diff --git "a/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/load_file.txt" "b/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/load_file.txt" @@ -0,0 +1,538 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf,len=537 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='13623v1 [hep-th] 31 Jan 2023 Unimodular Gravity in Covariant Formalism J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Klusoň† and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Matouš†‡ 1 † Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic ‡ North-Bohemian Observatory and Planetarium in Teplice, Koperníkova 3062, 415 01, Teplice, Czech Republic Abstract In this short note we study unimodular gravity in Weyl-De Donder formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We find corresponding Hamiltonian and study consequence of the unimodular constraint on the conjugate covariant momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We also find covariant Hamiltonian for Henneaux-Teitelboim unimodular action and study corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 1 Introduction and Summary Unimodular gravity was firstly introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Einstein in his paper [3] published in 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this work the unimodular constraint √−g = 1 was used as gauge fixing condition of general diffeomorphism in order to sim- plify calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then it was shown in [1, 2] that imposing this condi- tion before the variation of Einstein-Hilbert action leads to the traceless equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As we review below these equations of motion are classically equivalent to the general relativity equations of motion with cru- cial difference that the cosmological constant appears as integration con- stant rather than true cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This fact brings new hope how to solve cosmological constant problem which was however questioned in [4], 2 where it was argued that quantum corrections make the cosmo- logical constant ultraviolet sensitive in unimodular gravity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand it is important to stress that no definitive conclusions have been reached yet regarding this problem and unimodular gravity is still very inten- sively studied, for some works devoted to unimodular gravity, see for example [7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 1Email addresses: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Klusoň: klu@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Matouš: bmatous@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cz 2For review of unimodular gravity, see for example [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 1 One of the most interesting aspects of unimodular gravity is the number of physical degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Naively, unimodular constraint √−g = 1 re- duces the number of independent components of metric to nine which could suggest that the number of physical degrees of freedom is less than in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand unimodular gravity is invariant under restricted diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Taking these two aspects together we find that the num- ber of local physical degrees of freedom is the same as in ordinary general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This fact was proved with the help of the Hamiltonian analysis of unimodular gravity performed in [16, 17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand as was shown in these papers standard analysis of unimodular gravity based on D + 1 splitting of target space-time is rather non-trivial and shown complexity of the canonical analysis of systems with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then one could ask the question how unimodular gravity could be de- scribed in covariant canonical formalism that is known as Weyl-De Donder theory [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' The key point of this formulation is that we treat all partial derivatives as equivalent when we define conjugate momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For example, if we have scalar field φ with Lagrangian density in D +1 dimensional space- time equal to L = −1 2ηab∂aφ∂bφ − V (φ), we define the conjugate momentum as 3 πa = ∂L ∂∂aφ = −ηab∂bφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then covariant canonical Hamiltonian density is defined as H = πa∂aφ − L = −1 2πaηabπb + V (φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Clearly such a form of Hamiltonian density preserves diffeomorphism invari- ance of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This approach is known as multisymplectic field theory, see for example [29, 30, 31], for review, see [32] and for recent interesting application of this formalism in string theory, see [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is clear that such covariant canonical formalism is especially suitable for manifestly covariant theories as for example general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In fact, covariant canonical formalism of general relativity was found long time ago by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Hořava [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This analysis was recently generalized to the case of F(R) gravity in [37] and further elaborated in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this paper we apply this formalism for unimodular theory of gravity in D + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This is non-trivial task due to the well known complex- ity of canonical analysis of unimodular gravity in non-covariant formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 3We define ηab = diag(−1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=', 1), a, b = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 2 Further, it is also very interesting to study this system since it contains pri- mary unimodular constraint and it is non-trivial task how to deal with such systems in covariant canonical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In more details, we include this primary constraint to the action with corresponding Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we derive corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Using these equations of motion we find that the unimodular constraint implies another constraint on the canonical conjugate momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we show that this constraint is equivalent to the vanishing of the trace of the Christoffel symbols which is characteristic property of unimodular theory of gravity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This is nice and non-trivial result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand the Lagrange multiplier corresponding to the primary constraint cannot be determined as in non-covariant canon- ical formalism by imposing condition of the preservation of the secondary constraint due to the fact that the equations of motion for conjugate mo- menta are in the form of the divergence of these momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For that reason we determine this constraint in the same way as in the Lagrangian formalism when we calculate the trace of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As a result we obtain equations of motion that are traceless and that do not depend on the cos- mological constant which is in agreement with the Lagrangian formulation of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As the second step in our analysis we find covariant canonical formula- tion of Henneaux-Teitelboim formulation of unimodular gravity [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this case we again identify covariant Hamiltonian together with set of primary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we consider canonical form of the action and determine corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Solving these equations of motion we find that Lagrange multiplier is integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this case we repro- duce results well known from Lagrangian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' However we mean that this is nice and interesting application of the covariant canonical analysis to the constraint systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Let us outline our results and suggest possible extension of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We found covariant Hamiltonian formalism for unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' First of all we determined covariant Hamiltonian for general relativity action in D + 1 dimensions where we again introduced variable f ab = √−ggab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' At this place we would like to stress an importance of this result since it was not apri- ori known whether f ab is suitable for formulation of gravity in space-time of dimension different from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we imposed unimodular constraint using Lagrange multiplier method and then we studied corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We found that the consistency of the theory demands that the trace of conjugate momenta is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we showed that this is character- 3 istic property of unimodular gravity when we pass to Lagrangian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Final we found covariant Hamiltonian for Henneaux-Teltelboim formulation of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We identified primary constraints of the theory and then we studied equations of motion that follow from canonical form of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We showed that they precisely reproduce Lagrangian equations of motion that is nice consistency check of the covariant canonical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We mean that the analysis presented in this paper suggests that covariant Hamiltonian formalism is very close to Lagrangian formalism and in some situations the covariant Hamiltonian formalism is more suitable than La- grangian one, as for example study of thermodynamics properties of horizon [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is also clear that there are more systems that could be analysed with the help of covariant canonical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' One possibility is to study Weyl invariant gravity in this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Another possibility would be to perform analysis of theories of gravity with higher derivatives where the classical canonical analysis is very complicated, see for example [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We hope to return to these problems in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In the next section (2) we review properties of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='Then in section (3) we proceed to the co- variant canonical formulation of this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Finally in section (4) we perform covariant canonical formulation of Henneaux-Teltelboim unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 2 Brief Review of Unimodular Gravity In this section we review basic facts about unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For recent very nice and more detailed review, see for example [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Unimodular gravity is theory with the constraint √−g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Clearly such a condition has a consequence on allowed differomorphism transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In fact, let us consider general transformation of coordinates x′a = xa + ξa(x) (1) that implies inverse relation xa = x′a − ξa(x) ≈ x′a − ξa(x′) + O(ξ2) , (2) where a, b, c = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' , D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Under these transformation the metric gab trans- form as g′ ab(x) = gab(x) − ∂cgab(x)ξc(x) − gac(x)∂bξc(x) − ∂aξc(x)gcb(x) (3) 4 that implies following variation of metric δgab(x) = g′ ab(x) − gab(x) = −gac∂bxc − ∂aξcgcb − ∂cgabξc so that the variation of the square root of the determinant of metric is equal to δ � − det g = −(2∂aξa − ∂cgabgbaξc) � − det g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (4) In case of unimodular gravity this variation should vanish and hence we obtain following condition on ξa in the form ∇aξa = ∂aξa + 1 2gac∂dgcaξd = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (5) The most straightforward way how to find an action for unimodular gravity is to consider standard Einstein-Hilbert action with an unimodular constraint added S = 1 16π � dD+1x[√−g(R − 2¯Λ) + Λ(√−g − 1)] + Smatt , (6) where Λ is Lagrange multiplier whose variation ensures unimodular condition and where ¯Λ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Performing variation of the action (6) with respect to gab we obtain fol- lowing equations of motion 1 16π(Rab − 1 2gab(R − 2¯Λ + Λ)) = Tab , (7) where Tab is matter stress energy tensor defined as Tab = − 1 √−g δSmatt δgab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (8) The crucial point is that Λ is Lagrange multiplier that should be determined as a consequence of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' To do this we perform the trace of the equation (7) to express Λ as Λ = (1 − D) 1 + D R − 32π D + 1T + 2¯Λ , T ≡ gabTab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (9) 5 Inserting this result into (7) we obtain Rab − 1 D + 1gabR = 16π(Tab − 1 D + 1gabT) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (10) These equations of motion are trace-free and also most importantly they do not contain any information about cosmological constant ¯Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is important to stress that even equations of motion of general relativ- ity without unimodular constraint imposed split into 9 trace-free equations of motion and one additional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' To see this consider general relativity equations of motion Rab − 1 2gab(R − 2¯Λ) = 16πTab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (11) Taking the trace of this equation we can express R as R = 2 1 − D(16πT − (D + 1)¯Λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (12) Note that with the help of this equation we can rewrite (11) into trace-free form Rab − 1 D + 1Rgab = 16π(Tab − 1 D + 1Tgab) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (13) However we should again stress that (12) determines R as function of trace of matter stress energy tensor and true cosmological constant term in Einstein- Hilbert action while in case of unimodular gravity we express Λ-which is Lagrange multiplier and not constant, as function of R, T and ¯Λ, as follows from equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In order to check equivalence between unimodular gravity and ordinary general relativity we should be able to reproduce equation (12) in case of unimodular gravity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We can do this by following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Consider equations of motion (10) and rewrite them into the form Rab − 1 2gabR = 16π(Tab − 1 D + 1gabT) + 1 − D 2(D + 1)Rgab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (14) Now we apply covariant derivative on both sides of the equations above and using the fact that the covariant derivative of Einstein tensor Gab = Rab − 1 2gabR is zero we get 1 D + 1∇b(16πT − 1 − D 2 R) = 16π∇aTab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (15) 6 If we consider ordinary form of matter we obtain that divergence of stress energy tensor is zero as a consequence of matter equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then the right side of the equation above is zero and the left side can be easily integrated with the result R = 2 1 − D(16πT + Ω) , (16) where Ω now appears as true integration constant rather than the cosmolog- ical constant that was imposed in the theory by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In other words (16) is the last equation of motion of unimodular gravity and we fully recovered equivalence with general relativity however keeping in mind that we should still have to impose the condition √−g = 1 in the course of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Having performed basic review of unimodular gravity we proceed in the next section to its formulation in the covariant Hamiltonian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 3 Covariant Hamiltonian Formalism For D + 1 dimensional Unimodular Gravity In this section we find covariant Hamiltonian formalism for unimodular grav- ity in D + 1 formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As usual in the covariant formalism we split the Einstein-Hilbert action into bulk and boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Since this procedure is well known,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' see for example [35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 36] and also recent generalization to the case of F(R) gravity [37] we write immediately final result L = Lbulk + Lsurf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Lbulk = 1 16π √−g[Γh dkΓk ghggd − Γf fkΓk ghggh] + + 1 16π ¯Λ√−g + 1 16πλ(√−g − 1) ≡ ≡ Lquad + 1 16π ¯Λ√−g + 1 16πλ(√−g − 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Lsurf = 1 16π∂j[√−g(gikΓj ik − gijΓk ik)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (17) where Γa bc are Christoffel symbols Γa bc = 1 2gad(∂bgdc + ∂cgdb − ∂cgab) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (18) 7 and where ¯Λ is cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Note that the presence of the term with Lagrange multiplier allows us to treat all components of metric as in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now we are ready to proceed to the covariant Hamiltonian formulation of this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' The main idea of this formalism is to treat all derivatives of dynamical variables on the equal footing [27, 29, 35] which is sharp con- trast with the standard canonical formalism where the time coordinate has exceptional meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This is very attractive idea especially in the context of generally covariant theories since sometimes it is very difficult to perform D + 1 splitting of targe-space time and corresponding dynamical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In case of covariant canonical formalism of gravity we define conjugate momenta Mcmn to gmn in the following way Mcmn = ∂Lbulk ∂∂cgmn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (19) Note that the momenta are defined by bulk part of the Lagrangian density only as follows from the fact that equations of motion are derived by variation of the action when we fix metric and its derivative on the boundary, for careful discussion see [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then from (17) we obtain Mcmn = 1 32π √−g[gmkΓc kdgdn + gnkΓc kdgdm − −gmnΓc ghggh − Γf fk(gkmgcn + gkngcm) + gmngckΓf fk] (20) using δΓk gh δ∂cgmn = 1 4(gksδc g(δm s δn h + δn s δm h ) + +gksδc h(δm s δn g + δn s δm g ) − gksδc s(δm g δn h + δn g δm h )) (21) Then we could formulate covariant Hamiltonian formalism using canonical variales gab and Mcab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' However it turns out that the situation is much simpler when we introduce an alternative set of variables [35, 36] that are defined as f ab = √−ggab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (22) 8 Then it is easy to see that the conjugate momenta are defined by chain rule Nc ab = ∂Lquad ∂∂cf ab = ∂Lquad ∂(∂dgmn) ∂(∂dgmn) ∂(∂cfab) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (23) From (22) we see that f ab and gmn are related by point transformations so that ∂dgmn = ∂gmn ∂f ab ∂df ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (24) Then we have ∂(∂dgmn) ∂(∂cf ab) = ∂gmn ∂f ab δc d (25) and finally Nc ab = ∂Lquad ∂(∂cgmn)(−gmkBkl abgln) , (26) where Bkl ab = δgkl δf ab = (−f)− 1 D−1 �1 2(δk aδl b + δl aδk b ) − 1 D − 1f klfab � , (27) where we used the fact that − det f ≡ −f = (−g) D+1 2 (−g)−1 (28) and consequently √−g = (−f) 1 D−1 , gab = (−f)− 1 D−1f ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (29) Then using previous form of Mcmn we obtain Nc ab = ∂Lquad ∂(∂cgmn)(−gmkBkl abgln) = = − 1 32π[2Γc ab − Γf faδc b − Γf fbδc a] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (30) 9 Note that this relation does not depend on the number of space-time di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then in order to find corresponding Hamiltonian we should find inverse relation between Γa bc and Na bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Let us presume that it has the form Γc ab = ANc ab + B(Nd daδc b + Nd bdδc a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (31) Inserting (30) into (31) we obtain Nc ab = − 1 32π(2ANc ab + 2B(Nd daδc b + Nd bdδc a) − −(A + B(D + 2))Nf faδc b − (A + B(D + 2))Nf fbδc a) (32) using Γf fa = (A + B(D + 2))Nf fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Comparing left and right side we obtain that A and B are equal to A = −16π , B = −A D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (33) Then it is easy to find kinetic term of covariant Hamiltonian for D + 1 dimensional unimodular gravity in the form Hkin = ∂cf abNc ab − Lquad = 16π � Nb cdf daNc ab − 1 DNr raf abNs sb �� , (34) where we used the fact that ∂cf ab = ∂c √−ggab + √−g∂cgab = Γd dcf ab − Γa cdf db − Γb dcf da (35) together with the condition ∇cgab = 0 that implies ∂c √−g = Γd dc √−g , ∂cgab = −(Γa cdgdb + Γb cdgda) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (36) The final form of the covariant Hamiltonian for unimodular gravity con- tains terms with the unimodular constraint and true cosmological constant ¯Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then the phase-space form of the action has the form S = � dD+1x(Nc ab∂cfab−Hkin− 1 16π(−f) 1 D−1 ¯Λ− 1 16πλ((−f) 1 D−1 −1)) , (37) 10 where λ is Lagrange multiplier corresponding to unimodular constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' From the action above we determine corresponding equations of motion by per- forming variation with respect to f ab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Nc ab and λ δS = � dD+1x(δNc ab∂cfab + Nc ab∂cδfab − −δHkin δNc ab δNc ab − δHkin δf ab δf ab − − 1 16π(D − 1)(λ + ¯Λ)(−f) 1 D−1δf abfab − δλ((−f) 1 D−1 − 1)) = 0 (38) that implies following equations of motion ∂cf ab = δH δNc ab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (−f) 1 D−1 − 1 = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' −∂cNc ab = δH δf ab + λ 16π(D − 1)(−f) 1 D−1fab + ¯Λ 16π(D − 1)(−f) 1 D−1fab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (39) or explicitly ∂cf ab = 16π[Na cdf db + Nb cdf da − 1 D(f bdNs sdδa c + f adNs sdδb c)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' −∂cNc ab = 16π 2 (Nd caNc bd + Nd cbNc ad) − −16π D Nr raNs sb + λ 16π(D − 1)(−f) 1 D−1fab + ¯Λ 16π(D − 1)(−f) 1 D−1fab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (−f) 1 D−1 − 1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (40) Taking the trace of the second equation we can determine λ as λ = 16π(D − 1) (D + 1) (−∂cNc abf ab − 16πNd caf abNc bd + 16π D Nr raf abNs sb) − ¯Λ , (41) where we have took into account the equation on the fourth line in (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 11 Then the equations of motion for Nc ab have the form −∂cNc ab = 16π 2 (Nd caNc bd + Nd cbNc ad) − 16π D Nr raNs sb + + 1 (D + 1)(−∂jNj ikf ik − 16πNd cif ikNc kd + 16π D Nr rif ikNs sk)fab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (42) Clearly this equation is traceless and all dependence on the cosmological constant ¯Λ disappears which is an essence of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand one let us try to calculate the trace of the first equation that gives ∂cf abfab = 16π[Na cdf db + Nb cdf da − 1 D(f bdNs sdδa c + f adNs sdδb c)]fba (43) that can be simplified into the form ∂cf = 32π[D − 1 D ]Ns sc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now taking into account unimodular constraint we immediately get the con- dition Ns sc = 0 (44) that can be interpreted as secondary constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand the condition (44) seems to be too strong so that we should discuss it in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We begin with the recapitulation that unimodular gravity in the covariant Hamiltonian formalism is described by canonical conjugate variables f ab, Nc ab that are restricted by unimodular condition together with (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In order to find proper interpretation of the constraint (44) it is instructive to derive general relativity variables from f ab, Nc ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As the first step let us consider lin- ear combination of Nc ab that we denote as Γc ab and which is given by following prescription Γc ab = −16πNc ab + 16π D (Nd daδc b + Nd bdδc a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (45) This can be always done and we should again stress that Γc ab is not related to f ab at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Clearly Γc ab = Γc ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we define covariant derivative where Γc ab are coefficients of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Let us further define gab and its inverse gab in the following way gab = f ab(−f) 1 1−D , gab = fab(−f) 1 D−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (46) 12 Let us then define covariant derivative of gab as ∇cgab = ∂cgab + Γa cdgdb + Γb cdgda , (47) that, using (45), takes the form ∇cgab = (−f) 1 1−D × ×[∂cf ab − 16πNa cdf db − 16πNb cdf da + 16π D f bdNr drδa c + 16π D Nr drf daδb c] = 0 , (48) where we used the first equation in (40) that also implies ∂cf mnfmn = 32π D−1 D Ns sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now thanks to the equation ∇cgab = 0 we can express Γa bc in the form of Christoffel symbols Γa bc = 1 2gad(∂bgdc + ∂cgdb − ∂dgbc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (49) On the other hand let us return to the relation between Γa bc and Na bc that takes the form Γf fa = −32π D Nf fa (50) so that condition that Ns sa = 0 implies Γs sa = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (51) On the other hand from (49) we obtain Γf fc = 1 2gfd∂cgdf = ∂c det g = 0 (52) so that condition Ns sc = 0 is equivalent to unimodular condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is im- portant to stress that the fact that unimodular constraint implies Γs sa = 0 has not been appreciated too much with exception of recent interesting pa- per [10] where it was stressed that the equivalence between general relativity and unimodular gravity is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Rather, it was argued there that the natural geometry for unimodular relativity is equiprojective geometry [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We also see that the condition Ns sa = 0 emerges naturally in the covariant canonical formalism of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 13 4 Covariant Form of Unimodular Gravity In this section we perform covariant canonical formalism for Henneaux- Teitelboim formulation of unimodular gravity that has the form S = 1 16π � dD+1x√−g[R + λ(√−g − ∂aτ a)] , (53) where τ a is vector density and λ is Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now the equations of motion for λ implies √−g − ∂aτ a = 0 (54) while equation of motion for τ a leads to ∂aλ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (55) It is clear that the covariant Hamiltonian formulation of this theory is al- most the same as in previous case with difference that there is momentum conjugate to τ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Writting ∂aτ a = ∂bτ aδb a we obtain momentum conjugate to τ a to be equal to pb a = δL δ∂bτ a = − 1 16πλδb a (56) however this can be interpreted as primary constraints of the theory Gb a ≡ pb a + 1 16πλδb a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (57) In fact, the bare Hamiltonian is defined as HB = pb a∂bτ a + ∂cf abNc ab − L = = 16π[Nb cdf daNc ab − 1 DNr raf abNs sb] − 1 16πλ(−f) 1 D−1 (58) and we see that the dependence on momenta pν µ is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For that reason ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='we should consider Hamiltonian with primary constraints included ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='HT = 16π[Nb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf daNc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='ab − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='DNr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='raf abNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sb] − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλ(−f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='D−1 + Γa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='b(pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλδb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='(59) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='and consider corresponding equations of motion that arise from the variation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='of the canonical form of the action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='S = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='dD+1x(∂cf abNc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='ab + pa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='b∂aτ b − 16π[Nb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf daNc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='ab − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='DNr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='raf abNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sb] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλ(−f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='D−1 + Γa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='b(pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλδb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='(60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='so that the equations of motion have the form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='∂cf ab = 16π[Na ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf db + Nb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf da − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='D(f bdNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sdδa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='c + f adNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sdδb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='c)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' −∂cNc ab = 16π 2 (Nd caNc bd + Nd cbNc ad) − 16π D Nr raNs sb + λ (D − 1)(−f) 1 D−1fab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (−f) 1 D−1 + Γa a = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' ∂bτ a + Γa b = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' ∂apa b = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' pb a + 1 16πλδb a = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (61) If we combine the first and the second equation on the third line we find (−f) 1 D−1 = ∂aτ a (62) that has exactly the same form as equation (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We further perform partial derivative of the fourth equation on the third line and we obtain ∂bpb a = − 1 16π∂aλ (63) that using the third equation on the same line implies that ∂aλ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (64) This equation also shows that λ is constant and it can be interpreted as integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then it can be argued in the same way as in the pre- vious section that the equations (61) are equivalent to the Lagrangian equa- tions of Henneaux-Teitelboim gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In other words, covariant Hamiltonian description of Henneaux-Teiltelboim gravity is equivalent to corresponding Lagrangian description which is nice consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Acknowledgement: The work of JK is supported by the grant “Dualitites and higher order derivatives” (GA23-06498S) from the Czech Science Foundation (GACR).' metadata={'source': 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