Editing Special relativity (section)

==== Transformations of physical quantities between reference frames ====

Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.

The Lorentz transformation in standard configuration above, that is, for a boost in the ''x''-direction, can be recast into matrix form as follows:
<math display="block">\begin{pmatrix}
ct'\\ x'\\ y'\\ z'
\end{pmatrix} = \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}.
</math>

In Newtonian mechanics, quantities that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "[[four-vector]]s", in Minkowski spacetime. The components of vectors are written using [[tensor index notation]], as this has numerous advantages. The notation makes it clear the equations are [[manifestly covariant]] under the [[Poincaré group]], thus bypassing the tedious calculations to check this fact. 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. Recognizing other [[physical quantities]] as [[tensors]] simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used.

The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ''ct'' and spacelike component {{nowrap|1='''x''' = (''x'', ''y'', ''z'')}}, in a [[Covariance and contravariance of vectors|contravariant]] [[position vector|position]] [[four-vector]] with components:
<math display="block">X^\nu = (X^0, X^1, X^2, X^3)= (ct, x, y, z) = (ct, \mathbf{x} ).</math>
where we define {{nowrap|1=''X''<sup>0</sup> = ''ct''}} so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.<ref>Jean-Bernard Zuber & Claude Itzykson, ''Quantum Field Theory'', pg 5, {{isbn|0-07-032071-3}}</ref><ref>[[Charles W. Misner]], [[Kip S. Thorne]] & [[John A. Wheeler]], ''Gravitation'', pg 51, {{isbn|0-7167-0344-0}}</ref><ref>[[George Sterman]], ''An Introduction to Quantum Field Theory'', pg 4, {{isbn|0-521-31132-2}}</ref> Now the transformation of the contravariant components of the position 4-vector can be compactly written as:
<math display="block">X^{\mu'}=\Lambda^{\mu'}{}_\nu X^\nu</math>
where there is an [[Einstein notation|implied summation]] on <math>\nu</math> from 0 to 3, and <math>\Lambda^{\mu'}{}_{\nu}</math> is a [[matrix (mathematics)|matrix]].

More generally, all contravariant components of a [[four-vector]] <math>T^\nu</math> transform from one frame to another frame by a [[Lorentz transformation]]:
<math display="block">T^{\mu'} = \Lambda^{\mu'}{}_{\nu} T^\nu</math>

Examples of other 4-vectors include the [[four-velocity]] {{tmath|1= U^\mu }}, defined as the derivative of the position 4-vector with respect to [[proper time]]:
<math display="block">U^\mu = \frac{dX^\mu}{d\tau} = \gamma(v)( c , v_x , v_y, v_z ) = \gamma(v) (c, \mathbf{v} ). </math>
where the Lorentz factor is:
<math display="block">\gamma(v)= \frac{1}{\sqrt{1 - v^2/c^2 }} \qquad v^2 = v_x^2 + v_y^2 + v_z^2.</math>

The [[Mass in special relativity|relativistic energy]] <math>E = \gamma(v)mc^2</math> and [[relativistic momentum]] <math>\mathbf{p} = \gamma(v)m \mathbf{v}</math> of an object are respectively the timelike and spacelike components of a [[Covariance and contravariance of vectors|contravariant]] [[four-momentum]] vector:
<math display="block">P^\mu = m U^\mu = m\gamma(v)(c,v_x,v_y,v_z)= \left (\frac{E}{c},p_x,p_y,p_z \right ) = \left (\frac{E}{c}, \mathbf{p} \right ).</math>
where ''m'' is the [[invariant mass]].

The [[four-acceleration]] is the proper time derivative of 4-velocity:
<math display="block">A^\mu = \frac{d U^\mu}{d\tau}.</math>

The transformation rules for ''three''-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of ''four''-velocity and ''four''-acceleration are simpler by means of the Lorentz transformation matrix.

The [[four-gradient]] of a [[scalar field]] φ transforms covariantly rather than contravariantly:
<math display="block">\begin{pmatrix}
\dfrac{1}{c} \dfrac{\partial \phi}{\partial t'} &
\dfrac{\partial \phi}{\partial x'} &
\dfrac{\partial \phi}{\partial y'} &
\dfrac{\partial \phi}{\partial z'}
\end{pmatrix}
= \begin{pmatrix}
\dfrac{1}{c} \dfrac{\partial \phi}{\partial t} &
\dfrac{\partial \phi}{\partial x} &
\dfrac{\partial \phi}{\partial y} &
\dfrac{\partial \phi}{\partial z}
\end{pmatrix}
\begin{pmatrix}
\gamma & +\beta\gamma & 0 & 0\\
+\beta\gamma & \gamma & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix} ,</math>
which is the transpose of:
<math display="block">(\partial_{\mu'} \phi) = \Lambda_{\mu'}{}^{\nu} (\partial_\nu \phi)\qquad \partial_{\mu} \equiv \frac{\partial}{\partial x^{\mu}}.</math>
only in Cartesian coordinates. It is the [[covariant derivative]] that transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.

More generally, the ''co''variant components of a 4-vector transform according to the ''inverse'' Lorentz transformation:
<math display="block"> T_{\mu'} = \Lambda_{\mu'}{}^{\nu} T_\nu,</math>
where <math>\Lambda_{\mu'}{}^{\nu}</math> is the reciprocal matrix of {{tmath|1= \Lambda^{\mu'}{}_{\nu} }}.

The postulates of special relativity constrain the exact form the Lorentz transformation matrices take.

More generally, most physical quantities are best described as (components of) [[tensor]]s. So to transform from one frame to another, we use the well-known [[Tensor|tensor transformation law]]<ref>{{cite book |title = Spacetime and Geometry: An Introduction to General Relativity |author=Sean M. Carroll |publisher=Addison Wesley |date=2004 |isbn=978-0-8053-8732-2 |page=22 |url=https://books.google.com/books?id=1SKFQgAACAAJ}}</ref>
<math display="block">T^{\alpha' \beta' \cdots \zeta'}_{\theta' \iota' \cdots \kappa'} = \Lambda^{\alpha'}{}_{\mu} \Lambda^{\beta'}{}_{\nu} \cdots \Lambda^{\zeta'}{}_{\rho} \Lambda_{\theta'}{}^{\sigma} \Lambda_{\iota'}{}^{\upsilon} \cdots \Lambda_{\kappa'}{}^{\phi} T^{\mu \nu \cdots \rho}_{\sigma \upsilon \cdots \phi}</math>
where <math>\Lambda_{\chi'}{}^{\psi}</math> is the reciprocal matrix of {{tmath|1= \Lambda^{\chi'}{}_{\psi} }}. All tensors transform by this rule.

An example of a four-dimensional second order [[antisymmetric tensor]] is the [[relativistic angular momentum]], which has six components: three are the classical [[angular momentum]], and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order [[antisymmetric tensor]].

The [[electromagnetic field tensor]] is another second order antisymmetric [[tensor field]], with six components: three for the [[electric field]] and another three for the [[magnetic field]]. There is also the [[stress–energy tensor]] for the electromagnetic field, namely the [[electromagnetic stress–energy tensor]].