Recent content by darida

  1. darida

    A First Variation of Jacobi Operator

    <Moderator's note: Moved from a homework forum.> Homework Statement From this paper. Let ##L## be the Jacobian operator of a two-sided compact surface embedded in a three-maniold ##(M,g)##, ##\Sigma \subset M##, and defined by $$L(t)=\Delta_{\Sigma(t)}+ \text{Ric}( ν_{t} , ν_{t}...
  2. darida

    Derivative of Mean Curvature and Scalar field

    Further information (file attached, Appendix A, page 99): ∂_{t} = φ\vec{ν} So the derivation of φ with respect to t would be: \frac{dφ}{dt} = \frac{d}{dt} \left (\frac{1}{ν} \frac{∂}{∂t} \right ) \frac{dφ}{dt} = \frac{1}{ν} \frac{∂}{∂t} \left ( \frac{∂}{∂t} \right ) + \frac{∂}{∂t} \frac{d}{dt}...
  3. darida

    Derivative of Mean Curvature and Scalar field

    Homework Statement Page 16 (attached file) \frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ \frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ H = mean curvature of surface Σ A = the second fundamental of Σ ν = the unit normal vector field along Σ φ = the scalar field on three manifold M φ∈C^{∞}(Σ)...
  4. darida

    Components of The Electromagnetic Field Strength Tensor

    Never mind, just found the answer
  5. darida

    Components of The Electromagnetic Field Strength Tensor

    Ok now I get confused. So, I am trying to find the radial component of the magnetic field from the Hodge-dual of the Field Tensor, but then end up like this *F_{\mu\nu}=\frac{1}{2} \epsilon_{\mu\nu\lambda\rho}F^{\lambda\rho}= \begin{bmatrix} 0 B_x B_y B_z \\ -B_x 0 -E_z...
  6. darida

    Components of The Electromagnetic Field Strength Tensor

    Source: http://gmammado.mysite.syr.edu/notes/RN_Metric.pdf Section 2 Page: 2 Eq. (15) The radial component of the magnetic field is given by B_r = g_{11} ε^{01μν} F_{μν} Where does this equation come from? Section 4 Page 3 Similar to the electric charges, the Gauss's flux theorem for the...
  7. darida

    Electromagnetics in Spherical Symmetric Problem

    In a spherical symmetric problem the only nonzero components of the electric and the magnetic field are Er and Br Why?
  8. darida

    Reissner Nordstrom Metric

    Ah thank you, that's so simple >_<
  9. darida

    Reissner Nordstrom Metric

    I want to derive Reissner Nordstrom solution using this paper as a guide: http://gmammado.mysite.syr.edu/notes/RN_Metric.pdf, but I get confused by the Eq. (8). Why the Enstein's equation can be rewritten in that form and what is the physical meaning of the Eq. (8)?
  10. darida

    Integration of Ricci Scalar Over Surface

    Does this integration of Ricci scalar over surface apply in general or just for compact surfaces? ∫RdS = χ(g) where χ(g) is Euler characteristic. And could anybody give me some good references to prove the formula?
  11. darida

    Stability of Hawking Mass

    For example: Source: page 8
  12. darida

    Stability of Hawking Mass

    Thank you so much for the references. I want to know what we talk about when we talk about stability of Hawking energy? How can we know the Hawking energy is stable?
  13. darida

    Stability of Hawking Mass

    So, what should I do now?
  14. darida

    Stability of Hawking Mass

    What is stability of Hawking Mass and how to calculate it? Any references will be appreciated. Thanks
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