Gravito-electromagnetic analogies

Abstract

We reexamine and further develop different gravito-electromagnetic analogies found in the literature, and clarify the connection between them. Special emphasis is placed in two exact physical analogies: the analogy based on inertial fields from the so-called “1+3 formalism”, and the analogy based on tidal tensors. Both are reformulated, extended and generalized. We write in both formalisms the Maxwell and the full exact Einstein field equations with sources, plus the algebraic Bianchi identities, which are cast as the source-free equations for the gravitational field. New results within each approach are unveiled. The well known analogy between linearized gravity and electromagnetism in Lorentz frames is obtained as a limiting case of the exact ones. The formal analogies between the Maxwell and Weyl tensors are also discussed, and, together with insight from the other approaches, used to physically interpret gravitational radiation. The precise conditions under which a similarity between gravity and electromagnetism occurs are discussed, and we conclude by summarizing the main outcome of each approach.

Appendix A: Inertial forces—simple examples in flat spacetime

In Sect.  we have seen that the inertial forces felt in a given frame arise from two independent contributions of different origin: the kinematics of the observer congruence (that is, from the derivatives of the temporal basis vector of the frame, $\mathbf {e}_{\hat{0}}=\mathbf {u}$, where $\mathbf {u}$ is the observers’ 4-velocity), and the transport law for the spatial triads $\mathbf {e}_{\hat{i}}$ along the congruence. In order to illustrate these concepts with simple examples, we shall consider, in flat spacetime, the straightline geodesic motion of a free test particle (4-velocity $\mathbf {U}$), from the point of view of three distinct frames: a) a frame whose time axis is the 4-velocity of a congruence of observers at rest, but whose spatial triads rotate uniformly with angular velocity $\vec {\Omega }$; b) a frame composed of a congruence of rigidly rotating observers (vorticity $\vec {\omega }$), but carrying Fermi–Walker transported spatial triads ($\vec {\Omega }=0$); c) a rigidly rotating frame, that is, a frame composed of a congruence of rigidly rotating observers, carrying spatial triads co-rotating with the congruence $\vec {\Omega }=\vec {\omega }$ (i.e., “adapted” to the congruence, see Sect. 3.1). This is depicted in Fig. 2.

Fig. 2

A test particle in uniform motion in flat spacetime from the point of view of three different frames: (a) a frame composed of observers at rest, but carrying spatial triads that rotate with uniform angular velocity $\vec {\Omega }$; (b) a frame consisting of a congruence of rigidly rotating observers (vorticity $\vec {\omega }$), but each of them carrying a non-rotating spatial triad (i.e., that undergoes Fermi–Walker transport); (c) a rigidly rotating frame (a frame adapted to a congruence of rigidly rotating observers); the spatial triads co-rotate with the congruence, $\vec {\Omega }=\vec {\omega }$. Note: by observer’s rotation we mean their circular motion around the center; and by axes rotation we mean their rotation (relative to FW transport) about the local tetrad’s origin

In the first case there we have a vanishing gravitoelectric field $\vec {G}=0$, and a gravitomagnetic field $\vec {H}=\vec {\Omega }$ arising solely from the rotation (with respect to Fermi–Walker transport) of the spatial triads; thus the only inertial force present is the gravitomagnetic force $\vec {F}_{\mathrm{GEM}}=\gamma \vec {U}\times \vec {\Omega }$, cf. Eq. (42), with $\gamma \equiv -U^{\alpha }u_{\alpha }$. In the frame b), there is a gravitoelectric field $\vec {G}=\vec {\omega }\times (\vec {r}\times \vec {\omega })$ due the observers acceleration, and also a gravitomagnetic field $\vec {H}=\vec {\omega }$, which originates solely from the vorticity of the observer congruence. That is, there is a gravitomagnetic force $\gamma \vec {U}\times \vec {\omega }$ which reflects the fact that the relative velocity $v^{\alpha }=U^{\alpha }/\gamma -u^{\alpha }$ (or $\vec {v}=\vec {U}/\gamma$, in the observer’s frame, where $\vec {u}=0$) between the test particle and the observer it is passing by changes in time. The total inertial forces are in this frame

$$\begin{aligned} \vec {F}_{\mathrm{GEM}}=\gamma \left[ \gamma \vec {\omega }\times (\vec {r}\times \vec {\omega })+\vec {U}\times \vec {\omega }\right] . \end{aligned}$$

In the frame (c), which is the relativistic version of the classical rigid rotating frame, one has the effects of (a) and (b) combined: a gravitoelectric field $\vec {G}=\vec {\omega }\times (\vec {r}\times \vec {\omega })$, plus a gravitomagnetic field $\vec {H}=\vec {\omega }+\vec {\Omega }=2\vec {\omega }$, the latter leading to the gravitomagnetic force $2\gamma \vec {U}\times \vec {\omega }$, which is the relativistic version of the well known Coriolis acceleration, e.g., [112]. The total inertial force is in this frame

$$\begin{aligned} \vec {F}_{\mathrm{GEM}}=\gamma \left[ \gamma \vec {\omega }\times (\vec {r}\times \vec {\omega })+2\vec {U}\times \vec {\omega }\right] \end{aligned}$$

which is the relativistic generalization of the inertial force in e.g., Eq. (4.91) of [112]. Moreover, in this case, as discussed in Sects. 3.2.1 and 3.2.2, the $\Gamma _{\hat{j}\hat{k}}^{\hat{i}}$ in Eq. (43) are the connection coefficients of the (Levi–Civita) 3-D covariant derivative with respect to the metric $h_{ij}$ (defined by Eq. (59)) defined on the space manifold associated to the quotient of the spacetime by the congruence; $\vec {U}$ is the vector tangent to the 3-D curve (see Fig. 2c) obtained by projecting the particle’s worldline on the space manifold, and $\vec {F}_{\mathrm{GEM}}=\tilde{D}\vec {U}/d\tau$ is simply the covariant 3-D acceleration of that curve.