Editing Special relativity (section)

=== 4‑vectors ===
{{Main|Four-vector}}
Four‑vectors have been mentioned above in context of the energy–momentum {{nowrap|1=4‑vector}}, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, {{nowrap|1=4‑vectors}}, and more generally [[tensors]], greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are ''manifestly'' relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of [[Maxwell's equations]] in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the [[field strength tensor]] formulation.<ref name=Post_1962>{{cite book|title=Formal Structure of Electromagnetics: General Covariance and Electromagnetics|date=1962|publisher=Dover Publications Inc.|isbn=978-0-486-65427-0|author=E. J. Post}}</ref>

On the other hand, general relativity, from the outset, relies heavily on {{nowrap|1=4‑vectors}}, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such {{nowrap|1=4‑vectors}} even within a ''curved'' spacetime, and not just within a ''flat'' one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime.

==== Definition of 4-vectors ====
A 4-tuple, {{tmath|1=A=\left(A_{0}, A_{1}, A_{2}, A_{3}\right)}} is a "4-vector" if its component ''A<sub>i</sub>'' transform between frames according to the Lorentz transformation.

If using {{tmath|1=(ct, x, y, z)}} coordinates, ''A'' is a {{nowrap|1=4–vector}} if it transforms (in the {{nowrap|1=''x''-direction}}) according to
: <math>\begin{align}
  A_0' &= \gamma \left( A_0 - (v/c) A_1 \right)  \\ 
  A_1' &= \gamma \left( A_1 - (v/c) A_0 \right)\\
  A_2' &= A_2 \\ 
  A_3' &= A_3
\end{align} ,</math>
which comes from simply replacing ''ct'' with ''A''<sub>0</sub> and ''x'' with ''A''<sub>1</sub> in the earlier presentation of the [[#Lorentz transformations|'''Lorentz transformation.''']]

As usual, when we write ''x'', ''t'', etc. we generally mean Δ''x'', Δ''t'' etc.

The last three components of a {{nowrap|1=4–vector}} must be a standard vector in three-dimensional space. Therefore, a {{nowrap|1=4–vector}} must transform like {{tmath|1=(c \Delta t, \Delta x, \Delta y, \Delta z)}} under Lorentz transformations as well as rotations.<ref name="Schutz1985">{{cite book| last1=Schutz |first1= Bernard F. |title=A first course in general relativity|date=1985|publisher=Cambridge University Press|location=Cambridge, UK|isbn=0521277035|page=26}}</ref>{{rp|36–59}}

==== Properties of 4-vectors ====
* '''Closure under linear combination:''' If ''A'' and ''B'' are {{nowrap|1=4-vectors}}, then {{tmath|1=C = aA + aB}} is also a {{nowrap|1=4-vector}}.
* '''Inner-product invariance:''' If ''A'' and ''B'' are {{nowrap|1=4-vectors}}, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a {{nowrap|1=3-vector}}. In the following, <math>\vec{A}</math> and <math>\vec{B}</math> are {{nowrap|1=3-vectors}}:
*: <math>A \cdot B \equiv </math> <math>A_0 B_0 - A_1 B_1 - A_2 B_2 - A_3 B_3 \equiv </math> <math>A_0 B_0 - \vec{A} \cdot \vec{B}</math>
: In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in {{nowrap|1=3-space}}.
: Two vectors are said to be ''orthogonal'' if {{tmath|1= A \cdot B = 0 }}. Unlike the case with {{nowrap|1=3-vectors}}, orthogonal {{nowrap|1=4-vectors}} are not necessarily at right angles to each other. The rule is that two {{nowrap|1=4-vectors}} are orthogonal if they are offset by equal and opposite angles from the 45° line, which is the world line of a light ray. This implies that a lightlike {{nowrap|1=4-vector}} is orthogonal to ''itself''.
* '''Invariance of the magnitude of a vector:''' The magnitude of a vector is the inner product of a {{nowrap|1=4-vector}} with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which {{tmath|1= A \cdot A = 0 }}, while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval <math>c^2 t^2 - x^2</math> and the invariant length of the relativistic momentum vector {{tmath|1= E^2 - p^2 c^2 }}.<ref name="Morin2007" />{{rp|178–181}}<ref name="Schutz1985" />{{rp|36–59}}

==== Examples of 4-vectors ====
* '''Displacement 4-vector:''' Otherwise known as the ''spacetime separation'', this is {{nowrap|1=(''&Delta;t, &Delta;x, &Delta;y, &Delta;z''),}} or for infinitesimal separations, {{nowrap|1=(''dt'', ''dx'', ''dy'', ''dz'')}}.
*: <math>dS \equiv (dt, dx, dy, dz)</math>
* '''Velocity 4-vector:''' This results when the displacement {{nowrap|1=4-vector}} is divided by <math>d \tau</math>, where <math>d \tau</math> is the proper time between the two events that yield ''dt'', ''dx'', ''dy'', and ''dz''.
*: <math>V \equiv \frac{dS}{d \tau} = \frac{(dt, dx, dy, dz)}{dt/\gamma} = </math> <math>\gamma \left(1, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right) = </math> <math>(\gamma, \gamma \vec{v} ) </math>
{{multiple image
 | direction         = horizontal
 | width1            = 150
 | image1            = Momentarily Comoving Reference Frame.gif
 | caption1          = Figure 7-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.
 | width2            = 150
 | image2            = Lorentz transform of world line.gif
 | caption2          = Figure 7-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).
}}
: The {{nowrap|1=4-velocity}} is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.
: An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found that is momentarily comoving with the particle. This frame, the ''momentarily comoving reference frame'' (MCRF), enables application of special relativity to the analysis of accelerated particles.
: Since photons move on null lines, <math>d \tau = 0</math> for a photon, and a {{nowrap|1=4-velocity}} cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.
* '''Energy–momentum 4-vector:'''
*: <math>P \equiv (E/c, \vec{p}) = (E/c, p_x, p_y, p_z)</math>
: As indicated before, there are varying treatments for the energy–momentum {{nowrap|1=4-vector}} so that one may also see it expressed as <math>(E, \vec{p})</math> or {{tmath|1= (E, \vec{p}c) }}. The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy–momentum {{nowrap|1=4-vector}} is a conserved quantity.
* '''Acceleration 4-vector:''' This results from taking the derivative of the velocity {{nowrap|1=4-vector}} with respect to {{tmath|1= \tau }}.
*: <math>A \equiv \frac{dV}{d \tau} = </math> <math>\frac{d}{d \tau} (\gamma, \gamma \vec{v}) = </math> <math>\gamma \left( \frac{d \gamma}{dt}, \frac{d(\gamma \vec{v})}{dt} \right)</math>
* '''Force 4-vector:''' This is the derivative of the momentum {{nowrap|1=4-vector}} with respect to <math>\tau .</math>
*: <math>F \equiv \frac{dP}{d \tau} = </math> <math>\gamma \left(\frac{dE}{dt}, \frac{d \vec{p}}{dt} \right) = </math> <math> \gamma \left( \frac{dE}{dt},\vec{f} \right) </math>

As expected, the final components of the above {{nowrap|1=4-vectors}} are all standard {{nowrap|1=3-vectors}} corresponding to spatial {{nowrap|1=3-momentum}}, {{nowrap|1=3-force}} etc.<ref name="Morin2007" />{{rp|178–181}}<ref name="Schutz1985" />{{rp|36–59}}

==== 4-vectors and physical law ====
The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving {{nowrap|1=4-vectors}} rather than give up on conservation of momentum.

Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving {{nowrap|1=4-vectors}} require the use of tensors with appropriate rank, which themselves can be thought of as being built up from {{nowrap|1=4-vectors}}.<ref name="Morin2007" />{{rp|186}}
{{anchor|Acceleration}}