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Comparison of electromagnetic and gravitational radiation: What we can learn about each from the other
1. J. Belcher
and C. Koleci
, “Using animated textures to visualize electromagnetic fields and energy flow
,” e-print arXiv:0802.4034
3. D. A. Nichols et al., “Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: General theory and weak-gravity applications,” Phys. Rev. D 84, 124014 (2011).
4. A. Zimmerman, D. A. Nichols, and F. Zhang, “Classifying the isolated zeros of asymptotic gravitational radiation by tendex and vortex lines,” Phys. Rev. D 84, 044037 (2011).
5.We omit special relativistic effects here, such as the Lorentz factor. We assume that motions are slow compared to the speed of light.
6. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco, 1973), Sec. 36.1.
7. K. S. Thorne, R. H. Price, and D. A. MacDonald, Black Holes: The Membrane Paradigm (Yale U.P., New Haven, 1986).
8.In fully nonlinear general relativity the gravitoelectric and gravitomagnetic tensors are defined as projections of the Weyl tensor (equal to the Riemann tensor in source-free spacetime). For details see Ref. 3.
9.See Ref. 7, Eq. (5.21).
10.In general relativity these equations follow from the “Bianchi identities,” mathematical identities satisfied by the Riemann tensor. Since has no Newtonian equivalent these equations cannot be understood from simple Newtonian gravity theory.
11.The literature on general relativity often adopts units in which c = 1. As pointed out in Sec. I, we do not make that choice here. In accordance with Eq. (8), with our definitions has dimensionality 1/time2 and has dimensions 1/(timelength).
12.We emphasize this point because weak relativistic gravitational fields are often described with metric perturbations . These perturbations, like electromagnetic potentials, are mathematically useful, but they are subject to gauge transformations and hence do not directly represent physical effects.
13. B. Cabral and C. L. Leedom, “Imaging vector fields using line integral convolution,” in Proceedings of the SIGGRAPH 93, edited by J. T. Katiya (ACM Press, 1993), pp. 263–270.
14. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, Hoboken, 1998).
15.This ellipsoid can be drawn only if the three principal moments of inertia are all positive, but the fact that mass density is nonnegative does guarantee this. The ellipsoid cannot be used so simply for, say, because its tracelessness means that the sum of its principal values is zero.
16. T. Delmarcelle and L. Hesselink, “The topology of symmetric, second-order tensor fields,” in Proceedings of the Conference on Visualization 94, edited by D. Bergeron and A. Kaufman (IEEE Computer Society Press, Los Alamitos, CA, 1994), pp. 140–147.
17.This inclusion is made for simplicity to eliminate the mass monopole when the d→0 limit is taken. With the particle omitted, the argument would require eliminating the monopole moment by other means, such as a specific projection of only the quadrupole moment.
18.Although the static components are straightforward to compute with Eq. (6), this procedure is valid only for Cartesian components. To arrive at useful results those components must subsequently be transformed to the spherical basis and to functions of r and θ. In practice, it is much simpler to use the formalism of tensorially correct differentiation (“covariant differentiation”); see J. Romano and R. H. Price, “Why no shear in ‘Div, grad, curl, and all that’?,” Am. J. Phys. 80, 519–524 (2012).
19.In the gravitational case this follows from the fundamental definition of in general relativity. The tensor involves the components of a fourth rank curvature tensor with only a single time (i.e., 0) index. Such components reverse sign under time reversal. In static configurations these components must be unchanged under time reversal, hence they must be zero. The vanishing of for static configurations is also suggested by Eq. (10).
20.Due to the choice of sign in Eq. (6), is actually the negative of the gradient of g.
22.This issue is discussed in detail in J. W. Belcher and S. Olbert, “Field line motion in classical electromagnetism,” Am. J. Phys. 71, 220–228 (2003).
23.The second term on the right-hand-side of the first line accounts for the change in flux due to the motion of the curve C bounding the area A. The mathematics here for time changing electric flux is identical to that for the time changing magnetic flux in Faraday's law. See, e.g., Eq. (5.137) and the associated footnote in Ref. 14.
24.These fields are most simply computed using the vector spherical harmonics in Sec. 9.10 of Ref. 14.
26.Each curve in Fig. 7 corresponds to a particular eigenvalue. Each family represents a set of eigenvalues and of eigenvector components that vary smoothly from one curve to the adjacent curve. It turns out that the signs of the eigenvalues vary within a family. The eigenvalues for the top LIC in Fig. 7 are negative in both the near zone and the radiation zone, but positive in a part of the intermediate zone. The opposite applies to the bottom LIC.
27. E. Newman and R. Penrose, “An approach to gravitational radiation by a method of spin coefficients,” J. Math. Phys. 3, 566–578 (1962). In such a formalism all field theories consist of the same equations relating the scalars, and differ only in the number of scalars. There are three such (complex) scalars for electromagnetism describing the six components of E and B. There are five such (complex) scalars for the ten independent components of and . The two extra scalars are linear combinations of the components , and that carry information about radiation. (The latter two components vanish for our quadrupole example.) If the field equations that contain these extra two scalars are eliminated, then the field equations for the gravitational scalars—multiplied by r—are the same as those of electromagnetism.
28. R. H. Price, “Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields,” Phys. Rev. D 5, 2439–2454 (1972).
29.See Sec. 35.15 of Ref. 6.
30.The Landau-Lifschitz pseudotensor is constructed from the spacetime gradients of metric perturbations. The gravitoelectric and gravitomagnetic fields are constructed from the gradients of the gradients of the metric perturbations.
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