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 Incoming links: Fefferman--Graham coordinates

Fefferman--Graham coordinates Created 2026-09-24 Updated 2026-11-05

Fefferman--Graham coordinates put an asymptotically anti-de Sitter metric in the form

d s^{2} = L^{2} [d Z^{2} / Z^{2} + g_{ij} (Z, x) d x^{i} d x^{j} / Z^{2}]

Choose the Fefferman--Graham coordinates radial variable from

\frac{d Z}{Z} = \frac{d z}{z g ( z )} .

Near the boundary,

Z = z [1 + \frac{1 + μ ^{2}}{6} z^{3} + O (z^{4})], z = Z + O (Z^{4}) .

Consequently, through

O (Z^{2})

d s^{2} = \frac{L ^{2}}{Z ^{2}} [d Z^{2} - d τ^{2} + d X^{2} + d Y^{2} + O (Z^{3})],

A_{τ} = μL - μL Z + O (Z^{4}) .

By the holographic dictionary, the leading boundary value of

A_{τ}

is the source for charge density. Thus

μ

is the dimensionless holographic chemical potential; the coefficient of the normalizable linear term is proportional to the CFT charge density,

ρ = μL / (4 π G)

with the action normalization and outward-orientation convention used here.