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  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

    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)...
  3. darida

    Components of The Electromagnetic Field Strength Tensor

    Source: 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...
  4. 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?
  5. darida

    Reissner Nordstrom Metric

    I want to derive Reissner Nordstrom solution using this paper as a guide:, 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)?
  6. 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?
  7. darida

    Stability of Hawking Mass

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

    Ricci curvature

    Ansatz metric of the 4 dimensional spacetime: ds^2=a^2 g_{ij}dx^i dx^j + du^2 (1) where: Signature: - + + + Metric g_{ij} \equiv g_{ij} (x^i) describes 3 dimensional AdS spacetime i,j = 0,1,2 = 3 dimensional curved spacetime indices a(u)= warped factor u = x^D =...
  9. darida

    Christoffel Symbol

    Ansatz metric of the four dimensional spacetime: ds^2=a^2 g_{ab}dx^a dx^b - du^2 where: a,b=0,1,2 a(u)=warped factor Christoffel symbol of a three dimensional AdS spacetime: \Gamma^{c}_{ab}= \frac{1}{2} g^{cd}(∂_b g_{da} + ∂_a g_{bd} - ∂_d g_{ba}) Now how to find \Gamma^{a}_{b}?
  10. darida

    Weak field Geodesic equation

    Geodesic equation: m_{0}\frac{du^{\alpha}}{d\tau}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}= qF^{\alpha\beta}u_{\beta} Weak-field: ds^{2}= - (1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2}) Magnetic field, B is set to be zero. I want to find electric field, E, but don't know where to start, so...
  11. darida

    Poisson Bracket

    Homework Statement Show that Q_{1}=\frac{1}{\sqrt{2}}(q_{1}+\frac{p_{2}}{mω}) Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-\frac{p_{2}}{mω}) P_{1}=\frac{1}{\sqrt{2}}(p_{1}-mωq_{2}) P_{2}=\frac{1}{\sqrt{2}}(p_{1}+mωq_{2}) (where mω is a constant) is a canonical transformation by Poisson bracket test. This...
  12. darida

    Generating Function in Physics

    Could anybody give me some examples of generating function in physics, it's application, and it's use? Thank you
  13. darida

    Maple Maple - Error (in plots:-display)

    I tried this on maple: eps1 := [0, 1/5, 2/5, 3/5, 4/5, 1]: start1 := 2.3: base1 := 1.1: for i from 1 to 6 do R := start1*base1**0: points1[i] := [R, evalf(eval(mH, {M=1, r=R, epsilon = eps1[i]}))]: for a from 1 to 20 do R := start1*base1**a: points1[i] := points1[i], [R, evalf(eval(mH, {M=1...
  14. darida

    Hawking Mass in Schwarzschild Spacetime

    Homework Statement Metric signature: - + + + Schwarzschild metric: dS^{2}=-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}(sin\theta)^{2}d\phi^{2} Second fundamental form: h_{ij}=g_{kl}\Gamma^{k}_{ij}n^{l} where: i=1,2 j=1,2...