Editing Special relativity (section)

=== Geometry of spacetime ===
==== Comparison between flat Euclidean space and Minkowski space ====
{{see also|line element}}
[[File:Orthogonality and rotation.svg|thumb|350px|Figure 10–1. Orthogonality and rotation of coordinate systems compared between '''left:''' [[Euclidean space]] through circular [[angle]] ''φ'', '''right:''' in [[Minkowski spacetime]] through [[hyperbolic angle]] ''φ'' (red lines labelled ''c'' denote the [[worldline]]s of a light signal, a vector is orthogonal to itself if it lies on this line).<ref>{{cite book|title=Gravitation|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne |publisher=W.H. Freeman & Co|page=58|date=1973|isbn=978-0-7167-0344-0}}</ref>]]

Special relativity uses a "flat" 4-dimensional Minkowski space&nbsp;– an example of a [[spacetime]]. Minkowski spacetime appears to be very similar to the standard 3-dimensional [[Euclidean space]], but there is a crucial difference with respect to time.

In 3D space, the [[differential (infinitesimal)|differential]] of distance (line element) ''ds'' is defined by
<math display="block"> ds^2 = d\mathbf{x} \cdot d\mathbf{x} = dx_1^2 + dx_2^2 + dx_3^2, </math>
where {{nowrap|1=''d'''''x''' = (''dx''<sub>1</sub>, ''dx''<sub>2</sub>, ''dx''<sub>3</sub>)}} are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate ''X''<sup>0</sup> derived from time, such that the distance differential fulfills
<math display="block"> ds^2 = -dX_0^2 + dX_1^2 + dX_2^2 + dX_3^2, </math>
where {{nowrap|1=''d'''''X''' = (''dX''<sub>0</sub>, ''dX''<sub>1</sub>, ''dX''<sub>2</sub>, ''dX''<sub>3</sub>)}} are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a [[rotational symmetry]] of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig.&nbsp;10-1).<ref>{{cite book|title=Dynamics and Relativity|author1=J.R. Forshaw |author2=A.G. Smith |publisher=Wiley|page=247|date=2009|isbn=978-0-470-01460-8}}</ref> Just as Euclidean space uses a [[Euclidean metric]], so spacetime uses a [[Minkowski metric]]. {{Anchor|interval}}Basically, special relativity can be stated as the ''invariance of any spacetime interval'' (that is the 4D distance between any two events) when viewed from ''any inertial reference frame''. All equations and effects of special relativity can be derived from this rotational symmetry (the [[Poincaré group]]) of Minkowski spacetime.

The actual form of ''ds'' above depends on the metric and on the choices for the ''X''<sup>0</sup> coordinate.
To make the time coordinate look like the space coordinates, it can be treated as [[imaginary number|imaginary]]: {{nowrap|1=''X''<sub>0</sub> = ''ict''}} (this is called a [[Wick rotation]]).
According to [[Gravitation (book)|Misner, Thorne and Wheeler]] (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take {{nowrap|1=''X''<sup>0</sup> = ''ct''}}, rather than a "disguised" Euclidean metric using ''ict'' as the time coordinate.

Some authors use {{nowrap|1=''X''<sup>0</sup> = ''t''}}, with factors of ''c'' elsewhere to compensate; for instance, spatial coordinates are divided by ''c'' or factors of ''c''<sup>±2</sup> are included in the metric tensor.<ref>{{cite book |author=R. Penrose| title=The Road to Reality| publisher= Vintage books| date=2007 | isbn=978-0-679-77631-4| title-link=The Road to Reality}}</ref>
These numerous conventions can be superseded by using [[natural units]] where {{nowrap|1=''c'' = 1}}. Then space and time have equivalent units, and no factors of ''c'' appear anywhere.

==== 3D spacetime ====
[[File:Special relativity- Three dimensional dual-cone.svg|thumb|Figure 10–2. Three-dimensional dual-cone.]]

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space
<math display="block"> ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2, </math>
we see that the [[null geodesic|null]] [[geodesic]]s lie along a dual-cone (see Fig.&nbsp;10-2) defined by the equation;
<math display="block"> ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2 </math>
or simply
<math display="block"> dx_1^2 + dx_2^2 = c^2 dt^2, </math>
which is the equation of a circle of radius&nbsp;''c&nbsp;dt''.

==== 4D spacetime ====

If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:
<math display="block"> ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2 </math>
so
<math display="block"> dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2. </math>

[[File:Concentric Spheres.svg|thumb|Figure 10–3. Concentric spheres, illustrating in 3-space the null geodesics of a 4-dimensional cone in spacetime.]]
As illustrated in Fig.&nbsp;10-3, the null geodesics can be visualized as a set of continuous concentric spheres with radii&nbsp;=&nbsp;''c&nbsp;dt''.

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the [[star]]s and say "The light from that star that I am receiving is ''X'' years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance <math display="inline">d = \sqrt{x_1^2 + x_2^2 + x_3^2} </math> away and a time ''d''/''c'' in the past. For this reason the null dual cone is also known as the "light cone". (The point in the lower left of the Fig.&nbsp;10-2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)

The cone in the −''t'' region is the information that the point is "receiving", while the cone in the +''t'' section is the information that the point is "sending".

The geometry of Minkowski space can be depicted using [[Minkowski diagram]]s, which are useful also in understanding many of the [[thought experiment]]s in special relativity.