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# Gravitational Repulsion: An Oxymoron

The idea that plane waves propagating away from electromagnetic sources are gravitational waves because the coefficients of the indefinite metric are called "gravitational" potentials is nonsense. The idea, undoubtedly originated with Tolman, in his Relativity, thermodynamics and cosmology, pp. 272ff, and was picked up and capitalized on by numerous authors. In particular, Bonnor in "The gravitational field of light" concludes that "there is no evidence from my solutions that those waves carry energy away from the light pulses and beams which are their sources: the mass parameters of the later remained unchanged." So, what kind of waves are these? and why aren't spherical waves permitted where the sources can be pin-pointed? To call something gravitational just because you are using an indefinite metric with a propagation velocity equal to that of light does not make it gravitational.

Moreover, Bonnor also concludes that "parallel beams (or pulses) of light shining in the same sense [direction] do not interact". This is in contradiction with the second-order equation (i.e., a Weber-like equation) that was used to derive it. Like pairs of positive and negative charges, their fluxes in parallel and perpendicular directions contribute to the over-all force. It doesn't matter if Weber's exposition used the untenable Fechner hypothesis, the interpretation of longitudinal (proportional to the square of the velocity) and parallel (proportional to the acceleration) currents are bedrock to the interpretation of Weber's equation.

There appears a significant difference between treating the 4-momentum, p_a,whose pseudo-norm is

p_a p^a= g_(ab) p^a p^b = -mc^2, (1)

and the geodesic equations

m dp_a/ds = (1/2)dg_(bc)/dx^a p^b p^c, (2)

where g_(bc) is the metric, and x^a are the Schwarzschild coordinates (ct,r, theta, phi).

Choosing (1) over (2), we introduce the so-called conservation of energy

p_0= g_(00) p^0=- (1-2m/c^2 r)(dt/ds)=constant,

where "s" is proper time, and

p_(phi) =g_(phi,phi) p^(phi) r^2 d(phi)/ds=constant,

into (1) to obtain the geodesic orbit equation

(dr/d(phi))^2= r^4/h^2 { 2(E+m/r)- (h/r)^2(1-2m/c^2r)}.

Transforming to u=1/r, we come out with the Clairut equation

u"+u = (m/h^2) + 3m/c^2 u^2, (3)

which is the Einstein modified equation.

Rather, using (1), the orbital equation is

u" + [1-(m/hc)^2]u = m/h^2 +2m/c^2 u^2 + 3(m/c^2)u'^2/(1-2m/c^2 u), (4)

which even neglecting the last term on the left-hand side and the last term on the right-hand side does not give (3). For purely radial motion, the gravitational accerelation is

d^2r/dt^2= g [ 3(dr/dt/c)^2/(1-2m/c^2 r) - (1-2m/c^2 r), (5)

where g= m/r^2. Now, in a weak gravitational field, for any relative velocity

dr/dt > c/3^(1/2), (6)

"a particle at every point in the Schwarzschild field is repelled." McCruder, in "Gravitational repulsion in the Schwarzschild field" tells us that this result was first derived by Hilbert, and later confirmed by Treder.

From our last post, we know that the Gerber equation

F= g[1-3(dr/dt/c)^2+6 (r d^2r/dt^2)/c^2] (7)

gives the same numerical, and analytical, value of the advance of the perihelion of Mercury. Applying Weber's condition

(dr/dt)^2= r(d^2r/dt^2) (8)

to Gerber's equation (7), we get

F=g[1+3(dr/dt/c)^2] (9)

which destroys any possibility of gravity being repulsive.

What would be necessary--but not sufficient--is to reverse the signs of the longitudinal and parallel contributions in the force law. An example is given by Newton's law of centrifugal force [cf Seeing Gravity, p. 26

F=(h^2/r^3)[ 1+2(dr/dt)^2(v_s)^2-r(d^2/dt^2)/(v_s)^2], (10)

where v_s is the angular velocity. Applying Weber's condition, (8), to Newton's law (10), results in

F=(h^2/r^3)[1+(dr/dt)^2/(v_s)^2]. (11)

So there is no way that the inequality

v_s>dr/dt

would result in an attractive centrifugal force. Since curvature is built into Newton's equation, there is no way that it could ever "threaten" the stability of the motion. Would that have been possible would lead to a drastic reformulation of what physical laws are. McGruder's conclusion that gravitational repulsion "can only be detected by an observer whose meter sticks and clocks are not affected by gravity" is untenable, if only were it not for the fact that our measuring instruments are always affected by gravity, no matter how weak it may be.