Dimensionality, topology, energy, the cosmological constant and signature change

The most direct way of extracting a favoured classical space-time from the model gives R×S1×S2 in n=4.

“the most direct way of extracting a favoured classical space-time from our model gives R × S 1 × S 2 in n = 4”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

In the Kaluza-Klein scenario, the internal space is a product of two-spheres.

“the internal space is a product of two-spheres”

• Also stated in: § 1 Introduction · plain

  “The internal (compactified) space is a product of two-spheres.”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

The model is also consistent with R×S3×internal space in n=6,8,10,12.

“it is also consistent with R × S 3 × internal space in n = 6, 8, 10 or 12.”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

For any given Λ>0, the radii of the factor-spheres are uniquely determined by a_B = ((n_B-1)(n-2)/(2Λ))^{1/2}.

“Hence, for any given Λ > 0, the radii of the factor-spheres are uniquely determined to be a B = (n B -1)(n -2) 2Λ 1/2 B = 1, . . . m .”

In the Kaluza-Klein scenario, the observed spacelike slices are S3.

“in which case the observed spacelike slices are S 3”

• Also stated in: § 1 Introduction · plain

  “with spacelike slice Σ = S 3”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

If n>2 and at least one of the spheres is one-dimensional, the Einstein equations imply n_B=1 for all B and Λ=0.

“If n > 2 and at least one of the spheres is one-dimensional, the Einstein equations imply n B = 1 for all B = 1, . . .m, and Λ = 0.”

In that model, for large E, the global topology of spacelike slices is S1×S2.

“the global topology of spacelike slices being S 1 ×S 2 (hence predicting a universe of Kantowski-Sachs type)”

• Also stated in: § 1 Introduction · plain

  “the corresponding spacelike slice Σ being S 1 × S 2 , i.e. of Kantowski-Sachs type”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

In the simplified model allowing arbitrary products of spheres as Euclidean solutions, for large E, the favoured spacetime dimension is 4.

“In a simplified model, allowing for arbitrary products of spheres as Euclidean solutions, the favoured spacetime dimension is 4”

• Also stated in: § 1 Introduction · plain

  “It turns out that for large E the favoured dimension is n = 4”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

Among the configurations selected, those with large E are favoured.

“among the configurations selected already, those with large E are favoured”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

If the minimization is only approximate, there is some freedom for a Kaluza-Klein scenario.

“If the minimization is only approximate (caused by the details of some decoherence mechanism or by an effect at the wormhole scale), there is some freedom for a Kaluza-Klein scenario”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

In the generic case, the Einstein equations are equivalent to n_B-1/a_B^2 = 2Λ/(n-2) for B=1,...,m.

“In the generic case, as specified above, the Einstein equations are equivalent to the system of equations n B -1 a 2 B = 2Λ n -2 B = 1, . . . m .”

If n=1 or n=2, the Einstein equations lead to Λ=0.

“If n = 1 or n = 2, the Einstein equations lead to Λ = 0.”

Minimization at fixed but large E leads to low dimensions.

“minimization at fixed but large E leads to low dimensions”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

The value of the Euclidean action (2.1), when such a solution is inserted, is given by (4.7).

“The value of the Euclidean action (2.1), when such a solution is inserted, turns out to be (using R = 2nΛ/(n -2) for the total curvature scalar, which follows from contracting (2.2)) I = - V 8πG n 2Λ n -2 = - 1 8πG n n -2 2Λ (n-2)/2 m B=1 v n B (n B -1) n B /2 .”

In the Kaluza-Klein scenario, the total space-time dimension is 6, 8, 10 or 12.

“the total space-time dimension is 6, 8, 10 or 12.”

• Also stated in: § 1 Introduction · plain

  “total dimension n = 6, 8, 10 or 12.”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

In all configurations, the signature changing sphere is the one with the highest dimension.

“Moreover, in all configurations, the signature changing sphere is the one with the highest dimension.”

For large n, the dimension of the signature-changing sphere in K_n is p ≈ β√n + 1, with β = (2(1-ln 2))^{-1/2} ≈ 1.2765.

“we find the dimension of the signature changing sphere in K n for large n to be p ≈ β √ n + 1 (5.25) with β = (2(1ln 2)) -1/2 ≈ 1.2765.”

When expressed entirely in terms of Λ, E is given by (4.12).

“When expressed entirely in terms of Λ (using (4.4) and (4.5)), we obtain E = 1 16πG n v n A -1 (n -2) v n A (n A -1) 1/2 n -2 2Λ (n-3)/2 m B=1 v n B (n B -1) n B /2 . (4.12)”

Once (M,Σ), i.e. {n_1,...,n_m} and A, are fixed, the effective non-gravitational energy generated at Σ_A is given by E = (1/(8πG_n)) V_{Σ_A} Λ.

“Once (M, Σ) (i.e.{n 1 , . . .n m } and A) are fixed, the effective non-gravitational energy generated at Σ A is given by E = 1 8πG n V Σ A Λ ,(4.9)”

The complete Euclidean action is given by I = - (8π F^{1/(n-3)})/(2 κ_n (n-2)) (E/m_P)^{(n-2)/(n-3)}.

“The complete Euclidean action is given by I = - 8π F 1/(n-3) 2 κ n n -2 E m P (n-2)/(n-3) . (5.2)”

For n=5, F(S^2×S^3)=2√2π^3≈87.699.

“F ( S 2 × S 3 ) = 2 √ 2π 3 ≈ 87.699 (5.7)”

For n=4,...,15, the configurations K_n minimizing F are given by (5.9)-(5.20).

“Let us write down a list of the configurations K n minimizing F for n = 4, . . .15 K 4 = S 2 × S 2(5.9) K 5 = S 2 × S 3(5.10) K 6 = S 2 × S 4(5.11) K 7 = S 2 × S 2 × S 3(5.12) K 8 = S 2 × S 2 × S 4 (5.13) K 9 = S 2 × S 2 × S 5 (5.14 ) K 10 = S 2 × S 2 × S 2 × S 4 (5.15 ) K 11 = S 2 × S 2 × S 2 × S 5 (5.16 ) K 12 = S 2 × S 2 × S 2 × S 2 × S 4(5.17) K 13 = S 2 × S 2 × S 2 × S 2 × S 5(5.18) K 14 = S”

Due to the S^2 factor spheres in higher K_n, the relation between the cosmological constant and nucleation radius is Λ ∼ a_A^4 instead of Λ ∼ a_A^{-2}.

“Note that due to the bunch of S 2 -factor-spheres in the higher K n , the relation between the cosmological constant and the nucleation radius is not the standard one (Λ ∼ a A 4 instead of Λ ∼ a A -2 , see also equation (4.4)).”

The level spacing ΔE = E_{n-1}-E_n is of order E itself, ΔE≈E.

“Note that the level spacing ∆E ≡ E n-1 -E n is of the order of E itself, ∆E ≈ E.”

F(K_n)^{-1/(n-3)} → 1/√2 as n→∞.

“we note that F (K n ) -1/(n-3) → 1/ √ 2 as n → ∞”

For dimensions 16,...,42, the according configurations are products of two-spheres with an S^p, where p takes the values (6,5,6,7,6,7,6,7,6,7,8,7,8,7,8,7,8,9,8,9,8,9,8,9,8,9,10).

“For the dimensions (16, . . .42) the according configurations are products of two-spheres with an S p , where p takes the values (6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10).”

At fixed E, the asymptotic action is I_E(K_n) = -√2 κ_n n^{-2} (n-2)/(n-3) E/m_P (1+O(1/n)).

“I E (K n ) = - √ 2 κ n n -2 (n-2)/(n-3) E m P 1 + O 1 n . (6.1)”

If n>3, relation (4.12) can be used to express Λ in terms of E.

“If n > 3, the relation (4.12) can be used to express Λ in terms of E.”

For n=4, F(S^4)=9π^2/2≈44.4132.

“For n = 4 and n = 5 we find F ( S 4 ) = 9π 2 2 ≈ 44.4132 (5.4)”

Inverting the asymptotic expression up to first two orders, the favoured dimension for E≪m_P is n ≈ ln(m_P/E) + (1/(4β)) ln(m_P/E)^{1/2}.

“Inverting this expression up to the first two orders, it follows that the favoured dimension for E ≪ m P is n ≈ ln m P E + 1 4β ln m P E 1/2 .(6.12)”

• AssumptionThe authors state this as an assumption: κ_n = 1 for all n.

For small E, combining (6.1) and (6.12) gives, to leading order, I(E) ≈ -√2 ln(m_P/E)^{-1} E/m_P.

“For small E, (6.1) and (6.12) can be combined to give, to leading order, I(E) ≈ -√ 2 ln m P E -1 E m P , (7.1)”

I(E) is a monotonically decreasing function.

“This is a monotonically decreasing function.”

For n=4, F(S^2×S^2)=4π^2≈39.4784.

“F (S 2 × S 2 ) = 4π 2 ≈ 39.4784 (5.5)”

Inserting the result into (4.7), the action is expressed entirely in terms of the energy E, the dimensions {n_1,...,n_m}, and the number A of the sphere which induces the transition to a classical universe.

“Inserting the result into (4.7),we find the action expressed entirely in terms of the energy E, the dimensions {n 1 , . . ., n m } (characterizing the topology of M) and the number A of the sphere which induces the transition to a classical universe.”

For E≫m_P, I_E(K_4) > I_E(K_n) for all n≥5.

“For such a large value of E one has, clearly, I E (K 4 ) > I E (K n ) for all n ≥ 5.”

For any given configuration with n=3, there is a one-to-one correspondence between the cosmological constant and the nucleation energy.

“For any given configuration (with n = 3) this provides a one-to-one correspondence between the cosmological constant and the nucleation energy.”

The transition energies form a decreasing sequence.

“Thus we encounter a decreasing sequence of energy scales.”

As a consequence, E = (1/(8πG_n)) (v_{n_A-1}/v_{n_A}) a_A V Λ = - (v_{n_A-1}(n-2))/(2 v_{n_A} a_A) I.

“As a consequence, E = 1 8πG n v n A -1 v n A a A V Λ = - v n A -1 (n -2) 2 v n A a A I.(4.11)”

Decreasing E, the action of K_4 becomes equal to that of some higher K_n at a threshold E=E_4.

“Decreasing E, things can change only if at some threshold E = E 4 , the action of K 4 becomes equal to the action of some higher K n .”

For single-sphere configurations with large dimension, F(S_n) = 2^{3/2} π exp(n/2 - 1/4)(1+O(1/n)).

“F (S n ) = 2 3/2 π exp n 2 - 1 4 1 + O 1 n . (5.27)”

For n=5, F(S^5)=2048/27≈75.8519.

“F ( S 5 ) = 2048 27 ≈ 75.8519 (5.6)”

For n=5, F(S^2×S^3)=64.

“F (S 2 × S 3 ) = 64 . (5.8)”

Using Stirling's formula, F(K_n) = 2^{(n+2)/2} π exp(√n/(2β)) (1+O(1/√n)).

“F (K n ) = 2 (n+2)/2 π exp √ n 2β 1 + O 1 √ n . (5.26)”

The role of E being large at nucleation is formally related to the smallness of Λ in the standard wormhole theory.

“The role of E being large at nucleation is formally related to the smallness of Λ in the standard wormhole theory.”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

The according nucleation probability is P ∼ exp(-I).

“The according nucleation probability is P ∼ exp(-I).”

The configurations α(E) for small E have large n and represent the genuine multiple-dimensional quantum state of the early universe.

“In contrast, the configurations α(E) for small E have large n and represent the genuine multiple-dimensional quantum state of the early universe.”

The Euclidean action evaluated over the tunneling part is I/2, where I is the total action of the instanton.

“The Euclidean action evaluated over the tunneling part is I/2, where I is the total action of the instanton (the boundary term in (2.1) being obsolet just because the extrinsic curvature on Σ vanishes).”

In a model including wormholes, Λ should rapidly fall to zero once the size of the universe exceeds the wormhole scale.

“whereas in a model including wormholes it should rapidly fall to zero, once the size of the universe exceeds the wormhole scale [5]”

• cites [5] sourceNot checked — no full text of the source

As a consequence of the procedure, Λ effectively becomes dependent not only on the dimension but on the particular configuration itself.

“As a consequence of the procedure we suggest, Λ effectively becomes dependent not only on the dimension (as was anticipated before [7]) but on the particular configuration itself.”

• cites [7] sourceNot checked — no full text of the source

The configurations for large E will turn out to be four-dimensional.

“Moreover, the configurations for large E will turn out to be fourdimensional.”

E is precisely compensated by an amount of energy -E contained in the gravitational field.

“Hence, E is precisely compensated by an amount of energy -E contained in the gravitational field.”

The mechanism that drives the universe classical can be looked at as a process with energy balance 0 → E -E.

“Thus one can look at the mechanism that drives the universe classical as a process with energy balance 0 → E -E.”

Our results certainly become unreliable as E gets very small.

“Hence, our results certainly become unreliable as E gets very small.”

Minimization of the action is one basic principle to characterize the mechanism of emergence of classical time.

“Minimization of the action is certainly one basic principle to characterize this mechanism.”

Any such configuration describes a possible nucleation of the universe, where Σ and its data (h, φ) provide the initial conditions for the subsequent (classical) time evolution.

“Any such configuration describes a possible nucleation of the universe, where Σ and its data (h, φ) provide the initial conditions for the subsequent (classical) time evolution [9].”

• cites [9] sourceNot checked — no full text of the source

The configuration K^n with hypersurface Σ=S^{n-1} at t=0, t<0 half S^n, t>0 half H^n represents the quantum tunneling process occurring in the n=4 FRW minisuperspace model.

“This configuration represents the quantum tunneling process occuring in the n = 4 FRW minisuperspace model.”

Nucleation can be expected at large energies.

“This indicates that nucleation can be expected at large energies”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

Naively replacing minimization by maximization of I in our framework would obtain arbitrarily high nucleation dimensions to be favoured.

“Naively replacing the minimization by maximization of I in our framework, one would obtain arbitrarily high nucleation dimensions to be favoured.”

The appearance of arbitrarily large universes causes the need for a cutoff.

“It is in particular the appearance of arbitrarily large universes that causes the need for a cutoff [5]”

• cites [5] sourceNot checked — no full text of the source

From the point of view of dimensionality, the proposal reproduces a reasonable behaviour of the early universe.

“from the point of view of dimensionality, our proposal reproduces a reasonable behaviour of the early universe.”

• AssumptionThe authors state this as an assumption: Λ can be treated as an independent variable when the action is minimized.
• AssumptionThe authors state this as an assumption: The minimization of the Euclidean action at constant E is a valid alternative procedure.
• AssumptionRests on an assumption: The simplified model neglecting all matter contributions except the cosmological constant and admitting arbitrary products of spheres is adequate for the questions considered.

The universe becomes classical at large E.

“indicates that the unicerse becomes classical at large E.”

The mechanism that causes the universe to become classical at large scales might be related closer to the nucleation energy than to Λ.

“the mechanism that causes the universe to become classical (at large scales) might be related closer to the nucleation energy than to Λ.”

Nucleation, the emergence of a superposition of semiclassical universes, is probably related to decoherence, i.e. the actual selection of one universe that is classical at least with respect to large-scale variables such as dimension and topology.

“Nucleation (i.e. the emergence of a superposition of semiclassical universes) is probably related to decoherence [10]- [11] (i.e. the actual selection of one universe that is classical at least with respect to the large-scale variables such as dimension and topology).”

• cites [10] Gell-Mann M, 1990Not matched

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

• cites [11] Kobayashi SNot matched

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

The set of all M is not identical to the set of all (M,Σ).

“the set of all M is not identical to the set of all (M, Σ).”

The total energy of a closed universe is zero.

“However, in a sense, the total energy of a closed universe is zero.”

A real tunneling configuration is provided by an instanton (M, g, Φ), together with a symmetric hypersurface Σ.

“To summarize, a real tunneling configuration is provided by an instanton (M, g, Φ), together with a symmetric hypersurface Σ.”

• cites [9] sourceNot checked — no full text of the source

The Euclidean action may be expressed in terms of the variables (Λ,γ) or (E,α).

“The Euclidean action may thus likewise be expressed in terms of the variables (Λ, γ) or (E, α).”

In this sense, the minimum dimensionality of space-time is predicted to be 4.

“Hence, in this sence, the minimum dimensionality of space-time is predicted to be 4.”

The universe is of Kantowski-Sachs type.

“In other words, the universe is of Kantowski-Sachs type, a result that might be not very appealing (although consistent with observations).”

If n is fixed in addition to E, one can distinguish between different topologies.

“However, if n is fixed in addition, we can at least distinguish between different topologies.”

Any of our configurations M carries hypersurfaces Σ with zero extrinsic curvature (symmetric hypersurfaces), at which the universe can nucleate.

“Any of our configurations M carries hypersurfaces Σ with zero extrinsic curvature (i.e.symmetric hypersurfaces), at which the universe can nucleate.”

As a consequence of the proposal, small dimensions are favoured over large ones when the universe nucleates.

“We will confirm in Section 7 that -as a consequence of our proposal -the small dimensions are favoured over the large ones when the universe nucleates.”

If a wormhole-based model applies for energies E substantially larger than the wormhole scale, the predicted spatial slices are S^1 × S^2.

“In the case such a version of a wormhole-based model applies for energies E substantially larger than the wormhole scale, the prediction for the spatial slices is S 1 × S 2 .”

• AssumptionThe authors state this as an assumption: The non-gravitational degrees of freedom do not give rise to vacuum energies other than those represented by Λ.

In terms of classical signature change, one S^{n_A} induces the emergence of a Lorentzian time coordinate and a subsequent classical evolution of the corresponding spacelike slices, which are topologically products of S^{n_A-1} with the remaining unaffected spheres.

“In terms of classical signature change, one S n A induces the emergence of a Lorentzian time coordinate and a subsequent classical evolution of the corresponding spacelike slices (which are topologically products of S n A -1 with the remaining unaffected spheres).”

Since our configurations M are products of spheres, the possible hypersurfaces Σ are provided by the equators of the factor-spheres.

“Since our configurations M are products of spheres, the possible hypersurfaces Σ are provided by the equators of the factor-spheres.”

If decoherence details or a non-trivial matter sector induce nucleation around a configuration other than K_4, a modified action I ∼ -E^{(n-2)/(n-3)} ∼ -Λ^{-(n-2)/2} takes over the role of (7.3).

“If the details of the decoherence mechanism or some non-trivial matter sector induce nucleation around some other configuration than K 4 , a modified action I ∼ -E (n-2)/(n-3) ∼ -Λ -(n-2)/2 takes over the role of (7.3).”

• AssumptionThe authors state this as an assumption: κ_n = 1 for all n.

The most probable energy the actual classical universe will start with is E=∞.

“However, the answer will be E = ∞”

Since I(K_4) differs from I(S_4) only by a constant prefactor, the familiar δ(Λ)-peak is produced.

“Since it differs from I(S 4 ) only by a constant prefactor, the familiar δ(Λ)-peak is produced.”

If E plays the role of a quantity whose value is generic and only dynamically governed by a probability law, it appears unlikely that the n=3 case will contribute to the fluctuations between or succession of our various configurations M.

“Hence, if E plays the role of a quantity whose value is generic and only dynamically governed by a probability law, it appears unlikely that the n = 3 case will contribute to the fluctuations between (or succession of) our various configurations M.”

There is a clear tendency for F to be minimized by products of several S^2 with a higher dimensional S^p.

“Hence, there is a clear tendency for F to be minimized by products of several S 2 with a higher dimensional S p .”

The most probable configuration is the one minimizing F = (v_{n_A-1}/v_{n_A})(n_A-1)^{1/2} (n-2) ∏ v_{n_B}(n_B-1)^{n_B/2}.

“The most probable configuration is the one minimizing F = v n A -1 v n A (n A -1) 1/2 n-2 m B=1 v n B (n B -1) n B /2 . (5.1)”