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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

    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
  15. darida

    Ricci curvature

    Oh okay, but I've calculated both the 4-dimensional Ricci tensor and the 3-dimensional Ricci tensor separately. Here is my calculation:
  16. 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 =...
  17. darida

    Christoffel Symbol

    Oh okay
  18. darida

    Christoffel Symbol

    It's not in any articles/books. I just met someone who told me that :(
  19. darida

    Christoffel Symbol

    I don't know that's why I asked :confused: *edit: Well, one said that \Gamma^{\rho}_{\mu\nu} = ........\Gamma^{i}_{j} R_{\mu\nu} = ........R_{ij} where: \mu, \nu, \rho = (D + 1) dimensional curved spacetime indices R_{ij} = \Lambda_{D} g_{ij} \Lambda_{D} = cosmological constant
  20. darida

    Christoffel Symbol

    Ok I fix them:
  21. 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}?
  22. 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...
  23. darida

    Poisson Bracket

    Thank you!
  24. 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...
  25. 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
  26. 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...
  27. darida

    Hawking Mass in Schwarzschild Spacetime

    Ahh thank you so much, now I get it!
  28. darida

    Hawking Mass in Schwarzschild Spacetime

    I've checked it: |n^l|=g_{μ\nu}n^{μ}n^{\nu} |n^l|=g_{00}n^{0}n^{0}+g_{11}n^{1}n^{1}+g_{22}n^{2}n^{2}+g_{33}n^{3}n^{3} |n^l|=0+g_{11}n^{1}n^{1}+0+0 |n^l|=g_{11}n^{1}n^{1} |n^l|=\frac{r^2}{r^2-2Mr+q^2}(1)(1) |n^l|=\frac{r^2}{r^2-2Mr+q^2}
  29. darida

    Hawking Mass in Schwarzschild Spacetime

    Now with the same method I try to prove that the Hawking Mass in Reissner-Nordstrom spacetime is equal to (M-\frac{q^2}{2r}), but the result I got doesn't agree with it. Christoffel Symbols: \Gamma^{0}_{01}=\Gamma^{0}_{10}=-\frac{q^{2}+Mr}{r(r^{2}-2Mr+q^{2})}...
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