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We provide some general reasons why these statements should be questioned. Points i-iii have been shown to be incorrect in  and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. By direct calculation we show that Dirac's references to space-like surfaces are inessential and that such surfaces do not enter his calculations.
The secondary constraints resulting from the conservation of the primary constraints of Dirac are in fact different from the original constraints that Dirac called secondary also known as the "Hamiltonian" and "diffeomorphism" constraints. The Dirac constraints are instead particular combinations of the constraints which follow directly from the primary constraints.
Taking this difference into account we found, using two standard methods, that the generator of the gauge transformation gives diffeomorphism invariance in four-dimensional space-time; and this shows that points i-iii above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric gpv to lapse and shift functions and the three-metric gkm, which is not canonical.
This proves that point iv is incorrect. Points i-iii are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein's theory itself. The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings, but merely algebraic operations subjected to a regular and uniform rule of procedure.
Those who are fond of Mathematical Analysis will observe with pleasure Mechanics becoming one of its new branches and they will be grateful to me for having thus extended its domain.
The results previously obtained by others are reconsidered and classified as either "myth" or "reality" depending on whether they were obtained by what Lagrange called a regular and uniform rule of procedure, or by geometrical or some other reasonings. The results and conclusions constructed using such reasonings must be checked by explicit calculation without which they are meaningless and could be misleading, contradicting the rules of procedure and the essential properties of GR.
Originating more than half a century ago, the Hamiltonian formulation of GR is not a new subject. It began with advances in the Hamiltonian formulation of singular Lagrangians due to the pioneering works of Dirac  and Bergmann with coauthors [4, 5] on generalized constrained Hamiltonian dynamics1.
We always wondered what is the Rosenfeld contribution. Recently, due to an effort of Sal-. We restrict our discussion to the original Einstein metric formulation of GR. The first-order, affine-metric, form [17, 18] an English translation of this and a few other Einstein's papers can be found in  will be just briefly touched;but the analysis presented here can and must be extended to a affine-metric form and to other formulations e.
Einstein-Cartan formulation of GR. The relationship among these formulations has not been analyzed;and some authors have adopted to using the name "Dirac-ADM" or they refer to Dirac when actually working with the ADM Hamiltonian. This presumes equivalence of the Dirac and ADM formulations. These two, as we will demonstrate, are not equivalent. The Dirac conjecture , that knowing all the first-class constraints is sufficient to derive the gauge transformations, was made only after the appearance of  and became a well defined procedure only later [23, 24].
In the most general form with chains of constraints of any length and with application to field theories this procedure was considered for the first time by Castellani  for alternative approaches see ;problems or, at least, limitations of these approaches are discussed in . Deriving the gauge invariance of GR from the complete set of the first-class constraints should also be viewed as a crucial consistency condition that must be met by any Hamiltonian formulation of the theory;yet, this requirement did not attract much attention and it is not discussed in textbooks on GR, where a Hamiltonian formulation is presented e.
In books on constraint dynamics , even if such a procedure is discussed [14, 15], it is not applied to the Hamiltonian formulation of GR.
Recently this question was again brought to light by Mukherjee and Saha  who applied the method of  to the ADM Hamiltonian with the sole emphasis on presenting the method of deriving the gauge invariance,. In  the original article  and its English translation are accompanied by quite extensive and in many cases helpful comments, but we want to warn the reader that some of them reflect the opinion of the commentator, in particular, about the "crucial error" of Rosenfeld.
We will discuss this later. In  there appears a first complete derivation of the gauge transformations from the constraint structure of the ADM Hamiltonian. The expected transformation of the metric tensor is . In the literature on the Hamiltonian formulation of GR, the word "diffeomorphism" is often used as equivalent to the transformation 1 , which is similar to gauge transformations in ordinary field theories.
This meaning is employed in our article2. The expected transformation 1 does not follow from the constraint structure of ADM Hamiltonian  and a field-dependent and non-covariant redefinition of gauge parameters is needed3 to present the transformations of  in the form of 1 , i. The field-dependent redefinition of gauge parameters 2 goes back to work of Bergmann and Komar  where it was presented for the first time.
The same redefinition of gauge parameters 2 , but in a less transparent form, was obtained for the ADM Hamiltonian by Castellani  for the transformation of the g0p components of the metric tensor to illustrate his procedure for the construction of the gauge generators. This redefinition of gauge parameters was also discussed from different points of view in  in an attempt to justify the necessity of such changes or even make them compulsory.
We would like to return to footnote 1 about the Rosenfeld contribution at the dawn of the Hamiltonian analysis. From the beginning he restricts his interest to field-independent gauge parameters. This approach differs from what is advocated by Salisbury with coauthors . As an application of a general but 'not-yet' procedure, Rosenfeld considered a tetrad formulation of gravity and, according to Salisbury, made a "crucial error" that was corrected by Pons, Salisbury, and Shepley.
But the same "crucial error" emerges from Dirac's treatment of metric gravity;as we will demonstrate, the Dirac formulation allows one to restore gauge invariance without the need for field-dependent gauge parameters or the use of the field-dependent redefinition, i. Such an equivalence is leitmotif and cornerstone of the Rosenfeld treatment of gauge invariant systems.
Exactly the disappearance of this equivalence in the ADM formulation leads the authors of  to introduce the"diffeomorphism-induced gauge symmetry" to support ADM variables.
So, in such a case the "crucial error" of Rosenfeld is simply that he did not use ADM variables4. A common feature of different approaches [32, ] is that they consider only the ADM Hamiltonian  because this is a formulation that does not lead to 1. According to the conclusion of , the transformation 1 and the one with parameters that depend on the fields 2 are distinct.
In  this transformation is called the "specific metric-dependent diffeomorphism". The authors of  have a brief and ambiguous conclusion about 2 : "[it will] lead to the equivalence5 between the diffeomorphism and gauge transformations" and, at the same time, "demonstrate the unity of the different symmetries involved";these are contradictory statements.
Soon after appearance of , Samanta  posed the question "whether it is possible to describe the diffeomor-phism symmetries without recourse to the ADM decomposition". To answer this question, he derived the transformation 1 starting from the Einstein-Hilbert EH Lagrangian not the ADM Lagrangian and applying the La-grangian method for recovering gauge symmetries based on the use of certain gauge identities that appear in .
It is important that 1 follows exactly from this procedure without the need of a field-dependent and non-covariant redefinition of the gauge parameters, which would be necessary in [25, 32, ] where the ADM Hamiltonian is used. The question of the equivalence of 1 and 2 does not even arise in the approach of . In  the dif-. The conclusion of  that "the ADM splitting, which is essential for discussing diffeomor-phism symmetries, is bypassed" contradicts the obtained result.
Firstly, any feature that is "essential" cannot be "bypassed". Secondly, the transformations derived from the ADM Hamiltonian in  are not those of . It is not a "bypass" because the "destination" of having the invariance of 1 , is changed. Comparison of  and  allows us to conclude that ADM decomposition is not only inessential but incorrect because the EH Lagrangian leads directly to diffeomor-phism invariance without the need of any field-dependent and non-covariant redefinition of gauge parameters;and a correct Hamiltonian formulation should give the same result.
There is another statement in  that can also be found in many places "it is well known that this decomposition plays a central role in all Hamiltonian formulations of general relativity". This sentence combined with Hawking's "spiritual" statement forces one to conclude that the Hamiltonian formulation by itself contradicts the spirit of GR. This resonates with Pullin's conclusion  that "Unfortunately, the canonical treatment breaks the symmetry between space and time in general relativity and the resulting algebra of constraints is not the algebra of four dif-feomorphism".
We will show in this paper that the canonical formalism is in fact consistent with the diffeomorphism 1 when the Dirac constraint formalism is applied consistently and that the discrepancies between the ADM formalism and 1 can be explained. The difference of the results  and  which were obtained by different methods also implies the non-equivalence of the Lagrangian and Hamiltonian formulations. In all field theories e. Could this be a peculiar property of GR? Is GR a theory in which the Hamiltonian and Lagrangian formulations lead to different results or was a "rule of procedure" broken somewhere?
The gauge transformation of the metric tensor was derived using the method of  and, without any field-dependent redefinitions of gauge parameters, it gives exactly the same result as the Lagrangian approach of , as it should. In the Hamiltonian formulation of GR given in  the algebra of constraints is the algebra of "four diffeomorphism" in contradiction to the general conclusion of , which was based on a particular, ADM, formulation.
The procedure of passing to a Hamiltonian formulation in field theories based on the separation of the space and time components of the fields and their derivatives defined on the whole space-time, not on some hypersur-face is not equivalent to the separation of space-time into space and time.
For example, by rewriting the Einstein equations in components as was done before Einstein introduced his condensed notation , we do not abandon covariance even if it is not manifest.
The final result for the gauge transformation of the fields can be presented in covariant form when using the Hamiltonian formulation of ordinary field theories e. In any field theory, after rewriting its Lagrangian in components, the Hamiltonian formulation for singular Lagrangians follows a well defined procedure. Such a procedure is based on consequent calculations of the Poisson brackets PB of constraints with the Hamiltonian using the fundamental PBs of independent fields.
In the case of field theories they are. This is a local relation that does not presume any extended objects or surfaces. The canonical procedure does not itself lead to the appearance of any hypersur-faces; in  there are no references to such surfaces and the result is consistent with the Lagrangian formulation of . Such surfaces are either a phantom of interpretation.
The discussion of an interpretational approach is not on the main road of our analysis of the Hamiltonian formulations of GR. However, the routes of such an approach6 are quite interesting: one starting from the basic equations of the ADM formulation, according to , "would like to understand intuitively their geometrical and physical meaning and derive them from some first principles rather than by a formal rearrangement of Einstein's law".
By taking this approach, a formal rearrangement which is a "rule of procedure" is replaced by some sort of intuitive understanding. As a result, a new language is created which "is much closer to the language of quantum dynamics than the original language of Einstein's law ever was" . This language allows one "to recover the old comforts of a Hamiltonian-like scheme: a system of hypersur-faces stacked in a well defined way in space-time, with the system of dynamic variables distributed over these hypersurfaces and developing uniquely from one hyper-surface to another" .
Such an interpretation, although 'reasonable' from the point of view of classical Laplacian determinism, is hard to justify from the standpoint of GR . In GR, an entire spatial slice can only be seen by an observer in the infinite future  and an observer at any point on a space-like surface does not have access to information about the rest of the surface this is reflected in the local nature of 3 in field theories.
It would be non-physical to build any formalism by basing it on the development in time of data that can be available only in the infinite future and trying to fit GR into a scheme of classical determinism and nonrelativistic Quantum Mechanics with its notion of a wave function defined on a space-like slice. The condition that a space-like surface remains space-like obviously imposes restrictions on possible coordinate transformations, thereby destroying four-dimensional symmetry, and, according to Hawking, "it restricts the topology of space-time to be the product of the real line with some three-dimensional manifold, whereas one would expect that quantum gravity would allow all possible topologies of space-time including those which are not product" .
This restriction, imposed by the slicing of space-time, must be lifted at the quantum level ;but, from our point of view, avoiding it at the outset seems to be the most natural cure for this problem. A reader interested in such approaches can find more details in the articles we cite. The usual interpretation of the ADM variables, constraints, and Hamiltonian obviously contradicts the spirit of relativity.
With restrictions on coordinate transformations, which are imposed by such an interpretation, it is quite natural to expect something different from a diffeo-morphism transformation, as was found in [25, 32].
Any interpretation, whether or not it contradicts the spirit of GR, cannot provide a sufficiently strong argument to prove or disprove some particular result or theory, because an arbitrary interpretation cannot change or affect the result of formal rearrangements. The transformation, which is different from the diffeomorphism that follows from the ADM Hamiltonian is the result of a definite procedure [25, 32] and is based on calculations performed with their variables and their algebra of constraints.
In any case, it would likely be impossible to do so after we have gone beyond the point of no return on such a road. We must re-examine the derivation of ADM Hamiltonian right from the start. It is difficult to compare the results of  directly with those of ADM because some additional modifications of the original GR Lagrangian were performed by ADM and it is not easy to trace them according to the "rules of procedure".
We will start with the work of Dirac , where all modifications and assumptions are explicitly stated making it possible for them to be checked and analyzed.
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