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DOI 10.1140/epjc/s10052-016-4428-7 Regular Article - Theoretical Physics

Critical exact solutions for self-gravitating Dirac fields

Roberto Ciancia, Luca Fabbrib, Stefano Vignoloc

DIME Sez. Metodi e Modelli Matematici, Università di Genova, Piazzale Kennedy, Pad.D, 16129 Genova, Italy

Received: 12 June 2016 / Accepted: 29 September 2016

© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We consider the Einstein–Dirac field equations describing a self-gravitating massive neutrino, looking for axially symmetric exact solutions; in the search of general solutions, we find some that are specific and which have crit-ical features, such as the fact that the space-time curvature turns out to be flat and the spinor field gives rise to a van-ishing bi-linear scalarψψ = 0 with non-vanishing bi-linear pseudo-scalar iψγ5ψ = 0: because in quantum-field theory general computational methods are built on plane-wave solu-tions, for which the bi-linear pseudo-scalar vanishes while the bi-linear scalar does not vanish, then the solutions we found cannot be treated with the usual machinery of quantum-field theory. This means that for the Einstein–Dirac system there exist admissible solutions which nevertheless cannot be quantized with the common prescriptions; we regard this situation as yet another issue of tension between Einstein gravity and quantum principles. Possible ways to quench this tension can be seen either in enlarging the validity of quantum-field theory or by restricting the space of the solu-tions of the Einstein–Dirac system of field equasolu-tions.

1 Introduction

In modern physics, the field equations of many theories arise from an action functional where the Lagrangian embodies all the necessary information on the dynamics of one or more interacting fields. This information is encoded into the result-ing field equations, describresult-ing the physical features of the system.

Finding all exact solutions for a given system of field equa-tions has always been a task very difficult to accomplish: for example, solutions for two fields in interaction amount to just the Kerr solution of an electrodynamic field immersed in its

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own gravitational field, even if this is not entirely a complete solution since the charged mass distribution is thought to be condensed in the central singularity; complete solutions in which both fields and matter are considered have not been found until very recently. In [1], the system of Dirac fields in interaction with their own gravitational field has been consid-ered in the neutral massless case, and all exact solutions have been found: massless Dirac fields may be useful in describing the dynamics of the plasma we have when fermions have not yet acquired mass due to symmetry breaking in the very early universe. However, it is essential to consider also the system of Dirac fields in interaction with their own gravitational field for neutral but massive particles.

Such particles are neutrinos, thus quite elusive; neverthe-less, scattering amplitudes involving Z0-loops do take into account external legs in which two neutrino fields are present, and therefore having exact solutions in this case would have a considerable impact. In this paper, we will proceed by finding some of these solutions.

We will follow the same procedure of [1], although now due to the complexity of the task we will not find all exact solutions but only some remarkable instances: our achieve-ment may look incomplete, but we think the found solu-tions are nevertheless worth noticing in view of the tension between existing theories.

Indeed, one of the most important problems in fundamen-tal physics in modern times is the incompatibility between gravity and quantum-field theory (QFT); another aspect of this problem, or to some extent another problem altogether, is also the seemingly absolute separation of the two domains, with the standard model of particle physics looking totally disconnected from the standard model of cosmology: dis-crepancies range from the fact that gravity is not quantized nor renormalizable, to the fact that the standard model of par-ticle physics does not account for gravitational effects, and when it does the Higgs mechanism gives rise to a cosmolog-ical constant largely off the cosmologcosmolog-ical data [2].

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The list of incompatibilities giving rise to tensions between Einstein gravity and QFT is long, but to an atten-tive scrutiny some of the arguments arise from reasons that may have no general applicability: for example, according to the current wisdom encoded by the Wilson approach, in order for a theory to be renormalizable it must be such that the dynamical term should remain the leading term in the Lagrangian while going to smaller and smaller scales, but this approach contains the assumption that the interaction in question be expressed in terms of particles, an assump-tion that does not necessarily hold for gravity, not only for the fact that the graviton is still unobserved but also because gravity is not an interaction but the curvature of the space-time; another instance is the fact that gravity has never been quantized, and again here lies the hypothesis that everything in nature must be quantized, which is a hypothesis that starts to be questioned even among quantum-field theorists [3].

Of course, this does not mean that all of the incompatibili-ties are based on arguments that must be rejected as irrelevant but clearly the tension between these two domains is the most stringent if its rationale is given with the fewest additional hypotheses: here we present a source of tension that is based on no additional hypothesis at all.

More specifically, we will see that the Einstein–Dirac sys-tem of coupled field equations has exact solutions that do not respect some of the conditions needed for their quantum-field-theoretical treatment.

2 Notations and field equations

In the paper we use the notations of [1]. In particular, par-tial derivatives of a given function f(xh) are indicated by

fxh := ∂ f

∂xh. Latin and Greek indices run from 1 to 4. The metric tensor of the space-time is denoted by gi j, while any tetrad field associated with a given metric is denoted by = eμi dxi. In this way, one hasημν = gi jeiμeνj, where

ημν = diag(−1, −1, −1, 1) is the Minkowskian metric and

eiμ is the inverse matrix of eiμ. The relations eiμj = δij and eiμei

ν = δνμare then verified. Dirac matrices are denoted byγμ andi := eiμγμ as well asγ5 := iγ4γ1γ2γ3are defined. Also, the chiral representation of Dirac matrices is adopted, as well as natural units¯h = c = 8πG = 1 are used. The spinorial-covariant derivatives of a neutral Dirac field ψ are expressed as

Diψ = ψxi − iψ (1)

where the spinorial-connection coefficientsi are given by

i := − 1

gj hωiμνeμjeνk hk.

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In Eq. (2),ωiμνis the spin-connection, associated to a linear connectioni kj through the usual relation

 h

i j = ωiμνe h

μeνj+ ehμ∂ieμj. (3)

We will consider a Dirac field coupled to gravity in Einstein general relativity theory. The Dirac Lagrangian is

LD= i 2  ψiD iψ − Diψiψ  − mψψ (4)

where m is the mass of the spinor. Field equations are obtained by varying the Einstein–Hilbert plus the Dirac Lagrangian with respect to the metric and the spinor fields: they turn out to be the Einstein equations

Ri j− 1 2Rgi j = i 4  ψ(iDj)ψ − D( jψi)ψ  (5) and the Dirac equations

ihDhψ − mψ = 0 (6)

where Ri j and R denote the Ricci tensor and the curvature scalar associated with the Levi-Civita connection. The Ein-stein equations (5) can be written in the equivalent form Ri j = i 4  ψ(iDj)ψ − D( jψi)ψ  −m 4ψψgi j. (7)

3 Massive neutrino in Lewis–Papapetrou space-time In the following, we search for solutions of the coupled field Eqs. (5) and (6) under the assumption that the geometry of space-time is described by the Lewis–Papapetrou metric in the Weyl canonical coordinates, and the Dirac field represents a massive neutrino in its rest reference frame. The metric tensor is then expressed as

ds2= −B2(dρ2+ dz2) − A2(−W dt + dϕ)2+ ρ

2

A2dt 2(8)

where all functions A(ρ, z), B(ρ, z), and W(ρ, z) depend on theρ and z variables only, while the more general spinor field satisfying the given hypothesis of describing a matter distribution in the rest frame as a result is of the form

ψ = ⎛ ⎜ ⎜ ⎝ eiβ ei(β+μ) ei(β+φ) ei(β+φ+μ) ⎞ ⎟ ⎟ ⎠ (9)

where a priori, , β, μ, and φ are functions on all variables ρ, z, ϕ, t. The co-tetrad field associated to (8) is

e1= B dρ, e2= B dz, e3= A (−W dt + dϕ) ,

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and its dual is e1= 1 B ∂ρ, e2= 1 B ∂z, e3= 1 A ∂ϕ, e4= AW ρ ∂ϕ+ A ρ ∂t. (11) The momentum and spin four-vectors of the spinor field can, respectively, be expressed in terms of the frame (11) as

P= ψγμψ eμ=22+ 22e4 and

V = ψγμγ5ψ eμ=4 cos μ e1+ 4 sin μ e2+22−22e3,

(12) consistently with the requirement that P singles out the rest frame of the massive neutrino: in fact, the vector P is the velocity density, and (9) is simply the most general form of the spinor ensuring the spatial part of P vanishes.

By imposing the requirement that the vectors P and V both possess the same symmetries of the metric (8), namely L

∂ϕP = L∂t∂ P = 0 and L∂ϕ∂ V = L∂t∂ V = 0 (where L denotes Lie derivative), as well as by requiring that the spin vector V is orthogonal to the two Killing vectors∂ϕ and∂t, it necessarily follows that all the quantities, , μ, and φ have to be functions of the variablesρ and z only, and, moreover,  and  have to coincide, namely

 =  = (ρ, z), μ = μ(ρ, z) and φ = φ(ρ, z). (13) The check of (13) is straightforward taking the identities ψψ = 22+ 2cosφ, iψγ5ψ = −22+ 2sinφ

and−V2= P2=

ψψ2+iψγ ψ2 into account.

In order to perform covariant derivatives of the Dirac field and then to implement the field equations (5) and (6), we have to evaluate the spinorial connection. In this respect, employing Eqs. (2) and (3) together with the expression of the metric (8), it is seen that in our case the coefficients (2) induced by the Levi-Civita connection are

ρ= A 2Wργ3γ4 4ρ + Bzγ1γ2 2B (14a) z= − 2Bγ 1γ2+ A2Wz 4ρ γ 3γ4 (14b) ϕ= A 3W ρ 4ρ B γ 1γ4 2Bγ 1γ3+ A3Wz 4ρ B γ 2γ4 Az 2Bγ 2γ3 (14c) t = AW ρ 4B + W Aρ 2B  γ1γ3+AWz 4B + W Az 2B  γ2γ3 +  − ρ Aρ 2 A2B+ 1 2 A BA3W Wρ 4ρ B  γ1γ4 +  − ρ Az 2 A2BA3W Wz 4ρ B  γ2γ4. (14d)

Now, taking the conditions (13) into account, the field equations can be written in full. The Dirac equations (6) eval-uated for the metric (8) and for spinor (9), and decomposed in their real and imaginary parts, amount to eight real equations,

Bz 2B2− WzA2 4Bρ + z−βρ− φρ−μρ BA cos μ  Wβϕ+ βt  ρβϕcosμ

A − m cos φ cos μ + m sin φ sin μ = 0, (15a) Bρ 2B2 − WρA2 4Bρ + ρ+ βz+ φz+ μz B +  2Bρ + A sin μ  Wβϕ+ βt  ρ + βϕsinμ A

× m sin φ cos μ + m cos φ sin μ = 0, (15b) −Bz 2B2 − βρ+ φρ+ z BA cos μϕ+ βt  ρWzA2 4Bρ + βϕcosμ A

− m cos φ cos μ − m sin φ sin μ = 0, (15c) Bρ 2B2+ WρA2 4Bρ + ρ−βz−φz BA sin μ  Wβϕ+βt  ρ +  2Bρ +βϕsinμ

A + m sin φ cos μ − m cos φ sin μ = 0, (15d) −Bzcosμ + Bρsinμ  2B2 + A2sinμ − Wzcosμ 4BρA  Wβϕ+ βt  ρ + βϕ A +  sin μ 2Bρ

+−zcosμ+βρcosμ+μρcosμ+ρsinμ+μzsinμ+βzsinμ B − m cos φ = 0, (15e) −  Bρcosμ + Bzsinμ 2B2 − A2W ρcosμ + Wzsinμ 4Bρ cos μ 2Bρ − m sin φ

+βρsinμ+μρsinμ−μzcosμ−βzcosμ−ρcosμ−zsinμ

B =0, (15f) −A cos μ  Wβϕ+ βt  ρ + Bz 2B2 + βρ+ z Bβϕcosμ AWzA2 4Bρ

− m cos φ cos μ + m sin φ sin μ = 0, (15g) −A sin μ  ϕ+ βt  ρBρ 2B2 + βz− ρ B 2Bρβϕsinμ A + WρA2 4Bρ

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Analogously, the Einstein equations in the form (7) become BWρA22sinμ 2ρ − B 2φ ρcosμ − m B22cosφ +A 4W2 ρ 2ρ2 − Bρρ+ BzzB2 A 2 ρ A2 + Bz2+ B2ρ B2 + 2 Aρ + Bρ = 0, (16a) B A22Wzsinμ − Wρcosμ  4ρB2  φzcosμ + φρsinμ  2 + A4WzWρ 2ρ2 − 2 AzAρ A2 + Az + Bz = 0, (16b) ρ22β ρ+ μρ+ φρ 2 A = 0 (16c) −BWzA22cosμ 2ρ − B 2φ zsinμ + Wz2A4 2ρ2 − m B 22cosφBρρ+ BzzBρ B +B 2 ρ+ B2z B2 − 2 A2 z A2 = 0, (16d) ρ2(2β z+ μz+ φz) 2 A = 0, (16e) −A4Wρ2sinμ 2ρ B − m A 22cosφ −A 6W2 ρ+ Wz2  2ρ2B2 +A 2 ρ+ A2z− AAρρ− AAzz B2 − A Aρ ρ B2 + A4W z2cosμ 2ρ B = 0, (16f) ρ2A ρsinμ − Azcosμ A B + A4W2W ρsinμ − Wzcosμ 2ρ B + W A 6W2 ρ+ Wz2  2ρ2B2 + A2W ρρ+ Wzz 2B2 + ρ2β ϕ A + AW  Aρρ+ Azz− WA2 ρ+ A2z  + 2AAρWρ+ AzWz B2 + W A Aρ B2ρA2Wρ 2ρ B2 − 2sinμ 2B + m A 22W cosφ = 0, (16g) A4W22Wzcosμ − Wρsinμ  2ρ Bρ2A ρρ+ Azz A3B2 + 2ρW2  Azcosμ − Aρsinμ  A B + A2W W ρ− AAρW2 ρ B2 − Aρ A3Bρ2A 2W2 ρ+ Wz2  2B2 + ρ2A2 ρ+ A2z  A4B2 + ρ2  Wzcosμ − Wρsinμ 2BA6W2W2 ρ+ Wz2  2ρ2B2 − 4 AW  WρAρ+ WzAz B2 − 2ρ2βt AA2WW ρρ+ Wzz B2 + W 2A2 ρ+ A2z  B2 − AW2Aρρ+ Azz B2

+ W2Bsinμ+2A22cosφ− m A2W22cosφ = 0. (16h)

Note that the Einstein equationsρϕ and zϕ are identically verified.

4 Searching for solutions

To begin looking for solutions, from equations (16c) and (16e) we derive the relations

φρ = −2βρ− μρ, (17a)

φz = −2βz− μz. (17b)

Taking derivatives of (17a) and (17b) with respectϕ and t and taking (13), we obtainβρϕ = βρt = βzϕ = βzt = 0 from which

β(ρ, z, ϕ, t) = β(ρ, z) + ˆβ(ϕ, t). (18)

Inserting Eq. (18) into Eqs. (17a) and (17b) and omitting an inessential integration constant, we get

β(ρ, z) = −φ 2 − μ 2 (19) and thus β = −φ 2 − μ 2 + ˆβ. (20)

Inserting Eq. (20) into Eq. (15a) and taking derivatives with respect to t, we have

ˆβt t = −

ˆβϕtA2W+ ρ

A2 . (21)

Substituting Eq. (21) into the derivative with respect to t of the Dirac equation (15), it is easily seen that the function ˆβ has to satisfy ˆβϕt = 0, and thus, in view of (21), we have

ˆβ(ϕ, t) = −Et + V (ϕ) (22)

for some function V(ϕ). Inserting (20) and (22) into the Dirac equation (15) and taking derivatives with respect toϕ, we obtain Vϕϕ = 0, telling that V (ϕ) is linear. Setting to zero some integration constants, we conclude that

β = −φ 2 −

μ

2 − Et. (23)

Inserting Eq. (23) into the Dirac equations (15), it is imme-diately seen that Eqs. (15a) and (15b) are identical to Eqs. (15g) and (15h), respectively. Moreover, we can take of the form  = αe s 2 √ (24)

whereα is a constant and s(ρ, z) is a function. After inserting Eqs. (23) and (24) into the Dirac equations, we can subtract (15g) from (15c) and (15h) from (15d) and add (15g) to (15c) and (15h) to (15d) to obtain, respectively,

μρ = 2m B sin φ sin μ + sz and

μz = −2m B sin φ cos μ − sρ+ 1

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φρ = −2m B cos μ cos φ +2E A B cosρ μA 2W z 2ρ and φz = −2m B cos φ sin μ + 2E A B sinμ ρ + A2W ρ 2ρ . (25b) It is then a straightforward matter to verify that Eqs. (15e) and (15f) are automatically implied by Eq. (25) which represent (together with (13), (23) and (24)) all the content of Dirac equations. From (25b) we also get

Wρ= 4m Bρ cos φ sin μA2 − 4E B sinμ A + 2ρφz A2 and Wz = −4m Bρ cos μ cos φA2 + 4E B cosμ A − 2ρφρ A2 . (26) Returning to the Einstein equations, we insert Eq. (24) into the remaining six equations of (16), thus obtaining

WρA2α2essinμ 2ρ3 − α2esφ ρcosμ + m B cos φ ρ2 + A4Wρ2 2ρ2 − Bρρ+ Bzz B2 A2ρ A2 + Bρ2+ Bz2 B2 +2 Aρ ρ A + Bρ ρ B = 0, (27a)  WzA2α2 4ρ3 − α2φ ρ 2ρ2  essinμ −  WρA2α2 4ρ3 + α2φ z 2ρ2  es + cos μA4WρWz 2ρ2 − 2 AρAz A2 + Az ρ A+ Bz ρ B = 0, (27b) −WzA2α2escosμ 2ρ3 + Wz2A4 2ρ2 −α2es(φzsinμ + m B cos φ) ρ2 − Bρρ+ Bzz B +B 2 ρ+ Bz2 B2 − Bρ ρ B2 A2z A2 = 0, (27c) A4α2esW zcosμ − Wρsinμ  2ρ3B2 − A6W2 ρ + Wz2  2ρ2B2 − A  Aρρ+ Azz  B2 + A2ρ+ A2z B2 − m A2αρ22escosφ BA Aρ ρ B2 = 0, (27d) A ρα2 ρ AB2 + A4α2W W ρ 2ρ3B2 − α2 2ρ2B2  essinμ − A4α2W W z 2ρ3B2 + α2A z ρ AB2  escosμ + A 6WW2 ρ+ Wz2  2ρ2B2 + W A Aρ B2ρ +mWα2ρA22BescosφW  A2ρ+ A2z B2 + AW  Aρρ+ Azz B2 + 2 AWρAρ+ WzAz  B2 + A2  Wρρ+ Wzz 2B2 − WρA2 2ρ B2 = 0, (27e)  −A4W2α2 2ρ3B2 − 2Wα2A ρ ρ AB2 − α2W ρ 2ρ B2 + 2 ρ2B2  essinμ +  2Wα2A z ρ AB2 + A4W2α2W z 2ρ3B2 + α2W z 2ρ B2  escosμ ×  −m A2Wρ22αB2cosφ2Eρ ABα2+A22cosB φ  es +W2  A2 ρ+ A2z  B2 − A2W2 ρ+ Wz2  2B2 + ρ2A2 ρ+ A2z  A4B2 −ρ Aρ+ ρ2  Aρρ+ Azz A3B2 − AW2Aρρ+ Azz B2 −A2W  Wρρ+ Wzz B2 − 4 AWAρWρ+ AzWz B2 −A 6W2W2 ρ+ Wz2  2ρ2B2 + A2W W ρ− AW2Aρ ρ B2 = 0. (27f)

Defining the quantities

p:= 2 A

4W

A4W2− ρ2 and r:=

A4

A4W2− ρ2 (28)

from the combination (27d) + p(27e) + r(27f) we get the equation

2 A2α2es(−mρ cos φ + E A)

ρ BA4W2− ρ2 = 0, (29)

yielding the relation cosφ = E A

mρ. (30)

Making use of (30), Eq. (25) simplify to μρ = 2m B sin φ sin μ + sz and

μz = −2m B sin φ cos μ − sρ+ 1 ρ, (31a) φρ = −A 2W z 2ρ and φz = A2 2ρ . (31b)

For later use, we rewrite Eq. (25) also in the form μρ = 2B  m2ρ2− E2A2sinμ ρ + sz and μz = − 2Bm2ρ2− E2A2cosμ ρ − sρ+ 1 ρ, (32a) − E Aρ m2ρ2− E2A2 + E A ρm2ρ2− E2A2 = − A2Wz 2ρ and − E Az m2ρ2− E2A2 = A2Wρ 2ρ . (32b)

Going back to the remaining Einstein equations (27), multi-plying by 2ρA2B2the combination W (27d) + (27e), we get

4 A3(A ρWρ+AzWz) ρ + A4(W ρρ+Wzz) ρA 2α2essinμ ρ3 −A4W ρ ρ2 + 2 Aα2es(A

ρsinμ−Azcosμ)

ρ2 = 0.

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Let us now define the quantities v1(ρ, z) := A2α2essinμ ρ2 + A4Wρ ρ and v2(ρ, z) := − A2α2escosμ ρ2 + A4Wz ρ . (34)

Making use of Eq. (31), a direct check shows that Eq. (33) amounts to the identity (v1)ρ + (v2)z = 0 implying the existence of a function (ρ, z) satisfying v1 = z and

v2= −ρ. From these and (34) we derive

Wρ=−α 2essinμ ρ A2 + ρz A4 and Wz= α2escosμ ρ A2 − ρρ A4 (35) together with es = ρ  A4Wz+ ρρ  A2α2cosμ , (36a) sinμ = cosμ  −A4W ρ+ ρz  A4W z+ ρρ . (36b)

Equations (36) make sense under the condition cosμ = 0: this assumption is not restrictive, because under the condition cosμ = 0 there are no solutions of the field equations. The proof is long and we omit it for brevity.

We can rewrite the remaining Einstein equations (27), making use of (30), (31), and (36), according to

Wρz+ Wzρ 2ρ + A4W2 z 2ρ2 − B A3E Wz ρ2cosμB Eρ ρ A cos μBρρ+ Bzz B + Bρ2+ Bz2 B2 − 2 A2ρ A2 + 2 Aρ ρ A + Bρ ρ B = 0, (37a) −A4WρWz 2ρ2 +  Wzz− Wρρ  2ρ2 AρAz A2 +Az ρ A+ Bz ρ B = 0, (37b) −Wzρ+ Wρz 2ρ + A4Wρ2 2ρ2 − B A3E Wz ρ2cosμB Eρ ρ A cos μBρρ+ Bzz B + Bρ2+ Bz2 B2 − Bρ ρ B2 A2z A2 = 0, (37c) −A2Wρz 2ρ B2 − A5E Wz ρ2B cosμAEρ ρ B cos μAAρρ+ Azz  B2 + A 2 ρ+ A2z B2 − A Aρ ρ B2 + A2Wzρ 2ρ B2 = 0, (37d) AWρAρ+WzAz B2 + ρAρz− Azρ A3B2 + A2WW ρz−Wzρ 2ρ B2 − z 2 2+ W A5E W z ρ2 μ+ W AEρ ρ B cos μ+ W AAρρ+ Azz 2 − W  A2 ρ+ A2z  B2 + A2W ρρ+ Wzz 2B2 + W A Aρ ρ B2 = 0, (37e) A2W2W zρ− Wρz 2ρ B2 + 2ρWAzρ− Aρz A3B2 − ρEρ A3B cosμAWρ B cos μ2EρA5W2E Wz ρ2B cosμAE Wz B cosμ2 AW  WρAρ+ WzAz B2 + ρWzρ− Wρz 2 A2B2 − A AρW2 ρ B2 + Wz A2B2− ρ Aρ A3B2− AW2A ρρ+ Azz B2 − A2WW ρρ+ Wzz B2 −ρ2  Aρρ+ Azz A3B2 + ρ2A2 ρ+ A2z  A4B2 + W2A2 ρ+ A2z  B2 = 0. (37f) At this point of the discussion it is convenient to distinguish the two distinct cases E = 0 and E = 0.

A The case E = 0

In this case, we can search for solutions for A and W in terms of an unknown function T(ρ, z), in the form

A=  2ρmE

m4ρ2T2 ρ + 4E4

, (38a)

Tz= W. (38b)

Indeed, on the one hand Eqs. (38a) and (38b) can be inserted into (32b), showing that one equation is verified and the other gives the Laplace equation for T(ρ, z),

Tρρ+ Tzz+

Tρ

ρ = 0. (39)

The conclusion is that T(ρ, z) has to be harmonic and so, from (38b), W(ρ, z) must be harmonic too. Moreover, it seems physically meaningful to require that for large values ofρ and z the metric (8) has to reduce to Minkowskian space-time, and then, if W is regular anywhere (with respect to the adopted coordinates), we necessarily conclude that W = 0. Indeed, in such a circumstance Eq. (38b) implies that T is a function of the variableρ only, namely T = T (ρ). Moreover, going back to Eq. (32b), we see that A is a linear function of the variableρ only. Omitting an inessential integration con-stant, we have A(ρ) = c1ρ. Inserting W = 0 and A = c1ρ

into Eqs. (37d), (37e), and (37f), it is easily seen that the identitiesρ= z = 0 have to hold necessarily. But this is not possible due to Eq. (35). The conclusion follows that for W regular anywhere we have no solutions. Alternatively, the function W can be non-zero if and only if we give up its full regularity anywhere. This case is currently being studied.

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B The case E= 0

In this case, Eq. (32b) yield W constant and then W = 0. Also, from Eq. (30) we have φ = π/2 (or φ = 3π/2). Moreover, Eq. (32a) reduce to

μρ=2m B sin μ+sz and μz=−2m B cos μ − sρ+ 1 ρ.

(40) We can look for functions of the form A= ρeq(ρ,z)and B= eb(ρ,z)where q(ρ, z) and b(ρ, z) are unknown functions, and by employing these definitions, with (30) and (40), as well as W = 0 and φ = π/2, the Einstein equation (27) become

− bρρ− bzz2qρ ρ − 2qρ2+ bρ ρ = 0, (41a) −qz ρ − 2qρqz+ bz ρ = 0, (41b) − bρρ− bzzbρ ρ − 2qz2= 0, (41c) −ρ2e2q(ρ,z) e2b(ρ,z)  qρρ+ qzz+ qρ ρ  = 0, (41d) esα2sinμ 2ρ2e2b(ρ,z)+ esα2qρsinμ ρe2b(ρ,z)esα2qzcosμ ρe2b(ρ,z) = 0, (41e) − 1 e2q(ρ,z)e2b(ρ,z)  qρρ+ qzz+ qρ ρ  = 0. (41f)

We notice that (41d) and (41f) are equivalent and coincide with the Laplace equation for the unknown q(ρ, z), qρρ+ qzz+

ρ = 0. (42)

Moreover, directly from Eq. (41b) we get

bz = 2qρqz+ qz. (43)

Adding and subtracting Eqs. (41a) and (41c), we obtain, respectively, bρ = −qz2ρ + qρ2ρ + qρ (44) and 2bρρ+ 2bzz+ 2qρ ρ + 2qρ2+ 2q2z = 0. (45)

Making use of Eq. (42), it is a straightforward matter to verify that Eqs. (43) and (44) satisfy the conditions of the Schwarz theorem. Equations (43) and (44) are then solvable by quadra-tures, once the function q(ρ, z) is known. Also, Eq. (45) as a result is automatically verified if (42), (43) and (44) are. Finally, from (41e) we have

μ = arctan  2qzρ 2qρρ + 1  . (46)

The whole problem is now reduced to the solution of the Laplace equation (42). We discuss some particular cases:

• q(ρ, z) = c ln ρ For such a choice of the function q(ρ, z) we have immediately A= ρc+1, b=c2+ clnρ, μ = 0, and s= −2mc2ρ+c+1c2+c+1 +

lnρc2+c+1

c2+c+1 . Consequently, the metric assumes the form

ds2= −ρ2(c2+c)dρ2− ρ2(c2+c)dz2− ρ2(c+1)dϕ2+ ρ−2cdt2

(47) and the Dirac field as a result is

ψ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ αe2mρ(c2+c+1)−lnρ(c2+c+1) 2(c2+c+1)  ρ(c2+c)ρ e −iπ 4 αe2mρ(c2+c+1)−lnρ(c2+c+1) 2(c2+c+1)  ρ(c2+c)ρ e −iπ 4 αe2mρ(c2+c+1)−lnρ(c2+c+1) 2(c2+c+1)  ρ(c2+c)ρ e 4 αe2mρ(c2+c+1)−lnρ(c2+c+1) 2(c2+c+1)  ρ(c2+c)ρ e 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (48)

and a direct check on the Riemann tensor associated with (47) shows that values of the constant c for which we have asymptotic flatness do not exist. For c= −1 the Riemann tensor vanishes, so we obtain a space-time diffeomorphic to the Minkowskian one. Finally, for c = 0 we obtain q(ρ, z) = 0 so that A = ρ, μ = 0, B = 1 (without loss of generality),β = −π/4, and from Eqs. (40) we get s(ρ) = ln ρ − 2mρ. So summarizing all the results, for the metric (8) and the spinor field (9) we find the solutions, respectively, given by the Minkowskian metric in cylindrical coordinates, ds2= −  dρ2+ dz2  − ρ2 dϕ2+ dt2 (49)

and the Dirac field of the form

ψ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ αe−mρ √ρ e−iπ4 αe−mρ √ρ e−iπ4 αe−mρ √ρ eiπ4 αe−mρ √ρ eiπ4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (50)

(8)

for which the constraints given byψψ = 0 but iψγ5ψ = 0 hold identically.

• q(ρ, z) = −lnρ

2 +c1z+c2In this case we have b(ρ, z) =

c12ρ2

2 − lnρ

4 + c3andμ = π2, and so from (40) we get

sρ = ρ1 and sz = −2mec21 ρ2

2 +c3 ρ14

and a direct calculation shows that in this case the Schwarz theorem does not hold and then no solution exists.

• q(ρ, z) = 1

ρ2+z2 In such a circumstance, from Eqs. (43), (44), and (46) we deduce that b(ρ, z) =

2ρ4+4ρ2z2+2z4ρ2+z2ρ2 22+z2)52 and μ(ρ, z) = − arctan  2ρz 2+z2)32−2ρ2 

and inserting them into (40), we get two equations for the unknown s(ρ, z) and again this does not satisfy the integrability conditions.

• q(ρ, z) = cz

2+z2)32

The situation is identical to the last two cases, so given the explicit expression of the function q(ρ, z), we derive that of b(ρ, z) and those of the first derivatives μρ and μz of the function of μ(ρ, z) and inserting all the results into Eq. (40), we get the non-integrability of the equations.

5 Comments on the compatibility with QFT

In obtaining the previous results, we have found a special class of solutions, corresponding to the choice E= 0 and q(ρ, z) = c ln ρ giving rise to specific solutions which cor-respond to the Minkowskian metric in cylindrical coordinates and the Dirac field of the form (50) so that the curvature tensor vanishes and both scalar and pseudo-scalar bi-linear spinorial quantities vanish as well: what this means is that in looking for exact solutions of spinors in the presence of a curved space-time, we have stumbled upon spinors for which the space-time is flat and therefore for these spinors it becomes possible to talk about plane waves verifying the additional constraintsψψ = 0 and iψγ5ψ = 0. This situation gives

rise to a discussion of a possible interface between the present results and QFT, as we are going to show next.

In QFT, the most important quantity that we should be able to calculate is the scattering amplitude for a given process, because from it all cross sections and decay rates can be eval-uated [4]; the scattering amplitude of a process is obtained as a perturbative expansion given in terms of loop diagrams in which all information as regards the interaction is condensed in each vertex while every propagator is taken for free fields (for a general discussion of QFT and the calculation of scat-tering amplitudes see specifically chapter 4 of [4]); the fact that propagators are taken for free fields means that they are the Green functions of the free field equations and thus the

fields can be taken in the form of plane-wave solutions. As a prototypical example, in the calculation of the magnetic moment of the electron, we see that the scattering amplitude is given in terms of the following expression:

M ≈ q F1(k2)Aextμ (k)u(pμu(p) +

q 2mF2(k 2) × 1 2F ext μν(k)2iu(p)[γμ, γν]u(p) (51) where F1(k2) and F2(k2) are the form factors as functions

of the square momentum transfer k= p− p, while Aextμ (k) is the external electrodynamic potential with momentum k, and u(p) and u(p) are the initial and final electron states of momenta p and p calculated in momentum space; in this expression the spinor fields are taken in the form of plane-wave solutions of the free Dirac equation after a Fourier trans-formation. In the form of plane-wave solutions, however, one obtains i uγ5u = 0 identically and one sees that it is always possible to choose uu = 2m as a normalization condition assumed to calculate the completeness relationships that will be needed for computing the sum over all possible spin states (once again we refer to chapter 3 of [4]). Therefore in QFT the common paradigm of calculation dramatically uses the assumption of plane waves such that i uγ5u = 0 and with uu = 2m as normalization. This is precisely the issue we have been anticipating above as regards the interface between QFT and the solutions we have found here.

In order to give QFT a perturbative theory the condition of dealing with plane-wave solutions is necessary and such a condition is automatically verified by the solutions we found, and in this sense we can claim that our solution can naturally be treated within the perturbative methods of QFT; on the other hand, in QFT details of the calculations for many of the most important processes involve additional constraints such as i uγ5u= 0 and uu = 2m, which cannot be compat-ible with the constraintsψψ = 0 and iψγ5ψ = 0 of the solutions we have found here: this means that there are fields that cannot enter into the domain of applicability of QFT as we know it, and yet they are exact solutions of an accepted system of field equations. As a consequence, we regard this as a critical situation lying at the interface between QFT in its standard formulation and the Einsteinian theory of gravity coupled to the Dirac field.

Are there ways out of this situation? Of course, one of the possibilities would be to cope with the fact that QFT in its usual structure is powerful enough to cover most of the observed phenomenology but at the same time not general enough to include cases as those we found here: to this pur-pose we might wonder whether there exists a version of QFT that could perform calculations also for fields featured by the constraints i uγ5u= 0 and uu = 0 or, more in general, with-out the need of developing in plane waves. A complementary way out could be to keep QFT as it is, but finding a way so

(9)

that solutions like those we found above do not appear: to restrict the possible solutions it is necessary to endow the Einstein–Dirac system of field equations with more dynam-ical terms.

Clearly, this tension may also be resolved by a combina-tion of these two proposals.

6 Conclusion

In the present work, we have considered the system of Einstein–Dirac equations, looking for exact solutions, in axi-ally symmetric space-times. We have found some of them, which have the property of giving rise to a flat space-time and a non-trivial spinor field such thatψψ = 0 but iψγ5ψ = 0: the vanishing of the Riemann tensor was not imposed as a reasonable approximation but derived to be an exact result, and therefore we have demonstrated that despite everything interacting with its own gravity nevertheless there can exist matter which turns out to be free; this free matter field as a result is a type-3 spinor in the Lounesto classification [5], certainly non-standard although not singular [6,7]. Matter fields are treated in QFT by finding solutions in the form of plane waves: for them, the pseudo-scalar vanishes and the normalization condition ψψ = 2m is typically assumed. Instead, here we have found a flat space-time, in which cer-tainly plane waves can exist, but where there are massive spinors for whichψψ = 0 identically, and thus the normal-ization conditionψψ = 2m = 0 cannot hold. The solution we have found does not have at least some of the features that are necessary for a treatment in QFT; yet, they are solutions of the field equations of gravity coupled to matter: the solu-tions we found are admissible but nevertheless they cannot be treated quantum-field-theoretically at least in the form of QFT usually considered. This implies that the common pre-scriptions of QFT are not of total generality, which may be seen as yet another point of tension between quantization and gravity.

In order for this tension to be smoothed one then can con-sider the following two ways out: one possibility is to accept that QFT is not general enough to treat all types of particles; the other is to keep QFT as it is, and then go on in restricting the Einstein–Dirac field equations in some suitable way. For instance, this can be done by assuming that the Lagrangian of gravity be different from the Hilbert one: there is quite a list of theories of this type, from general modified grav-ity [8–10], to f(R)-gravity [11–13], non-minimally coupled models [14,15], and conformal theories [16]; in some situ-ations, some of these theories may also solve the problem

of gravitational renormalizability [17], although introducing unitarity violation and ghosts. If we retain the hypothesis of a least-order derivative, the other way out is to consider tor-sion, and because the partially conserved axial-current gives a coupling between the divergence of torsion and iψγ5ψ, torsion–spin interactions affect the dynamics of spinors, and such a dynamics will be restricted and consequently some solutions like those we found here may well be precluded.

It is remarkable how a simple specific solution may give us such an amount of information as regards physics: it is one more instance telling us that either our understanding of QFT is circumstantial and it should be generalized or that the field equations of the model should be empowered, for example by modifying the Hilbert Lagrangian or more appropriately by not neglecting the torsion of the space-time.

It is clearly impossible to speculate at the moment which one of these alternative scenarios is to be followed, although it is most likely it will be a combination of all.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP3.

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4. M.E. Peskin, D.V. Schröder, An Introduction to Quantum Field

Theory (Perseus Books, New York, 1995)

5. P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2001)

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