Lorentz transforms

[Schrödinger's hat]

Does anyone know where I might be able to find a general boost formula like this one written in spherical coordinates (ct,r,phi,theta)?

I really don't feel like expanding it all myself.

[mississippichem]

Does anyone know where I might be able to find a general boost formula like this one written in spherical coordinates (ct,r,phi,theta)?

I really don't feel like expanding it all myself.

 

hm...Lorentz Transforms in spherical polar coordinates...why what for Shroedinger's Hat?

 

don't answer that: just get it to work on my browser!

Edited May 18, 2011 by mississippichem

[Schrödinger's hat]

don't answer that: just get it to work on my browser!

 

I dun no 'bout dat type 'f fing.

I just do der maf an' alg'riv'ms.

You talk to da Cap'n if you wan' dat.

 

You c'd jus' upgrade browser.

[Royston]

Does anyone know where I might be able to find a general boost formula like this one written in spherical coordinates (ct,r,phi,theta)?

I really don't feel like expanding it all myself.

 

Lorentz transformations only make sense in local non-curved spacetime (i.e Euclidean), they are infintesimal sections on a manifold, where SR holds (part of the Lorentz group). It would make no physical sense to expand this set of linear equations into spherical coordinates...you really should start with a line element or path integral, then build a metric from that.

[Schrödinger's hat]

Hmm? I was still only thinking of Minkowski space. The transformations are the same no matter how you represent the coordinates.

I imagined it'd be rather a headache and involve a lot of affine transformations, rather than a straight matrix multiplication (possibly some trig/hyperbolic trig too). Rather like trying to do general 3D rotations and translations in cylindrical coordinates.

At any rate, I've given up that idea now and gone on to other methods.

Thanks anyway.

[Royston]

Oops, I meant line integral.

 

Hmm?

 

IOW, you define the space before you imbed (for want of a better word) the Lorentz transformations, for instance you could start with Cartesian spherical coordinates [math]x^2 + y^2 + z^2 = c^2[/math] expanding the transformations when the space is already defined (and the fact there's nothing physical about doing so) doesn't make much sense. Albeit, you've decided not to go down this route.

[md65536]

If you had an arbitrary vector (βr,βphi,βtheta), could you not apply a rotation matrix to rotate it onto the r axis, then apply the boost matrix for the Cartesian x axis, using r instead of x (they should be equivalent if the axises are aligned), and then apply an inverse of the rotation matrix to get it back to your original direction? Multiply these 3 matrices to calculate a general boost matrix?

 

My maths ain't so good, but I'm curious... would that work?

Edited May 22, 2011 by md65536