Editing Special relativity (section)

=== Physics in spacetime ===

==== 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]].

==== Metric ====

The [[metric tensor]] allows one to define the [[inner product]] of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the [[Minkowski metric]] ''η'' has components (valid with suitably chosen coordinates), which can be arranged in a {{nowrap|4 × 4}} matrix:
<math display="block">\eta_{\alpha\beta} = \begin{pmatrix}
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix} ,</math>
which is equal to its reciprocal, {{tmath|1= \eta^{\alpha\beta} }}, in those frames. Throughout we use the signs as above, different authors use different conventions – see [[Minkowski metric]] alternative signs.

The [[Poincaré group]] is the most general group of transformations that preserves the Minkowski metric:
<math display="block">\eta_{\alpha\beta} = \eta_{\mu'\nu'} \Lambda^{\mu'}{}_\alpha \Lambda^{\nu'}{}_\beta</math>
and this is the physical symmetry underlying special relativity.

The metric can be used for [[raising and lowering indices]] on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector ''T'' with another 4-vector ''S'' is:
<math display="block">T^{\alpha}S_{\alpha}=T^{\alpha}\eta_{\alpha\beta}S^{\beta} = T_{\alpha}\eta^{\alpha\beta}S_{\beta} = \text{invariant scalar}</math>

Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no {{math|Λ}} appears in its trivial transformation. The magnitude of the 4-vector ''T'' is the positive square root of the inner product with itself:
<math display="block">|\mathbf{T}| = \sqrt{T^{\alpha}T_{\alpha}}</math>

One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:
<math display="block">T^{\alpha}{}_{\alpha},T^{\alpha}{}_{\beta}T^{\beta}{}_{\alpha},T^{\alpha}{}_{\beta}T^{\beta}{}_{\gamma}T^{\gamma}{}_{\alpha} = \text{invariant scalars},</math>
similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one does not need to perform Lorentz transformations to determine the invariants.

==== Relativistic kinematics and invariance ====

The coordinate differentials transform also contravariantly:
<math display="block">dX^{\mu'}=\Lambda^{\mu'}{}_\nu dX^\nu</math>
so the squared length of the differential of the position four-vector ''dX<sup>μ</sup>'' constructed using
<math display="block">d\mathbf{X}^2 = dX^\mu \,dX_\mu = \eta_{\mu\nu}\,dX^\mu \,dX^\nu = -(c\,dt)^2+(dx)^2+(dy)^2+(dz)^2</math>
is an invariant. Notice that when the [[line element]] ''d'''''X'''<sup>2</sup> is negative that {{math|{{sqrt|−''d'''''X'''<sup>2</sup>}}}} is the differential of [[proper time]], while when ''d'''''X'''<sup>2</sup> is positive, {{math|{{sqrt|''d'''''X'''<sup>2</sup>}}}} is differential of the [[proper distance]].

The 4-velocity ''U''<sup>μ</sup> has an invariant form:
<math display="block">\mathbf U^2 = \eta_{\nu\mu} U^\nu U^\mu = -c^2 \,,</math>
which means 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. Differentiating the above equation by ''τ'' produces:
<math display="block">2\eta_{\mu\nu}A^\mu U^\nu = 0.</math>
So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.

==== Relativistic dynamics and invariance ====

The invariant magnitude of the [[four-momentum|momentum 4-vector]] generates the [[energy–momentum relation]]:
<math display="block">\mathbf{P}^2 = \eta^{\mu\nu}P_\mu P_\nu = -\left (\frac{E}{c} \right )^2 + p^2 .</math>

We can work out what this invariant is by first arguing that, since it is a scalar, it does not matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
<math display="block">\mathbf{P}^2 = - \left (\frac{E_\text{rest}}{c} \right )^2 = - (m c)^2 .</math>

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:
<math display="block">E_\text{rest} = m c^2.</math>

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

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 that 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:
<math display="block">F_\nu = \frac{d P_{\nu}}{d \tau} = m A_\nu </math>

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, that is, ''dp''/''dt'' while the four-force is defined by the rate of change of momentum with respect to proper time, that is, ''dp''/''dτ''.

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.