I Deriving the conformal Laplacian

  • I
  • Thread starter Thread starter Ssnow
  • Start date Start date
AI Thread Summary
The discussion revolves around deriving the conformal Laplacian, specifically the equation involving the metric g and its relation to the Laplacian and scalar curvature. The context includes the use of a normal frame where all Christoffel symbols are zero at point y. The goal is to express the Laplacian in a conformal manner, referencing Sniatycki's book on geometric quantization. The original poster, Ssnow, later indicates that they have resolved the problem. The exchange highlights the complexities of working with the conformal Laplacian in differential geometry.
Ssnow
Science Advisor
Messages
573
Reaction score
182
Hi to all!! I have a problem to derive the conformal laplacian

\sum_{m,n}g^{mn}(y)\partial_{q^m}\partial_{q^n}(\psi|\det{(g)}|^{1/4})(y)=\sum_{m,n}|det{g}|^{1/4}(\Delta \Psi -\frac{1}{6}R(y)\psi(y))

where $$g$$ is the metric associated to a Levi Civita connection in a normal frame, we have that in the point $y$ all Christoffel symbols are $$0$$ and $$\Gamma_{ij}^m,k(y)+\Gamma_{ki}^m,j(y)+\Gamma_{jk}^m,i(y)=0.$$

In practice the aim is to find the conformal expression of the Laplacian (Yamabe operator) ... the source is the book "Geometric quantization and quantum mechanics" Sniatycki J.
Thanks,
Ssnow
 
Physics news on Phys.org
I solved the problem, Ssnow
 
Hello! Let's say I have a cavity resonant at 10 GHz with a Q factor of 1000. Given the Lorentzian shape of the cavity, I can also drive the cavity at, say 100 MHz. Of course the response will be very very weak, but non-zero given that the Loretzian shape never really reaches zero. I am trying to understand how are the magnetic and electric field distributions of the field at 100 MHz relative to the ones at 10 GHz? In particular, if inside the cavity I have some structure, such as 2 plates...
Back
Top