Jackiw–Teitelboim gravity

Short description: Model of gravity with dilation

In theoretical physics, Jackiw–Teitelboim gravity, also known as JT gravity or the R=T model, is a theory of gravity with a dilaton in one spatial and one time dimension (1+1D). It was first formulated by Roman Jackiw and Claudio Teitelboim. [1] The theory is notable for being a toy model of quantum gravity that is exactly solvable, and it has found applications in understanding near-extremal black holes and the AdS/CFT correspondence.[2] [3]

The dynamics of JT gravity in the bulk are relatively simple, leading to a constant negative curvature spacetime (Anti-de Sitter space). [4] However, the theory possesses interesting dynamics on the boundary of this spacetime, which are described by the Schwarzian theory.[5] This boundary theory captures the low-energy behavior of the Sachdev-Ye-Kitaev (SYK) model, a model of quantum chaos.[2]

## Contents

* 1 Action
* 1.1 Boundary Terms
* 2 Schwarzian Boundary Theory
* 2.1 Derivation
* 3 References
* 4 See also

## Action

The action for Jackiw–Teitelboim gravity is defined on a 2-dimensional manifold M with a metric gμν and a dilaton field Φ. It is given by:

SJT=116πGN∫Md2x−gΦ(R+2)+Sboundary

where GN is the two-dimensional Newton constant, R is the Ricci scalar of the metric gμν, and the cosmological constant is chosen such that the equations of motion fix the spacetime to be constantly curved.

### Boundary Terms

For a manifold M with a boundary ∂M, it is necessary to include boundary terms in the action to have a well-posed variational problem. [6] The appropriate boundary terms for JT gravity are a Gibbons–Hawking–York boundary term (GHY) for the metric and a corresponding term for the dilaton field:

Sboundary=18πGN∫∂Mdth(ΦbK−Φb)

Here, h is the induced metric on the boundary, K is the extrinsic curvature, and Φb is the value of the dilaton at the boundary.[7] The second term is a counterterm that normalizes the action. [8]

The full action for Jackiw–Teitelboim gravity with boundary is therefore:

S=116πGN[∫Md2x−gΦ(R+2)+2∫∂MdthΦb(K−1)]

## Schwarzian Boundary Theory

A key feature of JT gravity is that its dynamics can be reduced to a theory on the one-dimensional boundary of the two-dimensional spacetime. This is achieved by integrating out the bulk fields (the metric gμν and the dilaton Φ), leaving an effective action for the boundary degrees of freedom.[9] The resulting effective action is the Schwarzian action, which describes the reparametrizations of the boundary.

### Derivation

The derivation of the Schwarzian boundary theory involves several steps:

1. Solving the bulk equations of motion: Variation of the JT action with respect to the dilaton Φ imposes the constraint R = -2, which means the bulk geometry is a slice of Anti-de Sitter space (AdS2). The equation of motion for the metric connects it to the dilaton's stress-energy tensor.[10]
2. Gauge fixing and boundary conditions: A convenient gauge choice is to fix the metric to be of the form ds2=dz2+dt2z2. The boundary of this space is located at z = ε for some small ε. The boundary curve is then parameterized by a time coordinate t(u), where u is a physical time on the boundary.
3. Evaluating the on-shell action: The bulk term of the action vanishes on-shell due to the equation of motion for the dilaton. The entire action is then given by the boundary term, evaluated on the solutions to the equations of motion.
4. The Schwarzian derivative: After carefully evaluating the boundary term with the imposed boundary conditions and gauge fixing, the action reduces to: Seff=−C∫du{tan⁡(πt(u)β),u}where C is a constant proportional to Φb/GN, β is the periodicity of the boundary time coordinate, and {f,u} denotes the Schwarzian derivative of a function f with respect to u: {f,u}=f‴(u)f′(u)−32(f″(u)f′(u))2

This effective action for the boundary reparametrization t(u) is the renowned Schwarzian action. [11] This theory describes the spontaneous and explicit breaking of conformal symmetry on the boundary, a feature also seen in the low-energy limit of the SYK model.[2]

## References

1. ↑ Jackiw, R. (1985). "Lower dimensional gravity". Nuclear Physics B 252: 343–356. doi:10.1016/0550-3213(85)90448-1. Bibcode: 1985NuPhB.252..343J. Teitelboim, C. (1983). "Gravitation and Hamiltonian structure in two spacetime dimensions.". Physics Letters B 126 (1–2): 41–45. doi:10.1016/0370-2693(83)90012-6. Bibcode: 1983PhLB..126...41T.
2. ↑ 2.0 2.1 2.2 Maldacena, J.; Stanford, D.; Yang, Z. (2016). "Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space". Progress of Theoretical and Experimental Physics 2016 (12): 12C104. doi:10.1093/ptep/ptw124.
3. ↑ Almheiri, A.; Polchinski, J. (2015). "Models of AdS2 backreaction and holography". Journal of High Energy Physics 2015 (11): 14. doi:10.1007/JHEP11(2015)014.
4. ↑ Garcia-Garcia, A. M.; Godet, Victor (2021). "Euclidean wormhole in the Sachdev-Ye-Kitaev model". Physical Review D 103 (4). doi:10.1103/PhysRevD.103.046014. Bibcode: 2021PhRvD.103d6014G. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.046014.
5. ↑ Iliesiu, Luca V. (2019). "On 2D gauge theories in Jackiw-Teitelboim gravity". arXiv:1909.05253 [hep-th].
6. ↑ Regge, T.; Teitelboim, C. (1974). "Role of surface integrals in the Hamiltonian formulation of general relativity". Annals of Physics 88 (2): 286–318. doi:10.1016/0003-4916(74)90404-7. Bibcode: 1974AnPhy..88..286R.
7. ↑ Brown, J. D.; York, J. W. (1993). "Quasilocal energy and conserved charges derived from the gravitational action". Physical Review D 47 (4): 1407–1419. doi:10.1103/PhysRevD.47.1407. PMID 10015717. Bibcode: 1993PhRvD..47.1407B.
8. ↑ Brown, A.R.; Gharibyan, H.; Lin, H.W.; Susskind, Leonard; Thorlacius, Larus; Zhao, Ying (2019). "Complexity of Jackiw-Teitelboim gravity". Physical Review D 99 (4): 046016. doi:10.1103/PhysRevD.99.046012. https://link.aps.org/doi/10.1103/PhysRevD.99.046012.
9. ↑ Alkalaev, Konstantin; Joung, Euihun; Yoon, Junggi (2022). "Schwarzian for colored Jackiw-Teitelboim gravity". Journal of High Energy Physics (9). doi:10.1007/JHEP09(2022)160. Bibcode: 2022JHEP...09..160A.
10. ↑ "Variation of action for Jackiw-Teitelboim (JT) gravity in order to get equations of motion". https://physics.stackexchange.com/questions/779878/variation-of-action-for-jackiw-teitelboim-jt-gravity-in-order-to-get-equatio.
11. ↑ Monti, Federico; Otness, Karl; Bronstein, Michael M. (2018). "The Schwarzian Theory - Origins". arXiv:1802.01572 [cs.LG].

## See also

* Dilaton § The dilaton in quantum gravity

* v
* t
* e

Theories of gravitation

Standard|

| Newtonian gravity (NG)|

* Newton's law of universal gravitation
* Gauss's law for gravity
* Poisson's equation for gravity
* History of gravitational theory

|

General relativity (GR)|

* Introduction
* History
* Mathematics
* Exact solutions
* Resources
* Tests
* Post-Newtonian formalism
* Linearized gravity
* ADM formalism
* Gibbons–Hawking–York boundary term

Alternatives to
general relativity|

| Paradigms|

* Classical theories of gravitation
* Quantum gravity
* Theory of everything

|

Classical|

* Einstein–Cartan
* Bimetric theories
* Gauge theory gravity
* Teleparallelism
* Composite gravity
* f(R) gravity
* Infinite derivative gravity
* Massive gravity
* Modified Newtonian dynamics, MOND
* AQUAL
* Tensor–vector–scalar
* Nonsymmetric gravitation
* Scalar–tensor theories
* Brans–Dicke
* Scalar–tensor–vector
* Conformal gravity
* Scalar theories
* Nordström
* Whitehead
* Geometrodynamics
* Induced gravity
* Chameleon
* Pressuron
* Degenerate Higher-Order Scalar-Tensor theories

Quantum-mechanical|

* Euclidean quantum gravity
* Canonical quantum gravity
* Wheeler–DeWitt equation
* Loop quantum gravity
* Spin foam
* Causal dynamical triangulation
* Asymptotic safety in quantum gravity
* Causal sets
* DGP model
* Rainbow gravity theory

Unified-field-theoric|

* Kaluza–Klein theory
* Dilaton
* Supergravity

Unified-field-theoric and
quantum-mechanical|

* Noncommutative geometry
* Semiclassical gravity
* Superfluid vacuum theory
* Logarithmic BEC vacuum
* String theory
* M-theory
* F-theory
* Heterotic string theory
* Type I string theory
* Type 0 string theory
* Bosonic string theory
* Type II string theory
* Little string theory
* Twistor theory
* Twistor string theory

Generalisations /
extensions of GR|

* Scale relativity
* Liouville gravity
* Lovelock theory
* (2+1)-dimensional topological gravity
* Gauss–Bonnet gravity
* Jackiw–Teitelboim gravity

Pre-Newtonian
theories and
toy models|

* Aristotelian physics
* CGHS model
* RST model
* Mechanical explanations
* Fatio–Le Sage
* Entropic gravity
* Gravitational interaction of antimatter
* Physics in the medieval Islamic world
* Theory of impetus

Related topics|

* Graviton

* v
* t
* e

Quantum gravity

Central concepts|

* AdS/CFT correspondence
* Causal patch
* Gravitational anomaly
* Graviton
* Holographic principle
* IR/UV mixing
* Planck scale
* Quantum foam
* Trans-Planckian problem
* Weinberg–Witten theorem

Toy models|

* 2+1D topological gravity
* CGHS model
* Jackiw-Teitelboim gravity
* Liouville gravity
* RST model
* Topological quantum field theory

Quantum field theory
in curved spacetime|

* Bunch–Davies vacuum
* Hawking radiation
* Semiclassical gravity
* Unruh effect

Black holes|

* Black hole complementarity
* Black hole information paradox
* Black-hole thermodynamics
* Bousso's holographic bound
* ER=EPR
* Firewall (physics)
* Gravitational singularity

Approaches|

| String theory|

* Bosonic string theory
* M-theory
* Supergravity
* Superstring theory

|

Canonical quantum gravity|

* Loop quantum gravity
* Wheeler-DeWitt equation

Euclidean quantum gravity|

* Hartle-Hawking state

Others|

* Causal dynamical triangulation
* Causal sets
* Noncommutative geometry
* Spin foam
* Group field theory
* Superfluid vacuum theory
* Twistor theory
* Dual graviton

Applications|

* Quantum cosmology
* Eternal inflation
* Multiverse
* FRW/CFT duality

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Jackiw–Teitelboim gravity. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Jackiw–Teitelboim_gravity&oldid=4369691"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template