Re: [Maxima-discuss] Spin connection tensor

[Viktor T. T. <vt...@vt...>]

Dear Daniel,

I cannot tell with certainty if what you did is correct, but it looks 
like you did everything by the book and got a sensible result; I don't 
see any obvious red flags. You used itensor/ctensor as intended, in a 
neat way: Found an itensor expression that could be correctly evaluated, 
converted it to ctensor, and then assigned values to what are normally 
itensor objects (ifg/ifr/ifri) but that's quite alright, since you were 
no longer using itensor at this point, you just wanted to evaluate the 
matrix terms in your SC. Your \omega appeared rather sparse but then 
again, that seems to be expected given that you only have time 
dependence, no dependence on the other coordinates. In short, I *think* 
you got it right, just don't quote me on this please :-)


Viktor


On 7/26/2025 6:52 AM, Daniel Volinski via Maxima-discuss wrote:
> 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 Bases
> Equation 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 calculate
> other things like curvature tensor, Ricci scalar, although there are other
> ways to calculate them. Eventually, I was planning to branch into
> Cartan'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 
> <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 
> <https://en.wikipedia.org/wiki/Spin_connection> 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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