## Sunday, March 6, 2016

### Properties of a charged rotating sphere in Maxwell-Chern-Simons theory

I calculate some properties of a charged rotating sphere in five dimensional Maxwell-Chern-Simons theory.
The set up
Charge is glued uniformly on a sphere, which is then made to rotate. If I use the coordinates \begin{align*} x_1 & = r \sin\theta\ \cos\phi\\ x_2 & = r \sin\theta\ \sin\phi\\ x_3 & = r \cos\theta\ \cos\psi\\ x_4 & = r \cos\theta\ \sin\psi \end{align*} then the sphere is the set of points $x_1^2 + x_2^2 + x_3^2 + x_4^2 =R^2$ and the source is \begin{equation*} J^{\mu}\partial_{\mu} = \rho \delta(r - R) \left( \partial_t + \omega \partial_{\phi} + \omega \partial_{\psi} \right) \end{equation*} with $\omega$ the angular frequency of the rotation and $\rho$ the charge density on the sphere. The total charge on the sphere is thus $q = 2 \pi^2 R^3 \rho$. More information about the set up can be found in a previous post.

Gauge potential
The gauge potential is \begin{equation*} A = f(r) dt + g(r) \left( \omega \sin^2\theta d \phi + \omega \cos^2\theta d \psi \right) \end{equation*} I do not know an analytical formula for $f(r)$ and $g(r)$, but I can write these functions as a series in the Chern-Simons parameter $\lambda$.

If $r < R$ then \begin{align*} g(r) =&\frac{1}{4} r^2 R \rho -\frac{1}{8} r^2 R^2 \lambda \rho ^2+\frac{3}{160} \lambda ^2 \rho ^3 \left(5 r^2 R^3+5 r^4 R^3 \omega ^2-9 r^2 R^5 \omega ^2\right)-\frac{3}{320} \lambda ^3 \rho ^4 \left(8 r^2 R^4+15 r^4 R^4 \omega ^2-33 r^2 R^6 \omega ^2\right)\\ &+\frac{3 \lambda ^4 \rho ^5 \left(2912 r^2 R^5+8400 r^4 R^5 \omega ^2-21648 r^2 R^7 \omega ^2+1890 r^6 R^5 \omega ^4-9072 r^4 R^7 \omega ^4+11151 r^2 R^9 \omega ^4\right)}{143360}\\ &-\frac{3 \lambda ^5 \rho ^6 \left(11880 r^2 R^6+48160 r^4 R^6 \omega ^2-142080 r^2 R^8 \omega ^2+23625 r^6 R^6 \omega ^4-128520 r^4 R^8 \omega ^4+181608 r^2 R^{10} \omega ^4\right)}{716800}\\ &+O\left(\lambda ^6\right) \end{align*} If $r > R$ then \begin{align*} g(r) =&\frac{\rho R^5}{4 r^2}-\frac{\lambda \left(\rho ^2 \left(3 r^2 R^6-2 R^8\right)\right)}{8 r^4}-\frac{3 \lambda ^2 \left(\rho ^3 \left(5 r^6 R^9 \omega ^2-20 r^6 R^7+20 r^4 R^9-5 r^2 R^{11}+R^{15} \left(-\omega ^2\right)\right)\right)}{160 r^8}\\ &+\frac{3 \lambda ^3 \rho ^4 \left(33 r^8 R^{10} \omega ^2-35 r^8 R^8-10 r^6 R^{12} \omega ^2+40 r^6 R^{10}-15 r^4 R^{12}-9 r^2 R^{16} \omega ^2+2 r^2 R^{14}+4 R^{18} \omega ^2\right)}{320 r^{10}}\\ &+O\left(\lambda ^4\right) \end{align*}
I write $h(r) = f'(r)$. If $r < R$ then \begin{align*} h(r) =&\frac{3}{8} r R^2 \lambda \rho ^2 \omega ^2-\frac{3}{8} r R^3 \lambda ^2 \rho ^3 \omega ^2+\frac{3}{160} \lambda ^3 \rho ^4 \left(20 r R^4 \omega ^2+15 r^3 R^4 \omega ^4-27 r R^6 \omega ^4\right)\\ &-\frac{9}{320} \lambda ^4 \rho ^5 \left(13 r R^5 \omega ^2+20 r^3 R^5 \omega ^4-42 r R^7 \omega ^4\right)\\ &+\frac{27 \lambda ^5 \rho ^6 \left(9240 r R^6 \omega ^2+22400 r^3 R^6 \omega ^4-53440 r R^8 \omega ^4+4550 r^5 R^6 \omega ^6-20160 r^3 R^8 \omega ^6+23121 r R^{10} \omega ^6\right)}{716800}\\ &+O\left(\lambda ^6\right) \end{align*} If $r > R$ then \begin{align*} h(r) =& \frac{R^3 \rho }{r^3}+\frac{3 R^{10} \lambda \rho ^2 \omega ^2}{8 r^7}-\frac{3 R^{11} \lambda ^2 \rho ^3 \left(3 r^2 \omega ^2-2 R^2 \omega ^2\right)}{8 r^9}-\frac{3 R^{12} \lambda ^3 \rho ^4 \left(-105 r^6 \omega ^2+120 r^4 R^2 \omega ^2-35 r^2 R^4 \omega ^2+15 r^6 R^2 \omega ^4-3 R^8 \omega ^4\right)}{160 r^{13}}\\ &+\frac{9 R^{13} \lambda ^4 \rho ^5 \left(-95 r^8 \omega ^2+140 r^6 R^2 \omega ^2-70 r^4 R^4 \omega ^2+12 r^2 R^6 \omega ^2+48 r^8 R^2 \omega ^4-20 r^6 R^4 \omega ^4-12 r^2 R^8 \omega ^4+6 R^{10} \omega ^4\right)}{320 r^{15}}\\ &+O\left(\lambda ^5\right) \end{align*}

The momentum of the electromagnetic field
The momentum of the electromagnetic field is defined by \begin{equation*} P^{\mu} = \int d^4 x \ T^{0 \mu} \end{equation*} Here, $\mu$ are Euclidean indices and $T^{\mu\nu}$ is the stress tensor \begin{equation*} T_{\mu \nu} = - F_{\mu}^{\ \ \beta} F_{\beta \nu} - \frac{1}{4} g_{\mu \nu} F^2 \end{equation*} A straightforward integration gives \begin{align*} P^0 = E =&\left(\frac{q^2}{8 \pi ^2 R^2}+\frac{q^2 \omega ^2}{16 \pi ^2}\right)-\frac{\left(q^3 \omega ^2\right) \lambda }{64 \left(\pi ^4 R^2\right)}+\left(\frac{3 q^4 \omega ^2}{512 \pi ^6 R^4}-\frac{9 q^4 \omega ^4}{1280 \pi ^6 R^2}\right) \lambda ^2\\ &+\left(-\frac{3 q^5 \omega ^2}{1280 \pi ^8 R^6}+\frac{81 q^5 \omega ^4}{10240 \pi ^8 R^4}\right) \lambda ^3+\left(\frac{39 q^6 \omega ^2}{40960 \pi ^{10} R^8}-\frac{1863 q^6 \omega ^4}{286720 \pi ^{10} R^6}+\frac{567 q^6 \omega ^6}{262144 \pi ^{10} R^4}\right) \lambda ^4+O(\lambda )^5\end{align*} The other components $P^i$ are zero.

The angular momentum of the electromagnetic field
The angular momentum of the electromagnetic field is defined by \begin{equation*} L^{\mu\nu} = \int d^3 x\ x^{\mu} \ T^{0\nu} - \mu\leftrightarrow\nu \end{equation*} A straightforward integration gives \begin{align*} L_{12} =& - L_{21} = L_{34} = - L_{43}\\ =&\frac{q^2 \omega }{32 \pi ^2}-\frac{\left(q^3 \omega \right) \lambda }{128 \left(\pi ^4 R^2\right)}+\left(\frac{3 q^4 \omega }{1024 \pi ^6 R^4}-\frac{3 q^4 \omega ^3}{1280 \pi ^6 R^2}\right) \lambda ^2\\ &+\left(-\frac{3 q^5 \omega }{2560 \pi ^8 R^6}+\frac{27 q^5 \omega ^3}{10240 \pi ^8 R^4}\right) \lambda ^3+\left(\frac{39 q^6 \omega }{81920 \pi ^{10} R^8}-\frac{621 q^6 \omega ^3}{286720 \pi ^{10} R^6}+\frac{1701 q^6 \omega ^5}{2621440 \pi ^{10} R^4}\right) \lambda ^4+O(\lambda)^5 \end{align*} The other components are zero.

Mathematica code
I write \begin{equation*} g(r) = \sum_{k,l} a_{kl}\ r^l \quad\text{for}\quad r \le R \end{equation*} and \begin{equation*} g(r) = \sum_{k,l} b_{kl}\ r^l \quad\text{for}\quad r \ge R \end{equation*} The following code calculates the coefficients $a_{kl}$ and $b_{kl}$
 Clear[a, b, gin, \[Rho], R, \[Omega], r, gout] 
a[0, 2] = \[Rho] R / 4 ;
a[0, l_] /; l != 2 := 0
a[k_, l_] /; l <= 1 := 0
a[k_, l_] /; (l >= k + 3 ) := 0
a[k_, l_] /; OddQ[l] := 0

a[k_, l_] /; l != 2 :=a[k, l] = 72 \[Omega]^2/(l^2 - 4) Module[{k1, k2, k3, l1, l2, l3}, Sum[k3 = k - 2 - k1 - k2; l3 = l + 2 - l1 - l2; a[k1, l1] a[k2, l2] a[k3, l3], {l1, 2, l + 2, 2}, {l2, 2, l + 2, 2}, {k1, 0, k - 2}, {k2, 0, k - 2}]]; 

b[0, 2] = \[Rho] R^5 / 4;
b[0, l_] /; l != 2 := 0
b[k_, l_] /; l <= 1 := 0
b[k_, l_] /; (l >= 3 k + 3 ) := 0
b[k_, l_] /; OddQ[l] := 0

b[k_, l_] /; l != 2 := b[k, l] = 1/(l^2 - 4) ( 12 b[k - 1, l - 2] R^3 \[Rho] + 72 \[Omega]^2 Module[{k1, k2, k3, l1, l2, l3}, Sum[ k3 = k - 2 - k1 - k2; l3 = l - 2 - l1 - l2; b[k1, l1] b[k2, l2] b[k3, l3], {l1, 0, l - 2, 2}, {l2, 0, l - 2, 2}, {k1, 0, k - 2}, {k2, 0, k - 2}]]); 

a[k_, 2] /; k != 0 := 1 / (4 R^2 ) (Sum[a[k, l] R^l ( - 2 - l), {l, 4, k + 2, 2}] + Sum[b[k, l]/ R^l ( 2 - l), {l, 4, 3 k + 2, 2}]) // Expand 

b[k_, 2] /; k != 0 := R^2/4 (Sum[a[k, l] R^l (-l + 2), {l, 4, k + 2, 2}] + Sum[b[k, l]/ R^l ( -l - 2), {l, 4, 3 k + 2, 2}]) // Expand 

 For[m = 0, m < 8, m++, 
ginfunctie[r_, R_, \[Rho]_, \[Lambda]_, \[Omega]_, m] = Sum[a[k, l] \[Lambda]^k r^l, {k, 0, m}, {l, 0, k + 2, 2}]; 
hinfunctie[r_, R_, \[Rho]_, \[Lambda]_, \[Omega]_, m] = 6 \[Lambda] \[Omega]^2 ginfunctie[r, R, \[Rho], \[Lambda], \[Omega], m]^2/r^3;  
goutfunctie[r_, R_, \[Rho]_, \[Lambda]_, \[Omega]_, m] = Sum[b[k, l] \[Lambda]^k/ r^l, {k, 0, m}, {l, 0, 3 k + 2, 2}]; 
houtfunctie[r_, R_, \[Rho]_, \[Lambda]_, \[Omega]_, m] = R^3 \[Rho]/r^3 + 6 \[Lambda] \[Omega]^2 goutfunctie[r, R, \[Rho], \[Lambda], \[Omega], m]^2/r^3; ]

Remark
I have written this post because I was not able to solve the problem analytically and perhaps these expansions are useful.