We compare the nature of electromagnetic fields and gravitational fields in linearized general relativity. We carry out this comparison both mathematically and visually. In particular, the “lines of force” visualizations of electromagnetism are contrasted with the recently introduced tendex/vortex eigenline technique for visualizing gravitational fields. Specific solutions, visualizations, and comparisons are given for an oscillating point quadrupole source. Among the similarities illustrated are the quasistatic nature of the near fields, the transverse 1/r nature of the far fields, and the interesting intermediate field structures connecting these two limiting forms. Among the differences illustrated are the meaning of field line motion and of the flow of energy.
REFERENCES
We omit special relativistic effects here, such as the Lorentz factor. We assume that motions are slow compared to the speed of light.
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.
See Ref. 7, Eq. (5.21).
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.
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.
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.
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.
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).
Due to the choice of sign in Eq. (6), is actually the negative of the gradient of g.
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.
These fields are most simply computed using the vector spherical harmonics in Sec. 9.10 of Ref. 14.
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.
See Sec. 35.15 of Ref. 6.
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.
