Editing Spacetime (section)

=== Time dilation and length contraction revisited ===
{{More citations needed section|date=March 2024}}
{{Main|Time dilation|Length contraction}}
[[File:Spacetime Diagrams Illustrating Time Dilation and Length Contraction.png|thumb|upright=1.5|Figure 3-3. Spacetime diagrams illustrating time dilation and length contraction]]

It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig.&nbsp;3-3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.

To reduce the complexity of the equations slightly, there are a variety of different shorthand notations for ''ct'':
: <math>\Tau = ct</math> and <math>w = ct</math> are common.
: One also sees very frequently the use of the convention <math>c = 1.</math>

[[File:Lorentz factor.svg|thumb|Figure 3–4. Lorentz factor as a function of velocity]]
In Fig. 3-3a, segments ''OA'' and ''OK'' represent equal spacetime intervals. Time dilation is represented by the ratio ''OB''/''OK''. The invariant hyperbola has the equation {{nowrap|1={{math|''w'' {{=}} {{radical|''x''<sup>2</sup> + ''k''<sup>2</sup>}}}}}} where ''k''&nbsp;=&nbsp;''OK'', and the red line representing the world line of a particle in motion has the equation ''w''&nbsp;=&nbsp;''x''/''β''&nbsp;=&nbsp;''xc''/''v''. A bit of algebraic manipulation yields <math display="inline">OB = OK / \sqrt{1 - v^2/c^2} .</math>

The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma <math>\gamma</math>:<ref name=Forshaw>{{cite book|last1=Forshaw|first1=Jeffrey|last2=Smith|first2=Gavin|title=Dynamics and Relativity|date=2014|publisher=John Wiley & Sons|isbn=978-1-118-93329-9|page=118|url=https://books.google.com/books?id=5TaiAwAAQBAJ|access-date=24 April 2017|language=en}}</ref>
: <math>\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} </math>

If ''v'' is greater than or equal to ''c'', the expression for <math>\gamma</math> becomes physically meaningless, implying that ''c'' is the maximum possible speed in nature. For any ''v'' greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.

In Fig. 3-3b, segments ''OA'' and ''OK'' represent equal spacetime intervals. Length contraction is represented by the ratio ''OB''/''OK''. The invariant hyperbola has the equation {{nowrap|1={{math|''x'' {{=}} {{radical|''w''<sup>2</sup> + ''k''<sup>2</sup>}}}}}}, where ''k''&nbsp;=&nbsp;''OK'', and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/''β''&nbsp;=&nbsp;''c''/''v''. Event&nbsp;A has coordinates
(''x'',&nbsp;''w'')&nbsp;=&nbsp;(''&gamma;k'',&nbsp;''&gamma;βk''). Since the tangent line through A and B has the equation ''w''&nbsp;=&nbsp;(''x''&nbsp;−&nbsp;''OB'')/''β'', we have ''&gamma;βk''&nbsp;=&nbsp;(''&gamma;k''&nbsp;−&nbsp;''OB'')/''β'' and
: <math>OB/OK = \gamma (1 - \beta ^ 2) = \frac{1}{\gamma}</math>

{{anchor|Lorentz transformations}}