Black-brane thermodynamics and field-theory temperature

Use d plus one bulk dimensions, with d boundary spacetime dimensions, the Lorentzian signature already stated and natural units throughout. The neutral planar Einstein solution modifies the Poincaré metric by a blackening function that vanishes at a radial horizon. The boundary time is normalized so the boundary representative is Minkowski space. Arbitrarily rescaling that time without translating temperature would change the apparent answer while leaving the underlying state description inconsistent.

Continue time to Euclidean signature and demand regularity at the horizon. The near-horizon radial-time plane must have no conical singularity, fixing the inverse-temperature period. This gives a temperature proportional to the inverse horizon coordinate. The derivation assumes a nonextremal regular horizon in the stated metric. Charged or higher-derivative solutions need their own near-horizon analysis rather than the neutral expression with an extra parameter inserted by analogy.

A planar horizon has infinite area when the parallel spatial volume is infinite. Divide by that same boundary coordinate volume to define entropy density. The induced horizon metric then gives an area density proportional to the radius-to-horizon-coordinate ratio raised to d minus one. The Einstein area law supplies the factor inverse four Newton constant. In natural units entropy is dimensionless; restoring Boltzmann constant changes the entropy units, not the geometric scaling.

For the neutral planar saddle, choose the standard flat-boundary zero-temperature pressure reference. The pressure and energy density obey the conformal equation of state, and the Helmholtz free-energy density is minus pressure at fixed volume with no charge. If a chemical potential is introduced, one must distinguish fixed-charge Helmholtz and fixed-potential grand ensembles and their boundary terms. Even a familiar saddle must be compared with other allowed saddles before declaring it globally dominant.

The temperature and entropy density give an enthalpy density equal to energy plus pressure. Differentiating with respect to the horizon parameter also verifies the neutral first law at fixed volume. Eliminating that parameter gives conformal temperature powers for the thermal densities. These are useful normalization checks: an omitted factor of four pi or a mismatch of boundary volumes will spoil identities even if each expression appears dimensionally plausible.

The planar neutral saddle has positive thermal heat capacity in this regime. This does not establish that every black hole has positive heat capacity or that every holographic system has the same phase diagram. Spherical boundaries and charged ensembles permit other structures, and finite-N or string corrections change leading coefficients. The function lab can compare a declared quadratic reference power but has no horizon, Euclidean period, ensemble or gravitational free energy and cannot determine a black-brane phase transition.