Jackiw–Teitelboim Gravity

Jackiw–Teitelboim gravity is the leading two-dimensional theory of the gravitational response of many near-extremal AdS2 throats. Its metric has no local graviton and remains locally AdS2, while a dilaton measures the transverse area inherited from the higher-dimensional black hole. Keeping the topological term, the dynamical dilaton, and the boundary terms separate explains how a locally fixed metric can nevertheless carry energy, entropy, and a nontrivial boundary mode.

Required background. Gauge Theory and String-Theory Preparation provides the effective-action and gauge-normalization language used below.

Helpful background. Quantum Gravity supplies effective-theory reasoning; Quantum Information supplies ensemble language; Semiclassical Gravity provides Euclidean horizon thermodynamics; and Black-Hole Information motivates later spectral questions.

The complete JT action

For a smooth two-dimensional Euclidean region $M$ with regulated boundary $\partial M$ and no corners, take

$$\begin{aligned} I_{\rm JT}={}&-\frac{\phi_0}{16\pi G_2} \left[\int_M\!\sqrt{g}\,R+2\int_{\partial M}\!\sqrt{h}\,K\right]\\ &-\frac{1}{16\pi G_2} \left[\int_M\!\sqrt{g}\,\phi\left(R+\frac{2}{L_2^2}\right) +2\int_{\partial M}\!\sqrt{h}\,\phi\left(K-\frac{1}{L_2}\right)\right]. \end{aligned}$$

This is the Euclidean convention $Z\simeq e^{-I_E}$, with $K=h^{\mu\nu}\nabla_\mu n_\nu$ for the outward unit normal. Gauss–Bonnet gives $\int\sqrt g R+2\int\sqrt h K=4\pi\chi(M)$, so the first line is $-S_0\chi(M)$ with $S_0=\phi_0/(4G_2)$. The second line is dynamical. The Gibbons–Hawking term makes the Dirichlet variational problem well posed, and the $1/L_2$ counterterm removes the leading cutoff divergence. Omitting either boundary contribution gives the wrong finite action even though the bulk equation may look correct.

The normalization follows Maldacena, Stanford, and Yang 2016, §3, Eqs. (3.7)–(3.15), pp. 6–8, translated so $G_2$ is explicit. A dimensional reduction determines $\phi_0$, the boundary value of $\phi$, and $G_2$; JT alone does not predict them.

Historically, the constant-curvature multiplier model was introduced by Teitelboim 1983, complete article, pp. 41–45 and Jackiw 1985, complete article, pp. 343–356. Jackiw’s open restatement writes the Lorentzian core as $I_1=\int d^2x\sqrt{-g}\,\eta(R-\Lambda)$ and derives its multiplier and metric equations in Jackiw 1993, §1, p. 1, Eq. (1), and §II, p. 2, Eqs. (3)–(4). These historical sources do not supply the separate $\phi_0$ sector, explicit $G_2$, Gibbons–Hawking term, counterterm, or near-extremal thermodynamic normalization in the displayed Euclidean action; MSY is the authority for that modern package.

Constant curvature and a responsive dilaton

Varying $\phi$ fixes

$$R=-\frac{2}{L_2^2}.$$

Thus every classical metric solution is locally AdS2. Varying the metric instead gives, in vacuum,

$$\nabla_\mu\nabla_\nu\phi-g_{\mu\nu}\nabla^2\phi +g_{\mu\nu}\frac{\phi}{L_2^2}=0.$$

For a Lorentzian matter action independent of $\phi$, use the site’s definition $T_{\mu\nu}=-2(-g)^{-1/2}\delta S_m/\delta g^{\mu\nu}$. The Lorentzian gravitational bulk action has positive coefficient $1/(16\pi G_2)$; varying it puts $-8\pi G_2T_{\mu\nu}$ on the right side of the displayed equation. Matter coupled directly to $\phi$ can also modify the curvature constraint. Thus stress changes the dilaton profile even when curvature remains constant. The topological constant $\phi_0$ must not be folded into $\phi$, because only the latter participates in this equation.

This division also clarifies the degrees of freedom. The curvature constraint removes local metric fluctuations, and diffeomorphisms remove coordinate changes in the interior. It does not remove the relative placement of a regulated boundary and a dilaton profile. Once the ultraviolet completion fixes the boundary clock and the large boundary value of the dilaton, transformations that move that boundary are physical. They carry the energy measured by the boundary Hamiltonian even though a local curvature detector still sees AdS2.

A convenient Lorentzian black-hole patch is

$$ds^2=-\frac{r^2-r_h^2}{L_2^2}dt^2+ \frac{L_2^2\,dr^2}{r^2-r_h^2}, \qquad \phi(r)=\phi_r\frac{r}{L_2^2}.$$

Direct substitution verifies both equations. Regularity of the Euclidean cigar fixes $T=r_h/(2\pi L_2^2)$. Although metrics with different $r_h$ are locally related, the regulated boundary and dilaton boundary condition distinguish their physical states.

Thermodynamics from the horizon and boundary

The Wald entropy and energy above extremality are

$$S=S_0+\frac{\phi_h}{4G_2} =S_0+\frac{\pi\phi_r}{2G_2}T, \qquad E=\frac{\pi\phi_r}{4G_2}T^2.$$

The energy follows by integrating $dE=T\,dS$ with $E(T=0)=0$; evaluating the renormalized on-shell action gives the same result. This agreement is an independent check on the counterterm, Euclidean period, and dilaton normalization. In a Reissner–Nordström reduction, matching the coefficient of $T$ to the four-dimensional area expansion fixes $\phi_r$ rather than leaving it adjustable.

The theory is valid below the throat gap, at leading order in the near-extremal expansion, with the ensemble and compact factors fixed. Higher-derivative operators, extra light Kaluza–Klein fields, or temperatures that probe the matching region add terms not contained in $I_{\rm JT}$. JT also supplies an effective thermodynamic entropy; the classical action by itself is not a microscopic count of extremal states.

One should also distinguish the classical saddle from its quantum path integral. Integrating over boundary reparametrizations produces loop corrections and a continuous spectral density within the effective theory, but neither operation determines which exact microscopic Hilbert space a particular higher-dimensional black hole has. That question requires the ultraviolet embedding and cannot be answered by relabeling $e^{S_0}$ as a degeneracy.

Boundary data and physical scope

In Poincaré coordinates with $ds^2=L_2^2(dt^2+dz^2)/z^2$, the regulated boundary has proper length $ds_{\partial}=L_2\,du/\epsilon$ and dilaton $\phi_b=\phi_r/\epsilon$. Here $u,z,\epsilon$, and $\phi_r$ have dimensions of length, while $\phi$ and $G_2$ are dimensionless. Moving that curve while preserving these conditions leaves a reparametrization $t=f(u)$ as the low-energy degree of freedom. Consequently the metric equation does not mean “nothing happens”: the boundary embedding and dilaton encode the response. The Schwarzian boundary mode is obtained by evaluating the same boundary term, not by adding an unrelated dynamics.

Exercises

Verify that the thermal solution satisfies the dilaton equation and use horizon regularity plus the first law to derive its entropy and energy above extremality.

Solution

For $\phi=\phi_r r/L_2^2$, the nonzero second derivatives combine so that $\nabla_\mu\nabla_\nu\phi-g_{\mu\nu}\nabla^2\phi+g_{\mu\nu}\phi/L_2^2=0$ component by component. Smoothness near $r=r_h$ makes Euclidean time periodic with $\beta=2\pi L_2^2/r_h$. The horizon dilaton is $\phi_h=\phi_r r_h/L_2^2=2\pi\phi_rT$, so Wald’s formula gives $S-S_0=\pi\phi_rT/(2G_2)$. Integrating $T\,dS$ then yields $E=\pi\phi_rT^2/(4G_2)$, fixing the integration constant at extremality. This independent thermodynamic integration agrees with the boundary-action calculation and therefore checks the sign of the counterterm as well as the temperature normalization.

References

• Jackiw, Roman. “Gauge Theories for Lineal Gravities.” arXiv:hep-th/9309082 (1993). Open manuscript.
• Jackiw, Roman. “Lower Dimensional Gravity.” Nuclear Physics B 252 (1985) 343–356. DOI:10.1016/0550-3213(85)90448-1.
• Maldacena, Juan, Douglas Stanford, and Zhenbin Yang. “Conformal Symmetry and Its Breaking in Two Dimensional Nearly Anti-de-Sitter Space.” Progress of Theoretical and Experimental Physics 2016 (2016) 12C104. arXiv:1606.01857.
• Teitelboim, Claudio. “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions.” Physics Letters B 126 (1983) 41–45. DOI:10.1016/0370-2693(83)90012-6.