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Section 3.7 Post-Newtonian Equations of Motion of Extended Bodies in Global Coordinates

Table 10 Section 3.7 Post-Newtonian Equations of Motion of Extended Bodies in Global Coordinates

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Figure 1.1. Sketch of the correspondence between the quantized energy levels of the real and effective conservative dynamics. n denotes the ‘principal quantum number’ (n = n,; + €+ 1, with nr = 0,1, ... denoting the number of nodes in the radial function), while @ denotes the (relative) orbital angular momentum (L? = ¢(¢ + 1) Kh”). Though the EOB method is purely classical, it is conceptually useful to think in terms of the underlying (Bohr-Sommerfeld) quantization conditions of the action variables Ip and J to motivate the identification between n and @ in the two dynamics. Section 1.3 EOB description of the conservative dynamics of two body systems Figure 1.1. Sketch of the correspondence between the quantized energy levels of the real and effective
Figure 1.1. Sketch of the correspondence between the quantized energy levels of the real and effective conservative dynamics. n denotes the ‘principal quantum number’ (n = n,; + €+ 1, with nr = 0,1, ... denoting the number of nodes in the radial function), while @ denotes the (relative) orbital angular momentum (L? = ¢(¢ + 1) Kh”). Though the EOB method is purely classical, it is conceptually useful to think in terms of the underlying (Bohr-Sommerfeld) quantization conditions of the action variables Ip and J to motivate the identification between n and @ in the two dynamics. Section 1.3 EOB description of the conservative dynamics of two body systems Figure 1.1. Sketch of the correspondence between the quantized energy levels of the real and effective
The ‘correspondence’ between the real and effective energy levels is illustrated in Fig. 1.1. At this stage, one needs to define a ‘dictionary’ between the real (relative) two-body dynamics, sum- marized in Eq. (1.3.10), and the effective one-body one, summarized in Eq. (1.3.14). As, on both sides, quantum mechanics tells us that the action variables are quantized in integers (reat = nh, Netr = nerrh, etc.) it is most natural to identify n = nem and € = Cem. One then still needs a rule for relating the two different energies Ey{""° and E «. Ref. [39] proposed to look for a general map between the real energy levels and the effective ones (which, as seen when comparing (1.3.10) and (1.3.14), cannot be directly identified because they do not include the same rest-mass contribution’), namely
The ‘correspondence’ between the real and effective energy levels is illustrated in Fig. 1.1. At this stage, one needs to define a ‘dictionary’ between the real (relative) two-body dynamics, sum- marized in Eq. (1.3.10), and the effective one-body one, summarized in Eq. (1.3.14). As, on both sides, quantum mechanics tells us that the action variables are quantized in integers (reat = nh, Netr = nerrh, etc.) it is most natural to identify n = nem and € = Cem. One then still needs a rule for relating the two different energies Ey{""° and E «. Ref. [39] proposed to look for a general map between the real energy levels and the effective ones (which, as seen when comparing (1.3.10) and (1.3.14), cannot be directly identified because they do not include the same rest-mass contribution’), namely
where A’ = dA/dr. As explained above the EOB metric function A(r) is defined by Padé resumming the Taylor-expanded result (1.3.12) obtained from the matching between the real and effective energy levels (as we were mentioning, one uses a similar Padé resumming for D(r) = A(r) B(r)). One similarly needs to resum i y» Le., the y component of the radiation reaction which has been introduced on the r.h.s. of Eq. (1.4.5). Section 1.4 EOB description of radiation reaction and of the emitted waveform during inspiral Several methods have been tried during the development of the EOB formalism to resum the radiatio reaction F, y (starting from the high-order PN-expanded results that have been obtained in the literature Here, we shall briefly explain the new, parameter-free resummation technique for the multipolar wave form (and thus for the energy flux) introduced in Ref. [78, 79] and perfected in [65]. To be precise, th new results discussed in Ref. [65] are twofold: on the one hand, that work generalized the € = m = resummed factorized waveform of [78, 79] to higher multipoles by using the most accurate current] known PN-expanded results [80-83] as well as the higher PN terms which are known in the test-mas limit [84, 85]; on the other hand, it introduced a new resummation procedure which consists in consic ering a new theoretical quantity, denoted as pem(x), which enters the (¢,m) waveform (together wit other building blocks, see below) only through its ¢-th power: hem o (pem(a))*. Here, and below, denotes the invariant PN-ordering parameter given during inspiral by x = (GMQ/ 3)/3,
where A’ = dA/dr. As explained above the EOB metric function A(r) is defined by Padé resumming the Taylor-expanded result (1.3.12) obtained from the matching between the real and effective energy levels (as we were mentioning, one uses a similar Padé resumming for D(r) = A(r) B(r)). One similarly needs to resum i y» Le., the y component of the radiation reaction which has been introduced on the r.h.s. of Eq. (1.4.5). Section 1.4 EOB description of radiation reaction and of the emitted waveform during inspiral Several methods have been tried during the development of the EOB formalism to resum the radiatio reaction F, y (starting from the high-order PN-expanded results that have been obtained in the literature Here, we shall briefly explain the new, parameter-free resummation technique for the multipolar wave form (and thus for the energy flux) introduced in Ref. [78, 79] and perfected in [65]. To be precise, th new results discussed in Ref. [65] are twofold: on the one hand, that work generalized the € = m = resummed factorized waveform of [78, 79] to higher multipoles by using the most accurate current] known PN-expanded results [80-83] as well as the higher PN terms which are known in the test-mas limit [84, 85]; on the other hand, it introduced a new resummation procedure which consists in consic ering a new theoretical quantity, denoted as pem(x), which enters the (¢,m) waveform (together wit other building blocks, see below) only through its ¢-th power: hem o (pem(a))*. Here, and below, denotes the invariant PN-ordering parameter given during inspiral by x = (GMQ/ 3)/3,
Figure 1.2. This figure illustrates the comparison (made in Refs. [66, 113]) between the (NR-completed) EOB waveform (Zerilli-normalized quadrupolar (2 = m = 2) metric waveform (1.5.2) with parameter- free radiation reaction (1.4.16) and with a5 = 0,ag = —20) and one of the most accurate numerical relativity waveform (equal-mass case) nowadays available [112]. The phase difference between the two is Ap < +£0.01 radians during the entire inspiral and plunge, which is at the level of the numerical error.
Figure 1.2. This figure illustrates the comparison (made in Refs. [66, 113]) between the (NR-completed) EOB waveform (Zerilli-normalized quadrupolar (2 = m = 2) metric waveform (1.5.2) with parameter- free radiation reaction (1.4.16) and with a5 = 0,ag = —20) and one of the most accurate numerical relativity waveform (equal-mass case) nowadays available [112]. The phase difference between the two is Ap < +£0.01 radians during the entire inspiral and plunge, which is at the level of the numerical error.
Figure 1.3. Close up around merger of the waveforms of Fig. 1.2. Note the excellent agreement between both modulus and phasing also during the ringdown phase.
Figure 1.3. Close up around merger of the waveforms of Fig. 1.2. Note the excellent agreement between both modulus and phasing also during the ringdown phase.
Figure 1.4. Comparison (made in [100]) between various analytical estimates of the energy-angular momentum functional relation and its numerical-relativity estimate (equal-mass case). The standard “Taylor-expanded” 3PN E(j) curve shows the largest deviation from NR results, especially at low j’s, while the two (adiabatic and nonadiabatic) 3PN-accurate, non-NR-calibrated EOB E(j) curves agree remarkably well with the NR one.
Figure 1.4. Comparison (made in [100]) between various analytical estimates of the energy-angular momentum functional relation and its numerical-relativity estimate (equal-mass case). The standard “Taylor-expanded” 3PN E(j) curve shows the largest deviation from NR results, especially at low j’s, while the two (adiabatic and nonadiabatic) 3PN-accurate, non-NR-calibrated EOB E(j) curves agree remarkably well with the NR one.
Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries The 6 constants C411, C412, C413, C211, C212, Co1 are still unknown. Their calculations will need an in- tricate treatment of spacelike infinity in a post-Newtonian setting. The both articles [41, 42] clearly indicate this intricate problem.
Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries The 6 constants C411, C412, C413, C211, C212, Co1 are still unknown. Their calculations will need an in- tricate treatment of spacelike infinity in a post-Newtonian setting. The both articles [41, 42] clearly indicate this intricate problem.
The next-to-leading order spin-orbit Hamiltonian reads, [29], and the leading order spin(1)-spin(1) dynamics is given by, going beyond linear order in spin, e.g. [47,
The next-to-leading order spin-orbit Hamiltonian reads, [29], and the leading order spin(1)-spin(1) dynamics is given by, going beyond linear order in spin, e.g. [47,
Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries
Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries
Finally, the next-to-leading order spin(1)-spin(1) dynamics reads, [31], Also this Hamiltonian goes beyond linear order in spin. For its derivation the Tulczyjew energy-
Finally, the next-to-leading order spin(1)-spin(1) dynamics reads, [31], Also this Hamiltonian goes beyond linear order in spin. For its derivation the Tulczyjew energy-
The leading order spin(1)-spin(2) expression of the spin-orbit center-of-energy vector renders zero. T] contribution to the center-of-energy vector from the spin(a)-spin(a) interaction can be found in [57]. To summarize, the following pN-Hamiltonians with spin are known fully explicitly, As examples, the up to the next-to-leading order spin-orbit center-of-energy vector is given by, [29, 56, 57), momentum tensor was needed one step beyond pole-dipole-particle approximation. All the previous Hamiltonians, spinless and spinning, were valid for both neutron stars and black holes with the excep- tion of the next-to-leading order spin(1)-spin(1) one which is valid for black holes only. Its generalization to neutron stars were achieved in [49, 50]. The next-to-leading order Hamiltonians found confimations through [49, 51-54], see [55]. Fe er i ee ee er ee ee oe
The leading order spin(1)-spin(2) expression of the spin-orbit center-of-energy vector renders zero. T] contribution to the center-of-energy vector from the spin(a)-spin(a) interaction can be found in [57]. To summarize, the following pN-Hamiltonians with spin are known fully explicitly, As examples, the up to the next-to-leading order spin-orbit center-of-energy vector is given by, [29, 56, 57), momentum tensor was needed one step beyond pole-dipole-particle approximation. All the previous Hamiltonians, spinless and spinning, were valid for both neutron stars and black holes with the excep- tion of the next-to-leading order spin(1)-spin(1) one which is valid for black holes only. Its generalization to neutron stars were achieved in [49, 50]. The next-to-leading order Hamiltonians found confimations through [49, 51-54], see [55]. Fe er i ee ee er ee ee oe
where 7 is the Euler number. The Inz-term has been calculated for the first time in [65, 66] and later verified in [28, 67]. The terms in e4 proportional to v* and v® have been obtained for the first time in [28] and were later confirmed by [68], and the term proportional to v? has been obtained only quite recently, [19]. The same holds with c, which has been derived in [42]. A numerical value of c; was known before, [41], coinciding, however, only in the first five digits, 153.88, with the analytic result. In the test-mass limit v = 0, the exact ISCO occurs for xjsco = 1/6 corresponding to the minimum
where 7 is the Euler number. The Inz-term has been calculated for the first time in [65, 66] and later verified in [28, 67]. The terms in e4 proportional to v* and v® have been obtained for the first time in [28] and were later confirmed by [68], and the term proportional to v? has been obtained only quite recently, [19]. The same holds with c, which has been derived in [42]. A numerical value of c; was known before, [41], coinciding, however, only in the first five digits, 153.88, with the analytic result. In the test-mass limit v = 0, the exact ISCO occurs for xjsco = 1/6 corresponding to the minimum
Up to 2pN order, these expressions read, [39, 71],
Up to 2pN order, these expressions read, [39, 71],
The given parametrization of the orbits is based on the 1pN parametrization by [72]. The latter parametriza tion is different from the one in [1] but identical to the one in [73] in the test-body limit. Through IpN order the parametrization given above plays an important role in current observations of binary pulsar systems, e.g. [74].
The given parametrization of the orbits is based on the 1pN parametrization by [72]. The latter parametriza tion is different from the one in [1] but identical to the one in [73] in the test-body limit. Through IpN order the parametrization given above plays an important role in current observations of binary pulsar systems, e.g. [74].
where the dimensionless spin-parameter a = Sc/GM? has the range a € [—1,1]. The canonica variables are (p,, L, Lz;1r, v, Q), where r and p,. denote the radial coordinate and its canonical conjugat« momentum, respectively, v being the true anomaly, i.e., the angle between the position vector r = rn and the direction of the ascending node N = (ez x L) /|ez x L|: cosv = n-N and sinv = n- W witt W =LxN/L. Qis the angle of the ascending node as measured from the x-axis of the non-rotatins orthonormal basis (ez,€,,e2): cosQ = N- e,; and sinQ = N - e,, measuring the precession of the orbital plane about L. which is tilted from the equatorial plane with the inclination angle 2, the angle between the background z-axis and the orbital angular momentum vector L, so Lz = Losi. The canonical Poisson-bracket relations read Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries
where the dimensionless spin-parameter a = Sc/GM? has the range a € [—1,1]. The canonica variables are (p,, L, Lz;1r, v, Q), where r and p,. denote the radial coordinate and its canonical conjugat« momentum, respectively, v being the true anomaly, i.e., the angle between the position vector r = rn and the direction of the ascending node N = (ez x L) /|ez x L|: cosv = n-N and sinv = n- W witt W =LxN/L. Qis the angle of the ascending node as measured from the x-axis of the non-rotatins orthonormal basis (ez,€,,e2): cosQ = N- e,; and sinQ = N - e,, measuring the precession of the orbital plane about L. which is tilted from the equatorial plane with the inclination angle 2, the angle between the background z-axis and the orbital angular momentum vector L, so Lz = Losi. The canonical Poisson-bracket relations read Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries
next step is to calculate the remaining integrals in the action (2.5.33), resulting in
next step is to calculate the remaining integrals in the action (2.5.33), resulting in
Hereof the two periastron shifts, AU and A®, the intrinsic and the one related with the precession of th orbital plane, respectively, result in the form
Hereof the two periastron shifts, AU and A®, the intrinsic and the one related with the precession of th orbital plane, respectively, result in the form
This quantity can also be deduced from [78] apart from the higher pN-order term linear in a which has not been treated therein. As @ deviates from L only at the order a?/c*, in the obtained results, C may always be replaced with L. These periastron advances agree with corresponding results in [39, 47] in the test-mass limit, apart from the higher pN-order term linear in a which does not appear therein. For the orbital period we get 2.5.3 Two-body systems with spinning components The motion of spins and precession of orbits of spinning binaries are important indicators for the grav- itational interaction. From an observational point of view, only the leading pN-order interactions are important, so in the following we will restrict ourselves to those interactions, particularly to the leading order spin-orbit and spin(1)-spin(2) interactions. Also, we shall restrict ourselves to the center-of-energy systems.
This quantity can also be deduced from [78] apart from the higher pN-order term linear in a which has not been treated therein. As @ deviates from L only at the order a?/c*, in the obtained results, C may always be replaced with L. These periastron advances agree with corresponding results in [39, 47] in the test-mass limit, apart from the higher pN-order term linear in a which does not appear therein. For the orbital period we get 2.5.3 Two-body systems with spinning components The motion of spins and precession of orbits of spinning binaries are important indicators for the grav- itational interaction. From an observational point of view, only the leading pN-order interactions are important, so in the following we will restrict ourselves to those interactions, particularly to the leading order spin-orbit and spin(1)-spin(2) interactions. Also, we shall restrict ourselves to the center-of-energy systems.
In the case of equatorial motion the resulting shift, say A®, is given by
In the case of equatorial motion the resulting shift, say A®, is given by
Post-Newtonian expansion of the metric tensor and the scalar field introduces a corresponding expan- sion of the stress-energy-momentum tensor of matter In order to simplify notations, we shall use the following abbreviations:
Post-Newtonian expansion of the metric tensor and the scalar field introduces a corresponding expan- sion of the stress-energy-momentum tensor of matter In order to simplify notations, we shall use the following abbreviations:
where = 7”, 0, is the D’Alembert (wave) operator of the Minkowski spacetime, h = hi; = 65"7hij, and the angular brackets around tensor indices in h<ijs = hij — 6i;h/3, denote the symmetric trace-free (STF) part of the spatial components of the metric tensor [the STF tensors which are explained in Chapter 5 of this book. They are also thoroughly discussed by Thorne [146], Blanchet and Damour [11] and in a review article by Poisson et al. [127]. Equations (3.3.30)-(3.3.33) are valid in the class of coordinates defined by the gauge condition (3.3.22).
where = 7”, 0, is the D’Alembert (wave) operator of the Minkowski spacetime, h = hi; = 65"7hij, and the angular brackets around tensor indices in h<ijs = hij — 6i;h/3, denote the symmetric trace-free (STF) part of the spatial components of the metric tensor [the STF tensors which are explained in Chapter 5 of this book. They are also thoroughly discussed by Thorne [146], Blanchet and Damour [11] and in a review article by Poisson et al. [127]. Equations (3.3.30)-(3.3.33) are valid in the class of coordinates defined by the gauge condition (3.3.22).
where y+: = Ox/Ot?, the post-Newtonian potential
where y+: = Ox/Ot?, the post-Newtonian potential
where fi) = ee vi = dw /du is a coordinate velocity of the body’s matter element with respect to the origin of the local coordinates w®. It is worth to emphasize that the integrals given in this section, are taken over a hypersurface of the coordinate time wu. It does not coincide with the hypersurface of the coordinate time ¢, which is used in the integrals (3.4.13)—(3.4.20) defining gravitational potentials in the global coordinates x®. This remark is important for making the post-Newtonian transformations of the potentials from one frame to another as it requires to use the Lie transport of the integrand function between the hypersurfaces [93, Section 5.2.3]. Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies
where fi) = ee vi = dw /du is a coordinate velocity of the body’s matter element with respect to the origin of the local coordinates w®. It is worth to emphasize that the integrals given in this section, are taken over a hypersurface of the coordinate time wu. It does not coincide with the hypersurface of the coordinate time ¢, which is used in the integrals (3.4.13)—(3.4.20) defining gravitational potentials in the global coordinates x®. This remark is important for making the post-Newtonian transformations of the potentials from one frame to another as it requires to use the Lie transport of the integrand function between the hypersurfaces [93, Section 5.2.3]. Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies
Section 3.4 Parametrized Post-Newtonian Coordinates Vector Ci; in (3.4.43)-(3.4.45) is the angular velocity of the kinematic rotation of the spatial axes of the local coordinates with respect to the global coordinates. We keep its contribution only up to quadratic terms which is sufficient for our goal. We also assume that C” has the post-Newtonian order of magnitude being comparable with the rate of the geodetic precession. The external solution contains monopole terms, Q = Q(u) and A = A(u), entering Ae and hes respectively. Function @ defines the unit of measurement of the coordinate time wu at the origin of the local coordinates, and function A defines the unit of measurement of spatial distances on the hypersuface of constant time wu. Physical meaning of the external multipoles @z can be understood if one writes down the Newtonian equation of motion of freely-falling test particle [93, 94]. It turns out that @; is an acceleration of the local frame with respect to the geodesic world line of the particle, and Qz (J > 2) describes the tidal force with multipolarity / (quadrupole, octupole, etc.).
Section 3.4 Parametrized Post-Newtonian Coordinates Vector Ci; in (3.4.43)-(3.4.45) is the angular velocity of the kinematic rotation of the spatial axes of the local coordinates with respect to the global coordinates. We keep its contribution only up to quadratic terms which is sufficient for our goal. We also assume that C” has the post-Newtonian order of magnitude being comparable with the rate of the geodetic precession. The external solution contains monopole terms, Q = Q(u) and A = A(u), entering Ae and hes respectively. Function @ defines the unit of measurement of the coordinate time wu at the origin of the local coordinates, and function A defines the unit of measurement of spatial distances on the hypersuface of constant time wu. Physical meaning of the external multipoles @z can be understood if one writes down the Newtonian equation of motion of freely-falling test particle [93, 94]. It turns out that @; is an acceleration of the local frame with respect to the geodesic world line of the particle, and Qz (J > 2) describes the tidal force with multipolarity / (quadrupole, octupole, etc.).
After fixing the gauge freedom as indicated above, the external metric tensor assumes in the local coordinates the following form
After fixing the gauge freedom as indicated above, the external metric tensor assumes in the local coordinates the following form
where Vz denotes a volume of the body B. This completes derivation of the metric tensor in the local coordinates. The Mixed Terms The mixed terms entering the metric tensor in the local coordinates are given as a particular solution of the inhomogeneous field equation (3.3.31) with the right side taken as a product of the internal and external solutions found on the previous step of the post-Newtonian iterations. Solving (3.3.31) yields where a double dot above @, denotes a second derivative with respect to time u. We have excluded the monopole, Q, and dipole, Q' terms from the non-linear part of the local metric tensor because they are absorbed to (yet unknown) Q and Q in the linear part of the metric (3.4.46). We could also decompose the product of the two sums in (3.4.49) into irreducible parts and absorb the STF piece of this decomposition to multipoles Q; (J > 2). This procedure is done later in the process of matching the internal and external solutions.
where Vz denotes a volume of the body B. This completes derivation of the metric tensor in the local coordinates. The Mixed Terms The mixed terms entering the metric tensor in the local coordinates are given as a particular solution of the inhomogeneous field equation (3.3.31) with the right side taken as a product of the internal and external solutions found on the previous step of the post-Newtonian iterations. Solving (3.3.31) yields where a double dot above @, denotes a second derivative with respect to time u. We have excluded the monopole, Q, and dipole, Q' terms from the non-linear part of the local metric tensor because they are absorbed to (yet unknown) Q and Q in the linear part of the metric (3.4.46). We could also decompose the product of the two sums in (3.4.49) into irreducible parts and absorb the STF piece of this decomposition to multipoles Q; (J > 2). This procedure is done later in the process of matching the internal and external solutions.
given by two equations [93, 101], is the (global) coordinate distance taken on the hypersurface of constant time ¢ between the point of matching, x’, and the origin of the local coordinates, 73 = x(t). ag ve oe ian ww - a a ae | wee Ps e i -y oe ae py aie ae
given by two equations [93, 101], is the (global) coordinate distance taken on the hypersurface of constant time ¢ between the point of matching, x’, and the origin of the local coordinates, 73 = x(t). ag ve oe ian ww - a a ae | wee Ps e i -y oe ae py aie ae
that describe the post-Newtonian transformation between the coordinate time wu of the local coordinates and the coordinate time ¢ of the global coordinates. The other functions entering (3.5.16), (3.5.17) are defined by algebraic relations
that describe the post-Newtonian transformation between the coordinate time wu of the local coordinates and the coordinate time ¢ of the global coordinates. The other functions entering (3.5.16), (3.5.17) are defined by algebraic relations
where the external scalar field @ coincides in this approximation with the external Newtonian potential U and is computed at the origin of the local coordinates, x’;(t), at the instant of time t. We remind that the scalar field perturbation is coupled with the factor 7 — 1, so that all physically-observed scalar- field effects must be proportional to this factor. We also notice that the external monopole (J = 0) and dipole (7 = 1) of the scalar field can not be removed from the observable gravitational effects by making coordinate transformation to a freely-falling frame because if a scalar field presents in one coordinate frame it must be present in any other coordinates. In other words, scalar fields do not obey the principle of equivalence. It was the primary reason why Einstein abandoned a pure scalar theory of gravity. External mass-type multipoles Q” for 1 > 2 are defined by the following equation ? where the dot denotes the time derivative with respect to time t. The external multipole moments @;, and Cy are analogues of Dixon’s multipole moments Aq,...a,;.7 and Bay...c;uv respectively [see (3.1.13) and (3.1.14)]. We shall use the above expressions for the external multipoles in derivation of equations of motion of extended bodies in next section.
where the external scalar field @ coincides in this approximation with the external Newtonian potential U and is computed at the origin of the local coordinates, x’;(t), at the instant of time t. We remind that the scalar field perturbation is coupled with the factor 7 — 1, so that all physically-observed scalar- field effects must be proportional to this factor. We also notice that the external monopole (J = 0) and dipole (7 = 1) of the scalar field can not be removed from the observable gravitational effects by making coordinate transformation to a freely-falling frame because if a scalar field presents in one coordinate frame it must be present in any other coordinates. In other words, scalar fields do not obey the principle of equivalence. It was the primary reason why Einstein abandoned a pure scalar theory of gravity. External mass-type multipoles Q” for 1 > 2 are defined by the following equation ? where the dot denotes the time derivative with respect to time t. The external multipole moments @;, and Cy are analogues of Dixon’s multipole moments Aq,...a,;.7 and Bay...c;uv respectively [see (3.1.13) and (3.1.14)]. We shall use the above expressions for the external multipoles in derivation of equations of motion of extended bodies in next section.
where the metric tensor perturbations hoo. loos hoi, hij and h, = hii in the local coordinates have been
where the metric tensor perturbations hoo. loos hoi, hij and h, = hii in the local coordinates have been
Thus, we define the post-Newtonian center of mass of the body by making use of (3.4.54) for the multipolar index 1 = 1. After some simplifications and re-arrangements of terms in the integrand, the conformal dipole moment reads Functional form of equations of motion of extended bodies in N-body system depends crucially on the choice of the reference point inside the body defining its center of mass. Position of the center of mass of the body is determined by the value of the internal dipole moment of the body. In scalar-tensor theory of sravity there are two possible definitions of the dipole moment: active dipole moment J’ and conformal dipole moment J’. It is difficult to foresee which definition is the best. Only after completing direct calculation of equations of motion it becomes clear that it is the conformal dipole moment yields the most optimal choice of the post-Newtonian center of mass of each body [93, 94, 158]. This is because after double differentiation with respect to time, only the conformal dipole moment leads to conservation of the individual body’s linear momentum, p’, while the post-Newtonian scalar or active dipole moments do not bear such a property.
Thus, we define the post-Newtonian center of mass of the body by making use of (3.4.54) for the multipolar index 1 = 1. After some simplifications and re-arrangements of terms in the integrand, the conformal dipole moment reads Functional form of equations of motion of extended bodies in N-body system depends crucially on the choice of the reference point inside the body defining its center of mass. Position of the center of mass of the body is determined by the value of the internal dipole moment of the body. In scalar-tensor theory of sravity there are two possible definitions of the dipole moment: active dipole moment J’ and conformal dipole moment J’. It is difficult to foresee which definition is the best. Only after completing direct calculation of equations of motion it becomes clear that it is the conformal dipole moment yields the most optimal choice of the post-Newtonian center of mass of each body [93, 94, 158]. This is because after double differentiation with respect to time, only the conformal dipole moment leads to conservation of the individual body’s linear momentum, p’, while the post-Newtonian scalar or active dipole moments do not bear such a property.
where the dot denotes the time derivative with respect to wu. After taking the derivative and using the local equations of motion of matter (3.6.3), we obtain: Until now the point a(t) has represented a location of the origin of the local coordinate system in the global coordinates taken at the time t. In general, the origin of the local coordinates may not coincide with the center of mass of the body B which can move with respect to the local coordinates with some velocity and acceleration. In order to be able to keep the center of mass of the body at the origin of the local coordinates we have to prove that for any instant of time the dipole moment defined by (3.6.10) and its time derivative that is, the linear momentum of the body) given by (3.6.13, can be kept equal to / The post-Newtonian linear momentum of the body, p’ , is defined as the first derivative of the dipole moment (3.6.10) with respect to the local time u, coordinates coincides with the center OF Mass OF the body, the dipole moment vanishes £ = VU The post-Newtonian definitions of mass and the center of mass of the body depend not only on the distribution of density of matter, its velocity and stresses inside the body but also on other terms describing mutual gravitational coupling of internal and external multipoles. Thorne and Hartle [147] believed that these terms introduce ambiguity to the definitions of mass and the center of mass as they can be either included or excluded to these definitions. We have shown [93, 94, 158] that this ambiguity is fictitious and does not exists. The coupling terms must be included to the definitions of mass, center of mass and other internal multipoles to achieve the most economic form of the equations of motion depending only on the mass-type and spin-type internal multipole moments of the bodies. It is also true that only under such definitions we can get EIH equations of motion in the limiting case of spherically- symmetric and non-rotating bodies. —_ oe 7 z a 7 ioe ow i4@ & «¢ Sf & aaah a =< sk: ee Qe,
where the dot denotes the time derivative with respect to wu. After taking the derivative and using the local equations of motion of matter (3.6.3), we obtain: Until now the point a(t) has represented a location of the origin of the local coordinate system in the global coordinates taken at the time t. In general, the origin of the local coordinates may not coincide with the center of mass of the body B which can move with respect to the local coordinates with some velocity and acceleration. In order to be able to keep the center of mass of the body at the origin of the local coordinates we have to prove that for any instant of time the dipole moment defined by (3.6.10) and its time derivative that is, the linear momentum of the body) given by (3.6.13, can be kept equal to / The post-Newtonian linear momentum of the body, p’ , is defined as the first derivative of the dipole moment (3.6.10) with respect to the local time u, coordinates coincides with the center OF Mass OF the body, the dipole moment vanishes £ = VU The post-Newtonian definitions of mass and the center of mass of the body depend not only on the distribution of density of matter, its velocity and stresses inside the body but also on other terms describing mutual gravitational coupling of internal and external multipoles. Thorne and Hartle [147] believed that these terms introduce ambiguity to the definitions of mass and the center of mass as they can be either included or excluded to these definitions. We have shown [93, 94, 158] that this ambiguity is fictitious and does not exists. The coupling terms must be included to the definitions of mass, center of mass and other internal multipoles to achieve the most economic form of the equations of motion depending only on the mass-type and spin-type internal multipole moments of the bodies. It is also true that only under such definitions we can get EIH equations of motion in the limiting case of spherically- symmetric and non-rotating bodies. —_ oe 7 z a 7 ioe ow i4@ & «¢ Sf & aaah a =< sk: ee Qe,
Translational equations of motion of the center of mass of body B with respect to the local coordinates w, which cover the body and its neighbourhood, are derived by making use of definition (3.6.13) of its linear momentum, p’. Differentiating (3.6.13) one time with respect to the local coordinate time wu, making use of the microscopic equations of motion (3.6.1)-(3.6.3), and integrating by parts to re-arrange a number of terms, one obtains [93, 158]
Translational equations of motion of the center of mass of body B with respect to the local coordinates w, which cover the body and its neighbourhood, are derived by making use of definition (3.6.13) of its linear momentum, p’. Differentiating (3.6.13) one time with respect to the local coordinate time wu, making use of the microscopic equations of motion (3.6.1)-(3.6.3), and integrating by parts to re-arrange a number of terms, one obtains [93, 158]
where M and M are the conformal and active gravitational masses of the body B and we again omitted terms which are proportional to the dipole moment J‘ and linear momentum p’ of the body like we did it in (3.6.16). The two masses, M and M, are not equal according to (3.6.9). The difference between them plays a role of a “charge” of the scalar field ¢ which couples with the dipole moment P’ = U,; of the external scalar field leading to the Dicke-Nordtvedt effect [52, 54, 153]. In general relativity, 1 = M, It is important to notice that the right side of (3.6.16) is the force exerted on the body B due to the gravitational coupling of its internal active multipole moments, J”, S” with the external multipole moments @r, Pz, Cr. The right side of (3.6.16) also contains the inertial force, J1@;, due to the non- geodesic motion of the origin of the local coordinates. Hence, in the most general case, p’ 4 0 which means that the center of mass of the body B accelerates with respect to the origin of local coordinates and the world line Z of the center of mass does not coincide with the world line W of the origin of the local coordinates. However, the external dipole moment Q; is yet unconstrained function of time and we can always reach consensus with the condition (3.6.15) by choosing @; in such a way that it compensates the gravitational coupling force. To eliminate acceleration of the world line Z with respect to W and to fix the center of mass of the body B at the origin of the local coordinates we, first, impose condition (3.6.15) and solve (3.6.16) with respect to Q;. It constrains the inertial force and yields
where M and M are the conformal and active gravitational masses of the body B and we again omitted terms which are proportional to the dipole moment J‘ and linear momentum p’ of the body like we did it in (3.6.16). The two masses, M and M, are not equal according to (3.6.9). The difference between them plays a role of a “charge” of the scalar field ¢ which couples with the dipole moment P’ = U,; of the external scalar field leading to the Dicke-Nordtvedt effect [52, 54, 153]. In general relativity, 1 = M, It is important to notice that the right side of (3.6.16) is the force exerted on the body B due to the gravitational coupling of its internal active multipole moments, J”, S” with the external multipole moments @r, Pz, Cr. The right side of (3.6.16) also contains the inertial force, J1@;, due to the non- geodesic motion of the origin of the local coordinates. Hence, in the most general case, p’ 4 0 which means that the center of mass of the body B accelerates with respect to the origin of local coordinates and the world line Z of the center of mass does not coincide with the world line W of the origin of the local coordinates. However, the external dipole moment Q; is yet unconstrained function of time and we can always reach consensus with the condition (3.6.15) by choosing @; in such a way that it compensates the gravitational coupling force. To eliminate acceleration of the world line Z with respect to W and to fix the center of mass of the body B at the origin of the local coordinates we, first, impose condition (3.6.15) and solve (3.6.16) with respect to Q;. It constrains the inertial force and yields
Multipolar expansion of the other external potentials is given by External vector potential is expressed in terms of the internal multipole moments as follows external potentials that are shown below. The “bad” multipoles, N% and R4, will not show up in the final form of equations of motion for the same reason — the mutual cancellation with similar terms. External vector potential is expressed in terms of the internal multipole moments as follows
Multipolar expansion of the other external potentials is given by External vector potential is expressed in terms of the internal multipole moments as follows external potentials that are shown below. The “bad” multipoles, N% and R4, will not show up in the final form of equations of motion for the same reason — the mutual cancellation with similar terms. External vector potential is expressed in terms of the internal multipole moments as follows
where we have also used index D to numerate the bodies in (3.7.16). One more potential Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies The above expansions contain a number of integrals depending on the internal structure of the bodies explicitly. They will mutually cancel out by similar terms after substitution of these expansions to equations of motion (3.5.26).
where we have also used index D to numerate the bodies in (3.7.16). One more potential Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies The above expansions contain a number of integrals depending on the internal structure of the bodies explicitly. They will mutually cancel out by similar terms after substitution of these expansions to equations of motion (3.5.26).
Chapter 3 Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies
Chapter 3 Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies
The multipolar tensor coefficients are expressed in terms of the internal (active) multipole moments and their derivatives. They also depend on velocities uv’, viz, Uc of the bodies and on their relative velocities,
The multipolar tensor coefficients are expressed in terms of the internal (active) multipole moments and their derivatives. They also depend on velocities uv’, viz, Uc of the bodies and on their relative velocities,
Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies
Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies
Four-dimensional scalar field external multipoles are defined by (3.5.27) and (3.8.31) and reads
Four-dimensional scalar field external multipoles are defined by (3.5.27) and (3.8.31) and reads
where a dot denotes a covariant derivative defined later in (3.8.41) and we have introduced the following where ¢ is the scalar field perturbation caused by external with respect to the body B masses. In the global coordinates ¢ = U defined in (3.5.24a).
where a dot denotes a covariant derivative defined later in (3.8.41) and we have introduced the following where ¢ is the scalar field perturbation caused by external with respect to the body B masses. In the global coordinates ¢ = U defined in (3.5.24a).
which provides the post-Newtonian momentum-velocity relation that has not been established in Dixon’s paper [62] to full extent. Partial progress in finding this relation has been made by Ehlers and Rudolph [67] but their momentum-velocity relation is incomplete and, furthermore, goes beyond the post-Newtonian approximation without justification. It is straightforward to check that the dynamical velocity (3.8.44) is orthogonal to the kinematic ve-
which provides the post-Newtonian momentum-velocity relation that has not been established in Dixon’s paper [62] to full extent. Partial progress in finding this relation has been made by Ehlers and Rudolph [67] but their momentum-velocity relation is incomplete and, furthermore, goes beyond the post-Newtonian approximation without justification. It is straightforward to check that the dynamical velocity (3.8.44) is orthogonal to the kinematic ve-
Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies Herein, functions ® and © without indices correspond to definitions (3.8.37) and (3.8.38) with indices
Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies Herein, functions ® and © without indices correspond to definitions (3.8.37) and (3.8.38) with indices
In the following we will frequently encounter such order symbols that we will often abbreviate by For the orbital velocity of the Earth about the Sun we have
In the following we will frequently encounter such order symbols that we will often abbreviate by For the orbital velocity of the Earth about the Sun we have
For the components of the Riemann curvature tensor one obtains We will now compute the differential geometrical quantities to post-Newtonian accuracy for celestial mechanical problems:
For the components of the Riemann curvature tensor one obtains We will now compute the differential geometrical quantities to post-Newtonian accuracy for celestial mechanical problems:
The following Theorem is not restricted to the static case. the symmetric Cartesian (every index takes the values x,y,z) tensor yi, ...i, 18 trace-free, i.e., a Cartesian STF-tensor. By induction one can prove that
The following Theorem is not restricted to the static case. the symmetric Cartesian (every index takes the values x,y,z) tensor yi, ...i, 18 trace-free, i.e., a Cartesian STF-tensor. By induction one can prove that
Note, that the conservation of angular momentum implies that the orbit is confined to a coordinate plane. Tleino nolar coordinates in the arhital nlane with For the PN Schwarzschild field the Lagrangian takes the form the specific energy € and the absolute value of the orbital angular momentum J = |J| can be written as
Note, that the conservation of angular momentum implies that the orbit is confined to a coordinate plane. Tleino nolar coordinates in the arhital nlane with For the PN Schwarzschild field the Lagrangian takes the form the specific energy € and the absolute value of the orbital angular momentum J = |J| can be written as
This implies that the post-Newtonian orbit is that of a precessing ellipse, the secular drift of the argument of periastron per revolution being given by
This implies that the post-Newtonian orbit is that of a precessing ellipse, the secular drift of the argument of periastron per revolution being given by
Theorem 4.5: From the tetrad condition (4.6.9) one infers that 4.6.4 The standard transformation between global and local coordinates The gravitational influence of external bodies is effaced and relation (4.6.10) is sometimes called weak effacement condition.
Theorem 4.5: From the tetrad condition (4.6.9) one infers that 4.6.4 The standard transformation between global and local coordinates The gravitational influence of external bodies is effaced and relation (4.6.10) is sometimes called weak effacement condition.
Figure 4.1. One global and N local coordinate systems are used for the description of the gravitational N-body problem (from [44]).
Figure 4.1. One global and N local coordinate systems are used for the description of the gravitational N-body problem (from [44]).
Figure 4.2. Three events of importance for the inversion of the coordinate transformation, ex, ez: and
Figure 4.2. Three events of importance for the inversion of the coordinate transformation, ex, ez: and
where, by convention, we are assuming that any term which contains an undefined tidal moment has to be replaced by zero. E.g., the term containing the factor (71 — 4)/(21 + 1) vanishes for | = 0. where, by convention, we are assuming that any term which contains an undefined tidal moment has Theorem 4.6: The external gravito-electric and gravito-magnetic fields, E and B, are completely deter- mined by the corresponding sets of tidal moments, G, and Hz, to post-Newtonian order. Explicitly they are given by
where, by convention, we are assuming that any term which contains an undefined tidal moment has to be replaced by zero. E.g., the term containing the factor (71 — 4)/(21 + 1) vanishes for | = 0. where, by convention, we are assuming that any term which contains an undefined tidal moment has Theorem 4.6: The external gravito-electric and gravito-magnetic fields, E and B, are completely deter- mined by the corresponding sets of tidal moments, G, and Hz, to post-Newtonian order. Explicitly they are given by
The tidal moments, GZ and Hf, can be split into two parts: a part resulting from the gravitational action of some external body A¥ E, and some inertial part related with the inhomogeneous part in the transformation of metric potentials: With similar techniques one can show that apart from simple gauge-terms, given by a single function A, also the external potentials, W and Wg, can be expressed in terms of the tidal-moments G'z and Hz, (remember that W(X* = 0) = W.(X*% = 0) = 0).
The tidal moments, GZ and Hf, can be split into two parts: a part resulting from the gravitational action of some external body A¥ E, and some inertial part related with the inhomogeneous part in the transformation of metric potentials: With similar techniques one can show that apart from simple gauge-terms, given by a single function A, also the external potentials, W and Wg, can be expressed in terms of the tidal-moments G'z and Hz, (remember that W(X* = 0) = W.(X*% = 0) = 0).
The global equations of motion in the gravitational N-body problem result from the local evolution
The global equations of motion in the gravitational N-body problem result from the local evolution
.7 The gravitational N-body problem
.7 The gravitational N-body problem
From this we get the Blanchet-Damour-Schafer (BDS) form [45] of the post-Newtonian Euler-equation: Since VG =1+ 2W/c? + Ou, the right hand side of the last equation reads:
From this we get the Blanchet-Damour-Schafer (BDS) form [45] of the post-Newtonian Euler-equation: Since VG =1+ 2W/c? + Ou, the right hand side of the last equation reads:
So far the central elements of Damour et al., [2] were exhibited and complemented with some so far unpublished material devoted to certain aspects of the whole issue. The DSX-formalism was further developed by three following papers [3-5]. a, In the second paper [3] the problem of translational Jaws of motion for a system of N arbitrarily composed and shaped, weakly self-gravitating, rotating deformable bodies are obtained at the first post- Newtonian approximation to Einstein’s theory of gravity. The full set of mass- and spin- multipole moments, M7, and S'z, was taken into account for each of the bodies. First complete and explicit results for the laws of motion of each body of the system were presented as an infinite series exhibiting the coupling of all the BD-moments to the post-Newtonian tidal moments, G; and Hz, felt by this body. Finally explicit expressions of these tidal moments in terms of BD-moments of the other bodies are derived. For a derivation of corresponding equations of motion assumptions about the time dependence of BD-moments for each of the bodies have to be made. A rigidly rotating multipole model, that leads to a closed set of dynamical equations of motion, was presented in Klioner et al., [47]. eet, = ueizecced cose wc oS ees coms = ees: lig weds = cds cee ecciiee cee eee ees = oc cece: sds: aise a oe ces Gees = ees
So far the central elements of Damour et al., [2] were exhibited and complemented with some so far unpublished material devoted to certain aspects of the whole issue. The DSX-formalism was further developed by three following papers [3-5]. a, In the second paper [3] the problem of translational Jaws of motion for a system of N arbitrarily composed and shaped, weakly self-gravitating, rotating deformable bodies are obtained at the first post- Newtonian approximation to Einstein’s theory of gravity. The full set of mass- and spin- multipole moments, M7, and S'z, was taken into account for each of the bodies. First complete and explicit results for the laws of motion of each body of the system were presented as an infinite series exhibiting the coupling of all the BD-moments to the post-Newtonian tidal moments, G; and Hz, felt by this body. Finally explicit expressions of these tidal moments in terms of BD-moments of the other bodies are derived. For a derivation of corresponding equations of motion assumptions about the time dependence of BD-moments for each of the bodies have to be made. A rigidly rotating multipole model, that leads to a closed set of dynamical equations of motion, was presented in Klioner et al., [47]. eet, = ueizecced cose wc oS ees coms = ees: lig weds = cds cee ecciiee cee eee ees = oc cece: sds: aise a oe ces Gees = ees
Figure 5.1. The Penrose diagram shows a worldline of the isolated system (binary star) originating at past timelike infinity J~ and ending at the future timelike infinity /*. Both light and gravity propagate along null geodesics going from the past null infinity J~ to the future null infinity J*. A particular light geodesic is emanating from the event (to,ao0) and ending at the event (t,a). Extrapolation of this light geodesic to the past null infinity defines the null wave vector k* = (k°,k) of the electromagnetic wave under consideration. Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields
Figure 5.1. The Penrose diagram shows a worldline of the isolated system (binary star) originating at past timelike infinity J~ and ending at the future timelike infinity /*. Both light and gravity propagate along null geodesics going from the past null infinity J~ to the future null infinity J*. A particular light geodesic is emanating from the event (to,ao0) and ending at the event (t,a). Extrapolation of this light geodesic to the past null infinity defines the null wave vector k* = (k°,k) of the electromagnetic wave under consideration. Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields
Cartesian tensors of the mass-type (mass) multipoles J< 4,s and spin-type (spin) multipoles $< 4,+ entirely describing gravitational field outside of an isolated astronomical system are always STF objects that can be checked by inspection of the definition following from the multipolar decomposition of the metric tensor perturbation hag [5, 56]. For this reason, to avoid the appearance of overcomplicated index notations we shall omit the angular brackets around the spatial indices of these (and only these) Cartesian tensors, that is we adopt: J4, = J<a,> and84, = $<a,>. We shall also use transverse (T) and transverse-traceless (TT) Cartesian tensors in our calculations
Cartesian tensors of the mass-type (mass) multipoles J< 4,s and spin-type (spin) multipoles $< 4,+ entirely describing gravitational field outside of an isolated astronomical system are always STF objects that can be checked by inspection of the definition following from the multipolar decomposition of the metric tensor perturbation hag [5, 56]. For this reason, to avoid the appearance of overcomplicated index notations we shall omit the angular brackets around the spatial indices of these (and only these) Cartesian tensors, that is we adopt: J4, = J<a,> and84, = $<a,>. We shall also use transverse (T) and transverse-traceless (TT) Cartesian tensors in our calculations
re JVU and §S; are the total mass and spin (angular momentum) of the system, and J4, and Sa, > two independent sets of mass-type and spin-type multipole moments, N' = «*/r is a unit vector ected from the origin of the coordinate system to the field point. Because the origin of the coordinate stem has been chosen at the center of mass, the expansions (5.2.5) — (5.2.8) do not depend on the iss-type dipole moment, J;, which is equal to zero by definition. We emphasize that in the linearised proximation the total mass JVl and spin 8; of the astronomical system are constant while all other iltipoles are functions of time which temporal behaviour obeys the equations of motion derived from > law of conservation of the stress-energy tensor of the system [1, 2]. Gravitational waves emitted by > system reduce its energy, linear and angular momenta. This effect does not appear in the linearised neral relativity but in higher order approximations we would obtain where the mass, spin, and linear ymentum of the system must be considered as functions of time like any other multipole. Higher-order avitational perturbations in the metric going beyond (5.2.5)-(5.2.8) are shown in a review paper by anchet [97]. They are not of concern in the present chapter. The canonical metric tensor (5.2.5)-(5.2.8) depends on the multipole moments J 4, (t—r) and 8 4, (t— taken at the retarded instant of time. The retardation is explained by the finite speed of propagation where we are, the so-called, gauge functions describing the freedom in the choice of coordinates cover- can. ing the manifold. The canonical perturbation, h2'3, obeys the homogeneous wave equation in vacuum
re JVU and §S; are the total mass and spin (angular momentum) of the system, and J4, and Sa, > two independent sets of mass-type and spin-type multipole moments, N' = «*/r is a unit vector ected from the origin of the coordinate system to the field point. Because the origin of the coordinate stem has been chosen at the center of mass, the expansions (5.2.5) — (5.2.8) do not depend on the iss-type dipole moment, J;, which is equal to zero by definition. We emphasize that in the linearised proximation the total mass JVl and spin 8; of the astronomical system are constant while all other iltipoles are functions of time which temporal behaviour obeys the equations of motion derived from > law of conservation of the stress-energy tensor of the system [1, 2]. Gravitational waves emitted by > system reduce its energy, linear and angular momenta. This effect does not appear in the linearised neral relativity but in higher order approximations we would obtain where the mass, spin, and linear ymentum of the system must be considered as functions of time like any other multipole. Higher-order avitational perturbations in the metric going beyond (5.2.5)-(5.2.8) are shown in a review paper by anchet [97]. They are not of concern in the present chapter. The canonical metric tensor (5.2.5)-(5.2.8) depends on the multipole moments J 4, (t—r) and 8 4, (t— taken at the retarded instant of time. The retardation is explained by the finite speed of propagation where we are, the so-called, gauge functions describing the freedom in the choice of coordinates cover- can. ing the manifold. The canonical perturbation, h2'3, obeys the homogeneous wave equation in vacuum
These functions, after they are substituted to equation (5.2.3), transform the canonical metric tensor perturbation to a remarkably simple form The most convenient choice simplifying the structure of the metric tensor perturbations, is given by the following gauge functions
These functions, after they are substituted to equation (5.2.3), transform the canonical metric tensor perturbation to a remarkably simple form The most convenient choice simplifying the structure of the metric tensor perturbations, is given by the following gauge functions
where Wa,, X4,, Yia,_,, and 2a, _, are Cartesian tensors depending on the retarded time. Their specific form is a matter of computational convenience (or the boundary conditions) for derivation and interpretation of observable effects but it does not affect the invariant quantities like the phase of elec- tromagnetic wave propagating through the field of the multipoles.
where Wa,, X4,, Yia,_,, and 2a, _, are Cartesian tensors depending on the retarded time. Their specific form is a matter of computational convenience (or the boundary conditions) for derivation and interpretation of observable effects but it does not affect the invariant quantities like the phase of elec- tromagnetic wave propagating through the field of the multipoles.
Equation (5.3.57) is linear with respect to the perturbation of the metric tensor, hag. Hence, it can be linearly decomposed in the equation for perturbations of the light-ray trajectory caused separately by mass and spin multipole moments. Substitution of the metric tensor (5.2.5) — (5.2.8) to Eq. (5.3.57) and replacement of spatial derivatives with respect to x’ with those with respect to parameters €' and T by making use of (5.3.30) yield the following linear superposition
Equation (5.3.57) is linear with respect to the perturbation of the metric tensor, hag. Hence, it can be linearly decomposed in the equation for perturbations of the light-ray trajectory caused separately by mass and spin multipole moments. Substitution of the metric tensor (5.2.5) — (5.2.8) to Eq. (5.3.57) and replacement of spatial derivatives with respect to x’ with those with respect to parameters €' and T by making use of (5.3.30) yield the following linear superposition
Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields Reference tetrad
Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields Reference tetrad
which is written down for the case of weak gravitational field, gag = Nag + hag. Here the unit spatial vectors a = (a',a”,a*), b = (b',b*,b°), and k = (k',k?,k®) are orthonormal in the Euclidean sense (6;j;a°b? = isa’ ko = = Oig b’k? = 0 and dija'ad = dij bb? = 047 kik) = 1) with vector k pointing to the direction of propagation of the light ray at infinity as given in (5.3.18). These basis vectors are convenient to track the changes in the parameters of the electromagnetic wave as it travels from the point of emission of light to the point of its observation. Let us define at each point of spacetime a coordinate basis of static observers t is worth noticing that the observer’s tetrad eG has two group of indices. The indices without round rackets run from 0 to 3 and are associated with time and space coordinates. The indices enclosed in the ound brackets numerate vectors of the tetrad and also run from 0 to 3. The coordinate-type indices of the strad have no relation to the tetrad indices. If one changes spacetime coordinates (passive coordinate ‘ansformation) it does not affect the tetrad indices while the coordinate indices of the tetrad change 1 accordance with the transformation law for vectors. On the other hand, one can change the tetrad ectors at each point in spacetime by doing the Lorentz transformation (active coordinate transformation) vithout changing the coordinate chart [106].
which is written down for the case of weak gravitational field, gag = Nag + hag. Here the unit spatial vectors a = (a',a”,a*), b = (b',b*,b°), and k = (k',k?,k®) are orthonormal in the Euclidean sense (6;j;a°b? = isa’ ko = = Oig b’k? = 0 and dija'ad = dij bb? = 047 kik) = 1) with vector k pointing to the direction of propagation of the light ray at infinity as given in (5.3.18). These basis vectors are convenient to track the changes in the parameters of the electromagnetic wave as it travels from the point of emission of light to the point of its observation. Let us define at each point of spacetime a coordinate basis of static observers t is worth noticing that the observer’s tetrad eG has two group of indices. The indices without round rackets run from 0 to 3 and are associated with time and space coordinates. The indices enclosed in the ound brackets numerate vectors of the tetrad and also run from 0 to 3. The coordinate-type indices of the strad have no relation to the tetrad indices. If one changes spacetime coordinates (passive coordinate ‘ansformation) it does not affect the tetrad indices while the coordinate indices of the tetrad change 1 accordance with the transformation law for vectors. On the other hand, one can change the tetrad ectors at each point in spacetime by doing the Lorentz transformation (active coordinate transformation) vithout changing the coordinate chart [106].
where t = t — t* and we used equation (5.3.25) explicitly while expressing the distance r = ry (T) as a function of time 7. The constant distance r; was introduced to make the argument of the logarithmic function dimensionless. This constant is not important for calculations as it always cancel out in final formulas. However, in the case of gravitational lensing it is convenient to identify the scale constant rz with the radius of the Einstein ring [62-65] “ In the case when mass and spin are constant, the integrals that we need to carry out are reduced to (1/r]'-4 and [1/r]!-?! which are formally divergent at the lower limit of integration at the past null infinity when time t + —oo. However, one must bear in mind that these integrals do not enter equa- tions of light-ray geodesics (5.3.57) alone but appear in these equations after taking at least one partial derivative with respect to either €' or 7 parameters. This differentiation effectively eliminates divergent parts of the integrals from the final result. Hence, in what follows, we drop out the formally divergent terms so that the integrals under discussion assume the following form
where t = t — t* and we used equation (5.3.25) explicitly while expressing the distance r = ry (T) as a function of time 7. The constant distance r; was introduced to make the argument of the logarithmic function dimensionless. This constant is not important for calculations as it always cancel out in final formulas. However, in the case of gravitational lensing it is convenient to identify the scale constant rz with the radius of the Einstein ring [62-65] “ In the case when mass and spin are constant, the integrals that we need to carry out are reduced to (1/r]'-4 and [1/r]!-?! which are formally divergent at the lower limit of integration at the past null infinity when time t + —oo. However, one must bear in mind that these integrals do not enter equa- tions of light-ray geodesics (5.3.57) alone but appear in these equations after taking at least one partial derivative with respect to either €' or 7 parameters. This differentiation effectively eliminates divergent parts of the integrals from the final result. Hence, in what follows, we drop out the formally divergent terms so that the integrals under discussion assume the following form
However, taking one more (a third) derivative eliminates all integrals from equation (5.4.15) completely. More specifically, we have Dealing with the double integrals of type [F'(t—1) / ri ~?] is more Sophisticated. We notice that taking the first and second derivatives from the double integral [F(t — r)/r]'~7! do not eliminate the integration along the light ray trajectory and making use of equations (5.4.12), (5.4.15) for explicit calculation of partial derivatives in the right part of equation (5.4.16) proves that all integrations disappear, that is
However, taking one more (a third) derivative eliminates all integrals from equation (5.4.15) completely. More specifically, we have Dealing with the double integrals of type [F'(t—1) / ri ~?] is more Sophisticated. We notice that taking the first and second derivatives from the double integral [F(t — r)/r]'~7! do not eliminate the integration along the light ray trajectory and making use of equations (5.4.12), (5.4.15) for explicit calculation of partial derivatives in the right part of equation (5.4.16) proves that all integrations disappear, that is
where H(p—q) is the Heaviside function defined by the expression (5.1.21), Ci(p,q) are the polynomial coefficient given by equation (5.1.20) and F)(t — r) denotes 0? F(t — r)/Ot” where r is considered as constant. Section 5.5 Gravitational Perturbations of the Light Ray
where H(p—q) is the Heaviside function defined by the expression (5.1.21), Ci(p,q) are the polynomial coefficient given by equation (5.1.20) and F)(t — r) denotes 0? F(t — r)/Ot” where r is considered as constant. Section 5.5 Gravitational Perturbations of the Light Ray
Integration of the light-ray propagation equations (5.3.60) — (5.3.63) is fairly straightforward. Perform- ing one integration of these equations with respect to time yields 5.5.2 Relativistic perturbation of the coordinate velocity of light
Integration of the light-ray propagation equations (5.3.60) — (5.3.63) is fairly straightforward. Perform- ing one integration of these equations with respect to time yields 5.5.2 Relativistic perturbation of the coordinate velocity of light
Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields
Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields
Section 5.5 Gravitational Perturbations of the Light Ray
Section 5.5 Gravitational Perturbations of the Light Ray
Figure 5.2. Astronomical coordinates used in calculation of light propagation. The origin of the coor- dinates is at the center-of-mass of the source of gravitational waves. The bundle of light rays is defined by a vector field k*. Vector K' = —k’ + O(c~*) is directed from observer towards the source of light. Vector Kis directed from the observer towards the source of gravitational waves. We define Ki = —N' = —a' /r, where x’ are the coordinates of observer with respect to the source of gravita- tional waves, and r = ||. The picture shows the plane of the sky being orthogonal to vector K’. Section 5.5 Gravitational Perturbations of the Light Ray
Figure 5.2. Astronomical coordinates used in calculation of light propagation. The origin of the coor- dinates is at the center-of-mass of the source of gravitational waves. The bundle of light rays is defined by a vector field k*. Vector K' = —k’ + O(c~*) is directed from observer towards the source of light. Vector Kis directed from the observer towards the source of gravitational waves. We define Ki = —N' = —a' /r, where x’ are the coordinates of observer with respect to the source of gravita- tional waves, and r = ||. The picture shows the plane of the sky being orthogonal to vector K’. Section 5.5 Gravitational Perturbations of the Light Ray
There exists a relation between the relativistic perturbations of the eikonal and the light-ray trajectory Equation (5.6.1) defines the light time delay effect in the global time ¢ of the ADM-harmonic reference frame. In order to convert it to observable proper time T’ of the observer, we assume for simplicity that the observer is in a state of free fall and moves with velocity V’ with respect to the reference frame of the isolated system. Transformation from the ADM-harmonic coordinate time ¢ to the proper time T’ is made with the help of the standard formula [1, 2] wave but also along the light rays citepfrolov. Both derivations lead, of course, to the same result
There exists a relation between the relativistic perturbations of the eikonal and the light-ray trajectory Equation (5.6.1) defines the light time delay effect in the global time ¢ of the ADM-harmonic reference frame. In order to convert it to observable proper time T’ of the observer, we assume for simplicity that the observer is in a state of free fall and moves with velocity V’ with respect to the reference frame of the isolated system. Transformation from the ADM-harmonic coordinate time ¢ to the proper time T’ is made with the help of the standard formula [1, 2] wave but also along the light rays citepfrolov. Both derivations lead, of course, to the same result
Without going into further details of the observational procedure we give an explicit expression fot the total angle of the light deflection a’. We have where the dot between the two vectors denotes the usual Euclidean scalar product. It is worth empha- sizing that the observed direction s’ to each source of light includes relativistic deflection of the light ray in the form of three perturbations. Two of them, a’ and 7’, depend only on the quantities taken at the point of observation, but 6", according to equation (5.5.35), is also sensitive to the strength of the gravitational field taken at the point of emission of light. This remark reveals that according to relation (5.6.24) a single gravitational wave signal may cause different angular displacements and/or time delays for different sources of light located at different distances from the source of gravitational waves even if the directions to the light sources are the same.
Without going into further details of the observational procedure we give an explicit expression fot the total angle of the light deflection a’. We have where the dot between the two vectors denotes the usual Euclidean scalar product. It is worth empha- sizing that the observed direction s’ to each source of light includes relativistic deflection of the light ray in the form of three perturbations. Two of them, a’ and 7’, depend only on the quantities taken at the point of observation, but 6", according to equation (5.5.35), is also sensitive to the strength of the gravitational field taken at the point of emission of light. This remark reveals that according to relation (5.6.24) a single gravitational wave signal may cause different angular displacements and/or time delays for different sources of light located at different distances from the source of gravitational waves even if the directions to the light sources are the same.
These expressions do not contain any explicit integration along the light ray trajectory because all ex- plicit integrals are either eliminated after taking partial derivatives with respect to the upper limit of the integrals or they are vanish because the numerical coefficient in front of them become nil.
These expressions do not contain any explicit integration along the light ray trajectory because all ex- plicit integrals are either eliminated after taking partial derivatives with respect to the upper limit of the integrals or they are vanish because the numerical coefficient in front of them become nil.
When deriving these equations we assumed that the time evolution of both the mass JV and the angular momentum 8° of the isolated system can be neglected. In opposite case, the right side of equations (5.6.50) and (5.6.51) would contain also time derivatives from the mass and the angular momentum. Terms of the order |€|/R in both formulas are produced by the time derivatives of vector k.
When deriving these equations we assumed that the time evolution of both the mass JV and the angular momentum 8° of the isolated system can be neglected. In opposite case, the right side of equations (5.6.50) and (5.6.51) would contain also time derivatives from the mass and the angular momentum. Terms of the order |€|/R in both formulas are produced by the time derivatives of vector k.
use of equations from section 5.3.3. The result is as follows
use of equations from section 5.3.3. The result is as follows
The Skrotskii effect due to the mass-type multipoles of the isolated system is given by
The Skrotskii effect due to the mass-type multipoles of the isolated system is given by
where Cx is given by equation (5.7.5) and ¢* is the time of the closest approach of the light ray to the barycenter of the isolated system. product of two parameters. Therefore, the convergence requires satisfaction of two conditions where the numerical value of the parameter ¢, must be taken for the smallest wavelength, Amin, in the spectrum of the gravitational radiation emitted by the isolated system. Conditions (5.7.10), (5.7.11) en- sure convergence of the post-Newtonian series in (5.7.8) and (5.7.9) respectively. If the source of light rays and observer are at infinite distances from the isolated system then « ~ 0 and €o ~ O, and the requirements (5.7.10), (5.7.11) are satisfied automatically, irrespective of the structure of the Fourier spectrum (5.4.5) of the gravitational radiation emitted by the isolated system. In a real astronomical practice such an assumption may not be always satisfied. In such cases it is more natural to avoid the
where Cx is given by equation (5.7.5) and ¢* is the time of the closest approach of the light ray to the barycenter of the isolated system. product of two parameters. Therefore, the convergence requires satisfaction of two conditions where the numerical value of the parameter ¢, must be taken for the smallest wavelength, Amin, in the spectrum of the gravitational radiation emitted by the isolated system. Conditions (5.7.10), (5.7.11) en- sure convergence of the post-Newtonian series in (5.7.8) and (5.7.9) respectively. If the source of light rays and observer are at infinite distances from the isolated system then « ~ 0 and €o ~ O, and the requirements (5.7.10), (5.7.11) are satisfied automatically, irrespective of the structure of the Fourier spectrum (5.4.5) of the gravitational radiation emitted by the isolated system. In a real astronomical practice such an assumption may not be always satisfied. In such cases it is more natural to avoid the
If we assume that the mass and angular momentum of the isolated system are conserved the asymp- totic expansions of integrals (5.4.1), (5.4.2) of the stationary part of the metric tensor have the following form Next several equations yield the estimates for the partial derivatives with respect to the parameters €' and 7 from functions of the retarded time (of gravity). In these estimates the numbers m and n depend on 1; we give the estimates from below for m and n for / > 1. post-Newtonian expansions of the metric tensor and/or observable effects and operate with the func- tions of the retarded time. It is important to notice that the retarded time s = ¢ — r which enters the result of the calculation of the is a characteristic of the Einstein equations of gravitational field, not the Maxwell equations. Therefore, measuring the effect of the gravitational deflection of light caused by time-dependent gravitational field allows us to measure the speed of gravity with respect to the speed of light. This type of experiments have been proposed in our paper [118] and successfully performed in 2003 [119]. There were several publications (see review [150]) arguing that the speed of gravity is irrelevant in the light-ray deflection experiments by moving bodies. Unfortunately, all the authors of those publications operated with the post-Newtonian expansion of the gravitational field which replaces the retarded time s = t — r of the gravitational field with the time ¢* of the closest approach of light to the light-ray deflecting body like in equation (5.7.6). This explains why the effect of the retardation of gravity was confused with the retardation of light in L180). wee i i wee me i
If we assume that the mass and angular momentum of the isolated system are conserved the asymp- totic expansions of integrals (5.4.1), (5.4.2) of the stationary part of the metric tensor have the following form Next several equations yield the estimates for the partial derivatives with respect to the parameters €' and 7 from functions of the retarded time (of gravity). In these estimates the numbers m and n depend on 1; we give the estimates from below for m and n for / > 1. post-Newtonian expansions of the metric tensor and/or observable effects and operate with the func- tions of the retarded time. It is important to notice that the retarded time s = ¢ — r which enters the result of the calculation of the is a characteristic of the Einstein equations of gravitational field, not the Maxwell equations. Therefore, measuring the effect of the gravitational deflection of light caused by time-dependent gravitational field allows us to measure the speed of gravity with respect to the speed of light. This type of experiments have been proposed in our paper [118] and successfully performed in 2003 [119]. There were several publications (see review [150]) arguing that the speed of gravity is irrelevant in the light-ray deflection experiments by moving bodies. Unfortunately, all the authors of those publications operated with the post-Newtonian expansion of the gravitational field which replaces the retarded time s = t — r of the gravitational field with the time ¢* of the closest approach of light to the light-ray deflecting body like in equation (5.7.6). This explains why the effect of the retardation of gravity was confused with the retardation of light in L180). wee i i wee me i
The relativistic time delay is given by The observable unit vector in the direction from the observer to the source of light is given by the following expression This section provides the reader with the asymptotic expressions for potentially observable relativistic effects by taking into account only the leading terms and neglecting all terms which are proportional to the parameter e.
The relativistic time delay is given by The observable unit vector in the direction from the observer to the source of light is given by the following expression This section provides the reader with the asymptotic expressions for potentially observable relativistic effects by taking into account only the leading terms and neglecting all terms which are proportional to the parameter e.
One more asymptotic expression is given for the integral taken from an STF partial derivative Taking into account only the leading order terms in (5.8.15), we obtain the asymptotic expansion Using this expression we can write the Taylor expansion for the functions of the retarded time s = t — 7
One more asymptotic expression is given for the integral taken from an STF partial derivative Taking into account only the leading order terms in (5.8.15), we obtain the asymptotic expansion Using this expression we can write the Taylor expansion for the functions of the retarded time s = t — 7
In this section we give expressions for the relativistic effects of the time delay, bending of light and the Skrotskii effect in the gravitational plane-wave approximation. In this approximation we neglect all terms of the order of 67, 62 and 52, and higher. For the time delay we have 5.8.2 Asymptotic expressions for observable effects
In this section we give expressions for the relativistic effects of the time delay, bending of light and the Skrotskii effect in the gravitational plane-wave approximation. In this approximation we neglect all terms of the order of 67, 62 and 52, and higher. For the time delay we have 5.8.2 Asymptotic expressions for observable effects
The observed astrometric direction from observer to the source of light in the plane gravitational-wave annroximation is The observed astrometric direction from observer to the source of light in the plane gravitational-wav
The observed astrometric direction from observer to the source of light in the plane gravitational-wave annroximation is The observed astrometric direction from observer to the source of light in the plane gravitational-wav
Figure 5.4. Relative configuration of observer (O), a source of light (S), and a localized system emitting of gravitational waves (D). The gravitational waves deflect light rays which are emitted at the moment to at the point S and received at the moment ¢ at the point O. The point E on the line OS corresponds to the moment of the closest approach of light ray to the system D. Notations for distances are OS = R, DO =r, DS =17r0, DE = d- the impact parameter, OF = 7 = rcos6, ES = 7 = 7 — R. The distance R is much smaller than both r and ro. There is no limitation on the impact parameter d which can be or may be not small as compared to all other distances.
Figure 5.4. Relative configuration of observer (O), a source of light (S), and a localized system emitting of gravitational waves (D). The gravitational waves deflect light rays which are emitted at the moment to at the point S and received at the moment ¢ at the point O. The point E on the line OS corresponds to the moment of the closest approach of light ray to the system D. Notations for distances are OS = R, DO =r, DS =17r0, DE = d- the impact parameter, OF = 7 = rcos6, ES = 7 = 7 — R. The distance R is much smaller than both r and ro. There is no limitation on the impact parameter d which can be or may be not small as compared to all other distances.
Covariant generalization of (7.2.12)—(7.2.14) is not quite straightforward because the covariant derivative of the metric tensor gv; = 0, and we cannot use it as a dynamic variable. In this case, we consider the set of the metric tensor, g,., the Christoffel symbols [,,,, and the Riemann tensor Rg), as a set of independent dynamic variables. The action is given by (7.2.1) where F = /—@f (guv TP uv,R” by) is a scalar density of weight +1. Variation of F is The order, in which the covariant derivatives are taken, is imposed by the procedure of the extracting the total divergence from the variation of the action in (7.2.2). The order of the derivatives is important because the covariant derivatives do not commute.
Covariant generalization of (7.2.12)—(7.2.14) is not quite straightforward because the covariant derivative of the metric tensor gv; = 0, and we cannot use it as a dynamic variable. In this case, we consider the set of the metric tensor, g,., the Christoffel symbols [,,,, and the Riemann tensor Rg), as a set of independent dynamic variables. The action is given by (7.2.1) where F = /—@f (guv TP uv,R” by) is a scalar density of weight +1. Variation of F is The order, in which the covariant derivatives are taken, is imposed by the procedure of the extracting the total divergence from the variation of the action in (7.2.2). The order of the derivatives is important because the covariant derivatives do not commute.
where fg = P= lq in accordance with definition (7.3.25). The potential energy of the scalar field, W = W/(®), is kept arbitrary.
where fg = P= lq in accordance with definition (7.3.25). The potential energy of the scalar field, W = W/(®), is kept arbitrary.
where we used the superposition € = €m + €, P = Pm + Pq. Second step is to transform the linear differential operator F’i,, in (7.5.37) to a more convenient form that will allow us to single out the gauge- dependent vector denoted by Linearized field equations for gravitational field variables, J,,,, are obtained from (7.5.28) after neglect- ing in its right side the non-linear source J,,,, and making a series of transformations to sort out similar terms. First, let us make use of equations (7.5.40) and (7.5.41) to find out
where we used the superposition € = €m + €, P = Pm + Pq. Second step is to transform the linear differential operator F’i,, in (7.5.37) to a more convenient form that will allow us to single out the gauge- dependent vector denoted by Linearized field equations for gravitational field variables, J,,,, are obtained from (7.5.28) after neglect- ing in its right side the non-linear source J,,,, and making a series of transformations to sort out similar terms. First, let us make use of equations (7.5.40) and (7.5.41) to find out
Here, H is a conformal Hubble parameter (7.4.4), a prime means a derivative with respect to 7, and we denoted big — FP | eB. q= (loo + lik) /2, len = liitleetlss, lays. = bag (1/3) dijlek, and the same index notations are applied to the tensor of energy-momentum %;; of the localized astronomical system, Deis = Liz — (1/3)bij Tex. These equations are clearly decoupled from one another, thus, demonstrating the advantage of the gauge sendition Be =0.
Here, H is a conformal Hubble parameter (7.4.4), a prime means a derivative with respect to 7, and we denoted big — FP | eB. q= (loo + lik) /2, len = liitleetlss, lays. = bag (1/3) dijlek, and the same index notations are applied to the tensor of energy-momentum %;; of the localized astronomical system, Deis = Liz — (1/3)bij Tex. These equations are clearly decoupled from one another, thus, demonstrating the advantage of the gauge sendition Be =0.
The Laplace-Beltrami operator for function q has been derived in (7.6.25). Now, we make use of equa- tions (7.6.25), (7.6.28a), (7.6.33) in calculating the right side of (7.6.32). After a significant amount of algebra, we find out that all terms explicitly depending on q and the gauge functions B® cancel out, so that equation for Vm becomes The Laplace-Beltrami operator for function ym is given in equation (7.6.28a) which is not gauge- invariant. Taking the covariant derivative from this equation and contracting it with &@* brings about the first term in the right side of equation (7.6.32),
The Laplace-Beltrami operator for function q has been derived in (7.6.25). Now, we make use of equa- tions (7.6.25), (7.6.28a), (7.6.33) in calculating the right side of (7.6.32). After a significant amount of algebra, we find out that all terms explicitly depending on q and the gauge functions B® cancel out, so that equation for Vm becomes The Laplace-Beltrami operator for function ym is given in equation (7.6.28a) which is not gauge- invariant. Taking the covariant derivative from this equation and contracting it with &@* brings about the first term in the right side of equation (7.6.32),
This equation can be also derived by the procedure being similar to that used in the previous subsection in deriving equation for Vm. We followed this procedure and confirm that it leads to (7.6.37) as expected. Equation (7.6.37) is clearly gauge-invariant. Besides Vq it depends on variable x and should be solved along with equation (7.6.29). which is obtained after subtraction of equation (7.6.13a) from (7.6.13b). Equation for Vq is derived directly from (7.6.36) and equations (7.6.34) and (7.6.29) for Vm and y respectively. We obtain, This equation can be also derived by the procedure being similar to that used in the previous subsection
This equation can be also derived by the procedure being similar to that used in the previous subsection in deriving equation for Vm. We followed this procedure and confirm that it leads to (7.6.37) as expected. Equation (7.6.37) is clearly gauge-invariant. Besides Vq it depends on variable x and should be solved along with equation (7.6.29). which is obtained after subtraction of equation (7.6.13a) from (7.6.13b). Equation for Vq is derived directly from (7.6.36) and equations (7.6.34) and (7.6.29) for Vm and y respectively. We obtain, This equation can be also derived by the procedure being similar to that used in the previous subsection
of the wave operators for the scalar, vector and tensor fields in the conformal spacetime and in arbi- trary coordinates. Notice that although the conformal spacetime coincides, in case of k = 0, with the Minkowski spacetime, the metric fg is not the diagonal Minkowski metric 7. unless the coordinates are Cartesian. Of course, the covariant derivative from a scalar must be understood as a partial derivative, thatis Pg = Fg.
of the wave operators for the scalar, vector and tensor fields in the conformal spacetime and in arbi- trary coordinates. Notice that although the conformal spacetime coincides, in case of k = 0, with the Minkowski spacetime, the metric fg is not the diagonal Minkowski metric 7. unless the coordinates are Cartesian. Of course, the covariant derivative from a scalar must be understood as a partial derivative, thatis Pg = Fg.
Similar procedure is applied to equation (7.6.37) and leads to a wave equation for function V, de-
Similar procedure is applied to equation (7.6.37) and leads to a wave equation for function V, de-
This is a wave equation with the speed of sound v, which determines the speed of propagation of the scalar perturbations in the dark matter considered as an ideal fluid. These perturbations can be interpreted as acoustic or sound waves of different wavelengths propagating in spacetime. Solution of homogeneous equation (7.7.37) describes the propagation of primordial scalar perturbations of dark matter. A particu- lar solution of the inhomogeneous equation (7.7.37) tells us how the perturbation of dark matter caused by the isolated astronomical system propagate. Dark matter and dark energy are described by scalar fields ® and V. The fields themselves are not gauge- invariant. Therefore, physical meaning have only the equations for the gauge-invariant perturbations of these fields which are Vn, Vy, and y. We consider, first, equation (7.6.34) for the gauge-invariant scalar Vn. We convert the covariant derivatives taken with respect to the background metric, gag, to the partial derivatives of the conformally-flat metric, fg and use equation (7.7.31a) for the Laplace- Beltrami operator along with the expressions for various cosmological parameters given in section 7.7.2. After arranging terms with respect to the powers of the Hubble parameter J, we obtain the scalar equation for function Vm describing the perturbations of dark matter,
This is a wave equation with the speed of sound v, which determines the speed of propagation of the scalar perturbations in the dark matter considered as an ideal fluid. These perturbations can be interpreted as acoustic or sound waves of different wavelengths propagating in spacetime. Solution of homogeneous equation (7.7.37) describes the propagation of primordial scalar perturbations of dark matter. A particu- lar solution of the inhomogeneous equation (7.7.37) tells us how the perturbation of dark matter caused by the isolated astronomical system propagate. Dark matter and dark energy are described by scalar fields ® and V. The fields themselves are not gauge- invariant. Therefore, physical meaning have only the equations for the gauge-invariant perturbations of these fields which are Vn, Vy, and y. We consider, first, equation (7.6.34) for the gauge-invariant scalar Vn. We convert the covariant derivatives taken with respect to the background metric, gag, to the partial derivatives of the conformally-flat metric, fg and use equation (7.7.31a) for the Laplace- Beltrami operator along with the expressions for various cosmological parameters given in section 7.7.2. After arranging terms with respect to the powers of the Hubble parameter J, we obtain the scalar equation for function Vm describing the perturbations of dark matter,
The reader can observe that equations (7.7.42a)-(7.7.42d) for linearized metric perturbations are decou- pled from each other. Moreover, equations (7.7.42b-(7.7.42d) are decoupled from the matter perturba- tions Vn, Xm, etc. Only equation (7.7.42a) for q is coupled with the matter perturbations governed by Post-Newtonian equations for gravitational perturbations in physical spacetime are (7.6.25), (7.6.27), (7.6.38) and (7.6.40). These equations are gauge-dependent. In order to fix the gauge we imposed the gauge conditions (7.5.64), (7.6.43). In this gauge, equations for the conformal metric tensor perturba- tions become
The reader can observe that equations (7.7.42a)-(7.7.42d) for linearized metric perturbations are decou- pled from each other. Moreover, equations (7.7.42b-(7.7.42d) are decoupled from the matter perturba- tions Vn, Xm, etc. Only equation (7.7.42a) for q is coupled with the matter perturbations governed by Post-Newtonian equations for gravitational perturbations in physical spacetime are (7.6.25), (7.6.27), (7.6.38) and (7.6.40). These equations are gauge-dependent. In order to fix the gauge we imposed the gauge conditions (7.5.64), (7.6.43). In this gauge, equations for the conformal metric tensor perturba- tions become
This equation describes propagation of the density perturbation of dark matter, dm, in the form of sound waves with velocity v;. Equation (7.8.6) is decoupled from any other perturbation and can be solved separately after the boundary conditions are specified. For this reason we call (7.8.6) master equation.
This equation describes propagation of the density perturbation of dark matter, dm, in the form of sound waves with velocity v;. Equation (7.8.6) is decoupled from any other perturbation and can be solved separately after the boundary conditions are specified. For this reason we call (7.8.6) master equation.
where the scale factor a in front of the integrals depends on the coordinates of the field point a = a [n(t,a)], and the functions in the integrand depend on the retarded time A particular solution of the wave equations satisfying condition (7.8.28), is the retarded integral [75. where 27? = (2°,a'), 7 = X° is the conformal time in isotropic coordinates connected to the coordinates x by a Lorentz boost 7 = n(27) = L°gx%, the null vector n® = {1,2'/r}, and r = 6;j;2'«? is the radial distance. This condition ensures that there is no infalling gravitational radiation arriving to the localized astronomical system from the future null infinity J‘. Effectively, it singles out the retarded solution of the wave equation. Whether the boundary condition (7.8.28) is valid or not, we do not know for sure because our knowledge of the universe is limited by the existence of the cosmological (also known as light or particle) horizon [81] that represents the boundary between the observable and the unobservable regions of the universe. Nonetheless, in case of spatially flat (k = 0) universe, the condition (7.8.28) seems to be highly plausible. ey EY CS Ae hs a ae Sea Cae et ee Sa dy MS ig tt eS A Se A Si a ey Me Bae a , because gravity propagates with finite speed. Equation (7.8.30) describes characteristic of the null cone in the conformal Minkowski spacetime that determines the causal nature of the gravitational field in the expanding universe with k = 0 []. Solutions (7.8.29a)—(7.8.29d) are Lorentz-invariant as shown by calculations in section 7.8.4.
where the scale factor a in front of the integrals depends on the coordinates of the field point a = a [n(t,a)], and the functions in the integrand depend on the retarded time A particular solution of the wave equations satisfying condition (7.8.28), is the retarded integral [75. where 27? = (2°,a'), 7 = X° is the conformal time in isotropic coordinates connected to the coordinates x by a Lorentz boost 7 = n(27) = L°gx%, the null vector n® = {1,2'/r}, and r = 6;j;2'«? is the radial distance. This condition ensures that there is no infalling gravitational radiation arriving to the localized astronomical system from the future null infinity J‘. Effectively, it singles out the retarded solution of the wave equation. Whether the boundary condition (7.8.28) is valid or not, we do not know for sure because our knowledge of the universe is limited by the existence of the cosmological (also known as light or particle) horizon [81] that represents the boundary between the observable and the unobservable regions of the universe. Nonetheless, in case of spatially flat (k = 0) universe, the condition (7.8.28) seems to be highly plausible. ey EY CS Ae hs a ae Sea Cae et ee Sa dy MS ig tt eS A Se A Si a ey Me Bae a , because gravity propagates with finite speed. Equation (7.8.30) describes characteristic of the null cone in the conformal Minkowski spacetime that determines the causal nature of the gravitational field in the expanding universe with k = 0 []. Solutions (7.8.29a)—(7.8.29d) are Lorentz-invariant as shown by calculations in section 7.8.4.
and y = 1/,/1— £? is the Lorentz factor. Equation (7.8.90) has two roots corresponding to the ad- vanced and retarded times. The root corresponding to the retarded-time solution of (7.8.90) is
and y = 1/,/1— £? is the Lorentz factor. Equation (7.8.90) has two roots corresponding to the ad- vanced and retarded times. The root corresponding to the retarded-time solution of (7.8.90) is
where the coefficients A,B,C of the quadratic form are,
where the coefficients A,B,C of the quadratic form are,
Related topics:
GeodesySpace GeodesyRelativistic cosmologyPhysical GeodesyGeneral Theory of RelativityOrbital/Celestial MechanicsClassical & Relativistic MechanicsPropagation of lightGeometrical GeodesyAtomic clocks
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