On the phase space in Double Field Theory
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Springer
Received: April 1, Revised: June 25, Accepted: July 8, Published: July 31,
2020 2020 2020 2020
Eric Lescanoa and Nahuel Mir´ on-Graneseb a
Instituto de Astronom´ıa y F´ısica del Espacio (IAFE-CONICET-UBA), Ciudad Universitaria, Pabell´ on IAFE, 1428 Buenos Aires, Argentina b Departamento de F´ısica, FCEyN, Universidad de Buenos Aires (UBA), Ciudad Universitaria, Pabell´ on 1, 1428 Buenos Aires, Argentina
E-mail: [email protected], [email protected] Abstract: We present a model of (double) kinetic theory which paves the way to describe matter in a Double Field Theory background. Generalized diffeomorphisms acting on double phase space tensors are introduced. The generalized covariant derivative is replaced by a generalized Liouville operator as it happens in relativistic kinetic theory. The section condition is consistently extended and the closure of the generalized transformations is still given by the C-bracket. In this context we propose a generalized Boltzmann equation and compute the moments of the latter, obtaining an expression for the generalized energymomentum tensor and its conservation law. Keywords: String Duality, Classical Theories of Gravity ArXiv ePrint: 2003.09588
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)239
JHEP07(2020)239
On the phase space in Double Field Theory
Contents 1 5 6
2 Relativistic Kinetic Theory 2.1 Basics 2.2 Lagrangian and Hamiltonian formulations 2.3 The relativistic Boltzmann equation
7 7 8 9
3 Double Field Theory 3.1 Double space and generalized fields 3.2 Generalized affine connection 3.3 Matter Lagrangian and the generalized energy-momentum tensor
11 11 12 13
4 Double Kinetic Theory 4.1 The double phase space 4.2 Generalized transfer equations and conservation laws 4.3 Applications
15 15 16 18
5 Outlook
19
A Conventions
21
B Closure
22
1
Introduction
The Einstein’s field equations, Gµν = Tµν
(1.1)
are fundamental relations which describe the dynamics of matter coupled to gravity in a Riemannian D-dimensional background (µ = 0, . . . , D − 1). The l.h.s. of the equation is given by the Einstein tensor, a divergenceless and symmetric tensor that depends on the geometric properties of the D-dimensional space-time and the r.h.s. is related to the matter and energy content of the system. Kinetic theory is the usual way to describe matter from microscopic principles. In this scheme the energy-momentum tensor is the second moment of the one-particle distribution function f = f [x, p] [1–4], Z √ µν T = p µ p ν f g dD p , (1.2)
–1–
JHEP07(2020)239
1 Introduction 1.1 Main results 1.2 Outline
with pµ the momentum and g the determinant of the metric tensor. The evolution of f is given by the relativistic Boltzmann equation pµ Dµ f = C[f ] ,
(1.3)
where C[f ] is the collision term and Dµ is the Liouville operator defined as Dµ = ∇µ − Γσµν pν
∂ . ∂pσ
(1.4)
δ ξ v µ ν = L ξ v µ ν + pρ
∂ξ σ (x) ∂vµ ν , ∂xρ ∂pσ
(1.5)
where Lξ is the Lie derivative acting on tensors defined as Lξ v µ ν = ξ σ
∂
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