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Question: Can you perform a conformal(metric) transformation with the Physics package?

Question:Can you perform a conformal(metric) transformation with the Physics package?

Hullzie16 333 Maple 2025
physics conformal
3 hours ago
0

Suppose I have a metric g, and I want to perform a conformal transformation g = exp(2Phi)*g, is there a straightforward way to do this for curvature quantitieies (Christoffel, Ricci Scalar etc)? I was able to do it rather easily for the Christoffel symbols, as seen below, but it required me making a substitution for each index pair. While this isn't horrible, it would be nice if there was a way to do it without this procedure. 

> 

restart;
> 

with(Physics):CompactDisplay(Phi(X)):

Phi(X)*`will now be displayed as`*Phi

 (1)
> 

Setup(metric=arbitrary):

_______________________________________________________

  

`Systems of spacetime coordinates are:`*{X = (x1, x2, x3, x4)}

  

`Default differentiation variables for d_, D_ and dAlembertian are:`*{X = (x1, x2, x3, x4)}

  

`The arbitrary metric in coordinates `*[x1, x2, x3, x4]

  

`Signature: `(`- - - +`)

  

_______________________________________________________

  

Physics:-g_[mu, nu] = Matrix(%id = 36893488152300428092)

 (2)
> 

Christoffel[~alpha,beta,nu,definition]

Physics:-Christoffel[`~alpha`, beta, nu] = (1/2)*Physics:-g_[`~alpha`, `~mu`]*(Physics:-d_[nu](Physics:-g_[mu, beta], [X])+Physics:-d_[beta](Physics:-g_[mu, nu], [X])-Physics:-d_[mu](Physics:-g_[beta, nu], [X]))

 (3)
> 

Simplify(eval((3),[g_[mu,beta]=exp(2*Phi(X))*g_[mu,beta],g_[mu,nu]=exp(2*Phi(X))*g_[mu,nu],g_[beta,nu]=exp(2*Phi(X))*g_[beta,nu]]))

Physics:-Christoffel[`~alpha`, beta, nu] = exp(2*Phi(X))*(Physics:-d_[nu](Phi(X), [X])*Physics:-g_[beta, `~alpha`]+Physics:-d_[beta](Phi(X), [X])*Physics:-g_[nu, `~alpha`]-Physics:-d_[`~alpha`](Phi(X), [X])*Physics:-g_[beta, nu]+Physics:-Christoffel[`~alpha`, beta, nu])

 (4)
> 

expand((4))

Physics:-Christoffel[`~alpha`, beta, nu] = (exp(Phi(X)))^2*Physics:-d_[nu](Phi(X), [X])*Physics:-g_[beta, `~alpha`]+(exp(Phi(X)))^2*Physics:-d_[beta](Phi(X), [X])*Physics:-g_[nu, `~alpha`]-(exp(Phi(X)))^2*Physics:-d_[`~alpha`](Phi(X), [X])*Physics:-g_[beta, nu]+(exp(Phi(X)))^2*Physics:-Christoffel[`~alpha`, beta, nu]

 (5)
 

 

Any thoughts appreciated, thank you!

MyConformal.mw
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