# Is the LIGO detector acceptance different for gravitational and for non-gravitational waves?

The famous signals recorded by the LIGO observatories are characterized as observations of passing gravitational waves, where at each observatory site:

Consider on the other hand a non-gravitational wave by the following description:

all particles which make up the Solar system exhibit a (tiny but non-zero) common acceleration (for each particle in addition to its individual "peculiar" acceleration due to expected processes and interaction with the constituents of the Solar system):

$${\bf a}_{\text{common}}[ \, {\bf r}, t \, ] := 10^{-10} \, {\rm m / s^2} \times \text{Sin}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}) / c) \, ] \times {\bf e}_a,$$

where space vector ${\bf r}$ and duration $t$ denote separations in reference to a suitable (helio-central-symmetric, non-rotating) system,

${\bf e}_p$ denotes the unit vector in direction of propagation of the wave through this system; for definiteness ${\bf e}_p \cdot \Delta_{\text{sites}} {\bf r}$ shall be similar to the corresponding value determined for event GW150914,

${\bf e}_a$ denotes a unit vector which, for definiteness, shall be exactly the direction of one arm of the "L"-shaped interferometer (at least at one LIGO site); and thus perpendicular to the "pull of the weights" of the LIGO test masses (at least at one site),

and $f := f[ \, t - ({\bf e}_x \cdot {\bf r}) / c \, ]$ denotes the wave frequency (in the suitable system) which shall vary as a function of the argument similar to the varying frequency of event GW150914, i.e. increasing roughly from 35 Hz to 250 Hz in the course of 0.2 seconds; and zero before this "chirp" as well as afterwards.

Such a wave is non-gravitational in the sense that

• any particle of negligible "peculiar" acceleration is still accelerating with non-zero ${\bf a}_{\text{common}}$ and is therefore not "free-falling" but "disturbed by the wave", and

• any particle exhibiting appreciable "peculiar" acceleration in a direction perpendicular to ${\bf e}_a$ is still not "moving freely at least along direction ${\bf e}_a$", but it is forced to move along that direction, according to ${\bf a}_{\text{common}}$.

My questions:

Is the probability for the LIGO observatories to register repeated changing of intereferometer arm length differences and to accept a corresponding signal (rather than to attribute "noise") due to a non-gravitational wave as specified
comparable (say, within one order of magnitude) to the corresponding acceptance probability for gravitational waves like those observed as event GW150914 ?

Are there any setup components involved (such as, perhaps, suitable actuators, in feedback loops) or measurement techniques applicable in the LIGO observatories which are expected to suppress the acceptance for non-gravitational waves, in comparison to the acceptance for gravitational waves ?

Note 1: An order-of-magnitude estimate of separation change

For two (suitably selected) constituents of the Solar system, $P$ and $Q$, with

${\bf a}[ \, P, t \, ] = {\bf a}_{\text{common}}[ \, t, {\bf r}[ \, P, t \, ] \, ] = 10^{-10} \, {\rm m / s^2} \times \text{Sin}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, P, t \, ]) / c) \, ] \times {\bf e}_a = {\bf \ddot r}[ \, P, t \, ]$,

${\bf a}[ \, Q, t \, ] = {\bf a}_{\text{common}}[ \, t, {\bf r}[ \, Q, t \, ] \, ] = 10^{-10} \, {\rm m / s^2} \times \text{Sin}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \times {\bf e}_a = {\bf \ddot r}[ \, Q, t \, ]$,

${\bf \dot r}[ \, P, t \, ] = 10^{-10} \, {\rm m / s^2} \times \frac{-1}{(2 \, \pi \, f)} \times \text{Cos}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, P, t \, ]) / c) \, ] \times {\bf e}_a,$

${\bf \dot r}[ \, Q, t \, ] = 10^{-10} \, {\rm m / s^2} \times \frac{-1}{(2 \, \pi \, f)} \times \text{Cos}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \times {\bf e}_a,$

$\| {\bf r}[ \, Q, t \, ] - {\bf r}[ \, P, t \, ] \| \approx {\bf e}_p \cdot ({\bf r}[ \, Q, t \, ] - {\bf r}[ \, P, t \, ]) \approx 4~{\rm km}$,

${\bf e}_a \cdot ({\bf r}[ \, Q, t \, ] - {\bf r}[ \, P, t \, ]) =$ $10^{-10} \, {\rm m / s^2} \times \frac{1}{(2 \, \pi \, f)^2} \times \Big(\text{Sin}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] - \text{Sin}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, P, t \, ]) / c) \, ] \Big) \approx$ $10^{-10} \, {\rm m / s^2} \times \frac{1}{(2 \, \pi \, f)^2} \times \Big(\text{Sin}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \, (1 - \text{Cos}[ \, 2 \, \pi \, f \, 4~{\rm km} / c \, ]) + \text{Cos}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \, \text{Sin}[ \, 2 \, \pi \, f \, 4~{\rm km} / c \, ] \Big).$

Considering $f = 100~{\rm Hz}$ fixed, and thus $2 \, \pi \, f \, 4~{\rm km} / c \approx 0.008$ therefore

${\bf e}_a \cdot ({\bf r}[ \, Q, t \, ] - {\bf r}[ \, P, t \, ]) \approx$

$10^{-10} \, {\rm m / s^2} \times \frac{1}{(2 \, \pi \, f)^2} \times \Big(\text{Sin}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \, \frac{1}{2} \, (2 \, \pi \, f \, 4~{\rm km} / c)^2 + \text{Cos}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \, (2 \, \pi \, f \, 4~{\rm km} / c) \Big) \approx$

$10^{-10} \, {\rm m / s^2} \times \frac{1}{(2 \, \pi \, f)^2} \times \Big(\text{Cos}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \, (2 \, \pi \, f \, 4~{\rm km} / c) \Big) =$

$10^{-10} \, {\rm m / s^2} \times \frac{1}{(2 \, \pi \, 100~{\rm Hz})} \, 4~{\rm km} / c \times \text{Cos}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ] \approx$

$2 \times 10^{-18} {\rm m} \times \text{Cos}[ \, 2 \, \pi \, f \, (t - ({\bf e}_p \cdot {\bf r}[ \, Q, t \, ]) / c) \, ].$

Note 2: An exact expression of acceleration involving the prescribed ${\bf a}_{\text{common}}$

The acceleration of Solar system constituent $P$ being exposed to the prescribed non-gravitational wave shall be specificly

$${\bf a}[ \, P, t \, ] := {\bf a}_{\text{common}}[ \, t, {\rm r}[ \, P, t \, ] \, ] + {\bf a}_{\text{SM}}[ \, P, t, \text{ all constituents of the Solar system up to } t \, ],$$

where

"${\bf a}_{\text{SM}}[ \, P, t, \text{ all constituents of the Solar system up to } t \, ]$" denotes the acceleration component imposed on $P$ at $t$ by all (other) constituents of the Solar system exactly as expected according to the Standard Model of particle properties, and the corresponding distribution of masses, charges, and fields, for the exact trajectories of all these constituents up to $t$.

This takes account of non-zero values ${\bf a}_{\text{common}}$ being prescribed to all constituents even prior to $t$; the Solar system "being affected" generally as a whole, and continuously, by the prescribed non-gravitational wave; the actual "peculiar motions" of the constituents being not "just" the sum of the prescribed disturbing ${\bf a}_{\text{common}}$ in addition to "undisturbed motion as usual", but generally being far more complicated.

However, particles which are "usually in free-fall" in the Solar system, and for which (correspondingly) the expected standard model acceleration component is still as good as negligible, ${\bf a}_{\text{SM}} \approx 0$, even though the other constituents were disturbed as prescribed, their resulting actual acceleration is supposed to be found exactly as the prescribed disturbing ${\bf a}_{\text{common}}$.

• What would generate such a "non-gravitational wave"? – ACuriousMind Sep 13 '17 at 20:42
• @ACuriousMind: "What would generate such a "non-gravitational wave"?" -- Interesting question (some might consider asking it explicitly on this site); but orthogonal to my question above. – user12262 Sep 13 '17 at 21:09

If every particle in a system experiences this acceleration in common, irrespective of mass, then the dynamics of this acceleration can be described by a metric tensor and geodesics along it, making the situation you describe indistinguishable from the free-falling condition in some geometry. You are trying to induce a background geometry and are specifying vectors, but this is contrary to the spirit of GR, and there is no requirement in GR that spacetime be homogenous or isotropic (and in fact, they are defintiely not in the case of plane wave gravitational wave geometries)

This metric may not obey Einstein's equation, but you haven't really shown how it wouldn't obey Einstein's equation. Certainly LIGO couldn't distinguish the two situations, since they are the same, at least at a facile level.

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Sep 15 '17 at 20:06
• @Jerry Schirmer: I'm sad to see that our comments have just been erased, for all intents and puposes. Since your most recent comments here had apparently been concerned with the statement of my OP question and especially its Note 2, I'd like to encourage you to repeat these comments above, directly following my question. – user12262 Sep 15 '17 at 20:23
• @user12262: any way you cut it, $a_{common}$ is an acceleration that obeys the equivalence principle, and if I understand what you're saying at all, it's the ananomolous effect you're trying to describe, while the "peculiar" motions are ordinary physics. Your $a_{common}$ IS a force that obeys the equivalence principle. – Jerry Schirmer Sep 16 '17 at 14:47
• @user12262: the whole point of ligo is that the ends of the interfermometer are both locally free-falling, but experience global relative acceleration as the gravitational wave moves through. How is this any differen than that? It's not about the magnitude of the effect, free falling means you can pick a sufficently small spacetime region where the effects vanish. If you have a bigger effect, you choose a smaller spacetime region to make it vanish. It's a thing in the spirit of the $\delta$-$\epsilon$ proofs from calculus. – Jerry Schirmer Sep 18 '17 at 0:10
• @user12262 your semantics don't change the essence of this, and neither will adding a bunch of formulae or calculating numbers. Adding spurious accelerations and saying that they're "non-gravitational" is something general relativity already has the mechanisms to deal with. All you're doing is changing the waveform. – Jerry Schirmer Sep 18 '17 at 17:13

Any acceleration to the solar system can only be imposed by gravitational interactions (barring an incoming explosive event, i.e. non gravitational energy creating the acceleration, but this will be one off). In physics as we have described and recorded it there are just the four fundamental forces.

So the question, IMO, reduces to "are gravitational waves produced by the solar system a source of error on the waves detected by LIGO"?* This reduces to "what is the quadrupole moment of the gravitational setup of the solar system".*

Reading this link (26.2.2) leads me to say that any gravitational waves from the asymmetries of the solar system would be of such large wavelength that they would not interfere with the waves detected by the design of LIGO.

In conclusion

Non gravitational waves by acceleration would come from explosive events. These a) might destroy the whole solar system, so no need to detect them, or b) if they affect only the sun (example the sun explodes ) we would certainly detect it with normal astronomical instruments. In addition the spread of the disturbance will be by gravity, as there is no other force,and the gravitational fields of the solar system are too weak to generate wavelengths detectable by LIGO

• anna v: "barring an incoming explosive event, i.e. non gravitational energy creating the acceleration" -- I didn't mean to bar from consideration whatever is deemed consistent with the prescribed kinematics. "These a) might destroy the whole solar system" -- $10^{-10}~{\rm m/s^2} \times \text{Cos}[ \, 2 \, \pi \, 100~{\rm Hz} \, t \, ]$, over a dozen oscillation periods, with wavelength $3'000~{\rm km}$ ?? Hardly seems destructive.[contd.] – user12262 Sep 14 '17 at 18:21
• anna v: "Any acceleration to the solar system can only be imposed by gravitational interactions" -- No; quite the opposite. Let's not confuse "acceleration" with "coordinate acceleration". The peculiar motion of the Moon, for instance, is (as good as) "free-falling", i.e. not accelerated. Accordingly, $\bf a$ is not to be confused with $\bf \ddot r$. (I should spell this out in the OP). "there are just the four fundamental forces." -- I accept this (and even have to use it in editing the OP) for the description of (the peculiar motions of constituents of) the Solar system. [contd.] – user12262 Sep 14 '17 at 18:22
• Specificly I require: $${\bf a} \text{ of Solar system constituent } P, \text{ at } t :=$$ $${\bf a}_{\text{common}}[ \, t, {\rm r}[ \, P, t \, ] \, ] +$$ $${\bf a} \text{ on } P \text{ precisely according to the Standard Model "four forces" ... }$$ $$\text{ ... due to all (other) Solar system constituents, on their respective trajectories up to } t.$$ (Obviously this includes the constituents of the LIGO detectors: test masses, electrons in photo diodes, actuators ...). But I don't require (nor would I rule out) that ${\bf a}_{\text{common}}$ was "plausibly imposed by known physics". – user12262 Sep 14 '17 at 18:23
• your a_common HAS to be imposed by the known forces, otherwise this question is not for this site, which deals with mainstream physics for its explanations. Angels and leprechauns and magic in general are not within the scope of mainstream physics. – anna v Sep 14 '17 at 18:25
• It is only gravitational effects that can generate waves at cosmic distances ( except maybe impacts from a supernova explosion); my basic argument is that solar size changes in gravity will produce very large wavelength gravitational waves, outside the LIGO scope because the masses involved in the solar system are very small with respect to the merging of two black holes. – anna v Sep 14 '17 at 18:35