top of page
Search

Einstein's Expression for the Advance of the Perihelion is Correct but for the Wrong Physical Reason

According to Wikipedia: "Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term of A/r^4, which accurately gives the observed excess turning rate of 43" every hundred years."

Although numerically correct, it does not come from a force component that is inversely proportional to the fourth power of distance. Rather it can be derived classically from Newton's theory of revolving orbits, and it involves a term of inverse cubic, A/r^3, and not forth order.

What we do need, however, if Droste's expression for the Schwarzschild velocity

(dr/dt)^2=(1-2m/c^2r)^2{1+A(1-2m/c^2r)},

where m=Gm, the gravitational parameter, and A is an arbitrary constant. In order that it vanish as r->infinity, A=-1. Then eliminating the time derivative using the conservation of angular momentum, h=r^2(dtheta/dt), we obtain the equation for the trajectory, dr/dthete. When the square of this term is added to r^2, we get the square of the rate of change of the arc length, s, with respect to the apsidal angle, theta.

The pedal is

p=r^2(dtheta/ds)=1/{1+(2mr/h^2)(1-2m/c^2r)^2}.

The gradient of the pedal p is the ratio of r to the radius of curvature, R,

dp/dr=r/R=1+(1-2m/c^2r)(m/h^2)(r-4m/c^2)

From these expressions, we calculate the central force as

F_S=(1/r^2){(m+(h^2-6m^2/c^2)(1/r)+8m^3/c^4r^4}.

Although the expression is exact, we can neglect the last term since it is proportional to the square of the Schwarzschild radius. The same order term is what GR would makes us retain in the equation of motion

u"+u=(1/hu)^2F(1/u).

where u=1/r, and the force is

F_E=(h/r)^2{m/h^2+3m/c^2r^4},

where the second term is only linear in the Schwarzschild radius and not quadratic, as in the above expression. According Newton's theory of revolving orbits, If the angle variables of two bodies are related by theta_1=n theta_2 and if the first particle executes an ellipse with a semi-latus rectum l=h^2/m,

1/r_1=(1+e cos theta_1)/l

the second body will execute elliptical motion with the same eccentricity, e, but according to

1/r_2=(1+e cos(n theta_2)/l,

If the elliptical orbit is "quiescent" in Newton's terminology, the body will rotate about the center of force by pi without a change in the radial motion. However, if the orbit is in "motion", the body will rotate about the center of force as it moves from one apsis to the other. The corresponding apsidal angle is pi/n.

Carrying through the same analysis as before, the inverse of the radius of curvature is found to be

1/R=(1/r)(1-n)+m^2/l^2

so that the force is

F_N=n^2m/r^2+(h^2/r^3)(1-n^2),

which is the characteristic Newtonian expression where the inverse cubic, rather than destroying the closed nature of the orbit, "merely" makes it precess.

Comparing the Schwarzschild force, F_S, with that of Newton's, F_N gives

n=6^{1/2}m/hc,

which apart from the numerical coefficient could be found on dimensional grounds. But the numerical coefficients lends overwhelming support that the apsidal angle is

pi/n= pi hc/6^{1/2} m,

and the shift in the perihelion is

pi n^2=6pi(m/hc)^2 per period.

Although this is the exact same expression as found by Einstein, it clearly shows that Newton's revolving orbital theory is at work, and not something that is merely chalked up to General Relativity.

Newton also emphasized that the radial motion for both orbits is exactly the same so that if the new orbit is viewed from axes rotating at a rate (n-1)d theta/dt, the two orbits would appear exactly the same. This rotation rate is constant only for circular orbits. For elliptical orbits, the rate increases as the body passes the perihelion and slows down near the aphelion.

If n is close to one the second orbit will be similar to the first but revolve more slowly about its center of force. The precession is in the same direction for n>1, while it precesses in the opposite direction for n<1. This is clearly the case in the shift in the perhelion of Mercury that is not explained by all the other contributions.