Editing Four-vector (section)

===Four-gradient===

Considering that [[partial derivative]]s are [[linear operator]]s, one can form a [[four-gradient]] from the partial [[time derivative]] {{math|∂}}/{{math|∂}}''t'' and the spatial [[gradient]] ∇. Using the standard basis, in index and abbreviated notations, the contravariant components are:

<math display="block">\begin{align}  
  \boldsymbol{\partial} & = \left(\frac{\partial }{\partial x_0}, \, -\frac{\partial }{\partial x_1}, \, -\frac{\partial }{\partial x_2}, \, -\frac{\partial }{\partial x_3} \right) \\
  & = (\partial^0, \, - \partial^1, \, - \partial^2, \, - \partial^3) \\
  & = \mathbf{E}_0\partial^0 - \mathbf{E}_1\partial^1 - \mathbf{E}_2\partial^2 - \mathbf{E}_3\partial^3 \\
  & = \mathbf{E}_0\partial^0 - \mathbf{E}_i\partial^i \\
  & = \mathbf{E}_\alpha \partial^\alpha \\
  & = \left(\frac{1}{c}\frac{\partial}{\partial t} , \, - \nabla \right) \\
  & = \left(\frac{\partial_t}{c},- \nabla \right) \\
  & = \mathbf{E}_0\frac{1}{c}\frac{\partial}{\partial t} - \nabla \\
\end{align}</math>

Note the basis vectors are placed in front of the components, to prevent confusion between taking the derivative of the basis vector, or simply indicating the partial derivative is a component of this four-vector. The covariant components are:

<math display="block">\begin{align}
  \boldsymbol{\partial} & = \left(\frac{\partial }{\partial x^0}, \, \frac{\partial }{\partial x^1}, \, \frac{\partial }{\partial x^2}, \, \frac{\partial }{\partial x^3} \right) \\
  & = (\partial_0, \, \partial_1, \, \partial_2, \, \partial_3) \\
  & = \mathbf{E}^0\partial_0 + \mathbf{E}^1\partial_1 + \mathbf{E}^2\partial_2 + \mathbf{E}^3\partial_3 \\
  & = \mathbf{E}^0\partial_0 + \mathbf{E}^i\partial_i \\
  & = \mathbf{E}^\alpha \partial_\alpha \\
  & = \left(\frac{1}{c}\frac{\partial}{\partial t} , \, \nabla \right) \\
  & = \left(\frac{\partial_t}{c}, \nabla \right) \\
  & = \mathbf{E}^0\frac{1}{c}\frac{\partial}{\partial t} + \nabla \\
\end{align}</math>

Since this is an operator, it doesn't have a "length", but evaluating the inner product of the operator with itself gives another operator:

<math display="block">\partial^\mu \partial_\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 = \frac{{\partial_t}^2}{c^2} - \nabla^2</math>

called the [[D'Alembert operator]].

==Kinematics==

=== Four-velocity ===
{{Main|Four-velocity}}

The [[four-velocity]] of a particle is defined by:

<math display="block">\mathbf{U} = \frac{d\mathbf{X}}{d \tau} = \frac{d\mathbf{X}}{dt}\frac{dt}{d \tau} = \gamma(\mathbf{u})\left(c, \mathbf{u}\right),</math>

Geometrically, '''U''' is a normalized vector tangent to the [[world line]] of the particle. Using the differential of the four-position, the magnitude of the four-velocity can be obtained:

<math display="block">\|\mathbf{U}\|^2 = U^\mu U_\mu = \frac{dX^\mu}{d\tau} \frac{dX_\mu}{d\tau} = \frac{dX^\mu dX_\mu}{d\tau^2} = c^2 \,,</math>

in short, the magnitude of the four-velocity for any object is always a fixed constant:

<math display="block">\| \mathbf{U} \|^2 = c^2 </math>

The norm is also:

<math display="block">\|\mathbf{U}\|^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,</math>

so that:

<math display="block">c^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,</math>

which reduces to the definition of the [[Lorentz factor]].

Units of four-velocity are m/s in [[International System of Units|SI]] and 1 in the [[geometrized unit system]].  Four-velocity is a contravariant vector.