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Special relativity

2007 Schools Wikipedia Selection. Related subjects: General Physics

   The special theory of relativity was proposed in 1905 by Albert
   Einstein in his article " On the Electrodynamics of Moving Bodies".
   Some three centuries earlier, Galileo's principle of relativity had
   stated that all uniform motion was relative, and that there was no
   absolute and well-defined state of rest; a person on the deck of a ship
   may be at rest in his opinion, but someone observing from the shore
   would say that he was moving. Einstein's theory combines Galilean
   relativity with the postulate that all observers will always measure
   the speed of light to be the same no matter what their state of uniform
   linear motion is.

   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. It
   yields the equivalence of matter and energy, as expressed in the famous
   equation E=mc^2, where c is the speed of light. Special relativity
   agrees with Newtonian mechanics in their common realm of applicability,
   in experiments in which all velocities are small compared to the speed
   of light.

   The theory was called "special" because it applies the principle of
   relativity only to inertial frames. Einstein developed general
   relativity to apply the principle generally, that is, to any frame, and
   that theory includes the effects of gravity. Special relativity doesn't
   account for gravity, but it can deal with accelerations.

   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. In particular, it states that the speed of light
   is the same for all observers, even if they are in motion relative to
   one another. 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. In particular, special
   relativity states that it is impossible for any material object to
   travel as fast as light.

   For history and motivation, see the article: history of special
   relativity

Postulates

    1. First postulate - Special principle of relativity - The laws of
       physics are the same in all inertial frames of reference. In other
       words, there are no privileged inertial frames of reference.
    2. Second postulate - Invariance of c - The speed of light in a vacuum
       is a universal constant (c) which is independent of the motion of
       the light source.

   The power of Einstein's argument stems from the manner in which he
   derived startling and seemingly implausible results from two simple
   assumptions that were founded on analysis of observations. An observer
   attempting to measure the speed of light's propagation will get the
   same answer no matter how the observer or the system's components are
   moving.

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).
     * Relativity of simultaneity — two events happening in two different
       locations that occur simultaneously to one observer, may occur at
       different times to another observer (lack of absolute
       simultaneity).
     * 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 travelling 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 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. (In this example, the
       observer would see the missile travel with a speed of 12/13 the
       speed of light.)
     * Inertia and momentum — as an object's velocity gets closer to the
       speed of light, it becomes more and more difficult to accelerate
       it.
     * Equivalence of mass and energy, E=mc^2 — mass and energy can be
       converted to one another, and play equivalent roles (e.g., the
       gravitational force on a falling apple is partly due to the kinetic
       energies of the subatomic particles it is made of).

Simultaneity

   From the first equation of the Lorentz transformation in terms of
   coordinate differences

          \Delta t' = \gamma \left(\Delta t - \frac{v \Delta x}{c^{2}}
          \right)

   it is clear that two events that are simultaneous in frame S
   (satisfying \Delta t = 0\, ), are not necessarily simultaneous in
   another inertial frame S' (satisfying \Delta t' = 0\, ). Only if these
   events are colocal in frame S (satisfying \Delta x = 0\, ), will they
   be simultaneous in another frame S'.

Time dilation and length contraction

   Writing the Lorentz Transformation and its inverse in terms of
   coordinate differences we get

          \Delta t' = \gamma \left(\Delta t - \frac{v \Delta x}{c^{2}}
          \right)
          \Delta x' = \gamma (\Delta x - v \Delta t)\,

   and

          \Delta t = \gamma \left(\Delta t' + \frac{v \Delta x'}{c^{2}}
          \right)
          \Delta x = \gamma (\Delta x' + v \Delta t')\,

   Suppose we have a clock at rest in the unprimed system S. Two
   consecutive ticks of this clock are then characterized by Δx = 0. 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

          \Delta t' = \gamma \Delta t \,

   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. This phenomenon is called time
   dilation.

   Similarly, suppose we have a measuring rod at rest in the unprimed
   system. In this system, the length of this rod is written as Δx. If we
   want to find the length of this rod as measured in the 'moving' system
   S', we must make sure to measure the distances x' to the end points of
   the rod simultaneously in the primed frame S'. 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':

          \Delta x' = \frac{\Delta x}{\gamma}

   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. This
   phenomenon is called length contraction or Lorentz contraction.

   These effects are not merely appearances; they are explicitly related
   to our way of measuring time intervals between 'colocal' events and
   distances between simultaneous events.

   See also the twin paradox.

Causality and prohibition of motion faster than light

   Diagram 2. Light cone
   Enlarge
   Diagram 2. Light cone

   In diagram 2 the interval AB 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. If it was
   possible for a cause-and-effect relationship to exist between events A
   and C, then logical paradoxes would result. For example, if A was the
   cause, and C the effect, then there would be frames of reference in
   which the effect preceded the cause. Another way of looking at it is
   that if there were a technology that allowed faster-than-light motion,
   it would also function as a time machine. Therefore, one of the
   consequences of special relativity is that (assuming causality is to be
   preserved as a logical principle), no information or material object
   can travel faster than light. 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. For example, if a constant force is applied to an object
   for a limitless amount of time, then integrating F=dp/dt gives a
   momentum that grows without bound, but this is simply because p = mγv
   approaches infinity as v approaches c. To an observer who is not
   accelerating, it appears as though the object's inertia is increasing,
   so as to produce a smaller acceleration in response to the same force.
   This behaviour is in fact observed in particle accelerators.

Composition of velocities

   If the observer in S\! sees an object moving along the x\! axis at
   velocity w\! , then the observer in the S'\! system, a frame of
   reference moving at velocity v\! in the x\! direction with respect to
   S\! , will see the object moving with velocity w'\! where

          w'=\frac{w-v}{1-wv/c^2}.

   This equation can be derived from the space and time transformations
   above. Notice that if the object is moving at the speed of light in the
   S\! system (i.e. w=c\! ), then it will also be moving at the speed of
   light in the S'\! system. Also, if both w\! and v\! are small with
   respect to the speed of light, we will recover the intuitive Galilean
   transformation of velocities: w' \approx w-v\!.

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 invariant mass m traveling at velocity v the energy
   and momentum are given (and even defined) by

          E = \gamma m c^2 \,\!

          \vec p = \gamma m \vec v \,\!

   where γ (the Lorentz factor) is given by

          \gamma = \frac{1}{\sqrt{1 - \beta^2}}

   where β is the velocity as a ratio of the speed of light. The term γ
   occurs frequently in relativity, and comes from the Lorentz
   transformation equations.

   Relativistic energy and momentum can be related through the formula

          E^2 - (p c)^2 = (m c^2)^2 \,\!

   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

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

          \vec p \approx m \vec v \,\!

   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:

          E_{rest} = m c^2 \,\!

   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 are meaningful.

   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. By looking at the difference in masses, one
   can predict which nuclei have extra stored energy that 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 have made it one of the most famous
   equations in all of science.

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. Defining such a quantity may
   sometimes be useful in that doing so simplifies a calculation by
   restricting it to a specific frame. For example, consider a body with
   an invariant mass m moving at some velocity relative to an observer's
   reference system. That observer defines the relativistic mass of that
   body as:

          M = \gamma m\!

   "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. The only mass that is frame independent
   is the invariant mass. When using the relativistic mass, the used
   reference frame should be specified if it isn't already obvious or
   implied. It also goes almost without saying that the increase in
   relativistic mass does not come from an increased number of atoms in
   the object. Instead, the relativistic mass of each atom and subatomic
   particle has increased.

   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!). "Relativistic mass" is also consistent with the concepts "time
   dilation" and "length contraction".

Force

   The classical definition of ordinary force f is given by Newton's
   Second Law in its original form:

          \vec f = d\vec p/dt

   and this is valid in relativity.

   Many modern textbooks rewrite Newton's Second Law as

          \vec f = M \vec a

   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

          \vec f = \gamma m \vec a + \gamma^3 m \frac{\vec v \cdot \vec
          a}{c^2} \vec v

   As seen from the equation, ordinary force and acceleration vectors are
   not necessarily parallel in relativity.

   However the four-vector expression relating four-force F^\mu\, to
   invariant mass m and four-acceleration A^\mu\, restors the same
   equation form

          F^\mu = mA^\mu\,

The geometry of space-time

   SR uses a 'flat' 4-dimensional Minkowski space, which is an example of
   a 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:

          ds^2 = dx_1^2 + dx_2^2 + dx_3^2

   where (dx[1],dx[2],dx[3]) 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:

          ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2

   If we wished to make the time coordinate look like the space
   coordinates, we could treat time as imaginary: x[4] = ict . In this
   case the above equation becomes symmetric:

          ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2

   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.
   Just as Euclidean space uses a Euclidean metric, so space-time uses a
   Minkowski metric. According to Misner (1971 §2.3), ultimately the
   deeper understanding of both special and general relativity will come
   from the study of the Minkowski metric (described below) rather than a
   "disguised" Euclidean metric using ict as the time coordinate.

   If we reduce the spatial dimensions to 2, so that we can represent the
   physics in a 3-D space

          ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2

   We see that the null geodesics lie along a dual-cone:

   image:sr1.jpg

   defined by the equation

          ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2

   or

          dx_1^2 + dx_2^2 = c^2 dt^2

   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:

   image:sr3.jpg

          ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2

          dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2

   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 d =
   \sqrt{x_1^2+x_2^2+x_3^2} 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".)

   image:sr1.jpg

   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:

          x^\nu=\left(t, x, y, z\right)

   That is, x^0 = t and x^1 = x and x^2 = y and x^3 = z. 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{\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}.

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:

          \eta_{\alpha\beta} = \begin{pmatrix} -c^2 & 0 & 0 & 0\\ 0 & 1 &
          0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}

   Its reciprocal is:

          \eta^{\alpha\beta} = \begin{pmatrix} -1/c^2 & 0 & 0 & 0\\ 0 & 1
          & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}

   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:

          \Lambda^{\mu'}{}_\nu = \begin{pmatrix} \gamma & -\beta\gamma/c &
          0 & 0\\ -\beta\gamma c & \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 t 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[i_1',i_2',...i_p'\right]}_{\left[j_1',j_2',...j_q'\righ
          t]} =
          \Lambda^{i_1'}{}_{i_1}\Lambda^{i_2'}{}_{i_2}...\Lambda^{i_p'}{}_
          {i_p}
          \Lambda_{j_1'}{}^{j_1}\Lambda_{j_2'}{}^{j_2}...\Lambda_{j_q'}{}^
          {j_q}
          T^{\left[i_1,i_2,...i_p\right]}_{\left[j_1,j_2,...j_q\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 co-ordinate system S to a primed system S', we calculate

          \begin{pmatrix} t'\\ x'\\ y'\\ z' \end{pmatrix} =
          x^{\mu'}=\Lambda^{\mu'}{}_\nu x^\nu= \begin{pmatrix} \gamma &
          -\beta\gamma/c & 0 & 0\\ -\beta\gamma c & \gamma & 0 & 0\\ 0 & 0
          & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} t\\ x\\
          y\\ z \end{pmatrix} = \begin{pmatrix} \gamma t- \gamma\beta
          x/c\\ \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 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.

Velocity and acceleration in 4D

   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 \\ \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
   4-vector are orthogonal.

Momentum in 4D

   The momentum and energy combine into a covariant 4-vector:

          p_\nu = m \cdot \eta_{\nu\mu} U^\mu = \begin{pmatrix} -E \\
          p_x\\ p_y\\ p_z\end{pmatrix}.

   where m is the invariant mass.

   The invariant magnitude of the momentum 4-vector is:

          \mathbf{p}^2 = \eta^{\mu\nu}p_\mu p_\nu = -(E/c)^2 + p^2 .

   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.

          \mathbf{p}^2 = - (E_{rest}/c)^2 = - (m \cdot c)^2 .

   We see that the rest energy is an independent invariant. A rest energy
   can be calculated even for particles and systems in motion, by
   translating to a frame in which momentum is zero.

   The rest energy is related to the mass according to the celebrated
   equation discussed above:

          E_{rest} = m c^2\,

   Note that the mass of systems measured in their centre of momentum
   frame (where total momentum is zero) is given by the total energy of
   the system in this frame. It may not be equal to the sum of individual
   system masses measured in other frames.

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.
   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{d p_{\nu}}{d \tau} = \begin{pmatrix} -{d E}/{d
          \tau} \\ {d p_x}/{d \tau} \\ {d p_y}/{d \tau} \\ {d p_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 in which
   case it is the negative of that rate of change times c^2. 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{d p}{d t} while the four force is defined by the rate of change
   of momentum with respect to proper time, i.e. \frac{d p} {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 the negative of the power
   transferred to that cell divided by the volume of the cell. This will
   be used below in the section on electromagnetism.

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. 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

   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 \\ J_x\\ J_y\\ J_z\end{pmatrix}

   The law of charge conservation 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 & -E_y & -E_z \\ E_x & 0 &
          B_z & -B_y \\ E_y & -B_z & 0 & B_x \\ E_z & B_y & -B_x & 0
          \end{pmatrix}

   The density of the Lorentz force f_\mu \! exerted on matter by the
   electromagnetic field becomes:

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

   Faraday's law of induction and Gauss's law for magnetism 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 & D_y & D_z \\
          -D_x & 0 & H_z & -H_y \\ -D_y & -H_z & 0 & H_x \\ -D_z & H_y &
          -H_x & 0 \end{pmatrix}

   Ampere's law and Gauss's law 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} \eta^{\nu\beta}
          F_{\alpha\beta}.

   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:

          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).

Status

   Special relativity is accurate only when gravitational potential is
   much less than c^2; in a strong gravitational field one must use
   general relativity (which becomes special relativity at the limit of
   weak field). At very small scales, such as at the Planck length and
   below quantum effects must be taken into consideration resulting in
   quantum gravity. However, at macroscopic scales and in the absence of
   strong gravitational fields, 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.

   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. See Status of special relativity for a more detailed
   discussion.

   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. It should be stated here that, contrary to many
       alternative claims, it said little about the invariance of the
       speed of light with respect to the source and observers velocity,
       as both source and observer were travelling together at the same
       velocity at all times.

   A number of experiments have been conducted to test special relativity
   against rival theories. These include:
     * Kaufman's experiment — electron deflection in exact accordance with
       Lorentz-Einstein prediction
     * Hamar experiment — no "ether flow obstruction"
     * 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.

   In addition, particle accelerators run almost every day somewhere in
   the world, and routinely accelerate and measure the properties of
   particles moving at near lightspeed. Many effects seen in particle
   accelerators are highly consistent with relativity theory and are
   deeply inconsistent with the earlier Newtonian mechanics.

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