REVIEW 5 minor 3 cited by
The Creation of Particles in an Expanding Universe
T0 review · 0 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read Particle creation in an expanding universe follows from unmodified quantum field theory and general relativity, as Leonard Parker's 1966 thesis establishes.
desk verdict Parker's 1966 thesis remains the clearest statement of the core mechanism of cosmological particle creation; the quantitative bounds are shakier but the central result stands. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the time-dependent Bogoliubov transformation: for a scalar field, the Fourier coefficient $a(\vec{k},t) = \alpha(k,t)^* A(\vec{k}) + \beta(k,t) A^\dagger(-\vec{k})$, with $|\beta(k,t)|^2$ giving the created-particle number per mode. The argument derives $\alpha$ and $\beta$ by comparing the exact field equation with the adiabatic (Liouville) approximation through an integral equation, yielding a convergent series and an upper bound $|\beta| \leq \sinh\int |S|\,dt$. To represent the particle number a static-like apparatus would measure during an expansion, the thesis removes the rapid oscillations of $a(\vec{k},t)$ using the second adiabatic approximation, producing operators $a^c_{\vec{k}}$ that are constant during a measurement and unique to order $H^3$. For massless non-zero-spin fields, the key is conformal invariance of the Penrose spinor equations, which keeps positive- and negative-frequency parts distinct.
What would settle it
Measure or compute the present cosmological creation rate of photons in an isotropically expanding Euclidean universe; the thesis predicts exactly zero for massless spin-1 fields, so any such creation attributable to expansion would falsify the no-creation theorem. For the quantitative bounds, an observed particle creation rate per unit volume exceeding the proton bound $10^{-64}\,\mathrm{gm\,cm^{-3}\,s^{-1}}$ and attributable to cosmic expansion would falsify the upper-bound claim.
Extended reading notes
Core claim
The central claim is that particle creation in an expanding universe follows from the unmodified general-relativistic Klein-Gordon and Dirac equations. The expansion makes the creation and annihilation operators evolve into superpositions of one another: an annihilation operator at a later time equals a linear combination of an annihilation and a creation operator at an earlier time, so the expectation value of the particle number in an initially empty state is positive. Quantitatively, the thesis places upper bounds on the absolute value of the present creation rate per unit volume: $10^{-105}\,\mathrm{gm\,cm^{-3}\,s^{-1}}$ for $\pi$-mesons, $10^{-69}$ for electrons, and $10^{-64}$ for protons, depending only on Hubble's constant, the present matter density, and the particle mass. The thesis also proves that massless fields of non-zero spin obey conformally invariant equations, so their positive- and negative-frequency parts never mix during a Euclidean expanding universe and no particles are created.
Load-bearing premise
The numerical upper bounds rest on identifying the operator $N_{\vec{k}}$ built by removing rapid oscillations from the field's Fourier coefficients with the particle number a static-like detector would count during a finite measurement interval; the thesis supports this identification only to order $H^3$ via the second adiabatic approximation, and it itself states that the concept of particle number during expansion is necessarily fuzzy.
Editorial extensions
If this is right
- The present-day creation rates are so small that direct detection is hopeless: less than one proton per litre of volume every $10^{30}$ years, and less than one $\pi$-meson per second in the entire observable universe.
- Creation occurs in pairs with zero net momentum and equal amounts of matter and antimatter, so the mechanism cannot explain the observed matter-antimatter asymmetry.
- In a radiation-dominated Friedmann universe ($R(t)\propto t^{1/2}$), massless minimally coupled mesons are created at exactly zero rate; in a matter-dominated universe ($R(t)\propto t^{2/3}$), infinitely massive mesons are created at exactly zero rate.
- Massless photons, neutrinos, and gravitons are not created by the expansion in an isotropically expanding Euclidean universe, provided quantization of higher-spin massless fields does not introduce complications.
- For slow expansions satisfying the Littlewood conditions, the created number per mode vanishes faster than any power of the small parameter (for example, like $e^{-1/\epsilon^2}$), so perturbative expansions in Hubble constant over mass fail.
Reading between the lines
- The same frequency-mixing mechanism later underpinned treatments of particle creation by black holes and by other time-dependent gravitational fields, suggesting the thesis's Bogoliubov method is a general template for gravitational particle production.
- The no-creation theorem for conformally invariant massless fields implies that in an exactly isotropic expansion, photons are not produced by this mechanism; any primordial photon background would need another source, such as quantum fluctuations of the metric itself.
- The thesis's bounds assume an initial vacuum and use the present matter density as a cap; the author notes the order of magnitude is unchanged for isotropic initial matter distributions, so the bounds are robust, but higher-order adiabatic approximations could reduce them further.
- A fully non-perturbative definition of observed particle number, possibly by averaging the oscillating operators over the measurement interval, remains open; the successive adiabatic approximations behave like an asymptotic series.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This manuscript is a retyped open-access edition of Leonard Parker's 1966 Harvard Ph.D. thesis, accompanied by a Foreword that places the work in historical context. The scientific content derives the behavior of quantized scalar and spin-1/2 fields in a Euclidean expanding universe from the covariant Klein-Gordon and Dirac equations. It constructs time-dependent mode operators satisfying canonical (anti)commutation relations at all times, obtains Bogoliubov coefficients α and β with |α|² − |β|² = 1, and derives exact results including pair creation, equality of charged and neutral creation rates, conservation of a momentum-type quantity, and a no-creation theorem for massless non-zero-spin fields via conformal invariance. For massive fields it defines an adiabatically smoothed particle-number operator N̄_k during a measurement and obtains upper bounds on present-day creation rates: 10⁻¹⁰⁵ gm cm⁻³ s⁻¹ for pion-mass scalars and 10⁻⁶⁹ / 10⁻⁶⁴ gm cm⁻³ s⁻¹ for electrons and protons. The central claim is that particle creation follows from unmodified quantum field theory plus general relativity.
Significance. Judged as a historical reissue, this is a seminal and self-contained contribution. The core Bogoliubov derivation is exact and proceeds from first principles: no parameter is fitted except the present matter density used as a variational constraint, and the derivation is fully written out for both bosons and fermions. The no-creation theorem for massless non-zero-spin fields is clearly stated and falsifiable. The manuscript is also valuable as an archival document because it makes a previously difficult-to-access dissertation freely available. The main caveat, explicitly acknowledged in Chapter V, is that the particle-number operator measured during expansion is defined through a second adiabatic approximation and a set of plausibility postulates rather than through a detailed detector model; this affects the numerical upper bounds but is independent of the exact Bogoliubov mixing that establishes particle creation.
minor comments (5)
- [Foreword] Page VI contains the typo "pionering," which should be "pioneering."
- [Chapter I, Introduction] Page 6 contains "anninilation," which should be "annihilation."
- [Chapter II, Section 9] The word "mesosns" appears and should be "mesons."
- [Chapter III, Section 8] "Krönecker" should be spelled "Kronecker."
- [Chapter V, Part A, Section 6] The identification of the operator N̄_k with the output of a static-like apparatus is heuristic rather than derived from a detector model; the thesis itself flags this limitation, but a brief editorial note connecting this identification to modern observer-dependent particle-number discussions would assist contemporary readers.
Circularity Check
No circular derivation: the creation effect follows from exact Bogoliubov mixing of mode operators; the quantitative bounds rest on an approximate detector identification but not on a fitted input or self-citation chain.
full rationale
The thesis is self-contained: it starts from the covariant Klein-Gordon and Dirac equations and the canonical commutation relations, then derives mode operators of the form a(k,t)=alpha* A + beta A^dagger and the identity |alpha|^2-|beta|^2=1 from the field equations. The initial-vacuum expectation value <0|a^dagger a|0>=|beta|^2 is therefore a computed consequence, not an input. The upper bounds in Chapter V use the observed matter density only as a constraint in a variational maximization (eqs. (45)-(51) and (104)-(111)); no parameter is fitted to the creation rate itself. The no-creation result for massless non-zero spin relies on Penrose's conformal invariance theorem, an external mathematical result, and the thesis explicitly refrains from quantizing higher-spin fields (Chapter IV). The only self-citation, L. Parker, Nuovo Cimento 40 (1965) 99, appears in a footnote comparing the first term of a series to an earlier approximation; it does not carry the derivation. The weakest point is the identification of N_k = a^{c dagger}_k a^c_k with the number a static-like apparatus would measure, stated in Chapter V.A.6: 'we feel justified in asserting that N_k corresponds to the particle number in the mode k which would be measured by a static-like apparatus.' This is an approximate physical interpretation, acknowledged by the thesis ('the very concept of the particle number during the expansion must necessarily be somewhat fuzzy'), not a circular reduction: altering the detector model could change the numerical bounds, but it would not erase the exact off-diagonal mixing |beta|^2 that constitutes the central creation prediction.
Assumptions & free parameters
free parameters (1)
- present matter density =
10^-5 cm^-3 (maximum observed number density of matter, used to maximize the bound)
assumptions (5)
- domain assumption The metric of a 3-dimensionally Euclidean expanding universe with line element ds^2 = -dt^2 + R(t)^2 (dx^j)^2 is a valid background for quantized matter fields.
- domain assumption Canonical commutation/anticommutation relations are imposed on the field and conjugate momentum at all times; the Fourier coefficients then satisfy creation/annihilation algebra.
- ad hoc to paper The adiabatic approximation with neglect of terms of order H^3 (second adiabatic approximation) yields the particle number operator for a single measurement.
- domain assumption The present number density of created particles cannot exceed the observed matter density of the universe (10^-5 cm^-3).
- standard math Conformal invariance of massless field equations implies no mixing of positive and negative frequency parts in a Euclidean expanding universe.
Cite this review
Pith. "Pith review of The Creation of Particles in an Expanding Universe." pith.science (2026). https://pith.science/paper/TIR5K2ZC
@misc{pith2026250705372,
author = {Pith},
title = {Pith review of: The Creation of Particles in an Expanding Universe},
year = {2026},
howpublished = {\url{https://pith.science/paper/TIR5K2ZC}},
note = {Machine review of arXiv:2507.05372}
}
read the original abstract
This document is the Ph.D. thesis of Leonard Parker, submitted to Harvard University in 1966. Over the decades, several generations of physicists have been introduced to the concept of particle creation by gravitational fields, a phenomenon that has become a cornerstone in exploring the interplay between gravitation and quantum theory. Yet, the foundational breakthrough that led to the prediction and understanding of this phenomenon remains unfamiliar to many. In the interest of historical accuracy and in recognition of a seminal contribution to physics, the thesis has been retyped and made it freely available as an open-access (arXiv) document. The reissued thesis is accompanied by a Foreword that places the work in its proper historical context. As the team responsible for this new edition, we (Antonio Ferreiro, Jos\'e Navarro-Salas, and Silvia Pla) hope that future generations will continue to draw inspiration from this pioneering text.
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Reference graph
Works this paper leans on
-
[1]
In addition, d dt a⃗k(t) approaches zero only ask−1, when k approaches infinity. This would lead to high energy divergence difficulties in the total creation rate, summed over all modes, ifaˆk(t)†a⃗k(t) were interpreted as the particle number operator
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[2]
We say that a functionG(H) is of orderH n, if and only iflimH→0 H −ℓG(H) is not zero for l ≥ n, and is zero forℓ < n. We write G(H) = O (H n) for "G(H) is of order H n." On the other handA ≈ B means "A is of roughly the same magnitude as B."
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[3]
in the limitH → 0 ), regardless of the relative magnitude of the expansion
An adiabatic approximation, in our case, is one whose cumulative error over long periods of time vanishes in the limit of an infinitely slow expansion of the universe (i.e. in the limitH → 0 ), regardless of the relative magnitude of the expansion
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[4]
These successive approximations are probably only possible because of the rela- tionship between the particle number and an adiabatic invariant which satisfies Littlewood’s theorem (cf. Chapter II)
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[5]
Therefore, we are justified in using the same symbolβc in each case
The nth adiabatic approximation reduces to the(n − 1)th when order H n is ne- glected. Therefore, we are justified in using the same symbolβc in each case. It is the same particle number which is approximated in each case
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[6]
Such terms were neglected in the equation preceding equation (15)
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[7]
Such a term was neglected in showing that (12) is a solution of eq. (10)
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[8]
1 ˙θ (t′) d dt′ n−1 S (t′) ˙θ (t′) [2j, t′]∗ # e−iθ(t′) , (F) or d dt βc = −(−i)n ˙θ(t)
Another way of looking at this heuristically is as follows. Just asβc in (25) is related to β(t) in (22) (or more preciselyζ(t)) by a partial integration,βc in the (n + 1)th adiabatic approximation will be related toβc in the nth adiabatic approximation by one partial integration. Thus, in thenth adiabatic approximation (n ≥ 1) we will have βc = − ∞X j=0 ...
Show all 35 references
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[9]
Apossiblecandidateforsuchanidealizednumberoperatoris( Av a⃗k(t) † Av a⃗k(t) , where Av denotes the time average over the interval∆t of measurement. Since the time involved in an accurate measurement of the particle number (near the mode⃗k ) must be much greater thanω(k, t)−1, ...
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[10]
These considerations are heuristic in nature. We have not rigorously proved that the behaviour cited actually occurs beyond the third adiabatic approximation, or even that the fourth and higher adiabatic approximations can be consistently carried out
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[11]
The proof is in part D
This statement is proved in connection with the corresponding work for the fermion field. The proof is in part D. It was felt better to include such details in the later sections, so as not to obscure the main arguments when they are first presented. 120 Summary In Chapters II...
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[14]
In the expanding universe with the metric ds2 = −dt2 + R(t)2 3X j=1 dxj 2 the fermion field may be written: ψ(⃗ x, t) = 1 (2πR(t))3/2 Z d3p r µ ω(k, t) X a,d a(a,d)(⃗ p, t)u(a,d)(⃗ p, t)eia(⃗ p·⃗ x− R t t0 dt′ω(k,t′)) . (55) The field excitation or quasi-particle creation and ...
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[15]
Thus, terms likeS(p, t)2 = O (H 2) and d dt S(p, t) = 104 O (H 2) are neglected
Since S(p, t) is of order H rather than H 2, our approximation this time consists in neglecting terms of order H 2 or higher during the interval ∆t of a single measurement of the fermion number. Thus, terms likeS(p, t)2 = O (H 2) and d dt S(p, t) = 104 O (H 2) are neglected. T...
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[16]
(68) The matrices β and γ5 are hermitian, and satisfy βγ 5 = −γ5β , β 2 = 1 , γ5 2 = 1 , and (69) σkβγ 5 = βγ 5σk , (k = 1, 2, 3) where σk = iβγ 5γk
From (42) and (65) of Chapter III, we have u(a,d)(−⃗ p, t) = βu(a,−d)(⃗ p, t) u(−a,d)(⃗ p, t) = −dγ5u(a,d)(⃗ p, t) . (68) The matrices β and γ5 are hermitian, and satisfy βγ 5 = −γ5β , β 2 = 1 , γ5 2 = 1 , and (69) σkβγ 5 = βγ 5σk , (k = 1, 2, 3) where σk = iβγ 5γk . As a cons...
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[17]
(79) Equation (78) is an immediate consequence of eqs
For the anti-commutation rules n ac (a,d)(⃗ p), ac (a′,d′) (⃗ p′)† o = δa,a′δd,d′δ(3) (⃗ p− ⃗ p′) (77) to hold to orderH, it follows from eq.(65) (as shown in Chapter III), that we must have to order H: X b D(b) (a)(p)c D(b) (−a)(p)c ∗ = 0 , (78) and X b | D(b) (a) (p)c|2 = 1 ...
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[18]
The particle number operators formed from theac(a, d)(⃗ p) and their adjoints satisfy the requirements corresponding to i), ii), and iii) of section A. Owing to the constancy of theac(a,d)(⃗ p) during the interval∆t of measurement, and to the resemblance of (72) to the field i...
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[19]
Thus we consider the D(a) (b) (p)c = D(a) (b) (p, t)c as functions of time
Now we consider how theD(a) (b) (p)c in two separate measurements are re- lated. Thus we consider the D(a) (b) (p)c = D(a) (b) (p, t)c as functions of time. Their first time-derivatives must be of orderH 2, so that the D(a) (b) (p, t)c may be regarded as time- independent duri...
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[20]
Using (81) and (65) we find that the expectation value of the observable number of fermions at timet in the mode ⃗ p, in the volume(LR(t))3 is ⟨Np(t)⟩ = D(1) (−1)(p, t)c 2
We will use the discrete representation for convenience. Using (81) and (65) we find that the expectation value of the observable number of fermions at timet in the mode ⃗ p, in the volume(LR(t))3 is ⟨Np(t)⟩ = D(1) (−1)(p, t)c 2 . (88) This refers to either one of the two spin...
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[21]
(86), or from (83) and (57)
We may now obtainD(1) (−1)(p, t) either from the series of part C, eq. (86), or from (83) and (57). We use the latter procedure (which is independent of the boundary conditions on R(t) and its derivatives ast → −∞). From (83), we have d dt D(a) (−a)(p, t)c = d dt D(a) (−a)(p, ...
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[22]
We have ¯DtN ≤ L3 π2 Z ∞ 0 dpp2 ¯DtNp
The expectation value of the absolute value of the observable creation rate in the volume(LR(t))3, summed over all modes and both spin quantizations will be denoted by ¯DtN . We have ¯DtN ≤ L3 π2 Z ∞ 0 dpp2 ¯DtNp . (97) As in the spin-zero case, we wish to maximize I = L3 π2 Z...
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[23]
According to (60) S(p, t) = 1 2 µ p˙R(t)/R(t) p2 R(t)2 + µ2
Now we consider the right side of (95). According to (60) S(p, t) = 1 2 µ p˙R(t)/R(t) p2 R(t)2 + µ2 . (101) Then d dt S(p, t) ω(p, t) = µ p 2 d dt ˙R(t)/R(t) p2 R(t)2 + µ2 3/2 ! d dt S(p, t) ω(p, t) = µ p 2R(t) p2 R(t)2 ˙R(t) R(t) 2 + ¨R(t) R(t) − µ2 2 ˙R(t) R(t) 2 − ¨R(t) R(t...
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[24]
Let us multiply the integrals in (98) and (99) byLR(t))−3, and change the variable of integration to x = p µR(t) . Then, using (103), our problem can be restated as follows: Maximize I ′ = 2 π2 Z ∞ 0 dxx2p ⟨Nx⟩H 2µ2 x (x2 + 1)3/2 , (104) under the constraint that J ′ = µ3 π2 Z...
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[25]
R. C. Tolman,Relativity, Thermodynamics, and Cosmology (Oxford, 1934) p. 371
1934
-
[26]
Bender and C
M. Bender and C. Itzykson, Revs. Mod. Phys.38 (1966) 333. vii BI. The Dirac Equation in General Relativity In this appendix we will develop the formalism of V. Bargmann, Berlin Akad. der Wiss. 1932, p. 346, for dealing with spinors in a Riemannian space. We will occasionally u...
1966
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[27]
First let us review the case of special relativity
Notation, and special relativity In this appendix we will use the notation of Bargmann (except that we setc = ℏ = 1). First let us review the case of special relativity. The fundamental tensor is˚gik where ˚g00 = −1 , ˚g11 = ˚g22 = ˚g33 = 1, ˚gik = 0 ( i ̸= k) . (1) The real c...
1957
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[28]
vierbein field
The γ-Matrices, and Covariance in General Relativity In General Relativity theγ-matrices are spacetime dependent 4 × 4 matrices which satisfy the covariant generalization of equation (4): γkγl + γlγk = 2gkl . (9) Note that theγk which satisfy (9) will generally be coordinate-d...
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[29]
(14) In order to form observable numbers, the generalized Pauli adjoint must be introduced
The Need for the Matrixα We have already pointed out that under a general change of the reference system,ψ xi and the γ-matrices transform as follows (with a coordinate dependentS): ψ → ψ′ = S−1ψ (13) γk → γ′k = S−1γlS ∂x′k ∂xl . (14) In order to form observable numbers, the g...
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[30]
(9) at each point has been chosen such that theγ-matrices form a differentiable matrix-field
The Partial Derivatives of theγ-Matrices Let us suppose that the gik are given, and the solutionγk of eq. (9) at each point has been chosen such that theγ-matrices form a differentiable matrix-field. consider two points P and P ′ separated by an arbitrarily small distance, suc...
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[31]
When S is coordinate-dependent then ∂lψ no longer has simple properties underS-transformation
Covariant Derivatives In special relativity,∂lψ transforms into ∂xm ∂x′l S−1∂mψ under a Lorentz transformation, since in that case S is coordinate-independent. When S is coordinate-dependent then ∂lψ no longer has simple properties underS-transformation. Nevertheless, it will ...
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[32]
(9) served as the starting point of the entire development, in much the same way as thegij are fundamental to the mathematics involving only tensors
Vanishing Covariant Derivatives, and the Norm The γi determined by eq. (9) served as the starting point of the entire development, in much the same way as thegij are fundamental to the mathematics involving only tensors. In order that theγi shall determine the properties of sp...
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[33]
Since |ψ|2 is a norm only in spin-space, and is a vector component in coordinate space, we postulate that under parallel transfer ofψ(P ) from P to P ′ the norm should change like the zeroth component of an ordinary 4-vector. That is, the norm ofψ=(P ′) at P ′ should be relate...
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[34]
(45) In special relativity this reduces to eq
The Covariant Form of the Dirac Equation The covariant generalization of the Dirac equation is taken to be γk∇kψ = µψ . (45) In special relativity this reduces to eq. (2). The covariance of (45) has already been demonstrated in section 2. The corresponding equation forφ† = −ψ†...
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[35]
The Dirac Equation for the Metric gij = diag(−1, R(t)2, R(t)2, R(t)2) In this metric given by g00 = −1 , g 11 = g22 = g33 = R(t)2 , g ij = 0 (i ̸= j) ( t = x0) , (47) a simple solution of equation (9) is given by the matrices γ0 = ˚γ0 , γ 1 = R(t)˚γ1 , γ 2 = R(t)˚γ2 , γ 3 = R(...
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[36]
Hence Max (|βc(0, t)| |αc(0, t)|) < 1
Then |αc(0, t)| = p 1 + |βc(0, t)|2 < p 3/2. Hence Max (|βc(0, t)| |αc(0, t)|) < 1 . Substituting this into (19) and using d dt S(0, t) ω(0, t) ≈ H 3 m2 ≈ 10−107 cm−1 , we have Max d dt |βc(0, t)|2 < 10−107 cm−1 . Substituting this inequality and (21) into (17), we finally obt...
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[1956]
79, 80, 81, and 86
pp. 79, 80, 81, and 86. A well known form of the interval for the expanding closed universe is1 ds2 = −dt2 + R(t)2 (dx2 + du2 + dv2 + dy2) , where x2 + u2 + v2 + y2 = 1 . (1) Inthisformitisclearthatthecoordinates x, u, v, yareconstrainedtothethree-dimensional hypersurfac...
Reviewed August 6, 2026 · model on record in the stance chip above.
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