r, which are the location of the black hole and cosmological horizon respectively. As the parameter a increases, keeping the cosmological constant fixed, the two positive zeros come closer. At some value of a, they collide. Approaching this value of a, the black hole and cosmological horizons are at nearly the same value of r. But the distance between them doesn't go to zero, because ƒ(r) is very small between the two zeros, and the square root of its reciprocal integrates to a finite value. If the two zeros of ƒ are at R + ε and R − ε taking the small ε limit while rescaling r to remove the ε dependence gives the Nariai solution. The form of ƒ near the almost-double-zero in terms of the new coordinate u given by r = R + u is: The metric on the causal patch between the two horizons reduces to which is the metric of . This form is local for an observer sandwiched between the black hole and the cosmological horizon, which reveal their presence as the two horizons at z = −R and z = R respectively. The coordinate z can be replaced by a global coordinate for the 1 + 1-dimensional de Sitter space part, and then the metric can be written as: In these global coordinates, the isotropy of de Sitter space makes shifts of the coordinate x isometries, so that it is possible to identify x with x + A, and make the space dimension into a circle. The constant-time radius of the circle expands exponentially into the future and the past, and this is Nariai's original form. Rotating one of the horizons in Nariai space makes the other horizon rotate in the opposite sense. This is a manifestation of Mach's principle in self-contained causal patches, if the cosmological horizon is included as "matter", like its symmetric counterpart, the black hole. Hawking temperature The temperature of the small and large horizon in the de Sitter–Schwarzschild can be calculated as the period in imaginary time of the solution, or equivalently as the surface gravity near the horizon. The temperature of the smaller black hole is relatively larger, so there is heat flow from the smaller to the larger horizon. The quantity which is the temperature of the black hole is hard to define, because there is no asymptotically flat space to measure it relative to.