Subject: The initial-value problem of GR: Fixing the gauge variables?
Date: Fri, 03 Dec 2004 17:39:06 +0200
From: Dimi Chakalov <>
To: James W York Jr <>
CC: Niall O'Murchadha <>,
     Arlen Anderson <>,
     Saul Teukolsky <>,
     Vincent Moncrief <>,
     Lazlo Szabados <>,
     Stephen R Lau <>,
     Harald P Pfeiffer <>,
     Gregory B Cook <>,
     Roger Blandford <>,
     Marc Kamionkowski <>,
     E Sterl Phinney <>,
     Katarzyna Grabowska <>,
     Jerzy Kijowski <>,
     Don Salisbury <>,
     Sergio Dain <>,
     Fred I Cooperstock <>,
     Christina Sormani <>
BCC: [snip]

Dear Professor York,

I have an immodest request for clarification the initial-value problem of general relativity. First, some history.

Fifty years ago, Einstein wrote in a letter to his friend M. Besso the
following [Ref. 1]:

"I consider it quite possible that physics cannot be based on the field concept, i.e., on continuous structures. In that case, *nothing* remains of my entire castle in the air, gravitational theory included, [and of] the rest of modern physics."

Thirty years ago, you and Prof. Niall O'Murchadha wrote your seminal paper on the initial-value problem of general relativity [Ref. 2], and six months ago you raised the issue again [Ref. 3]. In this latest paper, you wrote:

"I will make no suggestion here about how to fix gauge variables."

If possible, may I ask you for clarification of the crux of the problem and its possible solution by fixing the gauge variables,

I extend this request to all colleagues of yours. Also, please tell me about your proposal for canonical quantum gravity [ref [30] in Ref. 4].

The reason for quoting Einstein is my conviction that we need brand new ideas on the nature of continuum to solve the Cauchy problem,

and the problem of time,

Please see also [Ref. 5].

I think the first misleading step was the decomposition of Einstein's
equations into evolution equations and constraint equations [Ref. 6],
because the "evolution equations" do not, and cannot capture the generic dynamics of gravitational field [Ref. 7], as I tried to explain at

Perhaps it could be possible to elucidate the generic dynamics of
GR and hence solve the initial-value problem *only* by solving the
problem of continuum. Hence I'm trying to introduce a non-Archimedean object (called 'global mode of spacetime'), but perhaps there are other possibilities, say, by fixing the gauge variables. I very much hope to learn more from you and your colleagues.

The issue is very old, from Einstein's paper of November 15, 1915, which appeared on December 2, 1915. Eighty-nine years later, we have some "multi-fingered" time [Ref. 5], as modeled with your shift vectors and lapse functions (which are arbitrary and merely pertain to an arbitrary coordinate system). I'm inclined to suggest a very different interpretation of this "multi-fingered" time,

However, if you or some of your colleagues can solve the puzzle, by fixing the gauge variables "online", one-point-at-a-time, as they "evolve" on the Cauchy surface, I'll immediately drop my speculations and will study your papers on canonical quantum gravity, particularly your solution (if available) to the Hilbert space problem.

Please excuse my long email, and be assured that I will keep your feedback strictly private and confidential.

Kindest regards,

Dimi Chakalov


[Ref. 1] Abraham Pais, 'Subtle is the Lord...': The Science and the Life of Albert Einstein, Oxford University Press, New York, 1983, p. 467.

[Ref. 2] N. O'Murchadha and J. W. York, Jr., Initial-value problem of general relativity. I. General formulation and physical interpretation, Phys. Rev. D, 10, 428 (1974).

[Ref. 3] James W. York Jr., The Initial Value Problem Using Metric and Extrinsic Curvature, gr-qc/0405005 v1

"Einstein's theory of gravity permits the use of arbitrary non-singular
spacetime coordinates with a spacetime metric which is not specified
beforehand, but is the desired solution. This is what Einstein meant by his theory being "generally covariant." (...) Gauge freedom comes with a price, a corresponding restriction on the field values on a
constant-time hypersurface. A detailed discussion is found, for example, in [Tr]. These restrictions constitute what is called the initial value problem for the theory in question.

"In acceptable physical theories, if the initial data constraints are
satisfied at some time, then the remaining "evolution equations" carry the initial data forward (or backward) in time while preserving the
constraints. Thus one says that the "Cauchy problem" of the theory is well posed. That is, (1) the initial data constraints are satisfied and
(2) the evolution equations propagate the initial data in such a way
that the constraint continues to hold on every other spacelike
hypersurface. But a further problem also arises from gauge freedom.

"Theories with gauge freedom, such as electromagnetism and general relativity, are said to be both "overdetermined" and "underdetermined." They are overdetermined because there are constraints at each time that limit the freedom of the variables that are propagated, the dynamical variables. They are underdetermined because the gauge freedom means the equations of the theory cannot determine a fully unique solution. By gauge transformations, some of the variables can be changed. These changes do not alter the intrinsic physical meaning of a solution but they nevertheless can be vital in the description and recognition of the solution. That the problem of being overdetermined need be resolved at one time only (in principle), and that the gauge freedom in changing certain variables does not disturb either the feature just mentioned or the physical uniqueness of the problem are part and parcel of the well-posedness of a Cauchy problem.

"I will make no suggestion here about how to fix gauge variables. But there is something to say about just which variables are actually
"gauge" variables. One finds not the lapse and shift, but the
"densitized lapse" [Ash, An-Yo] and the shift."
[An-Yo] A. Anderson and J.W. York, Hamiltonian time evolution for general relativity, Phys. Rev. Letters, 81, 1154-1157 (1998).

[Ref. 4] A. Anderson and J.W. York, Hamiltonian time evolution for general relativity, Phys. Rev. Letters, 81, 1154-1157 (1998);

"The application of these ideas to canonical quantum gravity will appear elsewhere [30]."
[30] A. Anderson and J.W. York, in preparation.

[Ref. 5] Hamiltonian constraint,

"What does it mean? It means the Hamiltonian time flows maps points on the constrained subspace to points in the same orbit (generated by the constraints). Since physical observables are only defined after quotienting out the orbits, what this means is the Hamiltonian time flows map orbits to the SAME orbits. (...) (W)e can eliminate time since the time evolution of the orbits is trivial. This might sound bizarre physically, until you realize we don't really measure things at a particular (absolute) time. We only measure things relative to a dynamical clock. Even then, it still seems bizarre. (...) Sure, they both lie in the same orbit, but does this call into question the statement that states in the same orbit are physically indistinguishable? Perhaps.

"Note that for general relativity, though, we actually have infinitely many independent Hamiltonian constraints, one for each spatial point. This is because we have "multi-fingered" time where each spatial point has its own (nondynamical) "clock". The mathematics of this is covered by shift vectors and lapse functions. (The shift vectors are covered by the spatial diffeomorphism constraints).

[Ref. 6] Harald P. Pfeiffer, The initial value problem in numerical relativity, gr-qc/0412002, Sec. 2.

"Initial data forms the starting point for any evolution. For Einstein’s equations, the most widely used method to construct initial data is the conformal method, pioneered by Lichnerowicz [30] and extended to a more general form by York and coworkers [49,34,52]."

"Einstein's equations decompose into evolution equations and constraint equations for the quantities g_ij and K_ij . The *evolution equations* determine how g_ij and  K_ij are related between neighboring hypersurfaces, Eqs. (2.2) and (2.3)

"Cauchy initial data for Einstein’s equations consists of (gij ,Kij) on one hypersurface satisfying the constraint equations (2.4) and (2.5). After choosing lapse and shift (which are arbitrary and merely choose a specific coordinate system), Eqs. (2.2) and (2.3) determine (g_ij , K_ij) at later times. Analytically, the constraints equations are preserved under the evolution. In practice, however, during numerical evolution of Eqs. (2.2) and (2.3) or any other formulation of Einstein’s equations, many problems arise."

[Ref. 7] Robert R. Caldwell and Marc Kamionkowski, Echoes from the Big Bang, Scientific American, January 2001, pp. 38-43,

p. 39: "The fantastically rapid expansion of the universe immediately after the big bang should have produced gravitational waves. These waves would have stretched and squeezed the primordial plasma, inducing motions in the spherical surface that emitted the CMB radiation. These motions, in turn, would have caused redshifts and blueshifts in the radiation’s temperature and polarized the CMB. The figure here shows the effects of a gravitational wave traveling from pole to pole, with a wavelength that is one quarter the radius of the sphere."

Note: To explain the problem of continuum, as stressed by Einstein in 1954 [Ref. 1], consider the following task.

You have to cover the wall of your bathroom with tiles (classical task, R. Penrose tried it many years ago). The joints between tiles cannot be strictly zero, these gaps are needed for the tiles to be (i) countable and (ii) distinguishable. If we don't have the latter, we can't make a set of tiles.

Fine, but we also want to make a continuum of tiles at the scale of tables and chairs, such that all tiles will approach asymptotically zero size. Lee Smolin, for example, claims that we have 1099 such "tiles/atoms of volume" in every cubic centimeter of space, but I'm a bit skeptical.

The number is huge but COUNTABLE, even if it were in the range of Googles. We need to 'have our cake and eat it': in order to make a perfectly smooth continuum, the "number" of tiles should be UNcountable, but we also need to insert some joints between the tiles, in such a way that only one tile can be explicated at the scale of tables and chairs. This is the job of the putative 'global mode of spacetime', which resides in these joints. We call such a tile 'event', we model it with a mathematical point, and attach to it some point-like number. But the joints are totally hidden due to the "speed" of light.

For example, if we want to solve the inner product problem in quantum gravity, we have to explain how one of these tiles can be explicated at the scale of tables and chairs, say, in calculating the point-like values of physical quantities. But we do not operate with a 'set of tiles', firstly, and secondly -- to make a chain of distinguishable tiles/states of a physical system and hence explain the obvious dynamics of GR, we need the joints.

However, these joints should not be modeled with the lapse and shift, nor with the "densitized lapse" and the shift [Ref. 4]. This is again the case of the Buridan donkey paradox: before making an infinitesimal timelike displacement, the donkey needs to have this next step completely prepared and fixed by the bi-directional talk between matter and geometry on the whole Cauchy surface, which, on the other hand, requires this next infinitesimal step to be already completed.

Briefly, in order to move, you need to produce time, but in order to produce time, you need to move. Mother Nature has solved this Catch 22 type paradox, only we don't know how. Maybe She handles covariant derivatives differently. I hope to hear from James W. York and his colleagues on this issue.

See the paradox of [tiles --> zero], known as Thompson's Lamp paradox. I learned from Roland Omnes' quant-ph/0411201 v1, Sec 5, "Infinitesimal reduction", that Karel Kuchar has some ideas for solving the puzzle of macroscopic uniqueness of spacetime, although he has considered the case of pure gravity only. Qui vivra, verra.

If we cannot reveal the dynamics of GR, we will certainly fail in recovering the macro-world from quantum gravity. We don't need any preferred foliation/preferred time. What we need is hidden in the joints "between" the tiles. It isn't "nothing", it's just a non-Archimedean reality [Ref. 8].

The Archimedean axiom states that any given segment of a straight line can be surpassed by adding arbitrarily small segments of the same line [Ref. 9]. The reason why the joints between the tiles are non-Archimedean reality is based on the ability of these "gaps" to absorb infinitely many -- actual infinity -- possible states of physical systems, and to keep them in ONE state. This is conditio sine qua non for applying the principle of general covariance and for generating infinitely many possible, and equally physical, solutions for a given covariant equation from a known one by means of transformations within Einstein's GR. Hence all physical laws have a covariant form.

If you are curious about your non-Archimedean reality, check it with your own brain. More here.

If you're interested in the recent status of the initial value problem of GR, read the paper by Miguel Alcubierre [Ref. 10]: the lapse and shift do not refer to any observable, gauge-invariant entity. They represent our freedom in choosing the coordinate system, as we know from Einstein.

The Einstein equations provide us with some sort of "evolution" for the spatial metric and extrinsic curvature, but they do not say anything about the "evolution" of the lapse function and shift vector [Ref. 10]. Hence you may wonder, what is this "global time function  t " in the so-called 3+1 decomposition of Einstein's equations [Ref. 10]. It's like a skewer on which you stack 3-D hypersurfaces, which is why I call it 'BBQ Interpretation of GR'. Just imagine pieces of meat/3-D hypersurfaces on a skewer, which expand due to the heat/dark energy from the barbeque. But every well-educated gastronome knows that there must be a gap/joint between the pieces of meat/3-D hypersurfaces, represented with the lapse function [alpha] that measures the "proper time between adjacent hypersurfaces" [Ref. 10].

Isn't it obvious that the initial value problem is beyond the scope of the BBQ Interpretation of GR? You can't "fix" the gauge variables [Ref. 3], because you don't have access to them. You can't even touch them, because they are frozen to you: "the evolution equations preserve the constraints, that is, if they are satisfied initially they will remain satisfied at subsequent times." [Ref. 10] However, these "subsequent times" pertain to the expansion of the pieces of meat on your skewer, and the direction of their expansion is orthogonal to the 'global time' of the skewer, as we know from our BBQ experience.

You cannot map (i) the proper time of the evolution of the lapse function "between" adjacent hypersurfaces to (ii) the proper time of inflating the pieces of meat on your skewer. You know that there was inflation period of the universe, you have measured its effects imprinted on the CMB radiation [Ref. 7], you can even hear them. Sure. But you can't reveal the cosmological time in the BBQ Interpretation of GR, since it isn't there. A similar case is well-known from QM: many people have tried in the past to detect the hypothetical empty waves implied in QM, simply because we all have seen the diffraction pattern in the double-slit experiment. No way, there is no time parameter in QM, and we cannot map the intrinsic time of the dynamics of the quantum waves to the time parameter read by a clock. And yet many people believe, M. Alcubierre included [Ref. 10], that we can detect the gravitational waves, simply because we all have seen their pattern [Ref. 7] and because the cosmological time obviously exists. Yes, it does exist, but it is "orthogonal" to our local time of the inflating pieces of meat. And you can do nothing about it. It is not an observable, since it is gauge-dependent. Hence we do not know the dynamics of GR, because do not know the "time parameter" of GR, just as we don't know the "time parameter" of quantum waves. It may be frustrating, but that's the way it is.

Unless, of course, James W. York Jr. can fix the gauge variables "dynamically", while being confined inside the expanding pieces of meat (also known as 3-D hypersurfaces). However, he has to take a good "global look" at the whole skewer, which is impossible, I'm afraid. He is inevitably confined inside its inflating/expanding piece of meat (driven by the dark energy of the barbeque with constant acceleration, sources say), and hence the task boils down to having access simultaneously to the whole "multi-fingered" time -- in fact, infinitely-fingered time -- that pertains to every spatial point from the Cauchy surface. That's the only way for James W. York Jr. to handle infinitely many independent Hamiltonian constraints, one for each spatial point. [Ref. 5] This is also "the starting point of practically all of 3+1 numerical relativity", according to M. Alcubierre [Ref. 10].

Thus, the task of fixing the gauge variables "dynamically" can be defined as follows: The lapse and shift must be chosen dynamically as functions of the evolving geometry [Ref. 10], that is, we choose the coordinates "as we go" along both 3-D hypersurfaces and the skewer.

Only you don't have the skewer in the BBQ Interpretation of GR. Since you're fully confined inside your expanding 'piece of meat', all you can say about the skewer is that all spatial points from your piece of meat are "points" from the skewer as well. However, the skewer itself is a non-Archimedean reality, it can absorb infinitely many "points" from the 'pieces of meat', but the skewer itself it is not being stretched in conjunction with the stretching of the local 'pieces of meat'. (As Karel Kuchar eloquently put it, neither the ribs nor the fabric of the umbrella/skewer are being "appreciably stretched".) In other words, the proper time of the lapse that "separates" adjacent pieces of meat/hypersurfaces [Ref. 10] cannot be read by your clock. The latter can only interpret this proper time of the lapse as being "totally frozen" by its "donkian Hamiltonian".

My hunch is that the quantum-and-gravitational waves "live" on a null plane only, hence any inanimate clock would interpret their proper time as being zero. Just as the human self, these waves do not change along the time read by an inanimate clock, although all events read by the inanimate clock are also events of the non-Archimedean "evolution" of these waves (just as all spatial points from your 'pieces of meat' are "points" from the skewer as well; please see above.) Their true Hamiltonian has to be defined with the putative global mode of time -- the proper time of the skewer. It is a genuine non-Archimedean reality that lives "between" the points of the pseudo-Riemannian manifold of GR [Ref. 9]. All I can say to Einstein [Ref. 1] is this: your pseudo-Riemannian manifold may be embedded in a genuine non-Archimedean reality. What we denote with 'physical reality' is indeed included in this unique and utterly holistic non-Archimedean reality, and the effects from it begin to emerge from the scale of tables and chairs toward the worlds of the Large and the Small. An electron is "small" and a galaxy is "large" only with respect to the observers at the scale of tables and chairs, while an observer placed in the non-Archimedean reality (such as Einstein, perhaps) would find out that their metric is being altered in such a way that the size of the electron equals the size of the galaxy. Such kind of 'mutual penetration of the Large and the Small' should be bounded below by the Planck scale, and bounded above by some numerically finite but physically unattainable 'maximal volume of 3-D space'. I believe gravity should have applicable limits and hence should impose limit on the maximal volume of 3-D space, just as 'temperature' is bounded from below by the numerically finite but physically unattainable 'absolute zero temperature'. More about the puzzles of the 3-D space here, and the 'shape of space' (Graham Nerlich) at EPS 13 in July 2005.

NB: Please feel free to ignore all my speculations, just try to fix the gauge variables [Ref. 3]. If you succeed, I believe you will know how to define energy in GR, how to solve the Cauchy problem and the inner product problem, and will probably understand the nature of those empty waves called gravitational waves.

If James W. York Jr. and/or some of his colleagues finds any error in my notes above, or if some of them can fix the gauge variables [Ref. 3], I will not post here their rigorous mathematical proof, but will instead delete this sentence, and will also remove everything about the BBQ Interpretation of GR, as proposed today, 7 December 2004. Here's the idea in 1+1-D spacetime.

 {x  x x  x x  x} 
   \ | |  | | /
    {x x  x x}
     \ |  | /
      \|  |/
      {x  x}

Shortly "after" The Beginning at  {?} , we have the first 1-D space slice with just two spatial points, then we have the next slice with 4 spatial points, then the third slice with 6 points, etc. Obviously, the 1-D space is expanding along the skewer, and if we are 1-D creatures, we will notice that we have "time" as well: we can achieve an infinitesimal timelike displacement  inside our 1-D slice, hence will enjoy a genuine 1+1-D spacetime. However, each and every step of the cosmological time arrow along the skewer has two components, vertical and horizontal. Mother Nature makes this elementary increment of the cosmological time en bloc, while we can record only our horizontal displacement inside the 1-D slice, and only post factum, due to the so-called speed of light. Hence we're deeply puzzled, because we can look backwards along the horizontal direction of our ever-expanding spacetime, and discover the imprints from the Beginning in this horizontal 1+1-D slice [Ref. 7]. All horizontal slices are 1+1-D continuum of events, because, again, the vertical component of our displacement is hidden due to the speed of light. This is as it should be, as the lapse and shift represent our freedom along the vertical component, i.e. they are "gauge" functions.

Thus, we attribute our obvious dynamics to the horizontal component only, and try to detect some gravitational waves that we honestly believe have originated within the horizontal slices only. We also deeply believe that 13.7 billion years ago, as measured within the horizontal slices only, our universe has been in the state  {?} , since our infinitely-fingered time should have some "beginning". However, if we realize that there is a vertical component of that same infinitely-fingered time, the situation will become far more interesting.

And now, all we have to do is to choose the lapse and shift dynamically, as functions of the evolving geometry [Ref. 10], that is, we choose the coordinates "as we go" along both 1-D slice and the skewer. But can we fix these gauge variables [Ref. 3] without the vertical component of the dynamics of General Relativity?

D. Chakalov
December 3, 2004
Last update: December 7, 2004, 9:42:34 GMT

[Ref. 8] Branko Dragovich, math-ph/0306023, Sec. 2, Non-Archimedean Geometry,

"Recall that having two segments of straight line of different lengths a and b, where a < b, one can overpass the longer b by applying the smaller a some n-times along b. In other words, if a and b are two positive real numbers and a < b then there exists an enough large natural number n such that na > b. This is an evident property of the Euclidean spaces (and the field of real numbers), which is known as Archimedean postulate, and can be extended to the standard Riemannian spaces."

[Ref. 9] Diego Meschini et al., Geometry, pregeometry and beyond, gr-qc/0411053 v1.

[Ref. 10] Miguel Alcubierre, The status of numerical relativity, December 6, 2004, gr-qc/0412019 v1.

Report on plenary talk at the 17th International Conference on General Relativity and Gravitation (GR17), held at Dublin, Ireland, July 2004.

"2. The 3+1 decomposition

"The Einstein field equations are usually written in fully covariant form, with no distinction between space and time. This is elegant and mathematically powerful, but it is not very useful when one is interested in studying the evolution in time of a gravitational system starting from some appropriate initial data, the so called "initial value problem".

"It is well known from the seminal work of Choquet-Bruhat [8] that general relativity does allow an initial value formulation. Today there are three main procedures in which one can obtain such a formulation: the "Cauchy" or "3+1" formalism, the "conformal" formalism, and the "characteristic" formalism.

"In the 3+1 approach one introduces a global time function  t  whose levels sets are the hypersurfaces defining the foliation. One then defines three main ingredients: 1) The three-dimensional metric yij (i,j = 1, 2, 3) that measures distances within a given hypersurface, 2) the "lapse" function [alpha] that measures proper time between adjacent hypersurfaces, and 3) the "shift vector" [beta] that measures the relative speed between observers moving along the normal direction to the hypersurfaces, and those keeping constant spatial coordinates.

"The next step is to decompose the Einstein equations. Doing this one finds that they naturally split in two groups. One group involves no time derivatives and represents constraints that must be satisfied at all times. The "Hamiltonian constraint" is given by (...). The existence of the constraints implies, in particular, that one is not free to specify the 12 dynamical quantities { yij , Kij} as arbitrary initial conditions.

"It is important to mention that the Bianchi identities imply that the evolution equations preserve the constraints, that is, if they are satisfied initially they will remain satisfied at subsequent times. The equations just described are know as the Arnowitt-Deser-Misner equations, or ADM for short. They represent the starting point of practically all of 3+1 numerical
relativity. The reader interested in seeing how these equations are derived is referred to the original ADM article [13] or the classic review by York [14].
[14] J. W. York, in Sources of Gravitational Radiation, edited by L. L. Smarr (Cambridge University Press, Cambridge, UK, 1979), pp. 83-126.

"The ADM evolution equations introduced in the previous section are in fact highly non-unique. (...) This non-uniqueness of the evolution equations is well known. For example, the original equations of ADM [13] differ from those of York [14] precisely by the addition of a multiple of the Hamiltonian constraint. The reformulation of York can be shown to be better behaved mathematically [15] and has become the standard form used in numerical relativity.

"A key point that one has to worry about when studying the Cauchy problem is the well-posedness of the system of evolution equations, by which one understands that solutions exist (at least locally) and are stable in the sense that small changes in the initial data produce small changes in the solution.

"5. Gauge

"The Einstein equations provide us with evolution equations for the spatial
metric  yij  and extrinsic curvature  Kij . However, they do not say anything about the evolution of the lapse function [alpha] and shift vector [beta]. This is as it should be, as the lapse and shift represent our freedom in choosing the coordinate system, i.e. they are "gauge" functions. Choosing a good gauge is crucial when one solves the Einstein equations numerically, and one must choose carefully if one wants to avoid coordinate (and physical) singularities and to cover the interesting regions of spacetime. One can’t just specify the lapse and shift as a priori known functions of spacetime for a simple reason: Which functions are a good choice? The lapse and shift must therefore be chosen dynamically as functions of the evolving geometry, that is, we choose the coordinates as we go."