Subject: The initialvalue problem of GR: Fixing the
gauge variables?
Date: Fri, 03 Dec 2004 17:39:06 +0200 From: Dimi Chakalov <dimi@chakalov.net> To: James W York Jr <york@astro.cornell.edu> CC: Niall O'Murchadha <niall@ucc.ie>, Arlen Anderson <arley@physics.unc.edu>, Saul Teukolsky <saul@astro.cornell.edu>, Vincent Moncrief <vincent.moncrief@yale.edu>, Lazlo Szabados <szabados@konkoly.hu>, Stephen R Lau <lau@phys.utb.edu>, Harald P Pfeiffer <harald@astro.cornell.edu>, Gregory B Cook <cookgb@wfu.edu>, Roger Blandford <rdb@tapir.caltech.edu>, Marc Kamionkowski <kamion@tapir.caltech.edu>, E Sterl Phinney <esp@tapir.caltech.edu>, Katarzyna Grabowska <konieczn@fuw.edu.pl>, Jerzy Kijowski <kijowski@theta1.cft.edu.pl>, Don Salisbury <dsalisbury@austincollege.edu>, Sergio Dain <dain@aei.mpg.de>, Fred I Cooperstock <cooperstock@uvphys.phys.uvic.ca>, Christina Sormani <sormani@math.jhu.edu> BCC: [snip] Dear Professor York, I have an immodest request for clarification the initialvalue problem of general relativity. First, some history. Fifty years ago, Einstein wrote in a letter to his friend
M. Besso the
"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 initialvalue 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, http://Goddoesnotplaydice.net/Schwarz.html#freedom 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, http://Goddoesnotplaydice.net/Schwarz.html#addendum and the problem of time, http://Goddoesnotplaydice.net/points.html#Buridan_donkey Please see also [Ref. 5]. I think the first misleading step was the decomposition
of Einstein's
http://Goddoesnotplaydice.net/Kopeikin.html#note_2 Perhaps it could be possible to elucidate the generic
dynamics of
The issue is very old, from Einstein's paper of November 15, 1915, which appeared on December 2, 1915. Eightynine years later, we have some "multifingered" 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 "multifingered" time, http://Goddoesnotplaydice.net/Miller.html#transience However, if you or some of your colleagues can solve the puzzle, by fixing the gauge variables "online", onepointatatime, 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
References [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., Initialvalue 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, grqc/0405005 v1 "Einstein's theory of gravity permits the use of arbitrary
nonsingular
"In acceptable physical theories, if the initial data
constraints are
"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 wellposedness 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
[Ref. 4] A. Anderson and J.W. York, Hamiltonian
time evolution for general relativity, Phys. Rev. Letters, 81, 11541157
(1998);
"The application of these ideas to canonical quantum gravity
will appear elsewhere [30]."
[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 "multifingered" 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, grqc/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. 3843,
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 10^{99} 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 pointlike 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 pointlike 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 bidirectional 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' quantph/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 macroworld 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 nonArchimedean 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 nonArchimedean 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 nonArchimedean 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, gaugeinvariant 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 socalled 3+1 decomposition of Einstein's equations [Ref. 10]. It's like a skewer on which you stack 3D hypersurfaces, which is why I call it 'BBQ Interpretation of GR'. Just imagine pieces of meat/3D hypersurfaces on a skewer, which expand due to the heat/dark energy from the barbeque. But every welleducated gastronome knows that there must be a gap/joint between the pieces of meat/3D 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 wellknown 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 doubleslit 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 gaugedependent. 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 3D 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 "multifingered" time  in fact, infinitelyfingered 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 3D 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 nonArchimedean 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 quantumandgravitational 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 nonArchimedean "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 nonArchimedean reality that lives "between" the points of the pseudoRiemannian manifold of GR [Ref. 9]. All I can say to Einstein [Ref. 1] is this: your pseudoRiemannian manifold may be embedded in a genuine nonArchimedean reality. What we denote with 'physical reality' is indeed included in this unique and utterly holistic nonArchimedean 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 nonArchimedean 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 3D space'. I believe gravity should have applicable limits and hence should impose limit on the maximal volume of 3D space, just as 'temperature' is bounded from below by the numerically finite but physically unattainable 'absolute zero temperature'. More about the puzzles of the 3D 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+1D spacetime.
{x x x x x x}
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 infinitelyfingered time should have some "beginning". However, if we realize that there is a vertical component of that same infinitelyfingered 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 1D 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
[Ref. 8] Branko Dragovich, mathph/0306023, Sec. 2, NonArchimedean 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 ntimes 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,
grqc/0411053
v1.
[Ref. 10] Miguel Alcubierre, The status of numerical relativity, December 6, 2004, grqc/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 ChoquetBruhat [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 threedimensional metric y_{ij} (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 { y_{ij}
, K_{ij}} 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 ArnowittDeserMisner equations, or ADM
for short. They represent the starting point of practically all of 3+1
numerical
"The ADM evolution equations introduced in the previous section are in fact highly nonunique. (...) This nonuniqueness 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 wellposedness 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.
"The Einstein equations provide us
with evolution equations for the spatial
