Exponential Metric Fields arXiv:0710.5071v2 [gr-qc] 15 Jun 2010 Wytler Cordeiro dos Santos Abstract The Laser Interferometer Space Antenna (LISA) mission will use advanced technologies to achieve its science goals: the direct detection of gravitational waves, the observation of signals from compact (small and dense) stars as they spiral into black holes, the study of the role of massive black holes in galaxy evolution, the search for gravitational wave emission from the early Universe. The gravitational red-shift, the advance of the perihelion of Mercury, deflection of light and the time delay of radar signals are the classical tests in the first order of General Relativity (GR). However, LISA can possibly test Einstein’s theories in the second order and perhaps, it will show some particular feature of non-linearity of gravitational interaction. In the present work we are seeking a method to construct theoretical templates that limit in the first order the tensorial structure of some metric fields, thus the non-linear terms are given by exponential functions of gravitational strength. The Newtonian limit obtained here, in the first order, is equivalent to GR. so different from others fields, for example the electromagnetic field, the scalar field, etc., by themselves obey linear equations in a given spacetime, they form a nonlinear system when their mutual interactions are taken into account. The distinctive feature of the gravitational field is that it is self-interacting (as the YangMills field): it is non-linear even in the absence of other fields. This is because it defines the spacetime over which it propagates (Hawking & Ellis 1973). Linearized gravity is any approximation to General (0) (0) Relativity obtained from gµν = gµν + hµν (where gµν is any curved background spacetime) in Einstein’s equation and retaining only the terms linear in hµν (Wald 1984). The weakness of the gravitational field means in the context of general relativity that the spacetime is nearly flat. Small gravitational perturbations in Minkowski space can be treated in the simplest linearized version of General Relativity, Keywords Linearized gravity, Newtonian limit, Gravitational waves as describing a theory of a symmetric tensor field hµν propagating on at background spacetime. This theory is Lorentz invariant in the sense of Special Relativity. If one wants to obtain a solution of the nonlinear equations, it is necessary to employ an iterative method on approximate linear equations whose solutions are shown to converge in a certain neighbourhood of initial surface. It should be possible to avoid some of these difficulties of non-linearity by working in some spacetimes that shall be described in this paper. The proposal is that some metric fields can be separated into the parts carrying the dynamical information and those parts characterizing the coordinates system. In this proposal, terms of the coordinates system will have tensorial structure limited only in the first order. The tensors that will describe gµν have linear behavior. Naturally, there is a price that must be paid for the linear 1 Introduction The extreme difficulties which arise if one tries to draw physically important conclusions from the basic assumptions of Einstein’s theory are mainly due to the non-linearity of the field equations. Moreover, the fact that the spacetime topology is not given a priori, and the impossibility to integrate tensors over finite regions cause difficulties unknown in other branches of mathematical physics. Actually in this respect they are not Wytler Cordeiro dos Santos e-mail: wytler@fis.unb.br Universidade de Brası́lia, Faculdade do Gama, Brası́lia DF gµν = ηµν + hµν , (1) 2 tensors. The dynamic terms that carry the gravitational strength have exponential structure. The principal idea came from the basic principle that one should interpret (1) as separation between pure mathematical and physical terms in metric field tensor. In spite of the fact that ηµν plays a key role as empty flat and background spacetime of the Standard Model in the description of fundamental interactions, this background tensor metric ηµν is an object wholly mathematical and entirely geometrical, while hµν contains the physical information. The strength of gravity is tied in the components of hµν . The proposal of this paper is a working hypothesis to untie the strength of gravity from geometrical tensors. This proposal is valid for a family of metric field tensors gµν , and some basic examples such Newtonian limit and gravitational plane waves of low amplitudes are described. This paper is outlined as follows: in Sec. II, we present the basic mathematic concepts of the (quasi)idempotent tensors that compose the structure of metric fields approached in this work. In Sec. III, we propose how to link the strength of gravity with the tensors from Sec. II, then is defined a family of exponential metrics. In Sec. IV, we present some examples of these exponential metrics, such as: Yilmaz metric, circularly polarized wave and rotating bodies. In Sec. V, we present exponential metrics (‘adjoint metric field’) that are non-physical, but which help us to compute Christoffel symbols, and consequently the curvature tensors, Ricci tensors and determinant of metric field. In Sec. VI, we verify the Newtonian limit and also we obtain gravitational waves. In Sec. VII, we present a general conclusion. We assume spacetime (M, g) to be a C ∞ 4−dimensional, globally hyperbolic, pseudo-Riemannian manifold M with Lorentzian metric tensor g (whose components are gµν ) associated with the line element ds2 = gµν (x)dxµ dxν , assumed to have signature (+ − −−) (Landau & Lifschitz 1984). Lower case Greek indices refer to coordinates on M and take the values 0, 1, 2, 3. The relation between the metric field gµν and the material contents of spacetime is expressed by Einstein’s field equation, 8πG 1 Rµν − gµν R = 4 Tµν , 2 c (2) Tµν being the stress-energy-momentum tensor, Rµν the contract curvature tensor (Ricci tensor) and R its trace. In an empty region of spacetime we have Rµν = 0, such a region is called vacuum field. 2 Pure Mathematical terms of spacetime geometry The Minkowski flat spacetime (R4 , η), where the components of η are ηµν = diag(1, −1, −1, −1), is the simplest empty spacetime and it is in fact the spacetime of Special Relativity. One can obtain spaces locally identical to (R4 , η) but with different (large scale) topology properties by identifying points in R4 which are equivalent under a discrete isometry without a fixed point. Any local Lorentz frames is merely the statement that any curved space has the Minkowski flat space ‘tangent’ to it at any point. The Minkowski spacetime is the universal covering space for all such derived spaces (Hawking & Ellis 1973). In this sense it should be reasonable to one choose (R4 , η) as background spacetime and important compound piece of some spacetimes. In fact, the most straightforward approach to linear gravitation is realized in Minkowski spacetime. The conformal structure of Minkowski space is what one would regard as the ‘normal’ behavior of a spacetime at infinity. The metric tensor η from background Minkowski spacetime in any coordinates is an object wholly mathematical and entirely geometrical. No strength of gravity is linked up with the mathematical structure of η. Then, we choose this metric tensor as a principal descriptive piece of some spacetimes built below, and it does rise and lower the indices in the same way as in Special Relativity with ηµν η να = δµ α . The aim of this paper is to obtain some spacetimes (M, g) that their non-linearity are less hard, softer at least in a tensorial descriptive way. Therefore, it is defined only a symmetric tensor Υ, which like η is an object wholly mathematical and entirely geometrical. This (metric) tensor Υ that will be a piece of metric g can have non-static terms from spacetime in their components, however in this approach, this tensor purely will not have gravitational strength. The components Υµν of tensor Υ are raised and lowered by η µν and ηµν , η µν Υνα = Υµ α , Υµν η να = Υµ α , µν η Υνα η αβ = Υµβ . (3) In this context it is adopted the point of view, that Υ is a tensor on a background Minkowski spacetime, similar to deviation hµν from linearized version of general relativity (1). But instead one has the infinitesimal condition for |hµν | ≪ 1, it is accepted that the magnitude of Υµν can be equal to the magnitude of empty flat spacetime (|Υµν | ≈ |ηµν |). Moreover, it might be defined as an important mathematical relationship among 3 Υµν themselves, Υµν Υνρ = −2Υµ ρ . (4) The above equation is an important argument to shape some spacetimes that are described below. This equation will improve linearity in the tensorial sense. The contracting indices of tensor Υ by themselves show that Υµν are (quasi-)idempotent elements (Υ · Υ ∝ Υ; otherwise a factor −2 in the operation), and the equation (4) improves at least to a linear tensorial template of some tensor metric fields. It is possible to rise or lower indices of Υµν operating themselves, ρµ Υ Υµν Υ νσ ρσ = 4Υ . (5) One can also verify the expression Υµ ν Υνρ = −2Υµρ . The trace of Υ is obtained when ρ = µ in the expression (4), Υµν Υνµ = −2Υµ µ , (6) and realize derivative calculation of trace Υµ µ from (6) since ηµν = diag(1, −1, −1, −1), µ µν −2∂α Υµ = 2Υµν ∂α Υ , (7) g(Φ, x) = eΦ η + sinh(Φ)Υ, with components gµν = eΦ ηµν + sinh(Φ)Υµν . Υµν ∂α Υµν = Υµν ∂α Υµν = −∂α Υµ µ . (8) Only both these tensors, η from background Minkowski spacetime and Υ, will become the mathematical and geometrical basis to physical spacetimes described as follows. 3 Strength of gravity terms tied in spacetime geometry A dimensionless parameter Φ characterizing the strength of gravity at a spacetime point ℘ with coordinates (t, x) = (xα ) due to a gravitating source is the ratio of the potential energy, mϕN (due to this source), to the inertial mass-energy mc2 of a test body at ℘, i.e., ϕN (xα ) . c2 (9) Here ϕN (xα ) is the gravitational potential. For a point source with mass M in Newtonian gravity, Φ(xα ) = − GM , c2 r (10) where r is the distance to the source. So, for a nearly Newtonian system, we can use Newtonian potential for ϕN . (11) The inverse tensor is just g µν = e−Φ η µν − sinh(Φ)Υµν . (12) If the tensor Υ is diagonal, then the inverse tensor of g is g−1 (Φ, x) = g(−Φ, x) = e−Φ η − sinh(Φ)Υ. (13) From the above definitions it follows that gµν g να = δµ α , in fact gµν g thus, Φ(xα ) = To construct spacetimes with the basis η from background Minkowski spacetime and Υ defined in previous section, it is proposed to tie the strength of gravity Φ to theses tensors. While the tensor Υ can be a function of the coordinates (t, x) = (xα ), the strength of gravity Φ will be a function of the coordinates and also of the Newtonian gravitational constant G and of a parameter of mass M . Thus, I do propose: να =    Φ −Φ να να e ηµν + sinh(Φ)Υµν e η − sinh(Φ)Υ where the rightside is   δµ α + sinh(Φ) e−Φ Υµν η να − eΦ ηµν Υνα − sinh(Φ)Υµν Υνα that is δµ α i h α να Φ −Φ + sinh(Φ) −(e − e )Υµ − sinh(Φ)Υµν Υ α να
α 2 δµ − sinh (Φ) 2Υµ + Υµν Υ = (14) Using the expression (4) in the second term in the parenthesis (Υµν Υνα = −2Υµ α ) it results in, gµν g να = δµ α − sinh2 (Φ) [2Υµ α + (−2)Υµ α ] , (15) finally one did prove this identity gµν g να = δµ α . Because of equation (4), there are no quadratic values naturally in Υ and consequently the non-linearity of metric field will be less complicated. If one considers some spacetime (M, g) such as the same of g from definition (11), one can observe that the first term γµν = eΦ ηµν is a background spacetime conformally flat (where the Weyl tensor vanishes). In the previous section one has accepted that |Υµν | ≈ |ηµν |, but now because of sinh (Φ) (that can be small) multiplying this tensor Υ, it can be understood as (small) disturbance Sµν = sinh(Φ)Υµν , such as, gµν = γµν + Sµν , (16) 4 if one is dealing with the Einstein’s vacuum equation, the (small) disturbance that represents the gravitational wave can be separated from the conformally flat background spacetime γµν (Birrell & Davies 1982). 4 Some Examples 4.1 Yilmaz Metric As a first application of the metric field defined in previous sections, let us take Υµν to be: Υµν  0 0  = 0 0 0 2 0 0 0 0 2 0   0 0  0  , Υµ ν = 0 0 0 0 2 0 2 0 0 0 0 2 0  0 0  0 2 0 −2 0 0 0 0 −2 0  0 0  0 −2 and Υµν  0 0 = 0 0 (17) that Υµν Υνα = −2Υµ α , with trace given Υµν Υνµ = −2Υµ µ = −2TrΥ, that Tr(Υ) = −6 . Now we can display the tensor metric (11), gµν gµν = =    0 1 0 0 0 0 −1 0 0 0   + sinh(Φ) eΦ  0 0 0 −1 0  0 0 0 0 −1   Φ e 0 0 0 0 −e−Φ 0 0  ,  0 0 −e−Φ 0  0 0 0 −e−Φ 0 2 0 0 0 0 2 0  0 0 0 2 (18) GR: g00 = 1+Φ+ Φ2 3Φ3 + + ··· 2 16 g11 = −1 + Φ − 3Φ2 − ··· , 8 g11 coefficients differing only in the second order terms, while g00 differing in the third order. Both, Yilmaz and GR, give observational indistinguishable predictions for red-shift, light bending and perihelion advance, but the Yilmaz metric does not admit black holes. Citing this property and assuming that Yilmaz theory is correct, Clapp (Clapp 1973) has suggest that a significant component of quasar red-shift may be gravitational. Robertson (Robertson 1999a,b) has suggested that some neutron stars and black hole candidates may be like ‘Yilmaz stars’. Robertson argues that neutron star with mass ∼ 10M⊙ is found for Yilmaz metric while that an object of nuclear density greater than ∼ 2.8M⊙ should be a black hole in Schwarzschild metric. Ibison has tested Yilmaz theory by working out the corresponding Friedmann equations generated by assuming the Friedmann-Robertson-Walker cosmological metrics (Ibison 2006). There are a series of claims and counter-claims involving Fackerell (Fackerell 1996; Yilmaz 1994; Alley & Yilmaz 2000), and also Misner and Wyss (Wyss & Misner 1999; Misner 1999; Alley & Yilmaz 1999) about Yilmaz theory. At the present time both Yilmaz and Schwarzschild solutions give results in agreement with observation (Rosen 1974). However, it may be possible in the future, with LISA mission (NASA 2010; Baker et al. 2007), to distinguish between Yilmaz and Schwarzschild. since −eΦ + 2 sinh(Φ) = −e−Φ . Then the line element is, 4.2 Circularly Polarized Wave ds2 = eΦ c2 dt2 − e−Φ (dx2 + dy 2 + dz 2 ). Gravitational waves are one of the most important predictions of General Relativity. Now we can try a solution of gravitational waves in z direction, (19) This metric field (18,19) has been proposed by Yilmaz (Yilmaz 1958, 1992, 1976, 1982, 1973, 1977, 1997). In 2GM the case of a mass singularity, Φ = − 2 ≪ 1 we c r have the far-field metric, ds2 = (1 − 2GM 2GM 2 2 )c dt − (1 + 2 )(dx2 + dy 2 + dz 2 ). (20) c2 r c r This is to be contrasted with the Schwarzschild (in General Relativity, GR) line element in isotropic coordinates (Landau & Lifschitz 1984), Υµν Υµ ν ds =  1 + Φ/4 1 − Φ/4 2 2 2 c dt −  Φ 1− 4 4 2 2 2 2 2 2 (dr + r dθ + r sin θdφ ), (21) if we compare expansions with the line element of Yilmaz theory (19), Yilmaz: g00 = 1+Φ+ Φ2 Φ3 + + ··· 2 6 g11 = −1 + Φ − Φ2 + ··· 2 = 0 1 + cos ζ sin ζ 0  0 0 Υµα η αν =  0 0 0 sin ζ 1 − cos ζ 0 0 −1 − cos ζ − sin ζ 0  0 0 , 0 2 0 1 + cos ζ sin ζ 0 0 sin ζ 1 − cos ζ 0 0 − sin ζ −1 + cos ζ 0  0 0  0  −2 (22) and Υ 2 =  0 0 0 0 µν = Υαβ η αµ βν η  0 0 = 0 0  0 0 0 2 (23) so Υµν Υνα = −2Υµ α is verified for the tensor above and Tr(Υ) = −4. Then, if one chooses ζ = ωt − kz, the metric field is a solution for a gravitational plane 5 wave gµν = eΦ ηµν + sinh(Φ)Υµν , gµν = + eΦ 0  0 0  the presence of g00 and g33 as static terms, weakly pertubed in these coordinates.  0 0 0 − cosh(Φ) 0 0   0 − cosh(Φ) 0  0 0 −eΦ   0 0 0 0 0 cos ζ sin ζ 0  sinh(Φ)  0 sin ζ − cos ζ 0 . 0 0 0 2 4.3 Rotating Bodies (24) Where the first term can be the background spacetime (asymptotically flat) and the second term is the disturbance in this background or in other words the circularly polarized radiation hTµνT with amplitude sinh(Φ). The gravitational wave polarization is important from astrophysical and cosmological viewpoints. A binary system of two stars in circular orbit one around the other is expected to emit circularly polarized waves in the direction perpendicular to the plane of the orbit (Schutz 1990). Moreover, the Big Bang left behind an echo in the electromagnetic spectrum, the cosmic microwave background, but the Big Bang most likely also left cosmological gravitational waves that will be possible to observe with the help of LISA (NASA 2010; Baker et al. 2007). Since cosmological gravitational waves propagate without significant interaction after they are produced, once detected they should provide a powerful tool for studying the early Universe at the time of gravitational wave generation (Buonanno 2004). Various mechanisms for cosmological gravitational wave generation have been proposed, and many of these state that the cosmological gravitational wave are circularly polarized. T. Kahniashvili et al (Kahniashvili et al. 2005) argued that helical turbulence produced during a first-order phase transition generated circularly polarized cosmological gravitational waves. Other physicists have said that the parity violation due to the gravitational Chern-Simons term in superstring theory can produce the primordial gravitational waves with circular polarization (Lue et al. 1999; Choi et al. 2000; Alexander et al. 2006; Satoh et al. 2007; Saito et al. 2007). If we assume long distance from source, we can obtain plane wave solution with Φ ≪ 1 so that, gµν = = (1 + Φ)ηµν + ΦΥµν   1+Φ 0 0 0 −1 0 0  0   0  0 −1 0 0 0 0 −1 + Φ   0 0 0 0 sin ζ 0 0 cos ζ +Φ  0 sin ζ − cos ζ 0 0 0 0 0 the second term is the circularly polarized radiation hTµνT . Because of the far distance from source, we have The Kerr metric is important astrophysically since it is a good approximation to the metric of a rotating star at large distances. It is possible to obtain a kind Kerr metric from gµν = eΦ ηµν + sinh(Φ)Υµν , that in coordinates (t, x, y, z) the tensor Υµν is:  cosh2 Λ − sinh Λ cosh Λ sin φ   sinh Λ cosh Λ cos φ − sinh Λ cosh Λ sin φ sinh2 Λ sin2 φ − sinh2 Λ cos φ sin φ 0 0  0 0  0 sinh Λ cosh Λ cos φ − sinh2 Λ cos φ sin φ sinh2 Λ cos2 φ 0 0 (25) satisfying Υµν Υνρ = −2Υµ ρ with Tr(Υ) = −2. In this example we choose Υ33 = 0, but if the choice was Υ33 = 2, the above tensor still satisfy the algebra (4). One can change coordinates (t, x, y, z) to the BoyerLindquist coordinates (t, r, θ, φ), with spatial part as flat space in ellipsoidal coordinates, t x y z = = = = t, p 2 2 pr + a sin θ cos φ, 2 r + a2 sin θ sin φ, r cos θ, (26) The Minkowski tensor metric related to them is: 1 ηµν    0   =   0   0 0  0 2 2 cos2 θ − r +a r2 +a2 0 0 0 −r 2 − a2 cos2 θ 0 0 0
− r 2 + a2 sin2 θ 0       ,      (27) we are assuming that the angle φ from tensor Υµν of (25) can be the same angle from transformations (26). So, this coordinate transformations will become tensor (25) in:        Υµν = (−2)        cosh2 Λ 0 0 sinh Λ cosh ΛR sin θ 0 0 0 0 0 0 0 0 sinh Λ cosh ΛR sin θ 0 0 sinh2 ΛR2 sin2 θ               (28) √ where R = r2 + a2 . At the appendices, it is verified with details that the above tensor obeys the algebra (4) in the background Minkowski spacetime (27). Now we can choose a particular solution for this tensor choosing the geometric terms sinh Λ and cosh Λ: √ a sin θ r 2 + a2 sinh Λ = − and cosh Λ = , (29) ρ ρ 6 (0) with ρ2 = r2 + a2 cos2 θ (30) satisfying cosh2 Λ − sinh2 Λ = 1. The physical terms that contain the strength of gravity eΦ and sinh Φ can be: 1 Φ= Mr ≪ 1, + a2 ) (31) (r2 so that: eΦ ≈ 1 + Φ = 1 + Mr (r2 + a2 ) and sinh(Φ) ≈ Φ = Mr . + a2 ) (r2 We can compute each term of metric field gµν (for more details see appendices), g00 = g03 = g11 = g22 = g33 = Mr ∆ − a2 sin2 θ + 2 , ρ2 (r + a2 ) 2 2M ra sin θ g30 = , ρ2 2 2 ρ M rρ − + , ∆ ∆(r 2 + a2 ) 2 M rρ −ρ2 − 2 , r h + a2 i
2 2 2 − sin θ r + a2 − ∆a2 sin2 θ ρ2 4.4 Deformed Schwarzschild spacetime − M r sin2 θ, where we use the definition , ∆ = r2 − 2M r + a2 . (32) The metric tensor gµν in the matrix form is:  ∆−a2 sin2 θ      2M ra sin2 θ ρ2 ρ2 0 0 0 − ρ∆ 0 0 −ρ2 2M ra sin2 θ ρ2 0 0    +  0 2  with 0 0 Υµ ν = Υµα η αν 0 ρ2 0 0 M rρ ∆(r 2 +a2 ) 0 0 0 0 0 rρ − rM2 +a 2 0 2 0 −M r sin2 θ    .  (33) and The above first matrix is just ‘exact Kerr solution’. While the second matrix can be seen as deviation or deformity from ‘exact solution’. Then, it makes sense to see this deviation as approximate solution of (0) gµν = gµν + hµν , 1 In Another example can be given in spherical coordinates (t, r, θ, φ) where the components of metric tensor of Minkowski flat spacetime is: ηµν = diag(1, −1, −r2 , −r2 sin θ), with the respective inverse 1 η µν = diag(1, −1, − r12 , − r2 sin θ ). As we have seen before, let us describe a simpler (quasi-)idempotent tensor in spherical coordinates given by   0 0 0 0 0 2  0 0  Υµν =  (35) 2 2 0 0 r −r sin θ 0 0 −r2 sin θ r2 sin2 θ   0   0 i h 2 − sin2 θ (r 2 +a2 ) −∆a2 sin2 θ Mr (r 2 +a2 ) 2 where gµν is some known exact solution (that in this case is Kerr solution) and hµν is the perturbation. For Einstein’s vacuum equation it is possible to obtain explicitly the vacuum perturbation equations from an arbitrary exact solution (Wald 1984). Gravitational wave observations of extreme-massratio-inspirals (EMRIs) by LISA will provide unique evidence for the identity of the supermassive objects in galactic nuclei. It is commonly assumed that these objects are indeed Kerr black holes. K. Glampedakis and S. Babak argue that from the observed signal, LISA will have the potential to prove (or disprove) this assumption, by extracting the first few multipole moments of the spacetime outside these objects. The possibility of discovering a non-Kerr object should be taken into account when constructing waveform templates for LISA’s data analysis tools. They provide a prescription for building a ‘quasi-Kerr’ metric, that is a metric that slightly deviates from Kerr, and present results on how this deviation impacts orbital motion and the emitted waveform (Glampedakis & Babak 2006). this paragraph we assume G = 1 and c = 1. (34) Υµ ν = η µα Υαν  0 0 −1 sin θ  0 0   1  sin θ −1  0 0 −1  0 0  . sin θ −1 0 0 0 −2 = 0 0 0 0 0 0 0 −2 = 0 0 0 0 1 sin θ (36) (37) 7 Then Tr(Υ) = Υµ µ = −4. One can show that Υµν given by η µα Υαβ η βν is, Υµν  0 0 = 0 0 0 2 0 0 0 0 1 r2 1 − r2 sin θ 0 0    1  − r2 sin θ 1 r 2 sin2 θ (38) These examples suggest that the Tr(Υ) ∈ Z are constant numbers. Forward it will be necessary to calculate the derivative of Tr(Υ) in any situations, thus we shall assume that, ∂α Tr(Υ) = 0 (42) in this paper. that satisfies the algebraic relation (4), Υµα Υαν = =  0 0  0 0  0 0 0 4 0 0   0 2 − sin2 θ  0 −2 sin θ 2   0 0 0 0 0 −2 0 0  ν  −2  1  = −2Υµ . (39) 0 0 −1 sin θ 0 0 sin θ −1 From expression (11) we have the metric field gµν = eΦ ηµν + sinh ΦΥµν , it follows that, g00 grr gθθ gφφ gθφ = eΦ , 1 = − Φ, e = −r2 cosh Φ, = −r2 sin2 θ cosh Φ, = gφθ = −r2 sin θ sinh Φ, (40) in the case that Φ = − 2GM c2 r ≪ 1 with line element expanded in the first order of the strength of gravity, 2 ds = +   dr 2 2GM  − r 2 dθ 2 − r 2 sin2 θdφ2 c2 dt2 −  1− 2 c r 1 − 2GM 2 c r {z } | Schwarzschild spacetime   GM 4 r sin θdθdφ, (41) c2 This metric field is asymptotically flat, the components approach those of Minkowski spacetime in spherical coordinates. If only gθφ would be vanished, we could obtain the Schwarzchild spacetime. However, it is necessary the terms Υθφ = Υφθ = −r2 sin θ in the symmetric tensor of equation (35) for this tensor may obey the algebraic relation (4). Nevertheless, it raises a question: what kind of gravitating source should distort a spherically symmetric spacetime? The main purpose of this section is to illustrate some (quasi-)idempotent tensors Υ that must satisfy the algebraic relation (4). There are many works with discussions about Yilmaz metric field and their sources. However, this paper does not discuss the physics of gravitating sources of circularly polarized wave, rotating bodies and deformed Schwarzschild spacetime purposed here. In forthcoming work, it will be necessary to analyse principally the gravitating source that distort the static Schwarzschild solution. 5 Adjoint Metric Field, Christoffel Symbols and determinant 5.1 Adjoint Metric Field One can see that spacetime (11) is asymptotically flat. The Minkowski spacetime is the universal covering space for all such derived spacetimes, and in this sense it is possible to restore Minkowski spacetime in (11) if one turns off the gravitational strength in metric field, or in other words, if Φ = 0 then gµν = ηµν . However one can construct a kind of spacetime with hyperbolic cosine instead of hyperbolic sine that is not asymptotically flat, ğµν = eΦ ηµν + cosh(Φ)Υµν , (43) with respective inverse, ğ µν = e−Φ η µν + cosh(Φ)Υµν , (44) and with ğµν ğ να = δµ α since Υµν Υνα = −2Υµ α from equation (4). A map between metrics field (11) and (43) is obtained through partial derivative in Φ: ğµν = ∂gµν ∂Φ and ğ µν = − ∂g µν . ∂Φ (45) ∂ 2 gµν ∂ 2 g µν µν and g = . ∂Φ2 ∂Φ2 This ‘adjoint metric field’ (43) helps us to simplify the Christoffel symbols. But before, it is necessary to establish some relationships between g and ğ, also one has gµν = gµν ğνα = = + (eΦ ηµν + sinh(Φ)Υµν )(e−Φ η να + cosh(Φ)Υνα ) δµ α + [eΦ cosh(Φ) + e−Φ sinh(Φ)]Υµ α να sinh(Φ) cosh(Φ)Υµν Υ , (46) with Υµν Υνα = −2Υµ α and hyperbolic expressions eΦ = cosh(Φ) + sinh(Φ) and e−Φ = cosh(Φ) − sinh(Φ) we have: gµν ğ να = δµ α + Υµ α , (47) and, g µν ğνα = δ µ α + Υµ α . (48) 8 where, It is important to observe that: gµν ğ µν = g µν ğµν = Tr(η) + Tr(Υ). (49) = Γβ(1) µν Further we have: gµν Υνα = (eΦ ηµν + sinh(Φ)Υµν )Υνα = e−Φ Υµ α , (50) g µν Φ Υνα = e Υ µ α, (51) ğµν Υνα = −e−Φ Υµ α , (52) ğ µν Υνα = −eΦ Υµ α . (53) Nevertheless for this ‘adjoint metric field’, when there is not gravitational strength (Φ = 0) it is not possible to reorder Minkowski spacetime. Because if Φ = 0 then gµν = ηµν + Υµν . Unless all Υµν are vanished, one can not obtain Minkowski spacetime. Moreover, with Υµν 6= 0 a spacetime (M, ğ) has a different topology of physical spacetimes. For example, if one chooses Υµν from equation (17), the correspondent line element is ds̆2 = eΦ c2 dt2 + e−Φ (dx2 + dy 2 + dz 2 ), (54) and Γβ(2) µν = 1 sinh(Φ)g αβ (∂µ Υαν + ∂ν Υαµ − ∂α Υµν ). (61) 2 Since Γ is not a tensor, it can not have intrinsic geometrical meaning as measures of how much a manifold is curved. Below we shall compute intrinsic objects as Ricci tensor and scalar curvature. 5.3 Determinant g From identity (Schutz 1990), √ Γνµν = ∂µ (ln −g), = 5.2 Christoffel Symbols We want a manifold M with Levi-Civita connection, since the connection coefficient Γ is given by Γβµν = 1 αβ g (∂µ gαν + ∂ν gαµ − ∂α gµν ). 2 (55) = g αν ∂µ Υαν = + = + where we used (8) and (42), thus, √ 1 [Tr(η) + Tr(Υ)] ∂µ Φ = ∂µ (ln −g). (63) 2 (57) and we finally find,   √ Φ [Tr(η) + Tr(Υ)] . −g = exp 2 1 αβ h g ğαν ∂µ Φ + ğαµ ∂ν Φ − ğµν ∂α Φ 2 i sinh(Φ)(∂µ Υαν + ∂ν Υαµ − ∂α Υµν ) i h 1 (δ β ν + Υβ ν )∂µ Φ + (δ β µ + Υβ µ )∂ν Φ − gαβ ğµν ∂α Φ 2 1 sinh(Φ)gαβ (∂µ Υαν + ∂ν Υαµ − ∂α Υµν ). (58) 2 Let us consider Christoffel symbols as a combination of two terms: the first is dependent of ∂α Φ and the second is dependent of ∂α Υ, β(2) Γβµν = Γβ(1) µν + Γµν e−Φ η αν ∂µ Υαν − sinh(Φ)Υαν ∂µ Υαν e−Φ ∂µ Tr(Υ) − sinh(Φ)[−∂µ Tr(Υ)] = 0, Γνµν from which we obtain that Christoffel symbols are Γβ µν = = (56) that in term of (43) is ∂α gµν = ğµν ∂α Φ + sinh(Φ)∂α Υµν , 1h Tr(η)∂µ Φ + ∂µ Φ + Tr(Υ)∂µ Φ + Υα µ ∂α Φ 2 i −(δ α µ + Υα µ )∂α Φ 1 + sinh(Φ)g αν (∂µ Υαν + ∂ν Υαµ − ∂α Υµν ) 2 1 1 [Tr(η) + Tr(Υ)] ∂µ Φ + sinh(Φ)g αν ∂µ Υαν , 2 2 we note that the second term with g αν ∂µ Υαν is vanished, We have that, ∂α gµν = (eΦ ηµν + cosh(Φ)Υµν )∂α Φ + sinh(Φ)∂α Υµν , (59) (62) where g = det(gµν ), it is possible to obtain the determinant g by contracting the indices in (59) with β = ν and the equation (48): Γνµν that has signature (+ + ++).  1 β α δ ν δ µ + δ β µ δ α ν + δ α µ Υβ ν + δ α ν Υβ µ − gαβ ğµν ∂α Φ 2 (60) = (64) In fact, since ln(det gµν ) = Tr(ln gµν ). Also, one should √ use the identity dg = g g µν dgµν and to compute −g. One can verify the examples of above section in coordinates (t, x, y, z): Yilmaz metric: circularly polarized wave: rotating bodies: g = − exp {Φ [4 + (−6)]} = −e−2Φ g = − exp {Φ [4 + (−4)]} = −1 g = − exp {Φ [4 + (−2)]} = −e2Φ The natural volume element of manifold M with √ metric tensor (11), dv = −gd4 x, is invariant under 9 coordinate transformation. An action with Lagrangian L in spacetime (M, g) is given by, Z √ L −g d4 x, S= (65) M then we have,   Z Φ [Tr(η) + Tr(Υ)] d4 x. L exp S= 2 M For Newtonian limit, we must retain only the terms in the first order in the strength of gravity Φ, or in other words, we have ∂Φ∂Φ ≪ ∂∂Φ and Φ∂∂Φ ≪ ∂∂Φ , with this assumption we have the second term of (68) and (69) such as, 1 2∂µ ∂ν Φ + Υβ ν ∂µ ∂β Φ + Υβ µ ∂ν ∂β Φ 2
1 −ğµν g αβ ∂α ∂β Φ − [Tr(η) + Tr(Υ)]∂µ ∂ν Φ. 2 (1) Rµν = (66) 6 Newtonian Limit and Gravitational Waves 6.1 Newtonian Limit One can choose the simplest Υ, and it should be from (17) such as Tr(Υ) = −6, where the Ricci tensor is now given by (1) Rµν = 2∂µ ∂ν Φ + In the Newtonian limit one assumes that velocities are v ≪ 1, that gravitational potentials are near small, c their Minkowski values, 2 and that pressures or other mechanical stresses are negligible compared to the energy densities |P | ≪ ρc2 . The description of Einstein’s field equation (2) will use values from Christoffel symbols from equation (59), β(1) where this has two terms: Γµν dependent of ∂α Φ and β(2) Γµν dependent of ∂α Υ. In this section we shall verify the Einstein tensor Gµν = Rµν − 21 gµν R, only in the case β(2) that ∂α Υ = 0, then Γµν = 0, and in any coordinates system, the Ricci tensor is given by α(1) α(1) (1) β(1) α(1) β(1) Rµν = ∂α Γα(1) µν −Γµβ Γνα −∂ν Γµα +Γαβ Γµν . (67) (1)  1 ğµν ğ αβ − gµν g αβ ∂α Φ∂β Φ 2 1 + 2∂µ ∂ν Φ + Υβ ν ∂µ ∂β Φ + Υβ µ ∂ν ∂β Φ 2
−ğµν g αβ ∂α ∂β Φ , (68) = the second term of Ricci tensor is: α(1) Γµβ Γβ(1) να = 1 3 ∂µ Φ ∂ν Φ + (∂µ ΦΥγ ν + ∂ν ΦΥγ µ )∂γ Φ 2 2 1 1 − gµν g λγ ∂γ Φ ∂λ Φ + Υλ µ Υγ ν ∂γ Φ ∂λ Φ, 2 2 with (63) we can obtain the third term: 1 = [Tr(η) + Tr(Υ)]∂µ ∂ν Φ, 2 ∂ν Γα(1) µα (69) and the fourth is 1 [Tr(η) + Tr(Υ)] 2∂µ Φ∂ν Φ 4
β +∂µ ΦΥ ν ∂β Φ + ∂ν ΦΥβ µ ∂β Φ − ğµν g αβ ∂α Φ∂β Φ . (70) α(1) Γβα Γβ(1) µν = 2 Observe ηµν g µν 1 β 1 β 1 αβ Υ ν ∂µ ∂β Φ + Υ µ ∂ν ∂β Φ − ğµν g ∂α ∂β Φ. 2 2 2 (72) The scalar curvature is obtained by further contracting (1) indices R(1) = g µν Rµν , R(1) = 3g µν ∂µ ∂ν Φ + g µν Υβ ν ∂µ ∂β Φ, (73) g µν = (1 − Φ)η µν − ΦΥµν in terms of the first order in Φ, we find that the scalar curvature is R(1) = 3η µν ∂µ ∂ν Φ + η µν Υβ ν ∂µ ∂β Φ. (74) Since the Newtonian limit approach is in the background Minkowski spacetime, we find that R(1) = 32Φ + Υβµ ∂µ ∂β Φ. (75) From the field equations (2) we have Since ∂α Υ = 0 the first term of Rµν is given by ∂α Γα(1) µν (71) that gµν = eΦ ηµν +sinh(Φ)Υµν ≈ (1+Φ)ηµν +ΦΥµν = + Φ(ηµν + Υµν ) = ηµν + hµν and ≈ ηµν − Φ(ηµν + Υµν ) = ηµν − hµν . R=− 8πG T, c4 (76) in the situations where Newtonian theory can be applicable T ≈ ρc2 . Now we can combine equations (76) and (75) such as 32Φ + Υβµ ∂µ ∂β Φ = − 8πG ρ. c2 (77) We assume that velocities are small, then 2= ∂2 c2 ∂t2 − ∇2 ≈ −∇2 . (78) And in particular from (17) we have Υµν ⇒ 2δij , where δij is the Kronecker delta in R3 . The field equation (77) results in: −3∇2 Φ + 2δ ij ∂i ∂j Φ = − 8πG ρ, c2 (79) or ∇2 Φ = 8πG ρ. c2 (80) 10 2ϕN We can introduce Φ = 2 , where ϕN is the Newtoc nian potential. Thus we can obtain the Poisson’s equation of Newton’s law of gravitation ∇2 ϕN = 4πGρ, (81) in this way for a point source with mass M we have ϕN = − GM . r (2) Rµν = (82) 2GM Consequently Φ = − 2 . In this discussion that folc r lows (71) we assumed that Υµν is from (17), it results that the line element is given by (20) at thefirst order in Φ. Gravitational waves are one of the most important physical phenomena associated with the presence of strong and dynamic gravitational fields. Though such gravitational radiation has not yet been detected directly. There is strong indirect evidence for its existence around the famous binary pulsar PSR 1913+16 (Taylor & Weisberg 1982), that in 1974 it was discovered by R.A. Hulse and J.H. Taylor (Hulse & Taylor 1975), a discovery for which they were awarded the 1993 Nobel Prize. Here, for a description of plane gravitational waves we can assume that Φ is constant so ∂α Φ = 0, implying β(2) β(1) in Γµν = 0 in (59), thus we have Γβµν = Γµν given by equation (61): 1 sinh(Φ)g αβ (∂µ Υαν + ∂ν Υαµ − ∂α Υµν ). (83) 2 One can obtain the field equation writing the Ricci tensor in terms of Christoffel symbols above, α(2) 1 sinh(Φ) g αβ (∂α ∂µ Υβν + ∂α ∂ν Υβµ − ∂α ∂β Υµν ), 2 (87) with sinh(Φ) g αβ ≈ Φ[(1 − Φ)η αβ − ΦΥαβ ] = Φη αβ , one can obtain the Ricci tensor in Minkowski background spacetime, which this yields, (2) Rµν = 1 Φ (∂ β ∂µ Υβν + ∂ β ∂ν Υβµ − 2Υµν ), 2 (88) The gauge choice ∂ β Υβν = 0, simplifies Ricci tensor in: 6.2 Gravitational Waves Γβ(2) µν = The study of gravitational waves involves essentially the approximation of Einstein’s weak field equation, from these results one can obtain wave equation with low amplitudes, Φ ≪ 1. This can only be brought in the first order of the gravitational potential to find, α(2) (2) β(2) α(2) β(2) Rµν = ∂α Γα(2) µν −Γµβ Γνα −∂ν Γµα +Γαβ Γµν . (84) From equation (63) we have 1 Γα αν = [Tr(η) + Tr(Υ)]∂ν Φ = 0, such as, 2 α(2) (2) β(2) Rµν = ∂α Γα(2) µν − Γµβ Γνα , (89) One should observe that scalar curvature (2) R(2) = g µν Rµν is vanished for the first order in Φ, because R(2) = 1 1 µν η Φ 2Υµν = 2Tr(Υ) = 0, 2 2 since the trace of tensor Υ is a constant number, in accordance with equation (42). Then, for the field equation (2) the gravitational wave equation obtained in the vacuum is only, 2Υµν = 0. (90) It is necessary that the tensor Υµν satisfies the condition (4) and the above wave equation. Its shape should be, for example, the tensor (22) with ζ = κµ xµ = ωt − kz, a solution for gravitational plane waves with circular polarization traveling in the z direction, with amplitude Φ. 7 Conclusion (85) As shown in appendix, the The above Ricci tensor is given by, 1 (2) Rµν = sinh2 (Φ) ∂α Υαβ (∂µ Υβν + ∂ν Υβµ − ∂β Υµν ) 2 1 + sinh(Φ) g αβ (∂α ∂µ Υβν + ∂α ∂ν Υβµ − ∂α ∂β Υµν ) 2 1 − sinh2 (Φ)[g αγ g βλ ∂µ Υγβ ∂ν Υαλ 4 +2(g αγ g βλ − g αβ g γλ )∂β Υγµ ∂α Υνλ ]. 1 (2) Rµν = − Φ 2Υµν . 2 (86) This paper deals with a tensorial structure that assumes a (quasi-)idempotent feature to be able to improve at least the linear tensorial template of some tensor metric fields. It is clear that Einstein’s field equations are nonlinear, however, with these (quasi-)idempotent tensorial structure, without quadratic tensorial values, the nonlinearity becomes more moderate although there is a price to pay. The part that carries the dynamical information, the strength of gravity is tied to the tensorial 11 structure by exponential functions. In this approach the metric field can be characterized by a background spacetime conformally flat affected by a disturbance. We have approached some examples in this tensorial structure that results in exponential metric fields, we can point out as the main exponential metric obtained in this paper which has been extensively explored: the Yilmaz exponential metric (Yilmaz 1958, 1992, 1976, 1982, 1973, 1977, 1997; Clapp 1973; Robertson 1999a,b; Ibison 2006). H. Yilmaz has argued that in his theory, the gravitational field can be quantized via Feynman’s method (Yilmaz 1995; Alley 1995). Further, it has been found that the quantized theory is finite. Incidentally in the exponential metric fields approached in this work just as in the Yilmaz theory there are no black holes in the sense of event horizons, but there can be stellar collapse (Robertson 1999a,b). However, there are no point singularities. Interesting results obtained in this work from exponential metric fields are: circularly polarized wave; rotating bodies that in the first order is a deformation of Kerr metric and also we have a deformed static spherically symmetric spacetime. Many discussions around massive stellar objects have suggested, for example, that Kerr metric should be slightly deviated from Kerr. The possibility of discovering a non-Kerr object should be taken into account when constructing waveform templates for LISA’s data analysis tools (Glampedakis & Babak 2006; White 2006). The technological development is ripe enough so much so in the years to come we might be able to test the second order relativist-gravity effects and may lead to answers to some important questions of gravity. In this work, we have obtained a simple and general expression for the volume element of a manifold in coordinates (t, x, y, z) given in terms of strength of gravity and of traces of tensors η and Υ. It is possible that an analysis of any Lagrangian of field interacting with gravity will become easer. An interesting observation is the spacetime of circularly polarized plane wave, in this spacetime the volume element √ −g d4 x is the same of Minkowski spacetime, in this sense this gravitational radiation obtained from exponential metric field does not modify the volume element of background Minkowski spacetime where this plane wave travels onto. Moreover, it was purposed and verified the Newtonian limit as solution for Einstein’s equation, since we can assume that the trace of stress-energy tensor is T ≈ ρc2 . Other important solution of Einstein’s equation analysed in this paper was the plane gravitational wave for the empty space since we have considered the vanished stress-energy tensor to the first order in Φ. Both solved Einstein’s equations for Newtonian limit and plane gravitational wave propagating in the vacuum are cases that the strength of gravity is small, Φ ≪ 1. We have analysed the Newtonian limit in the case that ∂α Υ = 0, and analysed the plane gravitational wave considering the strength of gravity as a constant term, thus we had two indepen(1) dent Ricci tensors, Rµν which ∂α Υ = 0 (for Newtonian (2) limit) and Rµν which ∂α Φ = 0 (for plane gravitational wave). In a forthcoming work, an analysis of Einstein’s equations with both non-vanished ∂α Υ and ∂α Φ, will be considered. It is missing a discussion about quantities of physical interest in the solutions of Einstein’s equations which describe the exterior and interior gravitational field. Yilmaz has argued the existence of the matter part in the right-hand side of the field equations correspondent to field energy in the exterior. This paper lacks a discussion about the interior and the exterior field energies denoted by a total stress-energy tensor. An analysis about the total stress-energy tensor will be the object of a forthcoming study, where the physical consequences of terms of deformity in Kerr and Schwarzschild solutions could be analysed. We know that the dark energy and the dark matter problems are challenges to modern astrophysics and cosmology; as a typical example, we could mention the galactic rotation curves of spiral galaxies, that probably, indicates the possible failure of both Newtonian gravity and General Relativity on galactic and intergalactic scales. To explain astrophysical and cosmological problems with arguments against dark energy and dark matter many works have been devoted to the possibility that the Einstein-Hilbert Lagrangian, linear in the Ricci scalar R, should be generalized. In this sense, the choice of a generic function f (R) can be derived by matching the data and the requirement that no exotic ingredient have to be added (Allemandi et al. 2004; Barrow et al. 1983; Capozziello 2002; Capozziello et al. 2003, 2005; Carroll et al. 2004, 2005; Faraoni 2008; Flanagan 2003; Koivisto 2006; Nojiri et al. 2007, 2003a,b; Sotiriou et al. 2010). This class of theories when linearized exhibits others polarization modes for the gravitational waves, of which two correspond to the massless graviton and others such as massive scalar and ghost modes in f (R) gravity (Bellucci et al. 2009; Bogdanos et al. 2009). In this way, analyses in any order to f (R) gravity with ‘exponential metrics’ proposed in the present work could give a positive contribution to the debate of astrophysical and cosmological questions. Acknowledgements It is a pleasure to acknowledge many stimulating and helpful discussions with my colleagues and friends André L. A. Penna and Caio M. M. Polito. 12 A Details of Calculation for Rotating Bodies A.1 Multiplicative Properties of Υ in Boyer-Lindquist coordinates The metric tensor ηµν of Minkowski flat spacetime in Boyer-Lindquist coordinates is given by:  ηµν 1 0 0    r 2 +a2 (cos(θ))2  0 − r 2 +a2  =   0 0   0 0  0 0 0 2 −r2 − a2 (cos (θ)) 0 0
2 − r2 + a2 (sin (θ)) while the (quasi-)idempotent tensor Υµν from (25)  2 (cosh (Λ)) 0    0 0   Υµν = (−2)   0 0    √ sinh (Λ) cosh (Λ) r2 + a2 sin (θ) 0      ,     (A1) in the same coordinates is: 0 0 0 0  √ sinh (Λ) cosh (Λ) r2 + a2 sin (θ)    0   .  0   
2 2 (sinh (Λ)) r2 + a2 (sin (θ)) (A2) Let us verify that above tensor Υµν satisfies the algebraic relation (4). We begin with the inverse metric tensor of Minkowski space:  η µν 1 0 0    0 − 2 r22 +a2 2  r +a (cos(θ))  =   0 0    0 0 0 0 0 −1  2 − r2 + a2 (cos (θ)) 0 0 1 − (r2 +a2 )(sin(θ)) 2 such as Υµ ν e Υµ ν are calculated with contracting indices,  Υµ ν = η µα Υαν and      = (−2)       Υµ ν = Υµα η αν      = (−2)      2  (cosh (Λ)) 0 0 0 0 0 0 0 0 cosh(Λ) √ − sinh(Λ) r 2 +a2 sin(θ) 0 0      ,      (A3) √  sinh (Λ) cosh (Λ) r2 + a2 sin (θ)    0     0    2 − (sinh (Λ)) 2 (cosh (Λ)) 0 0 0 0 0 cosh(Λ) √ − sinh(Λ) r 2 +a2 sin(θ) 0 0 0 0 0 √ sinh (Λ) cosh (Λ) r2 + a2 sin (θ) 0 0 − (sinh (Λ)) 2 (A4)       .     (A5) 13 Notice that it follows the contravariant tensor Υµν given by:  2 cosh(Λ) √ (cosh (Λ)) 0 0 − sinh(Λ) r 2 +a2 sin(θ)     0 0 0 0  µν µα βν  Υ = η Υαβ η = (−2)   0 0 0 0    (sinh(Λ))2 cosh(Λ) √ 0 0 (r2 +a − sinh(Λ) 2 )(sin(θ))2 r 2 +a2 sin(θ)       .      (A6) Finally one can obtain that Υµν Υνα = −2Υµ α , this is proved by observing that, 2  Υµν Υνα      = (−2) · (−2)      (cosh (Λ)) 0 0 0 cosh(Λ) √ − sinh(Λ) r 2 +a2 sin(θ) 0 0 0 0 0 0 0 √ sinh (Λ) cosh (Λ) r2 + a2 sin (θ) 0 0 − (sinh (Λ)) 2        = −2Υµ α .     (A7) It is easy to see that the above tensor is the same of equation (A5) as claimed. A.2 Calculation of components Let us compute the components of metric tensor gµν = eΦ ηµν + sinh(Φ)Υµν for rotating bodies. First we may calculate g00 = eΦ η00 + sinh(Φ)Υ00 ≈ (1 + Φ)η00 + ΦΥ00 , Mr r 2 + a2 Mr , −2· 2 · 2 2 +a ) (r + a ) ρ2 √ r 2 + a2 Mr since Φ = 2 from (31), and also cosh Λ = from (29), 2 (r + a ) ρ g00 = 1 + Φ − 2Φ cosh2 Λ = 1 + g00 = 1− (r2 Mr Mr ρ2 − 2M r 2M r + + 2 = , 2 2 2 2 ρ (r + a ) ρ (r + a2 ) (A8) (A9) with ρ = r2 + a2 cos2 θ then, g00 = = = r2 + a2 cos2 θ − 2M r Mr r2 + a2 (1 − sin2 θ) − 2M r Mr + = + 2 2 2 2 2 ρ (r + a ) ρ (r + a2 ) 2 2 2 2 Mr r + a − 2M r − a sin θ + 2 ρ2 (r + a2 ) Mr ∆ − a2 sin2 θ + 2 , 2 ρ (r + a2 ) (A10) where ∆ = r2 − 2M r − a2 Calculation of g03 : g03 = = p −2 sinh(Φ) sinh Λ cosh Λ r2 + a2 sin θ !  √ 2   a sin θ 2M ra sin2 θ Mr r + a2 p 2 r + a2 sin θ = − , −2 2 2 r +a ρ ρ ρ2 again we used the definition (29). (A11) 14 Calculation of g22 :  g22 = −eΦ ρ2 = − 1 + Mr (r2 + a2 )  ρ2 = −ρ2 − M rρ2 r 2 + a2 (A12) Calculation of g33 : g33 = = = = = = = = = 2 2 2 −eΦ (r2 + a2 ) sin2θ − 2 sinh(Φ) sinh2 Λ(r  + a ) sin   θ2 2  Mr Mr a sin θ (r2 + a2 ) sin2 θ − 2 2 (r2 + a2 ) sin2 θ − 1+ 2 (r + a2 ) r + a2 ρ2 2M ra2 sin4 θ −(r2 + a2 ) sin2 θ − − M r sin2 θ ρ2 ρ2 (r2 + a2 ) sin2 θ + 2M ra2 sin4 θ − − M r sin2 θ ρ2 (r2 + a2 cos2 θ)(r2 + a2 ) sin2 θ + 2M ra2 sin4 θ − − M r sin2 θ ρ2 (r2 + a2 − a2 sin2 θ)(r2 + a2 ) sin2 θ + 2M ra2 sin4 θ − M r sin2 θ − ρ2  2  (r + a2 ) − a2 sin2 θ (r2 + a2 ) sin2 θ + 2M ra2 sin4 θ − − M r sin2 θ ρ2 −(r2 + a2 )2 sin2 θ + (r2 − 2M r + a2 )a2 sin4 θ − M r sin2 θ ρ2 i
− sin2 θ h 2 2 2 2 2 − ∆a sin θ − M r sin2 θ r + a ρ2 (A13) Calculation of g11 : g11 = −eΦ r2 ρ2 , + a2 (A14) we have that ∆ = r2 − 2M r − a2 can be given by: ∆ = ∆ = 2 r2 − 2M r− a2 = (r2 + a ) − 2M r 2M r 2 2 (r + a ) 1 − 2 , (r + a2 ) (A15) which implies that   1 2M r 1 1 − . = r 2 + a2 ∆ (r2 + a2 ) (A16) Now, the component g11 is given by:  2   2M r ρ2 ρ Mr Φ g11 = −e 2 1− 2 . =− 1+ 2 r + a2 (r + a2 ) ∆ (r + a2 ) Mr ≪ 1, the component g11 is given just in the first order: + a2   Mr 1− 2 (r + a2 ) M rρ2 + . ∆(r2 + a2 ) Hence we have that g11 = g11 = ρ2 ∆ ρ2 − ∆ − (A17) r2 (A18) (2) B Calculation of Ricci tensor Rµν The Ricci tensor from section VI, used to evaluated gravitational waves, is given by: α(2) α(2) (2) β(2) α(2) β(2) Rµν = ∂α Γα(2) µν − Γµβ Γνα − ∂ν Γµα + Γαβ Γµν . (B1) 15 Let Φ be constant. Then Γα αν = 1 [Tr(η) + Tr(Υ)]∂ν Φ are vanished such that 2 α(2) (2) β(2) Rµν = ∂α Γα(2) µν − Γµβ Γνα . (B2) Let us calculate the first term: ∂α Γα(2) µν = 1 1 sinh(Φ) ∂α g αβ (∂µ Υβν + ∂ν Υβµ − ∂β Υµν ) + sinh(Φ) g αβ (∂α ∂µ Υβν + ∂α ∂ν Υβµ − ∂α ∂β Υµν ). 2 2 (B3) If Φ is constant, then we have ∂α g αβ = sinh(Φ)∂α Υαβ , such that, ∂α Γα(2) µν = 1 1 sinh2 (Φ) ∂α Υαβ (∂µ Υβν + ∂ν Υβµ − ∂β Υµν ) + sinh(Φ) g αβ (∂α ∂µ Υβν + ∂α ∂ν Υβµ − ∂α ∂β Υµν ). (B4) 2 2 Let us calculate the second term from Ricci tensor (B2): α(2) Γµβ Γβ(2) να = = 1 1 sinh(Φ)g αγ (∂µ Υγβ + ∂β Υγµ − ∂γ Υµβ ) sinh(Φ)g βλ (∂α Υνλ + ∂ν Υαλ − ∂λ Υαν ) 2 2 1 sinh2 (Φ)g αγ g βλ [(∂µ Υγβ ∂α Υνλ − ∂µ Υγβ ∂λ Υαν ) + (∂β Υγµ ∂α Υνλ + ∂γ Υµβ ∂λ Υαν ) 4 +(∂β Υγµ ∂ν Υαλ − ∂γ Υµβ ∂ν Υαλ ) − (∂β Υγµ ∂λ Υαν + ∂γ Υµβ ∂α Υνλ ) + ∂µ Υγβ ∂ν Υαλ ]. The contracting indices α, β, γ and λ can be manipulated as following: α ↔ λ and β ↔ γ in the second term of each above parentesis, that result in α(2) Γµβ Γβ(2) να = 1 sinh2 (Φ)g αγ g βλ [ 2∂β Υγµ ∂α Υνλ − 2∂γ Υµβ ∂α Υνλ + ∂µ Υγβ ∂ν Υαλ ] 4 (B5) 1 sinh2 (Φ)[g αγ g βλ ∂µ Υγβ ∂ν Υαλ + 2(g αγ g βλ − g αβ g γλ )∂β Υγµ ∂α Υνλ ]. 4 (B6) or α(2) Γµβ Γβ(2) να = Finally we have the Ricci tensor given by: (2) Rµν = 1 sinh2 (Φ) ∂α Υαβ (∂µ Υβν + ∂ν Υβµ − ∂β Υµν ) 2 1 + sinh(Φ) g αβ (∂α ∂µ Υβν + ∂α ∂ν Υβµ − ∂α ∂β Υµν ) 2 1 − sinh2 (Φ)[g αγ g βλ ∂µ Υγβ ∂ν Υαλ + 2(g αγ g βλ − g αβ g γλ )∂β Υγµ ∂α Υνλ ]. 4 (B7) 16 References Alexander, S., Peskin, M. E. & Sheikh-Jabbari, M. M., 2006, Phys. Rev. Lett. 96, 081301, (arXiv:hep-th/0403069) Allemandi, G., Borowiec, A. & Francaviglia, M., 2004, Phys. Rev., D 70:103503, Alley, C.O. & Yilmaz, H., 1999, Refutation of C.W. 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