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Does the Weyl Tensor Describe Gravitational Radiation?

Updated: Feb 28, 2022

It has long been assumed that the near and far fields of gravity can be described by a matter dominated near field, accounted for by the Ricci tensor, and a gravitational radiation far field, accounted for by the "electric" component of the Weyl tensor. Exemplary of such is Hawking's paper on the "Gravitational radiation in an expanding universe." However, nothing could be further from the truth.


Under the misguided assumption that "accelerating masses radiate", Hawking claims that by "measuring relative accelerations of neighboring particles, [an observer] he may determine

R_{abcd}V^aV^c, where V^a is the velocity vector. The trace-free part of this represents the "electric components of the gravitational radiation"

E_{bd}=C_{abcd}V^aV^c.

The equation that is implied is the geodesic deviation equation, which determines the tidal force components, d^2W/dx^b dx^d, where W is the Newton's gravitational potential. That this represent tidal forces, the gravitational field must be static. Such a field can hardly represent gravitational radiation.


We recall, from a previous blog, Synge showed that if the gravitational field is static, then the sectional curvatures, K^i_j, in the geodesic deviation equation

d^2x^i/dt^2=-K^i_j x^j (1)

are constants and given by the second mixed derivatives of the gravitational potential. This he did by writing down the Hamiltonian,

H=(1/2)(p^i p_i+ K_{ij} x^i x^j),

and using the Hamiltonian equations,

dx^i/dt=dH/dp^i=p^i, and dp^i/dt=-dH/dx^i=- K^I_jx^j.

Differentiating the first equation with respect to t, and introducing the second leads directly to the geodesic deviation equation (1), provided the sectional curvatures are time-independent.

The sectional curvatures,

K_{ij}=E_{ij}=C_{iajb}V^aV^b,

coincide with the electric component of the Weyl tensor, C. The sectional curvatures, or equivalently, the electric component of the Weyl tensor, are given by the second mixed derivatives of the Newtonian gravitational potential. This ensures the existence of such a potential, or that the integral of p_i around a closed curve of element dx^i be an integral invariant. This certainly has nothing to do with gravitational radiation. Hawking's mistake was to take the geodesic deviation equation as an expression for an accelerating mass, and associate the sectional curvatures, or the electric component of the Weyl tensor with gravitational radiation that would be emitted from an accelerating mass. Nothing of the sort!


The other component of the Weyl tensor is known as the magnetic component,

H_{ij}=(1/2)e_{icd}C^{cd}_{bk}V^k,

where e is the spatial skew-symmetric permutation tensor. Since E, H and R_{ij}, the Ricci tensor, determine the Riemann tensor. So, at least in this sense, E and H should be complementary to the Ricci tensor. Yet, everyone in the field claims that

E_[ab]=-dH{ab}/dt

H_[ab]= dE_{ab}/dt

E^a_a=0

H^a_a=0,

which are the covariant analogs of Maxwell's equations far from their sources. The square brackets, [ab], indicates the commutator.


The first equation is the analog of Faraday's law, the second, that of the Maxwell displacement current, the third is Coulomb's law with no source, and the last is absence of magnetic charge. The first two equations, taken together, implies a wave solution for either component of the Weyl tensor. However, because E is tidal, it is independent of time and there is no wave solution. That there is no displacement current in the transverse Ampere equation means that there is no Faraday effect whereby a changing magnetic field creates a circulating current. Either both fields are time-independent, or they are not. What saves the Lorentz force from combining a static Coulomb field with the Lorentz force component, v X B, is the definition of the electric field as the sum of the gradient of a scalar potential and the time derivative of a vector potential. The latter is completely extraneous to gravitation; hence there is no induction in gravitation.





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