|From: Dimi Chakalov <firstname.lastname@example.org>
Subject: Modern Differential Geometry For Physicists 2nd Edn
Date: Wed, 21 Jun 2006 06:41:45 +0300
RE: Modern Differential Geometry
For Physicists 2nd Edn
Piotr Chrusciel: "I would be delighted
to see a new, extended, edition with
Dear Dr. Chrusciel,
Couldn't find such examples in your GR17 lecture "Recent results in mathematical relativity" either.
If you tell your students that a 3+1 decomposition of the gravitational field variables -- rather than maintaining manifest 4-covariance -- is the right approach, I don't believe they would find "Modern Differential Geometry For Physicists" helpful.
It seems to me that the Hamiltonian formulation of GR does not allow modification of GR, such that some dynamic "dark" energy would emerge, and since the latter is 18 times more than the stuff discussed in GR textbooks, I decided to share with you my concerns. I mean, if kids learn GR in a wrong way, I'm afraid no math can help them, even from Chris Isham.
If you know how to introduce dynamic
"dark" energy on some Cauchy hypersurface,
please do write me back. I would be delighted to read all
Measure Problem in Cosmology
Dear Professor Gibbons,
If we decide to abandon the anthropic or "top down" parapsychology, it seems to me that the solution to your problem (hep-th/0609095 v1, p. 20) requires the solution to the two generic problems in GR (hep-th/0609095 v1, p. 1), as suggested at
It shouldn't be surprising that the garbage we've swept under the rug from the outset has emerged at the end: "some new ingredient or dynamical principle is needed in the theory" (hep-th/0609095 v1, p. 20). The way I see it, the very Hamiltonian structure of GR has to be modified, by introducing a brand new degree of freedom (the so-called 'global mode of spacetime'). There is simply no other choice. I do believe your colleagues will agree, after reading carefully your outstanding paper.
I will be happy to elaborate, should you and/or your colleagues have questions on the proposal at the link above. The idea is so simple that even my 13-year old daughter was able to grasp it.
If necessary, I can also offer my comments on N. Turok's idea of "back-reaction of quantum fluctuations on the process of inflation itself",
as debunked by Akihiro Ishibashi and Bob Wald (gr-qc/0509108 v3).
and Neil Turok, The Measure Problem in Cosmology,
p. 1: "The problem of comparing different possible histories of the universe, and assigning a probability to each, is central to theoretical cosmology. We cannot expect a fundamental theory to predict precisely what we see today (...).
"There are several problems peculiar
to cosmology when attempting to construct a statistical theory. First,
there there is the problem of general covariance. There is no absolute
notion of space or time in general relativity:
these are properties of each particular classical solution of the field
equations. Second, the solutions generically possess singularities in the
past or the future, where the field equations break down in finite
p. 20: "... some new ingredient or
dynamical principle is needed in the theory, in order to explain why inflation
began. (...) The main conclusion we draw from this work is that the question
of why or how inflation started remains a deep mystery, and a great challenge
for fundamental theory. Until that question is answered, we should remain
cautious about claiming that cosmology’s classic puzzles are "solved"."
Note: May I add some remarks "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (On the hypotheses that form the foundations of geometry), after Bernhard Riemann's Inaugural Lecture in Göttingen. Back in 1854, he couldn't imagine that nearly 96 per cent from the stuff in the universe would turn out to be some "dark" effect of geometry.
If you follow the link above, you'll read the following claim: "This global/background time provides the fundamental clock that re-creates [X] -- not 24 times per second, as with the movie reel, but infinite number of time per second, as we know since the time of Zeno." Notice that the "duration" of the fundamental "tick" (fundamental timelike displacement), which can completely -- without any gaps! -- fill in any finite time interval from the cosmological time (as read by your wristwatch), is presented with our beloved "point": 'the grin of the cat without the cat', as observed by Alice. Put it differently, you've moved from physics to geometry, and are ready to study Chris Isham's lecture notes "Modern Differential Geometry For Physicists", in which you will learn in precisely what sense "a point x E X can be said to be 'near' to another point y E X" (Sec. 1.2.1, p. 3), and then will be convinced that "differentiation can be defined, thus opening up the very fruitful idea that the dynamical evolution of a physical system can be modelled by differential equations defined on the spacetime" (pp. 59-60 and Fig. 2.1).
But before you plunge into the math of the "block universe", recall Thompson's lamp paradox and ask yourself a simple question: what kind of "set" -- if any -- is 'the set of points' that can completely fill in any finite timelike interval? ["First, one needs a mathematically precise notion of the "set of points" that constitute spacetime (or that constitute a surface in ordinary geometry)", says Robert M. Wald in gr-qc/0511073 v1; see also Diego Meschini, Markku Lehto, and Johanna Piilonen, Geometry, pregeometry and beyond, gr-qc/0411053, Sec. 2.1 and footnote 12.]
Does this 'set of points' refer to some separable "space of states" comprised of, at most, a countably infinite elements, as in the case of the Hilbert space (cf. Mikio Nakahara's textbook below)? My answer is in the negative, hence I claim that we're dealing with a brand new entity -- a holistic non-Archimedean reality (check it out, with your brain, here and here; a brief outlook here). It carries the changing states of physical systems from one "point" E to the nearest "point" F (principle of locality), while preserving the 'sameness' of physical systems and the spacetime metric (the generalized distances between neighboring point-events E and F, see Diego Meschini et al., gr-qc/0411053, p. 6). There is no such thing in differential geometry, because the notion of the "set of points" that constitute spacetime, or that constitute a surface in ordinary geometry (cf. Bob Wald above), does not include these fundamental properties of this "set". They are exposed in Thompson's lamp paradox, and are widely known after Lucretius (concrete example here).
In math language, this holistic non-Archimedean reality makes the Hausdorff topological space connected (Modern Differential Geometry for Physicists, p. 61, footnote 1). We tacitly presume that it is "connected", but the phenomenon that "binds" the neighboring spacetime points is excluded from GR from the outset. There is no background structure whatsoever in Einstein's GR to facilitate the elementary timelike displacement -- from one "point" E to the nearest "point" F (cf. above). There is nothing "in between" these two neighboring spacetime points: no 'bare points' in GR are allowed. Hence we invoke a miracle to explain the dynamics of GR, and introduce this miracle by hand, as presumption that the Hausdorff topological space is indeed connected .
Of course, I could be all wrong, since Chris Isham claims that I "do not know enough theoretical physics to help with any research in that area." He delivered this bold statement on Wed, 23 Oct 2002 19:24:15 +0100, but hasn't so far provided any evidence whatsoever. I found this omission extraordinary, since I believe have been following his research program in quantum gravity (C. Isham, gr-qc/9310031), hence it shouldn't be difficult for him to find at least one error in my proposals.
The way I see it, the crucial difference is in the logic of propositions: Chris Isham denies the existence of 'potential reality' and subsequently 'potential point(s)', but suggests a modification of the truth values, by "neither true nor false but somewhere in between" (C. Isham, 2005). Thus, his approach is fully compliant with the "block universe" (George F. R. Ellis, gr-qc/0605049), while I suggest a clear differentiation of the 'past' vs the potential future of 'potential reality': what we call 'events' constitute only the local mode of spacetime, they are exclusively in the absolute past of the master/universal time arrow, and their truth value is either 'true' or 'false'. Also, in the local mode of spacetime, we're dealing with facts in the past, hence can always formulate mutually exclusive propositions, define an exhaustive set of such mutually exclusive propositions, such that their probabilities sum to unity, and feel "safe" with such local unitary physics: it will certainly look "time-reversible".
The potential reality, however, is open to 'the unknown unknown', and the rules of unitary evolution don't hold. It resides in the global mode of spacetime, by way of 'potential point(s)', hence its truth values are UNdecidable à la Gödel-- both 'true' AND 'false', or simply Yain. This is a slight modification of Jan Lukasiewicz' three-valued symbolic logic, after Aristotle [Ref. 1]. Bottom line here is the conjecture that the absolute past of the whole universe is being "enlarged" or rather "enriched" after each and every "tick" of the master/universal time arrow, much like the memory of the human brain. If you're interested, just follow the links.
NB: To prove me wrong or find an error, please try to introduce some dynamic "dark" energy on some Cauchy hypersurface (as I asked Piotr Chrusciel above), and send your manuscript to Chris Isham, regarding his paper "On the Emergence of Time in Quantum Gravity", gr-qc/9901024. I too would be delighted to read it. The way I see it, the "remnant" from the dynamic "dark" energy, left on a frozen Cauchy hypersurface, could only be (i) vanishing small, since it matches the "thickness" of the Cauchy hypersurface along the vertical axis of the "ladder", and (ii) constant (the cosmological constant [lambda] should be constant in time and space, cf. G F R Ellis and H van Elst, Cargèse Lectures 1998), i.e., it cannot be dynamical (R. Penrose, 2004, p. 777), because you've wiped out the "vertical" axis of the global mode of spacetime, as depicted with the ladder metaphor here. Hence you cannot introduce some "ideal endpoints", and the apparent causal structure of spacetime remains a total mystery: you need some sort of "boundary of spacetime" (see the ‘finite infinity’ proposal of George Ellis), and an "ideal endpoint" at the very "end" of the boundary (and at the "singularity" itself), fixed by the "context" of the Holon from the global mode of spacetime. Think of the Geroch-Kronheimer-Penrose "ideal endpoint" as the "endpoint" denoted with '2 min' from Thompson's lamp paradox. Now, all you have to do to resolve the Cauchy pathologies is to set such "context" on the Cauchy hypersurface, along the lines suggested by George Ellis (try a null surface though). Any time you look at your wrist watch, you observe a "parameter" of the dynamics of the gravitational field, which shouldn't be observable at all. So, you need the "context" of the Holon, which fixes the "ideal endpoints" in the local mode of spacetime, to explain this miracle.
In my opinion, the reason why numerical integration of the linearized ADM equations is possible for rather short times or for physically unrealistic symmetrical cases only (cf. Vasileios Paschalidis et al., gr-qc/0511075 v2) is that the 'linearized gravity' itself is an oxymoron. It may be possible to develop some custom-made "patches" only, in which the lapse function and the shift vector would be feedback-adjusted gauges (more here), but, since I "do not know enough theoretical physics to help with any research in that area" (C. Isham), I will patiently wait to hear from Piotr Chrusciel and his distinguished colleagues.
Last but no least, please bear in
mind the generic 'can of worms' in the Dirac-ADM
hypotheses: the treatment of 3-D space. We accept, after Einstein's
Hole Argument (Alan Macdonald, 2001), that
"the points occurring in the base sets of diffrerentiable manifolds with
which general relativity models spacetime should not be reified as physically
real" (Butterfield and Isham, 1999,
p. 33), yet we introduce a fixed background of bare points
with spatial relations in 3-D space: inside
vs outside. There is no absolute
time in Einstein's GR, but there is a fixed background of bare "points"
with fixed spatial properties. Put it differently,
the intrinsic property of classical 3-D space is that we can draw a sphere
with finite radius r at any point "here-and-now", such
that there will be a set of points inside the surface of the sphere,
and another set of points outside it, and then it is always possible
to vary r in such a way that an observer or object will
be either inside or outside the sphere, without losing its identity. Thus,
is bounded from below by the size of objects for which quantum effects
become significant, and bounded from above by the ever-increasing cosmological
horizon. Once you have such classical 3-D space, you can always play the
game 'catch a lion in Sahara' along the
axis r , and, because this 3-D space is isotropic and homogeneous,
you can orient r toward any direction. But how do we
get the point "here-and-now"?
Hence there is 'problem of time in quantum gravity' (K. Kuchar, 1992), but there is no 'problem of 3-D space', and no progress either. I hope Piotr Chrusciel and his distinguished colleagues can help, because I "do not know enough theoretical physics to help with any research in that area" (see above). In my humble opinion, the linearized ADM equations can display the dynamics of gravitational field no better than von Neumann's "collapse" can reveal the dynamics of quantum fields. If you take one snapshot from the bi-directional "talk", you may speculate, after Dirac-ADM, that it can be "decomposed" into 1-D time and 3-D space, but all you can achieve is to make Einstein spin in his grave like a helicopter.
If you wish to demonstrate how Mother Nature has "killed time" (J. Barbour, 2000; G F R Ellis, gr-qc/0605049), you should kill the 3-D space as well, by eliminating the possibility to move in 3-D space from 'inside' toward 'outside'. For if there were no difference between inside and outside, big and small, left and right, nothing can move, and the time will be indeed frozen, as claimed in GR textbooks (R. Geroch, 1978). But this "frozen time" is exactly the case of the global mode of spacetime, in which the potential states of physical systems (see above) are both 'before' and 'after', as well as both 'here' and 'there' and 'inside' and 'outside', like a superposition of |left glove inside the cage> and |right glove outside the cage>, in 'how to catch a lion in Sahara' game. (Notice that the clockwise "spin" depicted with your wristwatch in |left glove inside the cage> state will be in superposition with a "time-reversed" anti-clockwise "spin" in |right glove outside the cage> state.) That's what I mean by 'quantum spacetime'. Then, by "projecting" a frozen 3-D snapshot from it on the local mode of spacetime, we get an asymptotically flat 3-D space with CPT-symmetry (with some trenchant violations, such as the neutral kaon meson and the cosmic equator), and of course a lot of "dark" stuff from the global mode. This is also the proposed solution to the problem posed by Einstein [Ref. 2], since the "discrete" values of the potential states in the global mode are being converted into a perfect continuum of actualized states in the local mode: no "gaps" from the global mode exist in the local mode, hence the holistic effects from the global mode of spacetime are totally "dark".
To get started, please see the NB task above, and recall 'the embarrassment of richness' (the multiple choice problem) and 'the embarrassment of poverty' (the global problem of time), after Karel Kuchar [K. Kuchar, 1992, Sec. 2].
If all this sounds like philosophy, just consider the immediate implications for Gravitational Wave astronomy (A. Loinger, physics/0603214): the linearized ADM equations have completely obscured the fundamental difference between 'propagating through spacetime' vs 'oscillations of the "fabric" of spacetime itself', after Kip Thorne, hence LIGO Scientific Collaboration (395 scholars) are trying to detect GWs in the local mode of spacetime. Which reminds me of the famous saying by Confucius: "The hardest thing of all is to find a black cat in a dark room, especially if there is no cat." These 395 distinguished scholars don't know the true dynamics of GR, since we need quantum gravity to unravel it, but have a lot of cash to spend. Taxpayers' money, of course.
[Ref. 1] Bradley Dowden, TIME, The Internet Encyclopaedia of Philosophy, 2006, http://www.iep.utm.edu/t/time.htm
The first person to give a clear
presentation of the implications of treating declarative sentences as being
neither true nor false was the Polish logician Jan Lukasiewicz in 1920.
To carry out Aristotle's suggestion that future contingent sentences don't
yet have truth values, he developed a three-valued symbolic logic, with
all grammatical declarative sentences having the truth-values of True,
False, or else Indeterminate [T, F, or I]. Contingent sentences about the
future, such as Aristotle's prediction that there will be a sea battle
tomorrow, are assigned an I. Truth tables for the connectives of propositional
logic are redefined to maintain logical consistency and to maximally preserve
our intuitions about truth and falsehood. See Haack (1974) for more details
about this application of three-valued logic.
Chapter 4 contains a clear account
of Aristotle's argument for truth-value gaps, and its development in Lukasiewicz's
"Man kann gute Argumente dafür
anführen, daß die Realität überhaupt nicht durch ein
kontinuierliches Feld dargestellt werden könne. Aus den Quantenphänomenen
scheint nämlich hervorzugehen, daß ein endliches System von
endlicher Energie durch eine endliche Zahl von Zahlen (Quanten-Zahlen)
vollständig beschrieben werden kann. Dies scheint zu einer Kontinuums-Theorie
nicht zu passen und muß zu einem Versuch führen, die Realität
durch eine rein algebraische Theorie zu beschreiben. Niemand sieht aber,
wie die Basis einer solchen Theorie gewonnen
(There are good reasons to assume
that nature cannot be represented by a continuous field. From quantum theory
it could be inferred with certainty that a finite system with finite energy
can be completely described by a finite number of (quantum) numbers. This
seems not in accordance with continuum theory and has to render trials
to describe reality with purely algebraic means. However, nobody has any
idea of how one can find the basis of such a theory.)
10 Jun 2006 17:48:01 +0300
Dear Professor Nakahara,
I would like, if I may, to share with you my opinion on some assumptions regarding the Hilbert space [Ref. 1], and ask a question.
1. In a letter dated 13 November 1935, John von Neumann wrote to George David Birkhoff: "I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore" (quoted after Elemer E. Rosinger, quant-ph/0408191 v2). I don't know why John von Neumann had lost his faith in the Hilbert space, but the reason why I don't believe in it are explained at
My favorite presentation of QM is sketched in [Ref. 2].
2. I wonder if you can suggest a resolution of the problem of the "dynamics" of GR
within the fibre bundles formulation of GR.
The opinion of your colleagues will be greatly appreciated, too.
[Ref. 2] A. Ashtekar,
T.A. Schilling. Geometrical Formulation of
"The geometric formulation shows
that the linear structure which is at the forefront in text-book treatments
of quantum mechanics is, primarily, only a technical convenience and the
essential ingredients -- the manifold of states, the symplectic structure
and the Riemannian metric -- do not share this linearity."
Spacetime as a continuum with a singular global "point"
Dear Professor Tartaglia,
I read with great interest your latest
gr-qc/0601033 [Ref. 1]. I wish to
Regarding GR, it seems to me that we should include the "dark" effects of the so-called dynamic dark energy from the outset,
I dare to suggest a singular global "point" as the source of the so-called dynamic dark energy (DDE), which I call 'potential point',
The source of DDE cannot be a relativistic object,
since it is spanned "over" the whole universe as some transcendental tachyon, and "acts" on the whole 3-D hypersurface EN BLOC, without leaving any irregularities or clumps whatsoever.
That's why it is "dark". We don't have access to some now-at-a-distance reference frame, and cannot *directly* observe the perfectly smooth and omnipresent source of DDE, which springs from "empty space",
Otherwise we'd have witnessed all sorts of violations of the positive mass condition (objects with negative asymptotic mass), traversable wormholes, time warps, and other disasters,
I wonder if you or some your colleagues would agree.
"Our current vision of the cosmos,
especially in GR, is essentially
"The paradigm we are proposing here
considers a spacetime endowed with
"We may state it either way: telling
that the symmetry implies a zero
You wrote: "The theory of graviton interactions can be developed in direct analogy to that of electromagnetism. (...) With this background we can now examine various interesting reactions involving gravitons, as we detail below."
I'm afraid you've skipped all criticism, from Weyl to Loinger, to that "direct analogy".
As if your hypotheses are falsifiable.
Could you please show me one experimental/observational
Thank you very much in advance.
Regarding our previous email correspondence, perhaps you may wish to see
"dynamics" of the "observables" for the gravitational field
RE: Charles Torre, gr-qc/9404029
v1, 14 April 1994: "Dynamics takes place on the constraint surface [X]
defined by the Hamiltonian and momentum constraints H = 0 and Ha = 0. (...)
(T)here are essentially no local observables for vacuum spacetimes."
I very much respect your work.
If you're short of time, please see the flash movie at
Have a nice summer.
An infinity of roughly parallel surfaces
Dear Professor Salisbury,
It seems to me that Dirac's ides of introducing "an infinity of roughly parallel surfaces" [Ref. 1] can be revived,
Will be happy to elaborate.
Regarding the nature of observables in classical general relativity and your quantum gravity research,
please see my efforts at
"In a letter to Dirac dated October 9, 1959, Bergmann wrote "When I discussed your paper at a Stevens conference yesterday, two more questions arose, which I should like to submit to you: To me it appeared that because you use the Hamiltonian constraint HL to eliminate one of the non-substantive field variables K, in the final formulation of the theory your Hamiltonian vanishes strongly, and hence all the final variables, i. e. [XXX], [XXX], are "frozen", (constants of the motion). I should not consider that as a source of embarrassment, but Jim Anderson says that in talking to you he found that you now look at the situation a bit differently. Could you enlighten me?"
"Here is Dirac’s response, dated November 11, 1959: "If the conditions that you introduce to fix the surface are such that only one surface satisfies the condition, then the surface cannot move at all, the Hamiltonian will vanish strongly and the dynamical variables will be frozen.
"However, one may introduce conditions which allow an infinity of roughly parallel surfaces. The surface can then move with one degree of freedom and there must be one non-vanishing Hamiltonian that generates this motion. I believe my condition [XXX] is of this second type, or maybe it allows also a more general motion of the surface corresponding roughly to Lorentz transformations. The non-vanishing Hamiltonian one would get by subtracting a divergence from the density of the Hamiltonian."
[Regarding the last two sentences
from Dirac's letter, see Josep M. Pons, On Dirac's incomplete analysis
of gauge transformations,
v2, Sec. 7.]
Note: I wrote above that will be happy to elaborate. The crux of the matter, in my view, is to "distinguish between intrinsic time, on which variables might depend, and our initially arbitrarily chosen coordinate time" [Ref. 2].
In classical GR, these two kinds of "time" literally overlap, and the confusion is by no means resolved. I'm not aware of some "eventual quantum theory of gravity" proposed by Pons and Salisbury [ibid.], but in the version advocate on this web site the distinction is pretty obvious. If you're interested, just drop me a line.
Now, because everyone asks 'but where's da math?', let me stress that nobody was interested in my math when I offered it in December 2004, firstly, and secondly -- the math is highly non-trivial, since the main idea about 'pure geometry' is that it should be presented by a special mathematical object: not "zero", as in
[+bananas] + [-bananas] = 0 [bananas]
but as an omnipotent entity which has "zero intrinsic content". All you have to do is to embed it in that connected Hausdorff topological space discussed in Chris Isham's 'Modern Differential Geometry For Physicists'.
[Ref. 2] J. M. Pons and D. C. Salisbury, The issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity, gr-qc/0503013 v1:
"We present our conclusions in section
8, including possible implications of this work regarding an eventual quantum
theory of gravity.
"... we can and should distinguish
between intrinsic time, on which variables might depend, and our initially
arbitrarily chosen coordinate time."
In my view, the problem of the arrow
of time (p. 17) consists in establishing a substantial difference between
spacetime with 'potential future', and spacetime without it. It seems to
me that you and your colleagues explore only the latter case, while my
efforts are focused on
BTW it seems to me that the vector field "that points from past to future at each point of the universe" (p. 21) should be a bona fide gauge field. See how it works in the human brain at
Dark Energy (DDE) of ... what?
Dear Dr. Durrer,
I read with great interest and satisfaction your latest "No-go theorem for k-essence dark energy", astro-ph/0606584 v2, and tried to explain the puzzle of DDE to my 13-year old daughter as follows:
Suppose you accelerate a car, but the gauge fuel shows that you're actually gaining *more fuel* by accelerating the car. That's the ultimate 'free lunch' provided by DDE, only physicists cannot explain it.
I wonder if you and/or your colleagues would agree.
If you disagree, I wonder what could be the entity, call it [X], which (i) might solve the coincidence problem and (ii) perform "causally propagating fluctuations" (astro-ph/0606584 v2, p. 4). It seems to me that these two requirements are contradictory.
My reflections on the nature of DDE can be read at
I just tried to follow Paddy's logic in "Attempting the Impossible: Upanishads and bicycles",
Thanu Padmanabhan: "You can always specify what It is not; and if you eliminate all that is not, whatever remains is It -- something Holmes did appreciate in more prosaic matters."
Note added on December 1, 2006: see the efforts by Yungui Gong, Bin Wang, and Anzhong Wang in gr-qc/0611155 v1 (cf. below), and notice that they didn't even mention the crux of the matter: does the zero-point vacuum energy gravitate?
In fact, all people bypass this crucial issue, set by Einstein's cosmological "constant" since 1917. The result is a plethora of post hoc postulated "fields", and a total mess. The generalized Chaplygin gas (GCG) is not an appropriate model to approach the dynamic "dark" energy of the quantum vacuum. If you disagree, please write me back, and I shall elaborate.
Dear Professor Brans,
My efforts to speculate on the so-called
dynamic dark energy can be read
Regarding the geometrization of QM [Ref. 2], please see
[Ref. 1] Torsten
Asselmeyer-Maluga, Helge Rosé, Calculation of the
" Brans conjectured about sources
of gravity given by differential
[Ref. 2] Torsten Asselmeyer-Maluga, Helge Rose, Differential Structures - the Geometrization of Quantum Mechanics, gr-qc/0511089 v3.
p. 10: "Two questions remain: Where
does the imaginary unit really come
p. 12: "In the next paper we will
go further and describe something like
Theory of Gravity, gr-qc/0609033
Dear Professor Novello,
Please excuse me for this bulk email.
I am very much interested in your
"Spinor Theory of Gravity",
v1, and wonder if you believe that the dynamics of the two
Since Mark Hadley has argued that
spacetime is not time orientable, and
I extend my request for information on the dynamics of GR to all colleagues.
If possible, please send me a copy from your "Faked Inflation" (XIIth BSCG, Sept 19th).
The essential "constant" in Riemannian geometries
Dear Professor Papadopoulos,
I wonder how would the Dynamic Dark
Energy (DDE), as a global essential "constant", appear in the curvature-invariant
relations, such as (1.3)
Given the perfectly smooth and omnipresent character of DDE [Ref. 2], which makes it "global", I wonder what could be the additional essential constant that pertains *only* to DDE and makes it "dynamical" (the coincidence problem). Since DDE doesn't depend on the volume of space [Ref. 2], can you envisage its "dark" value in the case of, for example, Schwarzschild singularity, r = 0 ?
I will highly appreciate the opinions of your colleagues as well.
My efforts can be read at
[Ref. 1] G.O. Papadopoulos, On the
essential constants in Riemannian
2] The dynamical "dark" energy is a perfect fluid, which "provides
What's the Energy Density of the
Not surprisingly perhaps, Georgios Papadopoulos added "[still open issue]" to his gr-qc/0503096 v3, J. Math. Phys. 47, 092502 (2006). To be specific, he tackled "the problem of attributing a character to a constant (or a parameter) which may appear in the metric tensor field. Generally, there are two possibilities: this constant is either essential (i.e., a true degree of freedom) or spurious (i.e., absorbable with the help of a change in the coordinates)."
Then Georgios Papadopoulos suggested, in Sec. 2, a criterion "to check whether a constant, appearing in a metric tensor field, is essential or not", and offered a pedagogical example with a two-dimensional metric tensor field (Sec. 3).
It would be nice to see an implementation of Papadopoulos' criterion to the puzzle noted by T. Thiemann, and then proceed to the most puzzling "constant": see D. Giulini and N. Straumann, astro-ph/0009368 v1.
Recall the effort made long ago by James Anderson (James L. Anderson, Principles of Relativity Physics, Academic Press, New York, 1967), who suggested to define ‘general covariance’ as absence of what he calls ‘absolute structures’. The idea seems simple and straightforward (cf. Domenico Giulini, gr-qc/0603087 v1, emphasis added): "An agent which dynamically acts but is not acted upon may well be called ‘absolute’ -- in generalization of Newton’s absolute space. Such an absolute agent should be eliminated."
Is DDE an absolute structure/agent? It dynamically acts on the whole spacetime en bloc,but is not acted upon, hence we enjoy another 'essential constant': the fundamental timelike displacement, that is, "the existence of an intrinsic time interval associated to any timelike displacement" (T. Jacobson). Regardless of the fleeting content of the spacetime "points" that are "moved around" by the active diffeomorphism, the fundamental "tick" of the cosmological time arrow remains invariant. Thus, it seems that DDE is indeed an absolute structure/agent, only it also evolves "in time".
However, since Chris
Isham claims that I "do not know enough theoretical physics to help
with any research in that area", perhaps you may wish to see a professional
viewpoint from Lee Smolin, and another
finite universe without boundary
In your gr-qc/0611090 v1, you suggested that "one needs to revise his way of incorporating an equilibrating mechanism by adding a cosmological constant", and then went on with the following:
"Let us assume that it exists a state of stable internal (by definition there is nothing outside the Universe) equilibrium for the universe on the overall."
Perhaps you can't derive the relaxation time [tau] because it depends on the notion of 'nothing outside the Universe' w.r.t.w. we can define a closed system incorporating an equilibrating mechanism with dynamical cosmological "constant". The way I see it, the 'nothing outside the Universe' should make the closed system under investigation completely boundless. The task of reconciling these two requirements is outlined at
If you're interested, please write me back.
horizons, if any
Dear Professor Chrusciel,
Isn't it true that the ambiguities with the putative event horizon boil down to the non-existence of a gravitational field stress-energy in the right-hand side of the field equations, and subsequently the purely geometrical interpretation of gravity as some sort of "curvature" of spacetime? Since there is no uniquely defined energy density of the gravitational field inEinstein's GR, using nontensor quantities such as pseudo-tensors is like 'sweeping the garbage under the carpet'.
Please recall that Einstein never considered his GR a *complete* theory,
You may, of course, say 'faute de mieux, on couche avec sa femme', but do you *really* believe there is such thing as "event horizon"?
If yes, how would you disprove your belief? I mean, you're a mathematician, and can lay down your cards by stating the precise conditions under which you will reject the notion of 'event horizon' as a mathematical fiction/artifactdue to the essentially incomplete GR, as acknowledged by Einstein.
I will be very pleased to hear from you and from all colleagues of yours reading this email.
More info can be found at my web site. Please see also my CD ROM (work in progress) at
With kind regards,
A. Einstein, Born-Einstein Letters,
29 April 1924
Piotr Chrusciel. Black holes. Wed, 16 Jan 2002 09:26:22 GMT,
"Event horizons are a special case
of a family of objects called future horizons: by definition, these are
closed topological hypersurfaces H threaded by null geodesics, called
generators, with no future end points, and possibly with past end points.
At the latter, differentiability of H breaks down in general; a necessary
and suffcient condition for this breakdown has been given in . It seems
that most authors have been taking for granted that horizons are nice,
piecewise smooth hypersurfaces. This is, however, not the case, and examples
of nowhere C^1 horizons have been constructed in ."
Pankaj S. Joshi, Naresh Dadhich, Roy Maartens. Why do naked singularities
form in gravitational collapse?
[Ref. 3] Pankaj
S. Joshi. Gravitational Collapse: The Story so far.
Sure, by a simple coordinate transformation you can eliminate the gravitational field 'by hand', but is this trick a solution to the conflict of energy conservation and general covariance?
According to Gerard 't Hooft, I've totally missed the point (cf. below).
Subject: RE: Event horizons, if any
> Isn't it true that the ambiguities
with the putative event horizon
- Absolutely not. There should not
be such a thing as a `gravitational field stress-energy' at the r.h.s.
of Einstein's equations. In a co-moving frame there is no gravitational
field, hence no grav.stress-energy! In any other frame where you do have
effective grav. fields, you automatically get the correct grav. contribution
to the stress-energy from the Christoffel symbols in the covariant derivatives.
I've heard of more people who totally missed this point.
Note 1: Yeah, people sometimes miss the essential points, regrettably. For example, the point made by Levi-Civita back in 1917. Or the "point", taken literally, at which we "define" the equivalence principle using the time read by a physical clock. Then of course we cannot understand the so-called event horizon, and finally accept that we need quantum gravity. Quietly.
Meanwhile the general public is told that there is some "Giant Black Hole" right in the center of the galaxy RX J1242-11, and another one, this time much closer, in our Galaxy's center, the Sagittarius A*. We don't know what is a black hole, but supermassive black holes look much easier to explain:
"One idea is that an individual star-like black hole forms and swallows up enormous amounts of matter over the course of millions of years to produce a supermassive black hole. Another possibility is that a cluster of star-like black holes forms and eventually merges into a single, supermassive black hole. Or, a single large gas cloud could collapse to form a supermassive black hole."
Let's not forget the "intermediate-mass" black holes: "One explanation is that these sources are produced by intermediate-mass black holes that have masses a hundred or more times greater than the mass of the Sun. Such objects would have much larger event horizons (italics mine - D.C.), which would account for the larger sizes and lower temperatures associated with the quasisoft sources. Alternatively, they could be standard neutron stars or stellar black holes where the associated region of hot gas is for some as yet unknown reason much larger than usual."
These excerpts are not from National Enquirer but from the official site of Harvard-Smithsonian Center for Astrophysics (60 Garden Street, Cambridge, MA 02138). Now, for supermassive black holes there is even a Printable Field Guide, blackholes_supermassive.pdf. The relatively new story about the quasisoft sources is still a bit obscure, but if you are fluent in those quite ordinary stellar black holes, all you have to do is to find out why is the associated region of hot gas "much larger than usual." Perhaps because these new guys have "much larger event horizons", as measured with your tape.
If you wish to leave a comment, their email address is email@example.com. Well, they might be too busy to reply, but that's a different matter.
Piotr Chrusciel is probably also very busy, since he hasn't replied to my email of Thu, 17 Jan 2002 16:23:11 +0200 (see above). I asked him to lay down his cards by stating the precise conditions under which he would reject the notion of 'event horizon' as a mathematical fiction/artifact due to the essentially incomplete GR, as acknowledged by Einstein.
No reply so far. Two years of dark,
black-hole-like silence. I'm sure he is not treating GR as some
No way. He is a rock solid mathematical physicist, like Jean-Pierre
Luminet. Maybe he just didn't like my French. Which reminded me of
a famous statement by General Norman Schwarzkopf: Going to war without
the French is like going hunting without your accordion. But that's, of
course, totally unrelated. Just a free flight of my thoughts. Maybe there
is a (globally of effectively?) naked singularity
right in the center of my brain, and you never know what might pop out
from there, n'est-ce pas?
Do you know what is 'context'? See
"Sure, by a simple coordinate transformation you can eliminate the gravitational field 'by hand', but is this trick a solution to the conflict of energy conservation and general covariance?
"According to Gerard 't Hooft, I've totally missed the point (cf. below)."
Again, it's about the conflict b/w energy conservation and general covariance. I'll be happy to provide you some references, since November 1917.
On Mon, 8 Mar 2004 16:06:58 +0100,
Your current writings confirm my earlier opinion on your knowledge of GR.
On Thu, 11 Mar 2004 08:17:00 +0100,
"Hooft 't G." wrote:
Thank you. Perhaps you didn't have
the chance to look at my front web
Please pay attention to the proposed
interpretation of "how energy-momentum flows from matter to grav. fields
and back", which I
> So, yes, if you look at a spacetime
Not until you write your celebrated
paper "201 wrong theories for the
When you say "then", you imply something temporal. You do A , and then you get B . Take a look at a spacetime point x and transform to coordinates where the Levi-Civita connection vanishes at x , and then -- voila! -- there is no energy-momentum transport at that point.
But hold on: "But then, of course, this occurs elsewhere."
What animal would "occur elsewhere"? How fast it will end up "there"? In what reference frame you can measure the speed of this transportation?
What is the intrinsic time interval implied in those "moving points around"?
Moreover, why "of course"? You can say "of course" only if you can answer all the questions above. Try it.
Again, it's about the conflict between energy conservation and general covariance, as reiterated above.
If we were doing electrical engineering, for example, we can use all sorts of calculations, imaginary numbers included, but we will never think that our calculations correspond to the actual behavior of the electrical currents. Fine, but then we apply this pattern to GR, since all observables must be Diff(M)-invariant.
David Hilbert wrote back in December 1923 (Die Grundlagen der Physik, Mathematische Annalen, Heft 92, S.1-32, 1924): "I assert that for the general theory of relativity, i.e., in the case of general invariance of the Hamiltonian function, energy equations corresponding to the energy equations in orthogonally invariant theories do not exist at all. I could even take this circumstance as the characteristic feature of the general theory of relativity." And Max Born explicitly wrote that once he understood GR, he vowed never to work on it (cf. S. Deser, gr-qc/0301097).
Needles to say, Albert Einstein never considered his theory complete. But if you need some comfort, take a good deep look at "Introduction to General Relativity", lecture notes by 't Hooft, Gerard, available here. Pay special attention to Eq. (1.26), p. 8, and ponder on the following text (ibid., p. 28): "Now these equations strongly suggest a relationship between the tensors Tµv and Rµv , but we now have to be careful." Perhaps Gerard 't Hooft should have written 'very careful', since the issue is well-known from another textbook, by H. Weyl. See also the puzzle of "collapse" from G. Lemaître and C. Lanczos here. It's all due to a simple coordinate transformation, as stressed by David Hilbert in November 1915.
But Gerard 't Hooft feels happy, because "the matter-energy-momentum tensor is COVARIANTLY conserved". This is how "energy-momentum flows from matter to grav. fields and back".This is, sit venia verbo, sad.
Wait for "201 wrong theories for the cosmological constant",by 't Hooft, Gerard (under preparation, I hope), and don't forget the Printable Field Guide to SuperMassiveBlackHolesAndOtherMonsters, blackholes_supermassive.pdf. You will learn that "something massive is creating a powerful gravitational field". True. That's all we know, which is why I urged Prof. 't Hooft to write his celebrated paper "201 wrong theories for the cosmological constant",
Alternatively, try to read Prof.
> The point I raised has nothing
to do with dark matter or cosmological
1. This thread is about the ambiguities with the putative event horizon.
On April 6, 2001, I asked G. 't Hooft to sort them out:
"Since Gerard 't Hooft has clearly declared his intention to develop some fundamental theory, I think the first and foremost task should be to resolve the basic problems with the so-called black holes -- they might not exist at all (...)".
... just like those naked singularities might not exist at all. Otherwise we won't be here to read this.
I think it's a package of two misunderstandings,
black holes and naked singularities. See Gary Horowitz
and S. Shapiro & S. Teukolsky. The latter
reference is from 1991 (Saul Teukolsky was kind to send me his preprint
in October 1991). The latest reference is from T. Harada, January
2. My approach to this peculiar situation is to look at what I believe could be the crux of the misunderstanding: the conflict between energy conservation and general covariance. It does have some far-reaching implications, as explained by M. Weiss and J. Baez.
I specifically stressed that this conflict is the context of my question, but G. 't Hooft spotted something which he wanted to comment on, and decided to strike with "Absolutely not. There should not be such a thing as a `gravitational field stress-energy' at the r.h.s. of Einstein's equations."
Is this a solution to the conflict
between general covariance and energy conservation? If G. 't Hooft thinks
that he has solved the puzzle -- sorry, I'll terminate the discussion.
(Note added on March 15, 2004: G. 't Hooft replied by email of Mon, 15
Mar 2004 09:24:48 +0100 "I apologetically terminate this discussion." That's
sad, since I was hoping that he would suggest a solution to this very old
puzzle. Then I would have passed his startling insights to Matt Visser
and Carlos Barcelo, who explicitly wrote that the "conflict between quantum
physics and gravity is now becoming acute"; see their gr-qc/0205066
of May 16, 2002.)
3. As to his claim "The point
I raised has nothing to do with dark matter or cosmological constant" --
well, I'm not quite sure. Eliminating the gravitational field stress-energy
at the r.h.s. of Einstein's equation is a daunting
task. I'm not aware of any successful solution to it. My proposal leads
directly to the cosmological constant problems, as I tried to outline
More at GR17, again.
4. The reason why I repeatedly
urge G. 't Hooft to write his paper "201 wrong theories for the cosmological
constant" is that he kindly invited me to provide for a reference for the
201st wrong theory. The title of his unfinished paper was "200 wrong theories
for the cosmological constant", as he explained here.
It looks to me that Gerardus can read my thoughts, and has found errors
in my proposal. That's weird, since the only thing we agree upon in that
ESP is impossible. I have no idea how he
managed to read my mind and invite me to enrich his paper with the 201st
reference of a wrong theory for the cosmological constant.
5. I regret that Piotr
Chrusciel didn't reply in the past two years, but I hope he will explain
the latest developments in his study of the so-called event horizon at
The way I see it, the task is to replace the metaphysical belief in Penrose's
Cosmic Censorship Conjecture (CCC) with hard science. What else but quantum
gravity? Again, I hope to learn the opinion of Piotr Chrusciel at GR17
in Dublin. Qui vivra, verra.
In summary, (i) no continued gravitational "collapse" can produce some event horizon, as explained by Prof. Angelo Loinger, (ii) no continued deflation time can actually reach The Beginning, and (iii) no continued shrinking of a spacetime region can actually reach the so-called "point". Regarding (i), the task of CCC is taken care by the global mode of spacetime.