View source for Spacetime metrics explained for biologists

This is a reading course for the wikipedia article [[w:spacetime]].  Check also [[w:simple:spacetime]] for a lightning overview.
As the course progresses, the skeletons may be [[#Study notes|filled out]] by participants (see page history) and new resources may be spun off.

== Reading list==
Basic
*[[w:spacetime]] (suggested by [[User:JWSchmidt]])
Supplementary (suggested by [[User:Mu301|mikeu]])
*[[w:world line]] 
*[[w:Minkowski diagram]]
*[http://www.astro.ucla.edu/~wright/cosmolog.htm Ned Wright's Cosmology Tutorial]
*[http://www.phy.syr.edu/courses/modules/LIGHTCONE/index.html The Light Cone - an illuminating introduction to relativity]
More
*[http://authors.library.caltech.edu/9262/1/MORprl88.pdf Wormholes, Time Machines, and the Weak Energy Condition] by M. Morris, K. Thorne, and U. Yurtsever  (suggested by JWSchmidt)

==Motivating questions or ideas==
<!-- Wikiversians are welcome to refactor these discussion threads into DocumentMode -->

;Lorentz metric:
# At the Wikipedia article for [[w:World line#World lines in special relativity|world line]] the term "Lorentz metric" links to [[w:Spacetime|Spacetime]] and on that page there is this sentence, "This means the smooth [[Lorentz metric]] <math>g</math> has signature <math>\left(3,1\right)</math>."  The Wikipedia page for [[w:Lorentz metric|Lorentz metric]] links to [[w:Pseudo-Riemannian manifold#Lorentzian manifold|Pseudo-Riemannian manifold]] where it indicates that there are multiple Lorentzian metrics. That page also mentions the "Minkowski metric".  The Wikipedia page for [[w:Minkowski metric|Minkowski metric]] links to [[w:Minkowski space|Minkowski space]] where it says, "the term 'Minkowski space' is also used for analogues in any dimension: '''''n''+1 dimensional Minkowski space''' is a vector space or affine space of real dimension ''n''+1 on which there is an inner product or [[pseudo-Riemannian metric]] of signature (''n'',1)". I would like to know which of these metrics is typically used and I'd like an example of how it is used. --[[User:JWSchmidt|JWSchmidt]] 17:18, 10 February 2009 (UTC)
#* Our usual spacetime is (1+3)-dimensional  [i.e. n=3]. researchers are evenly split on the sign convention.  There is a rumour that physicists favour (+,-,-,-) and mathematicians favour (-,+,+,+).  This means that the square of distance from the point (t,b,c,d) [i.e. time=t, position= (b,c,d)] to the origin (0,0,0,0) is either (tt-bb-cc-dd) or (-tt+bb+cc+dd).  This structure <ref>i.e. [[w:inner product]]</ref> is called "Lorentz metric".  It differs from the usual [[w:Euclidean metric|Euclidean metric]] (i.e. Pythagoras theorem) in usual space by a change of sign from + to -.  [[User:Hillgentleman|Hillgentleman]] | //\\ |[[User talk:hillgentleman|Talk]] 03:09, 11 February 2009 (UTC)
#  My search for information about metrics started at [[w:Time travel|Time travel]] and then I went to [[w:Wormhole#Time travel|Wormhole]] and found the page section there on [[w:Wormhole#Metrics|metrics]]. Reference #8 on that Wikipedia page links to the article [http://authors.library.caltech.edu/9262/1/MORprl88.pdf Wormholes, Time Machines, and the Weak Energy Condition] by M. Morris, K. Thorne, and U. Yurtsever. Near the bottom of the second column on page 1446 there is an equation for a spacetime metric. Where does that metric come from and how does it relate to others like those mentioned above? --[[User:JWSchmidt|JWSchmidt]] 14:34, 11 February 2009 (UTC) 
#*  [[file:Coordonnees_polaires_plan.png|right|70px]] [[File:Cordonnees_spheriques.svg|right|70px]] In your reference, <math>ds</math> is the "infinitessimal distance" between two points which are "very very" close to each other on a curved space.  The formula "ds^2 = ..." is written in [[w:polar coordinates]] and in terms of [[w:differential forms]], and it defines a metric in a curved space ([[w:manifold]]) which describes the shape of the "wormhole" they are studying.  The simplest polar coordinate system is on the plane:  <math> r = (x^2 + y^2)^{1/2} , \theta = arctan(y/x)</math>.  You get the x and y back by <math> x=r cos\theta , y=r sin\theta</math>.  We were talking about a flat Minkowski space, whose metric can be written as <math> ds^2 = dt^2 - dx^2 - dy^2 - dz^2</math> .[[User:Hillgentleman|Hillgentleman]] | //\\ |[[User talk:hillgentleman|Talk]] 08:23, 12 February 2009 (UTC)

# Why is ...

== Dramatis Personae ==
=== Physics===
* [[spacetime]]
* [[Lorentz transformation]]
* [[speed of light]]
* [[frame of reference]]
* [[velocity]] and [[momentum]]
* [[Newton's law of motion]]

===Mathematics===
* [[vector space]]
* [[metric]]: ([[w:metric]])
* [[rotation]]
* [[matrix]]

==Special relativity==
===Basic principles===
* [[Speed of light]] is a universal constant, as is demonstrated by the [[w:Michelson-Morley experiment]]
* [[principle of relativity]]

===Phenomenology===
* [[dilation of time]]
* [[mass-energy equivalence]]
* [[causality]]

==Study notes==
<!-- Here is the beginning of the narrative which would become a full learning resource -->

Spacetime is just [[space]] and [[time]] put together <ref>in mathematics, it is just the [[direct sum]] of two vector spaces <math>\mathbb{R}^3 \oplus \mathbb{R}</math></ref>, i.e., the four-dimensional space given by the three "space"-directions and one "time"-direction.

The first question is, "Why do we do that?" 
A quick answer is then, "Spacetime is ''more'' than just space and time bundled together.  In fact, experiments show that spacetime is a more fundamental entity; you cannot really separate time from space. (These directions can "[[Lorentz transformation|interact]]".)  The first hints of that fact came from [[electromagnetism]]... (More later)"

The second question is: what is a [[metric in mathematics]]?  A quick answer is that it is a structure in a space with which you can measure the distance between two points.  On a straight surface like a flat piece of paper (ie 2-dimensional space), measuring distance is simple: [[Pythagorean theorem]] tells us that the distance between two points with coordinate (x,y) (p,q) is <math>\sqrt (x-p)^2 + (y-q)^2</math>  
On a curved surface like a sphere, measuring distance is [[geodesic|more involved]].

And fast forward to special relativity, as a direct consequence of the fundamental assumption that lightspeed is a universal constant, the [[Lorentz metric|metric]] of the four-dimensional spacetime is given by
...

==Homework==
#Expand the study notes above into useful learning resources.
#Calculate the Lorentzian (+,-,-,-) distance of the points (0,0,0,0) and (1,1,0,0) (i.e. two points which are 1 unit of time apart and 1 unit of space apart).

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==Footnotes==
<references/>

[[Category:Theory of relativity]]
[[Category:Spacetime]]