Hot spaces with positive cosmological constant in the canonical ensemble: de Sitter solution, Schwarzschild–de Sitter black hole, and Nariai universe

In a space with fixed positive cosmological constant <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi mathvariant="normal">Λ</a:mi></a:mrow></a:math>, we consider a system with a black hole surrounded by a heat reservoir at radius <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>R</d:mi></d:math> and fixed temperature <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>T</f:mi></f:math>, i.e., we analyze the Schwarzschild–de Sitter black hole space in a cavity. We use results from the Euclidean path integral approach to quantum gravity to study, in a semiclassical approximation, the corresponding canonical ensemble and its thermodynamics. We give the action for the Schwarzschild–de Sitter black hole space and calculate expressions for the thermodynamic energy, entropy, temperature, and heat capacity. The reservoir radius <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>R</h:mi></h:math> gauges the other scales. Thus, the temperature <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>T</j:mi></j:math>, the cosmological constant <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi mathvariant="normal">Λ</l:mi></l:math>, the black hole horizon radius <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>r</o:mi><o:mo>+</o:mo></o:msub></o:math>, and the cosmological horizon radius <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>r</q:mi><q:mi mathvariant="normal">c</q:mi></q:msub></q:math>, are gauged to <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>R</t:mi><t:mi>T</t:mi></t:math>, <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi mathvariant="normal">Λ</v:mi><v:msup><v:mi>R</v:mi><v:mn>2</v:mn></v:msup></v:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mfrac><y:msub><y:mi>r</y:mi><y:mo stretchy="false">+</y:mo></y:msub><y:mi>R</y:mi></y:mfrac></y:math>, and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mfrac><bb:msub><bb:mi>r</bb:mi><bb:mi mathvariant="normal">c</bb:mi></bb:msub><bb:mi>R</bb:mi></bb:mfrac></bb:math>. The whole extension of <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi mathvariant="normal">Λ</eb:mi><eb:msup><eb:mi>R</eb:mi><eb:mn>2</eb:mn></eb:msup></eb:math>, <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mn>0</hb:mn><hb:mo>≤</hb:mo><hb:mi mathvariant="normal">Λ</hb:mi><hb:msup><hb:mi>R</hb:mi><hb:mn>2</hb:mn></hb:msup><hb:mo>≤</hb:mo><hb:mn>3</hb:mn></hb:math>, can be split into three ranges. The first range, <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mn>0</kb:mn><kb:mo>≤</kb:mo><kb:mi mathvariant="normal">Λ</kb:mi><kb:msup><kb:mi>R</kb:mi><kb:mn>2</kb:mn></kb:msup><kb:mo><</kb:mo><kb:mn>1</kb:mn></kb:math>, includes York’s pure Schwarzschild black holes. The other values of <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:mi mathvariant="normal">Λ</nb:mi><nb:msup><nb:mi>R</nb:mi><nb:mn>2</nb:mn></nb:msup></nb:math> within this range also have black holes. The second range, <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi mathvariant="normal">Λ</qb:mi><qb:msup><qb:mi>R</qb:mi><qb:mn>2</qb:mn></qb:msup><qb:mo>=</qb:mo><qb:mn>1</qb:mn></qb:math>, opens up a folder containing Nariai universes, rather than black holes. The third range, <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mn>1</tb:mn><tb:mo><</tb:mo><tb:mi mathvariant="normal">Λ</tb:mi><tb:msup><tb:mi>R</tb:mi><tb:mn>2</tb:mn></tb:msup><tb:mo>≤</tb:mo><tb:mn>3</tb:mn></tb:math>, is unusual. One striking feature here is that it interchanges the cosmological horizon with the black hole horizon. The end of this range, <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi mathvariant="normal">Λ</wb:mi><wb:msup><wb:mi>R</wb:mi><wb:mn>2</wb:mn></wb:msup><wb:mo>=</wb:mo><wb:mn>3</wb:mn></wb:math>, only existing for infinite temperature, represents a cavity filled with de Sitter space inside, except for a black hole with zero radius, i.e., a singularity, and with the cosmological horizon coinciding with the reservoir radius. For the three ranges, for sufficiently low temperatures, which for quantum systems involving gravitational fields can be very high when compared to normal scales, there are no black hole solutions and no Nariai universes, and the space inside the reservoir is hot de Sitter. The limiting value <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:mi>R</zb:mi><zb:mi>T</zb:mi></zb:math> that divides the nonexistence from existence of black holes or Nariai universes, depends on the value of <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"><bc:mi mathvariant="normal">Λ</bc:mi><bc:msup><bc:mi>R</bc:mi><bc:mn>2</bc:mn></bc:msup></bc:math>. For each <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:mi mathvariant="normal">Λ</ec:mi><ec:msup><ec:mi>R</ec:mi><ec:mn>2</ec:mn></ec:msup></ec:math> different from one, for sufficiently high temperatures there are two black holes, one small and thermodynamically unstable, and one large and stable. For <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"><hc:mi mathvariant="normal">Λ</hc:mi><hc:msup><hc:mi>R</hc:mi><hc:mn>2</hc:mn></hc:msup><hc:mo>=</hc:mo><hc:mn>1</hc:mn></hc:math>, for any sufficiently high temperature there is the small unstable black hole, and the neutrally stable hot Nariai universe. Phase transitions can be analyzed, the dominant phase has the least action. The transitions are between Schwarzschild-de Sitter black hole and hot de Sitter phases and between Nariai and hot de Sitter. For small cosmological constant, the action for the stable black hole equals the pure de Sitter action at a certain black hole radius and temperature, and so the phases coexist equally. For <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mn>0</kc:mn><kc:mo><</kc:mo><kc:mi mathvariant="normal">Λ</kc:mi><kc:msup><kc:mi>R</kc:mi><kc:mn>2</kc:mn></kc:msup><kc:mo><</kc:mo><kc:mn>1</kc:mn></kc:math> the equal action black hole radius is smaller than the Buchdahl radius, the radius for total collapse, and the corresponding Buchdahl temperature is greater than the equal action temperature. So above the Buchdahl temperature, the system collapses and the phase is constituted by a black hole. For <nc:math xmlns:nc="http://www.w3.org/1998/Math/MathML" display="inline"><nc:mi mathvariant="normal">Λ</nc:mi><nc:msup><nc:mi>R</nc:mi><nc:mn>2</nc:mn></nc:msup><nc:mo>≥</nc:mo><nc:mn>1</nc:mn></nc:math> a phase analysis is also made.