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Gravitational Tidal Forces are Purely Irrotational: No Transverse Gravitational Waves

Updated: Feb 28

In a recent article, "Tidal forces are gravitational waves" Goswami & Ellis make this preposterous claim without a smidgen justification except for the desire to hook up with gravitational waves. More specifically, they claim that "whenever there is a time varying tidal interaction, the curls of both the electric and magnetic Weyl must be simultaneously non-zero which is necessary for obtaining a closed wave equation." Continuing they assert that "for the propagation of a Newtonian observable, we need a non-Newtonian variable."


This is pure idle wishful thinking. And, to add insult to injury, they assert out of the clear blue sky that tidal deformations must trave with the velocity of light between the two bodies."


The tidal forces in the outer Schwarzschild solution are well-known [Seeing Gravity]. They are determined from the will known Jacobi equation which describe the spreading of geodesics. The Jacobi equation for the deviation from a geodesic is

d^2X^i/dt^2-E^i_j X^j i,j=1,2,3

where E^i_j represents the electric part of the Weyl tensor. At most, but not in the Schwarzschild case, E^i_j can be a function only of the time "t". Synge was first to show that the Jacobi equation could be written in Hamiltonian form with coordinates X^i and momenta P^i. That the integral around a closed curve of P^i dX_i vanishes means that the gravitational tidal forces are necessary irrotational. This guarantees the existence of a scalar potential V, whose mixed second derivatives coincides with the sectional curvature K_{ij}, which satisfied the Poisson equation

k^i_i=4 pi G p=R^i_{0i0}=-R_{00}

where the four indices on R indicates the Riemann tensor, and the two indexed one, the Ricci tensor. Thus,

R_{00}=-4 pi G p, (1)

was the first attempt to generalize the Poisson equation. But, R_{00} is proportional to the sum p+3P, where P is the hydrostatic pressure. So this is not the Poisson equation. So we want a generalized Poisson equation which does not contain solely the density. And to get the full Einstein equations, pv^iv^j must be replaced by the energy-stress tensor, and the left-hand side follows from the conservation condition that the divergence of the Einstein tensor must vanish, just like the vanishing of the divergence of the energy-stress tensor T^i_j.


The existence of a rotational field would require additional terms in the Jacobi equation. These accelerations would be related to a solenoidal field in electromagnetism, and provide the second component in the Lorentz force involving the magnetic field. Since we know from all known solutions to Einstein's equations that the sectional curvatures are sufficient to determine the tidal forces, these terms must necessarily be absent in the Jacobi equation. In other words, there is no dual to the Faraday tensor.


What Synge showed was that the Jacobi equation could be written in Hamiltonian form

dX^i/dt=dH/dp^i and dP^i=-dH/dX^i,

where the Hamiltonian is

H=(1/2)(P^iP_i+K^i_jX_iX^j),

where the sectional curvatures K_i^j play the role of Hookean elastic constants. The Jacobi equation is obtained by differentiating one of the Hamilton equations a second time and using the other to obtain the second order equation

d^2x^i/dt^2+ K^i_jx^j=0,

related to geodesic deviant. This shows that the constant sectional curvatures,

K^{ij}=d^2H/dx^idx^j,

acts as elastic constants. Since the metric is symmetric, it can be diagonalized. It then results trace-free. The condition coincides with the vanishing of the eigenvalues of the Ricci tensor, which just happens to be Einstein's condition of emptiness.


So this condition is other than a condition for gravitational radiation. All motion is irrotational and there can be no axial vector fields.


A magnetic field cannot live in such an environment. The vanishing of the eigenvalues of the Ricci tensor is testimony to the fact that the sum of the partial curvatures in any given direction vanishes. This, it will be remembered, is Einstein's condition of emptiness. We would like to think of it as a force balance relation acting on an ellipsoid to achieve equilibrium.


But GEM, as it is known, makes the coefficients in the metric, time-dependent in addition to space dependent, and adds a cross term involving a "vector" potential. This supposedly describes a rotating system. But rotating systems do not need time dependencies to rotate. Godel's metric describes a rotating system, yet the magnetic component of the Weyl tensor, which supposedly accounts for rotations, as well as GWs, vanishes! So what does the magnetic component actually describe?


If the vector potential is time-independent, the gravitational force is a gradient, and this is what is encapsulated in the Lorentz force: a static Coulomb field and a motional field involving the magnetic force (i.e. the cross-product of velocity and magnetic field). Drawing on the analogy of Maxwell's equations and the corresponding equations involving gravity just won't do. Why should the gravitational potential become time-dependent in a theory that describes geodesic motion in weak fields?


If the vector potential is time-dependent, the sectional curvatures work out the same as if the vector potential were not there. To say that the curl of the Weyl electric field is nom zero needs some explaining, and not just jumping to a gravitational wave equation. Matching can be done with a non-rotating solutions like the Schwarzschild solution between outer and inner solutions, but no such luck with metrics involving rotating bodies. To such systems, you cannot even apply Einstein's principle of equivalence.


In conclusion, we would say that taking away the time dependencies of the coefficients in the indefinite metric, eliminating a fictitious vector potential, and being unable to justify the analogy with Maxwell's equations, leaves gravitational waves out in the cold.

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