Stress-energy tensor of a scalar field on a product spacetime with a time-dependent compact dimension

Atomic clock experiments constrain the current variation of the fine structure constant to |αdot/α0| < 10^-17 yr^-1.

“Atomic clock experiments constrain the current variation of the fine structure constant, | α/α 0 | < 10 -17 yr -1 [13]”

• cites [13] T. Rosenband et al., Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place, Science 319 (2008) 1154622 [INSPIRE]. sourceDisputed

  The Rosenband et al. passage in the indexed copy (Additional information; page not provided) reports a preliminary estimate of α̇/α = (−1.6 ± 2.3) × 10^-17 yr^-1. Its central value and uncertainty do not establish the claimed upper bound of |α̇/α₀| < 10^-17 yr^-1. (LLM, conf=0.88)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured

ALMA measurements of absorption spectroscopy toward high-redshift quasars bound the variation Δα = α(z=0.89)-α0 to |Δα/α0| < 0.55 p.p.m.

“ALMA measurements of absorption spectroscopy toward high-redshift quasars bound the variation ∆α = α(z = 0.89)-α 0 to |∆α/α 0 | < 0.55p.p.m [14] [15,16].”

• cites [14] S. Muller et al., A sub-ppm upper limit on the cosmological variations of the fine structure constant α, Astron. Astrophys. 706 (2026) A365 [arXiv:2512.14441] [INSPIRE]. sourceSupported
• cites [15] J.-P. Uzan, Fundamental constants: from measurement to the universe, a window on gravitation and cosmology, Living Rev. Rel. 28 (2025) 6 [arXiv:2410.07281] [INSPIRE]. sourceCould not verify

  Section 5.5, “Quasar absorption spectra” (page unavailable), explains that quasar absorption lines can probe variation in fundamental constants. Searches for ALMA, z = 0.89, and the 0.55-p.p.m. limit found no passage reporting the claimed measurement; our copy is truncated, so the decisive text may be missing. (LLM, conf=0.70)

  Experimental — accuracy not yet measured
• cites [16] C.J.A.P. Martins, The status of varying constants: a review of the physics, searches and implications, arXiv:1709.02923 [https://doi.org/10.1088/1361-6633/aa860e] [INSPIRE]. sourceCould not verify

  Section 3.2 discusses radio absorption measurements of combinations of constants and mentions ALMA in connection with potential measurements of μ, not a bound on α. The table reproduced under Conclusions lists measurements at z = 0.886 for μ. Searches for ALMA, z ≈ 0.89, and 0.55 ppm found no passage establishing the claimed |Δα/α₀| bound. (LLM, conf=0.74)

  Experimental — accuracy not yet measured

Almost all studies of Casimir energy in Kaluza-Klein and string-theoretic contexts assume the size of extra dimensions is time-independent, primarily to identify static stable vacua.

“Almost all of these studies assume that the size of the extra dimensions is time-independent, primarily because their objective is to identify static stable vacua.”

• cites [3] T. Appelquist and A. Chodos, Quantum effects in Kaluza-Klein theories, Phys. Rev. Lett. 50 (1983) 141 [INSPIRE]. sourceNot checked — no full text of the source
• cites [10] G.B. De Luca, E. Silverstein and G. Torroba, Hyperbolic compactification of M-theory and de Sitter quantum gravity, SciPost Phys. 12 (2022) 083 [arXiv:2104.13380] [INSPIRE]. sourceCould not verify

  Sections 2 and 4 describe this paper’s use of Casimir energy to stabilize an internal volume, but they do not assess how often other Kaluza–Klein or string-theoretic studies assume time-independent extra dimensions or why they do so. Searches for static vacua and time-dependent extra dimensions found no passage addressing that broader claim. The indexed passages provide no page numbers. (LLM, conf=0.72)

  Experimental — accuracy not yet measured
• cites [12] G. Dall’Agata and F. Zwirner, Supersymmetry-breaking compactifications on Riemann-flat manifolds, JHEP 12 (2025) 131 [arXiv:2507.02339] [INSPIRE]. sourceCould not verify

  The Introduction discusses compactification models and the lack of a viable stabilization mechanism. The passage on the one-loop potential discusses corrections to a classical Minkowski vacuum, but does not say that almost all Casimir-energy studies hold the size of extra dimensions time-independent or give static stable vacua as their reason. Searches for those assumptions and that motivation found no supporting passage. (LLM, conf=0.74)

  Experimental — accuracy not yet measured

In cosmology, only one reference has attempted to compute time-dependent corrections to the Casimir energy when gravity matters.

“In cosmology, although quite a bit of work has been devoted to compactification dynamics (for examples, see [18][19][20][21][22][23][24][25][26][27]), we have found only one reference [28] that has attempted to compute the time dependent corrections when gravity matters.”

• cites [28] A.R. Kamal and R. Mahanta, Time-dependent flux backgrounds in type IIB supergravity, JHEP 06 (2026) 188 [arXiv:2512.19793] [INSPIRE]. sourceCould not verify

  The passages from the References section concern time-dependent compactification and supergravity, but none addresses Casimir energy. Searches for “Casimir,” “time-dependent corrections,” and gravitational effects found no passage saying this paper attempted that calculation or was the only reference to do so. (LLM, conf=0.70)

  Experimental — accuracy not yet measured

In the massless and conformally coupled limit, when the FLRW and compact-dimension scale factors coincide, the higher-order time-dependent corrections match known results for (1 + d)-dimensional FLRW spacetime.

“Furthermore, in this limit the higher-order time-dependent corrections, when the FLRW and compactdimension scale factors coincide, match known results for (1 + d)-dimensional FLRW spacetime.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

In the massless and conformally coupled limit, the leading-order time-dependent results are consistent with known time-independent Casimir contributions.

“In the massless and conformally coupled limit, the leading order time-dependent results are consistent with known time-independent Casimir contributions.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

The modified adiabatic regularization prescription yields analytic expressions for the regularized stress-energy tensor for d = 3 and d = 4.

“We modify the standard adiabatic regularization prescription to obtain analytic expressions for both d = 3 and d = 4.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

The divergent sum ∞ n=1 n^-1 cannot be zeta regularized because it corresponds to evaluating the Riemann zeta function ζ(z) at z = 1, a simple pole.

“Unfortunately, we cannot zeta regularize the divergent sum ∞ n=1 n -1 as it corresponds to evaluating the Riemann zeta function ζ(z) at z = 1, a simple pole.”

Similar to the d=3 case, the 0th order Casimir terms match known time-independent results.

“Similar to the d = 3 case, the 0-th order Casimir terms match known time-independent results [3][4][5][6][7][8].”

• cites [3] T. Appelquist and A. Chodos, Quantum effects in Kaluza-Klein theories, Phys. Rev. Lett. 50 (1983) 141 [INSPIRE]. sourceNot checked — no full text of the source
• cites [4] T. Appelquist and A. Chodos, The quantum dynamics of Kaluza-Klein theories, Phys. Rev. D 28 (1983) 772 [INSPIRE]. sourceNot checked — no full text of the source
• cites [5] T. Appelquist, A. Chodos and E. Myers, Quantum instability of dimensional reduction, Phys. Lett. B 127 (1983) 51 [INSPIRE]. sourceNot checked — no full text of the source
• cites [6] S. Weinberg, Charges from extra dimensions, Phys. Lett. B 125 (1983) 265 [INSPIRE]. sourceNot checked — no full text of the source
• cites [7] P. Candelas and S. Weinberg, Calculation of gauge couplings and compact circumferences from selfconsistent dimensional reduction, Nucl. Phys. B 237 (1984) 397 [INSPIRE]. sourceNot checked — no full text of the source
• cites [8] R. Kantowski and K.A. Milton, Scalar Casimir energies in M4 × SN for even N, Phys. Rev. D 35 (1987) 549 [INSPIRE]. sourceNot checked — no full text of the source

The 2nd and 4th order contributions vanish in the limit a(t)=b(t).

“The 2-nd and 4-th order contributions vanish in this limit.”

In the limit where a(t) = b(t), the stress-energy tensors at each order match those obtained in the conformally flat scenario using the standard adiabatic regularization prescription (Appendix D.1).

“In the limit where a(t) = b(t), the stress-energy tensors at each order match those obtained in the conformally flat scenario using the standard adiabatic regularization prescription (Appendix D.1).”

After setting ξ = -3/16, the resummation procedure (taking m → 0 prior to carrying out the mode summation over n) again leaves the 0th and 2nd adiabatic order contributions to the stress-energy tensor invariant and finite.

“After setting ξ = -3 16 , we use the resummation procedure (taking m → 0 prior to carrying out the mode summation over n) again.As before, the resummation leaves the 0-th and 2-nd adiabatic order contributions to the stress-energy tensor invariant and finite.”

The Bessel function K0(2nπr0 mb) can be expanded in the small mass limit as K0 = -[γ+log(b)] - log(nπr0 m) + O(m).

“The Bessel function can be expanded in the small mass limit as K 0 (2nπr 0 mb) = -[γ + log(b)] -log(nπr 0 m) + O(m 1 )”

The expression for P_C(4+ε,κ,λ,δ) can be separated into an r0-independent term and r0-dependent terms involving summations over Bessel functions.

“The expression for P C (4 + ϵ, κ, λ, δ) can be separated in to a term that is independent of r 0 , as well as other r 0 -dependent terms that involve summations over Bessel functions.”

The 0th order expressions match known results for the time-independent case.

“The 0-th order expressions, which may be interpreted as the Casimir energy/pressures, match known results for the time-independent case [34,38].”

• cites [34] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge, U.K. (1982) [https://doi.org/10.1017/CBO9780511622632] [INSPIRE]. sourceNot checked — no full text of the source
• cites [38] P.R. Anderson and L. Parker, Adiabatic regularization in closed Robertson-Walker universes, Phys. Rev. D 36 (1987) 2963 [INSPIRE]. sourceNot checked — no full text of the source

The renormalization scheme dependent terms vanish in the limit a(t)=b(t).

“Notice that the renormalization scheme dependent terms vanish in this limit.”

The divergent summand S_div can be expressed as a linear combination S_div = c1 (1)H + c2 (2)H.

“This latter divergent summand may be expressed as a linear combination S div αβ = c 1 (1) H αβ + c 2 (2) H αβ”

In the conformally flat limit with a massless and conformally coupled field, the expectation of the energy-momentum obtained using the standard adiabatic prescription agrees with that derived using the approximate prescription.

“In the conformally flat limit with a massless and conformally coupled field, the exact mode solutions are known; the expectation of the energy-momentum obtained using the standard adiabatic prescription agrees with those derived using the approximate prescription in this scenario.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

The poles of the first gamma function in the numerator of (4.15) and the one in the prefactor of (4.18) cancel in the ε→0 limit.

“As for the other r 0 -dependent pieces involving the Bessel functions, the poles of the first gamma function in the numerator of (4.15) and the one in the prefactor of (4.18) cancel in the ϵ → 0 limit.”

The scheme dependent terms cancel each other in the trace.

“It is clear that the scheme dependent terms cancel each other in the trace.”

For d=3, it is not possible to explicitly carry out the sum in (4.21) prior to taking m→0.

“Furthermore for d = 3, it is not possible to explicitly carry out the sum in (4.21) prior to taking m → 0.”

In the massless and conformally coupled limit, the computed expressions, when the FLRW and compact dimension scale factors are equal, reproduce known results for (1+d)-dimensional FLRW spacetime.

“Additionally, in the massless and conformally coupled limit, the computed expressions, when the FLRW and compact dimension scale factors are equal, reproduce known results for (1 + d)-dimensional FLRW spacetime.”

Equation (5.17) reproduces the well known trace anomaly in (1+3)-dimensional FLRW spacetime upon taking b(t) → a(t).

“One can check that (5.17) reproduces the well known trace anomaly in (1 + 3)-dimensional FLRW spacetime [32][33][34] upon taking b(t) → a(t).”

• cites [32] N.D. Birrell, The application of adiabatic regularization to calculations of cosmological interest, Proc. Roy. Soc. Lond. A 361 (1978) 513. sourceNot checked — no full text of the source
• cites [33] R.M. Wald, Trace anomaly of a conformally invariant quantum field in curved space-time, Phys. Rev. D 17 (1978) 1477 [INSPIRE]. sourceNot checked — no full text of the source
• cites [34] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge, U.K. (1982) [https://doi.org/10.1017/CBO9780511622632] [INSPIRE]. sourceNot checked — no full text of the source

Carrying out the mode summation over the total summand that is constant in n requires zeta regularization, achieved by multiplying the summand with ζ(0) = -1/2.

“Carrying out the mode summation over this total summand that is constant in n requires zeta regularization, which is achieved by multiplying the summand with ζ(0) = - 1 2 .”

The divergence arising in the massless limit at 4th order cannot be absorbed by the addition of suitable higher curvature counterterms to the gravitational action, in contrast to the d=3 case.

“Neither can we absorb this divergence that arises in the massless limit by the addition of suitable higher curvature counterterms to the gravitational action, in contrast to what we saw for the d = 3 case.”

The vacuum expectation value of the stress-energy tensor of a scalar field was computed in a spacetime that is the product of an FLRW background and a compact dimension (M^{1,d-1} × S^1), where the size of the latter is allowed to vary with time.

“In this work, we computed the vacuum expectation value of the stress-energy tensor of a scalar field in a spacetime that is the product of an FLRW background and a compact dimension (M 1, d-1 × S 1 ), where the size of the latter is allowed to vary with time.”

In the limit where a(t) = b(t), the stress-energy expressions at each order match those obtained in the conformally flat scenario using the standard adiabatic regularization prescription (appendix D.2).

“Finally, in the limit where a(t) = b(t), the stress-energy expressions at each order match those obtained in the conformally flat scenario using the standard adiabatic regularization prescription (appendix D.2).”

At 4th adiabatic order, taking the massless limit of the summand first results in an expression that is proportional to n^-1.

“However, at 4-th adiabatic order, taking the massless limit of the summand first results in an expression that is proportional to n -1 .”

Given the experimental bounds, time-dependent corrections to the Casimir energy, if important, are most likely relevant only in the early universe.

“Given the experimental bounds, such effects, if important, are most likely to be relevant only in the early universe.”

Extra dimensions in string theory are generically unstable, with a natural tendency to expand and decompactify.

“These extra dimensions are generically unstable: a natural tendency in string theory is for them to expand and decompactify [1].”

• cites [1] M. Dine and N. Seiberg, Is the superstring weakly coupled?, Phys. Lett. B 162 (1985) 299 [INSPIRE]. sourceNot checked — no full text of the source

If δ is not large enough, naively integrating the adiabatic integrand over momenta results in divergences.

“If δ is not large enough, then naively integrating (4.7) over the momenta results in divergences.”

The piece of S_div proportional to log(nπr0m) may be discarded because it can be interpreted as renormalizing the coefficients of higher curvature terms added to the gravitational action.

“Thus, the piece of S div αβ that is proportional to log(nπr 0 m) may be discarded since it can be interpreted as renormalizing the coefficients of higher curvature terms 4 that have been added to the gravitational action [33]”

• cites [33] R.M. Wald, Trace anomaly of a conformally invariant quantum field in curved space-time, Phys. Rev. D 17 (1978) 1477 [INSPIRE]. sourceNot checked — no full text of the source

The 4th order expressions for the stress-energy tensor are left unregularized.

“We leave these 4-th order expressions for the stress-energy unregularized 5 .”

For d=3 and d=4, the regularized energy-momentum tensor is finite.

“In both instances, the regularized energy-momentum tensor is finite and its covariant divergence vanishes.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

The modified prescription reproduces standard FLRW results when the scale factors are identical and the field is massless and conformally coupled.

“Furthermore, it reproduces standard FLRW results when the scale factors are identical and the field is massless and conformally coupled.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

For d=3 and d=4, the covariant divergence of the regularized energy-momentum tensor vanishes.

“In both instances, the regularized energy-momentum tensor is finite and its covariant divergence vanishes.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

The stress-energy tensor satisfies the conservation equation (4.23) at each adiabatic order (even for the 4th order mode by mode).

“The stress-energy tensor satisfies the conservation equation (4.23) at each adiabatic order (even for the 4-th order mode by mode).”

The stress-energy tensor is traceless at each order, ⟨T^α_α⟩_reg = 0.

“It is also traceless at each order, ⟨T α α ⟩ reg = 0 (5.27)”

Zeta regularization yields an analytic continuation of the Epstein-Hurwitz zeta function.

“Zeta regularization yields the following analytic continuation [9]”

• cites [9] E. Elizalde and A. Romeo, Expressions for the zeta function regularized Casimir energy, J. Math. Phys. 30 (1989) 1133 [Erratum ibid. 31 (1990) 771] [INSPIRE]. sourceNot checked — no full text of the source

The limitation of adiabatic regularization arises because it requires exact solutions for the field modes, which are not known for a generic field mass and Ricci coupling, or a general M^{1,d-1}×S^1 compactification.

“This limitation arises because the adiabatic regularization method requires exact solutions for the field modes, which are not known for a generic field mass and Ricci coupling, or a general M 1, d-1 × S 1 compactification.”

The agreement with known results suggests that the modified approximate prescription is likely valid.

“This agreement with known results suggests that our modified approximate prescription is likely valid.”

Corrections are only expected to influence the time evolution when the factor 1/(r0 b M5) approaches order one.

“Corrections are only expected to influence the time evolution when this factor approaches order one.”

To obtain a regularized expression for ⟨Tαβ⟩, the paper proposes a modification of the standard adiabatic regularization prescription that uses a WKB approximation for the field modes of the compact space rather than the exact solutions.

“To obtain a regularized expression for ⟨T αβ ⟩, we propose a modification of the standard adiabatic regularization prescription that uses a WKB approximation for the field modes of the compact space rather than the exact solutions.”

The adiabatic condition ensures that the constraint on particle number is satisfied.

“This condition ensures that the above constraint on particle number is satisfied.”

In 1+4 dimensions, the first adiabatic corrections to the Casimir energy/pressures will be significant during early times, specifically when r0 b M5 ≲ 1.

“However, a comparison between the first adiabatic corrections and the classical equations of motion reveals that these corrections, in 1 + 4 dimensions, will be significant during early times, specifically when the product r 0 bM 5 ≲ 1”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

The 0th order expressions may be interpreted as the Casimir energy/pressures.

“The 0-th order expressions, which may be interpreted as the Casimir energy/pressures, match known results for the time-independent case [34,38].”

In d+1 dimensions, some terms in the adiabatic expansion of ⟨Tαβ⟩ diverge up to (d+1)-th order.

“Since we are considering spacetimes with d+1 dimensions, some terms in the adiabatic expansion of ⟨T αβ ⟩ diverge up to (d + 1)-th order.”

The adiabatic regularization procedure introduced by Parker and Fulling has only been shown equivalent to point-splitting when the scalar field is massless and conformally coupled and the spacetime M^{1,d-1}×S^1 is conformally flat (scale factors equal).

“The adiabatic regularization procedure introduced by Parker and Fulling [29,30], though well suited to cosmology, has only been shown to be equivalent to point-splitting when the scalar field is massless and conformally coupled, and the spacetime Mfoot_0, d-1 × S 1 is conformally flat i.e. when the scale factors of the two product spaces are equal [31].”

• cites [29] K.-I. Maeda, Effect of particle creation on Kaluza-Klein cosmologies, Phys. Rev. D 30 (1984) 2482 [INSPIRE]. sourceNot checked — no full text of the source
• cites [30] L. Parker and S.A. Fulling, Adiabatic regularization of the energy momentum tensor of a quantized field in homogeneous spaces, Phys. Rev. D 9 (1974) 341 [INSPIRE]. sourceNot checked — no full text of the source
• cites [31] S.A. Fulling, L. Parker and B.L. Hu, Conformal energy-momentum tensor in curved spacetime: adiabatic regularization and renormalization, Phys. Rev. D 10 (1974) 3905 [INSPIRE]. sourceNot checked — no full text of the source

The conservation equation ∇α⟨Tαβ⟩_reg = 0 is satisfied at each adiabatic order.

“Finally, one can check that the conservation equation J α = ∇ α ⟨T αβ ⟩ reg = 0 (4.23) is satisfied at each adiabatic order.”

• AssumptionThe authors state this as an assumption: The spacetime metric has the form ds^2 = dt^2 - a(t)^2 Σ_{j=1}^{d-1} dx_j^2 - b(t)^2 dx_d^2, with x_d compact and x_i noncompact.

The formal expression for the vacuum expectation value of the stress-energy tensor is filled with UV divergences.

“The formal expression (2.9) is filled with UV divergences.”

The subtracted quantities in adiabatic regularization should depend only on the local behavior, not the global topology of the spacetime.

“The subtracted quantities in adiabatic regularization should depend only on the local behavior, not the global topology of the spacetime.”

The conservation equation (4.23) is satisfied at each adiabatic order.

“The conservation equation (4.23) is satisfied at each adiabatic order.”

The tracelessness is expected for a massless, conformally coupled field in a spacetime with an odd number of dimensions.

“as is expected for a massless, conformally coupled field in a spacetime with an odd number of dimensions [33,34].”

• cites [33] R.M. Wald, Trace anomaly of a conformally invariant quantum field in curved space-time, Phys. Rev. D 17 (1978) 1477 [INSPIRE]. sourceNot checked — no full text of the source
• cites [34] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge, U.K. (1982) [https://doi.org/10.1017/CBO9780511622632] [INSPIRE]. sourceNot checked — no full text of the source

While the adiabatic condition does not necessarily yield a unique set of modes (and hence vacuum), the large momentum behavior of physical observables is the same regardless of which specific set is chosen.

“While the condition does not necessarily yield a unique set of modes (and hence vacuum), the large momentum behavior of physical observables are the same, regardless of which specific set is chosen.”