Editing Four-vector (section)

==Fundamental four-vectors==

===Four-position{{anchor|Position}}===

A point in [[Minkowski space]] is a time and spatial position, called an "event", or sometimes the '''position four-vector''' or '''four-position''' or '''4-position''', described in some reference frame by a set of four coordinates:

<math display="block"> \mathbf{R} = \left(ct, \mathbf{r}\right) </math>

where '''r''' is the [[three-dimensional space]] [[position vector]]. If '''r''' is a function of coordinate time ''t'' in the same frame, i.e. '''r''' = '''r'''(''t''), this corresponds to a sequence of events as ''t'' varies. The definition ''R''<sup>0</sup> = ''ct'' ensures that all the coordinates have the same [[physical dimension|dimension]] (of [[length]]) and units (in the [[SI]], meters).<ref name="e561">{{cite web | title=Details for IEV number 113-07-19: "position four-vector" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=113-07-19 | language=ja | access-date=2024-09-08}}</ref><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> These coordinates are the components of the ''position four-vector'' for the event.

The '''displacement four-vector''' is defined to be an "arrow" linking two events:

<math display="block"> \Delta \mathbf{R} = \left(c\Delta t, \Delta \mathbf{r} \right) </math>

For the [[differential (infinitesimal)|differential]] four-position on a world line we have, using [[Minkowski space#Minkowski tensor|a norm notation]]:

<math display="block">\|d\mathbf{R}\|^2 = \mathbf{dR \cdot dR} = dR^\mu dR_\mu = c^2d\tau^2 = ds^2 \,,</math>

defining the differential [[line element]] d''s'' and differential proper time increment d''τ'', but this "norm" is also:

<math display="block">\|d\mathbf{R}\|^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,,</math>

so that:

<math display="block">(c d\tau)^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,.</math>

When considering physical phenomena, differential equations arise naturally; however, when considering space and [[time derivative]]s of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the [[proper time]] <math>\tau</math>. As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the [[coordinate time]] ''t'' of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by (''cdt'')<sup>2</sup> to obtain:

<math display="block">\left(\frac{cd\tau}{cdt}\right)^2
  = 1 - \left(\frac{d\mathbf{r}}{cdt}\cdot \frac{d\mathbf{r}}{cdt}\right)
  = 1 - \frac{\mathbf{u}\cdot\mathbf{u}}{c^2} = \frac{1}{\gamma(\mathbf{u})^2} \,,
</math>

where '''u''' = ''d'''''r'''/''dt'' is the coordinate 3-[[velocity]] of an object measured in the same frame as the coordinates ''x'', ''y'', ''z'', and [[coordinate time]] ''t'', and

<math display="block">\gamma(\mathbf{u}) = \frac{1}{\sqrt{1 - \frac{\mathbf{u}\cdot\mathbf{u}}{c^2}}}</math>

is the [[Lorentz factor]]. This provides a useful relation between the differentials in coordinate time and proper time:

<math display="block">dt = \gamma(\mathbf{u})d\tau \,.</math>

This relation can also be found from the time transformation in the [[Lorentz transformation]]s.

Important four-vectors in relativity theory can be defined by applying this differential <math>\frac{d}{d\tau}</math>.

===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.

=== Four-acceleration ===

The [[four-acceleration]] is given by:

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

where '''a''' = ''d'''''u'''/''dt'' is the coordinate 3-acceleration. Since the magnitude of '''U''' is a constant, the four acceleration is orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:

<math display="block">\mathbf{A}\cdot\mathbf{U} = A^\mu U_\mu = \frac{dU^\mu}{d\tau} U_\mu = \frac{1}{2} \, \frac{d}{d\tau} \left(U^\mu U_\mu\right) = 0 \,</math>

which is true for all world lines. The geometric meaning of four-acceleration is the [[curvature vector]] of the world line in Minkowski space.