Pis’ma v ZhETF, vol. 109, iss. 10, pp. 705 - 706

May 25

Comment to the CPT-symmetic Universe: Two possible extensions

G. E. Volovik¹⁾

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

Landau Institute for Theoretical Physics, Russian Academy of Sciences, 119334 Moscow, Russia

Submitted

21 February 2019

Resubmitted 27 March 2019

Accepted 15 April 2019

DOI: 10.1134/S0370274X1910014X

In [1, 2] the antispacetime Universe was suggested as

The CPT-symmetric Universe has been obtained in

the analytic continuation of our Universe across the Big

the conformal time frame [1], where the metric in the

Bang singularity in conformal time. We consider two

spatially flat radiation-dominated era is:

different scenarios of analytic continuation. In one of

them the analytic continuation is extended to the tem-

g_µν = a²(τ)η_µν,

(1)

perature of the system. This extension suggests that if

where η_µν is the flat Minkowski metric; a(τ) is the scale

such analytic continuation is valid, then it is possible

factor and τ is the conformal time. Since the scale fac-

that the initial stage of the evolution of the Universe on

tor a(τ) ∝ τ, it was suggested that it can be analyt-

our side of the Big Bang was characterized by the neg-

ically transformed to the region τ < 0 before the Big

ative temperature. In the second scenario, the analytic

Bang. Then the terad fields, which are proporional to

continuation is considered in the proper time. In this

a(τ), also change sign under this analytic continuation,

scenario the Big Bang represents the bifurcation point

e^aµ(-τ) = -e^aµ(τ).

at which the Z₂ symmetry between the spacetime and

Now let us go further and extend the analytical con-

antispacetime is spontaneously broken.

tinuation to the temperature of the system. In the spa-

The extension of the Universe beyond the Big Bang

tially flat radiation-dominated era one has

using the analytic continuation across the singularity

has been considered for the radiation-dominated epoch

T (τ) ∝

, or β(τ) ∝ τ.

(2)

[1, 2]. In this analytic continuation, at which the scale

N (τ)

e₀₀(τ)

factor a(τ) changes sign at τ = 0, the gravitational

The analytic continuation of β(τ) suggests that in the

tetrads also change sign giving rise to what is called

Universe at τ < 0 the temperature is negative. The

the antispacetime. This means that the Universe on the

transition between the states with positive and nega-

other side of the Big Bang is the mirror image of the

tive temperatures occurs via the infinite T at τ = 0

Universe on our side of the Big Bang.

(β(0) = 0). The states with negative temperature are

Different types of the antispacetime obtained by the

unstable thermodynamically. This means that if the an-

space reversal P and time reversal T operations were

alytic continuation is really valid, then one of the two

considered ealier, including those where the determi-

states (before or after Big Bang) is thermodynamically

nant of the tetrads e changes sign [3-9]. Later the con-

unstable. It is more natural to assume, that the evolu-

sideration has been extended to thermal states, where

tion of the system from τ = -∞ to τ = 0 was equi-

possible analytic continuation of the temperature across

librium and corresponded to the positive temperature,

the transition from spacetime to antispacetime has been

T (τ < 0) > 0. This is because the system had enough

considered [10]. Here we discuss two scenarios of analytic

time to equilibrate before the collapse. So we should

contunuation across the Big Bang.

have the following analytic time dependence of temper-

ature:

T (τ) ∝ -

, or β(τ) ∝ -τ,

(3)

e₀₀(τ)

and thus on our side of the Big Bang the temperature

¹⁾e-mail: volovik@boojum.hut.fi

is negative, T (τ > 0) < 0. Of course, this may happen

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705

706

G. E. Volovik

only at the first stage of the evolution of our Universe,

the Big Bang the Universe may evolve as the quantum

i.e. immediately after the Big Bang. After some time

superposition of these two states, but due to rapid de-

the equilibration occurs, and the system returns again

coherence only one of the two states survives. Thus in

to the evolution with the equilibrium positive temper-

this analytic continuation the Z₂ (CPT) symmetry is

ature, T (τ > 0) > 0. The proper justification of this

spontaneously broken, as distinct from the scenario in

suggestion is beyond this comment.

[1] describing creation of two Universes both evolving to

Let us consider the analytic continuation in terms of

the future with the conservation of the CPT symmetry.

the physical proper time t. The metric for the radiation-

In conclusion, we extended the analytic continua-

dominated Universe is:

tion across the Big Bang proposed in [1] in two different

ways. The analytic continuation in the conformal time

g_µν = dt² - a²(t)dr², a²(t) ∝ t.

(4)

frame τ was extended to the temperature of the system.

√

This extension suggests that if the analytic continuation

Let us now assume that the scale factor a(t) ∼

t can

of the terad is valid, then it is possible that the initial

be analytically extended around the singularity at t = 0.

stage of the evolution of our Universe after the Big Bang

Then the spacetime (with positive spatial components of

was characterized by the negative temperature. The an-

tetrads) and antispacetime (with negative spatial com-

alytic continuation in the proper time t suggests that the

ponents of tetrads) can be connected by analytic con-

Big Bang is the bifurcation point of the second order

tinuation: by 2π rotation about t = 0. As distinct from

quantum transition from the Euclidean to Minkowski

scenario in [1], where the two Universes live together, in

the proper time scenario the spacetime and antispace-

signature, where symmetry between spacetime and an-

tispacetime is spontaneously broken.

time represent two different realizations of the Universe

I thank A. Starobinsky for criticism. This work

at t > 0, which exclude each other.

has been supported by the European Research Coun-

The Universe at t > 0 can be obtained by analytic

cil (ERC) under the European Union’s Horizon 2020

continuation from the negative t region, where the met-

research and innovation programme (Grant Agreement

ric has Euclidean signature. In this scenario, the Big

#694248).

Bang (at t = 0) represents the point of the phase transi-

tion from the Euclidean to Minkowski signature, which

is similar to that, say, in [11-14] (an example of such

1. L. Boyle, K. Finn, and N. Turok, Phys. Rev. Lett. 121,

transition in condensed matter can be found, e.g., in

251301 (2018).

[15]). At the Big Bang, the state with Euclidean signa-

2. L. Boyle, K. Finn, and N. Turok, arXiv:1803.08930.

ture transforms either to the spacetime or to antispace-

3. D. Diakonov, arXiv:1109.0091.

time (see Fig. 1), but not to the quantum superposition

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

104019 (2012).

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

Phys. D 21, 1242014 (2012).

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

064002 (2012).

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

Fig. 1. Bifurcation at Big Bang. Analytic continuation in

√

(2018).

physical proper time t: the scale factor a(t) ∝

t changes

8. G. E. Volovik, arXiv:1903.02418.

sign around the Big Bang point, transforming spacetime

9. S. N. Vergeles, arXiv:1903.09957.

into antispacetime. At t < 0 the scale factor is imaginary,

10. G. E. Volovik, Pis’ma v ZhETF

109,

(2019);

which corresponds to the metric with Euclidean signature.

arXiv:1806.06554.

Crossing the Big Bang from the t < 0 semiaxis, the Uni-

11. H. C. Steinacker, High Energ. Phys. 02, 033 (2018).

verse approaches either spacetime or antispacetime. In this

scenario, the Big Bang represents the bifurcation point at

12. A. Stern and Ch. Xu, Phys. Rev. D 98, 086015 (2018).

which the Z2 symmetry between the spacetime and anti-

13. A. D. Sakharov, Sov. Phys. Usp. 34, 409 (1991).

spacetime is spontaneously broken. The quantum super-

14. G. W. Gibbons and J. B. Hartle, Phys. Rev. D 42, 2458

position of the two states is not allowed in the macroscopic

(1990).

system

15. J. Nissinen and G.E. Volovik, JETP Lett. 106, 234

(2017).

of the two Universes. The latter is not allowed for the

16. M. Grady, hep-th/9409049.

macroscopic systems, which experience the spontaneous

symmetry breaking [16]. In principle, immediately after

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2019