Pis’ma v ZhETF, vol. 109, iss. 1, pp. 10 - 11

January 10

Negative temperature for negative lapse function

G. E. Volovik¹⁾

Low Temperature Laboratory, Aalto University, School of Science and Technology, FI-00076 AALTO, Finland

Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia

Submitted 25 October 2018

Resubmitted 6 November 2018

Accepted 8 November 2018

DOI: 10.1134/S0370274X19010028

Fermion dynamics distinguishes spacetimes having

“antispacetime”, the chirality of Weyl fermions changes:

the same metric g_µν , but different tetrads e_µa, and in

the left-handed fermions living in the spacetime trans-

particular, it distinguishes a lapse with negative sign,

form to the right-handed fermions in “antispacetime”.

N < 0 [1]. Here we show that the quasiequilibrium ther-

This transition experiences the nonanalytic behavior of

modynamic state may exist, in which the region with

the action at the crossing point. Here we discuss the

N < 0 has negative local temperature T(x) < 0, while

similar transition from “spacetime to antispacetime” by

the global Tolman temperature T₀ remains positive.

the time reversal and show that this transition may have

Tolman’s law [2] (see also [3]) states that in a static

analytical properties suggested in [8-10].

gravitational field with the shift Ni = 0, the locally

Let us assume that the lapse function N(x) is the an-

measured temperature T (x) obeys:

alytical function of the tetrad field. Then instead of the

T₀

conventional Tolman law, T(x) = T₀/|N(x)| in Eq.(1),

T (x) =

√

(1)

one would have the modified Tolman law

g₀₀(x)

|N(x)|

T₀

where T₀ is spatially constant in thermal equilibrium,

T (x) =

, N(x) = e₀₀(x) =

(2)

N (x)

e⁰⁰(x)

and N is the lapse function with g₀₀(x) = N2(x).

For negative e₀₀(x) (but still positive g₀₀(x)), the lo-

In the ADM parametrization with Nⁱ = 0, one has

cal temperature T(x) of fermions becomes negative, see

g_µνdx^µdx^ν = N²dt² - g_ikdxⁱdx^k.

In the effective gravity emerging for quasirelativistic

Fig. 1a.

quasiparticles in superfluids [4], T₀ is the conventional

Formation of negative temperature in the island with

negative e₀₀(x) can be explained in the following way.

temperature of the liquid as measured by external ob-

server. It is constant in space in thermal equilibrium.

For the fermions, the crossing e₀₀ = 0 corresponds to

The local T (x) is measured by the local “internal ob-

the change of the Hamiltonian H → -H. When the is-

server”, who uses quasiparticles for measurements.

land is formed, then immediately after formation one

has the state with inverse filling of the particle energy

Fermions interact with gravity via the tetrads in-

stead of the metric, g_µν = η^abe_aµe_bν . In terms of tetrads,

levels, which corresponds to the negative local temper-

one has N2 = g00 = (e00)2 = (e00)-2. The general

ature, T (x) < 0.

Though in general the negative temperature state

Lorentz transformations acting on fermions include two

discrete operations: the reversal of time, and parity

with inverse population is not in full equilibrium, in

transformation. Under time reversal we have T e₀₀ =

principle, it can be made locally stable, see, e.g., [11].

-e₀₀ and Tdet(e) = -det(e), and under parity trans-

Anyway, finally the state in the island relaxes to the

fully equilibrium state in Fig. 1b with positive temper-

formation - P e₀₀ = e₀₀ and P det(e) = -det(e). Corre-

spondingly, the fermionic vacuum has the four-fold de-

ature, T (x) = T₀/|e₀₀(x)| = T₀/|N(x)| > 0.

generacy.

So, while the dynamics in the negative lapse region

may correspond to the inverse arrow of time for fermions

In condensed matter the analog of parity transforma-

tion takes place in a topological Lifshitz transition, when

[1], the thermodynamics in this region may correspond

the chiral vacuum with Weyl nodes in the polar dis-

to the negative temperature.

torted superfluid³He-A[5, 6] crosses the vacuum state

As is demonstrated in [7] on example of the Weyl su-

of the polar phase with a degenerate fermionic tetrad,

perfluid, the action in terms of tetrads is non-analytic.

det(e) = 0 [7]. In this transition from “spacetime” to

For example, the action for the effective gauge field is

shown to be proportional to

√-g = |det(e)|. This is

¹⁾e-mail: volovik@boojum.hut.fi

contrary to the action proportional to det(e), which

Письма в ЖЭТФ том 109 вып. 1 - 2

2019

Negative temperature for negative lapse function

sideration beyond the Einstein general relativity. The

similar problem arises for the hypersurfaces, at which

the Newton constant changes sign [12].

In [1] three alternatives to the problem of antispace-

time were suggested: (i) Antispacetime does not exist,

and det (e) > 0 should constrain the gravity path inte-

gral; (ii) Antispacetime exists, but the action depends

on |det(e)|. (iii) Antispacetime exists and contributes

nontrivially to quantum gravity.

Our consideration suggests that the antispacetime

may exist with two possible realizations. The option (ii)

takes place in case of full equilibrium both in spacetime

and in antispacetime in Fig. 1b, and in this case the ac-

tion is non-analytic [7] together with conventional Tol-

man law in Eq. (1). The option (iii) with the analytic

behavior of the action takes place in the quasiequilib-

rium state in Fig. 1a with the negative temperature in

the island. This state is formed immediately after forma-

tion of the island, and obeys the analytic Tolman law

in Eq. (2). After relaxation to the full equilibrium the

nonanalytic behavior of the action and of the Tolman

law in Eq. (1) is restored.

This work has been supported by the European

Research Council (ERC) under the European Union’s

Horizon

2020

research and innovation programme

(Grant Agreement # 694248).

Fig. 1. Island of negative lapse function, N(x) = e00(x) <

0. (a) - After formation of the island, the metastable

Full text of the paper is published in JETP Letters

state with negative local temperature is formed accord-

journal. DOI: 10.1134/S002136401901003X

ing to the modified Tolman law, T (x) = T0/e00(x) < 0.

The global Tolman temperature T0 is constant in space,

T0 = const > 0. It is the temperature at infinity, where

1. M. Christodoulou, A. Riello, and C. Rovelli, Int. J. Mod.

e00(±∞) = 1. In this scenario, e00(x) crosses zero, while

Phys. D 21, 1242014 (2012).

and temperature T (x) crosses infinity. The negative tem-

2. R. Tolman, Relativity, Thermodynamics, and Cosmol-

perature state in the island is nonequilibrium, and finally

ogy, Oxford University Press, Oxford (1934).

it relaxes to the equilibrium state in Fig. b with posi-

3. J. Santiago and M. Visser, arXiv:1805.05583.

tive temperature obeying the conventional Tolman law,

4. G. E. Volovik, The Universe in a Helium Droplet,

T(x) = T0/|e00(x)| > 0

Clarendon Press, Oxford (2003).

5. R. Sh. Askhadullin, V. V. Dmitriev, P. N. Martynov,

A. A. Osipov, A.A. Senin, and A.N. Yudin, JETP Lett.

has been suggested in [8-10]. The nonanalytic behav-

100, 662 (2014).

ior of the action takes place when the boundary is

6. V. V. Dmitriev, A. A. Senin, A. A. Soldatov, and

crossed between two equilibrium degenerate states with

A. N. Yudin, Phys. Rev. Lett. 115, 165304 (2015).

different signs of det(e) or N(x). In both equilibrium

7. J. Nissinen and G. E. Volovik, Phys. Rev. D 97, 025018

states in Fig. 1b the temperature is positive, and the

(2018).

Tolman law is given by the nonanalytic equation (1).

8. D. Diakonov, arXiv:1109.0091.

The analytic action suggested in [8-10] can be real-

9. A. A. Vladimirov and D. Diakonov, Phys. Rev. D 86,

ized in the quasiequilibrium state in Fig. 1a, when the

104019 (2012).

boundary is crossed between the equilibrium state with

10. C. Rovelli and E. Wilson-Ewing, Phys. Rev. D 86,

N (x) > 0 and the nonequilibrium state in the island

064002 (2012).

with N(x) < 0 and negative T (x) < 0. In this case the

11. S. Braun, J. P. Ronzheimer, M. Schreiber, S. S. Hodg-

Tolman law is the analytic function of N(x) in Eq. (2),

man, T. Rom, I. Bloch, and U. Schneider, Science 339,

as well as the action.

52 (2013).

Note that the realization of the hypersurfaces, at

12. A. A. Starobinsky, Sov. Astron. Lett. 7, 36 (1981).

which det(e) crosses zero or infinity, may require con-

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2019