Re: [Maxima-discuss] Spin connection tensor

[Daniel V. <dan...@ya...>]

 Hi Viktor,
I was thinking about calculating the spin connector tensor the following way:I'm using the formula for the spin connection from:Sean Carroll, Spacetime and Geometry, Appendix J, Noncoordinate BasesEquation J.21, Page 483.
load(itensor)$
load(ctensor)$
iframe_flag:true$cframe_flag:true$ct_coords:[t,x,y,z]$
dim:length(ct_coords)$
depends(a,t)$
ishow(Eq:ω([μ,-a,b])=T([ν,-a])*S([-λ,b])*'icc2([μ,λ],[ν])-                     S([-λ,b])*idiff(T([λ,-a]),μ))$Eq:contract(canform(ev(Eq,icc2,ifc2,ifc1,ifb)))$
SC:ic_convert(Eq)$
lfg:matrix([-1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1])$
fri:matrix([1,0,0,0],[0,a,0,0],[0,0,a,0],[0,0,0,a])$
cmetric()$
depends([ifg,ifr,ifri],ct_coords)$
ifg:ident(dim)$ifr:fr$ifri:fri$array(ω,dim,dim,dim)$S:fr$T:fri$ev(SC)$
genmatrix(lambda([i,j],ω[1,i,j]),dim,dim);
cdisplay(ω,1)$

Is that correct?Thanks,
Daniel Volinski

    En domingo, 20 de julio de 2025, 17:04:23 GMT+3, Daniel Volinski via Maxima-discuss <max...@li...> escribió:  
 
  Hi Viktor,
Thank you for your input, comments and suggestions, very helpful.
What I was planning to do is to use the spin connection to calculateother things like curvature tensor, Ricci scalar, although there are otherways to calculate them.  Eventually, I was planning to branch intoCartan's structure equation.
Daniel Volinski

    En domingo, 20 de julio de 2025, 11:43:21 GMT+3, Viktor T. Toth <vt...@vt...> escribió:  
 
  
Well, it is certainly possible to do the basics, but I think the real question is, what exactly do you want to do?
 
 In itensor, for instance, we could do this:
 
 load(itensor)$
 imetric(g)$
 defcon(e,h,e)$
 components(g([m,n],[]),e([m],[a])*e([n],[b])*h([a,b],[]))$
components(w([m],[a,b]),e([n],[a])*'ichr2([s,m],[n])*e([],[s,b])+e([n],[a])*e([],[n,b],m))$
 declare(h,constant)$
 ishow(w([m],[a,b]))$
 
 or even this:
 
 ishow(rename(%,100))$
 ishow(ev(%,ichr2))$
 ishow(contract(rename(expand(%))))$
 
 but it's not exactly useful. Also, unless we are very careful, itensor gets easily confused when it comes to terms that have different raising/lowering rules for various indices.
 
 The ctensor package does have some support for a tetrad base. Run demo(tetrad), for instance, or also demo(ctensor4). There's also some support in itensor; try demo(itensor9). These might prove useful, depending on what you are trying to do.
 
 Eons ago, I also tried to document some of what is implemented in a preprint, at https://arxiv.org/pdf/cs/0503073 . Does not directly answer your question but it might help explore the implemented capabilities in these two packages.
 
 
 Viktor
 
 
 
 
 On 7/19/2025 3:40 PM, Daniel Volinski via Maxima-discuss wrote:
  
 
 Hi All, 
  I would like to calculate the Spin connection - Wikipedia using itensor/ctensor packages. Given a metric, is there any way to calculate it, or maybe it is some of the variables that already exist in the packages under some different name. 
  Thanks, 
  Daniel Volinski 
  
   
  
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