Electrodynamics on cosmological scales

When ξ is positive, equation (10) is equivalent to the Proca equation.

“When ξ is positive, this equation is equivalent to the Proca equation [34-36] and m A ≡ ξ 1 / 2 can be interpreted as the rest mass of photons.”

• cites [34] Ryder, L.H.: Quantum Field Theory, 2nd edn. Cambridge University Press, Cambridge (1996) sourceNot checked — no full text of the source
• cites [35] Tu, L.C., Luo, J., Gillies, G.T.: Rep. Prog. Phys. 68, 77 (2005) sourceNot checked — no full text of the source
• cites [36] Goldhaber, A.S., Nieto, M.M.: Rev. Mod. Phys. 82, 939 (2010) sourcePartly supported

  Section B (p. 5) describes the Proca modification of Maxwell’s equations as giving the photon a nonzero rest mass. The equations in our copy are not decoded, however, and the text does not identify the citing paper’s equation (10) or its parameter ξ. It therefore supports the general Proca–photon-mass connection, but not the stated equivalence under ξ > 0. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

m_A ≡ ξ^{1/2} can be interpreted as the rest mass of photons.

“When ξ is positive, this equation is equivalent to the Proca equation [34-36] and m A ≡ ξ 1 / 2 can be interpreted as the rest mass of photons.”

• cites [34] Ryder, L.H.: Quantum Field Theory, 2nd edn. Cambridge University Press, Cambridge (1996) sourceNot checked — no full text of the source
• cites [35] Tu, L.C., Luo, J., Gillies, G.T.: Rep. Prog. Phys. 68, 77 (2005) sourceNot checked — no full text of the source
• cites [36] Goldhaber, A.S., Nieto, M.M.: Rev. Mod. Phys. 82, 939 (2010) sourcePartly supported

  “Photon mass from the Proca equation” (pp. 5–6) says that Proca’s modification can give the photon a nonzero rest mass. The passages do not define ξ or establish that √ξ is that mass, so the specific identification is not supported by our copy. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

There is no observational or experimental evidence for gauge invariance of electromagnetic fields in a curved spacetime with a nonvanishing Ricci curvature.

“We also remark that there is no observational or experimental evidence for gauge invariance of electromagnetic fields in a curved spacetime with a nonvanishing Ricci curvature.”

• cites [36] Goldhaber, A.S., Nieto, M.M.: Rev. Mod. Phys. 82, 939 (2010) sourceCould not verify

  The discussion of gauge invariance (p. 9) calls the symmetry unobservable, but does not assess observational or experimental evidence for electromagnetic gauge invariance in curved spacetime. Searches for Ricci curvature and tests in curved spacetime found no passage addressing the claim’s specific setting. (LLM, conf=0.73)

  Experimental — accuracy not yet measured
• cites [48] Schwartz, M.D.: Quantum Field Theory and the Standard Model. The Cambridge University Press, Cambridge (2013) sourceNot checked — no full text of the source

If there exists a cosmological constant as current observations indicated, the κR^b_a A^a term in equation (33) gives rise to a photon mass m_A = (κΛ)^{1/2}, corresponding to λ_A ∼ 10^28 cm (the size of the cosmic horizon).

“For example, as mentioned in Sec. IV, if there exists a cosmological constant as current observations indicated [49, 50], the κR b a A a term in equation (33) gives rise to a photon mass m A = ( κ Λ) 1 / 2 , corresponding to λ A ∼ 10 28 cm (the size of the cosmic horizon).”

• cites [49] Dunkley, J., et al.: 2009. Astrophys. J. Suppl. 180, 306 (2009) sourceSupported
• cites [50] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud et al., arXiv:1502.01589 (2015) sourceSupported

The requirement Ω_A < Ω_m provides a first constraint on the value of y.

“The requirement that ω A ( z = 0) = Ω A &lt; Ω m provides the first constraint on the value of y”

• cites [49] Dunkley, J., et al.: 2009. Astrophys. J. Suppl. 180, 306 (2009) sourceCould not verify

  Section 3.2.2 discusses measurements involving Ω_m, but does not state Ω_A < Ω_m or connect that inequality to a constraint on y. Searches for the inequality and the proposed constraint found no passage addressing them in our copy; page numbers were not available. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [50] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud et al., arXiv:1502.01589 (2015) sourceCould not verify

  Section 5.2 reports a measured value of Ω_m, but it does not discuss Ω_A or a parameter y. Searches for the stated inequality and for Ω_A as a constraint on y found no passage establishing that relationship or calling it a first constraint. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

• AssumptionThe authors state this as an assumption: A proton has charge (1+y)|e| with |y|≪1.
• AssumptionThe authors state this as an assumption: The universe contains an equal number of electrons and protons.

For a closed universe with a nonzero net charge, Maxwell's equations always fail regardless of spacetime symmetry and charge distribution.

“For a closed universe with a nonzero net charge, Maxwell's equations always fail regardless of the spacetime symmetry and the charge distribution.”

• Also stated in: § II. FAILURE OF MAXWELL'S EQUATIONS ON COSMOLOGICAL SCALES · p. 6 · plain

  “Maxwell's equations cannot describe a closed universe with a nonzero net charge without any requirement on spacetime symmetry and charge distribution.”

Maxwell's equations cannot describe a homogeneous and isotropic universe with a uniformly distributed net charge, because the electromagnetic field tensor must vanish everywhere in such a universe.

“Maxwell's equations cannot describe a homogeneous and isotropic universe with a uniformly distributed net charge, because the electromagnetic field tensor in such a universe must be vanishing everywhere.”

• Also stated in: § II. FAILURE OF MAXWELL'S EQUATIONS ON COSMOLOGICAL SCALES · p. 4 · plain

  “This immediately implies that the inhomogeneous Maxwell equation (3) cannot describe a homogeneous and isotropic universe with a uniformly distributed net charge.”

Maxwell's equations imply that the total charge in a closed universe must always be zero, without relying on homogeneity, isotropy, or uniform charge distribution.

“Thus, Maxwell's equations imply that the total charge in a closed universe must always be zero . This statement does not rely on the assumption of homogeneity and isotropy of the universe and uniform distribution of the charge.”

The Proca-type modified Maxwell equation, which contains a photon mass term, can address the paradoxes.

“We consider two types of modified Maxwell equations, both can address the paradoxes.”

• Also stated in: § I. INTRODUCTION · p. 3 · plain

  “First, we show that an electromagnetic field equation with a photon mass term (i.e., the Proca equation) can address the paradoxes.”

For reasonable parameters, the modification does not affect existing experiments and observations.

“The results show that for reasonable parameters the modification does not affect existing experiments and observations.”

In a homogeneous and isotropic universe with a uniformly distributed net charge, Gauss's law is violated because E=0 everywhere while the charge density is nonzero.

“For a homogeneous and isotropic universe filled with a uniformly distributed charge, we have E = 0 everywhere since otherwise E would provide a preferred direction. Then Gauss's law is violated, since the right-hand side of equation (2) is nonzero.”

The Lagrangian (25)-(26) is invariant under the gauge transformation defined by equations (27).

“The Lagrangian is obviously invariant under the gauge transformation defined by”

For an electric charge uniformly distributed in a homogeneous and isotropic universe, the solution to equation (10) is given by equation (11).

“For instance, for an electric charge uniformly distributed in a homogeneous and isotropic universe, the solution to equation (10) is”

The electromagnetic field equation (10) can be derived from the action (15) with Lagrangian density L_EM defined by equation (16).

“The electromagnetic field equation (10) can be derived from the action”

Since μ^2 > 0, the potential V(φ) acquires a minimum at |φ| = φ_0 ≡ μ/λ^{1/2}.

“Since µ 2 &gt; 0, the potential V ( φ ) acquires a minimum at | φ | = φ 0 ≡ µ/λ 1 / 2 .”

Variation of S_EM (J_a=0) with respect to g_ab gives rise to the stress-energy tensor of electromagnetic fields.

“Variation of S EM ( J a = 0) with respect to g ab gives rise to the stress-energy tensor of electromagnetic fields [27]”

• cites [27] Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984) sourceNot checked — no full text of the source

Substituting equation (23) into equation (21) gives the usual equation for the Lorentz force (24).

“Substituting equation (23) into equation (21), we get the usual equation for the Lorentz force”

The Lagrangian in equation (30) describes a massive and real scalar field φ_1 interacting with a massive electromagnetic field A_a.

“Clearly, the Lagrangian in equation (30) describes a massive and real scalar field φ 1 interacting with a massive electromagnetic field A a .”

For a spacetime with R_ab = 0, equation (33) becomes the standard Maxwell equation.

“For a spacetime with R ab = 0, it becomes the standard Maxwell equation.”

The Maxwell equation in a flat spacetime can be expressed in terms of a symmetric tensor H_ab instead of the anti-symmetric tensor.

“To do so, we first show that the Maxwell equation in a flat spacetime can be expressed in terms of a symmetric tensor instead of the anti-symmetric tensor.”

For a spacetime with R_ab = Λ g_ab, equation (33) reduces to equation (10) with ξ = κ Λ.

“For a spacetime with R ab = Λ g ab , where Λ is the cosmological constant, the electromagnetic field equation (33) reduces to equation (10) with ξ = κ Λ.”

Equation (33) does not rely on the existence of another matter field, e.g., a scalar Higgs field.

“It also does not rely on the existence of another matter field, e.g., a scalar Higgs field.”

For a homogeneous and isotropic universe, the potential vector A_a = Φ(t)u_a automatically implies F_ab = 0.

“By the discussions in Sec. III, the potential vector A a = Φ( t ) u a automatically implies F ab = 0.”

Conservation of electric charge, ∇_a J^a = 0, implies that the gauge condition (39) has to be satisfied.

“Hence, the conservation of electric charge, ∇ a J a = 0, implies that the gauge condition”

The mass of the scalar field φ_1 is m = √2 μ.

“The mass of the scalar field φ 1 is m = √ 2 µ .”

Equation (33) does not introduce a new and small dimensional parameter, unlike the modified equation (10).

“It does not introduce a new and small dimensional parameter, unlike the modified equation (10).”

The Lagrangian (53) is identical to that in equation (32) with κ = -2, up to a boundary term which has no contribution to the action integral.

“It can be checked that the above Lagrangian is identical to that in equation (32) with κ = -2, up to a boundary term which has no contribution to the action integral.”

The electromagnetic field equation (51) can be derived from an action with Lagrangian density (53) by variation with respect to A_a.

“The electromagnetic field equation (51) can be derived from an action with a Lagrangian density”

For ħω ≫ m_A c^2, the energy spectrum of longitudinal photons should be given by equation (69).

“Hence, for ℏ ω ≫ m A c 2 , the energy spectrum of longitudinal photons should be”

For a given statistical state, the number of longitudinal photons is always suppressed by a factor m_A^2/2ω^2 relative to the number of transverse photons.

“For a given statistical state, the number of longitudinal photons are always suppressed by a factor m 2 A / 2 ω 2 relative to the number of transverse photons.”

The condition ω_A < Ω_r(1+z)^4 at z=1100 provides an even stronger constraint on y.

“An even stronger constraint on y comes from the condition of ω A &lt; Ω r (1 + z ) 4 at z = 1100 (the redshift of cosmic recombination), where we have used the fact that the density of radiation ρ r ∝ (1 + z ) 4 . Since ω A ∝ (1 + z ) 6 , we get”

• AssumptionThe authors state this as an assumption: A proton has charge (1+y)|e| with |y|≪1.
• AssumptionThe authors state this as an assumption: The universe contains an equal number of electrons and protons.
• AssumptionRests on an assumption: The radiation density scales as ρ_r ∝ (1+z)^4.

The requirement Ω_A < Ω_r provides a stronger constraint on y.

“A stronger constraint comes from the requirement that Ω A &lt; Ω r . Adopting Ω r = 4 . 2 × 10 -5 h -2 (including the massless neutrinos and the cosmic microwave background [18, 19]), we get”

• cites [18] Peebles, P.J.E.: Principles of Physical Cosmology. Princeton University Press, Princeton (1993) sourceNot checked — no full text of the source
• cites [19] Weinberg, S.: Cosmology. Oxford University Press, Oxford (2008) sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: A proton has charge (1+y)|e| with |y|≪1.
• AssumptionThe authors state this as an assumption: The universe contains an equal number of electrons and protons.

The requirement ρ_A < ρ_r at z=0 leads to χ < 2(Ω_r/3)^1/2 Ω_m^-1 d_H/λ_A ≈ 0.01 d_H/λ_A.

“The requirement of ρ A &lt; ρ r (the mean mass density of radiation) at z = 0 leads to χ &lt; 2(Ω r / 3) 1 / 2 Ω -1 m d H /λ A ≈ 0 . 01 d H /λ A .”

• AssumptionThe authors state this as an assumption: The universe has a mean net charge density ρ_e = χ ρ_m, where χ is the ratio of charge to mass.

For a universe containing an equal number of electrons and protons, the total net charge density at any redshift z is ρ_e = y ρ_b |e|/m_p.

“For a universe containing an equal number of electrons and protons, the total net charge density in the universe at any redshift z is given by ρ e = yρ b | e | /m p , where m p is the proton mass, and ρ b is the average mass density of baryons in the universe.”

• AssumptionThe authors state this as an assumption: A proton has charge (1+y)|e| with |y|≪1.
• AssumptionThe authors state this as an assumption: The universe contains an equal number of electrons and protons.

The requirement ρ_A < ρ_m at z=0 leads to χ < 2(3Ω_m)^-1/2 d_H/λ_A ≈ 2.1 d_H/λ_A.

“For example, the requirement of ρ A &lt; ρ m at z = 0 leads to χ &lt; 2(3Ω m ) -1 / 2 d H /λ A ≈ 2 . 1 d H /λ A .”

• AssumptionThe authors state this as an assumption: The universe has a mean net charge density ρ_e = χ ρ_m, where χ is the ratio of charge to mass.

By equation (68), when kT ≪ m_A c^2 we get P_s=1 : P_s=-1 : P_s=0 = 1 : 1 : 1.

“Then, by equation (68) we get P s =1 : P s = -1 : P s =0 = 1 : 1 : 1.”

The electromagnetic field density parameter scales with redshift as ω_A = Ω_A (1+z)^6.

“If the value of ω A at the present time (i.e., at z = 0) is denoted by Ω A , we find ω A = Ω A (1 + z ) 6 .”

The curvature-coupled term can naturally arise from the ambiguity in extending Maxwell's equations from a flat spacetime to a curved spacetime through the 'minimal substitution rule'.

“We show that the curvature-coupled term can naturally arise from the ambiguity in extending Maxwell's equations from a flat spacetime to a curved spacetime through the 'minimal substitution rule'.”

• Also stated in: § I. INTRODUCTION · p. 3 · plain

  “Because of this ambiguity in the 'minimal substitution rule', generalization of a law of physics in a curved spacetime is determined only up to addition of a term coupled to the spacetime curvature. Hence, it is natural to consider an electromagnetic field equation with a curvature-coupled term, since the field equation contains second-order derivatives of the potential vector.”

In a Ricci-flat spacetime, gauge invariance is restored.

“This means that in a Ricci-flat spacetime gauge invariance is restored.”

For electromagnetic fields, the presence of a curvature-coupled term in the field equation can represent interaction of electromagnetic fields with the spacetime curvature.

“For the case of electromagnetic fields, the presence of a curvature-coupled term in the field equation can represent interaction of electromagnetic fields with the spacetime curvature.”

The effect of the curvature-coupled term can be detectable or testable in astrophysical environments with high mass density or in the early universe when radiation drives the dynamics of cosmic expansion.

“However, we argue that, the field equations with a curvature-coupled term can be testable in astrophysical environments where the mass density is high or the gravity of electromagnetic radiations plays a dominant role in dynamics, e.g., the interior of neutron stars and the early universe.”

• Also stated in: § I. INTRODUCTION · p. 4 · plain

  “We find that, the effect of the curvature-coupled term can be detectable in an astrophysical environment with a high mass density, or in the early universe when radiation drives the dynamics of cosmic expansion.”

The 'minimal substitution rule' is essentially reflection of the principle of equivalence, which states that at any point in a spacetime it is possible to choose a free-fall frame where in a sufficiently small region the law of physics takes the form as in a flat spacetime.

“The 'minimal substitution rule' is essentially reflection of the principle of equivalence, which states that at any point in a spacetime it is possible to choose a free-fall frame where in a sufficiently small region the law of physics takes the form as in a flat spacetime [22, 27].”

• cites [22] Weinberg, S.: Gravitation and Cosmology. Wiley, New York (1972)Not matched

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

• cites [27] Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984) sourceNot checked — no full text of the source

If there is a cosmological constant in the universe, the curvature-coupled term in the electromagnetic field equation implies a photon mass determined by the cosmological constant.

“If there is a cosmological constant in the universe, the curvature-coupled term in the electromagnetic field equation implies a photon mass determined by the cosmological constant. In this sense we can say that the new electromagnetic field equation implies the Proca equation.”

A photon mass term in the electromagnetic field equation can naturally arise from spontaneous symmetry breaking and the Higgs mechanism in quantum field theory, where photons acquire mass by eating massless Goldstone bosons.

“This type of electromagnetic field equations can naturally arise from spontaneous symmetry breaking and the Higgs mechanism in quantum field theory, where photons acquire a mass by eating massless Goldstone bosons.”

• Also stated in: § I. INTRODUCTION · p. 3 · boosted

  “in quantum field theory it is well known that a gauge field can acquire a mass through spontaneous symmetry breaking and the Higgs mechanism [33, 34].”

• cites [33] Peskin, M.E., Schroeder, D.V.: An Introduction To Quantum Field Theory. Westview Press, New York (1995) sourceNot checked — no full text of the source
• cites [34] Ryder, L.H.: Quantum Field Theory, 2nd edn. Cambridge University Press, Cambridge (1996) sourceNot checked — no full text of the source

The presence of a term coupled to the spacetime curvature in the law of physics does not violate the principle of equivalence.

“Therefore, the presence of a term coupled to the spacetime curvature in the law of physics does not violate the principle of equivalence.”

A curvature-coupled term cannot be recovered by the 'minimal substitution rule', since it does not show up in a flat spacetime.

“Such a term cannot be recovered by the 'minimal substitution rule', since it does not show up in a flat spacetime.”

The two paradoxes indicate that Maxwell's equations need be modified to be applicable to the universe as a whole.

“The two paradoxes indicate that Maxwell's equations need be modified to be applicable to the universe as a whole.”

An electromagnetic field equation with a photon mass term is possible.

“So an electromagnetic field equation with a photon mass term is possible.”

When the standard Maxwell equation in a curved spacetime is expressed as a second-order differential equation of the potential vector, a curvature-coupled term also appears.

“In fact, when the standard Maxwell equation in a curved spacetime is expressed as a second-order differential equation of the potential vector (i.e., as a wave equation) a curvature-coupled term also appears [27], whose effects have been studied in [45, 46].”

• cites [27] Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984) sourceNot checked — no full text of the source
• cites [45] Tsagas, C.G.: Phys. Rev. Lett. 86, 5421 (2001) sourceNot checked — no full text of the source
• cites [46] Tsagas, C.G.: Class. Quantum Gravity 22, 393 (2005) sourceNot checked — no full text of the source

The modified Maxwell equations with a term coupling the electromagnetic field potential vector to the spacetime curvature tensor can address the paradoxes.

“The other type of modified Maxwell equations, which are more attractive in our opinions, contain a term with the electromagnetic field potential vector coupled to the spacetime curvature tensor.”

• Also stated in: § I. INTRODUCTION · p. 3 · plain

  “Since a non-empty universe must not be Ricci-flat, the paradoxes can be addressed by the inclusion of a curvature-coupled term in the electromagnetic field equation.”

• AssumptionRests on an assumption: A non-empty universe must not be Ricci-flat.

Maxwell's equations have a fundamental flaw when applied to the universe as a whole.

“In Sec. II we have shown that Maxwell's equations cannot describe a universe with a nonzero total charge, which indicates that Maxwell's equations have a fundamental flaw when they are applied to the universe as a whole.”

There is no known first principle to force the total charge in a universe to be exactly zero.

“However, there is no known first principle to force the total charge in a universe to be exactly zero. At least there is yet no such a principle that we know.”

An effective ξ A_a A^a term can naturally arise from spontaneous symmetry breaking by the Higgs mechanism if the electromagnetic field is coupled to a complex scalar field.

“Although ξ can just be a fundamental constant in nature, an effective ξA a term can naturally arise from spontaneous symmetry breaking by the Higgs mechanism if the electromagnetic field is coupled to a complex scalar field [33, 34].”

• cites [33] Peskin, M.E., Schroeder, D.V.: An Introduction To Quantum Field Theory. Westview Press, New York (1995) sourceNot checked — no full text of the source
• cites [34] Ryder, L.H.: Quantum Field Theory, 2nd edn. Cambridge University Press, Cambridge (1996) sourceNot checked — no full text of the source

When charge is conserved, the ξ-term in the stress-energy tensor, (1)T_EM,ab given by equation (20), has no effect on the dynamics of charged particles.

“Equation (24) indicates that when charge is conserved, the ξ -term in the stress-energy tensor, i.e., the (1) T EM ,ab given by equation (20), has no effect on the dynamics of charged particles.”

With field equation (10), the Type I Paradox described in Sec. II can be addressed.

“With the field equation (10), the Type I Paradox described in Sec. II can be addressed.”

Hence the Type II Paradox disappears too.

“Hence the Type II Paradox disappears too.”

The term (1)T_EM,ab affects the spacetime curvature through Einstein's field equations.

“However, the term (1) T EM ,ab affects the spacetime curvature through Einstein's field equations.”

Equation (33) can solve the Type I Paradox described in Sec. II.

“Let us check that equation (33) can solve the Type I Paradox described in Sec. II.”

Even if the 'minimal substitution rule' is insisted, different electromagnetic field equations can be obtained in a curved spacetime.

“So, even if the 'minimal substitution rule' is insisted, different electromagnetic field equations can be obtained in a curved spacetime.”

If a field equation in a flat spacetime contains second or higher order derivatives of a vector or a tensor, the corresponding field equation in a curved spacetime derived by the 'minimal substitution rule' is not unique.

“If a field equation in a flat spacetime contains second or higher order derivatives of a vector or a tensor, the corresponding field equation in a curved spacetime derived by the 'minimal substitution rule' is not unique.”

If the universe contains a cosmological constant, equation (33) implies the Proca-type equation with a photon mass determined by the cosmological constant.

“In this sense, we can say that the electromagnetic field equation (33) implies the Proca-type equation with a photon mass determined by the cosmological constant.”

The 'minimal substitution rule' cannot guarantee that the equation derived with it must be correct in physics in a curved spacetime.

“Although it can guarantee that the derived equation in a curved spacetime satisfies the principle of general covariance and returns to the correct equation in a flat spacetime, the 'minimal substitution rule' cannot guarantee that the equation derived with it must be correct in physics in a curved spacetime.”

For a homogeneous and isotropic universe, the Ricci curvature tensor R_ab must have the form R_ab = α(t)u_a u_b + β(t)h_ab.

“For a homogeneous and isotropic universe, the Ricci curvature tensor R ab must have a form of R ab = α ( t ) u a u b + β ( t ) h ab”

The modification to the Maxwell equation considered here does not affect the electrodynamics in a flat or Ricci-flat spacetime.

“This means that, unlike in the case of equation (10), the modification to the Maxwell equation considered here does not affect the electrodynamics in a flat or Ricci-flat spacetime.”

The equation of a conformally invariant scalar field cannot be derived from its version in a flat spacetime by the 'minimal substitution rule'.

“Since the term Rφ/ 6 is missing in a flat spacetime, the equation of a conformally invariant scalar field cannot be derived from its version in a flat spacetime (i.e., the Klein-Gordon equation for a massless scalar field) by the 'minimal substitution rule'.”

• cites [47] Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. The Cambridge University Press, Cambridge (1984) sourceNot checked — no full text of the source

With the electromagnetic field equation (33), the Type II Paradox is also addressed.

“Similar to the discussion in Sec. III, with the electromagnetic field equation (33), the Type II Paradox is also addressed.”

Since κ can be any number (or any function), the 'minimal substitution rule' can give rise to an infinite number of field equations in a curved spacetime corresponding to the Maxwell equation in a flat spacetime.

“Therefore, since κ can be any number (it can be any function in fact), the 'minimal substitution rule' can give rise to an infinite number of field equations in a curved spacetime corresponding to the Maxwell equation in a flat spacetime.”

Spontaneous symmetry breaking and the Higgs mechanism cannot be considered as the ultimate solution to the inconsistency problem in application of Maxwell's equations to a universe with a net charge.

“Although spontaneous symmetry breaking and the Higgs mechanism naturally give rise to an electromagnetic field equation in the form of equation (10), they cannot be considered as the ultimate solution to the inconsistency problem in application of Maxwell's equations to a universe with a net charge.”

There is a profound discontinuity as the mass of photons approaches zero: massive photons have three degrees of freedom, but massless photons have only two.

“There is a profound discontinuity as the mass of photons approaches zero: massive photons have three degrees of freedom, but massless photons have only two.”

• cites [56] Aragao de Carvalho, C., Rosa Jr., S.G.: J. Phys. A 13, 3233 (1980) sourceNot checked — no full text of the source
• cites [57] Mohling, F.: Statistical Mechanics: Methods and Applications. Wiley, New York (1983)Not matched

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

For massless photons, the statistical degeneracy factor is g = 2.

“For massless photons we have g = 2.”

The paradoxes arising from application of Maxwell's equations to a universe as described in Sec. II demand the inclusion of the curvature-coupled term in the electromagnetic field equation.

“The paradoxes arising from application of Maxwell's equations to a universe as described in Sec. II demands the inclusion of the curvature-coupled term in the electromagnetic field equation.”

The above arguments support the proposal of equation (33) as a candidate for the electromagnetic field equation in a curved spacetime.

“The above arguments support the proposal of equation (33) as a candidate for the electromagnetic field equation in a curved spacetime.”

Equation (10) indicates an intrinsic photon mass, if ξ is interpreted as a fundamental constant.

“Equation (10) indicates an intrinsic photon mass, if ξ is interpreted as a fundamental constant.”

Both types of modified Maxwell's equations discussed in previous sections have the unavoidable consequence that photons can have a nonzero rest mass.

“Both the two types of modified Maxwell's equations discussed in previous sections have an unavoidable consequence: photons can have a nonzero rest mass.”

Massive photons are described by the statistics of relativistic Bose gases.

“Massive photons are described by the statistics of relativistic Bose gases [56, 57].”

• cites [56] Aragao de Carvalho, C., Rosa Jr., S.G.: J. Phys. A 13, 3233 (1980) sourceNot checked — no full text of the source
• cites [57] Mohling, F.: Statistical Mechanics: Methods and Applications. Wiley, New York (1983)Not matched

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

Equation (33) indicates an effective photon mass as a function of the local spacetime curvature.

“Equation (33) indicates an effective photon mass as a function of the local spacetime curvature.”

Attempts in measuring or giving a limit on the photon mass by observing the difference in the arriving time of photons with different frequencies emitted by the same source are disfavored, since for any reasonable value of a photon mass the effect turns out to be extremely small.

“However, attempts in measuring or giving a limit on the photon mass by observing the difference in the arriving time of photons with different frequencies emitted by the same source are disfavored, since for any reasonable value of a photon mass the effect turns out to be extremely small [53-55].”

• cites [53] de Broglie, L.: La Mécanique Ondulatoire du Photon. Une Nouvelle Théorie de la Lumière. Hermann & Cie, Paris (1940)Not matched

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

• cites [54] Gintsburg, M.A.: Sov. Astron. 7, 536 (1964)Not matched

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

• cites [55] Kobzarev, I.Y., Okun, L.B.: Sov. Phys. Usp. 11, 338 (1968) sourceNot checked — no full text of the source

When ω ≫ m_A, the contribution of longitudinal photons to the total radiation is negligible.

“Therefore, when ω ≫ m A , the contribution of longitudinal photons to the total radiation is negligible.”

Because of the very weak interaction of longitudinal photons with matter, approaching a statistical equilibrium state of longitudinal photons in a cavity is extremely slow, with a time-scale comparable to the age of the universe.

“Because of the very weak interaction of longitudinal photons with matter, approaching to a statistical equilibrium state of longitudinal photons in a cavity is extremely slow, with a time-scale comparable to the age of the universe [59-61].”

• cites [59] Bass, L., Schrödinger, E.: Proc. R. Soc. Lond. A 232, 1 (1955) sourceNot checked — no full text of the source
• cites [60] Goldhaber, A.S., Nieto, M.M.: Rev. Mod. Phys. 43, 277 (1971) sourceNot checked — no full text of the source
• cites [61] Stueckelberg, E.C.G.: Helv. Phys. Acta 30, 209 (1957)Not matched

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

The value of g depends on how the longitudinal photons interact with matter.

“However, the value of g depends on how the longitudinal photons interact with matter.”

According to [60], the practical impossibility for longitudinal photons to get equally partitioned with transverse photons in energy is due to the very large skin depth of longitudinal photons compared to that of transverse photons (with a ratio ≥ ω^2/m_A^2).

“According to [60], the practical impossibility for longitudinal photons to get equally partitioned with transverse photons in energy is due to the very large skin depth of longitudinal photons compared to that of transverse photons (with a ratio ≥ ω 2 /m 2 A ) [62, 63].”

• cites [60] Goldhaber, A.S., Nieto, M.M.: Rev. Mod. Phys. 43, 277 (1971) sourceNot checked — no full text of the source
• cites [62] Gertsenshtein, M.E., Solovei, L.G.: JETP Lett. 9, 79 (1969)Not matched

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

• cites [63] Kroll, N.M.: Phys. Rev. Lett. 26, 1395 (1971) sourceNot checked — no full text of the source

It is quite general that the contribution of m_A^2 to the radiation field is ∼ (m_A^2/k^2) × the dominant 1/r term as indicated in the dipole radiation case.

“Finally, we argue that it is quite general that the contribution of m 2 A to the radiation field is ∼ ( m 2 A /k 2 ) × the dominant 1 /r term as indicated in the dipole radiation case (eqs. 63 and 64).”

We cannot imagine any kind of electromagnetic radiation generated in lab or astrophysical conditions where longitudinal photons can have a detectable contribution.

“We cannot imagine any kind of electromagnetic radiation generated in lab or astrophysical conditions where longitudinal photons can have a detectable contribution.”

When kT ≫ m_A c^2, the rest mass of photons can be ignored and the statistics approaches that of massless photons.

“When kT ≫ m A c 2 , the rest mass of photons can be ignored and the statistics approaches that of massless photons.”

When photons have an energy ħω ≫ m_A c^2, they interact with matter very weakly.

“Here we demonstrate that when photons have an energy ℏ ω ≫ m A c 2 they interact with matter very weakly by considering the electromagnetic radiation generated by an electric dipole p oscillating with a frequency ω .”

In a homogeneous and isotropic universe, the electromagnetic field tensor F_ab must vanish everywhere.

“As discussed in Secs. II and III, in a homogeneous and isotropic universe we must have F ab = 0 everywhere.”

• AssumptionThe authors state this as an assumption: The universe is homogeneous and isotropic.

When there is a cosmological constant Λ, the electromagnetic field equation (33) contains a photon mass term with m_A = (κΛ)^1/2, corresponding to a Compton wavelength comparable to the Hubble distance.

“In Sec. IV we have shown that when there is a cosmological constant Λ in the universe, the electromagnetic field equation (33) contains a photon mass term with m A = ( κ Λ) 1 / 2 , corresponding to a Compton wavelength comparable to the Hubble distance.”

Even if the outside of an object is vacuum and Ricci-flat, the nonvanishing curvature-coupled term in the interior can still affect the structure of the electromagnetic field outside the object.

“Therefore, even if the outside of an object is vacuum and Ricci-flat, the nonvanishing curvature-coupled term in the interior of the object can still affect the structure of the electromagnetic field outside the object.”

When kT ≪ m_A c^2, the statistics of massive photons become identical to that of a non-relativistic Bose gas with a degeneracy factor g = 3.

“Hence, when kT ≪ m A c 2 , the statistics of massive photons become identical to that of a non-relativistic Bose gas with a degeneracy factor g = 3.”

If an observer measures the electric field outside a charged ball and interprets the result with standard Gauss's law, the electric field appears as being generated by a total charge Q_ps.

“If an observer measures the electric field outside a charged ball and interprets the result with the standard Gauss's law, he will find that the electric field appears as being generated by a total charge”

If an observer measures the magnetic field outside a star and interprets the result with standard Ampere's circuital law, the magnetic field appears as being generated by a total current density inside the star.

“Similarly, if an observer measures the magnetic field outside a star and interprets the result with the standard Amp` ere's circuital law, he will find that the magnetic field appears as being generated by a total current density inside the star given by”

In a homogeneous and isotropic universe, there is no electromagnetic force on the cosmological scale.

“However, as we have already demonstrated, in a homogeneous and isotropic universe F ab must vanish everywhere so there is no electromagnetic force on the cosmological scale.”

• AssumptionThe authors state this as an assumption: The universe is homogeneous and isotropic.

The above constraint on y is consistent with the result obtained in lab experiments if λ_A ∼ d_H ∼ 10^28 cm.

“The above constraint on the value of y is consistent with the result obtained in lab experiments if we take λ A ∼ d H ∼ 10 28 cm [31].”

• cites [31] Bressi, G., et al.: Phys. Rev. A 83, 052101 (2011) sourceNot checked — no full text of the source

The stress-energy tensor of the electromagnetic field arising from the net charge causes an attractive force, instead of a repulsive force according to Einstein's field equation.

“In fact, the stress-energy tensor of the electromagnetic field arising from the net charge causes an attractive force, instead of a repulsive force according to Einstein's field equation.”

• AssumptionThe authors state this as an assumption: The universe has a mean net charge density ρ_e = χ ρ_m, where χ is the ratio of charge to mass.

The stress-energy tensor of the electromagnetic field in a homogeneous and isotropic universe corresponds to a perfect fluid with mass density ρ_A and pressure P_A.

“corresponding to a perfect fluid with mass density ρ A and pressure P A defined by”

• AssumptionThe authors state this as an assumption: The universe is homogeneous and isotropic.

Equation (33) indicates that local curvature of the background spacetime can affect electrodynamics.

“Equation (33) indicates that local curvature of the background spacetime can affect the electrodynamics.”

In a strongly curved spacetime, the effect of the curvature-coupled term may be measurable on scales comparable to the curvature radius of the spacetime.

“So, we can expect that, in a strongly curved spacetime, the effect of the curvature-coupled term may be measurable on scales comparable to the curvature radius of the spacetime.”

The parameter χ remains constant as the universe evolves.

“Since ρ m ∝ (1 + z ) 3 , by equation (73) the parameter χ remains a constant as the universe evolves.”

• AssumptionThe authors state this as an assumption: The universe has a mean net charge density ρ_e = χ ρ_m, where χ is the ratio of charge to mass.

A constraint on the parameter χ can be converted to a constraint on the charge difference between electrons and protons, the charge of neutrons and neutrinos, and the asymmetry of matter and anti-matter.

“Hence, a constraint on the parameter χ can be converted to a constraint on the charge difference between electrons and protons, the charge of neutrons and neutrinos, and the asymmetry of matter and anti-matter [29, 30, 37].”

• cites [29] Sengupta, S., Pal, P.B.: Phys. Lett. B 365, 175 (1996) sourceNot checked — no full text of the source
• cites [30] Caprini, C., Ferreira, P.G.: J. Cosmol. Astropart. Phys. 02, 006 (2005) sourceNot checked — no full text of the source
• cites [37] Lyttleton, R.A., Bondi, H.: Proc. R. Soc. Lond. A 252, 313 (1959) sourceNot checked — no full text of the source

In the total stress-energy tensor of electromagnetic fields, only the (0)T_EM,ab part is accessible to experiments based on electromagnetic methods.

“In the total stress-energy tensor of electromagnetic fields, only the (0) T EM ,ab part is accessible to experiments based on electromagnetic methods.”

Objects clearly satisfying the criterion r_c ≲ r_A include neutron stars and the early universe.

“Objects clearly satisfying this criterion include neutron stars and the early universe.”

For a closed universe, Maxwell's equations always fail if the total charge in the universe is nonzero, without any requirement on spacetime symmetry and charge distribution (Type II Paradox).

“For a closed universe (i.e., a universe with a compact spatial section), Maxwell's equations always fail if the total charge in the universe is nonzero, without any requirement on spacetime symmetry and charge distribution (Type II Paradox).”

The places where the effect of the curvature-coupled term in the new field equation and the Lagrangian can be detectable include the neighbor and interior of neutron stars, and the early universe.

“The places include the neighbor and interior of neutron stars, and the early universe.”

Equation (87) indicates that the curvature term in field equation (33) will lead to measurable effects if r_c ≲ r_A.

“Equation (87) indicates that the curvature term in the field equation (33) will lead to measurable effects if r c ≲ r A .”

In the very early universe, the curvature-coupled term in the electromagnetic field equation should affect all electromagnetic processes on scales of the curvature radius of the early universe.

“The curvature-coupled term in the electromagnetic field equation should affect all the electromagnetic processes on scales of the curvature radius of the early universe.”

• AssumptionRests on an assumption: In the very early time, the universe has a very high mass density and a very small curvature radius.

In a strongly curved spacetime, the conservation of (0)T_EM,ab is affected by the presence of a pseudo-charge current.

“In a spacetime region that is strongly curved, the conservation of (0) T EM ,ab is affected by the presence of a pseudo-charge current, according to equation (89).”

Due to the high mass density inside a neutron star, a pseudo-charge current arising from the curvature-coupled term can affect the structure of the magnetic field of a neutron star.

“Due to the high mass density present inside a neutron star, a pseudo-charge current arising from the curvaturecoupled term can affect the structure of the magnetic field of a neutron star.”

• AssumptionRests on an assumption: A typical neutron star has radius r ≈ 10^6 cm and overall mass density ρ ≈ 5×10^14 g cm^-3.

For a neutron star, the curvature-coupled term in the electromagnetic field equation can produce detectable effects on its inside and outside magnetic fields.

“Therefore, for a neutron star, we expect that the curvature-coupled term in the electromagnetic field equation can produce detectable effects on its inside and outside magnetic fields.”

• AssumptionRests on an assumption: A typical neutron star has radius r ≈ 10^6 cm and overall mass density ρ ≈ 5×10^14 g cm^-3.

Maxwell's equations cannot describe the electrodynamics of a homogeneous and isotropic universe with a uniformly distributed net charge, since in such a universe the electromagnetic field must be vanishing everywhere (Type I Paradox).

“Maxwell's equations cannot describe the electrodynamics of a homogeneous and isotropic universe with a uniformly distributed net charge, since in a homogeneous and isotropic universe the electromagnetic field must be vanishing everywhere (Type I Paradox).”

• AssumptionThe authors state this as an assumption: The universe is homogeneous and isotropic.

Even where r_A ≪ r_c, the magnitude of (1)T_EM,ab can be comparable to that of (0)T_EM,ab.

“Even in a place where r A ≪ r c (hence the curvature terms in eq. 40 are small), the magnitude of (1) T EM ,ab can be comparable to that of (0) T EM ,ab .”

In the early universe when the gravity of radiation drives the expansion of the universe, the dark electromagnetic energy may play a very important role.

“In the early universe when the gravity of radiation drives the expansion of the universe, the dark electromagnetic energy may play a very important role.”

In the early time when the universe is small, the curvature-coupled term in the electromagnetic field equation affects all electromagnetic processes on scales comparable to the curvature radius of the universe.

“In the early time when the universe is small, the curvature-coupled term in the electromagnetic field equation affects all the electromagnetic processes on scales comparable to the curvature radius of the universe.”

• AssumptionRests on an assumption: In the very early time, the universe has a very high mass density and a very small curvature radius.

If the interior of the object is strongly curved due to a large mass density, the effect of the pseudo-charge current density arising from the curvature-coupled term may be testable in measurements of the outside electromagnetic field.

“If the interior of the object is strongly curved due to the presence of a large mass density, the effect of the pseudo-charge current density arising from the curvature-coupled term may be testable in measurements of the outside electromagnetic field.”

The stress-energy tensor (0)T_EM,ab interacts with both the charge current J^a and the pseudo-charge current J^a_ps.

“In contrast, the stress-energy tensor (0) T EM ,ab (eqs. 19 and C4) interacts with both the charge current J a and the pseudo-charge current J a ps , since by equations (B5) and (33) we have”