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Print version ISSN 0103-9733
Braz. J. Phys. vol.40 no.4 São Paulo Dec. 2010
Instituto de Física de São Carlos, Universidade de São Paulo CEP 369, 13560-970 São Carlos, SP, Brazil
This paper deals with the emission of gravitational radiation in the context of a previously studied metric nonsymmetric theory of gravitation. The part coming from the symmetric part of the metric coincides with the mass quadrupole moment result of general relativity. The one associated to the antisymmetric part of the metric involves the dipole moment of the fermionic charge of the system. The results are applied to binary star systems and the decrease of the period of the elliptical motion is calculated.
Keywords: dipole radiation, nonsymmetric.
In previous papers  we have studied a metric nonsymmetric theory of gravitation. In the flat space linear approximation [1-I] the antisymmetric part of the metric satisfies Maxwell's type vacuum equations, describing then a spin-1 field. Furthermore, the theory was shown to be free of ghost negative-energy modes when expanded about a Riemannian curved space, being outside of the ill-behaved nonsymmetric theories analyzed by Damour, Deser and McCarthy .
After establishing the field equations in [1-I] their solution for a point source mass was obtained in [1-II], together with its consequences for the motion of test particles and light. The theory was shown to be consistent with the four classical solar tests of general relativity (GR). Next in  the conservation laws associated to the theory were studied. In  we have proved the analogue of the GR static theorem, that is, that the field outside of a time-dependent spherically symmetric source is necessarily static. More recently  a post-Newtonian approximation of the theory was developed with the purpose of application to pulsar binary star systems.
The sources of the field are the energy-momentum-stress tensor Tαβ and the fermionic current density Sα found, for instance, in the description of the interior of stars (electrons, protons and neutrons).
In this work we investigate the emission of gravitational radiation predicted by the theory. It is shown that the emitted radiation consists of two parts, one coming from the mass quadrupole of the source and the other from its fermion dipole moment. The first one comes from the symmetric part of the metric, which coincides with the GR result, and the other from the antisymmetric part.
The results are considered for a binary pulsar system. As it is known in GR  the internal motion of the system are ellipses described by its components and the loss of energy by emission of radiation produces a decrease of the orbital period of the elliptical motion of the system. This is an astronomical observed effect and we want to analyze. With the results obtained in  we calculate the contribution of the dipole radiation to the GR result for the secular decrease of the orbital period. This contribution contains the fermionic charges of the pulsar and of its companion. Information about these quantities for particular binaries can also be obtained by analyzing the contribution to the GR values for the precession of the periastron and for the Doppler-red-shift parameter. What their contributions will be is a topic for future investigation.
In Sec. II we present the field equations. In Sec. III we use the energy- momentum-stress pseudotensor derived in  to calculate the rate of emitted radiation. After elaborating the weak-field expansion in Sec. IV the rate of radiation, the luminosity, is obtained. In Sec. V the results are applied to a binary system and the secular decrease of the orbital period of the pulsar is calculated. In Sec. VI we present a summary of our conclusions and highlight future work.
2. FIELD EQUATIONS
The field equations of the theory are [1-1]
The notation ( ) and [ ] designates symmetric and antisymmetric parts. In the first equation K = 8πG as usual and
symmetric because the second term is (see just after (2.10)). Λ is the cosmological constant and T = gρσTρσ. In equation (2.2), Γ[αβ] is the curl of the vector Γα = Γµ[αµ] which acts then as a vector potential. As it is defined up to a gradient we can choose the gauge
In (2.3) and (2.4) we use the notation X = with g=det(gαβ) and gαβ is the inverse of gαβ as defined by
¿From (2.4) we have
the equation of continuity for the fermionic current, saying that
is a constant. This is the fermion charge of the system. From (2.4) its dimension is of square-length.
Equation (2.3) can be solved for the symmetric part of the connection (1-I) giving ,
where sαβ, symmetric and with determinant s, is the inverse of g(αβ) as defined by sαβg(αγ) = δγβ . In deriving Eq. (2.10) we come across the relation , which can be re-obtained from that equation. We then see that the second term on the right of (2.5) is in fact symmetric.
3. THE LUMINOSITY
The total energy-momentum-stress divergence equation in the theory is 
Tµν is the upper indices matter stress tensor, with which the down indices Tαβ is related to by
and tασ is the generalized gravitational stress pseudotensor.
Integration of (3.1) for α = 0 in a volume containing the localized source gives
where S is the surface that involves the volume V and ni is the normal to the surface with area element dA. The first term is the rate of decrease of the total energy P0 ,
Taking S outside the localized matter source, in the radiation zone, equation (3.6) says that the rate of decrease is given by the flux of t0i, which is the generalized gravitational Poynting vector. Then the rate at which the system looses energy due to gravitational waves generated by the internal motion of its components is given by
where, with the sources located around the origin, the gravitational waves energy rate is given by
Here r = |r| is the radius of S at infinity and ni = xi/r is its normal. This is the generalized luminosity formula. From (13)
From (3.9) we see that this quantity needs to be calculated only to order r-2 at infinity. Therefore, as it is quadratic in the fields we will need the fields only to order r-1. Also, only to order G to have the luminosity to that order.
As for P0 the integrand in (17) contains the two terms
where we have neglected here the Λ term as in GR. With the fields calculated to order G we will have also the total energy P0 to that order.
For a star binary system that we are interested in, the internal motion are ellipses described by its components and the loss of energy P0 will produce a decrease of the orbital period P of the elliptical motion of the system. To obtain it we go to the weak-field expansion in the next section.
4. WEAK-FIELD EXPANSION
We now expand the field equations about a flat space-time background by putting
where hαβ is the Minkowski metric diag(+1,-1,-1,-1) and |hαβ| << 1. Then to linear order we have, dispensing with O(h2),
where h = ηαβh(αβ). From (7)
where indices are raised and lowered with η i.e., hλα = hαβhλβ. Thence,
For the inverse of g(αβ) we have
From here the linear part of (10) is
because ln(s/g) is of second order in the fields. In fact, we have the relations g-1 = εαβγδεµνρσ gαµgβνgγρgδσ/4! and s-1 = εαβγδεµνρσ gαµgβνgγρgδσ/4!. Writing gαβ = g(αβ) + g[αβ] we find, to lowest order, g-1 = s-1- h[µν]h[µν]/2 or s/g = 1+h[µν]h[µν]/2. Therefore, ln(s/g) = h[µν]h[µν]/2 and (4.6) follows. Then,
Separating the symmetric and antisymmetric contributions in (10) we have to the considered order
The luminosity in (3.9) splits in the two corresponding terms
The first contribution is
In the Appendix we show that this contribution reproduces the GR mass quadrupole result.
is the traceless mass quadrupole moment to be taken at the retarded time t - r.
Let us consider now the second contribution
As we are outside the sources and at infinity equation (2) tell us that equation (32) can be written
For a system of particles T[µν] is null. Thence, the right-hand-side of (2) to first order in G is, from (15), equal to kT[0i](0) = kµ0νηµiT[µν](0) = 0. Therefore, the equation reduces to
in all space. Equation (2.4) to first order is
Taking the divergence of (39) and using the second we obtain, from (6),
The solution is
Far away we have, with t* = t - r being the retarded time with respect to the origin around which the source is located, we have to dipole order
where [S\dot]α = dSα/d t*. Therefore the time component is, with (9),
where F is the fermion charge and
is the fermionic dipole moment. For the space component we have
From (2.9), xiSµ,µ = 0 or xiS0,0+(xiSj),j-Si = 0. Thence
Taking (44) and (46) in the time-space component of (39) we have, noting that ∂i[p\dot]j(t*) =[p\dot]jni,
For the space-space component
With these expressions (38) gives, after contraction with nir2,
As the angular integral of ninj is equal to 4πδij/3 we find
where we have taken Λ < 0 in agreement with [1-I]
5. THE BINARY PULSAR SYSTEM
We consider now a binary system and make use of the total energy of the system obtained in ,
where r = | x1(t)-x2(t)| is the distance between the particles.
For the bound elliptical motion of the system we have
where V is the constant velocity of the center of mass of the system and
is its internal energy, a being the semimajor axis of the relative elliptical motion. From Kepler\'s third law,
As argued in GR, due to the emission of radiation the ellipsis will be deformed with time variation of a and P. Differentiation of the above relations and eliminating the term da/dt it follows, with dEI/dt = dP0/dt from (53),
Therefore, with (3.8) the decrease of the period P due to the emission of radiation is given by,
Let r1 and r2 be the positions of the pulsar and of its companion measured from the center of mass of the system, and r = r1-r2. We then have m1r1+m2r2 = 0 and in terms of the total mass M = m1+m2,
¿From here the fermionic dipole moment of the system is, p = F1r1+F2r2 or
where µ = m1m2/M is the reduced mass and
is the dipole parameter.
The Keplerian orbit for the system, with eccentricity e is given by
where φ is the angle between r and the periastron direction. The rate of energy lost by the system is given by the sum of the results in equations (35) and (51). The first one gives the GR result , the average over one period being ,
For the second one the average is
Using (62) we obtain
With these results the secular decrease of the orbital period P is give by
Due to the presence of the cosmological constant the contribution of the dipole is not expected to be important for a determination of the fermion charges of a given system. For this purpose we will then analyze in a third paper the decrease of the precession of the periastron of the binary system and of the Doppler-red-shift parameter.
Giving sequence to the study of a metric nonsymmetric theory of gravitation we have discussed the emission of gravitational radiation. The out put power of emission contains two terms, one involving the symmetric part of the metric and the other its antisymmetric part. The first one agrees with the mass quadrupole result of GR and the other is a dipole emission due to the fermionic current of the system.
The results have been applied to a star binary system and the decrease of the orbital period of the system has been calculated. The dipole contribution contains the fermion charges Fp and Fc of the pulsar and its companion. As the dipole part is proportional to the cosmological constant the result of the calculation indicates that the dipole part is negligible. Additional information is then needed for the determination of the fermion charges of a given system. This can be obtained by analyzing their contribution to the GR values for the decrease of the precession of the periastron of the binary system, and of the Doppler-red-shift parameter. What their contributions will be is a topic for future work.
 S. Ragusa, Phys. Rev. D 56, 864 (1997). [ Links ]. The term 2Da/3 of equation (6.7) is here replaced by Ga; Gen. Relat. Gravit. 31 275 (1999). [ Links ] These papers will be referred to as I and II, respectively.
 T. Damour, S. Deser, and J. McCarthy, Phys. Rev. D 45, R3289 (1992); 47, 1541 (1993). [ Links ]
 S. Ragusa, Braz. J. Phys. 35, 1020 (2005). [ Links ]
 S. Ragusa and F. Bosquetti, Braz. J. Phys. 36, 1223 (2006). [ Links ]
 S. Ragusa, submitted.
 See for instance, N Straumann, General Relativity and Relativistic Astrophysics, Springer-Verlag 1984. [ Links ]
 P. C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963). [ Links ]
(Received on 15 January, 2010)
Equation (1) gives to lowest order,and neglecting the contribution of the cosmological constant as in the corresponding equation in GR,
where (αβ) is the matter tensor of order zero, with no interactions, of special relativity. By contracting with ηαβ and going back, we obtain
where in the last step we have gone to the upper indices energy tensor by (13). From here contraction with ηανηµβ gives
With equations (28) and (29), we have
and from here,
Substituting in (73),
From this we have the free matter conservation law
and adopting the Hilbert gauge
The solution of this equation with an outgoing boundary condition is
At points far from the source (r >> d=linear dimension of source, ) and keeping only the first term of the expansion of the integrand we have
at the retarded time t - r relative to the origin. We can now calculate the first contribution to the luminosity in (34). Using in (31) the inverse of (78),
where Θ = ηαβΘαβ, equation (35) follows.