The Cosmic Defect theory (CD), which is presented elsewhere in this
conference, introduces in the standard Einstein-Hilbert Lagrangian an elastic
term accounting for the strain of space-time viewed as a four-dimensional
physical continuum. In this framework the Ricci scalar acts as the kinetical
term of the strain field whose potential is represented by the additional
terms. Here we are presenting the linearised version of the theory in order to
analyze its implications in the weak field limit. First we discuss the recovery
of the Newtonian limit. We find that the typical static weak field limit
imposes a constraint on the values of the two parameters (Lame coefficients) of
the theory. Once the constraint has been implemented, the typical gravitational
potential turns out to be Yukawa-like. The value for the Yukawa parameter is
consistent with the constraints coming from the experimental data at the Solar
system and galactic scales. We then come to the propagating solutions of the
linearised Einstein equations in vacuo, i.e. to gravitational waves. Here,
analogously with other alternative or extended theories of gravity, the
presence of the strain field produces massive waves, where massive (in this
completely classical context) means subluminal. Furthermore longitudinal
polarization modes are allowed too, thus lending, in principle, a way for
discriminating these waves from the plane GR ones.Comment: Proceedings of 'Invisible Universe International Conference', Paris,
  June 29- July 3, 200

Radicella, Ninfa

Tartaglia, Angelo

English

arXiv

PORTO Publications Open Repository TOrino

Massive gravitational waves from the Cosmic Defect theory
N. Radicella and A. Tartaglia
Dipartimento di Fisica, Politecnico di Torino and INFN sez. Torino
Abstract. The Cosmic Defect theory (CD), which is presented elsewhere in this conference, introduces in the standard
Einstein-Hilbert Lagrangian an elastic term accounting for the strain of space-time viewed as a four-dimensional physical
continuum. In this framework the Ricci scalar acts as the kinetical term of the strain field whose potential is represented by
the additional terms. Here we are presenting the linearised version of the theory in order to analyze its implications in the
weak field limit. First we discuss the recovery of the Newtonian limit. We find that the typical static weak field limit imposes
a constraint on the values of the two parameters (Lamé coefficients) of the theory. Once the constraint has been implemented,
the typical gravitational potential turns out to be Yukawa-like. The value for the Yukawa parameter is consistent with the
constraints coming from the experimental data at the Solar system and galactic scales. We then come to the propagating
solutions of the linearised Einstein equations in vacuo, i.e. to gravitational waves. Here, analogously with other alternative
or extended theories of gravity, the presence of the strain field produces massive waves, where massive (in this completely
classical context) means subluminal. Furthermore longitudinal polarization modes are allowed too, thus lending, in principle,
a way for discriminating these waves from the plane GR ones.
Keywords: Alternative models of gravity, gravitational waves
PACS: 04.30.-w, 04.50.Kd
FRAMEWORK
The Lagrangian density we want to analyse is
L = R +
1
2
Cµνρσ εµν ερσ ,
where the second term on the r.h.s. is a potential term, in the form of an elastic potential whose field is the strain tensor
εµν
.
= 12 (gµν −ηµν), and the Ricci scalar is computed by means of the observed metric gµν .
The strain is ascribed to a cosmic point-like defect, this is why we called the theory the Cosmic Defect theory [1] In
our analogy we are interested in an isotropic medium; in this case the elastic constants take a simple form:
Cµνρσ = λ ηµν ηρσ + µ(ηµρηνσ + ηµσ ηνρ).
They now depends on two parameters only, the so-called Lamé coefficients λ and µ .
The Elastic potential then translates in
V =
1
2
[
λ ε2 + 2µεµνεµν
]
,
where, following the structure of the elastic coefficients, the covariant version of the ε tensor is obtained by lowering
the indices with the total metric g.
The action integral is then
S =
∫
d4x [R +V ]
√−g. (1)
By varying the action in eq.(1) w.r.t. the metric gµν that is the only dynamical field we obtain the Elastic Einstein
Equations (EEE):
Gµν = T eµν (2)
where
T eµν = λ ε
[(
ε
4
+
1
2
)
gµν − εµν
]
+ µ
[
εµν +
1
2
gµνεαβ εαβ −2εµαεαν
]
.
1128
WEAK FIELD LIMIT
In order to look for the gravitational waves of this theory we must linearize it around the Minkowski space-time, to
which the theory reduces locally when the defect is not so strong.
It means that the metric can be written as
gµν = ηµν + εhµν ε ≪ 1,
where the perturation metric hµν is driven by ε . The strain tensor becomes
εµν =
1
2
[
gµν −ηµν
]
=
1
2
εhµν ,
where we use the coordinate invariance of the theory in order to fix the flat metric to be the Minkowski one (i.e.
Diag(−1,1,1,1)). The strain tensor represents the perturbation itself. Looking at the EEE, they are:
Gµν = Teµν , (3)
where
Teµν = λ ε
[(
ε
4
+
1
2
)
gµν − εµν
]
+ µ
[
εµν +
1
2
gµνεαβ εαβ −2εµαεαν
]
. (4)
First of all we should look at the linearized Einstein tensor, that, from
Rαβ γδ ≃ ∂γΓαβ δ − ∂δ Γαβ γ ,
reduces to
Gαβ ≃
1
2
[
∂γ ∂β hγα + ∂ γ∂α hβ γ −hαβ − ∂α∂β h−ηαβ ∂γ∂ δ hγδ + ηαβh
]
,
where indices are raised and lowered by means of the Minkowski metric and h is the trace of the perturbation1.
The linearised Elastic energy-momentum tensor becomes
T eµν ≃
λ
2
εηµν + µεµν .
The linearised EEE in vacuo reduce to
[
∂γ∂β hγα + ∂ γ∂α hβ γ −hαβ − ∂α ∂β h −ηαβ ∂γ∂ δ hγδ + ηαβ h
]
− µ
(
hµν +
λ
2µ hηµν
)
= 0
(5)
In order to investigate more deeply these equations we rewrite them by using their divergence and trace. The
divergence, thanks to the contracted Bianchi identities, gives a relation between the divergence of the perturbation
and that of its trace:
hµν;ν =−
λ
2µ h
;µ . (6)
By tracing eq.(5) one gets
2
(
1 + λ
2µ
)
h− µ
(
1 + 2λµ
)
h = 0. (7)
The trace of the perturbation becomes a dynamical field. This degree of freedom is always ghostlike, regardless the
combination of the parameters [2, 3] (and it could be tachyonic, too): this makes the theory pathological from the
quantum point of view. To avoid this behaviour we must constraint our parameters so that 1 + λ/2µ = 0. This choice
reduces our linearised Elastic energy-momentum tensor to the Pauli-Fierz mass term [4, 5], which actually leads to
1 It is worthwhile here to be precise about the sign convention on Riemann and Ricci tensors. In this paper we use the Riemann tensor written
above, where the upper index is the first one, and we contract the first and the third index in Riemann to obtain Ricci.
1129
different predictions from those of General Relativity, no matter how small the graviton mass is [6, 7]. This is what is
called van Dam-Veltman-Zakharov discontinuity [8, 9], whose existence, accordingly to some authors, can be ascribed
to the explicit breaking of the gauge invariance by the mass term so that it can be cured if the mass is generated by
the compactification of a higher dimensional theory [10] or to the evaluation of the linearised theory around flat
backgrounds [11].
The debate on massive gravity is still open and we choose the Fierz-Pauli mass term as the linearised version of the
CD theory.
Coming back to eq.(7), the choice λ = −2µ makes the trace of the perturbation to vanish in vacuum and now eq.(6)
translates in what would have been the Lorentz gauge in General Relativity. This really simplifies the linearised
Einstein tensor that reduces to −hµν/2 so that the equations one finally gets are:
(+ µ)hµν = 0 (8)
hµν;ν = 0 (9)
h = 0. (10)
Gravitational waves
The set of eqs.(8, 9, 10) represent propagating massive waves and plane waves are solutions of this equations:
hµν = αµνeiκβ x
β
. (11)
Designating the propagation direction as the z axis, the wave vector is κ = (ω ,0,0,ck) and from the eq.(11) we obtain
the dispersion relation:
ω =±c
√
k2− µ
The waves are subluminal, which is commonly referred to as being "massive" (µ < 0, consistently with the
cosmological limit). Let us now look at which are the polarisation modes of this spacetime. From eqs.(8,9,10), we
have 5 dynamical degrees of freedom but this does not mean that the expect the same number of polarisation modes.
In a metric theory of gravity there can be at most six polarisation modes, as shown in [12]. The analysis can be
performed by looking at the geodesic deviation equation, that states which is the displacement between a pair of free-
falling particles when a gravitational wave arrives. The three-acceleration depends on the "electric" components of the
Riemann tensor (Ri0k0). It can be shown that there are six algebraically independent components of the Riemann tensor
by using the Newman-Pensore formalism. First, one choose a complex null basis, the so-called null tetrad (k, l,m,m¯),
that is related to a cartesian system {t,x,y,z} by
k = 1√
2
(1,0,0,1), l = 1√
2
(1,0,0,−1),
m =
1√
2
(0,1, i,0), m¯ = 1√
2
(0,1,−i,0).
We remember that we have chosen to orient the axes so that the wave travels in the +z direction, and, u being
u = t−z/c, the k vector is proportional to ∇u. Then it is possible to split the Riemann tensor into irreducible parts [14]:
the Weyl tensor (Ψ0,Ψ1,Ψ2,Ψ3,Ψ4), the traceless Ricci tensor (Φ00,Φ01,Φ11, Φ12,Φ22,Φ02, that are five complex
scalars) and the Ricci scalar (Λ).
When considering plane waves only some of them are different from zero and, among these, six are independent. The
ones that are helicity (s) eigenstates under rotations about the z axis are
Ψ2,Φ22 → s = 0,
Ψ3, ¯Ψ3 → s = 1,
Ψ4, ¯Ψ4 → s = 2.
1130
We recall that these six wave amplitudes are observer dependent but there are some invariant statements that are true
for all standard observers2 if they are true for anyone. These statements constitute the E(2) classification of waves,
based on the Petrov type of the Weyl tensor [13].
These are related to the Riemann tensor, in cartesian coordinates, as follows [12]:
Ψ2(u) = −16Rz0z0(u),
Ψ3(u) = −12Rx0z0(u)+
i
2
Ry0z0(u),
Ψ4(u) = −Rx0x0(u)+ Ry0y0(u)+ 2iRx0y0(u),
Ψ22(u) = −(Rx0x0(u)+ Ry0y0(u)) .
What we measure in a detection experiment is the relative acceleration of test masses, that is the six "electric"
components of the Riemann tensor. One can express these informations in the so-called driving-force matrix
Si j(t) = R0i0 j.
In general there are eight unknowns, six polarisations and two direction cosines, but if one can establish the direction
of the gravitational wave by other information, i.e. the k direction is known, the six elements of the Riemann tensor
are sufficient to determine the amplitudes of the gravitational waves.
Let us now apply this approach to our theory, where eqs.(8,9,10) must be satisfied. The second set, applied to a
perturbation that propagates in the positive z direction, shows that the h0µ modes are proportional to the hµz ones. The
null trace condition, instead, allows us to express the h00 or the hzz = ω2h00/k2 to the hxx and hyy modes. In our case
we still have all the six polarisation modes, as can be seen by computing the linearised Riemann tensor.
Coming back to the driving force matrix, it can be expresse in terms of the basis polarization matrices in the z direction:
Si j(t) =
6
∑
r=1
p(z,t)re(z)ri j,
where the amplitudes pr(z,t) are real and the index r runs over the six modes. The polarization tensor has the form
[12]:
e(z)1i j = −6

 0 0 00 0 0
0 0 1

 , e(z)2i j =−2

 0 0 10 0 0
1 0 0

 ,
e(z)3i j = 2

 0 0 00 0 1
0 1 0

 , e(z)4i j =−12

 1 0 00 −1 0
0 0 0

 ,
e(z)5i j =
1
2

 0 1 01 0 0
0 0 0

 , e(z)6i j =−12

 1 0 00 1 0
0 0 0

 ; (12)
the first tensor being related to Ψ2, the second and the third to the real and imaginary part of Ψ3, the two next are those
that correspond to Ψ4 and the last one is relative to the scalar Φ22 mode.
These modes can play a role in discriminating among theories of gravity, and in particular they can leave a signature
on the CMB anisotropies [15, 16]
Static weak field limit
When we want to reduce to Newtonian limit fields and eventual sources are taken to be static .
2 To determine standard observers each observer sees the wave travelling in the z-dir and measures the same frequency for a monochromatic wave.
1131
Looking at eq.(8) we get a screened Poisson equation:
∇2hµν =−µhµν , (13)
whose solution is a Yukawa-like potential.
OBSERVATIONAL CONSTRAINTS
The only parameter left in our theory is µ . In order to quantify it we may have recourse to the fit of the type Ia
supernovae luminosity which we presented in [1]. There we found a value for the bulk modulus; it was B ∼ 10−52
m−2. From this result we have
|µ | ∼ 10−51 m−2.
In order to compare it with the upper bounds that we find in literature it is worthwhile to rewrite our "mass" parameter
by using the Planck constant h¯ and the speed of light c so to get it with dimensions of a mass:
mg =
√
|µ | h¯
c
≃ 6 ·10−66kg.
We know that General Relativity passes all Solar System tests so that this immediately provides an upper limit for the
µ parameter that determines the Yukawa-like fall off of eq.(13) [17]. Moreover, the absence of this effect at the galaxy
cluster level provides the limit we were able to find [18]:
mg ≤ 2 ·10−65kg.
Other limits come from the dispersion in gravitational waves since, if the graviton had a rest mass, the decay rate of
an orbiting binary would be affected. As the decay rates of binary pulsars agree very well with GR, the errors in their
agreements provide a limit on the graviton mass [19, 20, 21] but this limit is dramatically less restrictive than the one
from the Yukawa potential. There are a lot of work on similar effects on the timing of a pulsar signal propagating
in a gravitational field [22], or on the measurements of dispersion in gravitational waves using interferometers or by
observing gravitational radiation from in-spiralling orbiting binaries [23, 24, 25, 26, 27].
Finally, an exhaustive review on "massive" gravitons has been done by Goldhaber and Nieto [28].
REFERENCES
1. A. Tartaglia and N. Radicella. A tensor theory of space-time as a strained material continuum. ArXiv:0903.4096, 2009.
2. D.G. Boulware and S. Deser. Can gravitation have a finite range? Phys. Rev. D, 3:3368–3382, 1972.
3. S. Deser. Ann. Israel Phys. Soc., 9:77–87, 1990.
4. M. Fierz. Helv. Phys. Acta, 12:3, 1939.
5. M. Fierz and W. Pauli. On relativistic wave-equations for particles of arbitrary spin in an electromagnetic Þeld. Proc. Roy. Soc.
London, A, 173:211, 1939.
6. M. Carrera and D. Giuliani. Classical analysis of the van dam - veltman discontinuity. arXiv:gr-qc/0107058, 2001.
7. S. Deser and B. Tekin. Newtonian counterparts of spin 2 massless discontinuities. Class. Quant. Grav., 18:L171, 2001.
8. H. van Dam and M. Veltman. Massive and mass-less yang-mills and gravitational fields. Nucl. Phys. B, 22:397–411, 1970.
9. V.I. Zakharov. Linearized gravitation theory and the graviton mass. JETP Lett., 12:312, 1970.
10. G. Gabadadze Deffayet, G.R. Dvali and A. I. Vainshtein. Nonperturbative continuity in graviton mass versus perturbative
discontinuity. Phys. Rev. D, 65:044026, 2002.
11. M. Porrati. Fully covariant van dam-veltman-zakharov discontinuity, and absencethereof. Phys. Lett. B, 534:209, 2002.
12. D.L. Eardley, D.M Lee and A.P. Lightman. Gravitational-wave observations as a tool for testing relativistic gravity. Phys. Rev.
D, 8:3308, 1973.
13. A.Z. Petrov. Sci. Nat. Kazan State University, 114:55, 1954.
14. Newman. An approach to gravitational radiation by a method of spin coefficients. J. Math. Phys., 3:566–578, 1962.
15. D. Bessada and O.D. Miranda. Cmb polarization in theories of gravitation with massive gravitons. Class. Quantum Grav.,
26:045005, 2009.
16. D. Bessada and O.D. Miranda. Cmb anisotropies induced by tensor modes in massive gravity. JCAP, 003, 2009.
17. C. Talmadge, J.P. Berthias, R.W. Hellings, and E.M. Standish. Model-independent constraints on possible modifications of
newtonian gravity. Phys. Rev. Lett., 61:1159–1162, 1988.
18. A.S. Goldhaber and M.M. Nieto. Mass of the graviton. Phys. Rev. D, 9:1119, 1974.
1132
19. T. Damour and J.H. Taylor. On the orbital period change of the binary pulsar psr 1913+16. Astrophys. J., 366:501–511, 1991.
20. J.H. Taylor, A. Wolszzan, T. Damour, and J.M. Weisberg. Experimental constraints on strong-field relativistic gravity. Nature,
355:132–136, 1992.
21. L.S. Finn and P.J. Sutton. Bounding the mass of the graviton using binary pulsar observations. Phys. Rev. D, 65:044022, 2002.
22. D. Baskarav, A.G. Polnarev, M.S. Pshirkov, and K.A. Postnov. Limits on the speed of gravitational waves from pulsar timing.
Phys. Rev. D, 78:044018, 2008.
23. D.I. Jones. Bounding the mass of the graviton using eccentric binaries. Astrophys. J., 618:L115–L118, 2005.
24. C.M. Will. Bounding the mass of the graviton using gw observations of insparalling compact binaries. Phys. Rev. D,
57:2061–2068, 1998.
25. S.L. Larson and W.A. Hiscock. Using binary stars to bound the mass of the graviton. Phys. Rev. D, 61:104008, 2000.
26. C. Cutler and S.L. Hiscock, W.A.and Larson. Lisa, binary stars and the mass of the graviton. Phys. Rev. D, 67:024015, 2003.
27. C.M. Will. Bounding the mass of the graviton with gravitational waves: Effect of spin precessions in massive black hole
binaries. arXiv:0906.3602, 2009.
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1133


A tensor theory of space-time as a strained material continuum.

An approach to gravitational radiation by a method of spin coefﬁcients.

binary stars and the mass of the graviton.

Bounding the mass of the graviton using binary pulsar observations.

Bounding the mass of the graviton using eccentric binaries.

Bounding the mass of the graviton using gw observations of insparalling compact binaries.

Bounding the mass of the graviton with gravitational waves: Effect of spin precessions in massive black hole binaries. arXiv:0906.3602,

Can gravitation have a ﬁnite range?

Classical analysis of the van dam - veltman discontinuity. arXiv:gr-qc/0107058,

Cmb anisotropies induced by tensor modes in massive gravity.

Cmb polarization in theories of gravitation with massive gravitons.

Experimental constraints on strong-ﬁeld relativistic gravity.

Fully covariant van dam-veltman-zakharov discontinuity, and absencethereof.

Gravitational-wave observations as a tool for testing relativistic gravity.

Israel Phys.

Limits on the speed of gravitational waves from pulsar timing.

Linearized gravitation theory and the graviton mass.

Massive and mass-less yang-mills and gravitational ﬁelds.

Model-independent constraints on possible modiﬁcations of newtonian gravity.

Newtonian counterparts of spin 2 massless discontinuities.

Nonperturbative continuity in graviton mass versus perturbative discontinuity.

On relativistic wave-equations for particles of arbitrary spin in an electromagnetic Þeld.

On the orbital period change of the binary pulsar psr 1913+16.

Photon and graviton mass limits. arXiv:0809.1003,

Using binary stars to bound the mass of the graviton.

Massive gravitational waves from the Cosmic Defect theory

N. Radicella

A. Tartaglia

Jean-Michel Alimi

André Fuözfa

Crossref

RADICELLA, NINFA

TARTAGLIA, Angelo

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

03 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEMassive gravitational waves from the Cosmic Defect theory / Radicella N.; Tartaglia A.. - In: AIP CONFERENCEPROCEEDINGS. - ISSN 0094-243X. - STAMPA. - 1241(2010), pp. 1128-1133. ((Intervento presentato al convegnoInvisible universe tenutosi a Parigi (Francia) nel 20 giugno - 3 luglio 2009.OriginalMassive gravitational waves from the Cosmic Defect theoryPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2381234 since:AIPMassive gravitational waves from the Cosmic Defect theoryN. Radicella and A. TartagliaDipartimento di Fisica, Politecnico di Torino and INFN sez. TorinoAbstract. The Cosmic Defect theory (CD), which is presented elsewhere in this conference, introduces in the standardEinstein-Hilbert Lagrangian an elastic term accounting for the strain of space-time viewed as a four-dimensional physicalcontinuum. In this framework the Ricci scalar acts as the kinetical term of the strain field whose potential is represented bythe additional terms. Here we are presenting the linearised version of the theory in order to analyze its implications in theweak field limit. First we discuss the recovery of the Newtonian limit. We find that the typical static weak field limit imposesa constraint on the values of the two parameters (Lamé coefficients) of the theory. Once the constraint has been implemented,the typical gravitational potential turns out to be Yukawa-like. The value for the Yukawa parameter is consistent with theconstraints coming from the experimental data at the Solar system and galactic scales. We then come to the propagatingsolutions of the linearised Einstein equations in vacuo, i.e. to gravitational waves. Here, analogously with other alternativeor extended theories of gravity, the presence of the strain field produces massive waves, where massive (in this completelyclassical context) means subluminal. Furthermore longitudinal polarization modes are allowed too, thus lending, in principle,a way for discriminating these waves from the plane GR ones.Keywords: Alternative models of gravity, gravitational wavesPACS: 04.30.-w, 04.50.KdFRAMEWORKThe Lagrangian density we want to analyse isL = R +12Cµνρσ εµν ερσ ,where the second term on the r.h.s. is a potential term, in the form of an elastic potential whose field is the strain tensorεµν.= 12 (gµν −ηµν), and the Ricci scalar is computed by means of the observed metric gµν .The strain is ascribed to a cosmic point-like defect, this is why we called the theory the Cosmic Defect theory [1] Inour analogy we are interested in an isotropic medium; in this case the elastic constants take a simple form:Cµνρσ = λ ηµν ηρσ + µ(ηµρηνσ + ηµσ ηνρ).They now depends on two parameters only, the so-called Lamé coefficients λ and µ .The Elastic potential then translates inV =12[λ ε2 + 2µεµνεµν],where, following the structure of the elastic coefficients, the covariant version of the ε tensor is obtained by loweringthe indices with the total metric g.The action integral is thenS =∫d4x [R +V ]√−g. (1)By varying the action in eq.(1) w.r.t. the metric gµν that is the only dynamical field we obtain the Elastic EinsteinEquations (EEE):Gµν = T eµν (2)whereT eµν = λ ε[(ε4+12)gµν − εµν]+ µ[εµν +12gµνεαβ εαβ −2εµαεαν].1128WEAK FIELD LIMITIn order to look for the gravitational waves of this theory we must linearize it around the Minkowski space-time, towhich the theory reduces locally when the defect is not so strong.It means that the metric can be written asgµν = ηµν + εhµν ε ≪ 1,where the perturation metric hµν is driven by ε . The strain tensor becomesεµν =12[gµν −ηµν]=12εhµν ,where we use the coordinate invariance of the theory in order to fix the flat metric to be the Minkowski one (i.e.Diag(−1,1,1,1)). The strain tensor represents the perturbation itself. Looking at the EEE, they are:Gµν = Teµν , (3)whereTeµν = λ ε[(ε4+12)gµν − εµν]+ µ[εµν +12gµνεαβ εαβ −2εµαεαν]. (4)First of all we should look at the linearized Einstein tensor, that, fromRαβ γδ ≃ ∂γΓαβ δ − ∂δ Γαβ γ ,reduces toGαβ ≃12[∂γ ∂β hγα + ∂ γ∂α hβ γ −hαβ − ∂α∂β h−ηαβ ∂γ∂ δ hγδ + ηαβh],where indices are raised and lowered by means of the Minkowski metric and h is the trace of the perturbation1.The linearised Elastic energy-momentum tensor becomesT eµν ≃λ2εηµν + µεµν .The linearised EEE in vacuo reduce to[∂γ∂β hγα + ∂ γ∂α hβ γ −hαβ − ∂α ∂β h −ηαβ ∂γ∂ δ hγδ + ηαβ h]− µ(hµν +λ2µ hηµν)= 0(5)In order to investigate more deeply these equations we rewrite them by using their divergence and trace. Thedivergence, thanks to the contracted Bianchi identities, gives a relation between the divergence of the perturbationand that of its trace:hµν;ν =−λ2µ h;µ . (6)By tracing eq.(5) one gets2(1 + λ2µ)h− µ(1 + 2λµ)h = 0. (7)The trace of the perturbation becomes a dynamical field. This degree of freedom is always ghostlike, regardless thecombination of the parameters [2, 3] (and it could be tachyonic, too): this makes the theory pathological from thequantum point of view. To avoid this behaviour we must constraint our parameters so that 1 + λ/2µ = 0. This choicereduces our linearised Elastic energy-momentum tensor to the Pauli-Fierz mass term [4, 5], which actually leads to1 It is worthwhile here to be precise about the sign convention on Riemann and Ricci tensors. In this paper we use the Riemann tensor writtenabove, where the upper index is the first one, and we contract the first and the third index in Riemann to obtain Ricci.1129different predictions from those of General Relativity, no matter how small the graviton mass is [6, 7]. This is what iscalled van Dam-Veltman-Zakharov discontinuity [8, 9], whose existence, accordingly to some authors, can be ascribedto the explicit breaking of the gauge invariance by the mass term so that it can be cured if the mass is generated bythe compactification of a higher dimensional theory [10] or to the evaluation of the linearised theory around flatbackgrounds [11].The debate on massive gravity is still open and we choose the Fierz-Pauli mass term as the linearised version of theCD theory.Coming back to eq.(7), the choice λ = −2µ makes the trace of the perturbation to vanish in vacuum and now eq.(6)translates in what would have been the Lorentz gauge in General Relativity. This really simplifies the linearisedEinstein tensor that reduces to −hµν/2 so that the equations one finally gets are:(+ µ)hµν = 0 (8)hµν;ν = 0 (9)h = 0. (10)Gravitational wavesThe set of eqs.(8, 9, 10) represent propagating massive waves and plane waves are solutions of this equations:hµν = αµνeiκβ xβ. (11)Designating the propagation direction as the z axis, the wave vector is κ = (ω ,0,0,ck) and from the eq.(11) we obtainthe dispersion relation:ω =±c√k2− µThe waves are subluminal, which is commonly referred to as being "massive" (µ < 0, consistently with thecosmological limit). Let us now look at which are the polarisation modes of this spacetime. From eqs.(8,9,10), wehave 5 dynamical degrees of freedom but this does not mean that the expect the same number of polarisation modes.In a metric theory of gravity there can be at most six polarisation modes, as shown in [12]. The analysis can beperformed by looking at the geodesic deviation equation, that states which is the displacement between a pair of free-falling particles when a gravitational wave arrives. The three-acceleration depends on the "electric" components of theRiemann tensor (Ri0k0). It can be shown that there are six algebraically independent components of the Riemann tensorby using the Newman-Pensore formalism. First, one choose a complex null basis, the so-called null tetrad (k, l,m,m¯),that is related to a cartesian system {t,x,y,z} byk = 1√2(1,0,0,1), l = 1√2(1,0,0,−1),m =1√2(0,1, i,0), m¯ = 1√2(0,1,−i,0).We remember that we have chosen to orient the axes so that the wave travels in the +z direction, and, u beingu = t−z/c, the k vector is proportional to ∇u. Then it is possible to split the Riemann tensor into irreducible parts [14]:the Weyl tensor (Ψ0,Ψ1,Ψ2,Ψ3,Ψ4), the traceless Ricci tensor (Φ00,Φ01,Φ11, Φ12,Φ22,Φ02, that are five complexscalars) and the Ricci scalar (Λ).When considering plane waves only some of them are different from zero and, among these, six are independent. Theones that are helicity (s) eigenstates under rotations about the z axis areΨ2,Φ22 → s = 0,Ψ3, ¯Ψ3 → s = 1,Ψ4, ¯Ψ4 → s = 2.1130We recall that these six wave amplitudes are observer dependent but there are some invariant statements that are truefor all standard observers2 if they are true for anyone. These statements constitute the E(2) classification of waves,based on the Petrov type of the Weyl tensor [13].These are related to the Riemann tensor, in cartesian coordinates, as follows [12]:Ψ2(u) = −16Rz0z0(u),Ψ3(u) = −12Rx0z0(u)+i2Ry0z0(u),Ψ4(u) = −Rx0x0(u)+ Ry0y0(u)+ 2iRx0y0(u),Ψ22(u) = −(Rx0x0(u)+ Ry0y0(u)) .What we measure in a detection experiment is the relative acceleration of test masses, that is the six "electric"components of the Riemann tensor. One can express these informations in the so-called driving-force matrixSi j(t) = R0i0 j.In general there are eight unknowns, six polarisations and two direction cosines, but if one can establish the directionof the gravitational wave by other information, i.e. the k direction is known, the six elements of the Riemann tensorare sufficient to determine the amplitudes of the gravitational waves.Let us now apply this approach to our theory, where eqs.(8,9,10) must be satisfied. The second set, applied to aperturbation that propagates in the positive z direction, shows that the h0µ modes are proportional to the hµz ones. Thenull trace condition, instead, allows us to express the h00 or the hzz = ω2h00/k2 to the hxx and hyy modes. In our casewe still have all the six polarisation modes, as can be seen by computing the linearised Riemann tensor.Coming back to the driving force matrix, it can be expresse in terms of the basis polarization matrices in the z direction:Si j(t) =6∑r=1p(z,t)re(z)ri j,where the amplitudes pr(z,t) are real and the index r runs over the six modes. The polarization tensor has the form[12]:e(z)1i j = −6 0 0 00 0 00 0 1 , e(z)2i j =−2 0 0 10 0 01 0 0 ,e(z)3i j = 2 0 0 00 0 10 1 0 , e(z)4i j =−12 1 0 00 −1 00 0 0 ,e(z)5i j =12 0 1 01 0 00 0 0 , e(z)6i j =−12 1 0 00 1 00 0 0 ; (12)the first tensor being related to Ψ2, the second and the third to the real and imaginary part of Ψ3, the two next are thosethat correspond to Ψ4 and the last one is relative to the scalar Φ22 mode.These modes can play a role in discriminating among theories of gravity, and in particular they can leave a signatureon the CMB anisotropies [15, 16]Static weak field limitWhen we want to reduce to Newtonian limit fields and eventual sources are taken to be static .2 To determine standard observers each observer sees the wave travelling in the z-dir and measures the same frequency for a monochromatic wave.1131Looking at eq.(8) we get a screened Poisson equation:∇2hµν =−µhµν , (13)whose solution is a Yukawa-like potential.OBSERVATIONAL CONSTRAINTSThe only parameter left in our theory is µ . In order to quantify it we may have recourse to the fit of the type Iasupernovae luminosity which we presented in [1]. There we found a value for the bulk modulus; it was B ∼ 10−52m−2. From this result we have|µ | ∼ 10−51 m−2.In order to compare it with the upper bounds that we find in literature it is worthwhile to rewrite our "mass" parameterby using the Planck constant h¯ and the speed of light c so to get it with dimensions of a mass:mg =√|µ | h¯c≃ 6 ·10−66kg.We know that General Relativity passes all Solar System tests so that this immediately provides an upper limit for theµ parameter that determines the Yukawa-like fall off of eq.(13) [17]. Moreover, the absence of this effect at the galaxycluster level provides the limit we were able to find [18]:mg ≤ 2 ·10−65kg.Other limits come from the dispersion in gravitational waves since, if the graviton had a rest mass, the decay rate ofan orbiting binary would be affected. As the decay rates of binary pulsars agree very well with GR, the errors in theiragreements provide a limit on the graviton mass [19, 20, 21] but this limit is dramatically less restrictive than the onefrom the Yukawa potential. There are a lot of work on similar effects on the timing of a pulsar signal propagatingin a gravitational field [22], or on the measurements of dispersion in gravitational waves using interferometers or byobserving gravitational radiation from in-spiralling orbiting binaries [23, 24, 25, 26, 27].Finally, an exhaustive review on "massive" gravitons has been done by Goldhaber and Nieto [28].REFERENCES1. A. Tartaglia and N. Radicella. A tensor theory of space-time as a strained material continuum. ArXiv:0903.4096, 2009.2. D.G. Boulware and S. Deser. Can gravitation have a finite range? Phys. Rev. D, 3:3368–3382, 1972.3. S. Deser. Ann. Israel Phys. Soc., 9:77–87, 1990.4. M. Fierz. Helv. Phys. Acta, 12:3, 1939.5. M. Fierz and W. Pauli. 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Massive gravitational waves from the Cosmic Defect theory

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