Editing Four-vector (section)

===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>.