Understanding GR Perturbations: J. Albert's Guide

[rawsilk]

Hi all,

I have recently completed two courses in general relativity and am well versed in things like the ray chaudhuri equation, tetrads, etc. I had to give a 2hr talk on gravitational radiation to my class and so understand GWs at some relatively respectable level. What I am inquiring about is the formulation of the perturbation metric in a precise sense. What literature normally says is, $$g_{ab} = \eta_{ab} + h_{ab},$$ where $$h_{ab}\ll \eta_{ab}$$ for non-zero elements. What I want to know is whether there is a more precise definition of "small". More to the point is there a more fundamental point of view for using perturbation theory and when it is valid in differential geometry or math in general. Feel free to use big words ;)

J. Albert

 

[dextercioby]

Actually this is not entirely true. We usually express the perturbation from a known scenario (metric in GR, fundamental states in QM) in terms of a parameter which contains the <smallness>. So the h ab is small but the radiation field can be <normal>.

$$g_{\mu_\nu} = \eta_{\mu\nu} + \lambda h_{\mu\nu}$$.

so that $|\lambda| <<1$ and the components of h have the same order of magnitude as the components of h.

 

[rawsilk]

I have seen that form too in quantum perturbation theory and Gr lit. Thanks for your input, I'm just curious if there are any other constraints on the perturbation. Of course it must live in the same space as the background metric and can only be specified after choosing a coord map. But certainly there must be other constraints on when it may be used or else there wouldn't exist things like second order perturbation theories. It arises from a Taylor approximation to be sure.

 

[NanakiXIII]

What the smallness of the metric perturbation means physically is somewhat application-dependent. If you're doing a post-Newtonian expansion, that's an expansion in orders of $v$, the typical velocities in your gravitational system. If you're dealing with gravitational radiation far away from any source, you can do a post-Minkowskian expansion, which means you expand in $G$, the gravitational constant. When calculating actual waveforms from e.g. inspiraling compact binaries, what is done is taking these two expansions, further expanding them into multipoles and then matching the two (the post-Newtonian expansion only works near the source and the post-Minkowskian one only far away, but there is some overlap where you can match them.)

 

[sardar]

Hello J. Albert,

Thank you for reaching out and sharing your interest in general relativity and gravitational radiation. It's great to hear that you have completed two courses in the subject and are well-versed in concepts like the ray chaudhuri equation and tetrads.

In regards to your inquiry about the formulation of the perturbation metric, you are correct that the common definition is $$g_{ab} = \eta_{ab} + h_{ab}$$ where $$h_{ab}$$ is considered small compared to $$\eta_{ab}$$. However, the precise definition of "small" can vary depending on the context and application. In general, "small" refers to a quantity that is significantly smaller than another quantity, but there is no specific threshold that applies universally. In the context of perturbation theory, "small" typically refers to terms that can be neglected without significantly affecting the overall solution.

In terms of a more fundamental point of view for using perturbation theory, it can be seen as a method for approximating solutions to complex systems by breaking them down into simpler, more manageable parts. In the case of general relativity, perturbation theory can be used to study small deviations from a background metric, such as in the case of gravitational waves.

In differential geometry and mathematics in general, perturbation theory is a powerful tool that allows us to approximate solutions to nonlinear systems. It is based on the idea of linearizing a system around a known solution and then using small perturbations to improve the approximation. The validity of perturbation theory depends on the specific problem at hand and the size of the perturbations involved. In general, it is valid when the perturbations are small enough to not significantly affect the overall solution.

I hope this helps provide some insight into the formulation and application of perturbation theory in general relativity. If you have any further questions, please don't hesitate to ask. Best of luck with your studies and future research.

Sincerely,