A nonsingular, anisotropic universe in a black hole with torsion and particle production

May combine two claims.

The interior of a Schwarzschild black hole can be represented by the Kantowski-Sachs metric, which describes a closed, anisotropic universe.

“The interior of a Schwarzschild black hole can be represented by the Kantowski-Sachs metric (KS) [29].This metric describes a closed, anisotropic universe [30,31].”

• cites [29] Popławski, N.J.: Phys. Lett. B 690, 73 (2010) sourceCould not verify

  The citation [29] is attached to the statement that the Schwarzschild interior can be represented by the Kantowski–Sachs metric; the closed, anisotropic-universe description is cited separately. Our copy discusses the Kerr–Newman metric in Section IV (p. 5) and black-hole mass limits in Section V (p. 6), but searches for Kantowski–Sachs and the Schwarzschild interior found no passage addressing the representation attributed to this paper. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [30] Popławski, N.J.: Phys. Lett. B 727, 575 (2013) sourceNot checked — no full text of the source
• cites [31] Popławski, N.: Classical Physics: Spacetime and Fields. arXiv:0911.0334 sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

A recent evidence suggests that the universe may be closed.

“A recent evidence suggests that the universe may be closed [1].”

• cites [1] Di Valentino, E., Melchiorri, A., Silk, J.: Nat. Astron. 4, 196 (2020) sourceSupported

May combine two claims.

The collapsing matter undergoes a nonsingular bounce and then expands as a new, closed universe whose total energy is zero.

“Instead, the collapsing matter undergoes a nonsingular bounce [21] and then expands as a new, closed universe [4,5,22] whose total energy is zero [23].”

• cites [21] Sciama, D.W.: Rev. Mod. Phys. 36, 463 (1964) sourceNot checked — no full text of the source
• cites [4] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Lett. B 216, 272 (1989) sourceNot checked — no full text of the source
• cites [5] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Rev. D 41, 383 (1990) sourceNot checked — no full text of the source
• cites [22] Sciama, D.W.: Rev. Mod. Phys. 36, 1103 (1964) sourceCould not verify

  Citation [23] is attached specifically to the claim that the new universe’s total energy is zero. The passages in our copy (labeled Abstract; no page given) contain erratum corrections, not a discussion of a universe’s total energy. Searches for zero energy and gravitational energy in a closed universe found no relevant passage. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [23] Hehl, F.W., Datta, B.K.: J. Math. Phys. 12, 1334 (1971) sourceNot checked — no full text of the source

May combine two claims.

Both entropy increase and isotropization can be generated by cosmic inflation.

“Both processes can be generated by cosmic inflation [36].”

• cites [36] Popławski, N.: arXiv:1304.0047 sourceCould not verify

  The metric-affine gravity section says post-bounce dynamics explains the universe’s large-scale isotropy without needing cosmic inflation. It does not say whether inflation can generate isotropy, and the passages found for entropy and inflation do not address entropy increase. An alternative explanation is not a contradiction of the claim that inflation can generate both processes. (LLM, conf=0.75)

  Experimental — accuracy not yet measured

May combine two claims.

Isotropization would last during inflation, after which it would end and the anisotropy of the universe would grow again during the radiation-dominated era and the matter-dominated era.

“Isotropization would last during inflation, after which it would end and the anisotropy of the universe would grow again during the radiation-dominated era and the matter-dominated era [37].”

• cites [37] Nomura, K., Shirafuji, T., Hayashi, K.: Prog. Theor. Phys. 86, 1239 (1991) sourceCould not verify

  The passages on spinning particles and torsion, including §3.2 (page not supplied), discuss spin currents and equations of motion, not cosmological evolution. Searches for isotropization, inflation, and anisotropy during the radiation- and matter-dominated eras found no passage addressing the claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

May combine two claims.

The shear scalar diverges at η=0 for vacuum and at η=π/2.

“It diverges at η = 0 for vacuum, and at η = π/2.”

The quote only partly states this claim.

At η=0, the dust Kantowski-Sachs solution corresponds to a disk singularity.

“At η = 0, we have X = κm, Y = A, Ẋ = -B/(2A), Ẏ = 0, a = (κmA 2 ) 1/3 , and ȧ = -B/[6A 1/3 (κm) 2/3 ].In vacuum, this instant corresponds to a disk singularity.”

May combine two claims.

The right-hand side of equation (30) is positive, indicating that the shear scalar grows with decreasing a faster than ∼ a^-6.

“The right-hand side in this equation is positive, which indicates that the shear scalar grows with decreasing a faster than ∼ a -6 .”

May combine two claims.

Equation (11) shows that if YŸ ≫ 1 then X is approximately proportional to Y, and then a is also approximately proportional to Y and satisfies aä ≫ 1, which holds during volume inflation.

“Equation (11) shows that if Y Ÿ ≫ 1 then X is approximately proportional to Y .Then a is also approximately proportional to Y and satisfies aä ≫ 1, which holds during volume inflation.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Kopczyński and Kuchowicz found that a spatially flat, homogeneous, anisotropic universe of Bianchi type I has no cosmological singularity if the effect of torsion is greater than that of shear.

“Kopczyński [40] and Kuchowicz [41] found that a spatially flat, homogeneous, and anisotropic universe of the Bianchi type I [16] has no cosmological singularity if the effect of torsion is greater than that of shear.”

• cites [40] Trautman, A.: Symp. Math. 12, 139 (1973)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [41] Trautman, A.: Nat. Phys. Sci. 242, 7 (1973) sourceNot checked — no full text of the source

May combine two claims.

At extremely high densities, the negative corrections from the spin-torsion coupling violate the strong energy condition and manifest as repulsive gravity that may prevent the formation of a gravitational singularity in a black hole.

“At extremely high densities, much greater than nuclear density, the negative corrections from the spintorsion coupling in (1) violate the strong energy condition and manifest themselves as repulsive gravity that may prevent the formation of a gravitational singularity in a black hole.”

• cites [21] Sciama, D.W.: Rev. Mod. Phys. 36, 463 (1964) sourceNot checked — no full text of the source

May combine two claims.

The effective energy density and pressure of a spin fluid are given by ε = ε - α n_f^2 and p = p - α n_f^2.

“Consequently, the effective energy density and pressure of a spin fluid are given by ε = ϵ -αn 2 f , p = p -αn 2 f ,(1)”

• cites [5,[18][19][20]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

Tafel found that avoidance of singularities in Einstein-Cartan theory may also occur for flat, homogeneous universes of Bianchi types I-VIII.

“Tafel [42] found that the avoidance of singularities in EC may also occur for flat, homogeneous universes of the Bianchi types I-VIII.”

• cites [42] Kopczyński, W.: Phys. Lett. A 39, 219 (1972) sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

A physical law that turns a black hole into a bridge to a new universe must avoid the black-hole singularity.

“A physical law that turns a black hole into a bridge to a new universe must avoid the black-hole singularity.”

Classified as probably not a substantial claim.

Such a closed universe might have originated as a baby universe from a bounce in the interior of a black hole existing in a parent universe.

“Such a universe might have originated as a baby universe from a bounce in the interior of a black hole existing in a parent universe [2][3][4][5].”

• cites [2] Novikov, I.D.: J. Exp. Theor. Phys. Lett. 3, 142 (1966)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [3] Pathria, R.K.: Nature 240, 298 (1972) sourceNot checked — no full text of the source
• cites [4] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Lett. B 216, 272 (1989) sourceNot checked — no full text of the source
• cites [5] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Rev. D 41, 383 (1990) sourceNot checked — no full text of the source

May combine two claims.

A universe born in a black hole would be located on the other side of the event horizon and connected to the parent universe through an Einstein-Rosen bridge.

“A universe born in a black hole would be located on the other side of the event horizon of the black hole and connected to the parent universe through an Einstein-Rosen bridge [3].”

• cites [3] Pathria, R.K.: Nature 240, 298 (1972) sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A closed, homogeneous, and anisotropic universe is either of Bianchi type IX or described by the Kantowski-Sachs metric.

“A closed, homogeneous, and anisotropic universe is either of the Bianchi type IX [16] or described by the KS metric.”

• cites [16] Popławski, N.: arXiv:1910.10819; arXiv:1912.02173Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Without further phenomena, a Kantowski-Sachs universe would be singular and oscillatory, and would not exceed the size of the parent black hole.

“Without further phenomena, such a universe would be singular and oscillatory [32], and would not exceed the size of the parent black hole.”

• cites [32] Eisenhart, L.P.: Non-Riemannian Geometry. American Mathematical Society, Providence (1927) sourceNot checked — no full text of the source

May combine two claims.

If our universe was formed in a black hole, then it must have increased its entropy and isotropized in the past to match the currently observed FLRW universe.

“If our universe was formed in a black hole, then it must have increased its entropy [5,33,34] and isotropized in the past to match the currently observed FLRW universe [35].”

• cites [5] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Rev. D 41, 383 (1990) sourceNot checked — no full text of the source
• cites [33] Schrödinger, E.: Space–Time Structure. Cambridge University Press, Cambridge (1954)Not checked yet

  Not checked in the free run.

• cites [34] Schouten, J.A.: Ricci-Calculus. Springer, Berlin (1954) sourceNot checked — no full text of the source
• cites [35] Hehl, F.W., McCrea, J.D.: Found. Phys. 16, 267 (1986) sourceNot checked — no full text of the source

May combine two claims.

In Einstein-Cartan theory, torsion is determined by the field equations and the resulting torsion tensor is proportional to the spin tensor of fermionic matter.

“Instead, torsion is determined by the field equations obtained from varying the action with respect to the torsion tensor [6][7][8].The resulting torsion tensor is proportional to the spin tensor of fermionic matter.”

• cites [6] Smolin, L.: Class. Quantum Grav. 9, 173 (1992) sourceNot checked — no full text of the source
• cites [7] Hawking, S.: Black Holes and Baby Universes and other Essays. Bantam Dell, New York (1993)Not checked — no full text of the source
• cites [8] Stuckey, W.M.: Am. J. Phys. 62, 788 (1994) sourceNot checked — no full text of the source

May combine two claims.

Torsion provides a mechanism for preventing a singularity, but this mechanism alone is not sufficient.

“Consequently, torsion provides a mechanism for preventing a singularity.However, this mechanism alone is not sufficient.”

May combine two claims.

Without torsion, the left-hand side of (32) would be absent and the inequality would not be satisfied, resulting in a singularity.

“Without torsion, the left-hand side of ( 32) would be absent and this inequality would not be satisfied, resulting in a singularity.”

May combine two claims.

After the bounce in Y, the universe expands and Y reaches values larger than 2GM; since the universe is closed, Y eventually stops increasing and starts decreasing, and the universe contracts to another bounce.

“After the bounce in Y , the universe expands and Y reaches values larger than 2GM .Since the universe is closed, Y eventually stops increasing and starts decreasing, and the universe contracts to another bounce.”

May combine two claims.

The universe may have several bounces and cycles until it reaches the size at which the cosmological constant becomes dominant and prevents the next contraction; then the cycles end and the universe begins indefinite expansion.

“The universe may have several bounces and cycles until it reaches the size at which the cosmological constant becomes dominant and prevents the next contraction.Then the cycles end and the universe begins indefinite expansion [5,33].”

• cites [5] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Rev. D 41, 383 (1990) sourceNot checked — no full text of the source
• cites [33] Schrödinger, E.: Space–Time Structure. Cambridge University Press, Cambridge (1954)Not checked yet

  Not checked in the free run.

May combine two claims.

If β is too big, the right-hand side of (38) could become positive, causing temperature to grow with increasing a, leading to eternal inflation.

“When a increases and if β is too big, then the right-hand side of ( 38) could become positive.In this case, the temperature would grow with increasing a, which would lead to eternal inflation [5].”

• cites [5] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Rev. D 41, 383 (1990) sourceNot checked — no full text of the source

May combine two claims.

If β is big enough, particle production will increase T quickly enough to make the right-hand side of (36) negative while keeping the left-hand side positive, preventing a cosmological singularity and generating a bounce.

“If β is big enough, then particle production will increase T quickly enough to make the right-hand side of ( 36) negative while keeping the left-hand side positive.This increase will prevent a cosmological singularity and generate a bounce.”

• AssumptionThe authors state this as an assumption: The spin fluid in the early universe is created by quantum particle production.

May combine two claims.

If the number of fermions in a black hole is constant, then the fermion number density grows with decreasing a as ∼ a^-3, while the shear scalar grows faster than ∼ a^-6, so n_f^2 growing as ∼ a^-6 could not satisfy (32) at sufficiently small a and a singularity would form.

“If the number of fermions in a black hole is constant, then the fermion number density grows with decreasing a according to ∼ a -3 .Since the shear scalar σ 2 grows with decreasing a faster than ∼ a -6 , n 2 f growing as ∼ a -6 could not satisfy (32) at sufficiently small values of a and a singularity would form.”

May combine two claims.

The scenario is only approximate: the interior Schwarzschild metric and therefore the Kantowski-Sachs metric is never reached if a nonsingular bounce occurs.

“This scenario is only approximate.Its purpose was to show how torsion and particle production could avoid a singularity even if the shear is present.In reality, the interior Schwarzschild metric and therefore the KS metric, which has topology R × S 2 , is never reached if a nonsingular bounce occurs.”

May combine two claims.

When the universe enters the cosmological-constant-dominated era, the anisotropy would decrease and the closed universe would become asymptotically isotropic.

“When the universe enters the cosmologicalconstant-dominated era, the anisotropy would decrease and the closed universe would become asymptotically isotropic.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

The Kantowski-Sachs metric should be replaced with a more general Bianchi IX form that could allow the universe to asymptotically tend to the closed FLRW geometry with topology S^3.

“It should be replaced with a more general Bianchi IX form, that could allow the universe to asymptotically tend to the closed FLRW geometry with topology S 3 .”

May combine two claims.

Torsion and particle production may together violate the strong energy condition and give the universe one or more nonsingular bounces.

“We showed that torsion and particle production may together violate the strong energy condition and give the universe one or more nonsingular bounces.”

• AssumptionThe authors state this as an assumption: The spin fluid in the early universe is created by quantum particle production.

May combine two claims.

To avoid a singularity, n_f^2 must grow faster than ∼ a^-6 and fermions must be produced in a black hole.

“Therefore, to avoid a singularity, n 2 f must grow faster and fermions must be produced in a black hole.”

May combine two claims.

Because of particle production, which increases entropy, each cycle of expansion and contraction lasts longer and reaches a larger maximum of Y than the preceding cycle.

“Because of particle production, which increases entropy, each cycle of expansion and contraction lasts longer and reaches a larger maximum of Y than the preceding cycle [5,33].”

• cites [5] Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Rev. D 41, 383 (1990) sourceNot checked — no full text of the source
• cites [33] Schrödinger, E.: Space–Time Structure. Cambridge University Press, Cambridge (1954)Not checked yet

  Not checked in the free run.

May combine two claims.

If the right-hand side of (38) is slightly less than 1, then T would be approximately constant, H would be approximately constant, and the mean scale factor a would grow almost exponentially, generating inflation.

“If the right-hand side of ( 38) is slightly lesser than 1, then T would be approximately constant.Accordingly, H would be also approximately constant and the mean scale factor a would grow almost exponentially, generating inflation.”

• AssumptionThe authors state this as an assumption: The spin fluid in the early universe is created by quantum particle production.

May combine two claims.

Particle production in an expanding phase can generate a finite period of inflation, during which the universe produces enormous amounts of matter and entropy, and isotropizes.

“Furthermore, particle production in an expanding phase can generate a finite period of inflation, during which the universe produces enormous amounts of matter and entropy, and isotropizes.”

• AssumptionThe authors state this as an assumption: The spin fluid in the early universe is created by quantum particle production.

May combine two claims.

After this period, the effects of torsion would become small and the universe would smoothly enter the radiation-dominated expansion, followed by the matter-dominated expansion, and then by the cosmological-constant-dominated expansion.

“After this period, the effects of torsion would become small and the universe would smoothly enter the radiation-dominated expansion, followed by the matter-dominated expansion, and then by the cosmological-constant-dominated expansion.”