section_path: Relativistic/Four-vector formulation
section_name: Four-vector formulation
paragraph_text: Four-vector formulation
  
  In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example R for position. The expression for the four-momentum depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of proper time, τ, defined by
  
  c 2 d τ 2 = c 2 d t 2 − d x 2 − d y 2 − d z 2, {\displaystyle c^{2}{\text{d}}\tau ^{2}=c^{2}{\text{d}}t^{2}-{\text{d}}x^{2}-{\text{d}}y^{2}-{\text{d}}z^{2}\,,}
  
  is invariant under Lorentz transformations (in this expression and in what follows the (+ − − −) metric signature has been used, different authors use different conventions). Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by √ −1; or by keeping time a real quantity and embedding the vectors in a Minkowski space. In a Minkowski space, the scalar product of two four-vectors U = (U, U, U, U) and V = (V, V, V, V) is defined as
  
  U ⋅ V = U 0 V 0 − U 1 V 1 − U 2 V 2 − U 3 V 3. {\displaystyle \mathbf {U} \cdot \mathbf {V} =U_{0}V_{0}-U_{1}V_{1}-U_{2}V_{2}-U_{3}V_{3}\,.}
  
  In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by
  
  U ≡ d R d τ = γ d R d t, {\displaystyle \mathbf {U} \equiv {\frac {{\text{d}}\mathbf {R} }{{\text{d}}\tau }}=\gamma {\frac {{\text{d}}\mathbf {R} }{{\text{d}}t}}\,,}
  
  and the (contravariant) four-momentum is
  
  P = m 0 U, {\displaystyle \mathbf {P} =m_{0}\mathbf {U} \,,}
  
  where m is the invariant mass. If R = (c t, x, y, z) (in Minkowski space), then
  
  P = γ m 0 (c, v) = (m c, p). {\displaystyle \mathbf {P} =\gamma m_{0}\left(c,\mathbf {v} \right)=(mc,\mathbf {p})\,.}
  
  Using Einstein's mass–energy equivalence, E = m c, this can be rewritten as
  
  P = (E c, p). {\displaystyle \mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)\,.}
  
  Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.
  
  The magnitude of the momentum four-vector is equal to m c:
  
  ‖ P ‖ 2 = P ⋅ P = γ 2 m 0 2 (c 2 − v 2) = (m 0 c) 2, {\displaystyle \|\mathbf {P} \|^{2}=\mathbf {P} \cdot \mathbf {P} =\gamma ^{2}m_{0}^{2}\left(c^{2}-v^{2}\right)=(m_{0}c)^{2}\,,}
  
  and is invariant across all reference frames.
  
  The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting m = 0 it follows that
  
  E = p c. {\displaystyle E=pc\,.}
  
  In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles.
  
  The four-momentum of a planar wave can be related to a wave four-vector
  
  P = (E c, p →) = ℏ K = ℏ (ω c, k →) {\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)=\hbar \mathbf {K} =\hbar \left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)}
  
  For a particle, the relationship between temporal components, E = ħ ω, is the Planck–Einstein relation, and the relation between spatial components, p = ħ k, describes a de Broglie matter wave.