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Pushing the Limits of Time Beyond the Big Bang Singularity: Scenarios for the Branch Cut Universe

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Pushing the Limits of Time Beyond the Big Bang Singularity: Scenarios for the Branch Cut Universe Received 05 april 2021; Revised 11 april 2021; Accepted 17 april 2021 DOI: xxx/xxxx ARTICLE TYPE Pushing the limits of time beyond the Big Bang singularity: Scenarios for the branch cut universe César A. Zen Vasconcellos*1,2 | Peter O. Hess3,4 | Dimiter Hadjimichef1 | Benno Bodmann5 | Moisés Razeira6 | Guilherme L. Volkmer1 1Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil In this contribution we identify two scenarios for the evolutionary branch cut uni- 2International Center for Relativistic verse. In the first scenario, the universe evolves continuously from the negative Astrophysics Network (ICRANet), Pescara, Italy complex cosmological time sector, prior to a primordial singularity, to the positive 3Universidad Nacional Autónoma de one, circumventing continuously a branch cut, and no primordial singularity occurs Mexico (UNAM), México City, México in the imaginary sector, only branch points. In the second scenario, the branch cut 4Frankfurt Institute for Advanced Studies (FIAS), J.W. von Goethe University and branch point disappear after the realisation of the imaginary component of the (JWGU), Hessen, Germany complex time by means of a Wick rotation, which is replaced by the thermal time. In 5 Unversidade Federal de Santa Maria the second scenario, the universe has its origin in the Big Bang, but the model con- (UFSM), Santa Maria, Brazil 6Laboratório de Geociências Espaciais e templates simultaneously a mirrored parallel evolutionary universe going backwards Astrofísica (LaGEA), Universidade Federal in the cosmological thermal time negative sector. A quantum formulation based on do PAMPA (UNIPAMPA), Caçapava do the WDW equation is sketched and preliminary conclusions are drawn. Sul, Brazil Correspondence KEYWORDS: *Av. Bento Gonçalves, 9500 - Agronomia, Big Bang, General Relativity, Friedmann’s Equations, Big Bounce, WdW Equation Porto Alegre - RS, 91501-970. Email: [email protected] 1 THE ILLUSION OF TIME Time in physics is usually considered a fundamental vari- able, defined by its measurement as the reading of a clock. In Newton’s conception of the clockwork universe, evolving like opposition to this view, John Weeler and Bryce DeWitt devel- a mechanical perfect clock and whose movements of its gears oped in 1967 the so called WdW equation (DeWitt, 1967) arXiv:2105.08108v1 [gr-qc] 17 May 2021 are governed by the laws of physics, with an inherent pre- based on the audacious idea of physics without time, a theo- dictability, prevailed for more than three centuries, until a retical framework that sought to combine quantum mechanics revolutionary concept emerged, thanks to the genius mind of and general relativity, representing a step towards a consistent Hermann Minkowski(Minkowski, 1915), with profound con- theory of quantum gravity. sequences for our current understanding of its structure and More recently, Carlo Rovelli affirmed that the flow of time evolution. is an illusion and that our naive perception of it doesn’t corre- According to his view, instead of being considered sepa- spond to physical reality (Rovelli, 2019), a vision that is in tune rate entities (though intimately related), space and time were with Albert Einstein’s perception of time2. Carlos Rovelli also combined into a single continuum entity, the spacetime1. recently revisited the idea of ‘physics without time’ (Rovelli, 2Albert Einstein in a letter to the family of Michele Besso, his collaborator and closest friend, once wrote: “Now he has departed from this strange world a little 1In Minkowski’s own words: “Henceforth space by itself and time by itself are ahead of me. That means nothing. People like us, who believe in physics, know doomed to fade away into mere shadows, and only a kind of union of the two will that the distinction between past, present, and future is only a stubbornly persistent preserve an independent reality” (Minkowski, 1915). illusion.” (Einstein, 2020). 2 César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer 2004, 2015; Rovelli & Smerlak, 2011) bearing in mind that, where ̄̂ denotes the time-ordering operator. In the continuum in accordance with the second law of thermodynamics, for- limit, we write the amplitude .xbtbðxata/ as a path integral ward in time is the direction in which entropy increases, and in xb which we gain information, so the flow of time is a subjective iA.x/_` .xbtbðxata/ ≡ Dxe : (3) feature of the universe, not an objective part of physical real- Ê x ity3. In this realm, in which the observable universe does not a show time reversal symmetry, events, rather than particles or This equation is the corresponding Feynman’s formula for fields, are the basic constituents of the universe, and the task of the quantum-mechanical amplitude (1) and represents the sum physics would be to describe the relationship between events. over all paths in configuration space with a phase factor con- These conceptions present some points of contact with a taining the action A[x]. line of thought we recently developed (Vasconcellos, Had- jimichef, Razeira, Volkmer, & Bodmann, 2020; Vasconcellos 2.2 Wick rotation in statistical and quantum et al., 2021) where we applied the tools of singular semi- mechanics Riemannian geometry to push the limits of general relativity and time beyond a primordial singularity, giving rise to a In statistical mechanics, the quantum partition function Z.T /, branch cut universe. In this contribution we sought to identify which contains all information about the thermodynamical evolutionary scenarios for the branch cut universe. equilibrium properties of a quantum system, reads *Ĥ _k T *H. ̂p;̂x/_k T Z.T / ≡ T r e B ≡ T r e B : (4) 2 WICK ROTATION OF In this expression, T r.F̂/ denotes the trace of the operator COSMOLOGICAL TIME F̂ = e*H. ̂p;̂x/_kB T ; (5) Wick rotation is a well known theoretical method that encapsu- and kB is the Boltzmann constant. For a N-particle sys- lates a connection between quantum mechanics and quantum tem described by the Schrödinger equation for instance, the statistical mechanics and in another ground relativistic field quantum-statistical system refers to a canonical ensemble. theory in Minkowski spacetime manifolds and Euclidean field The quantum statistical partition function Z.T / may be theory in Riemannian spacetime manifolds. related to the quantum-mechanical time evolution operator ZQM .tb * ta/ i t t Ĥ ` Z t t T r Û t ; t T r e* . b* a/ _ ; 2.1 Path integral formalism QM . b * a/ ≡ . b a/ = (6) The path integral formalism (Feynman & Hibbs, 1965) by making an analytical continuation of the time interval tb *ta describes the quantum transition amplitudes of the unitary time to the negative imaginary value using a Wick rotation: ̂ evolution operator, U.tb; ta/ (a representation of the abelian tb * ta → *i`_kBT: (7) group of time translations), between the localised quantum In quantum mechanics and quantum field theory, the Hamilto- mechanical states of a particle (x ; t ) to (x ; t ), with x and t a a b b nian density acts as the generator of the Lie group of time denoting space and time Cartesian coordinates. The matrix ele- H translations while in statistical mechanics the Hamiltonian ments of the quantum time evolution amplitudes, using bra’s represents a Boltzmann weight in an ensemble. The contour- ( x ) and ket’s ( x ) notation, read ê bð ð aë rotation from the real to the imaginary time-axis, results in a ̂ .xbtbðxata/ = êxbðU.tb; ta/ðxaë tb > ta ; (1) correspondence between the imaginary time component and For a system with a time-independent Hamiltonian operator, the inverse of the temperature, T and it can be understood as a Ĥ , the time evolution operator is simply realisation of the imaginary component of time. * i Ĥ .t *t / ̂ ̂ ` b a U.tb; ta/ = ̄e ; (2) 2.3 Euclidean quantum gravity Euclidean quantum gravity refers to a quantum theory of Rie- mannian manifolds in which the quantisation of gravity occurs 3In general relativity, the reading of a clock is not given by the time variable t, but is instead expressed by a line integral depending on the gravitational field, in a Euclidean spacetime, generated by means of a Wick computed along the clock’s world-line , given as rotation. The corresponding gravitational path integral in the t presence of a field may be expressed as T = g .x; t/ dx dx ; Ê ù d4x g R Z = D[g]D[]e∫ ð ð : (8) (see Rovelli (2015) for more details). Ê César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer 3 Additional assumptions imposed to the manifolds as com- represents a dimensionless thermodynamical connection pactness, connectivity and boundaryless (no singularities), between the energy density .t/ and pressure p.t/ of a perfect make this formulation a strong candidate for overcoming the fluid thus enabling the fully description of the equation of state limitations presented by general relativity in the domain of (EoS) of the system. Positive pressure corresponds to > 3_2, strong gravity, more precisely, the elimination of singulari- negative pressure to < 3_2 and for a universe dominated by ties in extreme physical conditions. There are other techniques a cosmological constant, → 0. that sought to overcome these limitations of general relativity. In the limit in which the dimensionless thermodynamical Among these, we highlight the pseudo-complex general rel- connection obeys .t/ → = constant, the integral (11) ativity, a very powerful technique based on pseudo-complex reduces to spacetime coordinates (Hess, 2017; Hess & Boller, 2020; *1 ln[.t/_0] = *2 lim .t/ dln.ln [ .t/]/ Hess, Schäfer, & Greiner, 2015) with observational predictions .t/→ Ê given in (Schönenbach et al., 2014). ≃ *2 ln.ln*1[ .t/]/ ⇒ ln .ln*1[ .t/]/*2 ⇒ .t/ ≃ 0 ; (13) ln*2[ .t/] 3 COSMOGRAPHY IN AN UNIVERSE which corresponds to an analytically continued expression for WITH A BRANCH CUT the density of the branch cut universe.
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