A system of mechanics applicable at high velocities (approaching the velocity of light) in the absence of gravitation; a generalization of Newtonian mechanics, due almost entirely to Albert Einstein (1905). Its fundamental postulates are that the velocity of light c is the same for all observers, no matter how they are moving; that the laws of physics are the same in all inertial frames; and that all such frames are equivalent. On this basis, no object may have a velocity in excess of the velocity of light; and two events which appear simultaneous to one observer need not be so for another. The system gives laws of mechanics which reproduce those of common sense Newtonian mechanics at low velocities, and is well supported experimentally, especially in particle physics. Generalized special relativity, incorporating gravitation, is called general relativity. The constancy of the velocity of light is supported experimentally (eg the MichelsonMorley experiment), and implied by Maxwell's equations of electromagnetism, which express the velocity of light in terms of simple fundamental electric and magnetic constants. Newtonian mechanics is impossible to reconcile with this, since it produces velocity addition rules which fail for light and Maxwell's equations. Einstein's solution (1905) was to insist that the velocity of light really is the same for all observers, and that the form of Maxwell's equations is the same for all. This then implied that the rules of mechanics needed to be rewritten. The new system of mechanics derived by Einstein is consistent only if old notions of basic quantities such as mass, energy, space, and time are modified; it also incorporates a velocity addition rule. Special relativistic mechanics is fully supported by observation.
The special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies".
This theory has a variety of surprising consequences that seem to violate common sense, but that have been verified experimentally. Special relativity overthrows Newtonian notions of absolute space and time by stating that distance and time depend on the observer, and that time and space are perceived differently, depending on the observer.
The theory was called "special" because it applies the principle of relativity only to inertial frames.
Although special relativity makes relative some quantities, such as time, that we would have imagined to be absolute based on everyday experience, it also makes absolute some others that we would have thought were relative. Special relativity reveals that c is not just the velocity of a certain phenomenon - light - but rather a fundamental feature of the way space and time are tied together.
For history and motivation, see the article: history of special relativity
Postulates
Main article: Postulates of special relativity
First postulate - Special principle of relativity - The laws of physics are the same in all inertial frames of reference.Lack of an absolute reference frame
The principle of relativity, which states that there is no stationary reference frame, dates back to Galileo, and was incorporated into Newtonian Physics. Special relativity is formulated so as to not assume that any particular frame of reference is special;
Consequences
Einstein has said that all of the consequences of special relativity can be found from examination of the Lorentz transformations.
These transformations, and hence special relativity, lead to different physical predictions than Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counter-intuitive:
Time dilation — the time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship travelling near the speed of light and returns to discover that his twin has aged much more). Composition of velocities — velocities (and speeds) do not simply 'add', for example if a rocket is moving at ⅔ the speed of light relative to an observer, and the rocket fires a missile at ⅔ of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer.Reference frames, coordinates and the Lorentz transformation
Relativity theory depends on "reference frames".
An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a "point" in space-time. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.
For example, the explosion of a firecracker may be considered to be an "event".
In relativity theory we often want to calculate the position of a point from a different reference point.
Suppose we have a second reference frame S', whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity with respect to S along the axis.
Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always moving with respect to some other reference frame.
Let's define the event to have space-time coordinates in system S and in S'. Then the Lorentz transformation specifies that these coordinates are related in the following way:
where is called the Lorentz factor and is the speed of light in a vacuum.
The and coordinates are unaffected, but the and axes are mixed up by the transformation.
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
Simultaneity
From the first equation of the Lorentz transformation in terms of coordinate differences
it is clear that two events that are simultaneous in frame S (satisfying ), are not necessarily simultaneous in another inertial frame S' (satisfying ).
Time dilation and length contraction
Writing the Lorentz Transformation and its inverse in terms of coordinate differences we get
and
Suppose we have a clock at rest in the unprimed system S. If we want to know the relation between the times between these ticks as measured in both systems, we can use the first equation and find
This shows that the time Δt' between the two ticks as seen in the 'moving' frame S' is larger than the time Δt between these ticks as measured in the rest frame of the clock.
Similarly, suppose we have a measuring rod at rest in the unprimed system. In other words, the measurement is characterized by Δt' = 0, which we can combine with the fourth equation to find the relation between the lengths Δx and Δx':
This shows that the length Δx' of the rod as measured in the 'moving' frame S' is shorter than the length Δx in its own rest frame.
These effects are not merely appearances;
See also the twin paradox.
Causality and prohibition of motion faster than light
In diagram 2 the interval AB is 'time-like';
The interval AC in the diagram is 'space-like'; On the other hand, the logical situation is not as clear in the case of general relativity, so it is an open question whether or not there is some fundamental principle that preserves causality (and therefore prevents motion faster than light) in general relativity.
Even without considerations of causality, there are other strong reasons why faster-than-light travel is forbidden by special relativity.
Composition of velocities
If the observer in sees an object moving along the axis at velocity , then the observer in the system, a frame of reference moving at velocity in the direction with respect to , will see the object moving with velocity where
This equation can be derived from the space and time transformations above. Also, if both and are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities:
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.
There are a couple of (equivalent) ways to define momentum and energy in SR.
Given an object of invariant mass m traveling at velocity v the energy and momentum are given (and even defined) by
where γ (the Lorentz factor) is given by
where β is the velocity as a ratio of the speed of light.
Relativistic 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.
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.
Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:
Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy.
This formula becomes important when one measures the masses of different atomic nuclei.
Relativistic mass
Introductory physics courses and some older textbooks on special relativity sometimes define a relativistic mass which increases as the velocity of a body increases. According to the geometric interpretation of special relativity, this is often deprecated and the term 'mass' is reserved to mean invariant mass and is thus independent of the inertial frame, i.e., invariant.
Using the relativistic mass definition, the mass of an object may vary depending on the observer's inertial frame in the same way that other properties such as its length may do so. That observer defines the relativistic mass of that body as:
"Relativistic mass" should not be confused with the "longitudinal" and "transverse mass" definitions that were used around 1900 and that were based on an inconsistent application of the laws of Newton: those used f=ma for a variable mass, while relativistic mass corresponds to Newton's dynamic mass in which p=Mv and f=dp/dt.
Note also that the body does not actually become more massive in its proper frame, since the relativistic mass is only different for an observer in a different frame.
Physics textbooks sometimes use the relativistic mass as it allows the students to utilize their knowledge of Newtonian physics to gain some intuitive grasp of relativity in their frame of choice (usually their own!).
Force
The classical definition of ordinary force f is given by Newton's Second Law in its original form:
and this is valid in relativity.
Many modern textbooks rewrite Newton's Second Law as
This form is not valid in relativity or in other situations where the relativistic mass M is varying.
This formula can be replaced in the relativistic case by
As seen from the equation, ordinary force and acceleration vectors are not necessarily parallel in relativity.
However the four-vector expression relating four-force to invariant mass m and four-acceleration restors the same equation form
The geometry of space-time
SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time.
The differential of distance(ds) in cartesian 3D space is defined as:
where (dx1,dx2,dx3) 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:
If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary: x4 = ict . In this case the above equation becomes symmetric:
This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space.
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 the 4-dimensional cone:
This null dual-cone represents the "line of sight" of a point in space. (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'.
The geometry of Minkowski space can be depicted using Minkowski diagrams, which are also useful in understanding many of the thought-experiments in special relativity.
Physics in spacetime
Here, we see how to write the equations of special relativity in a manifestly invariant form. The position of an event in spacetime is given by a contravariant four vector whose components are:
That is, x = x and x = z. Subscripts are covariant indices which also range from zero to three as with the spacetime gradient of a field φ:
Metric and tranformations of coordinates
Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any inertial reference frame) as:
Its reciprocal is:
Then we recognise that co-ordinate transformations between inertial reference frames are given by the Lorentz transformation tensor Λ. For the special case of motion along the x-axis, we have:
which is simply the matrix of a boost (like a rotation) between the x and t coordinates. Also, β and γ are defined as:
More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy:
where there is an implied summation of and from 0 to 3 on the right-hand side in accordance with the Einstein summation convention.
All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well known tensor transformation law
Where is the reciprocal matrix of .
To see how this is useful, we transform the position of an event from an unprimed co-ordinate system S to a primed system S', we calculate
which is the Lorentz transformation given above.
The squared length of the differential of the position four-vector constructed using
is an invariant.
The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact.
Velocity and acceleration in 4D
Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector Uμ is given by
Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another. Uμ also has an invariant form:
So all velocity four-vectors have a magnitude of c. Given this, differentiating the above equation by τ produces
So in relativity, the acceleration four-vector and the velocity 4-vector are orthogonal.
Momentum in 4D
The momentum and energy combine into a covariant 4-vector:
where m is the invariant mass.
The invariant magnitude of the momentum 4-vector is:
We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
We see that the rest energy is an independent invariant.
The rest energy is related to the mass according to the celebrated equation discussed above:
Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame.
Force in 4D
To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. The covariant version of the four-force is:
where is the proper time.
In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing in which case it is the negative of that rate of change times c2.
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector.
Relativity and unifying electromagnetism
The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
Electromagnetism in 4D
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity.
The charge density and current density are unified into the current-charge 4-vector:
The law of charge conservation becomes:
The electric field and the magnetic induction are now unified into the (rank 2 antisymmetric covariant) electromagnetic field tensor:
The density of the Lorentz force exerted on matter by the electromagnetic field becomes:
Faraday's law of induction and Gauss's law for magnetism combine to form:
Although there appear to be 64 equations here, it actually reduces to just four independent equations.
The electric displacement and the magnetic field are now unified into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor:
Ampere's law and Gauss's law combine to form:
In a vacuum, the constitutive equations are:
Antisymmetry reduces these 16 equations to just six independent equations.
The energy density of the electromagnetic field combines with Poynting vector and the Maxwell stress tensor to form the 4D stress-energy tensor. It is the flux (density) of the momentum 4-vector and as a rank 2 mixed tensor it is:
where is the Kronecker delta.
The conservation of linear momentum and energy by the electromagnetic field is expressed by:
where is again the density of the Lorentz force.
Status
Main article: Status of special relativity
Special relativity is accurate only when gravitational potential is much less than c) and thus accepted by the physics community.
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.
Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields).
Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) - thus Newtonian mechanics can be considered as a special relativity of slow moving bodies.
A few key experiments can be mentioned that led to special relativity:
The Trouton-Noble experiment showed that the torque on a capacitor is independent on position and inertial reference frame — such experiments led to the first postulate The famous Michelson-Morley experiment gave further support to the postulate that detecting an absolute reference velocity was not achievable.A number of experiments have been conducted to test special relativity against rival theories.
Explanations of special relativity
The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus. Relativity An introduction to special relativity at the undergraduate level, without calculus. Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University. Special relativity theory made intuitive A new approach to explain the theoretical meaning of Special Relativity from an intuitive geometrical viewpoint Special Relativity Stanford University, Helen Quinn, 2003 Relativity: the Special and General Theory, available freely at Project Gutenberg, by Albert Einstein Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics. Caltech Relativity Tutorial A basic introduction to concepts of Special and General Relativity, requiring only a knowledge of basic geometry. Understanding Special Relativity - The theory of special relativity in an easily understandable way. The origins of Einstein's special theory of relativity - A historical approach to the study of the special theory of relativity. Relativity in its Historical Context The discovery of special relativity was inevitable, given the momentous discoveries that preceded it.
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