Special relativity

For a more accessible and less technical introduction to this topic, see Introduction to special relativity.

USSR postage stamp dedicated to Albert Einstein

Special relativity (SR) (also known as the special theory of relativity) is a physical theory of measurement in inertial frames of reference proposed by Albert Einstein and others. Starting from simple yet far-reaching postulates, the theory derives the Lorentz transformation and its sometimes counterintuitive conclusions regarding the observer-dependence of certain quantities, such as speed, distance, and the time interval. Experiments have extensively vindicated the theory, and it forms the basis of many other successful physical theories such as relativistic quantum field theory and general relativity.

The theory incorporates two postulates, namely Galilean invariance and the universality of the speed of light. The first postulate, Galilean invariance, states that the laws of physics do not single out a privileged inertial reference frame; hence any convenient inertial frame may be used to describe a physical process, and any description of nature must remain agnostic of the specific inertial frame used. According to the second postulate, the speed of light is privileged: any inertial observer measures this quantity to be the same. This postulate contradicts the previously held theory (Galilean transformation) that all speeds depend on the observer. This postulate is motivated by the fact that electrodynamics predicts a certain value of the speed of light, despite not referencing any particular reference frame (in accordance with Galileo's principle). Einstein took this to mean that, rather than abandoning Galilean relativity, one must modify the Galilean transformation to take account of this.

These two postulates have far-ranging consequences, which experiments have verified extensively. Starting from these postulates, one can derive what modifications Galilean transformation needs to be consistent with the postulates. The resulting transformation is the Lorentz transformation. According to this, spatial and temporal coordinates must be mixed (somewhat analogously to rotation) when converting from one inertial frame to another. Hence it is inconvenient to consider space and time separately; one considers the two to be aspects of a combined entity, spacetime. It follows that certain quantities that pertain to space or time alone are not invariant, that is, different observers may observe these quantities to have different values. These quantities include length, time intervals, speed, and even the notion of simultaneity: one observer may observe an event to precede another, while another observer may observe a different order of the events. Special relativity also predicts a relation between mass and energy, namely that the mass of an object equals its energy as observed by an observer moving along with the body. The well-known equation E=mc² summarizes this prediction. Experiments have verified the wide-ranging predictions of special relativity to an extraordinary precision.

Special relativity supersedes and corrects the older theory, Newtonian mechanics (including the Galilean transformation), in the realm where bodies have speeds comparable to the speed of light. When all bodies have speeds far smaller than the speed of light (as in most everyday phenomena), Newtonian mechanics remains valid, and special relativity agrees with Newtonian mechanics.

The word special in the name of the theory means that the principle of relativity applies only to inertial observers. When one considers gravitation as well, one may apply the principle of relativity to any observer, including accelerating ones. This forms the basis of general relativity, a later theory also due to Einstein. According to general relativity, locally inertial frames exist at every point in spacetime, even in the presence of gravitation. In these frames special relativity remains valid up to first order. When gravity is weak, higher-order effects are negligible.

Postulates

Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results... How, then, could such a universal principle be found?

— Albert Einstein: Autobiographical Notes

Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:

• The Principle of Relativity – The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.
• The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body." (from the preface). That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.

It should be noted that the derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (which are made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.

Following Einstein's original presentation of special relativity, , many different sets of postulates have been proposed in various alternative derivations. However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the Principle of Relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.

Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.

Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:

The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws...

Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.

From the principle of relativity alone without assuming the constancy of the speed of light, i.e. using the isotropy of space and the symmetry implied by the principle of special relativity, one can show that the space-time transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.

Mass-energy equivalence

See also: Mass-energy equivalence

In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for E = mc². In his 1905 paper, Einstein used, besides his two principal postulates, the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy/momentum of light transforms like the energy/momentum of massless particles, which was known to be true from Maxwell's equations. The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905. Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong. Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.

Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass-energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.

Status of special relativity

Predictions

Main article: Consequences of special relativity

From the Lorentz transformation, a number of interesting effects regarding non-invariant physical quantities emerge. The non-invariance means that different observers observe dilated or contracted quantities relative to other observers. These effects include:

• 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 traveling near the speed of light and returns to discover that his or her twin sibling has aged much more).
• Relativity of simultaneity – the time coordinate changes during a Lorentz boost, in constrast to the Galilean transformation. This means two events that appear simultaneous in one inertial frame appear non-simultaneous (have different time coordinates) in another. In particular, this means action at a distance, which crucially relies on simultaneity, is impossible.
• Lorentz contraction – the dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).
• Composition of velocities – velocities (and speeds) do not simply 'add', for example if a rocket is moving at 2⁄3 the speed of light relative to an observer, and the rocket fires a missile at 2⁄3 of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. (In this example, the observer would see the missile travel with a speed of 12⁄13 the speed of light.)

Main article: Mass-energy equivalence

• Energy and momentum – in light of the Lorentz transformation, the nonrelativistic formulas for energy and momentum need to be modified and combined into a covariant four-momentum. In particular, this implies a relation between energy and (rest) mass: the mass of a system equals its energy in the centre-of-momentum frame, often summarized as E=mc².
• Causality

A Lorentz transformation cannot change a timelike four-vector to a spacelike or null one. In other words, a slower-than-light object must remain slower than light. (Likewise a faster-than-light object, known as a tachyon, must remain faster than light; these, however, are theorized not to exist.) Lightlike objects remain at the speed of light. This means that no material object may send signals superluminally (faster than light). It is less obvious, but nevertheless true, that other methods that do not involve sending a particle cannot enable superluminal transmission of information. This is known as causality.

See also: Causality and Tachyonic antitelephone

Domain of validity

Special relativity is accurate only when the absolute value of the gravitational potential is much less than c², that is, when spacetime is nearly flat. In a strong gravitational field, one must use general relativity. General relativity reduces to special relativity in the weak-field limit. Near the Planck length, quantum effects come into play, and one must take quantum gravity into consideration. Some quantum gravity theories, in fact, predict Lorentz violation at very small scales.

In the limit of small speeds (or equivalently in the limit $c\to \infty$), special relativity reduces to Newtonian mechanics. Most objects in everyday experience do not travel at relativistic speeds; hence relativistic effects are not especially notable, and nonrelativistic mechanics suffices.

Special relativity exhibits mathematical self-consistency. It is an integral part of modern physical theories, such as quantum field theory and general relativity (in the limiting case of negligible gravitational fields).

Experimental confirmation

Main article: Status of special relativity

Prior to Einstein's 1905 paper, several experiments had been conducted which, in retrospect, support special relativity. (Of these, Einstein was only aware of the Fizeau experiment before 1905.)

• The Trouton–Noble experiment showed that the torque on a capacitor is independent of position and inertial reference frame.
• The Michelson-Morley experiment further supported the impossibility of detection of an absolute reference velocity.(Contrary to some claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer travelled together at the same velocity at all times.)
• The Fizeau experiment measured the speed of light in moving media, confirming the relativistic addition of colinear velocities.

Subsequently, a number of experiments have tested special relativity against rival theories. These include:

• Kaufmann-Bucherer-Neumann experiments – electron deflection in approximate agreement with Lorentz-Einstein prediction.
• Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations
• Rossi-Hall experiment – relativistic effects on a fast-moving particle's half-life
• Experiments to test emitter theory demonstrated that the speed of light is independent of the speed of the emitter.
• Hammar experiment – no "aether flow obstruction"

Today, special relativity is experimentally tested to extremely high degree of accuracy (10⁻²⁰) and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors. In addition, particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light; the observed behaviour is completely consistent with special relativity and inconsistent with Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles.

Relativistic kinematics

Spacetime

Main article: Minkowski space

Special relativity adopts a flat 4-dimensional Euclidean spacetime equipped with a non-positive-definite metric, known as Minkowski space. An event is an idealized occurrence that is localized in spacetime, such that it can be labelled with coordinates in some reference frame. In other words, it is a point in spacetime. The position of an event in spacetime is given by a contravariant four vector whose components are:

$x^{\nu }=\left(ct,x,y,z\right)$

where $x^{1}=x$ and $x^{2}=y$ and $x^{3}=z$ as usual. We define $x^{0}=ct$ so that the time coordinate has the same dimension of distance as the other spatial dimensions; in accordance with the general principle that space and time are treated equally, so far as possible. Superscripts are contravariant indices in this section rather than exponents except when they indicate a square. Subscripts are covariant indices which also range from zero to three as with the spacetime gradient of a field φ:

$\partial _{0}\phi ={\frac {1}{c}}{\frac {\partial \phi }{\partial t}},\quad \partial _{1}\phi ={\frac {\partial \phi }{\partial x}},\quad \partial _{2}\phi ={\frac {\partial \phi }{\partial y}},\quad \partial _{3}\phi ={\frac {\partial \phi }{\partial z}}.$

A reference frame is an observational perspective in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes and the from which the time coordinate of events can be measured using a 'clock' (any reference device with uniform periodicity).

The Minkowski metric, η, given in components (valid in any inertial reference frame) as:

$\eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}$

which is equal to its reciprocal, $\eta ^{\alpha \beta }$, in those frames. (Note that we may adopt another sign convention, with all signs flipped. Both conventions are common; here we shall adopt the former.)

Lorentz transformation

Main article: Lorentz transformation

Diagram 1. Changing views of spacetime along the world line of a rapidly accelerating observer. In this animation, the vertical direction indicates time and the horizontal direction indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower quarter of the diagram shows the events that are visible to the observer, and the upper quarter shows the light cone- those that will be able to see the observer. The small dots are arbitrary events in spacetime. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates.

Coordinate 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:

$\Lambda ^{\mu '}{}_{\nu }={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}$

which is simply the matrix of a boost (like a rotation) between the x and ct coordinates. Where μ' indicates the row and ν indicates the column. Also, β and γ are defined as:

$\beta ={\frac {v}{c}},\ \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}.$

More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy:

$\eta _{\alpha \beta }=\eta _{\mu '\nu '}\Lambda ^{\mu '}{}_{\alpha }\Lambda ^{\nu '}{}_{\beta }\!$

where there is an implied summation of $\mu '\!$ and $\nu '\!$ from 0 to 3 on the right-hand side in accordance with the Einstein summation convention. The Poincaré group is the most general group of transformations which preserves the Minkowski metric and this is the physical symmetry underlying special relativity.

All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law

$T_{\left[j_{1}',j_{2}',\dots ,j_{q}'\right]}^{\left[i_{1}',i_{2}',\dots ,i_{p}'\right]}=\Lambda ^{i_{1}'}{}_{i_{1}}\Lambda ^{i_{2}'}{}_{i_{2}}\cdots \Lambda ^{i_{p}'}{}_{i_{p}}\Lambda _{j_{1}'}{}^{j_{1}}\Lambda _{j_{2}'}{}^{j_{2}}\cdots \Lambda _{j_{q}'}{}^{j_{q}}T_{\left[j_{1},j_{2},\dots ,j_{q}\right]}^{\left[i_{1},i_{2},\dots ,i_{p}\right]}$

Where $\Lambda _{j_{k}'}{}^{j_{k}}\!$ is the reciprocal matrix of $\Lambda ^{j_{k}'}{}_{j_{k}}\!$.

To see how this is useful, we transform the position of an event from an unprimed coordinate system S to a primed system S', we calculate

${\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix}}=x^{\mu '}=\Lambda ^{\mu '}{}_{\nu }x^{\nu }={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}={\begin{pmatrix}\gamma ct-\gamma \beta x\\\gamma x-\beta \gamma ct\\y\\z\end{pmatrix}}$

which is the Lorentz transformation given above. All tensors transform by the same rule.

The squared length of the differential of the position four-vector $dx^{\mu }\!$ constructed using

$\mathbf {dx} ^{2}=\eta _{\mu \nu }\,dx^{\mu }\,dx^{\nu }=-(c\cdot dt)^{2}+(dx)^{2}+(dy)^{2}+(dz)^{2}\,$

is an invariant. Being invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. Notice that when the line element $\mathbf {dx} ^{2}$ is negative that $d\tau ={\sqrt {-\mathbf {dx} ^{2}}}/c$ is the differential of proper time, while when $\mathbf {dx} ^{2}$ is positive, ${\sqrt {\mathbf {dx} ^{2}}}$ is differential of the proper distance.

The Lorentz transformations (spatial rotation and Lorentz boost) form a group, known as the Lorentz group. This, combined with spatiotemporal translation, forms the Poincaré group.

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. Also in constructing such equations we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.

Light cone

In two spatial dimensions, the null geodesics ($ds^{2}=0$) lie on double cone:

By analogy, the null geodesics in three spatial dimensions form a 4-dimensional hypercone:

This cone represents the "line of sight" of a point in space: the past half is the region of spacetime at which a given observer receives signals from the past, whereas the future half is the region of spacetime occupied by lightlike signals sent from the observer. For example, the starlight one observes at a given moment (ignoring atmospheric aberrations, etc.) lie on this cone. These cones are known as light cones.

A Minkowski diagram visually depicts the geometry of Minkowski space; these are useful when reasoning about special relativity.

Velocity and acceleration

Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector U^μ is given by

$U^{\mu }={\frac {dx^{\mu }}{d\tau }}={\begin{pmatrix}\gamma c\\\gamma v_{x}\\\gamma v_{y}\\\gamma v_{z}\end{pmatrix}}$

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. U^μ also has an invariant form:

${\mathbf {U} }^{2}=\eta _{\nu \mu }U^{\nu }U^{\mu }=-c^{2}.$

So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. The acceleration 4-vector is given by $A^{\mu }=d{\mathbf {U} ^{\mu }}/d\tau$. Given this, differentiating the above equation by τ produces

$2\eta _{\mu \nu }A^{\mu }U^{\nu }=0.\!$

So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal.

Relativistic dynamics

Further information: [[:Mass in special relativity and Conservation of energy]]

As defined in nonrelativistic mechanics, the momentum and energy depend on the velocity, which is not Lorentz-invariant. To generalize these quantities, one must modify them with relativistic corrections. There are two general approaches: one is to add Lorentz factors to preserve Newtonian formulas, such as F=ma. Another is to construct Lorentz-covariant quantities, such as four-vectors, from non-covariant quantities. An example of the former approach is the relativistic mass. For example, in special relativity, the conserved momentum must be modified as

$\mathbf {p} =m{\frac {d(\gamma \mathbf {x} )}{dt}}=\gamma ^{3}m{\frac {d\mathbf {x} _{\parallel }}{dt}}+\gamma m{\frac {d\mathbf {x} _{\perp }}{dt}}.$

where m denotes the invariant (rest) mass and where $\mathbf {x} _{\parallel }$ and $\mathbf {x} _{\perp }$ denote transverse and longitudinal parts of the position respectively. To preserve the nonrelativistic equation p=mv, one defines the so-called transverse (or relativistic) and longitudinal masses $m_{\perp }=\gamma m$ and $m_{\parallel }=\gamma ^{3}m$; one may conceive of this as a non-isotropic mass. These quantities are not Lorentz-invarant scalars. In this article we shall use mass to refer to the rest mass, clarifying where necessary.

Alternatively, we may take the covariant route. By switching to the Lorentz-covariant proper-time derivative, we can absorb the Lorentz factor γ differently:

$\mathbf {p} =m{\frac {d\mathbf {x} }{\mathrm {d} \tau }}$

where τ denotes the proper time. Likewise, the kinetic energy of a point particle is

$E=\gamma mc^{2}.$

Again using the same manoeuvre, we have

$E=mc^{2}{\frac {dt}{d\tau }}.$

Since t and x combine to form the four-position x^μ, it is clear that E and p combine to form the four-momentum p^μ:

$p^{\mu }=mc^{2}{\frac {dx^{\mu }}{d\tau }}.$

Note that the four-momentum's magnitude is simply the mass. This quantity is constructed completely from Lorentz-invariant quantities; hence it is fully Lorentz-covariant.

Mass

Main article: Mass–energy equivalence

(Rest) mass is not additive: the mass of a composite system in general differs from the sum of the masses of its constituents, since it incorporates, in addition to the rest masses of its constituents, the kinetic and potential energy as measured in the centre-of-momentum frame. Hence conservation of (rest) mass fails relativistically. For instance, an electron and a positron, both massive, can annihilate into a pair of photons, which lack mass. The additive quantity is the four-momentum, that is, energy and momentum. Hence conservation of energy and momentum hold in special relativity. In addition, since the relativistic mass is simply the energy (times some constant factors), it is conserved as well.

For massless particles, m is zero. The relativistic energy-momentum equation still holds, however, and by substituting m with 0, the relation E = pc is obtained; when substituted into Ev = c²p, it gives v = c: massless particles (such as photons) always travel at the speed of light.

A massless particle (for example, a photon) can nevertheless contribute to the total invariant mass of a system, since some or all of its momentum is cancelled by another particle, causing a contribution to the system's invariant mass due to the photon's energy. For single photons this does not happen, since the energy and momentum terms exactly cancel.

Looking at the above formula for invariant mass of a system, one sees that, when a single massive object is at rest (v = 0, p = 0), there is a non-zero mass remaining: m_rest = E/c². The corresponding energy, which is also the total energy when a single particle is at rest, is referred to as "rest energy". In systems of particles which are seen from a moving inertial frame, total energy increases and so does momentum. However, for single particles the rest mass remains constant, and for systems of particles the invariant mass remain constant, because in both cases, the energy and momentum increases subtract from each other, and cancel. Thus, the invariant mass of systems of particles is a calculated constant for all observers, as is the rest mass of single particles.

The mass of systems and conservation of invariant mass

For systems, the inertial frame in which the momenta of all particles sums to zero is called the center of momentum frame. In this special frame, the relativistic energy-momentum equation has p = 0, and thus gives the invariant mass of the system as merely the total energy of all parts of the system, divided by c²

$m=\sum E/c^{2}$

This is the invariant mass of any system which is measured in a frame where it has zero total momentum, such as a bottle of hot gas on a scale. In such a system, the mass which the scale weighs is the invariant mass, and it depends on the total energy of the system. It is thus more than the sum of the rest masses of the molecules, but also includes all the totaled energies in the system as well. Like energy and momentum, the invariant mass of closed systems cannot be changed so long as the system is closed (no mass or energy allowed in or out), because the total relativistic energy of the system remains constant so long as nothing can enter or leave it.

An increase in the energy of such a system which is caused by translating the system to an inertial frame which is not the center of momentum frame, causes an increase in energy and momentum without an increase in invariant mass. E = mc², however, applies only to closed systems in their center-of-momentum frame where momentum sums to zero.

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."

Einstein was not referring to closed (isolated) systems in this remark, however. For, even in his 1905 paper, which first derived the relationship between mass and energy, Einstein showed that the energy of an object had to be increased for its invariant mass (rest mass) to increase. In such cases, the system is not closed (in Einstein's thought experiment, for example, a mass gives off two photons, which are lost).

Mass of closed (isolated) systems

In a closed system (i.e., in the sense of a totally isolated system) the total energy, the total momentum, and hence the total invariant mass are conserved. Einstein's formula for change in mass translates to its simplest ΔE = mc² form, however, in non-closed systems in which energy, and thus invariant mass, is allowed to escape (for example, as heat and light). Einstein's equation shows that such systems must lose mass, in accordance with the above formula, in proportion to the energy they lose to the surroundings. Conversely, if one can measure the differences in mass between a system before it undergoes a reaction which releases heat and light, and the system after the reaction when heat and light have escaped, one can estimate the amount of energy which escapes the system. In both nuclear and chemical reactions, such energy represents the difference in binding energies of electrons in atoms (for chemistry) or between nucleons in nuclei (in atomic reactions). In both cases, the mass difference between reactants and (cooled) products measures the mass of heat and light which will escape the reaction, and thus (using the equation) give the equivalent energy of heat and light which may be emitted if the reaction proceeds.

In chemistry, the mass differences associated with the emitted energy are around one-billionth of the molecular mass. However, in nuclear reactions the energies are so large that they are associated with mass differences, which can be estimated in advance, if the products and reactants have been weighed (atoms can be weighed indirectly by using atomic masses, which are always the same for each nuclide). Thus, Einstein's formula becomes important when one has measured the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have stored energy that can be released by certain nuclear reactions, providing important information which was useful in the development of nuclear energy and, consequently, the nuclear bomb. Historically, for example, Lise Meitner was able to use the mass differences in nuclei to estimate that there was enough energy available to make nuclear fission a favorable process. The implications of this special form of Einstein's formula have thus made it one of the most famous equations in all of science.

Because the E = mc² equation applies only to isolated systems in their center of momentum frame, it has been popularly misunderstood to mean that mass may be converted to energy, after which the mass disappears. However, popular explanations of the equation as applied to systems include open systems for which heat and light are allowed to escape, when they otherwise would have contributed to the mass (invariant mass) of the system.

Historically, confusion about mass being "converted" to energy has been aided by confusion between mass and "matter", where matter is defined as fermion particles. In such a definition, electromagnetic radiation and kinetic energy (or heat) are not considered "matter." In some situations, matter may indeed be converted to non-matter forms of energy (see above), but in all these situations, the matter and non-matter forms of energy still retain their original mass.

For closed/isolated systems, mass never disappears in the center of momentum frame, because energy cannot disappear. Instead, this equation, in context, means only that when any energy is added to, or escapes from, a system in the center-of-momentum frame, the system will be measured as having gained or lost mass, in proportion to energy added or removed. Thus, in theory, if an atomic bomb were placed in a box strong enough to hold its blast, and detonated upon a scale, the mass of this closed system would not change, and the scale would not move. Only when a transparent "window" was opened in the super-strong plasma-filled box, and light and heat were allowed to escape in a beam, and the bomb components to cool, would the system lose the mass associated with the energy of the blast. In a 21 kiloton bomb, for example, about a gram of light and heat is created. If this heat and light were allowed to escape, the remains of the bomb would lose a gram of mass, as it cooled. However, invariant mass cannot be destroyed in special relativity, but only moved from place to place. In this thought-experiment, the light and heat carry away the gram of mass, and would therefore deposit this gram of mass in the objects that absorb them.

Force

Noncovariant force

In special relativity, Newton's second law does not hold in its form F = ma, but it does if it is expressed as

$\mathbf {F} ={\frac {d\mathbf {p} }{dt}}$

where p is the momentum as defined above ($\mathbf {p} =\gamma m\mathbf {v}$) and "m" is the invariant mass. Thus, the force is given by

$\mathbf {F} =m{\frac {d(\gamma \,\mathbf {v} )}{dt}}=m\left({\frac {d\gamma }{dt}}\,\mathbf {v} +\gamma {\frac {d\mathbf {v} }{dt}}\right).$

Carrying out the derivatives gives

$\mathbf {F} ={\frac {\gamma ^{3}mv}{c^{2}}}{\frac {dv}{dt}}\,\mathbf {v} +\gamma m\,\mathbf {a}$

which, taking into account the identity $v{\tfrac {dv}{dt}}=\mathbf {v} \cdot \mathbf {a}$, can also be expressed as

$\mathbf {F} ={\frac {\gamma ^{3}m\left(\mathbf {v} \cdot \mathbf {a} \right)}{c^{2}}}\,\mathbf {v} +\gamma m\,\mathbf {a} .$

If the acceleration is separated into the part parallel to the velocity and the part perpendicular to it, one gets

$\mathbf {F} ={\frac {\gamma ^{3}mv^{2}}{c^{2}}}\,\mathbf {a} _{\parallel }+\gamma m\,(\mathbf {a} _{\parallel }+\mathbf {a} _{\perp })\,$

$=\gamma ^{3}m\left({\frac {v^{2}}{c^{2}}}+{\frac {1}{\gamma ^{2}}}\right)\mathbf {a} _{\parallel }+\gamma m\,\mathbf {a} _{\perp }\,$

$=\gamma ^{3}m\left({\frac {v^{2}}{c^{2}}}+1-{\frac {v^{2}}{c^{2}}}\right)\mathbf {a} _{\parallel }+\gamma m\,\mathbf {a} _{\perp }\,$

$=\gamma ^{3}m\,\mathbf {a} _{\parallel }+\gamma m\,\mathbf {a} _{\perp }\,.$

Consequently in some old texts, γ³m is referred to as the longitudinal mass, and γm is referred to as the transverse mass, which is the same as the relativistic mass. See mass in special relativity.

Four-force

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. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.

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. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:

$F_{\nu }={\frac {dp_{\nu }}{d\tau }}={\begin{pmatrix}-{d(E/c)}/{d\tau }\\{dp_{x}}/{d\tau }\\{dp_{y}}/{d\tau }\\{dp_{z}}/{d\tau }\end{pmatrix}}$

where $\tau \,$ 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 (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e. ${\frac {dp}{dt}}$ while the four force is defined by the rate of change of momentum with respect to proper time, i.e. ${\frac {dp}{d\tau }}$.

In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.

Kinetic energy

The Work-energy Theorem says the change in kinetic energy is equal to the work done on the body, that is

$\Delta K=W=\int _{\mathbf {r} _{0}}^{\mathbf {r} _{1}}\mathbf {F} \cdot d\mathbf {r}$

(Click [show] for intermediate steps)

$=\int _{t_{0}}^{t_{1}}{\frac {d}{dt}}(\gamma m\mathbf {v} )\cdot \mathbf {v} dt$ $=\left.\gamma m\mathbf {v} \cdot \mathbf {v} \right|_{t_{0}}^{t_{1}}-\int _{t_{0}}^{t_{1}}\gamma m\mathbf {v} \cdot {\frac {d\mathbf {v} }{dt}}dt$ $=\left.\gamma mv^{2}\right|_{t_{0}}^{t_{1}}-m\int _{v_{0}}^{v_{1}}\gamma v\,dv$ $=m\left(\left.\gamma v^{2}\right|_{t_{0}}^{t_{1}}-c^{2}\int _{v_{0}}^{v_{1}}{\frac {2v/c^{2}}{2{\sqrt {1-v^{2}/c^{2}}}}}\,dv\right)$ $=\left.m\left({\frac {v^{2}}{\sqrt {1-v^{2}/c^{2}}}}+c^{2}{\sqrt {1-v^{2}/c^{2}}}\right)\right|_{t_{0}}^{t_{1}}$ $=\left.{\frac {mc^{2}}{\sqrt {1-v^{2}/c^{2}}}}\right|_{t_{0}}^{t_{1}}$ $=\left.{\gamma mc^{2}}\right|_{t_{0}}^{t_{1}}$

$\displaystyle =\gamma _{1}mc^{2}-\gamma _{0}mc^{2}.$

If in the initial state the body was at rest (γ₀ = 1) and in the final state it has speed v (γ₁ = γ), the kinetic energy is K = (γ − 1)mc², a result that can be directly obtained by subtracting the rest energy mc² from the total relativistic energy γmc².

Application in cyclotrons

The application of the above in cyclotrons is immediate:

${\frac {dW}{dt}}=mc^{2}{\frac {d\gamma }{dt}}$

In the presence of a magnetic field only, the Lorentz force is:

$\mathbf {F} =q\mathbf {v\times B}$

Since:

${\frac {dW}{dt}}=\mathbf {F\cdot v} =0$

it follows that:

${\frac {d\gamma }{dt}}=0$

meaning that γ is constant, and so is v. This is instrumental in solving the equation of motion for a charge particle of charge q in a magnetic field of induction B as follows:

$\mathbf {F} ={\frac {d\gamma m_{0}\mathbf {v} }{dt}}=\gamma m_{0}{\frac {d\mathbf {v} }{dt}}$

On the other hand:

$\mathbf {F} =q\mathbf {v\times B}$

Thus:

$\gamma m_{0}{\frac {d\mathbf {v} }{dt}}=q\mathbf {v\times B}$

Separating by components, we obtain:

$qBv_{y}=\gamma m_{0}{\frac {dv_{x}}{dt}}$

$-qBv_{x}=\gamma m_{0}{\frac {dv_{y}}{dt}}$

$0=\gamma m_{0}{\frac {dv_{z}}{dt}}$

The solutions are:

$v_{x}=r\omega \cos(\omega t)\$

$v_{y}=-r\omega \sin(\omega t)\$

$\omega ={\frac {qB}{\gamma (v_{0})m_{0}}}$

By integrating one more time with respect to t the differential equations above we obtain the equations of motion: a circle of radius $r={\frac {\gamma (v_{0})m_{0}v_{0}}{qB}}$ in the plane z=constant, where $v_{0}$ is the initial speed of the particle entering the cyclotron. Notice that this calculation ignores the Abraham-Lorentz force which is the reaction to the emission of electromagnetic radiation by the particle. If the speed is held constant by applying an electric field, then the magnitude of the acceleration is constant, $a={\frac {{v_{0}}^{2}}{r}}\,,$ but its direction keeps changing in a cyclotron. The jerk is proportional with the second time derivative of speed:

${\frac {d^{2}v_{x}}{dt^{2}}}=-r\omega ^{3}\cos(\omega t)$

${\frac {d^{2}v_{y}}{dt^{2}}}=r\omega ^{3}\sin(\omega t)$

Because the jerk is directed opposite to the velocity, the Abraham-Lorentz force tends to slow the particle down. Note that the Abraham-Lorentz force is much smaller than the Lorentz force:

$\mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} =-{\frac {\mu _{0}q^{4}B^{2}}{6\pi c\gamma ^{2}{m_{0}}^{2}}}\mathbf {v} \,$

${\frac {F_{rad}}{F_{Lorentz}}}={\frac {\mu _{0}q^{3}B}{6\pi c\gamma ^{2}{m_{0}}^{2}}}$

so, it can be ignored in most computations.

Classical limit

Notice that γ can be expanded into a Taylor series for ${\frac {v^{2}}{c^{2}}}<1$, obtaining:

$\gamma =\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {(2k-1)v^{2}}{2kc^{2}}}=1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}+{\frac {3}{8}}{\frac {v^{4}}{c^{4}}}+{\frac {5}{16}}{\frac {v^{6}}{c^{6}}}+\ldots$

and consequently

$E-mc^{2}={\frac {1}{2}}mv^{2}+{\frac {3}{8}}{\frac {mv^{4}}{c^{2}}}+{\frac {5}{16}}{\frac {mv^{6}}{c^{4}}}+\ldots ;$

$\mathbf {p} =m\mathbf {v} +{\frac {1}{2}}{\frac {mv^{2}\mathbf {v} }{c^{2}}}+{\frac {3}{8}}{\frac {mv^{4}\mathbf {v} }{c^{4}}}+{\frac {5}{16}}{\frac {mv^{6}\mathbf {v} }{c^{6}}}+\cdots .$

For velocities much smaller than that of light, one can neglect the terms with c² and higher in the denominator. These formulas then reduce to 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.

Relativistic electrodynamics

Main article: Classical electromagnetism and special relativity

Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation-speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.

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. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.

Electromagnetism in 4D

Main article: Covariant formulation of classical electromagnetism

Maxwell's equations in the 3D form are already consistent with the physical content of special relativity. But we must rewrite them to make them manifestly invariant.

The charge density $\rho \!$ and current density $[J_{x},J_{y},J_{z}]\!$ are unified into the current-charge 4-vector:

$J^{\mu }={\begin{pmatrix}\rho c\\J_{x}\\J_{y}\\J_{z}\end{pmatrix}}.$

The law of charge conservation, ${\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0$, becomes:

$\partial _{\mu }J^{\mu }=0.\!$

The electric field $[E_{x},E_{y},E_{z}]\!$ and the magnetic induction $[B_{x},B_{y},B_{z}]\!$ are now unified into the (rank 2 antisymmetric covariant) electromagnetic field tensor:

$F_{\mu \nu }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&B_{z}&-B_{y}\\E_{y}/c&-B_{z}&0&B_{x}\\E_{z}/c&B_{y}&-B_{x}&0\end{pmatrix}}.$

The density, $f_{\mu }\!$, of the Lorentz force, $\mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B}$, exerted on matter by the electromagnetic field becomes:

$f_{\mu }=F_{\mu \nu }J^{\nu }.\!$

Faraday's law of induction, $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$, and Gauss's law for magnetism, $\nabla \cdot \mathbf {B} =0$, combine to form:

$\partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu }=0.\!$

Although there appear to be 64 equations here, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0=0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.

The electric displacement $[D_{x},D_{y},D_{z}]\!$ and the magnetic field $[H_{x},H_{y},H_{z}]\!$ are now unified into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor:

${\mathcal {D}}^{\mu \nu }={\begin{pmatrix}0&D_{x}c&D_{y}c&D_{z}c\\-D_{x}c&0&H_{z}&-H_{y}\\-D_{y}c&-H_{z}&0&H_{x}\\-D_{z}c&H_{y}&-H_{x}&0\end{pmatrix}}.$

Ampère's law, $\nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}$, and Gauss's law, $\nabla \cdot \mathbf {D} =\rho$, combine to form:

$\partial _{\nu }{\mathcal {D}}^{\mu \nu }=J^{\mu }.\!$

In a vacuum, the constitutive equations are:

$\mu _{0}{\mathcal {D}}^{\mu \nu }=\eta ^{\mu \alpha }F_{\alpha \beta }\eta ^{\beta \nu }\,.$

Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define $F^{\mu \nu }\,$ by

$F^{\mu \nu }=\eta ^{\mu \alpha }F_{\alpha \beta }\eta ^{\beta \nu }\,$

the constitutive equations may, in a vacuum, be combined with Ampère's law etc. to get:

$\partial _{\beta }F^{\alpha \beta }=\mu _{0}J^{\alpha }.\!$

The energy density of the electromagnetic field combines with Poynting vector and the Maxwell stress tensor to form the 4D electromagnetic stress-energy tensor. It is the flux (density) of the momentum 4-vector and as a rank 2 mixed tensor it is:

$T_{\alpha }^{\pi }=F_{\alpha \beta }{\mathcal {D}}^{\pi \beta }-{\frac {1}{4}}\delta _{\alpha }^{\pi }F_{\mu \nu }{\mathcal {D}}^{\mu \nu }$

where $\delta _{\alpha }^{\pi }$ is the Kronecker delta. When upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field.

The conservation of linear momentum and energy by the electromagnetic field is expressed by:

$f_{\mu }+\partial _{\nu }T_{\mu }^{\nu }=0\!$

where $f_{\mu }\!$ is again the density of the Lorentz force. This equation can be deduced from the equations above (with considerable effort).

History

Main article: History of special relativity

The principle of relativity, which states that there is no preferred inertial reference frame, dates back to Galileo, and was incorporated into Newtonian Physics. In the late 19^th century, it was realized that Maxwell's equations predict a single value of the speed of electromagnetic waves; this was subsequently confirmed. This appeared to single outa preferred reference frame, contradicting Galilean relativity. This led physicists to suggest that the universe was filled with an entity called the aether, which would act as the medium through which these waves travelled. The aether was thought to constitute an absolute reference frame against which speeds could be measured. In order to be consistent with the observed universe, the aether was postulated by necessity to have some extraordinary properties: it was sufficiently elastic that it could support electromagnetic waves, and those waves could interact with matter, yet it offered very little resistance to bodies passing through it.

Various experiments, including the Michelson-Morley experiment, indicated that the Earth was always 'stationary' relative to the aether–something that was difficult to explain, since the Earth is in orbit around the Sun. Einstein's solution was to discard the notion of an aether and an absolute state of rest. Special relativity is formulated so as to not assume that any particular frame of reference is special; rather, in relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in a vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.

To explain the null result, Henri Poincaré and Hendrik Lorentz derived versions of what is now known as the Lorentz transformation on various assumptions, such as that the interatomic spacing was electromagnetic and therefore stretched or contracted according to Maxwell's equations. Einstein, in 1905, published his seminal paper on special relativity, where he demonstrated that the Lorentz transformation could be derived from simple postulates alone if one allows modifications of kinematics and dynamics.

See also

People: Arthur Eddington | Albert Einstein | Hendrik Lorentz | Hermann Minkowski | Bernhard Riemann | Henri Poincaré | Alexander MacFarlane | Harry Bateman | Robert S. Shankland | Walter Ritz

Relativity: Theory of relativity | History of special relativity | principle of relativity | general relativity | Fundamental Speed | frame of reference | inertial frame of reference | Lorentz transformations | Bondi k-calculus | Einstein synchronisation | Rietdijk-Putnam Argument

Physics: Newtonian Mechanics | spacetime | speed of light | simultaneity | physical cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis | Lorentz ether theory | Moving magnet and conductor problem | Shape waves| Relativistic heat conduction

Maths: Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors | split-complex number | Relativity in the APS formalism

Philosophy: actualism | conventionalism | formalism

Paradoxes: Twin paradox | Ehrenfest paradox | Ladder paradox | Bell's spaceship paradox

Experiments: Kennedy–Thorndike experiment | Trouton–Rankine experiment | Michelson–Morley experiment | Hafele–Keating experiment | Ives–Stilwell experiment | Rossi–Hall experiment

References

Textbooks

• Einstein, Albert (1920). Relativity: The Special and General Theory.
• Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1-56731-136-9
• Freund, Jűrgen (2008) Special Relativity for Beginners - A Textbook for Undergraduates World Scientific. ISBN 981-277-160-3
• Robert Geroch (1981). General Relativity From A to B. University of Chicago Press. ISBN 0-226-28864-1
• Grøn, Øyvind (2007). Einstein's General Theory of Relativity. New York: Springer. ISBN 978-0-387-69199-2. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Logunov, Anatoly A. (2005) Henri Poincaré and the Relativity Theory (transl. from Russian by G. Pontocorvo and V. O. Soleviev, edited by V. A. Petrov) Nauka, Moscow.
• Charles Misner, Kip Thorne, and John Archibald Wheeler (1971) Gravitation. W. H. Freeman & Co. ISBN 0-7167-0334-3
• Post, E.J., 1997 (1962) Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications.
• Wolfgang Rindler (1991). Introduction to Special Relativity (2nd ed.), Oxford University Press. ISBN13: 9780198539520; ISBN 0198539525
• Wolfgang Rindler (2006). Relativity: Special, General, and Cosmological (2nd ed.), Oxford University Press. ISBN13: 9780198567325; ISBN 0198567324
• Schutz, Bernard F. A First Course in General Relativity, Cambridge University Press. ISBN 0-521-27703-5
• Silberstein, Ludwik (1914) The Theory of Relativity.
• Taylor, Edwin, and John Archibald Wheeler (1992) Spacetime Physics (2nd ed.). W.H. Freeman & Co. ISBN 0-7167-2327-1
• Tipler, Paul, and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman & Co. ISBN 0-7167-4345-0

Journal articles

• Alvager et al. (1964) "Test of the Second Postulate of Special Relativity in the GeV region," Physics Letters 12: 260.
• Darrigol, Olivier (2004) "[The Mystery of the Poincaré-Einstein Connection]," Isis 95(4): 614-26.
• Mitchell Feigenbaum (2008) "The Theory of Relativity - Galileo's Child."
• Gulevich, D. R., et al. (2008) "Shape waves in 2D Josephson junctions: Exact solutions and time dilation," Phys. Rev. Lett. 101: 127002.
• Rizzi, G., et al., (2005) "Synchronization Gauges and the Principles of Special Relativity," Found. Phys. 34: 1835-87.
• Will, Clifford M. (1992) "Clock synchronization and isotropy of the one-way speed of light," Physics Review D 45: 403-11.
• Wolf, Peter, and Petit, Gerard (1997) "Satellite test of Special Relativity using the Global Positioning System," Physics Review A 56(6): 4405-09.

External links

Original works

• Zur Elektrodynamik bewegter Körper Einstein's original work in German, Annalen der Physik, Berne 1905

Special relativity for a general audience (no math knowledge required)

• Wikibooks: Special Relativity
• 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.
• Einstein Online Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.
• Lawrence Sklar (1977). Space, Time and Spacetime. University of California Press. ISBN 0520031741.
• Lawrence Sklar (1992). Philosophy of Physics. Westview Press. ISBN 0813306256.

Special relativity explained (using simple or more advanced math)

• Caltech Relativity Tutorial A basic introduction to concepts of Special and General Relativity, requiring only a knowledge of basic geometry.
• Greg Egan's Foundations.
• The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus.
• Motion Mountain, Volume II - A modern introduction to relativity, including its visual effects.
• The Origins of Einstein's Special Theory of Relativity - A historical approach to the study of the special theory of relativity.
• Problem Solving for Special Relativity Cases A page that explains how to solve problems in special relativity.
• MathPages - Reflections on Relativity A complete online book on relativity with an extensive bibliography.
• Relativity An introduction to special relativity at the undergraduate level, without calculus.
• Relativity: the Special and General Theory at Project Gutenberg, by Albert Einstein
• Special Relativity Stanford University, Helen Quinn, 2003
• 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.
• Understanding Special Relativity The theory of special relativity in an easily understandable way.
• Relativity Calculator - Learn Special Relativity Mathematics Mathematics of special relativity presented in as simple and comprehensive manner possible within philosophical and historical contexts.
• An Introduction to the Special Theory of Relativity (1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus." (130 pp; pdf format)

Visualization

• Raytracing Special Relativity Software visualizing several scenarios under the influence of special relativity.
• Real Time Relativity The Australian National University. Relativistic visual effects experienced through an interactive program.
• Spacetime travel A variety of visualizations of relativistic effects, from relativistic motion to black holes.
• Through Einstein's Eyes The Australian National University. Relativistic visual effects explained with movies and images.
• Warp Special Relativity Simulator A computer program to show the effects of traveling close to the speed of light.
• Animation clip visualizing the Lorentz transformation.
• Original interactive FLASH Animations from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.

Other

• "Einstein Was Right (Again): Experiments Confirm that E= mc²" A recent direct measurement of Einstein's famous equation accurate to "four-tenths of 1 part in 1 million".
• Relativity in its Historical Context The discovery of special relativity was inevitable, given the momentous discoveries that preceded it.