tag:blogger.com,1999:blog-2269351477810212131.post6524156650251859736..comments2024-03-22T08:22:47.170-07:00Comments on Copasetic Flow: Spheres, Special Relativity, and Rotationsantigrav_kids KD0FNRhttp://www.blogger.com/profile/08273077706643157078noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-2269351477810212131.post-11216183299484586962013-02-17T03:32:59.137-08:002013-02-17T03:32:59.137-08:00Thanks Jonah!
I've seen the Wick rotation a l...Thanks Jonah!<br /><br />I've seen the Wick rotation a lot over on the Casimir side of my work. Both Sommerfeld, in the second volume of his physics course, and Moller have excellent examples where they treat the complex rotation angle as real. In Sommerfeld's case, he shows very nicely how the E and B fields rotate into each other as the velocity of a travelling charge is changed.antigrav_kids KD0FNRhttps://www.blogger.com/profile/08273077706643157078noreply@blogger.comtag:blogger.com,1999:blog-2269351477810212131.post-84395649146261297742013-02-15T21:16:04.893-08:002013-02-15T21:16:04.893-08:00Great post! The fact that a sphere always looks th...Great post! The fact that a sphere always looks the same after Lorentz transform always puzzled me. Thanks for explaining it so nicely!<br /><br />I didn't explain this in my post on the topic, because I felt it was too technical for my blog, but there's a transformation called a Wick rotation we use to go between Lorentzian and Euclidian spacetimes. <br /><br />http://en.wikipedia.org/wiki/Wick_rotation<br /><br />The idea is that you rotate through the complex plane to change the lorentzian signature to a Euclidean one. You've probably seen a matrix for Lorentz transforms that takes the form <br /><br />[ cosh(phi) -sinh(phi) ]<br />[ -sinch(phi) cosh(phi) ],<br /><br />where phi is called the boost parameter, and<br /><br />phi = cosh^{-1}(gamma).<br /><br />The secret is that the hyperbolic trig functions are just trig functions of an imaginary angle.<br /><br />cosh(x) = cos(ix)<br />sinh(x) = -isin(ix)<br /><br />So you're absolutely right that Lorentz transforms are a rotation. The secret is that if the norm square of timelike vectors is negative, those vectors are complex. Furthermore, the angle of "rotation" for a Lorentz boost is imaginary.Jonahhttp://www.thephysicsmill.comnoreply@blogger.com