Black holes contradict the Nernst-Planck ( N/ P) version of the 3rd law of thermodynamics, but agree with its unattainability (U) version. This happens without contradiction, because the N/ P and U versions are not equivalent, namely, N/ P implies U but U does not imply N/ P. So, black holes obey the weaker version of the 3rd law, but not the stronger one.
It is commonly believed that the Nernst-Planck (N/P) version of the 3rd law of thermodynamics and the unattainability (U) version are equivalent [
In Section 2, we show, with two examples, how N/P Þ U; moreover, the left hand side of the implication needs to include the 1st and the 2nd laws of thermodynamics. It is clear that the above amounts to −U Þ −N/P but does not imply that U Þ N/P [
The considerations for the Schwarzschild and Kerr black holes are reserved to Sections 3 and 4. In Section 3, the thermodynamics of the Schwarzschild black hole immediately illustrates the violation of N (or N/P) and simultaneously the fulfillment of U [
Through the use of two kinds of systems, one hydrostatic and the other magnetic, we show how, by the well known zig-zag processes, the N and obviously also the N/P versions of the 3rd law together with the 1st and 2nd laws, imply the unattainability version of the 3rd law.
Consider the picture in
a → b , c → d , e → f , …, are isothermal compresions which, from the “TdS” equation (consequence of the 2nd. law) T Δ S = C p Δ T − α T V Δ p [
At the same time it is clear that a cooling to T = 0 is possible in a finite number of zig-zag’s if S ( 0 + , p 1 ) ≠ S ( 0 + , p 2 ) i.e. if N does not hold.
Consider a system of N spins 1/2, each with magnetic moment μ in the presence of an external magnetic field B. The picture of the entropy S as a function of T, B, and N is analogous to that in
S ( T , B , N ) = N ( l n ( 2 C h x ) − x T h x )
where x = μ B T [
laws of thermodynamics as well the canonical ensamble of equilibrium statistical mechanics. In this case S 0 = S ( 0 + , B , N ) = 0 . The vertical parts of the zig-zag’s ( a → b , …) are magnetizing isothermals ( Δ B > 0 ), while the horizontal parts ( b → c , …) are demagnetizing adiabatics ( Δ B < 0 ). It is easy to verify that in
each isothermal, Δ S | T , Δ B > 0 = − μ T ( 1 2 T h ( μ B T ) + μ B T C h 2 h ( μ B T ) ) Δ B < 0 (entropy descends), while in each adiabatic, Δ T | S , Δ B < 0 = − T | Δ B | B < 0 (temperature descends). Again, an infinite quantity of each time smaller zig-zag’s is needed to arrive at T = 0 + .
It is well known that for the Schwarzschild black hole of mass M and horizon radius 2M, the entropy S = A 4 and the Hawking temperature T = κ 2 π , where A is the horizon area and κ is the surface gravity, are given by
T = 1 8 π M ≡ T S c h w . (1)
and
S = 4 π M 2 ≡ S S c h w . (2)
respectively. So, for T → 0 + , M → + ∞ and therefore S → ∞ at the absolute zero. The last result implies the violation of N or N/P, and at the same time the fulfillment of U, due to the impossibility for a black hole to reach an infinite amount of mass or energy in any finite time, let it be proper or measured at r = ∞ [
For a Kerr black hole of mass M and angular momentum J ( 0 ≤ J < M 2 ) in Boyer-Lindquist coordinates, the temperature and entropy at the event horizon
r + = M + M 2 − ( J M ) 2 are respectively given by [
T ( M , J ) = 1 − ( J / M 2 ) 2 4 π M ( 1 + 1 − ( J / M 2 ) 2 ) (3)
and
S ( M , J ) = 2 π M 2 ( 1 + 1 − ( J / M 2 ) 2 ) . (4)
At J = 0 both quantities are continuous ( C 0 ) and reproduce the Schwarzschild values T ( M ,0 ) = T S c h w . and S ( M ,0 ) = S S c h w . . Since
( ∂ T ∂ J ) M ( M , J ) = − J 4 π M 5 × 1 ( 1 + 1 − ( J / M 2 ) 2 ) 2 1 − ( J / M 2 ) 2 (5)
and
( ∂ S ∂ J ) M ( M , J ) = − 2 π J / M 2 1 − ( J / M 2 ) 2 , (6)
T and S have also continuous first derivatives ( C 1 ) for 0 ≤ J < M 2 with
( ∂ T ∂ J ) M ( M ,0 + ) = 0 − and ( ∂ S ∂ J ) M ( M ,0 + ) = 0 − . (7)
At J = M 2 both the event horizon at r + and the Cauchy horizon at r − coincide:
r + = r − = M , (8)
the black hole region disappears, formally reaching the so called “extreme black hole”, with
T ( M , M 2 ) = 0 and S ( M , M 2 ) = 2 π M 2 = S S c h w . 2 , (9)
and the first derivatives in (5) and (6) diverge:
( ∂ T ∂ J ) M ( M , ( M 2 ) − ) = − ∞ and ( ∂ S ∂ J ) M ( M , ( M 2 ) − ) = − ∞ . (10)
In other words, in the interval J ∈ [ 0, M 2 ] , T and S belong to C 0 but not to C 1 , with − ∞ < ( ∂ T ∂ J ) M , ( ∂ S ∂ J ) M < 0 for 0 < J < M 2 (see
On the other hand,
( ∂ S ∂ M ) J ( M , J ) = 4 π M × 1 + 1 − ( J / M 2 ) 2 1 − ( J / M 2 ) 2 = 1 T ( M , J ) , (11)
which is C 0 and C 1 at J = 0 since ( ∂ S ∂ M ) J ( M ,0 ) = 8 π M and
∂ ∂ J ( ( ∂ S ∂ M ) J ) M ( M , J ) = 4 π J M 3 × 1 ( 1 − ( J / M 2 ) 2 ) 3 (12)
with ∂ ∂ J ( ( ∂ S ∂ M ) J ) M ( M ,0 ) = 0 , but divergent at J = ( M 2 ) − with ( ∂ S ∂ M ) J ( M , ( M 2 ) − ) = + ∞ , and therefore neither C 0 nor C 1 since ∂ ∂ J ( 1 T ( M , J ) ) M ( M , ( M 2 ) − ) = + ∞ (see
I.e., T ( M , J ) is monotonously decreasing with J from 1 8 π M at J = 0 , to 0 + at J = ( M 2 ) − , at constant M.
It can be easily verified that
∂ ∂ J ( ( ∂ S ∂ M ) J ) M ( M , J ) = ∂ ∂ M ( ( ∂ S ∂ J ) M ) J ( M , J ) = 4 π J M 3 × 1 ( 1 − ( J / M 2 ) 2 ) 3 (13)
holds for all J ∈ [ 0, M 2 ) , while at J = M 2 ,
∂ 2 S ∂ J ∂ M | J = ( M 2 ) − = ∂ 2 S ∂ M ∂ J | J = ( M 2 ) − = + ∞ . (14)
The breaking down of the Maxwell-type relation ∂ 2 S ∂ J ∂ M = ∂ 2 S ∂ M ∂ J and therefore the analyticity of S as a function of ( M , J ) at J = M 2 , is the indication of
a phase transition into a naked singularity occurring at T = 0 + and therefore of the unattainability (U) of this value of the absolute temperature. At the same time, the M-dependent value of the entropy at T = 0 (or T = 0 + ) given by (9), shows that the N or N/P version of the 3rd. law is violated.
As a function of T at fixed M, J is given by J ( T ) = M 2 1 − 8 π M T 1 − 4 π M T .
Finally, we study S as a function of T for fixed M. From (3) and (4)
S ( T ) = 2 π M 2 1 − 4 π M T , 0 < T ≤ 1 8 π M (15)
with S ( 0 + ) = 2 π M 2 , S ( 1 8 π M ) = 4 π M 2 , and
( ∂ S ∂ T ) M ( T ) = 8 π 2 M 3 ( 1 − 4 π M T ) 2 (16)
which equals 8 π 2 M 3 at T = 0 + and 16 π 2 M 3 at T = 1 8 π M (see
We review and emphasize the fact that the Schwarzschild and Kerr black holes obey the weaker (unattainability U) version of the 3rd law of thermodynamics but not the stronger one (Nernst N or Nernst-Planck N/P). There is no contradiction in this fact since both the U and the N (or N/P) versions are not equivalent.
The author thanks for hospitality to the Instituto de Astronomía y Física del Espacio (IAFE) of the Universidad de Buenos Aires and CONICET, Argentina, where part of this work was done. Also, the author thanks Ernesto Eiroa at IAFE and Josué G. M. Trujillo at UASLP, México, for useful discussions, and Oscar Brauer at the University of Leeds, UK, for the drawing of the Figures.
The author declares no conflicts of interest regarding the publication of this paper.
Socolovsky, M. (2023) Black Holes and the Third Law of Thermodynamics Revisited. Journal of High Energy Physics, Gravitation and Cosmology, 9, 499-505. https://doi.org/10.4236/jhepgc.2023.92039