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For a wave of the form, say, $$ ds^2 = -dt^2 + [1 - A\sin(\omega(t-z))]dx^2 + [1 + A\sin(\omega(t-z))]dy^2 + dz^2$$

how should one orient the detector (that works by monitoring the distance between two free flying masses) to register the largest possible response?

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    $\begingroup$ With just two masses it is almost impossible. You typically need three (or more) freefalling masses as in LIGO or LISA. $\endgroup$ – mmeent Jan 23 '18 at 15:15
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    $\begingroup$ @mmeent That's only in practice. In principle — which is what we deal with here — there are no other sources of signal in the universe, so a two-mass system is perfectly fine. Save your practical details for engineering.SE! ;) $\endgroup$ – Mike Jan 23 '18 at 18:16
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The question is actually slightly harder after the intial edit. LIGO has a well known antenna pattern that shows that it is most sensitive to gravitational waves coming from directly above or below, so you would first have to put LIGO in the $x$-$y$ plane, because the wave is moving in the $z$ direction. Moreover, this particular wave also has greatest "contrast" between the $x$ and $y$ axes, so you would just need to align the two legs of LIGO with the $x$ and $y$ axes. More specifically, this is a "plus-polarized" wave, which means that it doesn't change any lengths along the diagonals — which means that if LIGO's legs were along the diagonals it wouldn't see anything. Also, the sensitivity is the same whether you're pointing up or down, or whether one leg is aligned with $+x$ or $-x$, etc. So there are actually eight "optimal" orientations.

After the edit, however, you're not talking about LIGO any more, but about a hypothetical detector that has just one "leg". In this case, the antenna pattern is clearly rotationally symmetric about that leg's axis, with a maximum orthogonal to the leg. So you would want to place the detector in the $x$-$y$ plane — and again along either the $x$ or $y$ axis.


Antenna patterns arise because of how these detectors work. (Regular E&M antennas also have antenna patterns.) The hypothetical single-leg detector is, presumably, only sensitive to changes in its length — as opposed to its width. The gravitational waves here are moving along the $z$ direction, but are only changing lengths of things along the $x$ and $y$ directions. So if this detector's single leg were along the $z$ axis, it just wouldn't register the waves. As you tilt the detector down along the $x$ axis, for example, the gravitational waves would have a larger "component" along the detector, so the detector becomes more sensitive. Similarly along the $y$ axis — the relative sign difference just changes the phase of the effect, not its size. However, you can also see that a detector in the $x$-$y$ plane that's at some 45$^\circ$ angle will be stretched by the $x$ component of the metric just as much as it's squeezed by the $y$ component (or vice versa), resulting in no net change to the length of the leg.

In LIGO, there are two legs and they measure the difference in the length-changes (strain) between the two legs. The reason for that is just because there are so many things that can cause a change in the lengths of the legs, but fewer things that change the difference between the lengths of the legs. Gravitational waves will change that difference, so that's (one of the reasons) why interferometers are used. The analysis of the antenna pattern is similar to the above, but complicated because you need to keep track of the two legs and make sure you only measure the differences.

I don't know of any really good references for this, but this note from LIGO works through the details. For the single-leg detector, just set $\vec{v}=0$ in equation (2.1), for example.

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  • $\begingroup$ Can you explain why we have to align the two legs with the $x$ and $y$ axes? I've never heard about antenna patterns. If I could show this wave only has plus polarization, would that imply this orientation? $\endgroup$ – user174295 Jan 23 '18 at 15:58
  • $\begingroup$ I've updated my answer to explain it a bit. Specifically: yes, since this wave only has a plus polarization, the optimal orientation of your single-leg detector would necessarily be along the $x$ or $y$ axes. $\endgroup$ – Mike Jan 23 '18 at 18:14

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