# The Hamiltonian formulation of General Relativity: myths and reality

###### Abstract

A conventional wisdom often perpetuated in the literature states that: (i) a 3+1 decomposition of space-time into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [Proc. Roy. Soc. A 246 (1958) 333] and of ADM [Arnowitt, Deser and Misner, in “Gravitation: An Introduction to Current Research” (1962) 227] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i-iii) have been shown to be incorrect in [Kiriushcheva et al., Phys. Lett. A 372 (2008) 5101] 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. In addition, we show that his assumption , used to simplify his calculation of different contributions to the secondary constraints, is unwarranted; yet, remarkably his total Hamiltonian is equivalent to the one computed without the assumption . 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 from the primary constraints. Taking this difference into account we found, using two standard methods, 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 to lapse and shift functions and the three-metric , 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.

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“On ne trouvera point de Figures dans cet Ouvrage. Les méthodes que j’y expose ne demandent ni constructions, ni raisonnemens géométriques ou mécaniques, mais seulement des opérations algébriques, assujéties à une marche régulière et uniforme. Ceux qui aiment ľ Analyse, verront avec plaisir la Mécanique en devenir une nouvelle branche, et me sauront gré ď en avoir étendu ainsi le domaine.”

J. L. Lagrange, “Mécanique Analytique” (1788)

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.

## I Introduction

We begin our paper with words written more than two centuries ago by Lagrange in the preface to the first edition of the “Mécanique Analytique” Lagrange because they express our standpoint in analyzing of the Hamiltonian formulation of General Relativity (GR). 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 work of Dirac on generalized (constrained) Hamiltonian dynamics Dirac-1 .

We restrict our discussion to the original Einstein metric formulation of GR. The first-order, metric-affine, form Einstein will be just briefly touched; but the analysis presented here can and must be extended to a metric-affine form and to other formulations.

In chronological order (which is also ranked inversely in popularity) the Hamiltonian formulation of GR was considered by Pirani, Schild, and Skinner (PSS) Pirani , Dirac Dirac , and Arnowitt, Deser, and Misner (ADM) ADM and references therein. The relationship among these formulations has not been analyzed; and some authors have adopted to using the name “Dirac-ADM” or 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 Diracbook , that knowing all the first-class constraints is sufficient to deduce the gauge transformations, was made only after the appearance of Pirani ; Dirac ; ADM and became a well defined procedure only later Castref1 ; Castref2 . The application of such a procedure to field theories was considered for the first time by Castellani Castellani (for alternative approaches see HTZ ; Novel ; Novel-1 ). 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.g. Gravitation ; Waldbook ). In books on constraint dynamics Kurt ; Gitmanbook ; HTbook , even if such a procedure is discussed HTbook , it is not applied to the Hamiltonian formulation of GR. Recently this question was again brought to light by Mukherjee and Saha Saha who applied the method of Novel to the ADM Hamiltonian with the sole emphasis on presenting the method of deriving the gauge invariance, not on the results themselves. In Saha 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 Landau

(1) |

where is the gauge parameter and the semicolon “” signifies the covariant derivative. 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 article.^{1}^{1}1In
mathematical literature the term diffeomorphism refers to a mapping from one
manifold to another which is differentiable, one-to-one, onto, with a
differentiable inverse. The expected invariance (1) does not follow
from the constraint structure of ADM Hamiltonian and a
field-dependent and non-covariant redefinition of gauge
parameters is needed^{2}^{2}2More detail on the derivation of (2)
is given in the last Section where application of Castellani’s procedure to
the ADM Hamiltonian is reexamined ( and
are gauge parameters of ADM formulation). to present
the transformations of Saha in the form of (1), i.e.

(2) |

The field-dependent redefinition of gauge parameters (2) goes back
to work of Bergmann and Komar Bergmann 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
Castellani for the transformation of the 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 SalSund ; PonsSS ; Pons ; PonsS , the most
recent derivation is in Saha . A common feature of these different
approaches is that they only consider the ADM Hamiltonian. According to the
conclusion of Bergmann , the transformation (1) and the one
with parameters that depend on the fields (2) are distinct. In PonsSS this transformation is called the “specific metric-dependent diffeomorphism”. The authors of
Saha have a brief and ambiguous conclusion about (2):
“[it will] lead to the equivalence^{3}^{3}3Here and everywhere in this article the Italic in
quotations is ours. 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 Saha , Samanta Samanta posed the question “whether it is possible to describe the diffeomorphism 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 Lagrangian method for recovering gauge symmetries based on the use of certain gauge identities that appear in Gitmanbook . It is important that (1) follows exactly from this procedure without the need of field-dependent and non-covariant redefinition of the gauge parameters, which would be necessary in Castellani ; Saha where the ADM Hamiltonian is used. The question of the equivalence of (1) and (2) does not even arise in the approach of Samanta . In Samanta the diffeomorphism transformations were also derived by applying the same method to the first-order, affine-metric, formulation Einstein of GR. The conclusion of Samanta that “the ADM splitting, which is essential for discussing diffeomorphism 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 Saha are not those of Samanta . It is not a “bypass” because the “destination” of having the invariance of (1) is changed.

The conclusion about the results of Saha and Samanta should be that the ADM decomposition is inessential and incorrect because it does not lead to diffeomorphism invariance. This discrepancy between these two recent results vindicates Hawking’s old statement Hawking “the split into three spatial dimensions and one time dimension seems to be contrary to the whole spirit of relativity”, the more recent statements of Pons Pons : “Being non-intrinsic, the 3+1 decomposition is somewhat at odds with a generally covariant formalism, and difficulties arise for this reason”, and Rovelli Rovelli : “The very foundation of general covariant physics is the idea that the notion of a simultaneity surface all over the universe is devoid of physical meaning”.

There is another statement in Samanta 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 Pulin 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 diffeomorphism”. 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 Saha and Samanta which were obtained by different methods also implies the non-equivalence of the Lagrangian and Hamiltonian formulations. In all field theories (e.g., Maxwell or Yang-Mills) the Hamiltonian and Lagrangian formulations give the same result for gauge invariance, so for GR to differ seems unnatural. 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?

Recently, in collaboration with Racknor and Valluri KKRV , we demonstrated that, by following the most natural first attempt of PSS Pirani and by applying the rules of procedure Dirac-1 ; Diracbook ; Kurt ; Gitmanbook ; HTbook ; Castellani , the Hamiltonian formulation of GR (without any modifications of the action or change of variables) leads to consistent results. The gauge transformation of the metric tensor was derived using the method of Castellani and, without any field-dependent redefinitions of gauge parameters, it gives exactly the same result as the Lagrangian approach of Samanta , as it should. In the Hamiltonian formulation of GR given in KKRV the algebra of constraints is the algebra of “four diffeomorphism”, in contradiction to the general conclusion of Pulin which was based on the 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 hypersurface) is not equivalent to 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. In addition, such explicit separation of the space and time components and the derivatives of the fields does not affect space-time itself and is not to be associated with any 3+1 decomposition, slicing, splitting, foliation, etc. of space-time. 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.g., Yang-Mills, Maxwell), as well as in GR KKRV . 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

(3) |

This is a local relation that does not rely on any extended objects or surfaces. Again, as with separation into components, this locally defined canonical PB does not affect space-time and is not related to space-like surface or any other hypersurface because (3) is zero for in a whole space-time and there is no information in (3) that, using mathematical language, can allow one to classify two separate points as points on a particular space-like surface or on any surface. The canonical procedure does not itself lead to the appearance of any hypersurfaces; in KKRV there are no references to such surfaces and the result is consistent with the Lagrangian formulation of Samanta . Such surfaces are either a phantom of interpretation or canonical procedure was abandoned by their introduction.

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
approach^{4}^{4}4We have to confess that we found hard to understand
approaches which are not analytical and, to avoid any misinterpretations, we
will merely quote their advocates. A reader interested in this approach can
find more details in the articles we cite. are quite interesting: one
starting from the basic equations of the ADM formulation, according to
HKT , “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” HKT . This language allows
one “to recover the old comforts of a Hamiltonian-like
scheme: a system of hypersurfaces stacked in a well defined way in space-time,
with the system of dynamic variables distributed over these hypersurfaces and
developing uniquely from one hypersurface to another” Kuchar . Such an interpretation, although ‘reasonable’ from the point
of view of classical Laplacian determinism, is hard to justify from the
standpoint of GR Ann33 . In GR, an entire spatial slice can only be
seen by an observer in the infinite future Ann34 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” Hawking . This restriction, imposed by the slicing of space-time, must
be lifted at the quantum level Thiemann ; but, from our point of view,
avoiding it at the outset seems to be the most natural cure for this problem.

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 diffeomorphism transformation, as was found in Castellani ; Saha .

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 different from diffeomorphism that follows from the ADM Hamiltonian is the result of a definite procedure Castellani ; Saha and is based on calculations performed with their variables and their algebra of constraints. From the beginning we will not use the language of 3+1 dimensions, so as to avoid the necessity of getting ourselves “out of space and back into space-time” Smolin at the end of the calculations. 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 reexamine the derivation of ADM Hamiltonian right from the start.

It is difficult to compare the results of KKRV 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 Dirac , where all modifications and assumptions are explicitly stated making it possible for them to be checked and analyzed. In addition, Dirac’s canonical variables are components of the metric tensor which are the same as those used in KKRV where diffeomorphism invariance was derived directly from the Hamiltonian and constraints. Moreover, in GKK two Hamiltonian formulations, based on the linearized Lagrangians of Pirani ; KKRV and Dirac , were considered. Despite there being different expressions for the primary and secondary constraints, these two formulations have the same algebra of PBs among the constraints, and with the Hamiltonian, therefore, they have the same gauge invariance. This is exactly what one can expect in the case of full GR, provided one makes no deviation from canonical procedure. In analyzing the ADM formulation we will follow a different path. We will not start from the GR Lagrangian, but instead compare the final results of Dirac and ADM and try to determine what deviation from the canonical procedure lead to the transformations found in Castellani ; Saha which are distinct from those of (1).

In the next Section we shall thoroughly reexamine the Dirac derivation of the GR Hamiltonian Dirac with emphasis on the effect of his modifications of the action and of the other simplifying assumptions he makes. In particular, we will investigate whether space-like surfaces actually play any role in his derivation, or if they just serve as an illustration which can be completely disregarded from the standpoint of the canonical procedure, as in KKRV . In Section III, using Castellani’s procedure and the results of Section II, we derive the transformations of the metric tensor. The result is the same as those found in Samanta and KKRV . The same result is obtained by application of the method used in Saha to Dirac’s Hamiltonian, which illustrates the equivalence of these two methods. Some peculiarities of such methods, that cannot be seen in ordinary field theories, are briefly discussed and related to the peculiarities of diffeomorphism invariance as it compares to the gauge invariance in ordinary theories. Finally, we consider the ADM Hamiltonian formulation of GR. In the last Section IV we demonstrate that the ADM formulation follows from Dirac’s by a change of variables. The canonicity of this change of variables (the ADM lapse and shift functions) is analyzed. Based on this analysis, the general and more restrictive criteria for a canonical transformation in the case of singular gauge invariant theories are discussed.

## Ii Analysis of Dirac derivation

In KKRV the GR Hamiltonian, constraints, closure of the Dirac procedure, and the diffeomorphism transformation of the metric tensor were derived without any reference to space-like surfaces, the use of any 3+1 decomposition of space-time, or slicing, splitting, foliation, etc., as well as without modifications of the Lagrangian or the introduction of any new variables. (The canonical variables of KKRV are components of the metric tensor.) Dirac, when considering the Hamiltonian formulation of GR in Dirac , also used the metric tensor as a canonical variable; but he made frequent references to space-like surfaces. If such surfaces, which according to Hawking contradict the whole spirit of General Relativity, are the part of Dirac’s calculations, then one has to expect transformations different from diffeomorphism and similar to the one found in Saha from the ADM Hamiltonian. Our main interest is to find out, by following all the steps of Dirac’s derivation of the Hamiltonian, the place where (if anywhere) space-like surfaces enter his derivation or where (if anywhere) his approach deviates from a regular and uniform rule of canonical procedure. If there is no deviation, one should then obtain the diffeomorphism invariance (1), the same as found in KKRV . This would resemble what happens in linearized GR, as discussed in GKK .

In Dirac , Dirac started the Hamiltonian formulation from the
“gamma-gamma” part of the Einstein-Hilbert
(EH) Lagrangian (Eq. (D8))^{5}^{5}5We will refer on Dirac equations quite
often and use the convention, Eq. (D#), to mean equation # from
Dirac . (e.g., see Landau ; Carmeli )

(4) |

where

(5) |

The same Lagrangian was used in Pirani and KKRV . This is a Lagrangian of a local field theory in four(or any)-dimensional space-time, and space-like surfaces or any other hypersurfaces are not intrinsic to such a formulation.

(6) |

where are momenta conjugate to . The exact form of can be found in Pirani ; KKRV (Greek subscripts run from to and Latin ones from to where is the dimension of space-time).

In addition to eliminating the second order derivatives of the metric tensor present in the Ricci scalar in passing from the EH Lagrangian to its gamma-gamma part (4) so that Carmeli

(7) |

Dirac made an additional change to the Lagrangian in order to eliminate the second term in (6). The modified Lagrangian is obtained by adding two total derivatives which are non-covariant (Eq. (D15))

(8) |

This change does not affect the equations of motion, but leads to simple primary constraints (Eq. (D14))

(9) |

It was shown in GKK , that the linearized version of the modified
(8) and unmodified Lagrangians (4), despite leading to
different expressions for the constraints and the Hamiltonian, result in the
same constraint structure, the same number of first-class constraints, and the
same gauge invariance, which is the linearized version of diffeomorphism. This
is what one can also expect in the case of full GR. According to
Dirac , the simplification (9) “can be
achieved only at the expense of abandoning four-dimensional
symmetry” which is obviously correct for this modification
of the Lagrangian (8); yet Dirac’s further conclusion that
“four-dimensional symmetry is not a fundamental property of
the physical world” is too strong and has to be clarified.
Of course, four-dimensional symmetry of the Lagrangian is destroyed by the
modification (8); but this change does not affect the equations of
motion, which are the same as the Einstein equations. Consequently, for the
equations of motion, not only four-dimensional symmetry is preserved, but also
general covariance.^{6}^{6}6The term “four-dimensional” symmetry used by Dirac probably reflects
the fact that the gamma-gamma part of the Lagrangian, quadratic in first order
derivatives, is not generally covariant after the elimination of terms with
second order derivatives in the full EH Lagrangian (7).
If four-dimensional symmetry is preserved in the equations of motion, which
are invariant under general coordinate transformations, then diffeomorphism
should be recovered in the course of the Hamiltonian analysis, as in
KKRV .

The new Lagrangian differs from the original one (4) only for terms linear in the time derivatives of a metric (i.e. ‘velocities’), the parts responsible for the simplification of the primary constraints. We then have

(10) |

where the numbers in brackets indicate the order in velocities (for the Hamiltonian and constraints it will indicate the order in momenta). The exact form of is given by Eq. (D18).

This Lagrangian is used to pass to the Hamiltonian

(11) |

With the modification of (8) the part of the Lagrangian , as was shown by Dirac, can be written as

(12) |

where is the Christoffel symbol

(13) |

and

(14) |

with

(15) |

Note, that in the second order formulation, , , and are just short notations and none of them denote a new and/or independent variable.

Some comments about (12) are in order. The careful reader will definitely wonder how the parts of the Lagrangian which are quadratic and linear in velocities can have contributions without velocities, ; the direct calculation of does not have such contributions (see Dirac’s unnumbered equation preceding (D19))

(16) |

Dirac completed this square, leading to the compact form of (12). Working with (16) instead of (12), will of course not change the results and actually has no calculational advantage. However, we keep (12) so as to compare our calculations with those of Dirac.

The in (12) (explicitly given by (24)) is independent of the velocities. The only part of (11) that has dependence on is

(17) |

Performing the variation , we obtain (see (D18-D21))

(18) |

Equation (18) is easy to solve for due to the invertability of

(19) |

where the inverse to in any space-time dimension (except ) is

(20) |

This result gives

(21) |

After substitution of (21) into (17) (note that (18) can be solved for thus making the calculations shorter) we obtain the total Hamiltonian

(22) |

where (the canonical part of the Hamiltonian) is given by Eqs. (D33, D34)) as,

(23) |

with (Eq. (D19)) and (Eq. (D8)):

(24) |

(25) |

Note that the second term of (23), the part linear in the momenta, arises only after some rearrangement. The direct substitution of into (the only part of (11) that leads to terms linear in the momenta) gives

(26) |

which after integration by parts and using leads to

(27) |

The first term of (27) can be written in the form given by Dirac (D41)

(28) |

with

(29) |

We note that in obtaining the expression for the Hamiltonian (23), all direct calculations with the initially modified Lagrangian (12) were performed by Dirac without any reference to space-like surfaces or any additional restrictions or assumptions.

The next step in the canonical procedure is to find the time development of the primary constraints and see if there are any secondary constraints (or -equations in Dirac’s terminology). PBs among the primary constraints are obviously zero, . The PBs of the primary constraints (9) with the total Hamiltonian (22) are

(30) |

where we keep Dirac’s convention for the fundamental PB (Eq. (D11)),

(31) |

According to Dirac, “the second term of (D33) [ in our (23)] is very complicated and a great deal of labour would be needed to calculate it directly” and instead of performing the variation , he uses some arguments (see (D23-D27)) related to the displacements of surfaces of constant time, and thus he infers that the Hamiltonian “must be of the form” (see (D28))

where and are independent of the . Dirac’s arguments are very general and independent of the particular form of the Lagrangian, i.e. they have no connection with his initial modifications of leading to . And, even in the linearized case GKK , without these modifications, the secondary constraints have a dependence on ; this dependence also happens in full GR KKRV . In any case, the explicit form of the constraints cannot be found using such arguments and explicit calculations are needed; one has to use a well defined rule of procedure to find them, i.e. we must calculate . Dirac performed these calculations using an additional simplifying assumption (see below) and this result has to be analyzed and compared to what follows from direct calculations.

According to Dirac, there are no contributions from to a vector constraint ( in Dirac’s notation) which presumably comes from the time development of the corresponding primary constraint (30). Furthermore, , which comes from the time development of the primary constraint, , can be calculated with the additional simplifying assumption , which gives (Eq. (D36)):

(32) |

As a result, all of , along with the biggest part of , is dropped from his calculations. According to Dirac Dirac , the equation for “must hold also when does not vanish”. It is important to check this assumption by direct calculation because if the result of is the same as that of Dirac’s, then the simplifying assumption of (32), along with any references to surfaces of constant time, has nothing to do with his final result. In such a case, Hawking’s criticism of formulations based on the introduction of space-like surfaces, which is in contradiction with the whole spirit of General Relativity and restricts topology of space-time Hawking , cannot be applied to the Dirac analysis of GR. This also means that the transformations (1) should be derivable in the Dirac Hamiltonian formulation, as was done in the Lagrangian formulation Samanta or for the Hamiltonian formulation obtained in KKRV .

If the results following from the assumption of (32) are different from those where the assumption is not made, then we cannot use (32) as an extra condition in the midst of the calculations and we have to go back to the original Lagrangian to introduce this condition from the outset. This is the rule followed in ordinary constraint dynamics; all imposed constraints must be solved at the Lagrangian level, or added to the Lagrangian using Lagrange multipliers, before performing a variation and/or considering the Hamiltonian formulation.

For example, when Chandrasekhar considers the Hamiltonian for Schwarzschild space-time he, first of all, writes the Lagrangian using this metric and only then passes to the Hamiltonian formulation Chand . Similarly, the condition (32) corresponds to a particular coordinate system, one which is static Landau ; Carmeli ; and, of course, the momenta , which are conjugate to the eliminated variables cannot appear in such a formulation. Note that the initial modification of the Lagrangian (8) is irrelevant in a static coordinate system as the last two terms in (8) are zero when is zero. For field theories, especially generally covariant ones, there is an additional restriction: the unambiguous canonical formulation must be performed without explicit reference to ambient space-time by making an a priori choice of a particular coordinate system or subclass of coordinate systems Isham , i.e. without destroying the main feature of a theory from the beginning.

To find out whether or not Dirac’s formulation is correct or any reference to surfaces of constant time and the simplifications of (32) [or (D36) of Dirac ] are relevant to his actual results, we perform a “great deal of labour” to find the functional derivatives separately for each contribution of and to compare the results with those obtained by Dirac.

For in (25), we find that

(33) |

where the ’s are combinations of terms of different order in (note that the terms of first and zero orders in cancel)

(34) |

and

(35) |

When , the result (33), is considerably simplified (this is because or equal zero if at least one index is zero):

(36) |

According to Dirac, this is the only source of contributions to the scalar constraint and he constructed it using the simplifying assumption of (32) and later concluded that it “must hold also when does not vanish”. Let us check this assertion by explicitly separating all space and time indices in (36)

(37) |

Some terms in (37) have explicit dependence on the space-time components of the metric tensor and these components will disappear only if condition (32) is imposed. For there are even more such components. Even with condition (32), the result is not zero and this part of the Hamiltonian, , contributes to the vector constraint.

Now let us find contributions coming from the second part, . After a rearrangement of the terms given in (24) into a form which is more suitable for calculation, we obtain

(38) |

For the part we calculate

(39) |

The variation obviously produces contributions which are only third and second order in as in (33). For terms of third order we find

(40) |

and in second order we have two contributions: the first proportional to

(41) |

and the second with an index on

(42) |

Note, that we cannot present the part quadratic in (41, 42) in a compact form, where terms with derivatives are a common factor, because of the mixture of four and three indices, which is the result of the original noncovariant modification (8) of the Lagrangian. When performing these calculations we have to consider all possible combinations separately.

It is not difficult to confirm that is not zero, even with assumption (32), there are contributions to both the constraints and . Consequently, Dirac’s conjecture, if made separately for and , is not correct; but, when both parts are combined, the contribution of zeroth order to the secondary constraint is greatly simplified

(43) |

The -part is the same with or without condition (32) and is given by (43) with (it is zero when (32) is imposed). Frequently (43) is written in a different form which is based on the following observation: if in the expression for the four-dimensional Ricci scalar

(44) |

we keep only the spatial indices and change the covariant component of to or, equivalently, impose the conditions (32), we obtain the expression shown in square brackets of (43), which is often called .

Equation (43) gives contributions to the secondary constraints of zeroth order in the momenta . There are obviously contributions to . Dirac’s vector constraint, , does not have such contributions, so it is not directly related to the time development of the corresponding primary constraint (we will discuss this later).

For , the equation (43) has to be compared to the corresponding expression of Dirac’s (D39):

(45) |

where (D38)

(46) |

is a part of full expression (5) where after passing to “form” only the terms cubic in are present. Terms quadratic and linear in are neglected, which results from the simplifying assumption because all non-cubic terms have either or the derivatives . In his final expressions (45) Dirac keeps , not , which is consistent with his statement that this has to be true without the simplifying assumption which removes the difference between and . In addition, we keep in all equations. Dirac used and (or, probably, now more familiar notation for and (or ) for ) which are connected by or .

By differentiating the second term of (45), it is not difficult to derive the relation

(47) |

Dirac’s scalar constraint () is not the result of a direct calculation of . This difference is not important for the proof of closure of the Dirac procedure and one can always consider linear combinations of constraints. For Castellani’s procedure (or any other procedure) for finding gauge transformations we have to be careful with such redefinitions as we will demonstrate in the next Section.

Until now, we have been concerned with the most complicated contributions to the secondary constraints which are zeroth order in the momenta. Let us now consider the contributions to all orders. In the other two orders we obtain (using (23))

(48) |

(49) |

Note, as there are no contributions linear in the momenta to the scalar constraint, but unless we impose (32).

was already calculated in (43). For the full scalar constraint, , the relation (47) is preserved in all orders

(50) |

The vector constraint has non-zero contributions in all orders of the momenta unless (32) is imposed. Before we continue to compare our direct calculations with those of Dirac, let us try to present the canonical Hamiltonian as a linear combination of the secondary constraints we calculated above.

We approach this problem by considering different orders in the momenta. The highest order is the second and the result is easily obtained from the first terms of (23) and (48)

(51) |

(using ).

By considering (27), which is equivalent to the second terms of (23) and (49), we have in first order

(52) |

and are of the same form and we anticipate is also in this form. Unfortunately, this is not obvious and we have to perform some calculations to show it. To preserve the structure found in (51, 52), we will demonstrate that

(53) |

Note, that for given by (27) we also obtain (52) only up to a total spatial derivative. Using (25), (38), and (43) we have

(54) |

This equation demonstrates that the relations found for and are also valid for and the canonical Hamiltonian can be written in terms of as

(55) |

Of course, this is correct up to total temporal (see (8)) and spatial (see (8), (27), and (54)) derivatives. The modification of the initial Lagrangian (8) was proposed by Dirac while (54) is obtained in the course of preserving relations found among contributions of higher order in the momenta to the constraints and the Hamiltonian. It would be very difficult to guess (54) without knowing the final result. Such an additional integration appearing in (54), is very often performed at the Lagrangian level. For example, in the book by Gitman and Tyutin Gitmanbook , in addition to Dirac’s (8) (which are and first term of of Eq. (4.4.12) in Gitmanbook ), the integrations of (54) were performed at the Lagrangian level (the second and third terms of ). The integrations of (54) can be derived only in the course of the Hamiltonian procedure, but such integrations (if they are known) are also correct when applied to the Lagrangian because (going back to Dirac’s derivation) it is clear that was constructed before the elimination of the velocities (i.e., at the Lagrangian level).

How is this covariant form of (55) (which is equivalent to what was found in KKRV ) related to Dirac’s expression for the Hamiltonian? Are they equivalent? The relationship between scalar constraints and was found in (50); we now consider the relation between the vector constraints.

Let us inspect the form of our constraints calculated to different orders appearing in (48), (49), and (43). There are simple relations between the contributions of different orders to and :

that allow one to write (to all orders)

(56) |

with

(57) |

Solving (56) for gives a combination of the constraints and which were originally calculated from the time development of the corresponding primary constraints.

In terms of this combination of constraints and , we obtain a different form of the canonical Hamiltonian

(58) |

This form of is easy to compare with Dirac’s, because his vector constraint is simply related to

(59) |

For we find

(60) |

Equation (59), together with (47), demonstrates the equivalence of the two different forms of given in (55) and (58) to Dirac’s canonical Hamiltonian

(61) |

We would like to emphasize that Dirac’s constraints are not a direct result of the time development of the primary constraints which produce ((50) and (60)). The only place known to us where this is stated is in the book by Gitman and Tyutin (Eq. (4.4.19) of Gitmanbook ); but Dirac’s particular combinations of constraints and the corresponding form of the Hamiltonian are usually used.

The linear approximation of gives exactly the constraints of linearized GR GKK . In the linearized case there is no difference between and ; therefore linearized gravity can provide little “guidance” to full GR, in contrast to what was emphasized by ADM in ADM-1 . Any such guidance has to be taken cautiously.

To demonstrate closure of the Dirac procedure, any form of the canonical Hamiltonian (61) is suitable as they are all equivalent; and any linear combination of constraints can be used for this purpose (e.g., and ). When using Castellani’s procedure to derive the gauge transformations generated by first-class constraints, we have to consider those secondary constraints that directly follow from the corresponding primary ones and the PBs of secondary constraints with the total Hamiltonian, not just with its canonical part (this is also discussed in the next Section).

All of Dirac’s secondary constraints have a zero PB with the primary
constraints. In constraint dynamics this means that Lagrange multipliers
cannot be found at this stage. As the PB of the secondary constraints with the
canonical part of the Hamiltonian is zero or proportional to constraints, the
procedure is closed. This is exactly the case here when we are taking into
account the algebra^{7}^{7}7This algebra is called “hypersurface deformation algebra” or “Dirac
algebra” and can be found in many places, e.g.
Castellani ; Saha . of PBs among Dirac’s combinations of the secondary constraints:

(62) |

When dealing with the “covariant” secondary constraints , the multipliers are again not determined, but now we have

(63) |

The closure of Dirac’s procedure is obviously preserved when using the covariant constraints because and Dirac’s constraints are simply related by (50) and (60). This can also be shown by direct calculation of without any reference to Dirac’s combinations of constraints and their algebra. These calculations are long and to perform them we found it more convenient to work in the intermediate stages with and . This allows us to sort out terms uniquely, and at the final stage we can express the result, using (56), in terms of covariant constraints. The details of such calculations will be given in KKX . We arrive to the following PBs of with the canonical part of the Hamiltonian

(64) |

and