Davies Points in Black Hole Thermodynamics

• Davies Points are specific loci in black-hole thermodynamics where the heat capacity diverges, indicating critical phase transitions and distinct response behavior.
• They are classified into Type I (extremal) with a 3/2-order transition and Type II (inflection) with a 4/3-order transition based on temperature derivative analyses and fractional Ehrenfest methods.
• Thermodynamic topology and the tunneling interpretation show that Davies Points correspond to topological defects and vanish nonthermal corrections in Hawking radiation, underscoring their multifaceted significance.

Searching arXiv for recent and foundational papers on Davies points in black-hole thermodynamics. arXiv search query: "Davies point black hole thermodynamics" Davies points are distinguished loci in black-hole thermodynamics at which the heat capacity diverges. Historically, they were identified with second-order phase transitions by analogy with ordinary thermodynamics, because the divergence occurs in a second-order response function. In more recent treatments, however, Davies points have acquired a more differentiated meaning: in Reissner–Nordström–Anti–de Sitter (RN–AdS) thermodynamics they split into qualitatively distinct singularities with different fractional Ehrenfest orders, in thermodynamic topology they appear as simple zeros of a common vector field carrying winding number $w_{\rm D}=-1$, and in the tunneling picture they are the points at which the nonthermal correction to Hawking emission vanishes (Wang et al., 2022, Hazarika et al., 2024, Torrente-Lujan, 9 Jun 2026, La, 2010).

1. Definition and thermodynamic characterization

In black-hole thermodynamics, a Davies point is defined as a point in parameter space at which the heat capacity diverges. With Helmholtz free energy $F=M-TS$, entropy $S$, and Hawking temperature $T$, the heat capacity is written as

$C := T\left(\frac{\partial S}{\partial T}\right),$

so a Davies point is equivalently characterized by $1/C=0$ (Hazarika et al., 2024).

For a one-parameter black-hole branch, the same condition may be expressed through derivatives of the temperature. In the canonical ensemble of RN–AdS black holes at fixed charge $Q$,

$$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$

with $r_+$ the horizon radius (Wang et al., 2022). In the topological formulation based on a common vector field, the first component satisfies

$$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$

so the zero set includes both $F=M-TS$0 and $F=M-TS$1; the former is the Hawking–Page point, while the latter is the Davies point (Torrente-Lujan, 9 Jun 2026).

The classical Ehrenfest reading is that the first derivative of $F=M-TS$2 with respect to the order parameter remains finite while the second derivative changes sign or diverges, so the heat-capacity divergence indicates a continuous, second-order phase transition (Hazarika et al., 2024). A central refinement introduced later is that the divergence of $F=M-TS$3 alone does not determine the detailed singularity structure of the free energy, and therefore does not by itself distinguish all Davies points (Wang et al., 2022).

2. RN–AdS realization and the two kinds of Davies points

For the four-dimensional RN–AdS black hole, the metric function is

$F=M-TS$4

with cosmological constant $F=M-TS$5, and the horizon is at $F=M-TS$6. The entropy and temperature are

$F=M-TS$7

The heat capacity at fixed charge is

$F=M-TS$8

so the divergence condition is

$F=M-TS$9

This explicitly exhibits the Davies points as stationary points of $S$0 (Wang et al., 2022).

In RN–AdS thermodynamics, two qualitatively distinct Davies points occur. When $S$1, the discriminant is positive and there are two distinct real roots,

$S$2

which correspond to a local maximum and a local minimum of $S$3. Each satisfies $S$4 with $S$5. These are the Type I, or extremal, Davies points (Wang et al., 2022).

In the limiting case $S$6, the two extrema coalesce at

$S$7

and one additionally has $S$8. The temperature then has an inflection point rather than an ordinary extremum. This is the Type II Davies point, identified as the critical or inflection Davies point (Wang et al., 2022).

The distinction is thermodynamically significant because both types produce $S$9, yet one is associated with ordinary local extremality of the temperature and the other with coalescence of extrema into a higher-order inflection. This difference becomes decisive in the fractional classification of the transition order.

3. Fractional Ehrenfest classification

A generalized Ehrenfest scheme based on fractional derivatives provides a finer classification of Davies-point singularities than the standard integer-order analysis. In this framework one studies the Helmholtz free energy $T$0 in the canonical ensemble using Caputo fractional derivatives $T$1, rather than only the integer derivatives $T$2. The key properties quoted in the RN–AdS analysis are

$T$3

A jump discontinuity in $T$4 at $T$5 signals a phase transition of order $T$6 (Wang et al., 2022).

For a Type I extremal Davies point, define the dimensionless deviations

$T$7

Solving the equation of state $T$8 near the Davies point yields a double-valued expansion

$T$9

and the free energy expands as

$C := T\left(\frac{\partial S}{\partial T}\right),$0

The leading non-analytic term is therefore proportional to $C := T\left(\frac{\partial S}{\partial T}\right),$1, so

$C := T\left(\frac{\partial S}{\partial T}\right),$2

As $C := T\left(\frac{\partial S}{\partial T}\right),$3, $C := T\left(\frac{\partial S}{\partial T}\right),$4 is continuous for $C := T\left(\frac{\partial S}{\partial T}\right),$5, has a finite jump at $C := T\left(\frac{\partial S}{\partial T}\right),$6, and diverges for $C := T\left(\frac{\partial S}{\partial T}\right),$7. By definition, the transition is of order $C := T\left(\frac{\partial S}{\partial T}\right),$8 (Wang et al., 2022).

For the Type II inflection point at $C := T\left(\frac{\partial S}{\partial T}\right),$9, define

$1/C=0$0

and reparametrize the approach to criticality by

$1/C=0$1

Solving $1/C=0$2 then gives a single-valued expansion

$1/C=0$3

with free energy

$1/C=0$4

The first non-analytic term is proportional to $1/C=0$5, and

$1/C=0$6

Hence $1/C=0$7 is continuous for $1/C=0$8, has a finite jump at $1/C=0$9, and diverges for $Q$0. The transition is therefore of order $Q$1 (Wang et al., 2022).

Davies-point type	Local structure of $Q$2	Fractional order
Type I	Local maximum or minimum	$Q$3
Type II	Inflection point	$Q$4

This classification shows that the usual statement that Davies points are “second-order” is not maximally informative: the same heat-capacity divergence can arise from distinct non-analytic exponents in the free-energy expansion.

4. Thermodynamic topology, winding number, and Davies scales

A separate line of work reformulates Davies points through thermodynamic topology. In the unified $Q$5-mapping approach, one introduces a two-component vector field on the $Q$6-plane,

$Q$7

with normalization $Q$8 and topological current

$Q$9

The zeros of $$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$0 therefore localize the thermodynamic critical points. Since

$$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$1

the condition $$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$2 implies either $$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$3 or $$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$4, identifying the Hawking–Page point and the Davies point within a single vector field (Hazarika et al., 2024).

The associated winding number of an isolated zero,

$$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$5

distinguishes the two. For the Davies-type critical point one finds

$$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$6

while for the Hawking–Page point one finds

$$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$7

In the 2026 common-vector-field formulation on the auxiliary $$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$8-plane, the same conclusion follows from

$$C_Q \equiv T\left(\frac{\partial S}{\partial T}\right)_Q \to \infty \quad \Longleftrightarrow \quad \frac{\partial T}{\partial r_+}=0,$$9

together with the stability condition

$r_+$0

which selects the minimum of $r_+$1 and the maximum of $r_+$2 on the relevant branch (Torrente-Lujan, 9 Jun 2026).

Selected explicit Davies-point locations in AdS examples are as follows (Hazarika et al., 2024):

System	Davies point	Additional information
Schwarzschild–AdS	$r_+$3	$r_+$4
RN–AdS, fixed $r_+$5	$r_+$6	$r_+$7
Kerr–AdS, fixed $r_+$8	$r_+$9 solves $$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$0	for $$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$1, $$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$2

The 2026 formulation further packages the Davies and Hawking–Page zeros into signed first moments

$$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$3

with $$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$4 or $$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$5. For a single Davies/Hawking–Page pair,

$$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$6

but the first moments are nonzero. Normalizing by the Davies-point coordinates defines the dimensionless ratios

$$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$7

where $$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$8 is the Davies scale. In four-dimensional Schwarzschild–AdS with reduced entropy $$\varphi^S = \frac{2F}{S}\,\partial_S F = -\,2F\,\partial_S T = -\,2F\,\frac{T}{C_Y},$$9, one has

$F=M-TS$00

which gives

$F=M-TS$01

Exactly the same values arise for the grand-canonical RN–AdS family because the reduced shape of $F=M-TS$02 is unchanged. The corresponding dimensionless barrier,

$F=M-TS$03

is $F=M-TS$04 in four dimensions, while for charged non-rotating AdS black holes in $F=M-TS$05 spacetime dimensions,

$F=M-TS$06

For Kerr–AdS at fixed angular velocity, the winding signs remain $F=M-TS$07 and $F=M-TS$08, while the normalized dipole ratios and barrier receive no $F=M-TS$09 corrections and deform only at order $F=M-TS$10 (Torrente-Lujan, 9 Jun 2026).

5. Tunneling interpretation of the Davies critical point

In the Parikh–Wilczek tunneling picture, Hawking radiation is treated as a quantum tunneling process with emission probability

$F=M-TS$11

For $F=M-TS$12, the effective inverse temperature expands as

$F=M-TS$13

so the emission rate becomes

$F=M-TS$14

The first factor is the purely thermal Boltzmann contribution, while the second is a nonthermal correction controlled by the specific heat $F=M-TS$15 (La, 2010).

This yields a direct physical meaning for the Davies critical point. The sign of $F=M-TS$16 determines whether the nonthermal contribution enhances or suppresses emission: if $F=M-TS$17, the correction exceeds unity and emission is enhanced; if $F=M-TS$18, it is less than unity and emission is suppressed. At the Davies point,

$F=M-TS$19

so the coefficient of $F=M-TS$20 vanishes and the emission becomes exactly thermal,

$F=M-TS$21

The Davies point therefore separates an enhanced-emission phase from a suppressed-emission phase (La, 2010).

For the Reissner–Nordström black hole, the non-extremal Davies solution is

$F=M-TS$22

For Kerr,

$F=M-TS$23

and in Kerr–Newman the Davies line satisfies

$F=M-TS$24

When emitted quanta also carry charge $F=M-TS$25 and angular momentum $F=M-TS$26, the rate still factorizes into a thermal piece and a nonthermal remainder, and the $F=M-TS$27 term in the exponent continues to carry the factor $F=M-TS$28. It therefore vanishes at the same Davies point even when charge and angular momentum emissions are included (La, 2010).

6. Conceptual significance and common misconceptions

A recurrent misconception is that every Davies point is exhaustively characterized by the statement “the heat capacity diverges, therefore the transition is second-order.” The RN–AdS analysis shows that this is incomplete. The condition $F=M-TS$29 implies $F=M-TS$30, but it does not distinguish a local extremum of $F=M-TS$31 from a higher-order inflection. Under the generalized Ehrenfest scheme, the former is a $F=M-TS$32-order transition and the latter is a $F=M-TS$33-order transition (Wang et al., 2022).

A second misconception is that the Davies point and Hawking–Page point are thermodynamically unrelated. In the common-vector-field approach, they are zeros of the same field, but they carry opposite topological charges: $F=M-TS$34 The Davies point is thus not merely a divergence of a response function; it is also a topological defect in thermodynamic state space, and in the elementary AdS branch it is the top of the free-energy barrier separating thermal AdS from the large black hole (Hazarika et al., 2024, Torrente-Lujan, 9 Jun 2026).

A third misconception is that the physical meaning of the Davies point is obscure. In the tunneling picture it is sharply defined: it is the locus where the leading nonthermal back-reaction correction to Hawking radiation disappears, so the radiation is exactly thermal at the transition point and changes from enhancement to suppression as one crosses it (La, 2010).

Taken together, these results place Davies points at the intersection of response-function singularities, non-analytic free-energy structure, thermodynamic topology, and Hawking-emission dynamics. This suggests that “Davies point” is not a single uniform notion but a family of related critical loci whose common signature is a divergent heat capacity, while their detailed order, topology, and dynamical interpretation depend on the underlying thermodynamic geometry and ensemble.