Gravitational Waves
Yanbei Chen
The following is a link to the full pdf presentation.
After Yanbei gave his talk, there was a question as to why the graviton has spin 2. Yanbei has kindly provided, in “simple terms”, the following explanation:
* In field theory, the value of angular momentum J is an indicator of how a field transforms when we apply a spatial rotation. J itself has three components, (Jx, Jy, Jz). Although a general field does not have any particular feature when we apply rotations, it can always be decomposed into terms, each with a distinctive transformation feature.
– For example, a scalar field, i.e., a function F(x,y,z), can be decomposed into a sum of f_lm(r)*Y_lm(theta,phi), where r, theta, phi are the spherical polar coordinates, and Y_lm are the spherical harmonics. The spherical harmonics transform with distinctive patterns under rotation. Y_lm has angular momentum of l, and z- component angular momentum of m. Here l and m are integers, with |m| <=l.
– A simpler example is that in two dimensions, F(x,y) can always be written as a sum over f_m(r)*exp(i*m*phi), where m will be the corresponding z-direction angular momentum.
– As a special case, in both of the above two examples, if m=1, then the function returns to its original value when we rotate around the z axis for 360 degrees. For a general m > 1, the function returns to its value after 360/m degrees.
* The above examples are for “orbital angular momentum”, where we have a scalar function of (x,y,z), and we would like to classify how it transforms under rotation. These fields all have “spin zero”. Spin is defined for fields that have multiple components — for which we would like to understand how these components transform when we apply rotations.
– Vector fields are the simplest multiple-component fields. For them, if we rotate along any axis for 360 degrees, they become themselves. This indicates that they have spin 1. Because electromagnetic fields are vector fields, they have spin 1.
– For most general tensor fields, spins can be larger than 1. Gravitons are quanta for gravitational waves, which are represented by perturbations to the metric tensor. Since the metric tensor is second rank and symmetric (essentially a quadratic form), it is invariant after rotation of 180 degrees. This means it should have spin of 2. [Under more careful considerations, one has to use the symmetric, and trace-free part of the metric perturbation.]
* For the most general tensor field, it must be characterized by both L and S.
