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The Origin of Apsidal Motion in General Relativity

Updated: Jun 18, 2021

One of the three fundamental tests of General Relativity is the calculation of the precession of the planet Mercury that could not be explained by conventional Newtonian theory. It resulted in the addition of 43 arcseconds/century that was missing from the other phenomena that was accounted for by Newtonian theory. General relativity could also account for the rotation of the line of apsides in close binaries such as the one discovered by Taylor and Hulse, PSR1913+16.


However, all this arises at the expense of the violation of the conservation of angular momentum in euclidean space. Consider a spherically symmetric metric


ds^2=-F(r)dt^2+R^2(r)dr^2+r^2d\theta^2,

where all the metric coefficients depend on the radial coordinate, r, and are independent of the time coordinate. The lagrangian is given by

L=(1/2)(-F(dt/ds)^2+R^2(dr/dt)^2+r^2(d\theta/ds)^2.


Since t and \theta are cyclic variables the Euler-Lagrange equations imply that


F^2(dt/ds)= E,

and

r^2(d\theta/ds)=h,

are first integrals of the motion. That is, both are conserved in time.


The first looks like the conservation of the energy, E, while the second, appears to be the conservation of the angular momentum, h. However, these conservation laws should be expressed in coordinate time, t, and not in the proper, or local, time s. Dividing the second by the first gives

(r/F)^2(d\theta/dt)^2=h/E,

which is neither the conservation of angular momentum, nor the energy. In fact, it is the origin of the small, corrections to Keplerian dynamics as viewed in the rotation of the line of apsides in close binaries and the advance of the perihelion of Mercury, both of which cannot be explained by Newton's inverse square law.


To proceed, it is advantageous to introduce an auxiliary variable G(\rho)=r, such that

R dr=R G' d\rho=d\rho,

i.e., G'=1/R, where the prime indicates the derivative with respect to the new radial coordinate, rho. Introducing this into the metric, transforms it into

ds^2=-F^2dt^2+d\rho^2+G^2d\theta^2.


Now, if we use Einstein's condition of "emptiness", the metric coefficients are not all independent. In fact, it is easy to show that from the conditions of the vanishing of the eigenvalues of the Ricci tensor, the following condition holds

G'=F

This condition immediately implies the vanishing of Gaussian curvature,

G"/G=G'F'/GF=F'/G. (I)

But, the other conditions obtained from the vanishing of the eigenvalues of the Ricci tensor only are satisfied by a Newtonian potential, m/G, in which case

G'=(1-2m/G)^(1/2)=F.

It is, so to speak, a quirk of the Newtonian potential. The other conditions that have to be fulfilled are

G"/G+F'G'/FG-(1/G^2-G'^2/G^2)=0

2G"/G+F"/F=0

F"/F+F'G"/FG=0.


Eliminating F'G'/FG from the second and third conditions, results in

2G"/G+F"/F=0,

which is only satisfied by an inverse radial potential. It is not satisfied by a metric having a

constant value of the curvature. On could say that this is due to the "flatness" of the problem.


But, if the situation were truly flat, there would be no gravitational potential at all. We could have equally expressed the vanishing of the Gaussian curvature as

F"/F+F'G'/GF=0 (II)

However neither (1) nor (Ii) are consequences of the metric

ds^2=-F^2dt^2+G^2d\theta^2,

at \rho=constant, since neither F nor G is a function of the laboratory, or coordinate, time variable.


In fact, the appearance of an independent time variable is worrisome. Both F and G are implicit functions of t through the variable \rho, i.e., dG/dt=G'd\rho/dt and dF/dt=F'd\rho/dt.

Since all motions are required to follow geodesics, d^2t/ds^2=d^2\theta/ds^2=0 [A Besse,

Einstein Manifolds, p. 105, Eqn (3.42c)], the conservation of energy would require F'=0. while the conservation of angular momentum G'=0, both of which are blatantly false.


Thus, we are forced to conclude that the conservation of energy, E, has no meaning, and the conservation of angular momentum is violated in Euclidean space which selects coordinate time, and not proper time.


The appearance of G' in the orbital equations of motion has a long history. If R=1/(1-2m/G)^(1/2), as in the outer solution of the Schwarzschild problem, G'=(1-2m/G)^(1/2), The modified orbital equation in terms of the inverse variable, u=1/\rho, is

d^2u/d\theta^2+G'u=m/h^2,

where h has the value G^2d\theta/ds, since that is a conserved quantity. Sommerfeld [see McCrea, Relativity Physics, pp 29-31] obtained the expression

d^2u/d\theta^2+(1-(m/ch)^2)u=m/h^2,

by replacing G with h^2/m, on the strength of Kepler's III law. He used it to explain the hyperfine structure in hydrogen. It is represented "to a good approximation by the corresponding ellipse rotating in such a way as to produce this advance of the 'perihelion'".


Unfortunately, it presents only on sixth the advance of the perihelion of the planet Mercury which Ives obtained by considering a velocity dependent mass [JOSA, 38 (1948) 413]. It was previously used by Sommerfeld, which McCrea considered that "when used with proper caution Sommerfeld's theory still provides some sort of a rough physical picture of the way in which the dependence of the mass of an electron upon its velocity will affect its energy levels." Again, this is a red-herring, and the effect depends on the non-conservation of the angular momentum.


The G'-factor is easily obtained from the following consideration. Consider the spin of a star in the z-direction while the binary rotates solely in the plane perpendicular to this direction. The spherical surface is

dS^2=dx_1^2+dx_2^2+dz^2=dr^2+r^2d\theta^2+dz^2,

which is acted upon by the constraint,

x_1^2+x_2^2+x_3^2=c^2/w^2,

where c is the speed of light, and w the orbital speed. The constraint implies

dz=-rdr,

and since r^2+z^2=c^2/w^2, the metric can be written as

dS^2=dr^2/(1-w^2r^2/c^2)+rd\theta^2.

Introducing G, we find,

G'=(1-w^2G^2/c^2)^(1/2).

If we now introduce Kepler's III law, the rotational speed is replaced by

w^2=nm/R^3,

where R is the distance between the binary stars and R/n, is the center of gravity. According to this the pulsar axis of rotation would precess by an amount (w a/c)^2 per orbital period, where we have set G equal to the semi-major axis of the orbit, a. General relativity increases this value by a factor 3pi/(1-e^2), where e is the eccentricity. For PSR1913+16 this amounts to about 2 degrees/year.


Finally, if we set F=(1-nmG^2/R^3)^(1/2), the condition for the vanishing of the Gaussian curvature,

m/R^2-w^2(R/n)=0,

expresses the force balance law between tidal and centrifugal forces with R/n representing the center of gravity. Neither proper time nor Riemann curvature have a role in this two-dimensional analysis of "internal" tidal forces proportional to G^2, as opposed to "external" tidal forces that are proportional to 1/G^3. In the two-dimensional case, there are no tangential curvatures in the Ricci tensor {Petersen, Riemann Geometry, p. 69.]



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