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- Newsgroups: sci.physics
- Path: sparky!uunet!email!news
- From: gast@next.ben-fh.tuwien.ac.at (Gast)
- Subject: Re: A long-winded primer on four-vectors (part 1)
- Message-ID: <1992Jul22.123454.20826@email.tuwien.ac.at>
- Sender: news@email.tuwien.ac.at
- Nntp-Posting-Host: next.ben-fh.tuwien.ac.at
- Organization: Technical University of Vienna
- References: <1992Jul21.182314.14031@husc3.harvard.edu>
- Date: Wed, 22 Jul 1992 12:34:54 GMT
- Lines: 70
-
- In article <1992Jul21.182314.14031@husc3.harvard.edu>
- mcirvin@husc8.harvard.edu (Mcirvin) writes:
-
- > Likewise, a four-vector in the xt plane (t = time!) transforms
- > like this under a velocity boost with "rapidity parameter" phi:
- >
- > x -> x' = x cosh phi + t sinh phi
- > t -> t' = x sinh phi + t cosh phi
- >
- > where by x and t, I mean the corresponding vector components.
- >
- > Cosh phi is the famous time dilation factor, gamma, which
- > is 1/sqrt(1-v^2/c^2), with v the relative velocity. Sinh phi
- > is gamma times v/c, or just gamma*v if c = 1. Quantities
- > such as velocity, energy, and momentum can be expressed in terms
- > of four-vectors, making notation more compact and easy to
- > handle, and making it obvious when something is a relativistic
- > invariant and when it isn't.
-
- The article was a prety good overview about the subject of four-vectors.
- Yet, two short remarks:
-
- Remark 1:
- If you don't set c=1, the coordinates of a point in a Minkowsky-space are
- (if any doubt, consider the units):
-
- a
- x = (ct,x,y,z) (a is an upper index)
-
- Therefore the transfromation above has to be
-
- x -> x' = x cosh phi + ct sinh phi
- t -> t' = x/c sinh phi + t cosh phi
-
- Remark 2:
- Because of the metric there are two kinds of coordinates neccessary. For
- example the space-vector is given as
-
- a
- contravariant: x = (ct, x, y, z) (a is an upper index)
-
- covariant: x = (ct,-x,-y,-z) (a is a lower index)
- a
-
- The relationship between a contravariant and a covariant vector is given
- as
- b
- A = g .A where g is the described metric tensor
- a ab ab
-
- The dot product is aways defined as the product of a covariant and a
- contravariant vector, which explains the minus sign:
-
- a a
- A.B := A .B = A .B
- a a
-
- For example the absolut square of the space-vector is:
- a
- x .x = (ct).(ct) + (x).(-x) + (y).(-y) + (z).(-z)
- a
-
- a 2 2 2 2 2
- x .x = c t - x - y - z
- a
-
- I hope this two remarks are comprehensible, too.
-
- --
- Harry
-