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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}...
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}...
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)...
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...
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...
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)?
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?
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?
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}?
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...
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...
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...
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})}...