Gravitational Waves: Difficulty Level of Study?

[kent davidge]

I just want to know, is the study (say, at an elementary - to intermediate level) of gravitational waves much more difficult than the electromagnetic waves?

 

[kent davidge]

I know I asked a similar question back on april, but this time I would like to have a comparison between electromagnetic and gravitational waves...

 

[berkeman]

I just want to know, is the study (say, at an elementary - to intermediate level) of gravitational waves much more difficult than the electromagnetic waves?

I know I asked a similar question back on april, but this time I would like to have a comparison between electromagnetic and gravitational waves...

What can you tell us about how LIGO works? Can you think of a way of making a mini-LIGO? What might be some of the issues with doing that?

 

[Orodruin]

It is unclear to me if you are asking about the difficulties of studying to understand gravitational waves from a theoretical perspective or about the practical difficulties of experimentally studying the gravitational wave phenomenon.

 

[kent davidge]

about the difficulties of studying to understand gravitational waves from a theoretical perspective

that ⬆

 

[vanhees71]

The starting point are the linearized Einstein equations for the free gravitational field. It's only a bit more complicated than Maxwell's equations, because the gravitational field is a massless rank-2 tensor field while the em. field is a massless vector (rank-1 tensor) field.

The treatment is analogous in both cases, using gauge invariance to fix the gauge to make life as easy as possible. In the em. case the four-vector potential $A_{\mu}$ is only determined up to a gradient, i.e., $A_{\mu}$ and $A_{\mu}'=A_{\mu}+\partial_{\mu} \chi$ with an arbitrary scalar field $\chi$ are physically equivalent. To fix the gauge completely for free fields you can impose the gauge conditions $\partial_{\mu} A^{\mu}=0$ and also $A^0=0$ (radiation gauge). That leaves you with two independent field-degrees of freedom, which physically have the meaning of the two independent polarization states (e.g., you can choose the helicity basis, i.e., left- and right-circular polarized modes of the em. field to describe all possible polarization states of em. waves as superposition of those).

For the linearlized Einstein equations the gauge invariance is nothing else than the general covariance under all transformations of the coordinates. The symmetric 2nd-rank tensor field has 10 components, and one can show that fixing the gauge completely also in this case leaves you with only two independent "polarization degrees of freedom", which you can again choose as the helicity eigenmodes.

This follows also from the general representation theory of the Poincare group: Massless irreducible fields have only two indepenent "polarization degrees of freedom", with a convenient basis being given by the helicity eigenstates. The helicity can only be $0$ (scalar fields), $\pm 1$ (vector field like the em. field), $\pm 2$ (rank-2 tensor field like the gravitational field), etc.

 

[kent davidge]

Thanks, @vanhees71 . That was a good overview. I wonder if you wrote some manuscript containing this subject?

 

[vanhees71]

Not yet ;-)). I think the best source to start with GR is Landau&Lifshitz Vol. 2.