We all know that GR needs modification at the microscopic scale but there are some attempts to modify GR in the classical regime as well, known by the name of "Modified Gravity". As far as I understand, these attempts can be possibly justified in the classical mode of thinking for the reason that there is some amount of guess-work involved in deriving the Einstein field equations. Thus, one might want to try out another internally consistent guess and get a modified classical theory of gravity.

Now, according to Weinberg's book Gravitation and Cosmology (Page 152), the reason for the guess-work involved in determining the Einstein field equations is because we need to apply the Equivalence Principle (or the Principle of General Covariance) on the behavior of the gravitational fields themselves if we want to determine the field equations. In order to do so, we go to a frame in which the field is extremely feeble and if we know how the field behaves in such a frame, we will transform the equations to a general frame and we will have the general form of field equations. But, sadly, we don't have enough information about the behavior of extremely feeble gravitational fields apriori. We only have Newton's law which holds for the static kind of feeble fields but not for generic (but yet feeble) fields. This is the reason why we first need to guess what would be the equations governing the feeble fields and then we generalize them by performing a general coordinate transformation. Now, this guess-work can be eliminated and the classical theory of gravity can be described without any indeterminacy whatsoever (or so it appears to me) if we know the (special) relativistically covariant law that governs generic feeble fields.

This excites me to ask whether we have gathered enough experimental data about the feeble fields via studying the feeble gravitational waves at the LIGO. If such is the case then we can determine the classical theory of gravity with cent percent certainty. It goes without saying that such a step will put an end to the possibility of a debate over whether there is dark matter or not. So, my question is, do we have any possibility of gathering enough experimental data about the feeble fields via studying the feeble gravitational waves at the LIGO?

I feel that it might be of a general interest to directly read what Weinberg has said in the context of whether the field equations for the strong fields can be easily determined using a general coordinate transformation if we know the truly precise covariant equations determining the weak fields.

Gravitation & Cosmology by Weinberg, Chapter 7, Section 1

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    $\begingroup$ @RobJeffries Isn't GR certain to break-down at small enough distance scales? We need a quantum theory at that scale, right? $\endgroup$ – Feynmans Out for Grumpy Cat Oct 17 '17 at 21:33
  • $\begingroup$ The LIGO detections test strong gravitational fields, not weak ones. I guess I just don't understand the question (or what connection GR has with a wavelength of light). $\endgroup$ – Rob Jeffries Oct 17 '17 at 23:08
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    $\begingroup$ @RobJeffries No, it has nothing to do with the wavelength of light. I think it is a standard terminology to use "UV" to denote the microscopic physics and "Infrared" to denote the macroscopic physics. I understand that LIGO detects gravitational fields coming from regions with intense gravitational fields but the fields that it actually detects are extremely weak. If it were strong fields that the LIGO were detecting then its job would have been pretty easy. Correct me if I am wrong. Thanks for your interest and time. $\endgroup$ – Feynmans Out for Grumpy Cat Oct 17 '17 at 23:13
  • $\begingroup$ @RobJeffries In order to avoid possible confusion, I have edited the question. Have a look. $\endgroup$ – Feynmans Out for Grumpy Cat Oct 18 '17 at 3:52
  • $\begingroup$ These may be of interest. arxiv.org/abs/1710.06168 arxiv.org/abs/1710.06394 $\endgroup$ – Rob Jeffries Oct 18 '17 at 21:35

Rather than the IR effects of gravity proving GR through the equivalence principle (and whether strengthened in the kilonova detection) I argue below that the IR argument seems not convincing, but the speed of light of gravity is more convincing.

Seems to me that your IR question reduces to whether weak, or linearized (since weak) gravitational fields, obey the principle of equivalence. Different versions of the equivalence principle but to go with the simplest is whether inertial mass and gravitational mass are the same. That is, whether these weak gravitational waves affect different masses the same way.

The fact is that we observe those GWs affecting the space shrinking and dilating in the LIGO/VIRGO detectors the same amounts, since we get amplitudes on all 3 pairs consistent with the source distance in the case of the kilonova detection at about 40 Mpsec. So it's changing space the same way. You'll probably ask for the accuracies on equal inertial and gravitational masses, and clearly you'd have to calculate.

It also seems to me that since the principle of equivalence holds pretty accurately for static or pseudostatic fields, the IR limit is already getting probed that way, anyway, adding some continuity arguments from quantum field theory to cover the step from pseudostatic to IR.

Still, your statement that if the principle is true in weak fields then easily extrapolate to strong fields by invoking covariance is quite a jump. It is really for purposes of testing what happens in strong gravity that GWs from black holes, which confirm some of the predicted black hole properties and GW creation predictions of GR, are of great interest.

The parametrized post Newtonian approximations may have something to say on this also that perhaps others can relate, and it applies to weak gravity. The accuracies and which tests are needed to eliminate all possible explanations of dark matter with modified gravity are covered in the literature, and it's not clear to me that the IR accuracy of the principle of equivalence improving would change things.

More importantly, the kilonova detection proved that the speed of GWs (gravitational waves) was the same as that of light to within $10^{-15}$. This may put some constraints in modified gravity theories, and it is the first time that that equality has been shown in measurements. BTW! The value comes from the approx 2 seconds time difference between the GW and gamma ray detection times, yielding about one part in 65 million years. That's approx $10^7$ seconds in one year, and approx. $10^8$ years to get the 1 part in $10^{15}$ accuracy number. it'll be interesting to see the calculations as to how much more constrained are the modified gravity theories. Just recently some articles were talking about this, if found, placing additional limits on modified gravity. See eg https://arstechnica.com/science/2017/02/theoretical-battle-dark-energy-vs-modified-gravity/. See that the kilonova detection measurements places more limits in modified gravity as an explanation for dark energy at https://phys.org/news/2017-02-quest-riddle-einstein-theory.html

It also seems to place strong limits on Verlinde's modified gravity theory for dark matter arising out of interactions with dark energy. See https://phys.org/news/2017-02-quest-riddle-einstein-theory.html

It's be interesting to see a strict review of what modified gravity theories remain after kilonova.

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    $\begingroup$ "Still, your statement that if the principle is true in weak fields then easily extrapolate to strong fields by invoking covariance is quite a jump. " Yes, I don't understand that either: I assume that's the OP's rendering of Weinberg's quotation (which I don't have and will have to look up); I can't imagine Weinberg saying that. I also agree that the light vs gravity speed experimental result is simply stunning. One could argue that Hulse Taylor gave some indirect experimental confirmation, since the spindown rate depends on $c$, but I'm still saying "OMG" about the latest GWs. $\endgroup$ – WetSavannaAnimal Oct 19 '17 at 1:23
  • $\begingroup$ Thanks for your reply, I still need to think about whether it completely resolves my question or not. I will leave more detailed comments once I am done the thinking. In the meantime, I have added the snapshot of the exact words of Weinberg for I feel it might be of a general interest based the comment by @WetSavannaAnimalakaRodVance. $\endgroup$ – Feynmans Out for Grumpy Cat Oct 19 '17 at 20:44
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    $\begingroup$ @WetSavannaAnimalakaRodVance: I believe I remember this passage, and remember that the Weinberg GR book is written with the idea of GR as a classical field theory. the argument is that you start with perturbative gravity, and then look for the simplest possible theory with this perturbative limit (there is some other constraint, but I forget what it is). You end at GR, but I don't think you can rule out auxilary field theories like Brans-Dicke or TeVeS or modified Einstein's equation theories like $f(R)$ gravity or anything like that out via this method. $\endgroup$ – Jerry Schirmer Oct 19 '17 at 21:03

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