STRANGE MISCONCEPTIONS OF GENERAL RELATIVITY

G. ’t Hooft

Note - this page is not a weblog. I won’t post reactions from others, and the page won’t be growing to infinite length. It will be adjusted whenever I find time and inspiration to improve my wording.

Physicists who write research papers, lecture notes and text books on the subject of General Relativity - like me - often receive mails by amateur scientists with remarks and questions. Many of these show a genuine interest in the subject. Their requests for further explanations, as well as their descriptions of deeper thoughts about the subject, are often interesting enough to try to answer them, and sometimes discussions result that are worthwhile.

However, there is also a group of people, calling themselves scientists, who claim that our lecture notes, text books and research papers are full of fundamental mistakes, thinking they have made earth shaking discoveries themselves that will upset much of our conventional wisdom.

Indeed, it often happens in science that a minority of dissenters try to dispute accepted wisdom. There’s nothing wrong with that; it keeps us sharp, and, very occasionally, accepted wisdom might need modifications. Usually however, the dissenters have it totally wrong, and when the theory in question is Special or General Relativity, this is practically always the case.  Fortunately, science needs not defend itself. Wrong papers won’t make it through history, and totally ignoring them suffices. Yet, there are reasons for a sketchy analysis of the mistakes commonly made. They are instructive for students of the subject, and I also want to learn from these mistakes myself, because making errors is only human, and it is important to be able to recognize erroneous thinking from as far away as one can ...

Examples of the themes that we regularly encounter are:
- "Einstein’s equations for gravity are incorrect",
- "Einstein’s equivalence principle is incorrect or not correctly understood",
- "Black holes do not exist",
- "Einstein’s equations have no dynamical solutions",
- "Gravitational waves do not exist",
- "The Standard Model is wrong",
- "The Big Bang never occurred; Hubble's Law for the cosmic red shift is being mis-understood",
- "Cosmic background radiation does not exist",
and so on.

When confronted with claims of this sort, my first reaction is to politely explain why they are mistaken, attempting to identify the erroneous ideas on which they must be based. Occasionally, however, I thought that someone was just reporting things he had read elsewhere, and my response was more direct: "Never have I seen so much nonsense in one single package ..." or words of similar nature. This, of course, was a mistake, because these had been the thoughts of that person himself. When other correspondents also continued to defend concoctions that I thought to have extensively exposed as unfounded, I again felt tempted to use more direct language. So now I am a villain.
A curious thing subsequently happened. A handful of people with seriously flawed notions of general relativity apparently joined forces, and are now sending me more and more offensive emails, purportedly exposing my "stupidity" and collecting more "scientific" arguments to back their views.

They find some support from ancient publications by famous physicists; in the first decades of the 20^th century, indeed, Karl Schwarzschild, Hermann Weyl, and even Albert Einstein, had misconceptions about the theory, which at that time was brand new, and these pioneers indeed had not yet grasped the full implications. They can be excused for that, but today’s professional scientists know better.

As for my "stupidity", my own knowledge of the theory does not come from blindly accepting wisdom from text books; text books do contain mistakes, so I only accept scientific facts when I fully understand the arguments on which they are based. I feel no need whatsoever to defend standard scientific wisdom; I only defend the findings of which I have irrefutable evidence, and it so happens that most of these are indeed agreed upon by practically all experts in the field.

The mails I have sent to my "scientific opponents" appear to be a waste of time and effort, so now I use this site to carefully explain where their arguments go astray. Rather than trying to bring them to their senses (which would be about as effective as trying to bring Jehovah’s Witnesses to their senses), I rather address students who might otherwise be misled by what they read on the Internet. The people whose "ideas" I will discuss will be denoted by single initials, for understandable reasons
   [Note: To be sure, I do not want to expose people by name, excusing them of making basic mistakes, but some of them don't see things that way; in his own blog, Mr. C. reacts on this page by identifying all "friends" he found here. My apologies about that]
From their reactions it became clear that analyzing someone’s mistaken train of thought is far from easy. What exactly are the blind spots? I try to spot these, but I receive furious responses that only suggest that the blind spots must be elsewhere. Where do their incorrect assertions come from? Of course, the mathematical equations at those points are missing, so I start guessing. I had to modify some of the guesses I made earlier on this page; actually, I prefer to explain how the math goes, and why the physical world is described by it.

This is not intended as a scientific article, since after all, the math can be obtained from many existing text books. Sadly, these text books are "dismissed" as being "erroneous". Clearly, therefore, I won’t be completely successful. To the students I insist: most of the text books being criticized by those folks are actually very good, although it always pays to be critical, and whatever you read, check it with your own common sense.

Here come some of the crazy assertions concerning General Relativity, and my responses.

"Einstein’s equations for gravity are incorrect,
they have no dynamical solutions, and do not imply gravitational waves as described in numerous text books."

Mr. L. makes this claim, and now he basically refers to a paper that he once managed to get published in a refereed journal. It is clear to me that the referee in question must have been inattentive. It happens more often that incorrect papers appear in refereed journals. Science is immune to that; false papers are simply being ignored, and so is this one; it is not being referred to by professional scientists (Spires mentions only one reference that is not by the author himself).
Dynamical solutions means solutions that depend non-trivially on space as well as time. Numerous of such solutions are being generated routinely in research papers, but most of them require some sort of approximation techniques. The gravitational waves emitted by binary pulsars are typical examples. The procedure to obtain these solutions, using routines to solve Einstein’s equations, is well-known and described in the text books. L. notes that approximations are not exact, and exact solutions do not exist.
Approximations are of course used in many branches of physics. Some are reasonably accurate, some may be questionable. In the case of gravitational waves emitted by time-dependent massive objects, the approximations used are extremely accurate, and furthermore, any doubt can be removed by producing the next term in the approximation, which in many of these examples turns out to be completely negligible. L. does not have the mathematical abilities to do such calculations.

It so happens that also exact, analytical solutions exist that depend non-trivially on space and time. I showed L how these solutions can be obtained in meticulous detail. In order to present and discuss a special example, one can simply assume cylindrical symmetry. This symmetry assumes that the solution is invariant under the transformations z → z + a and φ → φ + b, where z is the third space coordinate and φ the angle between x and y. The dependence on the radius r in the x-y plane, and on time t may be anything. In a beautiful paper, Weber and Wheeler found the complete solution, where the physical degrees of freedom are functions of r and t that turn out to obey simple equations. Not surprisingly, one finds that the solutions take the form of Bessel functions, but even more to the point, one also finds wave packets in the r-t plane. Not many physical systems have cylindrical symmetry, but that’s not the point. The point is now that these solutions contradict L’s claim.

What does L say about this? "I have proven that dynamical solutions do not exist, so your solution is wrong". What is wrong about it? First, he ignores the wave packets and focuses on the plane wave solutions. These have infinite extension in space and time and represent infinite energy. That, indeed, is problematic in gravity. If the energy in a given region with linear dimensions R exceeds R in natural units, a black hole is formed so that space-time undergoes a subtle change in topology. This might arguably be called unacceptable. The problem is manifest in our explicit solutions (the non-linear integral describing the function γ diverges), and this is why it is important to use wave packets instead. The wave packages are identical to the ones in Maxwell theory, and since they represent only finite amounts of energy (per unit of length in the z direction), these solutions are indeed legitimate. I showed  L how to construct explicit, analytical examples of such wave packets. For all such configurations, the γ  integral converges.
Yet, L insists: "I have proven that dynamical solutions do not exist, so your solution is wrong. It violates causality". What? To me, causality means that the form of the data in the future, t > t₁, is completely and unambiguously dictated by their values and, if necessary, time derivatives in the past, t = t₁. So, I constructed the complete Green function for this system and showed it to Mr. L. This function gives the solution at all times, once the solution and its first time derivative is given at t = t₁, which is a Cauchy surface. Causality is obeyed.

 "You are a mathematician, not a good physicist", L then says. "You can’t add a physical source, so your solution is not causal", L says, without explaining what that means. My guess is that what he means is that cylindrically symmetric sources "are unphysical". Indeed, you won’t find many such sources in the universe, but that’s totally besides the point. The point is that any kind of sources might occur in Nature, and the solution I am discussing is the one that can be constructed without any need for approximations, if  the source would happen to have cylindrical symmetry. This particular case thus disproves L’s claim. His complaint that the solution I wrote down explicitly is the sourceless one is strange; just like in Maxwell’s theory, you can add as many cylindrically symmetric sources as you like, and indeed, this may be an instructive exercise for students. In my solutions, one might assume the sources to be at the boundary of the system. But you can also read the solution as follows: it describes how any kind of cylindrically symmetric ingoing gravitational wave converges to the centre, and smoothly bounces outward again. It is a dynamical solution disproving L. He stubbornly continues claiming that I don’t understand wave packets, and illustrates this by writing down an expression that does not obey decent boundary conditions. My solutions obey the boundary conditions as required. Remember that there might be (weak) sources at the boundary. Cylindrically symmetric wave packets are generated by cylindrically symmetric sources. Unlike wave packets that are only functions of  x - t , these wave packets are functions of  r  and  t  that tend to spread out in space. There is nothing wrong or unphysical about that.
"That’s because you don’t understand the equivalence principle, that would have implied that gravitational fields carry no energy", L continues. Now this is perhaps the real reason of his beliefs. Apparently, he fails to understand where the energy in a gravitational wave packet comes from, thinking that it is not given by Einstein’s equations, a misconception that he shares with Mr. C. Due to the energy that should exist in a gravitational wave, gravity should interact with itself. Einstein’s equation should have a term describing gravity’s own energy. In fact, it does. This interaction is automatically included in Einstein’s equations, because, indeed, the equations are non-linear, but neither L nor C appear to comprehend this.
One way to see how this works, is to split the metric g_μν into a background part, g^o_μν, for which we could take flat space-time, and a dynamical part: substitute in the Einstein-Hilbert action: g_μν = g^o_μν + g¹_μν . The dynamical part,  g¹_μν ,  is defined to include all the ripples of whatever gravitational wave one wishes to describe. Just require that the background metric g^o_μν obeys the gravitational equations itself; one can then remove from the Lagrangian all terms linear in  g¹_μν. This way, one gets an action that starts out with terms quadratic in  g¹_μν, while all its indices are connected through the background field g^o_μν. This is because both  g^o_μν and g¹_μν transform as true tensors under a coordinate transformation; all terms in the expansion in powers of g¹_μν  are therefore separately generally invariant. The stress-energy-momentum tensor can then be obtained routinely by considering infinitesimal variations of the background part, just like one does for any other type of matter field;  the infinitesimal change of the total action (the space-time integral of the Lagrange density) then yields the stress-energy-momentum tensor. Of course, one finds that the dynamical part of the metric indeed carries energy and momentum, just as one expects in a gravitational field. As hydro-electric plants and the daily tides show, there’s lots of energy in gravity, and this agrees perfectly with Einstein’s original equations. In spite of DC calling it "utter madness", this procedure works just perfectly. L and C shout that this stress-energy-momentum tensor is a "pseudotensor". Indeed, its transformation properties are subtle, and one might wish to claim that splitting g_μν in a background part and a dynamical part is "unphysical". But then, indeed, one should accept the fact that the notion of energy is observer dependent anyway. An observer who is in free fall in a gravitational field may think there’s no energy to be gained from gravity.
Actually, one can define the energy density in different ways, since one has the freedom to add pure gradients to the energy density, without affecting the total integral, which represents the total energy, which is conserved. Allowing this, one might consider the Einstein tensor G_μν itself to serve as the gravitational part of the stress-energy-momentum tensor, but there would be problems with such a choice. The definition using a background metric (which produces only terms that are quadratic in the first derivatives) is much better, and there’s nothing wrong with a definition of energy, stress and momentum that’s frame dependent, as long as energy and momentum are conserved. In short, if one wants only first derivatives, either frame dependence or background metric dependence are inevitable. 

L furthermore claims that the "established theory" uses Einstein’s equivalence principle incorrectly, as if there were several versions of it. Pauli had it all wrong, according to him, which is his explanation of our "mistakes".

When all other arguments fail, L accuses me of doing "undergraduate physics". Indeed, our discussions rarely transcend that level, but there’s nothing wrong with undergraduate physics. Another interesting accusation one gravitational dissenter threw at me was that I am "at the wrong side of the history of science". Well, we’ll see about that.

"Black holes do not exist,
they are solutions of the equation for the Ricci tensor R_μν = 0, so they cannot carry any mass. And what is usually called a "horizon" is actually a physical singularity."

C and L appear to be friends, but it dawned on me that the mistakes they make are different ones. Mr. C attacks some generally accepted notions about black holes. It appears that the introduction of test particles is inadmissible to him. A test particle, freely falling in a gravitational field, feels no change in energy and momentum, and mathematically, we describe this situation in terms of comoving coordinate frames. This does not fit in C’s analysis, so, test particles are forbidden. A test particle is an object with almost no mass and almost no size, such as the space ship Cassini orbiting Saturn. C calls the use of almost "poetry", but in fact this is a notion that can be defined in all mathematical rigor, as we learn in our math courses. C is "self taught", so he had no math courses and so does not know what almost means here, in terms of carefully chosen limiting procedures.

Mr. C. adds more claims to this: In our modern notation, a radial coordinate  r  is used to describe the Schwarzschild solution, the prototype of a black hole. "That’s not a radial distance!", he shouts. "To get the radial distance you have to integrate the square root of the radial component g_rr of the metric!!" Now that happens to be right, but a non-issue; in practice we use  r  just because it is a more convenient coordinate, and every astrophysicist knows that an accurate calculation of the radial distance, if needed, would be obtained by doing exactly that integral. "r  is defined by the inverse of the Gaussian curvature", C continues, but this happens to be true only for the spherically symmetric case. For the Kerr and Kerr-Newman metric, this is no longer true. Moreover, the Gaussian curvature is not locally measurable so a bad definition indeed for a radial coordinate. And why should one need such a definition? We have invariance under coordinate transformations. If so desired, we can use any coordinate we like. The Kruskal-Szekeres coordinates are an example. The Finkelstein coordinates another. Look at the many different ways one can map the surface of the Earth on a flat surface. Is one mapping more fundamental than another?
"The horizon is a real singularity because at that spot the metric signature switches from (+,-,-,-) to (-,+,-,-)", C continues. This is wrong. The switch takes place when the usual Schwarzschild coordinates are used, but does not imply any singularity. The switch disappears in coordinates that are regular at the horizon, such as the Kruskal-Szekeres coordinates. That’s why there is no physical singularity at the horizon.
But where does the black hole mass come from? Where is the source of this mass? R _μν = 0 seems to imply that there is no matter at all, and yet the thing has mass! Here, both L and C suffer from the misconception that a gravitational field cannot have a mass of its own. But Einstein’s equations are non-linear, and this is why gravitational fields can be the source of additional amount of gravity, so that a gravitational field can support itself. In particle theories, similar things can happen if fields obey non-linear equations, we call these solutions "solitons". A black hole looks like a soliton, but actually it is a bit more complicated than that.
The truth is that gravitational energy plus material energy together obey the energy conservation law. We can understand this just as we have explained it for gravitational waves.  And now there is a thing that L and C fail to grasp: a black hole can be seen to be formed when matter implodes. Start with a regular, spherically symmetric (or approximately spherically symmetric) configuration of matter, such as a heavy star or a star cluster. Assume that it obeys an equation of state. If, according to this equation of state, the pressure stays sufficiently low, one can calculate that this ball of matter will contract under its own weight. The calculation is not hard and has been carried out many times; indeed, it is a useful exercise for students. According to Einstein’s equations, the contraction continues until the pressure is sufficiently high to stop any further contraction. If that pressure is not high enough, the contraction continues and the result is well-known: a black hole forms. Matter travels onwards to the singularity at  r = 0, and becomes invisible to the outside observer. All this is elementary exercise, and not in doubt by any serious researcher. However, one does see that the Schwarzschild solution (or its Kerr or Kerr-Newman generalization) emerges only partly: it is the solution in the forward time direction, but the part corresponding to a horizon in the past is actually modified by the contracting ball of matter. All this is well-known. An observer cannot look that far towards the past, so he will interpret the resulting metric as an accurate realization of the Schwarzschild metric. And its mass? The mass is dictated by energy conservation. What used to be the mass of a contracting star is turned into mass of a "ball of pure gravity". I actually don’t care much about the particular language one should use here; for all practical purposes the best description is that a black hole has formed.
But has it really? Isn’t it so that the collapsing star hangs out forever at the horizon? Well, in terms of the Schwarzschild coordinates, this is formally true! The Schwarzschild solution is the asymptotic limit of the solution in the forward time direction. At finite times, the region behind the horizon does not exist. However, for this analysis, one can better use the Eddington-Finkelstein coordinates, where one does notice that the future part of the horizon does exist. This discussion is compounded a bit because the construction of the maximal extension of spacetime is subtle, and it is certainly not understood by C. Think of a map of the North Pole of the Earth, where it could be that coordinates were chosen such that they cannot be extended across the equator. Formally, the equator is then a horizon. But nobody who’s walking on the equator has any trouble with that.
These self proclaimed scientists in turn blame me of "not understanding functional analysis". Indeed, L maintains that there is a difference between a mathematical calculation and its physical interpretation, which I do not understand. He makes a big point about Einstein’s "equivalence principle" being different from the "correspondence principle", and everyone, like me, who says that they in essence amount to being the same thing, if you want physical reality to be described by mathematical models, doesn’t understand a thing or two. True. Nonsensical statements I often do not understand. What I do understand is that both ways of phrasing this principle require that one focuses on infinitesimally tiny space-time volume elements.
Sadly, even my co-laureate, Tini Veltman, every now and then displays his complete ignorance about black holes. As with others, his problem is that "you can't  have a transition where a topologically trivial space, such as a neutron star, suddenly changes into a black hole"... "Doesn't a black hole have a singularity? If I throw a peanut into a neutron star, will it suddenly generate a singularity?" As is the case with others, he does not realize that the singularity, formally, lies in the infinite future. Einstein's equations can be completely solved to show exactly how such a singularity (which is hidden behind the horizon anyway) forms. Particularly these equations become easy if you assume spherical symmetry. Most importantly, such a (simple) calculation reveals that the only material properties you need to know about to predict what you will see, are regular ones. I can make a black hole out of mobile phones, which will still be working according to ordinary physical laws while they plummet through the horizon formed by their own gravitational fields (yes, I need very many phones ...)

"You can’t have massive objects near a black hole;
and you can’t have multiple black holes orbiting one another"

Again, C is speaking. He has a problem with the notion of test particles, which are objects whose mass (and/or charge) is negligible for all practical purposes, so that they can be used as probes to investigate the properties of field configurations. Again, this is a question of making valid approximations in physics. A space ship such as the Cassini probe near Saturn, has mass, but it is far too light to have any effect on the planets and moons that it observes, so, its orbit is a geodesic as long as its engines are switched off. No physicist is surprised by these facts, but for C, approximations are inexcusable. For him, the Cassini probe cannot exist.
Astrophysicists studying black holes routinely make the same assumptions. A valid question is: could the tiny effects of probes such as Cassini have explosive consequences for black holes or other solutions to Einstein’s equations? You don’t have to be a superb physicist - but you must have better intuitions than C - to conclude that such things do not happen. A black hole surrounded by matter, such as a space probe, or, more to the point, an accretion disk, is described not by the equation R _μν = 0, but by equations such as R _μν = tiny, where "tiny" is the material source of gravity that obeys its own equations. Similarly, the gravitational, tidal, effects of another nearby black hole can slightly modify the boundary conditions. To deduce the consequences of such perturbations one can use perturbation expansion - alas, that was the thing refuted by C. But if one does anyway, one finds that a tiny gravitational wave is spread around by this source, partly being absorbed and partly scattered by the black hole, without any catastrophe happening anywhere. Indeed, if you suspect this approximation to be too crude (quod non), you may always check the original equations. You will find: R’ _μν - tiny  = (much tinier), and again compute the ripples spread around by the much tinier source. This is what perturbation theory is about. The expansion parameter, Newton’s constant, is conveniently small in all practical cases, so nothing will happen. Only when the perturbative corrections are not small, gravitational instabilities may arise. We know what that is, if we may rely on our primitive physical intuitions: you then may end up with real black holes. Interesting, unlikely, but no problem whatsoever for the established theory.

Any doubts about these facts are removed once the existence and properties of the Green functions for the linearized theory have been established. These Green functions can then be used to study systematic expansions to obtain the solutions of the complete, non-linear theory, to any required accuracy. Good theoretical physicists completely control the proper use of Green functions. Neither L nor C seem to have heard about Green functions. I did construct them, and found that, provided due attention is paid to the gauge freedom in the use of coordinates, these functions are well-behaved.

A third player, DC, strongly supports L and C, but his claims are too opaque for me to even address. In a previous version of this site, I addressed his theories about the 9-11 events, but I removed this part; this site should be about General Relativity only.

It’s casting pearls to the swine; my crowd of friends, E., L, C and DC being principal representatives, do not show the slightest willingness to revise their views, and they will probably react with further twists, as they did in the past. Those will usually end up in my spam filter (their mails are too numerous and too offensive to me), but if anyone else has questions, I would be obliged to respond.

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Addendum 17 and 22 /3/2010:

I emphasize that any modification of Einstein’s equations into something like  R _μν  - 1/2 R g_μν = κ( T_{μν (matter)}  +  t _{μν (grav)} )   where  t _{μν (grav)}   would be something like a "gravitational contribution" to the stress-energy-momentum tensor, is blatantly wrong.   Writing such a proposal betrays a complete misunderstanding of what General Relativity is about. The energy and momentum of the gravitational field is completely taken into account by the non-linear parts of the original equation. This can be understood and proven easily, as I explained in the main text.  Note that a freely falling observer experiences no gravitational field and no energy-momentum transfer; hence there cannot be a covariant tensor such as  t _{μν (grav)} .

By the way, there is a point where I happen to agree with DC: the wars in Irak and Afghanistan are big economic boosters for the weapon industry, but from a humanitarian and political point of view they are big mistakes (I apologise for this remark in a site devoted to General Relativity).

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Addendum 12/7/2010

"Much of today’s physics is based on errors and fraud",
Einstein had it totally wrong, and so on. Indeed, how could I forget the most vociferous player of the anti-Gerard-’t-Hooft club: Mr.E., who not only advocates a complete revision of General Relativity, but also Elementary Particle Physics and even electro-magnetism. Nearly all of present-day Theoretical Physics is based on unbelievable errors, according to E. He was quiet for some time, but now asks me to admit the errors of my ways, resign, step down as Chief Editor of Foundations of Physics, and return my Nobel prize. His arguments are difficult to follow because of a murky notation, which presumably explains how he could go so astray in the first place. For example, he asserts that "the Riemannian connection field is antisymmetric, not symmetric, in its two lower indices", which implies that we (Einstein together with many others) "forgot to consider torsion in our field equations", and that, "of course, the connection field should be antisymmetric, because commutators are antisymmetric, not symmetric!"  Well, I’m sorry for E, but a connection field isn’t a commutator. One could say that the only commutator involved is the commutator between two partial derivatives, which vanishes. In Particle Physics, one may regard the gauge vector potential as a connection field, and that’s antisymmetric in two of its indices (for orthogonal gauge groups) or anti-hermitean (in other cases). In gravity, one may complement the connection field with a connection for the local Lorentz group, which is nearly but not quite antisymmetric, because the Lorentz group is not quite orthogonal. It’s the Riemann curvature field that may be regarded as a commutator, and it is antisymmetric in its last two indices. So, how does E "prove" that the Riemannian connection field is antisymmetric? His arguments in his paper 139, eqs. (9)-(12), can be summarized as follows. To be proven: A=B. Proof:

A=B+(A-B)
A=B+...
A=B

A proof that he concludes with a triumphant "Q.E.D."

Fact is that, if the Riemannian connection field were to be chosen antisymmetric, it could not serve as a connection field at all, as its coordinate transformation rule  contains an important term  ∂ ²x^λ/ ∂x^μ ∂x^ν , which is symmetric under interchange of the indices μ and ν . But E continues, suggesting to harvest free energy out of vacuum fluctuations or things such. O, yes, according to E. the photon has a mass of  5.10^-41 kg, which would give electromagnetic fields a range of no more than 7 mm, indeed a drastic modification of the Standard Model.

Addendum 18/8/2010

"gravitational waves do not exist"
Here is the plane gravitational wave for beginners. You must have read somewhere what a metric tensor is. A solution to Einstein’s equations is:

g _μν (t,x,y,z) = diag (  -c², 1+ ε sin ω(z/c-t), 1- ε sin ω(z/c-t), 1  )    +   O(ε² ω² |x|² ) .

In typical LIGO experiments, the amplitude ε  is very small, something like 10^-20 or smaller (and indeed dimensionless), and the frequency ω is typically of the order of 1000 Hz. The energy density of the wave is (C c²/G) ε² ω² ,  where C is some numerical coefficient like 1/8π² , c is the velocity of light, and G is Newton’s constant (you might be surprised that Newton’s constant occurs in the denominator, but that’s how it is, very much energy, or mass density, is needed to make a gravitational field). The last term, O(ε² ω² |x|² ) , may be hard to understand for beginners. It says that, actually, the plane wave isn’t plane, it gets corrections that grow quadratically with distance. These corrections, which I do not spell out in detail, come from the non-linear terms in Einstein’s equations. For trained physicists, however, it is easy to understand their physical origin: the gravitational wave carries energy, therefore has a tiny mass density, and therefore generates a gravitational field by itself. For LIGO, this is totally negligible: only if |x| becomes hundreds of millions of light years, this term becomes bothersome, but the actual gravitational source is not that far away, and/or does not shine for so many millions of years, so we never have a wave that stretches that far. To be precise: in practice you only need the linear parts of Einstein’s equations, and you have to build wave packets by superimposing these plane waves. Then, these wave packets don’t stretch that far, and the bothersome correction term never becomes large.

By superimosing plane waves, one can construct cylindrical waves, or wave packets. As described earlier, exact solutions can be written down then, and they allow one to write the non-linear correction term in closed form.

To see the effect that a wave of this sort has when passing three masses, suspended in an L shaped configuration, you have to compute the connection field  Γ, to discover that the only relevant components are Γ ⁱ ₀₀, and they all vanish to order ε, because g₀₀ does not oscillate. So, in the coordinate frame used, the masses do not move . If the two arms of the L scape are in the x and the y direction, an interferometer will notice that their relative length oscillates, an oscillation that will show up in the interference pattern. That’s how LIGO works.

Addendum July 2016
O, now how do they react in the announcements by the LIGO team that a couple of gravitational wave signals have now finally been detected, the first of which being a very bright one that could hardly be mistaken for anything else? DC has his answer ready: "GW150914 is a FRAUD". This certainly is an easy way out.

Addendum 9/1/2012

I thought my arguments should have been crystal clear, but, instead, they dug their trenches deeper. They now tell each other that the final proof of my stupidity is there. Indeed, even more allies were found. Another Mr. L, call him AL, has written a paper entitled "Wrong idées fixes", arXiv:physics/0403092v1. What he calls "idées fixes" are black holes, gravitational radiation and a few other, equally well-established consequences of general Relativity. Then come the author’s counter arguments, for which indeed the title of the paper would have been more appropriate.

I resist temptation to repeat what I said about black holes and gravitational waves, but, concerning the latter, I can add the following further clarifications: the metric tensor has 10 independent degrees of freedom. Indeed, they are not all independently observable; you can always go to a coordinate frame where some of these degrees of freedom vanish. But not all at the same time, since you have only 4 coordinates to modify. Of the 6 remaining degrees of freedom, only 2 propagate, the two helicities of the gravitational wave. The situation is very similar to that of the Maxwell equations. The vector potential field has 4 degrees of freedom. One of these can be adjusted any way you like by a local gauge transformation. Of the remaining 3 degrees of freedom, only two propagate: the two helicities of the photon. The fact that in gravity 4, and in the Maxwell case 1 of the surviving degrees of freedom do not propagate, has to do with the possibility to perform further coordinate adjustments at infinity.

In gravity, just as in the Maxwell case, you can consider a smooth background solution to the equations, and then superimpose a rapidly oscillating disturbance on top of that. The disturbance obeys its own equations. Due to the fact that, in gravity, the equations are non-linear, there will be a tiny correction to the linear equations, and this precisely produces the gravitational field disturbances caused by the wave.

I’m afraid your IQ has to surpass a certain threshold to understand how this works out, and why this exactly represents the stress, energy and momentum of the waves (or wave packet). Curiously, one can say that "we are lucky": starting from Einstein’s equations, it could have been that Newton’s constant G turned out to be negative. Gravity would be a repulsive force but that does not seem to contradict anything. Until you compute the energy of gravitational waves: that energy would have been negative, which would be somewhat disturbing from a physical point of view.

Curiously, L accuses me of understanding mathematics but not the underlying physics. I need help, he shouts. Well, here is a question he might bother helping me with. He claims that experiments have shown that the gravitational effect due to an electric field are repulsive. Have they? How large are these effects? Any gravitational effect of an electric field is so weak that a test mass needs to be observed for hours to see its reaction, and its motion then becomes macroscopic only if the energy density of this electric field surpasses that of a detonating atomic bomb. Indeed, looking up the experiment cited, the authors mention a slight discrepancy with General Relativity: the effect they claimed was factors one million trillion bigger (and it wasn’t about electric fields but the graviational gyroscopic effect). Oops. How sure is Mr. L that he got his numbers right?

12/09/2014: After a recent repetition of the same discussion on the web, C reacted again. It now appears that, thank goodness, he might be equiped with a rudimentary sense of humor. He writes a "paper" with the following abstract:

I extend my thanks to Professor Gerardus ’t Hooft, Nobel Laureate in Physics, for making more widely known my work on black hole theory, big bang cosmology, and Einstein’s General Theory of Relativity, by means of his personal website, and for providing me thereby with the opportunity to address the subject matter - supported by extensive references to primary sources for further information - in relation to his many comments, by means of this dedicated paper. The extensive mathematical appendices herein are not prerequisite to understanding the text.

Indeed they are not. The text reiterates much of the nonsense we saw before, ornamented with numerous citations out of context. Just because gravity is non-linear, you can't have more than one black hole in the entire universe, is one of the messages. In a systematic perturbation expansion one can compute the interactions, due to non-linearity, between black holes. This, however, is something he does not want to hear about. You can have one apple, but because its smell is a non-linear function of its distance, you can never have more than one apple in the entire universe.
Big Bang Theory is creationism, is another message. What's the alternative? A steady state universe? Those are now known to be unstable, and truckloads of observational eveidence would have to be denied. And what's the alternative to black holes? Perhaps even Mr. C can solve the equations as to what happens when a large spherical body made of dust collapses under its own weight. Take sand that remains pressureless up to its normal density σ. At higher densities, you can take the pressure as high as you want. Take an initial spherical configuration with radius R and uniform density ρ<<σ. What happens when (8πGR²/(3c²))³ρ²σ>>1? It's an undergraduate problem, but let's see what C thinks of it. Will he notice that it's way inside the horizon that ρ starts to exceed σ? Even if that goes off like a nuclear bomb, it's far too late, the debris won't come out.
Central singularity? Yes, it's physical for an observer who travels inside the black hole, since he will be killed by it. Outside observers don't notice a thing. Again, whether or not you still want to call that physical is a linguistic problem. Not my field of interest, just like the question whether the r coordinate is a radius, a non-issue on which he wastes six pages.

Amusingly, there is a crowd of interesting guys, I guess all members of the viXra community, who all condemn me for my views on real science, and my description of BAD theoretical physicists. C is a member, and it turns out there are more vociferous people there. Remarkably, in their e-mails I am accused of exactly the shortcomings they suffer from themselves - I could not have described those better. I will patiently wait for them sending in their solutions of the undergraduate problem sketched above. Long before that, undoubtedly, I will receive more insulting emails.

Addendum, January 2017.
Since the above was written, LIGO's gravitational signal was followed by more such signals. LIGO's brief announcements contained the usual popular explanations of what gravitational waves are, and that they go with the speed of light. Mr. C has not moved an inch from his earlier positions: "gravitational waves do not exist, and if they do, they can go with any speed they like, the equations haven't been applied correctly."
Indeed, the linearised equations were used, which are accurate up to 20 decimal places or so. But C's point now is that one can add fluctuating waves to the coordinates as well, and these can go with any speed. "With the speed of thought". True, but you still get the superposition of an observable part of the solution, which still goes with the speed of light, and an unobservable part. The unobservable part, the coodinate artifacts, are solutions to the homogeneous equations, so they have no source, and since the detection device can be handled with the same fluctuating coordinates, they also leave no signal. C forgets that you can de the same thing with Maxwell's solutions that produce light waves. You can make a gauge transformation that oscillates also with the speed of thought. It leaves no signal on our retina, or in our mobile phones. Ans does light exist? Do mobile phones work? Does light go with the speed of light?

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