Lorentzian Holographic Correlators and Infalling Conditions

A Lorentzian bulk solution does not define a boundary correlator until the state-preparation contour, operator ordering, horizon condition, and boundary normalization are fixed. For a thermal retarded function in a stationary black-brane saddle, regularity on the future horizon selects the infalling solution. The renormalized ratio of normalizable response to nonnormalizable source then gives $G_R$; choosing the outgoing solution gives the advanced response instead.

Required background. Euclidean Preparation and Lorentzian State Dictionaries supplies the state map, and Lorentzian Witten Diagrams and Real-Time Orderings fixes operator ordering.

Helpful background. Retarded, Advanced, and Keldysh Bases fixes response conventions; Boulware, Hartle–Hawking, and Unruh States explains why horizon regularity is state-dependent.

Infalling waves at a future horizon

Consider a scalar with action

$$S=\frac12\int d^{d+1}x\,\sqrt{\lvert g\rvert} \left(g^{MN}\partial_M\phi\,\partial_N\phi-m^2\phi^2\right)$$

in the site convention $(+,-,\ldots,-)$ and a metric whose $t$–$r$ sector is $ds^2=f(r)dt^2-dr^2/f(r)$, with

$$f(r)=4\pi T(r-r_h)+O\!\left((r-r_h)^2\right).$$

For $\phi=e^{-i\omega t+i\mathbf k\cdot\mathbf x}\psi(r)$, the two near-horizon solutions are

$$\psi_{\mathrm{in/out}} \sim (r-r_h)^{\mp i\omega/(4\pi T)}.$$

The tortoise coordinate obeys $r_*\sim(4\pi T)^{-1}\log(r-r_h)$. Hence $\phi_{\mathrm{in}}\sim e^{-i\omega(t+r_*)}$ is smooth in ingoing Eddington–Finkelstein time $v=t+r_*$ and crosses the future horizon. This causal prescription is the central step in the real-time recipe of Son and Starinets 2002.

Near the AdS boundary,

$$\psi(r)=A(\omega,\mathbf k)\,r^{\Delta-d} +B(\omega,\mathbf k)\,r^{-\Delta}+\cdots, \qquad \Delta(\Delta-d)=m^2L^2.$$

After adding local counterterms, the source is $A$ and the response is proportional to $B$. In the standard scalar quantization,

$$G_R(\omega,\mathbf k) =(2\Delta-d)\frac{B}{A}+G_{\mathrm{contact}},$$

up to the declared normalization of the bulk action and Fourier transform. Contact terms can shift the real polynomial part but not the nonlocal pole structure or spectral density.

Radial flux and the spectral sign

The conserved Klein–Gordon flux for a real radial equation is

$$\mathcal F =i\sqrt{\lvert g\rvert}\,g^{rr} \left(\psi^*\partial_r\psi-\psi\partial_r\psi^*\right).$$

For positive frequency, an infalling mode carries flux into a future horizon. With the convention

$$\rho(\omega,\mathbf k)=-2\,\operatorname{Im}G_R(\omega,\mathbf k),$$

the sign of $\rho$ for $\omega>0$ follows from that absorbed flux after the overall action normalization is fixed. This provides a check independent of reading coefficients from an asymptotic series.

As the first application, solve the radial equation from $r_h$ to the boundary with the infalling Frobenius series as initial data. Normalize $A=1$, extract $B$, add the holographic counterterms, and verify: analyticity of $G_R$ in the upper half $\omega$-plane, nonnegative absorption in the chosen convention, and causal support in time. The Lorentzian variational prescription and its relation to Euclidean preparation are developed systematically by Skenderis and van Rees 2009.

Adversarial horizon prescriptions

Repeat the calculation with the outgoing exponent. The radial flux reverses and the poles move to the analytic structure appropriate to $G_A$, not $G_R$. Relabeling this answer “retarded” is exposed by at least one of three tests: it is analytic in the wrong half-plane, its spectral sign is reversed, or its Fourier transform has advanced rather than retarded support.

Euclidean regularity is not a substitute unless the Euclidean cap and analytic continuation prepare the same thermal state and ordering. Likewise, a horizonless saddle has no infalling boundary condition to inherit; its normal-mode prescription must be derived from its own interior boundary conditions.

Validity window and handoff

The calculation is a linear response result in a specified classical saddle, channel, state, and time-translation-invariant regime. Taking $N\to\infty$ before late time produces a continuous spectral response and irreversible absorption. It does not determine exact finite-$N$ recurrences or prove that every state with the same exterior metric has the same correlator.

Thermal and Nonequilibrium QFT owns the general response theory, and Mathematical QFT owns rigorous analyticity and spectral statements. This chapter supplies the black-background realization; later pages test its quasinormal and late-time limits.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality: Prescription, Renormalization and Examples.” Journal of High Energy Physics 2009, 085 (2009). doi:10.1088/1126-6708/2009/05/085.
• Son, Dam T., and Andrei O. Starinets. “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications.” Journal of High Energy Physics 2002, 042 (2002). doi:10.1088/1126-6708/2002/09/042.