Let me start with a cautious warning: I was never able to understand Einstein's General Relativity (GR), hence everything said here could be wrong. Let me outline my reflections on Einstein's GR, hoping that the reader will help.

Despite my sincere and laborious efforts in the past thirty-two years, I'm still deeply puzzled by the discrepancy between the calculation effectiveness of GR and its wonky assumptions, such as the general covariance. I think the situation strongly resembles Quantum Mechanics (QM): as a calculation tool, it too works amazingly well, despite its lousy mathematical structure [1]. We are perhaps inclined to confess, after R. Feynman, that nobody understands QM nor the transition from the world of tables and chairs to the quantum realm [2], but Einstein's GR is a classical theory, and hence we expect that can follow the line of reasoning in it and discover some corresponding process taking place in Nature 'out there', which leads to some alleged "spacetime curvature" that would act back on matter, telling it how to move [3].

What kind of energy could be attributed to something that is pure geometry? That I was never able to understand. Here's why.

To begin with, let's recall that Albert Einstein was not at all happy with the principle of general covariance. He has been struggling with its inevitable implications for full three years, from 1912 to 1915: "The requirement of general covariance takes away from space and time the last remnant of physical objectivity." See his seminal paper quoted above, and the painful history of the early days of Einstein's GR, in C. Rovelli's book [4], Sec. 2.2.5, and in T. Sauer's essay [5].

From the physical point of view, there is something very spooky in Einstein's GR. As stated eloquently by Richard Feynman in his Lectures on Physics: "Is it not possible that perhaps gravitation is due simply to the fact that we do not have the right coordinate system?" The problem is not that of semantics, but reflects the basic principle of general covariance. Not only are we refused to consider "the last remnant of physical objectivity", but also the most important issue in Einstein's GR: the energy-components of the gravitational field. They cannot be described mathematically by a tensor, as shown by Hermann Weyl back in 1922 [6].

Hence we encounter a strange kind of geometrical reality that somehow cannot be directly measured nor localized. It is a genuine "non-local" entity [7],[8],[9], and the best we could make of it requires imposing the condition of some fictional 'isolated system', as stressed by T. Levi-Civita back in 1917 [10].

Now, how do we circumvent all these discrepancies? Specifically, how does the bi-directional "talk" [3] occur? The textbook lore is "we see only covariant derivatives" [11].

Fine, but how do we actually see them? Post factum only. Once the complete dynamics of the coupled geometry-matter system is completely worked out, we can introduce the notion of spacetime [12]. The preliminary bi-directional "talk" [3] is completely hidden. Nothing in Einstein's GR can sneak into this totally dark gap.

And here comes the real big mess. In the modern approach to Einstein's GR, called geometrodynamics, general relativity describes the evolution of the 3-geometry in time [13].

In what time? It's like trying to lift up yourself by pulling up your hair. We have "time" and 3-D space only after the complete dynamics of the coupled geometry-matter system has been completely worked out. Meanwhile we all live in some dark gaps. This is certainly unacceptable for any classical theory that is supposed to describe the world 'out there'.

From logical point of view, it is equally unacceptable to embed a genuine Catch 22 in the heart of Einstein's GR: we can introduce the notion of spacetime only after we complete the bi-directional "talk", but in order to initiate this same bi-directional "talk", we need its end-product. We should not have any evolution whatsoever, because this self-acting bundle cannot move, cannot "fly", as John Wheeler used to say. Unless, of course, it "flies" through the gaps.

Besides, Mother Nature certainly does not calculate covariant derivatives, watching carefully for those "outlawed" non-covariant ones, as explained eloquently by Gerard 't Hoof [11]. She should somehow "know" or rather anticipate what could be the next subject of the bi-directional "talk", and work it out quietly, while we all (Gerard 't Hoof included) remain in these gaps.

  Let me sum up with my explanation of the bi-directional "talk" of matter and geometry [3], which I offered to my 11-year old daughter. Suppose you do in a china shop, and the manager there tells you that there is a white dancing elephant in his shop, but it's sort of invisible. Even more: each and every piece of fragile china requires fixing of its state by the white dancing elephant, so he has to constantly dance and jump around. Of course, he doesn't break anything, for the simple reason that the states of all pieces of china that you look at (in the good old 3-D space) have been already fixed by this white elephant before you look at them, so you cannot actually see the dancing elephant. Moreover, in case you worry, you can easily go to any place in the shop and make this white dancing elephant vanish completely (by a simple co-ordinate transformation, of course [6]). Theoretically, the white dancing elephant may produce all sorts of nasty things (such as gravitational collapse & naked singularity, whichever you prefer), but we can easily eliminate them (again, by a simple co-ordinate transformation, [14]). The only thing you might be bothered with is the twin brother of the white elephant, which is sort of "opposite" to the white one, so we call him "dark" [15]. Perhaps he is strong exactly as his white twin brother (which is why we enjoy the much-needed asymptotically flat spacetime in the shop), but since we cannot actually observe any elephant whatsoever, it's safe to shop. Besides, ...

Here my daughter interrupted me and said, - 'Daddy, are you pulling my leg?' I said, - 'Not at all, darling, I'm just trying to explain to you the nature of gravity. Honestly.' Obviously, I wasn't quite successful. Let me try to do better, by going straight to the dark gaps.

These gaps may not be dark at all. They are utterly needed for the phenomenon of transience [16], which is the crux of the notions of time and space.

First, recall that the metaphysical idea of 'time' refers to some object which exists and changes its state 'in time'. We can think of changes in time if we have at least two "successive" states separated by a gap in which the object does not exist. This is the metaphysics of the phenomenon of transience [16]. If there were no gaps of alleged "non-existence", there would be only one single state of the object, which will not be allowed to change 'in time'. Instead of 'time', we would be living in one instant stretched to infinity, which is nothing but 'eternity', after St. Augustine. And since we can explain the notion of time only with the notion of space, and vice versa, these gaps of the phenomenon of transience are needed for the notion of space as well. In fact, we operate with a single entity called 'spacetime', after Hermann Minkowski. However, the gaps of spacetime (also the Gaps of Zen) are hidden in the theory of relativity from the outset.

In order to identify these gaps, we need a reference object: the instant 'now'. Not just one instant 'now', but a sequence of ordered instants 'now', separated by these gaps of transience. This is impossible in the theory of relativity [17], and strictly forbidden in GR [18]. These gaps are unique, they pertain simultaneously to all spacelike surfaces, and we know that such an absolute reference frame or ether is banned in the theory of relativity.

All we can do to unravel these gaps is to show that they are needed to find the Holy Grain of modern theoretical physics: quantum gravity. Since these gaps pertain to literally everything in the universe, the obvious choice will be to focus on something that covers the whole universe as well: the cosmological time.

What is the elementary increment of the cosmological time? What is the corresponding energy that drives the whole universe through these elementary increments/steps of cosmological time? They have to be well-hidden, of course. Which brings us to the question posed above, about the nature of gravitational energy in the left-hand side of Einstein's equation.

I've tried to explain here the whole bundle of tasks, elaborating on the Born probability rule. What I haven't done so far is to speculate on the nature of the energy-components of the gravitational field, and why we should be able to make them zero (on paper), as stressed by Hermann Weyl [6].

Are the energy-components of the gravitational field zero indeed? The answer is 'partially a yes and partially a no'. Or 'Jain', in German. They are not zero (both positive and negative) only in the virtual gaps of spacetime, and 'almost zero' inside the spacetime created "after" the passage of the cosmological time through these gaps. The numerical value of 'almost zero' matches that of the cosmological constant, [lambda], introduced by Albert Einstein. It may look like a tiny little thing, but is shrouded with tons of unsolved mysteries.

Going back to the issue raised by Hermann Weyl [6], we should be a bit more precise by stressing that the energy-components of the gravitational field cannot be strictly zero, but can only tend asymptotically toward zero. (Currently, we're some 13.7 billion years away from the inflation stage, so the remnant from the primordial gravitational energy should be vanishing small.) That is, 'almost zero'. But not exactly zero. For to be exactly zero, the gravitational field would have to be a totally neutral geometrical entity lacking any ability to "talk back" to matter.

Thus, we may obtain a qualitative estimate of the energy pertaining to 'something that is pure geometry' (see above): it should be vanishing small, trending asymptotically toward zero inside 4-D spacetime. We certainly don't know how much energy is being stored inside the gaps, but the portion from this energy that is being squeezed inside 4-D spacetime is definitely 'very small'. We don't know how gravity couples or 'talks back' to quantum fields, but because gravity couples to all forms of energy, we are confident that some form of coupling does occur. Also, we are firmly confident that, in Einstein's GR, this spooky energy-from-pure-geometry must be converted into some "normal" physical from of energy, and hence it will show up in the right-hand side: only matter can act on matter. Put it differently, the amazing 'common denominator' of geometry and matter, which permits shifting between 'apples and oranges' (geometry and matter), is provided by the astonishing correspondence between any kind of energy and mass, from another Einstein's equation, E = cm². The result from this conversion of energy-from-pure-geometry into "normal" physical energy shows in the non-linear dynamics of matter fields, making all matter fields "self-interacting". Hence in order to obtain a qualitative estimate of the energy being squeezed through the 'dark gaps', we should examine the "size" of these gaps. Not the whole pool of virtual gravitational energy that might reside inside the gaps, but only some tiny little portion that can generate 'spacetime curvature' inside 4-D spacetime, which, in turn, will 'talk back' to the matter fields in the right-hand side of Einstein's equation.

(This pool of virtual gravitational energy is the only possible solution to the dynamic adjustment of the so-called dark energy, in every stage of the evolution of the universe. Recall the coincidence problem and the interpretation by R. Penrose: "any non-constancy in [lambda] would have to be accompanied by a compensating non-conservation of the mass-energy of the matter", in The Road to Reality, Jonathan Cape, London, 2004, p. 777. See also "the subtle balance observed today between ordinary matter and the cosmological constant" in [Julien Lesgourgues, astro-ph/0409426 v1, p. 57 and Sec. 2.4.4], and "we live right in the middle of the spike of the delta function. If the dark energy had appeared earlier it would have interfered with structure formation: if later, we would still be unaware of it" in [Paul Frampton, astro-ph/0409166 v3]. Hence there is no need for any ad hoc scalar field to model the inflation: all we need is the pool of virtual gravitational energy.)

If so, the size of the gaps should correspond to the "thickness" of 3-D hypersurface, that is, to the "size" of an "event". It is not 'exactly zero', as we know since the time of Zeno, but is a dynamic entity running toward zero. Any time we 'look at it' with differential or tensor calculus, it looks as if it had indeed reached zero size. Why? Because we can calculate only things that we see in our past light cone. Everything there can be observed post factum only, and everything is cast on a perfectly continuous fabric of zero-size "events". The gaps are not there, however, and we cannot calculate their "size" being confined inside spacetime continuum. But since it is indeed a continuum, the size of the gaps inside spacetime could only be 'tending asymptotically toward zero' or, for short, 'almost zero'.

Hence my hunch is that the "thickness" of 3-D hypersurface is 'almost zero' due to the infinitesimal gaps of transience.

To explain this, lets take a closer look at Matthew Frank's Einstein's Equation in Pictures [19]. We start with "a unit timelike vector v at a point p" (ibid.), which points to ... where? To the gaps, of course! To obtain Einstein's equation, we have to shrink its footing -- not the "timelike" vector itself but just its 3-D space footprint with diameter 2r -- by instructing it to 'get down to zero'. This is how we obtain "points" in differential calculus, after G.W. Leibnitz. Or a circumference of a circle, after Archimedes. The Angels are in the details of the infinitesimal. It's very simple indeed. Or is it?

I suppose now we can understand why the energy-components of the gravitational field could "neither form a tensor nor are they symmetrical" [6]. If they were a tensor, they would be a physical stuff, like all physical fields. But then the gaps of the transience would be physically exposed, there would be a physical absolute reference frame, and we would never had the joy of studying Einstein's GR. Actually, there will be physically observable "negative energy" and we all will be dead.

Of course, all this should not imply that I have come any closer to understanding Einstein's GR. The interplay of the twin elephants, gravity and anti-gravity, is beyond my very limited math skills. Two years ago, on November 3, 2002, I suggested some very general ideas, which did not spark any interest so far. I wonder why. Maybe because I didn't use math. Unlike John von Neumann, for example, I'm not good in math. But here is what he wrote to George David Birkhoff, in a letter dated 13 November 1935: "I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore" (quoted after [1]). I believe Einstein would have agreed that sometimes imagination is more important than math.

All I wanted to say is that we can perhaps recover the true remnant of physical objectivity, which Albert Einstein considered lost in 1916. The path has been clearly outlined by Chris Isham [20],[21]: "space and time are such crucial categories for thinking about, and describing, the empirical world, that it is bound to be ferociously difficult to understand their emerging, or even some aspects of them emerging, from 'something else'."

What else but the gaps of transience? If we only find the right path, it won't be difficult at all, thanks to the astonishing effectiveness of mathematics in the natural sciences. Raffiniert ist der Herrgott, aber boshaft ist Er nicht! (Albert Einstein).

To finish this metaphysical exercise, let me try to get real and face some bold facts. I was very much hoping to deliver a talk at a seminar in Imperial College in November 2002, but it somehow didn't work out. The subject of my talk was "About points, if any". I haven't stop working on this subject, and have compiled a number of simple geometrical ideas. These are ideas about the pre-geometrical "dynamics" of the emergence of spacetime [21]. By trying then on my brain, they worked amazingly well and smooth. However, this purely introspective observation cannot constitute an evidence that I've hit the right path toward quantum gravity. The real problems and challenges are in the following.

Perhaps we can develop a complete theory of quantum gravity only by restoring the true remnant of physical objectivity in both Einstein's GR and quantum mechanics [22]. Quantum gravity should unify the missing physical reality in both theories, hence the first-off task is to find it. It seems to me that there are two 'missing pieces' from the jigsaw puzzle of quantum gravity, which are common to Einstein's GR and quantum mechanics.

The first missing piece deals with the notions of 'spacetime' and 'quantum state', and I believe we can recover the physical reality by introducing a new kind of reality, potential reality, acting as reference object. It is "outside" our 4-D spacetime, since it resides in the "gaps". It is also a holistic object and provides the full 'context' for 'quantum state out there' and 'spacetime curvature'. Just as the context of a sentence defines the meaning of all words in it (=matter, fields, etc.), and all words are "self-coupled" by their common context (=gravity). Only this 'context' is UNspeakable, just like a Platonic idea. Hence we can envisage a solution to the Hilbert space problem: the evolution of 'quantum state of X' can be uniquely defined only with respect to our new reference object/context. The 'quantum state of X' can be entirely 'here-and-now' in each and every "point" from its trajectory (no tails problems nor cat states), because the reference instants at which we define probabilities are supplied by our new reference object/context: all "words" in the "sentence" are fixed in the "background" of their 'context', called potential reality. Otherwise we just cannot make unitary dynamics of our 'quantum state of X', nor specify any finite time interval from our cosmological time.

The second missing piece is the puzzle of the localization of 'quantum state of X'. In both Einstein's GR and quantum mechanics, we encounter the non-local and 'self-interacting' nature of the "words" due to their "context". See the case of the 'dark twin elephant' here, and, regarding quantum mechanics, recall that we cannot, even in principle, locate the 'quantum state of X' on the light cone immediately prior the measurement. It's not, cannot, and should not be there [22]. It is still "spread around" in the "context" (recall Wheeler's smoky dragon); it is still an "isolated system" waiting patiently to be measured with our inanimate devices [10]. We can speculate extensively about such quantum system immediately prior the measurement, we can speculate about the complex phase of its quantum wave [2], but 'at the end of the day' we are forced to apply the Born rule, and wipe out the true quantum dynamics completely. We cannot restore it within the theory of relativity, both special and Einstein's GR, since we don't yet have the new reference object/context, called potential reality. Not yet.

Again, I could be all wrong. Or even 'not even wrong'. One of the leading experts in quantum gravity, whose opinion I deeply and honestly respect, stated that I "do not know enough theoretical physics to help with any research in that area."

I wonder if you agree. Please be very frank. I can take it.

If you prefer hard-core science, perhaps you may wish to disregard my reflections on Einstein's GR and wait for gravitational waves. Or better wait for microscopic black holes, either from LHC or from the Pierre Auger Cosmic Ray Observatory in Argentina (cf. Karen Wright, Black Holes Made Here (Deliveries begin in 2007), Discover, June 2004, pp. 62-63). Jonathan L. Feng has calculated that the Auger observatory can indeed detect showers of microscopic black holes, up to ten such showers a year -- "black holes appearing over your head, for free" (ibid., p. 63). That's certainly better than spending billions of dollars and euro, taxpayers' money. Another prominent scientist, MIT's artificial-intelligence guru Rodney A. Brooks predicted that by 2020 implants will let us carry out thought-activated Google searches (Rodney Brooks, Toward a Brain-Internet Link, Technology Review, October 31, 2003). But what if our thought-activated Google searches are hit by showers of microscopic black holes? Well, I certainly "do not know enough theoretical physics to help with any research in that area."
 
 

D. Chakalov
Saturday, September 18, 2004
Final version: Thursday, September 30, 2004, 23:02:18 GMT
 
 

References and notes

1. Elemér E. Rosinger, What is wrong with von Neumann's theorem on "no hidden variables", quant-ph/0408191 v2.

"As it turns out, in von Neumann's above theorem the assumption (2.4) on the real-linearity of the function E in (2.1) is physically questionable."

In this context, see: A. Ashtekar, T.A. Schilling. Geometrical Formulation of Quantum Mechanics, gr-qc/9706069

"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."
 

2. Chen Ning Yang, Square Root of Minus One, Complex Phases and Erwin Schrödinger, in: Schrödinger: Centenary Celebration of a Polymath. (Proceedings of a conference held at Imperial College to celebrate the centenary of the birth of Erwin Schrödinger, March 31 - April 3, 1987.) Ed. by Clive W. Kilmister, Cambridge University Press, New Rochelle, 1987.
 

3. "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve", John Wheeler.

4. Carlo Rovelli, Quantum Gravity, Cambridge U. Press, Cambridge, 2004. (Draft version available from here. I noticed that the draft version is quite recent, it has been updated even with the forthcoming monograph by T. Thiemann.)

See Sec. 1.1.3, The physical meaning of general relativity; also Sec. 2.2.5, General Covariance ("the meaning of the coordinates"): "Reality is not made by particles and fields on a spacetime: it is made by particles and fields (including the gravitational field), that can only be localized with respect to one another. No more fields on spacetime: just fields on fields. Relativity has become general."

See also Sec. 2.4.4, Table 2.1: Times. The first row of this amazing table is devoted to the human brain, but the notion of time in the brain is denoted with a question mark, [?]. Regardless from this genuine gap in Rovelli's knowledge of the human brain, he has managed to complete his book, denouncing the notion of time at the most fundamental level; see the last row of this table. How has he managed to write a whole book in the total absence of time at the level of quantum gravity? Relationally, of course!

Strangely enough, none of the 385 references in the draft version of Rovelli's book points to Claus Kiefer's monograph bearing the same title, Quantum Gravity. None of the well-known papers by Claus Kiefer are mentioned either. Perhaps Carlo Rovelli enjoys a very good memory, actually. "But success, I think, can only be granted by scrupulous intellectual honesty", says C. Rovelli. But how are we supposed to exercise our 'scrupulous intellectual honesty'? Relationally?

I hope this strange omission has been fixed in the printed version of Rovelli's book, and all issues of quantum gravity raised by Claus Kiefer have been duly addressed, with scrupulous intellectual honesty.
 

5.Tilman Sauer, The Relativity of Discovery: Hilbert's First Note on the Foundations of Physics, physics/9811050. See Sec. 3.2, The Concept of Energy, and Refs. 1 and 90 therein.
 

6. Hermann Weyl (1922), Space-Time-Matter, Fourth Edition, translated by Henry L. Brose, Dover Publications, New York, 1951, p. 270:

"And yet, physically, it seems devoid of sense to introduce the  t^k as energy-components of the gravitational field, for these quantities neither form a tensor nor are they symmetrical. In actual fact, if we choose an appropriate co-ordinate system, we may take all the  t^k  at one point vanish; it is only necessary to choose a geodesic co-ordinate system."
 

7. Roger Penrose, The Road to Reality, Jonathan Cape, London, 2004. See p. 458, the "gravitational contribution to energy-momentum", and pp. 467-468. As acknowledged by R. Penrose (p. 468), the trick of cooking a conservation equation applies (i) "where it may be assumed that the spacetime is asymptotically flat", and (ii) "in an exact way in the limit when the system becomes completely spatially isolated from everything else".

Well, this 'completely spatially isolated system' sounds like astrology to me, but, as I stated in the beginning, I could be all wrong.
 

8. Katarzyna Grabowska and Jerzy Kijowski, Canonical gravity and gravitational energy, in: Differential Geometry and Its Applications, Proc. Conf., Opava (Czech Republic), August 27-31, 2001, Silesian University, Opava, 2001, pp. 261-274; pdf file available from here.

"There is a lot of ambiguities in the definition of gravitational energy. A textbook version of the Legendre transformation, which is often used to derive Hamiltonian formalism from the Lagrangian field theory, leads to a somewhat paradoxical result: gravitational energy vanishes modulo boundary terms. The same textbook version of the Canonical Field Theory (used, e.g., as a starting point for second quantization of Electrodynamics) is only "volume sensitive" but not "boundary sensitive". This means that boundary phenomena are simply neglected. But here, in Gravity Theory, neglecting boundary terms means neglecting everything. Some authors improve this version of Canonical Gravity by imposing extra requirements on the energy functional in the asymptotically flat case (see, e.g., A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific, Singapore, 1991).

"In this way gravitational Hamiltonian is defined as "zero + boundary corrections". These corrections are, however, often obtained not by a universal procedure, well defined for any field theory (e.g., electrodynamics), but via ad hoc improvements, which make no sense outside of Gravity Theory."

These ad hoc "improvements" make no sense to me at all. It's like mixing apples with oranges, since in Einstein's GR the gravitational "field", unlike all physical fields, is a geometrical entity.
 

9. Carl Hoefer, Energy Conservation in GTR, Stud. Hist. Phil. Mod. Phys. 31(2), 187 (2000); see pp. 193-195, pdf file available from here. Just a few quotes:

p. 191: "Intuitively, if energy-momentum is really being conserved locally, then when one integrates up it should be conserved over regions as well. Since this fails, we have to fall back on a different understanding of what (1) represents.
...
p. 193: "It is called the gravitational stress-energy pseudo-tensor, and its non-tensorial nature means that there is no well-defined, intrinsic 'amount of stuff' present at any given point.

Regarding GWs, see p. 196: "If energy is not conserved quite generally, there is no need to make up a story about where it has gone when a system loses it."

See also footnote 11, p. 195.

 

10. Tullio Levi-Civita, On the analytic expression that must be given to the gravitational tensor in Einstein's theory, Rendiconti della Reale Accademia dei Lincei, vol. 26, 381 (1917); physics/9906004

"The nature of ds2 is always such as to balance all mechanical actions; in fact the sum of the energy tensor and of the inertial one identically vanishes.

"One is naturally led to associate this proposition with d’Alembert's principle "the lost forces (i.e. directly applied forces and inertial ones) balance each other". The equilibrium expressed by (10’) is just the most complete occurrence that can be conceived from the mechanical standpoint. In fact, not only the total force applied to each single element comes to vanish, but also stresses, energy flow and energy density (by taking inertia into account through Aik) behave in this way.

"It is clear that this total lack of mechanical entities pertains to isolated systems.
...

"In fact, by virtue of (10’) or, if one likes, of the generalised d’Alembert’s principle, when the energy tensor Tik vanishes, the same occurrence must happen to the gravitational tensor Aik. This fact entails total lack of stresses, of energy flow, and also of a simple localisation of energy."
 

11. G. 't Hooft, Introduction to General Relativity, version 8/4/2002
See p. 8: "the total energy-momentum tensor", Eq. (1:26); also p. 22: "Thus we see that if the Riemann curvature vanishes a coordinate frame can be constructed in terms of which all geodesics are straight lines and all covariant derivatives are ordinary derivatives. This is a flat space."
p. 28: Eq. (7:12)
p. 29: Eqs. (7:26) and (7:27).
p. 31: "we see only covariant derivatives"
p. 33: "non-covariant derivatives are outlawed"
 

12. Alejandro Perez, Introduction to Loop Quantum Gravity and Spin Foams, gr-qc/0409061 v1

"The gravitational interaction is fundamentally different from all the other known forces. The main lesson of general relativity is that the degrees of freedom of the gravitational field are encoded in the geometry of spacetime. The spacetime geometry is fully dynamical: in gravitational physics the notion of absolute space on top of which ‘things happen’ ceases to make sense. The gravitational field defines the geometry on top of which its own degrees of freedom and those of matter fields propagate. This is clear from the perspective of the initial value formulation of general relativity, where, given suitable initial conditions on a 3-dimensional manifold, Einstein’s equations determine the dynamics that ultimately allows for the reconstruction of the spacetime geometry with all the matter fields propagating on it. A spacetime notion can only be recovered a posteriori once the complete dynamics of the coupled geometry-matter system is worked out. Matter affects the dynamics of the gravitational field and is affected by it through the non trivial geometry that the latter defines (see for instance Theorem 10.2.2 in R.M. Wald, General relativity, Chicago U. Press, Chicago, 1984). General relativity is not a theory of fields moving on a curved background geometry; general relativity is a theory of fields moving on top of each other [C. Rovelli, Quantum Gravity, Cambridge U. Press, Cambridge, 2004].
 

13. Bryan Kelleher, Scale-invariant gravity: Spacetime recovered, gr-qc/0307091 v3; Class. Quant. Grav. 21 (2004) 483-495.

"As formulated by Einstein, the natural arena for gravity as represented by general relativity is spacetime. We have a purely 4-dimensional structure and the 4-geometry reigns. The reformulation of the theory in canonical dynamical form by Dirac [1] and Arnowitt, Deser and Misner (ADM) [2] led away from the 4-dimensional picture and placed the emphasis more on the 3-geometry. The configuration space is superspace and general relativity describes the evolution of the 3-geometry in time (geometrodynamics)."
 

14. Andre Gsponer, More on the early interpretation of the Schwarzschild solution, physics/0408100

"Hence, by introducing rather than removing a singular term, Lanczos did the "opposite" of what Lemaître was apparently the first to have done. Furthermore, generalizing from the case of the Schwarzschild metric, Lanczos concluded his paper by stressing that [6, p. 539]:

"this example proves how little one might conclude from a singular behavior of the g_ij functions about a real singularity of the field, for it can possibly be eliminated by a coordinate transformation."
-- 
[6] Cornelius Lanczos, Ein vereinfachendes Koordinatensystem für
Einsteinschen Gravitationsgleichungen, Phys. Zeit. 23 (1922) 537-539.
 

15. Lawrence M. Krauss, Cosmological Antigravity, Scientific American, January 1999, pp. 53-59.

pp. 54-55: "In the general theory of relativity, the source of gravitational forces (whether attractive or repulsive) is energy. Matter is simply one form of energy. But Einstein's cosmological term is distinct. The energy associated with it does not depend on position or time -- hence the name "cosmological constant". The force caused by the constant operates even in the complete absence of matter and radiation. Therefore, the source must be a curious energy that resides in empty space (also known as 'dark gaps' -- D.C.)."

p. 59: "The universe is either open or filled with an energy of unknown nature".
 

16. Abner Shimony, Implications of Transience for Spacetime Structure, in: S.A. Huggett, L.J. Mason, K.P. Tod, S.T. Tsou, and N.M.J. Woodhouse (eds.), The Geometric Universe: Science, Geometry, and the Work of Roger Penrose. Oxford: Oxford University Press, 1998, pp. 161-172.

"Even more problematic is the role of transience in physical theory. Classical mechanics, special relativity, and general relativity differ profoundly in their assumptions about spacetime structure, but in all three the structure is characterized without any reference to the slipping away of the present moment into the past."
 

17. N. David Mermin. What Is Quantum Mechanics Trying to Tell Us? quant-ph/9801057

"The notion of now -- the present moment -- is immediately evident to consciousness as a special moment of time (or a brief interval -- of order perhaps a few tenths of a second). It seems highly plausible to me that your now overlaps with my now or, if you are very far away from me, with a region space-like separated from my now. On the other hand, I can conceive of it not working this way -- that your now is two weeks behind or fifteen minutes ahead of my now. In that case when we have a conversation each of us is talking to a mindless hulk. (...) Physics has nothing to do with such notions. It knows nothing of now and deals only with correlations between one time and another. The point on my world-line corresponding to now, obvious as it is to me, cannot be identified in any terms known to today's physics."
 

18. J. Butterfield and C.J. Isham (1999). Spacetime and the Philosophical Challenge of Quantum Gravity, gr-qc/9903072

[Section Time in General Relativity]: "When we turn to classical general relativity, the treatment of time is very different. Time is not treated as a background parameter, even in the liberal sense used in special relativity, viz. as an aspect of a fixed, background spacetime structure. Rather, what counts as a choice of a time (i.e. of a timelike direction) is influenced by what matter is present; (as is, of course, the spatial metrical structure). The existence of many such times is reflected in the fact that if the spacetime manifold has a topology that enables it to be foliated as a one-parameter family of spacelike surfaces, this can generally be done in many ways -- without any subset of foliations being singled out in the way families of inertial reference frames are singled out in special relativity. From one perspective, each such parameter might be regarded as a legitimate definition of (global) time. However, in general, there is no way of selecting a particular foliation, or a special family of such, that is 'natural' within the context of the theory alone. In particular, these definitions of time are in general unphysical, in that they provide no hint as to how their time might be measured or registered."
 

19. M. Frank, Einstein's Equation in Pictures, gr-qc/0203100, March 15, 2002.

Matthew Frank: "Einstein's equation says that energy is the curvature of space. What does this mean?"
 

20. C.J. Isham (1993). Prima Facie Questions in Quantum Gravity, gr-qc/9310031

"IV. Both general relativity and standard quantum theory appear only in certain limiting situations in the context of a theory that starts from radically new perspectives. Very little is known about potential schemes of this type or, indeed, if it is necessary to adopt such an iconoclastic position in order to solve the problem of quantum gravity. However, the recurring interest in such a possibility is based on the frequently espoused view that the basic ideas behind general relativity and quantum theory are fundamentally incompatible and that any complete reconciliation will necessitate a total rethinking of the central categories of space, time and matter."
 

21. C.J. Isham, J. Butterfield (1999). On the Emergence of Time in Quantum Gravity, gr-qc/9901024

 "The usual tools of mathematical physics depend so strongly on the real-number continuum, and its generalizations (from elementary calculus 'upwards' to manifolds and beyond), that it is probably even harder to guess what non-continuum structure is needed by such radical approaches, than to guess what novel structures of dimension, metric etc. are needed by the more conservative approaches that retain manifolds. Indeed, there is a more general point: space and time are such crucial categories for thinking about, and describing, the empirical world, that it is bound to be ferociously difficult to understand their emerging, or even some aspects of them emerging, from 'something else'."
 

22. Erwin Schrödinger, "Die gegenwärtige Situation in der Quantenmechanik", Naturwissenschaften 23, pp. 807-812; 823-828; 844-849 (1935). Translated by John D. Trimmer,

Sec. 8, Theory of Measurement, Part One,
http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html#sect8

"The rejection of realism has logical consequences. In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has. But then what does it mean?"

Sec. 9, The Psi-function as Description of State,
http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html#sect9

"The rejection of realism also imposes obligations. (...) Therefore if a system changes, whether by itself or because of measurements, there must always be statements missing from the new function that were contained in the earlier one. In the catalog not just new entries, but also deletions, must be made."

Sec. 15, Natural Law or Calculating Device?
http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html#sect15

"That "sharp time" is an anomaly in Q.M. and that besides, more or less independent of that, the special treatment of time forms a serious hindrance to adapting Q.M. to the relativity principle, is something that in recent years I have brought up again and again, unfortunately without being able to make the shadow of a useful counterproposal."

This translation appeared as Section I.11 of Part I of Quantum Theory and Measurement, ed. by John A. Wheeler and Wojciech H. Zurek, Princeton University Press, Princeton, New Jersey, 1983.