Gravitational wave time variation

[PeterDonis]

I mean on what mathematical model is based

I don't know that there is one. But you proposed a heuristic that is perfectly reasonable; it's just a different one from the standard one.

Unless you are blurring the distinction between heuristics and mathematical theory.

I suppose I am. A heuristic doesn't necessarily require a detailed mathematical theory if all you're using it for is a conceptual understanding. If you're trying to use it to make quantitative predictions, that's something different.

The point I was making is not that there is some detailed alternative model that you can use to quantitatively predict results like the LIGO observations. The point was simply that the standard linearized description of GWs is not "the way things really are"; it's just a heuristic description that, when modeled mathematically, makes predictions that are accurate enough for our purposes. So thinking about it as though "the length between the arms is changing" is what is "really happening" with a GW, and nothing else is going on, is wrong. The best way we have of saying "what is really happening", in the context of GR, is that the spacetime geometry has "ripples" in it, and how those "ripples" look to a particular detector depends on the relationship between the ripples and the detector.

 

[RockyMarciano]

I understand your view, now. Thanks.

 

[pervect]

I believe that a "local inertial frame" would cover an area considerably larger than the Ligo apparatus, though it would be finite in extent.

Here is what I get specifically.

Gravity waves can be + or x polarized, for simplicity we'll assume the wave is purely + polarized, and that it is moving in the +z direction. Then we can approximate the metric as:

$$-dt^2 + [1+2h(t-z) ]dx^2 + [1-2h(t-z)]dy^2 + dz^2$$

here h() is an arbitrary function, representing an arbitrary strain, the same dimensionless number reported by Ligo.

To first order we can approximate this metric as an orthonormal basis of one-forms

$$\omega_1 = dt \quad \omega_2 = [1+h(t-z)]dx \quad \omega_3 = [1-h(t-z)]dy \quad \omega_4 = dz$$

(GRtensor uses indices from 1 to 4, and it's too likely to induce errors for me to attempt change the notation to a more natural and readable 0-3).

One might write instead $\omega_1 = \sqrt{1+2h(t-z)}$ if one prefers, but the end result will be the same to the first order which is all that the metric is good for.

To check this, it's convenient to assume $h(u) = a \sin \Omega u$ where $u = t-z$, and $\Omega$ is just an angular frequency, which we've chosen to capitalize to avoid confusing it with the basis one forms $\omega_i$. Then the Einstein tensor G is zero to the first order - it has components of order $a^2$ which we can neglect.

We can find the Riemann

$$R_{1212} = -R_{1224} = R_{2424} = a \Omega^2 \sin \, \Omega u/ (1 + a \sin \, \Omega u ) \approx a \Omega^2 sin \, \Omega u$$
$$-R_{1313} = R_{1334} = -R_{3434} = a \Omega^2 \sin \, \Omega u / (1 - a \sin \, \Omega u ) \approx a \Omega^2 sin \, \Omega u$$

And we use the relation between the Riemann and Fermi-normal coordinates (MTW, pg 332) to find an approximate expression for the metric in Fermi-normal coordinates, coordinates which best approximate the "laser local lab frame". There are a bunch of assorted cross terms which would be tedious and error-prone to write out explicitly, the interesting term is the metric coefficient of dt^2

$$ [-1 - a \Omega^2 (x^2 - y^2) \sin \Omega u] \, dt^2$$

The gradient of this of this gives the tidal force (for small x and y). I believe that the other terms are not important for understanding the physics (I suppose I could be overlooking something, but I don't think so.)

[add]
The above was all in geometric units, a physical interpretations would be that the coefficient of "time dilation" dt^2 is proportional
$$1 + 4 \pi^2 \, a \frac{x^2 - y^2} { L^2}$$

where L is the wavelength of the gravitataional wave.

It can be seen that for large x and y the metric becomes ill behaved, and that the metric should be thought of only as being valid for "small x and y". However, due to the extremely small value of a in Ligo, on the order of $10^{-21}$, the region covered will include the whole apparatus - probably the whole solar system.

 

[PeterDonis]

I believe that a "local inertial frame" would cover an area considerably larger than the Ligo apparatus

As I said before, this can't possibly be the case as you state it, because by definition spacetime curvature in a local inertial frame is negligible, and a gravitational wave is a wave of spacetime curvature, so spacetime curvature can't possibly be negligible over the area covered by LIGO if LIGO detects a GW.

What you are constructing are basically Fermi normal coordinates centered on the LIGO detector (at the base of the arms). It's certainly true that such coordinates can cover a "world tube" that is more than wide enough to encompass the entire LIGO apparatus. But Fermi normal coordinates are not the same as a local inertial frame. In a local inertial frame the metric is $-dt^2 + dx^2 + dy^2 + dz^2$, by definition. There can't be any correction terms. If there are, it isn't a local inertial frame.

 

[RockyMarciano]

After a conversation with one of the detection paper signing authors from LIGO he has convinced me that the analysis using varaible coordinate light speed is not tenable because it is not physical(in exactly the same way the coordinate light speeds found in cosmology are not considered to break special relativity and allow superluminal signaling, a variable coordinate velocity cannot be considered a physical invariant leading to a measurable signal and if it was it would lead to a non-detection and/or to the possibility of superluminal signaling. So the only possible analysis consistent with a non-null detection is the one that considers the laser speed as constant at all times and locations in the arms whether the GW is passing or not.

 

[PeterDonis]

a variable coordinate velocity cannot be considered a physical invariant

This is true; coordinate-dependent quantities are not physical invariants. But that is equally true of the coordinate speed of light in the coordinates the LIGO authors prefer to use. The fact that that coordinate speed happens to be $c$ at all values of the coordinates doesn't mean it's an invariant.

The actual invariants in the LIGO detectors are the observables: the signals that are shown in the graphs. Basically, these are amplitudes of interference patterns in the detectors vs. time. So it is an invariant fact that, when the laser beams go down the two perpendicular arms and are reflected at the mirrors, they interfere with each other when they return. The LIGO authors' interpretation of this--that the speed of light is constant and the interference pattern is therefore due to changes in the lengths of the arms because of the passage of a GW--is certainly an easy interpretation to visualize and work with; but that doesn't make it an invariant.

in exactly the same way the coordinate light speeds found in cosmology are not considered to break special relativity and allow superluminal signaling, a variable coordinate velocity cannot be considered a physical invariant leading to a measurable signal and if it was it would lead to a non-detection and/or to the possibility of superluminal signaling

Um, what? Variable coordinate light speeds cannot allow superluminal signaling (which is true), therefore variable coordinate light speeds cannot be right because they would allow superluminal signaling? This is self-contradictory.

Actually, variable coordinate speeds of light are commonly used for interpretation in cosmology, and nobody bats an eye--precisely because they don't lead to superluminal signaling, so they're not a problem. The same is true for any coordinates in which the coordinate speed of light is variable (another common example is Schwarzschild coordinates). It just so happens that the most convenient coordinates for cosmology--comoving coordinates in FRW spacetime--lead to variable coordinate speeds of light; whereas it just so happens that the most convenient coordinates for LIGO analysis are coordinates in which the coordinate speed of light is $c$ everywhere. But that doesn't make different coordinates in which the coordinate speed of light is variable "wrong"; it just makes them less convenient for this particular problem.

 

[RockyMarciano]

The actual invariants in the LIGO detectors are the observables: the signals that are shown in the graphs. Basically, these are amplitudes of interference patterns in the detectors vs. time. So it is an invariant fact that, when the laser beams go down the two perpendicular arms and are reflected at the mirrors, they interfere with each other when they return. It just so happens that the most convenient coordinates for cosmology--comoving coordinates in FRW spacetime--lead to variable coordinate speeds of light; whereas it just so happens that the most convenient coordinates for LIGO analysis are coordinates in which the coordinate speed of light is $c$ everywhere. But that doesn't make different coordinates in which the coordinate speed of light is variable "wrong"; it just makes them less convenient for this particular problem.

I agree with this but I'm having a really hard time seeing how it applies to the LIGO detection case.
So if the observables for the detection of a GW here in the LIGO experiment are the difference in phase(different times of flight for each laser path originated in the difference in arms lengths), I would like to see a mathematical treatment using coordinates in which the coordinate speed of light is variable, even if less convenient, where the observable described in the previous paragraph is recovered, even if just in principle if the mathematical treatment in those coordinates is very involved. Can you show something like this?
I've looked for such a treatment and haven't been able to find anything. Everywhere I looked I just found the linearized gravity harmonic gauge condition(under many names:de Donder, Lorenz, Fock...) with constant c in the interferometer arms.And I myself can't figure out how to reproduce the detection observable with changing c as it seems evident that it would lead to no difference in times of flight even if there was a change in length(the change in c would cancel the change in length in both arms).

 

[PeterDonis]

if the observables for the detection of a GW here in the LIGO experiment are the difference in phase

Yes, this is a direct observable.

(different times of flight for each laser path originated in the difference in arms lengths)

But this is not. Attributing the difference in phase to "different times of flight originating in the difference in arms lengths" is already adopting a particular set of coordinates.

I would like to see a mathematical treatment using coordinates in which the coordinate speed of light is variable, even if less convenient, where the observable described in the previous paragraph is recovered, even if just in principle if the mathematical treatment in those coordinates is very involved. Can you show something like this?

No. As I think I said earlier in this discussion, I'm not aware of anyone having actually done such an analysis. But that doesn't mean there is no such analysis. Nor does it change the fact that talking about different times of flight originating in different arm lengths is adopting a particular set of coordinates.

I myself can't figure out how to reproduce the detection observable with changing c as it seems evident that it would lead to no difference in times of flight even if there was a change in length(the change in c would cancel the change in length in both arms).

Heuristically, the kind of model I'm thinking of would have no change in the coordinate lengths of the arms; the variation in the coordinate speed of light over a constant arm length would lead to a variation in coordinate times of flight, and therefore a phase shift.

 

[pervect]

After a conversation with one of the detection paper signing authors from LIGO he has convinced me that the analysis using varaible coordinate light speed is not tenable because it is not physical(in exactly the same way the coordinate light speeds found in cosmology are not considered to break special relativity and allow superluminal signaling, a variable coordinate velocity cannot be considered a physical invariant leading to a measurable signal and if it was it would lead to a non-detection and/or to the possibility of superluminal signaling. So the only possible analysis consistent with a non-null detection is the one that considers the laser speed as constant at all times and locations in the arms whether the GW is passing or not.

I would take the position that the coordinates where the laws of physics are as close to Newtonian as possible (which I regard as being the most physical because that's where my physical intuition is the strongest) are Fermi-normal coordinates, and point out that those are NOT the coordinates that the Ligo paper used.

I would also admit that using Fermi-normal coordinates would be a giant pain in the rear to use, though less of a pain if one is willing to make certain approximations.

I went through some rather detailed discussion and derivations about an approximate conversion to such coordinates and the resulting metric, but I don't think it "got through", so I don't see much point in rehashing it.

I'll quote at the end of this post a bit from the literature about the power of Fermi-normal coordinates and their "physical interpretation" from http://arxiv.org/abs/0901.4465

The point of this is this: my assumption that anyone who is searching for "physicalness" in coordinates would be well advised to consider Fermi-normal coordinates. "Physicality", a rather vague term, does seems to be the goal of the OP here, so that's why I suggested them. I hoped that the mention of the name alone would be sufficeint, but it seems not. So I'll go into some backgroud with some quotes from the literature about what these coordinates are, and their claims to "physicallity" (which are also in the literature).

http://arxiv.org/abs/0901.4465][/PLAIN]
Nowadays the Fermi normal coordinates are usually - although improperly - called Fermi coordinates. In experimental gravitation, Fermi normal coordinates are a powerful to ol used to describe various experiments: since the Fermi normal coordinates are Minkowskian to first order, the equations of physics in a Fermi normal frame are the ones of special relativity, plus corrections of higher order in the Fermi normal coordinates, therefore accounting for the gravitational field and its coupling to the inertial effects. Additionally, for small velocities v compared to light velocity c, the Fermi normal coordinates can be assimilated to the zeroth order in (v/c) to classical Galilean coordinates. They can be used to describe an apparatus in a “Newtonian” way (e.g. [1, 3, 8, 10]), or to interpret the outcome of an experiment (e.g. [11] and comment [21], [5, 6, 15, 17]). In these approaches, the Fermi normal coordinates are considered to have a physical meaning, coming from the principle of equivalence (see e.g. [18]), and an operational meaning: the Fermi normal frame can be realized with an ideal clock and a non extensible thread [29]. This justifies the fact that they are used to define an apparatus or the result of an experiment in terms of coordinate dependent quantities.

So in conclusion I think that Fermi-Normal coordinates, which are NOT the ones used in the Ligo analysis, might provide some insight into the interpretation process of the results. I'd also say that they aren't the simplest to use mathematically, and that the easiest approach is to analyze the problem in the coordinates that Ligo used, first, then convert to Fermi-normal coordinates to aid in the physical interpretation. The tools needed to convert coordinates involve knowledge of diffeomorphisms, and the tensor transformation laws.

I'll also say that when using generalized coordinates, such as GR does, coordinates are not necessarily chosen for any "physical" significance at all, and that this takes some getting used to. The metric is what converts physically non-significant coordinates into physically significant distances and times, via the mechanism of the invariant Lorentz interval. The process goes like this: coordinates go into the metric, which generates the Lorentz interval from coordinate displacements. The Lorentz interval is independent of the observer, and can be further converted into proper time intervals and proper distance intervals when one chooses an "observer".

 

[RockyMarciano]

But this is not. Attributing the difference in phase to "different times of flight originating in the difference in arms lengths" is already adopting a particular set of coordinates.

Fair enough. But this misses the point I'm trying to make in relation to gravitational radiation and linearized GR in the absence of a static spacetime situation. Here the coordinate condition acts like a gauge condition. Using the gauge terminology you are referring to coordinates sets choices in the context of this choice fixing a gauge(in this case the gauge refers to GL4 Diff(M) invariance under arbitrary coordinate transformations) like what's seen for instance in the Schwarzschild solution. But here we have to deal with a coordinate condition that is a gauge condition that doesn't fix a gauge: the Lorenz gauge condition(harmonic condition), a partial gauge, must be used due to the indeterminacy of the EFE. In other words the equation $\Box h_{\mu\nu}=0$ is only valid in the Lorenz gauge condition(i.e. harmonic coordinate condition)
Under these circumstances the gauge condition rules out the different set of coordinates you mention below, that measures variation in coordinate speed of light over arm length, which would be just a change of coordinates that is routinely performed when using interferometers for other applications (for instance when used to measure refractive changes), because we are talking about a certain gauge condition that takes advantage of the degrees of freedom left by the partial gauging to leave the coordinate t unaffected by any coordinate transformation and be able to concentrate in change of length in the arms.

Heuristically, the kind of model I'm thinking of would have no change in the coordinate lengths of the arms; the variation in the coordinate speed of light over a constant arm length would lead to a variation in coordinate times of flight, and therefore a phase shift.

Again, this could only be done in a static situation, where the existence of a timelike killing vector allows you to fix the gauge. Note that in order to measure change in coordinate speed of light you would have to have a wavelength change in the laser to keep the frequency constant, otherwise you cannot detect phase shift, and such wavelength change is not compatible with no change in arm length. This shows the usefulness of the partial Lorenz gauge in gravitational radiation detection, it allows you to ignore the change in wavelength that comes with the stretching of the interferometer's arms.

 

[PeterDonis]

Here the coordinate condition acts like a gauge condition.

Yes, it does. So what? It's still a coordinate condition, and there is still no requirement that one has to adopt any particular coordinate condition. It's a convenience, not a physical necessity.

here we have to deal with a coordinate condition that is a gauge condition that doesn't fix a gauge

All this means is that one single condition doesn't pick out a unique coordinate chart; it only picks out a family of coordinate charts (harmonic coordinates), and you have to impose a further condition to pick out one particular chart in that family. None of this changes what I was saying at all.

Under these circumstances the gauge condition rules out the different set of coordinates you mention below

No, it doesn't, because a gauge condition is not a physical condition; it's a choice of coordinates. As I said above, there is no requirement to adopt any particular coordinate condition. It's a convenience, not a physical necessity.

this could only be done in a static situation, where the existence of a timelike killing vector allows you to fix the gauge.

The existence of a timelike KVF does not require you to pick a particular coordinate chart or impose a particular gauge condition. Nor does it enable you to use a particular coordinate chart or fix a particular gauge condition that you couldn't fix in its absence. It just makes it convenient to adopt a particular choice of coordinates/gauge. But that's a convenience, not a physical necessity.

in order to measure change in coordinate speed of light you would have to have a wavelength change in the laser to keep the frequency constant, otherwise you cannot detect phase shift

The phase shift is an invariant; if it's present in one coordinate chart, it's present in any coordinate chart.

such wavelength change is not compatible with no change in arm length.

It might not be compatible with no change in arm proper length, given a particular definition of what "proper length" means. (But even that is not uniquely determined, because it requires a choice of simultaneity convention, i.e., a coordinate choice.) But I was talking about coordinate length; a wavelength change is perfectly compatible with no change in coordinate arm length.

 

[pervect]

I ran across a rather interesting paper. "Fermi-normal, optical, and wave-synchronous coordinates for spacetime with a plane gravitational wave", http://www.phys.utb.edu/downloads/0264-9381_31_8_085006.pdf

Abstract
Fermi-normal (FN) coordinates provide a standardized way to describe the
effects of gravitation from the point of view of an inertial observer. These
coordinates have always been introduced via perturbation expansions and
were usually limited to distances much less than the characteristic length
scale set by the curvature of spacetime. For a plane gravitational wave this
scale is given by its wavelength which defines the domain of validity for
these coordinates known as the long-wavelength regime. The symmetry of this
spacetime, however, allows us to extend FN coordinates far beyond the long-
wavelength regime. Here we present an explicit construction for this long-range
FN coordinate system based on the unique solution of the boundary-value
problem for spacelike geodesics. The resulting formulae amount to summation
of the infinite series for FN coordinates previously obtained with perturbation
expansions. We also consider two closely related normal-coordinate systems:
optical coordinates which are built from null geodesics and wave-synchronous
coordinates which are built from spacelike geodesics locked in phase with the
propagating gravitational wave. The wave-synchronous coordinates yield the
exact solution of Peres and Ehlers–Kundt which is globally defined. In this
case, the limitation of the long-wavelength regime is completely overcome,
and the system of wave-synchronous coordinates becomes valid for arbitrarily
large distances. Comparison of the different coordinate systems is done by
considering the motion of an inertial test mass in the field of a plane gravitational
wave.

It talked a bit about gauge conditions, but I didn't see an explicit reference to which gauge condition Fermi-normal coordinates might satisfy. That's an interesting question for the weak-field formulation of the theories, where gauge conditions are defined, that I don't know the answer to.

Gauge conditions were mentioned, for instance It was mentioned that TT gauge was useful for analyzing gravitational waves with laser interferometry.

Themost important point I think is that gauge conditions and/or coordinate choices cannot matter to experimental results. It seems to me like that basic point is getting lost somewhere in the math :(.

 

[RockyMarciano]

I ran across a rather interesting paper. "Fermi-normal, optical, and wave-synchronous coordinates for spacetime with a plane gravitational wave", http://www.phys.utb.edu/downloads/0264-9381_31_8_085006.pdf

It talked a bit about gauge conditions, but I didn't see an explicit reference to which gauge condition Fermi-normal coordinates might satisfy. That's an interesting question for the weak-field formulation of the theories, where gauge conditions are defined, that I don't know the answer to.

I've been trying to argue that the only gauge condition that the linearized gravity formulation of gravitational radiation(no other available to my knowledge) is consistent with is the harmonic (Lorenz) gauge. This is a well known fact( you can see it explicitly stated for intance in Ruffini's textbook "Basic concepts in relativistic astrophysics" pages 148-150 accesible in google books), that doesn't mean observables depend on a certain gauge any more than the fact that in the relativistic formulation of EM the Lorenz gauge being the only one compatible with a vanishing divergence of the 4-current does.

Gauge conditions were mentioned, for instance It was mentioned that TT gauge was useful for analyzing gravitational waves with laser interferometry.

The TT gauge is an additional gauge choice overimposed to the Lorenz gauge that makes the analysis easier.

Themost important point I think is that gauge conditions and/or coordinate choices cannot matter to experimental results. It seems to me like that basic point is getting lost somewhere in the math :(.

That basic point is of course unavoidable. I share the sentiment that it gets lost somewhere in the math. My posts are trying to find it back. I'm not getting much help so far.
There is an added problem in the GR case with respect to the EM case with which the analogy about gauge conditions is usually drawn. The main difference, besides the fact that Maxwell's equations are already linear so there's no need for a previous linearization of the equations like in GR, is that in EM one can formulate the wave equation without the potentials, that are the subject of the gauge freedom, it can be done just with the observable fields E, B with or without sources. This fact alone already ensures that the gauge conditions don't affect physical observables.

I would like to see exactly how this works in GR for gravitational radiation. The linearization is based in the gauge freedom of the metric (indeterminacy of the EFE on the metrics) that takes the role of the potentials in the EM analogy. In the equation $g_{\mu\nu}=η_{\mu\nu}+h_{\mu\nu}$ the decomposition between the flat background and the metric perturbation $h_{\mu\nu}$ is not uniquely determined and therefore there is not only gauge invariance in relation to the well known arbitrariness of the coordinates in which the equation is valid but also about the possible $h_{\mu\nu}$ metrics. Now it can be shown that the only way to obtain a wave equation from the linearized EFE is through the Lorenz gauge condition(that in GR also happens to be a coordinate condition). But I don't see exactly how to construct the wave equation based on the observable field, that in this case is the tidal field that is supposed to oscllate and propagate and that is represented by the Weyl curvature that appears in the absence of the Einstein tensor, with a derivation that uses the linearized EFE with Einstein tensor.

I guess it's got to do with how the Newtonian tidal force is represented in the geodesic deviation in the weak field. That in itself already demands the use of the Lorenz gauge.

 

[PeterDonis]

I don't see exactly how to construct the wave equation based on the observable field, that in this case is the tidal field that is supposed to oscllate and propagate and that is represented by the Weyl curvature that appears in the absence of the Einstein tensor, with a derivation that uses the linearized EFE with Einstein tensor.

No, the "field" that is analogous to $E$ and $B$ in electrodynamics is actually the Riemann tensor, not the Weyl tensor. The Weyl tensor is the part of the Riemann tensor that is not directly connected to the source; the Einstein tensor is the part that is (via the EFE). But you need both parts to construct the wave equation, just as you need the full EM field tensor $F$, which is the Lorentz-covariant representation of the field, to construct the EM wave equation, not just the part of $F$ that is not directly connected to the source.

(It can be difficult to see this parallel between GR and EM because we are not used to splitting up the EM field $F$ into parts analogous to the Weyl and Einstein tensors. We are used to splitting it up into $E$ and $B$, but that split is coordinate-dependent, whereas the split of the Riemann tensor into the Weyl and Einstein tensors is not. I'm not actually sure exactly what split of $F$ would correspond to the split of Riemann into Weyl and Einstein.)

 

[vanhees71]

I've not tried this, but could it be that the split of $F_{\mu \nu}$ you are thinking about for a "quasi-point-like particle" is the split in an instantaneous boosted Coulomb field (i.e., in the momentaneous rest frame of the particle you have just a Coulomb field), which contains the relation of the field to the sources (charge-current distribution four-vector) and the rest, which is a free radiation field?

 

[PeterDonis]

could it be that the split of $F_{\mu \nu}$ you are thinking about for a "quasi-point-like particle" is the split in an instantaneous boosted Coulomb field (i.e., in the momentaneous rest frame of the particle you have just a Coulomb field), which contains the relation of the field to the sources (charge-current distribution four-vector) and the rest, which is a free radiation field?

I see what you're saying: at any point the total field is a superposition of the Coulomb field due to the presence of sources at that point, and the radiation field describing propagation of the field from sources elsewhere. I'm not sure how this split would correspond to Maxwell's Equations, though; there's no way to split up $F$ itself into a part that only appears in the source-free equations, and a part that only appears in the equations with source.

 

[vanhees71]

No, what I mean is the field from a (in general accelerated) charge. Forget the notorious radiation-reaction problem. Then the four-potential is given by the Lienard-Wichert potentials and the electromagnetic field by the corresponding derivatives (see, e.g., Landau-Lifshitz vol. II, where this is derived in the most beautiful way). The electromagnetic field splits in a part that depends only on the velocity of the particle (taken at the "regarded time") and goes like $1/r^2$ with distance (for $r \rightarrow \infty$) and a part that's proportional to the acceleration that goes like $1/r$. The first term is indeed the instantaneously boosted Coulomb field (having of course both electric and magnetic components if $\vec{v} \neq 0$).

I'm not sure, however, whether this split in the electromagnetic fields in a "boosted Coulomb" and a "radiative" part is analogous to the split of the Riemann tensor in Weyl and Einstein part.

 

[pervect]

I've been trying to argue that the only gauge condition that the linearized gravity formulation of gravitational radiation(no other available to my knowledge) is consistent with is the harmonic (Lorenz) gauge. This is a well known fact( you can see it explicitly stated for intance in Ruffini's textbook "Basic concepts in relativistic astrophysics" pages 148-150 accesible in google books), that doesn't mean observables depend on a certain gauge any more than the fact that in the relativistic formulation of EM the Lorenz gauge being the only one compatible with a vanishing divergence of the 4-current does.

The TT gauge is an additional gauge choice overimposed to the Lorenz gauge that makes the analysis easier.

My position is that gauge choices are just that - a choice. When you're trying to solve the linearized equations, the Lorenz gauge is the obvious choice, because it makes the equations easier to solve and it's also the one you'll find in your textbooks. That doesn't mean that it would be incorrect and/or impossible in principle to do the analysis in some other gauge, just difficult and non-standard.

However, I also argue that the result of this gauge choice are coordinates that are not necessarily the best choice for interpreting the results, though it's obviously the best choice for solving the problem.

A related point I want to make, which I did a bit of checking my memory of this point, is that the gauge choice is equivalent to a coordinate choice. See for instance http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec10.pdf

Indeed, small deviations from $g_{\mu\nu} = \eta_{\mu\nu}$ may arise either because spacetime is perturbed from Minkowski, or because we perturbed the coordinate system (or both). So we must understand the implications of perturbing the coordinate system, or making a small gauge transformation.

.

Thus an infinitesimal change of coordinates in which the “grid” is displaced by the vector ξ changes the metric perturbation according to [[eq 17]]

$$\Delta h_{\mu\nu} = -\xi_{\mu,\nu} - \xi_{\nu,\mu}$$

Now, we already assume that $g_{\mu\nu} \approx \eta_{\mu\nu}$ to do linearized theory, but it's easiest to analyze the results when all the partial derivatives of $g_{\mu\nu}$ with respect to the coordinates $\frac {\partial g_{\mu\nu}}{ \partial x^{\rho}}$are zero, because this implies that all the Christoffel symbols are zero. We can't make this happen everywhere, but we can make it happen in a small local region, and then we can better understand the results physically in this small local region.

The issue is that the gauge choice that we made to make the linearized equations easy to solve doesn't have the property that $\frac {\partial g_{\mu\nu}}{ \partial x^{\rho}}=0$, and thus makes the physical interpretation more difficult. So we either need to deal with understanding the physical effects of the non-zero Christoffel symbols at an intuitive level (which is possible but not something suited to a brief post), or we need to convert from the coordinates implied by our gauge choice (which we did not because it was mandated, but because it was so much easier) to a more convenient set of coordinates. Because we know how tensors transform when we change coordinates, it's not all that difficult (at least in principle) to work the problem out in one set of coordinates, and change to another, easier-to-interpret set.

The set of coordinates with zero Christoffel symbols has a history and a name - it's just a special case of the Fermi-Normal coordinates I was referring to earlier. The special case is that all the Christoffel symbols are zero only for an observer with zero proper acceleration. If you have an accelerated observer, you can only make some of the Christoffel symbols zero (and you have to deal with something rather like the 'fictitious forces' you get in Newtonian physics for an accelerated frame), but when you have an observer whose proper acceleratio is zero, i.e. a geodesic observer, and you use Fermi-Normal coordinates, the Christoffel symbols all vanish - not everywhere, but in some small local region.T

 

[PeterDonis]

I'm not sure, however, whether this split in the electromagnetic fields in a "boosted Coulomb" and a "radiative" part is analogous to the split of the Riemann tensor in Weyl and Einstein part.

I'm not sure either. For one thing, the Weyl tensor is not purely "radiative"; for example, the Weyl curvature in the vacuum region around a static, spherically symmetric mass does not "propagate" anywhere, it's static; but it's still Weyl curvature.

 

[RockyMarciano]

My position is that gauge choices are just that - a choice. When you're trying to solve the linearized equations, the Lorenz gauge is the obvious choice, because it makes the equations easier to solve and it's also the one you'll find in your textbooks. That doesn't mean that it would be incorrect and/or impossible in principle to do the analysis in some other gauge, just difficult and non-standard.

Yes, in general they are just a choice by definition. But GR is not your typical gauge theory, and I already mentioned the indeterminacy problem of the EFE above that leads to GR not having a unique solution to the initial value problem(this also happens in electrodynamics with the key difference that in GR the metrics don't act on the space like the potentials in electrodynamics but define the space where the dynamics take place). In the particular case of gravitational radiation the harmonic condition acts as an initial condition constraint equation, it is mathematically unavoidable to obtain a wave equation and in which a further gauge transformation(TT gauge) allows solutions with two polarizations perpendicular between them and wrt the propagation direction of the waves. It is equivalent to the coordinate condition used in cosmology in the 3+1 formulation, that it also works as a constraint equation for the initial value problem. In as much as the gauges are used as initial conditions they are no longer a choice in that sense(cf. Straumann 2.8) though they of course exploit the gauge freedom of the theory to be able to use them.