the observers as to the nature of reality. Or, the nature of the universe must not change for an observer if their inertial state changes.

Every physical theory should look the same mathematically to every inertial observer.

2. Second postulate (invariance of c)

The speed of light in vacuum, commonly denoted c, is the same to all inertial observers, is the same in all directions, and does not depend on the velocity of the object emitting the light. This postulate has been verified experimentally. When combined with the First Postulate, this Second Postulate is equivalent to stating that light does not require any medium (such as "aether") in which to propagate.

Mathematical formulation of the postulates

In 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)

Every physical law is invariant under inertial co-ordinate transformations. Thus, if an object in spacetime obeys the mathematical equations describing a physical law in one inertial frame of reference, it must necessarily obey the same equations when using any other inertial frame of reference.

2. Second Postulate (Invariance of c)

There exists an absolute constant <math>0 < c < \infty<math> with the following property. If A, B are two events which have co-ordinates <math>(x_1,x_2,x_3,t)<math> and <math>(y_1,y_2,y_3,s)<math> in one inertial frame <math>F<math>, and have co-ordinates <math>(x'_1,x'_2,x'_3,t')<math> and <math>(y'_1,y'_2,y'_3,s')<math> in another inertial frame <math>F'<math>, then

<math>\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + (x_3-y_3)^2} = c(s-t)<math> if and only if <math>\sqrt{(x'_1-y'_1)^2 + (x'_2-y'_2)^2 + (x'_3-y'_3)^2} = c(s'-t')<math>.

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>. 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. 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:

If A, B are two events which have co-ordinates <math>(x_1,x_2,x_3,t)<math> and <math>(y_1,y_2,y_3,s)<math> in one inertial frame <math>F<math>, and have co-ordinates <math>(x'_1,x'_2,x'_3,t')<math> and <math>(y'_1,y'_2,y'_3,s')<math> in another inertial frame <math>F'<math>, then <math>s-t = s'-t'<math>. Furthermore, if <math>s-t=s'-t'=0<math>, then

<math>\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + (x_3-y_3)^2} = \sqrt{(x'_1-y'_1)^2 + (x'_2-y'_2)^2 + (x'_3-y'_3)^2}<math>.

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 relativity

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 unreproducable experimental error. General relativity is still insufficiently confirmed by experiment to exclude certain alternative theories of gravitation such as the Brans-Dicke theory.

Consequences of special relativity

Special relativity has several consequences that struck many people as bizarre, among which are:

The time lapse between two events is not invariant from observer to another, but is dependent on the relative speeds of the observers' reference frames. (See Lorentz transformation equations)
Two events that occur simultaneously in different places in one frame of reference may occur at different times in another frame of reference (lack of simultaneity).
The dimensions (e.g. length) of an object as measured by one observer may differ from the results of measurements of the same object made by another observer. (See Lorentz transformation equations)
The twin paradox concerns a twin who flies off in a spaceship travelling near the speed of light. When he returns he discovers that his twin has aged much more rapidly than he has (or he aged more slowly).
The ladder paradox involves a long ladder travelling near the speed of light and being contained within a smaller garage.

Lack of an absolute reference frame

Special 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 19^th 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 energy

In 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

<math>E = \gamma m c^2\,<math>

<math> p = \gamma m v \,<math>

where γ (the Lorentz factor) is given by

<math>\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}<math>

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

<math> E \approx m c^2 + \begin{matrix} \frac{1}{2} \end{matrix} m v^2 \,<math>

<math> p \approx m v \,<math>

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 mass

It is often stated that in special relativity the mass of a body increases as its velocity increases. 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

<math>M = \gamma m \,<math>

Since γ increases with velocity so does the relativistic mass. This definition is 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 rest mass become equal.

Neither definition is right or wrong, it is simply a matter of convenience. However, many physicists dislike the concept of the relativistic mass because it is not a scalar. In other words, the relativistic mass along one axis may not be the same as the relativistic mass along another axis. Furthermore 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.

Simultaneity and causality

Special 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.)

light cone

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 relativity

SR 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

Michelson-Morley experiment - ether drift
Hamar experiment - obstruction of ether flow
Trouton-Noble experiment - torque on a capacitor
Kennedy-Thorndike experiment - time contraction
Forms of the emission theory experiment

External links

Our sister project, Wikiversity, provides a course on Special relativity .

Reflections on Relativity (http://www.mathpages.com/rr/rrtoc.htm) A complete online book on relativity
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Special_relativity.html (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Special_relativity.html)
Brane World Mach Principles and the Michelson-Morley experiment (http://www.mathpreprints.com/math/Preprint/paultrr/20040119/1/Evaluation_of_Brane_World_Mach_Principles.pdf)
Why Hyperspace & Dual Reference frames (http://doc.cern.ch//archive/electronic/other/ext/ext-2004-121.pdf)
Special Relativity: a matter of perspective (http://home.earthlink.net/~descubes/Relativity/Relat1.html) Special relativity presented as a natural 4D extension of perspective
Petites exp�riences de pens�e (http://fr.wikipedia.org/wiki/Relativit%C3%A9_restreinte#Petites_exp.C3.A9riences_de_pens.C3.A9e) : five interesting thought experiments about special relativity quoted in the French wikipedia (unfortunately in French).
Special relativity theory made intuitive (http://spoirier.lautre.net/en/relativity.htm) : a new approach to explain the theoretical meaning of Special Relativity from an intuitive geometrical viewpoint

General subfields within physics
Classical mechanics \| Condensed matter physics \| Continuum mechanics \| Electromagnetism \| General relativity \| Particle physics \| Quantum field theory \| Quantum mechanics \| Solid state physics \| Special relativity \| Statistical mechanics \| Thermodynamics

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