4.4.1 Perturbative Treatment of the Periastron Advance

Since (4.4.17) is linear in the unknown function Δu, its solution is the linear combination

Let us study the form of each Δun. The first contribution Δu0 is uninteresting and irrelevant since, for an arbitrary constant k, we have:

so that replacing u0 → u0 + Δu0 amounts to an unobservable renormalization of the Keplerian parameters a and e. Next we consider Δu2. Here a solution of the inhomogeneous equation is:

Also the perturbation u0 → u0 + Δu2 is essentially unobservable, since Δu2(φ) is periodic of period 2π . Indeed by replacing u0 → u0 + Δu2 we obtain a new orbit which is a slightly deformed ellipsis where the aphastro and periastro continue to occur at φ = 0 and φ = π as in the Keplerian case. This can be appreciated by looking at Figs. 4.13, 4.14, 4.15. In order to emphasize the extremely small effect of the perturbation we have chosen an ultra narrow and eccentric Keplerian orbit with parameters:

and in Fig. 4.13 we have compared the plot of the unperturbed function u0(φ) with the perturbed one u0(φ) + Δu2(φ). The resulting new orbit is plotted in Fig. 4.14 and compared with the unperturbed one. The periodic character of the perturbation implies that the new orbit looks like a deformed ellipsis of slightly changed shape and size. We can compare the actual shape of the perturbed orbit with a purely Keplerian ellipsis by defining a renormalized semilatus rectum and eccentricity via the formulae:

aR = r(π ) + r(0)

2

(4.4.24)

r(π ) − r(0) eR = +

r(π ) r(0)

where

1

r(φ) = (4.4.25) u0(φ) + Δu2(φ)

Fig. 4.13 Comparison of the unperturbed solution u0(φ) (thin line) and of the perturbed one

u0(φ) + Δu2(φ) (thick line) in the case of Keplerian parameters a = 8m, e = 0.7. As one sees, even in this extreme case of an ultra-narrow and ultra-relativistic orbit the perturbation modifies very slightly the shape of the function without changing its behavior and its periodicity

Fig. 4.14 Comparison of the Keplerian orbit of parameters a = 8m, e = 0.7 (thin line) with the perturbed orbit obtained by replacing

u0 → u0 + Δu2 (thick line)

Fig. 4.15 Comparison between a Keplerian orbit of parameters a = 8m, e = 0.7 perturbed by the periodic perturbation Δu2 (thick line) and a Keplerian orbit of parameters aR = 5.97683m, eR = 0.624945 (thin line) that, by definition, has the periastron and aphastron located at the same distances from the center of gravity as the perturbed orbit. This comparison shows the deviation from a purely elliptic shape

In Fig. 4.15 we have plotted the Keplerian orbit with parameters a = aR = 5.97683m, e = eR = 0.624945, together with the perturbed orbit of Keplerian parameters a = 8m, e = 0.7. By definition in these two orbits the aphastron and the

Fig. 4.16 Comparison of the unperturbed solution u0(φ) (thin line) and of the perturbed one

u0(φ) + Δu1(φ) (thick line) in the case of Keplerian parameters a = 70m, e = 0.6. The shift in period of the perturbed solution is evident from the plot

Fig. 4.17 Comparison of the unperturbed Keplerian orbit of parameters a = 70m,

e = 0.6 (thin line) with the orbit perturbed by

u0(φ) + Δu1(φ) (thick line). The phenomenon of the periastron advance is evident

by the exact solutions is essentially due to the non-periodic perturbation Δu1 of (4.4.26) as we have claimed.

Equation (4.4.31) is the final perturbative formula predicted by General Relativity for the periastron anomaly and constitutes one of the classical tests of the theory proposed by Einstein himself.

In the case of the solar system we have m = m6 = GM6/c2 where M6 is the solar mass. The same formula can also be applied to binary stellar systems. In this case we have:

where m1,2 denote the masses of the two companion stars.

It is interesting to insert the numerical values of the Keplerian parameters in the

Since the period of Mercury is TMercury = 0.24 years it follows that in one terrestrial century we have 100/0.24 = 416.6 orbits of Mercury. On the other hand, if we

Fig. 4.18 The effective radial potential for the motion of massless particles in Schwarzschild geometry. It displays a maximum at

r = 3m but no minima. In this

figure the plot is given for

L = 5

The difference with the massive case emerges when we enforce the new constraint (4.5.2) which yields the relation:

is the light-like counterpart of the massive effective potential (4.3.9). We can immediately note the differences. While the centrifugal barrier r12 is common to both

cases, the Newtonian attraction − 1r is absent in the light-like case. This corresponds to the obvious fact that in Newton’s theory photons do not feel gravity since they are massless. Yet the short range attractive correction due to General Relativity− r13 applies both to photons and massive particles. It is this correction which is responsible for the bending of light-rays in strong gravitational fields like that generated by the sun. This phenomenon is the light-like counterpart of the periastron

advance and constitutes another classical test of General Relativity. The plot of the potential Veffphoton(r) is displayed in Fig. 4.18. As it is evident from its shape there

is a maximum at r = 3m but no minima. Hence, in the case of photons there are no stable circular orbits: only open orbits are available that correspond to the aforementioned bending of light-rays. We study such a phenomenon by writing the exact differential equation of the orbit. Proceeding in full analogy with our previous treatment of the time-like geodesics we trade the derivatives with respect to the affine

Fig. 4.19 The Newtonian trajectory of a photon is a straight line and the impact parameter is the minimal distance from the center of mass reached along such a trajectory

and taking a further dφd derivative of (4.5.16) we find:

which is solved either by u = 0 or by:

The first possibility corresponding to circular orbits is excluded since, as we have already observed, the effective potential has no minima. Hence we are left with (4.5.18) which is the light-like counterpart of (4.4.10). Just as in the massive case the differential equation (4.5.18) can be either treated numerically or perturbatively, when the Newtonian approximation (4.5.12) applies. In order to emphasize the new relativistic phenomenon implied by Schwarzschild geometry, namely the gravitational bending of light rays, we have considered the extreme situation provided by a photon impinging on a center of mass with a tiny impact parameter, namely

and choosing some non-vanishing incidence angle (φ0 = π/5 in our example) we have numerically solved (4.5.18) with initial conditions:

The resulting photon trajectory is displayed in Fig. 4.20. As one sees, rather than proceeding undisturbed on a straight line the photon makes a turn around the center of mass and emerges at an angle different from the one of incidence. The exact relativistic orbit is compared with the corresponding Newtonian one in Fig. 4.21.

Fig. 4.20 The trajectory of a photon in Schwarzschild geometry with impact parameter b = 8.5m and incidence angle φ0 = π5

Fig. 4.21 Comparison between the Newtonian trajectory (thick line) and the exact relativistic trajectory (thin line) of a photon that impinges a center of mass with impact parameter

b = 8.5m and incidence angle

φ0 = π5

The perturbative treatment of (4.5.18) follows the same lines as the perturbative treatment of orbit equation for the massive particle (4.4.10). We set

In the post-Newtonian approximation (4.5.12) we obtain the following differential equation for the perturbation:

which admits the following particular solution:

Hence, choosing our reference frame orientation in such a way that φ0 = 0, we can write the perturbed solution:

Fig. 4.22 Comparison between the Newtonian and post-Newtonian perturbed trajectory of a photon impinging on a center of mass with impact parameter

b = 30m and incidence angle

φ0 = 0

The corresponding trajectory is plotted in Fig. 4.22 for an impact parameter b = 30m. In the same figure we have also plotted the corresponding Newtonian trajectory, namely a straight-line with the same minimal distance from the center of mass. The angle δ shown in Fig. 4.22 is one half of the total deflection angle of light rays predicted by General Relativity:

The angle δ can be evaluated from the post-Newtonian perturbed solution (4.5.25) with the following reasoning. The incoming and outgoing photons are at infinite distance from the center of mass, namely at r = ∞ u = 0. If we fix φ0 = 0, the zeros of the unperturbed function u0(φ) (see (4.5.22)), are at:

The zeros of the perturbed function will instead occur at:

where δ 1 is small. Hence we can use the power series expansion of trigonometric functions and write:

Inserting (4.5.30) into (4.5.26) we conclude that General Relativity predicts the following total deflection angle