Spin-Weighted Spherical Harmonics

​
degree s
0
1
2
3
4
5
6
7
8
degree l
0
1
2
3
4
5
6
7
8
order m
0
Spin-weighted spherical harmonics are generalizations of standard spherical harmonics and like them are complex functions on the sphere. The multipole expansions of an electromagnetic field can be expressed in terms of spin-weighted spherical harmonics. They are particularly important in the theory of gravitational waves.

Details

For spin weight
s=0
, the spin-weighted spherical harmonics become identical to the spherical harmonics. The case of spin weight
s=2
is important for describing gravitational waves. In this Demonstration you can choose different values of the spin weight
s
to see the angular distribution in space for different
l
and
m
modes.
1) Introductory level reference:
http://en.wikipedia.org/wiki/Spin-weighted_spherical _harmonics
2) Intermediate level reference:
J. J. G. Scanio, "Spin-Weighted Spherical Harmonics and Electromagnetic Multipole Expansions," Am. J. Phys., 45(2), 1977 pp. 173–178.
3) Advanced level reference:
J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, "Spin-
s
Spherical Harmonics and
ð
," J. Math. Phys., 8(11), 1967 pp. 2155–2161.

External Links

Gravitational Wave (ScienceWorld)
Spherical Harmonic (Wolfram MathWorld)

Permanent Citation

Satya Mohapatra, Stephen Wolfram
​
​"Spin-Weighted Spherical Harmonics"​
​http://demonstrations.wolfram.com/SpinWeightedSphericalHarmonics/​
​Wolfram Demonstrations Project​
​Published: February 24, 2009