Anti-de Sitter geometry and its conformal boundary

Let L be the AdS curvature radius, z a positive radial coordinate and x the d coordinates parallel to the boundary. Take all these coordinates to have length units, with light speed set to one. The warp factor is dimensionless, so the line element has length-squared units. At any fixed positive z, the induced parallel metric is conformal to Minkowski space. The negative time sign specifies a Lorentzian geometry; a Euclidean treatment requires an explicit continuation.

The radial coordinate is not one more boundary spatial direction. Constant-z slices supply a convenient cutoff description, while the putative boundary theory lives in one fewer spacetime dimension. In ten-dimensional string examples there may also be a compact internal manifold; the AdS factor can then be described by a lower-dimensional effective gravity theory. Counting those compact directions as extra boundary coordinates would confuse dimensional reduction with the AdS boundary construction.

As z approaches zero the physical metric diverges in these coordinates. Multiplying by the compensating conformal factor gives a finite representative of the boundary metric. Choosing a different defining coordinate or Weyl factor changes that representative while preserving the boundary conformal class. Thus the boundary data naturally couple to a conformal structure, with scale and anomaly subtleties handled in the field theory and renormalized gravitational action.

The boundary is at infinite proper distance along a constant-time radial curve. Integrating the radial line element gives a logarithm, so an arbitrarily small cutoff cannot be interpreted as a nearby material wall. At the same time, causal and optical statements are not determined by this spatial distance alone. The factor relating conformal metrics preserves local null cones, but global causal structure and boundary conditions still require the full spacetime and its chosen patch or covering.

Rescaling z and every parallel coordinate by the same positive factor leaves the Poincaré line element unchanged. This bulk isometry acts as a dilatation on boundary coordinates, motivating part of the conformal symmetry match. Curvature is nevertheless nonzero: scalar invariants are fixed by dimension and L. A large L relative to a microscopic string or Planck length can make curvature corrections small; merely rescaling coordinate labels cannot achieve that physical hierarchy.

The Einstein equation relates the constant curvature to a negative cosmological constant in a specified normalization. These formulas describe the model's AdS background rather than a measurement of the cosmological constant in nature. The spacetime lab performs a flat Lorentz transformation of one event and can check local signature-related coordinate arithmetic. It has no curvature, AdS radius, radial warp factor, global patch or conformal boundary, so it cannot simulate this geometry or test a holographic correspondence.