Abstract
In this short note we investigate canonical formalism for General Relativity which is formulated with the metric \(f^{ab}=(-g)^\alpha g^{ab}\). We find corresponding Hamiltonian and we show that constraint structure is the same as in the standard formulation. We also analyze another model when the spatial part of metric \(h^{ij}\) is related with the new one by relation \(a^{ij}=(\det h_{ij})^\beta h^{ij}\) and we argue that it corresponds to the gauge fixed version of the General Relativity formulated with the metric \(f^{ab}=(-g)^\alpha g^{ab}\).
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1 Introduction and summary
Dynamical variable of General Relativity is a metric with components \(g_{ab}\). They are natural variables for formulation of Riemann geometry and corresponding quantities as for example a scalar curvature that is fundamental part of Einstein-Hilbert action. Properties of this action were carefully examined by T. Padmanabhan, K. Parattu and B.R. Majhi in [1] and discussed in more details in [2]. However it was stressed here that it is possible to define new variables \(f^{ab}=\sqrt{-g}g^{ab}\) which could be even more appropriate for a description of dynamics of gravity. An importance of these variables was already stressed in [3,4,5] and were recently presented in an important paper [2]. In fact, they have significant meaning in covariant canonical formulation [6, 7] of General Relativity [8]Footnote 1 and its generalization [11,12,13,14,15,16,17]. Further, it was nicely shown in [1] that these variables have a nice thermodynamic interpretation in the emergent gravity paradigm that claims that gravity is emergent from some unknown more fundamental theory. Then we could conjecture that \(f^{ab}\) variables are more fundamental than \(g_{ab}\) and study a consequence of this hypothesis. In particular, it would be interesting to formulate canonical formalism for these variables. By canonical formalism we mean a conventional formalism based on \(D+1\) splitting of space time [18], for review see [19]. The goal of this paper is to investigate this question.
More precisely, we consider a theory with the metric \(f^{ab}\) that is related to \(g^{ab}\) by a point transformation \(f^{ab}=(-g)^\alpha g^{ab}\), where parameter \(\alpha \) can be an arbitrary number and our goal is to study dependence of the theory on \(\alpha \).Footnote 2 Then in order to find canonical formulation of theory for \(f^{ab}\) variable we should again perform \(D+1\) splitting of \(f^{ab}\) when we introduce variables \(M,a^{ij},M^i\) whose precise definitions will be given in the next section and which are related to similar splitting of \(g^{ab}\) metric in terms of \(N,h_{ij},N^i\). With the help of these relations we will be able to find corresponding conjugate momenta. During this procedure we also find a primary constraint that relates \(a^{ij}\) with M and which is a consequence of the fact that M is a dynamical variable as opposite to the lapse N whose conjugate momentum is primary constraint of the theory in the original formulation. On the other hand performing standard manipulation we obtain Hamiltonian that has similar form as the standard one. Then the requirement of the preservation of the primary constraints leads to an emergence of \(D+1\) secondary constraints which are Hamiltonian constraint together with D spatial diffeomorphism constraints. Finally we study stability of these constraints. It turns out that it is useful to express them in terms of the original variables as composite objects from \(a^{ij},M\) and conjugate momenta. Then it turns out that the constraints and Poisson brackets between them have the same form as in General Relativity.
We also consider the case when we use new set of variables for spatial metric \(h_{ij}\) only. In this case the situation is simpler than in the more general case since lapse function does not change. We determine corresponding Hamiltonian and constraint structure that has again the same form as in General Relativity. Finally we argue that this Hamiltonian can be derived by gauge fixing of the primary constraint in the model with dynamical M.
Let us outline our results. We investigate general Theory formulated in terms of new variable \(f^{ab}\) and study their constraint structure. We show that compared to the original case the Hamiltonian is more complicated and introducing new variables does not bring new benefits for the theory. In other words while \(f^{ab}\) variable has significant meaning in the thermodynamics interpretation of the theory and in the covariant canonical formalism standard Hamiltonian formalism is naturally formulated in terms of original metric \(g_{ab}\) and conjugate momenta.
This paper is organized as follows. In the next Sect. 2 we introduce new variable \(f^{ab}\) and determine corresponding Hamiltonian. In the Sect. 3 we study stability of the primary constraints and determine constraint structure of theory. Finally in Sect. (4) we introduce new variable for spatial part of the metric only and study corresponding Hamiltonian.
2 Hamiltonian formalism for \(f^{ab}\) metric
As we wrote in the introduction the goal of this paper is to study a canonical structure of General Relativity action which is expressed in terms of variable \(f^{ab}=(-g)^\alpha g^{ab}\) in \(D+1\) dimensions, where \(a,b=0,1,\dots ,D\). This form of the metric is generalization of the relation \(f^{ab}=\sqrt{-g}g^{ab}\) that was introduced long time ago by Einstein and whose importance in the covariant canonical formalism was stressed recently by Padmanabhan and his colleagues in [1]. An unanswered question remains what is the form of the canonical structure for the General Relativity formulated with the variable \(f^{ab}\). The goal of this paper is to find Hamiltonian for \(f^{ab}\) and conjugate momenta.
In the first step of our analysis we find inverse relations between \(g^{ab}\) and \(f^{ab}\). From definition above we get
With the help of these results we find inverse relation
In order to find canonical formulation of theory we again presume \(D+1\) decomposition of metric \(g_{ab}\). Explicitly, in case of metric \(g_{ab}\) we introduce lapse function \(N=1/\sqrt{-g^{00}}\) and the shift function \(N^i=-g^{0i}/g^{00}\). In terms of these variables we write the components of the metric \(g_{ab}\) as
where \(h_{ij}\) is non-singular spatial D-dimensional metric with inverse \(h^{jk}\) so that \(h_{ij}h^{jk}=\delta _i^k\).
Let us presume the same decomposition of \(f^{ab}\) and its inverse \(f_{ab}\)
where \(a_{ij}a^{jk}=\delta _i^k\) and where \(M^i=a^{ij}M_j\). Then if we firstly compare \(g^{00}\) with \(f^{00}\) we obtain
As the next step we consider relation between \(g^{0i}\) and \(f^{0i}\) and we get
that implies equality of the shift functions
Finally we proceed to the relation between \(g^{ij}\) and \(f^{ij}\) that allows us to find relation between \(h^{ij}\) and \(a^{ij}\) in the form
In summary, we have following relations between original and new variables
Now we are ready to proceed to the definition of the momenta conjugate to \(M,M^i\) and \(a_{ij}\). Note that the action for gravity in \(D+1\) dimensions has the form
where R(g) is scalar curvature. In order to find canonical action we use \(D+1\) decomposition of R
and where r is scalar curvature defined with \(h_{ij}\) and \(\nabla _i\) is covariant derivative compatible with the metric \(h_{ij}\). Note that the divergence terms in (11) can be ignored in the action (10). Finally we introduced metric \(\mathcal {G}^{ijkl}\) and its inverse \(\mathcal {G}_{ijkl}\) that have the form
From their definition we obtain useful relations
As the first step we proceed with the momentum conjugate to M and we get
where \(\mathcal {L}\) is Lagrangian density defined as \(S=\int d^{D+1}x\mathcal {L}\) and where we used the fact that \(f^{ab}\) and \(g^{ab}\) are related by point transformations so that
and also the fact \(h^{ij}h_{jk}=\delta ^i_k\) so that
In case of the momenta conjugate to \(M^i\) we also get that they are primary constraints
as follows from the fact that \(M^i=N^i\) and from the fact that the action (10) does not depend on time derivative of \(N^i\).
Finally we proceed to the momentum conjugate to \(a^{ij}\) and we obtain
Taking the trace of this equation we get
that implies primary constraint of the theory
Now we are ready to proceed to the definition of the canonical Hamiltonian
where \(\mathcal {H}_T\) is kinetic part of Hamiltonian and where we used the fact that
Then using previous relations we obtain explicit form of \(\mathcal {H}_T\) in the form
where
Finally we consider last part of the Hamiltonian and after some calculations we get
where \(\mathcal {H}_m\) is defined as
In order to find final form of \(\mathcal {H}_m\) given in (26) we firstly performed integration by parts in (25) and used the relation between \(\pi _{ij},p_M\) and \(K_{ij}\) given in (18). In summary we get Hamiltonian density \(\mathcal {H}=\mathcal {H}_T+M^i\mathcal {H}_i\) and set of primary constraints \(\pi _i \approx 0,\mathcal {D}\approx 0\). In the next section we will study stability of these constraints.
3 Stability of primary constraints
General theory of systems with constraints says that primary constraints should be preserved during the time evolution of system [20] which means that
where \(H=\int d^D\textbf{x}\mathcal {H}=\int d^D\textbf{x}(\mathcal {H}_T+M^m\mathcal {H}_m)\) and where the symbol \(\approx 0\) means that this equation hold on the surface where all constraints are valid.
We start with the constraints \(\pi _m\approx 0\) where the requirement of their preservations implies secondary constraints
using canonical Poisson bracket
In case of the constraint \(\mathcal {D}\approx 0\) the situation is more involved. Recall that \(\mathcal {D}\approx 0\) has explicit form
This constraint has following Poisson brackets with canonical variables
that implies
We further have
Collecting these terms together we finally obtain
Further, since \(\left\{ \mathcal {D}(\textbf{x}),\mathcal {H}_m(\textbf{y})\right\} =0\) we immediately find that
so that requirement of the preservation of the primary constraint \(\mathcal {D}\approx 0\) implies secondary constraint
At this stage we identified \(D+1\) secondary constraints \(\mathcal {H}_i\approx 0 \, \mathcal {H}_T\approx 0\) together with \(D+1\) primary constraints \(\pi _i\approx 0 \, \mathcal {D}\approx 0\). Now we should check stability of secondary constraints. To proceed to these calculations it is convenient to express \(\mathcal {H}_i\) in different way. We start with following Poisson bracket
which has almost canonical form. In order to have Poisson brackets in the canonical form we introduce \(\tilde{\Pi }^{ij}\) defined as
that has following Poisson brackets
Note that \(\tilde{\Pi }^{ij}\) is related to \(K_{ij}\) by following formula
Then it is easy to see that the spatial diffeomorphism constraint can be written as
Clearly this diffeomorphism constraint has the same form as spatial diffeomorphism constraint derived in General Relativity. Further, since the Poisson bracket between \(\tilde{\Pi }^{ij}\) and \(h_{kl}\) (39) has the same form as in case of General Relativity(up to sign) we find that the Poisson brackets between two smeared forms of diffeomorphism constraints has the form
where \(\textbf{T}_S(X^i)\) is defined as
Let us now turn our attention to the Hamiltonian constraint \(\mathcal {H}_T\). In fact, using \(\tilde{\Pi }^{ij}\) we can write Hamiltonian constraint in the form
where N and h are composed from canonical variables \(a^{ij}\) and M. Then it is clear that the Poisson brackets between smeared form of diffeomorphism constraints and Hamiltonian constraint has the same form as in General Relativity. Explicitly, we have
This result makes an analysis of General Relativity with \(a^{ij}\) and \(\pi _{ij}\) as canonical variables complete. We see that introducing these new variables leads only to an emergence of new constraint \(\mathcal {D}\approx 0\) which replaces original constraint \(p_N\approx 0\). Then the remaining constraint structure is completely the same. Further, from the form of the Hamiltonian and diffeomorphism constraint it is hardly to see that they would simplify resulting Hamiltonian. For that reason we mean that introducing new variables does not bring new benefits for theory.
4 New set of alternative variables
In this section we derive canonical formulation for gravity when we introduce spatial metric \(a^{ij}\) related to \(h^{ij}\) in the form
while N and \(N^i\) remain the same. As in the second section we derive inverse relation between \(h^{ij}\) and \(a^{ij}\) in the form
so that momentum conjugate to \(a^{ij}\) is equal to
As in previous section we find Hamiltonian in the form
Taking the trace of the relation for \(\pi _{ij}\) we obtain
and finally we get
Then the Hamiltonian constraint has the form
where
Then using the fact that
it is easy to see that the Poisson bracket between \(\Pi ^{ij}\) and \(h_{ij}=a_{ij} a^{\frac{\beta }{\beta D-1}}\) has the canonical form (up to sign)
Note that the Hamiltonian can be written in the form
where
Now as in the case of General Relativity \(\pi _N,\pi _i\) which are momenta conjugate to N and \(N^i\) are primary constraints of the theory. Then the requirement of their preservation implies that \(\mathcal {H}_T\approx 0 \, \mathcal {H}_i\approx 0\) are secondary constraints that have the same form as in the case of General Relativity and also thanks to the Poisson brackets (54) we get that the Poisson brackets between these constraints are the same as in General Relativity. In other words \(\mathcal {H}_T\approx 0,\mathcal {H}_i\approx 0\) are first class constraints.
Now we show that this theory can be derived from the theory studied in Sect. 2 when we fix the gauge symmetry \(\mathcal {D}=\pi _{ij}a^{ij}-\frac{\alpha -1}{2\alpha }p_MM\approx 0\). Let us fix this gauge symmetry by introducing gauge fixing function \(\mathcal {F}\equiv M-K\approx 0 \, K=\textrm{const}\). Then the Poisson bracket between \(\mathcal {D}\) and \(\mathcal {F}\) is non-zero and they are second class constraints that can be explicitly solved. Solving \(\mathcal {D}\) for \(p_M\) and M we get
Inserting this result into \(\tilde{\Pi }^{ij}\) defined in (38) we get that we should identify \(\beta \) with \(\alpha \) as
For \(K=1\) we find that \(h^{ij}=a^{ij}a^{-\frac{\alpha }{\alpha (D+1)-1}}=a^{ij}a^{-\frac{\beta }{\beta D-1}}\) and the correspondence is exact.
As the final point of this section we express Hamiltonian constraint in terms of physical variables \(a^{ij}\) and \(\pi _{ij}\)
We see that generally this Hamiltonian constraint has similar form as in case of the original variables. On the other hand we hardly see any simplification introducing new variables \(f^{mn}\) defined above.
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Notes
There are several reasons why to prefer \(f^{ab}=\sqrt{-g}g^{ab}\) instead of \(f_{ab}=\sqrt{-g}g_{ab}\) in four dimensions that were discussed in more details in [1]. For example, \(f^{ab}\) scales linearly with \(g_{ab}\rightarrow \Omega g_{ab}\) which is crucial for replacement of \(g_{ab}\) with \(f^{ab}\) as was shown in [1]. It is clear that this nice property does not hold for general \(\alpha \) and for \(D\ne 3\) however we still follow this prescription in order to make a contact with previous results. Of course it is possible in principle to use an alternative scaling \(f_{ab}=(-g)^{\alpha }g_{ab}\) but we will not discuss this case in the present paper.
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This work is supported by the grant “Dualitites and higher order derivatives” (GA23-06498 S) from the Czech Science Foundation (GACR).
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Klusoň, J. Canonial analysis of general relativity formulated with the new metric \(f^{ab}=(-g)^{\alpha }g^{ab}\). Gen Relativ Gravit 56, 71 (2024). https://doi.org/10.1007/s10714-024-03258-0
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DOI: https://doi.org/10.1007/s10714-024-03258-0