Would the vast and seemingly diffuse clouds of dark matter floating around our galaxy (and most others) absorb gravitational waves? Is this perhaps why we haven't detected any yet?
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It is very difficult to detect gravitational waves because gravity is such a weak force. For the same reason though it's very difficult to dampen gravitational waves. We know that dark matter behaves similar to 'normal matter' for what its equation of state and coupling is concerned, so it won't be able to absorb gravitational waves any more efficiently, which is essentially not at all. (Also: The expression 'gravity waves' describes a phenomenon seen in cloud formation and has nothing to do with General Relativity as Google will tell you.) |
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To start this discussion I present a few facts about gravity waves. A weak linear gravity wave is a perturbation on a background metric $\eta_{ab}$ with the total metric $$ g_{ab}~=~\eta_{ab}~+~h_{ab}. $$ The Ricci curvature to first order is $$ R_{ab}~=~{1\over 2}\Big(\partial_c\partial_a{h^c}_b~+~\partial_c\partial_b{h^c}_a~-~\partial_a\partial_bh~-~\partial_c\partial^ch_{ab}\Big). $$ The harmonic gauge $\partial_c{h^c}_a~=~1/2\partial_a h$ gives the Einstein field equation $$ \partial^c\partial_ch_{ab}~-~\frac{1}{2}\eta_{ab}\partial^c\partial_ch~=~\frac{16\pi G}{c^4}T_{ab}, $$ for the traceless metric term ${\bar h}_{ab}~=~h_{ab}~-~(1/2)\eta_{ab}h$ with the simple wave equation $$ \partial^c\partial_c{\bar h}_{ab}~=~\frac{16\pi G}{c^4}T_{ab}. $$ This gravity wave interacts with a set of test masses by inducing a quadrupolar motion. Let us suppose we have two such masses. These masses are on independent geodesics which will deviate from each other according to the variation of a vector connecting the masses $x^a$ by the equation $$ \frac{d^2x^a}{ds^2}~=~{R^a}_{bcd}X^cU^bU^d. $$ For weak gravity we can set $U^b~\simeq~(1,~0,~0,~0)$, a pure time directed 4-vector and the geodesic deviation equation is approximately $$ \frac{d^2x^a}{ds^2}~\simeq~{R^a}_bX^b. $$ One can then as an exercise input the Ricci curvature into this equation. Now let us assume there is a connecting spring between the two masses so that $$ \frac{D^2x^a}{ds^2}~\simeq~\frac{d^2x^a}{ds^2}~-~{R^a}_bX^b, $$ where now the equation describes a deviation between two nongeodesic moving particles. Since the perturbing force is a spring we then have $D^2x^a/ds^2~=~-kx^a$, which is just the spring equation familiar from Newton’s second law of motion. To address this question about dark matter interacting with gravity waves we think of this spring as the mutual interaction between particles. The spring constant for DM is very small, for DM is extremely weakly interacting. The result is that any heat which might be generated by gravity waves, $E~=~(3/2)\langle kx^2\rangle$ $=~kT$ will be very small. So using DM as a way of detecting gravity waves will likely prove to be frustrating. |
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