# 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 filled out by participants (see page history) and new resources may be spun off.

Basic

Supplementary (suggested by mikeu)

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## Motivating questions or ideas

Lorentz metric
1. At the Wikipedia article for world line the term "Lorentz metric" links to Spacetime and on that page there is this sentence, "This means the smooth Lorentz metric ${\displaystyle g}$  has signature ${\displaystyle \left(3,1\right)}$ ." The Wikipedia page for Lorentz metric links to Pseudo-Riemannian manifold where it indicates that there are multiple Lorentzian metrics. That page also mentions the "Minkowski metric". The Wikipedia page for Minkowski metric links to 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. --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 [1] is called "Lorentz metric". It differs from the usual Euclidean metric (i.e. Pythagoras theorem) in usual space by a change of sign from + to -. Hillgentleman | //\\ |Talk 03:09, 11 February 2009 (UTC)
2. My search for information about metrics started at Time travel and then I went to Wormhole and found the page section there on metrics. Reference #8 on that Wikipedia page links to the article 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? --JWSchmidt 14:34, 11 February 2009 (UTC)
• In your reference, ${\displaystyle ds}$  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: ${\displaystyle r=(x^{2}+y^{2})^{1/2},\theta =arctan(y/x)}$ . You get the x and y back by ${\displaystyle x=rcos\theta ,y=rsin\theta }$ . We were talking about a flat Minkowski space, whose metric can be written as ${\displaystyle ds^{2}=dt^{2}-dx^{2}-dy^{2}-dz^{2}}$  .Hillgentleman | //\\ |Talk 08:23, 12 February 2009 (UTC)
1. Why is ...

## Study notes

Spacetime is just space and time put together [2], 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 "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 ${\displaystyle {\sqrt {(}}x-p)^{2}+(y-q)^{2}}$  On a curved surface like a sphere, measuring distance is more involved.

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

## Homework

1. Expand the study notes above into useful learning resources.
2. 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).

## Footnotes

1. i.e. w:inner product
2. in mathematics, it is just the direct sum of two vector spaces ${\displaystyle \mathbb {R} ^{3}\oplus \mathbb {R} }$