# Search results

1. ### 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. ### 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. ### 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. ### Components of The Electromagnetic Field Strength Tensor

Never mind, just found the answer
5. ### 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...

Thank you!
7. ### 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...
8. ### 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?
9. ### Reissner Nordstrom Metric

Ah thank you, that's so simple >_<
10. ### 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)?
11. ### 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?
12. ### Stability of Hawking Mass

For example: Source: page 8
13. ### 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?
14. ### Stability of Hawking Mass

So, what should I do now?
15. ### Stability of Hawking Mass

What is stability of Hawking Mass and how to calculate it? Any references will be appreciated. Thanks
16. ### 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:
17. ### 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 =...

Oh okay
19. ### Christoffel Symbol

It's not in any articles/books. I just met someone who told me that :(
20. ### 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
21. ### Christoffel Symbol

Ok I fix them:
22. ### 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}?
23. ### 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...

Thank you!
25. ### 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...
26. ### Generating Function in Physics

Could anybody give me some examples of generating function in physics, it's application, and it's use? Thank you
27. ### 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...
28. ### Hawking Mass in Schwarzschild Spacetime

Ahh thank you so much, now I get it!
29. ### 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}
30. ### 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})}...