"Einstein, and the Most Beautiful of All Theories" - BY Avik Dubey

 

The British theoretical physicist Paul Dirac (1902–1984) one of the founders of quantum mechanics, once wrote:

“There was difficulty reconciling the Newtonian theory of gravitation with its instantaneous propagation of forces with the requirements of special relativity, and Einstein working on this difficulty was led to a generalization of his relativity — which was probably the greatest scientific discovery that was ever made.”

General Relativity is generally recognized as a theory of exceptional beauty. Several tests, along the years, confirmed the consistency of the theory. I will describe one of the tests, which correctly explained the “anomalous” precession of the perihelion of Mercury (see link), which Newton’s theory of gravitation failed to predict.

 

                                                  

 Figure 1: The figure shows the perihelion precession of Mercury.

The Problem with Newton’s Theory

The precession (or rotation) of the perihelion (the point in the orbit of a planet that is nearest to the Sun) has a variety of causes. Two of them are:

·        The presence of the other planets that causes a perturbation of one another’s orbit, which is the leading cause

·        The oblateness of the Sun (see figure) which is significantly less relevant

 

                                                    

Figure 2: The figure shows a sphere of radius compressed to an oblate ellipsoid (source).

The perihelion rate of the precession of Mercury does not agree with the prediction of Newton’s theory of gravity. This anomaly was noticed by the French astronomer and mathematician Urbain Le Verrier. The final measurement, by Simon Newcomb in 1882, estimated that the actual precession rate disagreed with Newton’s prediction by 43 degrees. Many ad hoc solutions were proposed, but none of them worked.

Using General Relativity to Calculate the Perihelion Precession of Mercury

The Schwarzschild solution is the solution of the Einstein field equations that describe the geometry of the vacuum spacetime around the Sun. In other words, the Schwarzschild metric is the metric of the Solar system caused by the spacetime curvature generated by the Sun. It is valid when you can:

·        Treat the Sun as a non-rotating object

·        Neglect the gravitational field originating from the other planets of the Solar system.

The Schwarzschild solution has the following line element:

                                     

Equation 1: The line element of the Schwarzschild  solution which describes the geometry of the vacuum spacetime around the Sun.

 

The parameter R = 2M is known as the Schwarzschild radius. The coordinates r, θ, and φ are spherical coordinates, as illustrated in Fig. 3.

                          

                           Figure 3: Spherical coordinates (source).

Note that from the isotropy of the metric, we always have θ = π/2 (the orbits are restricted to the equatorial place). In fact, according to the two-body problem (in our case, the bodies are the Sun and the planet), the motion of a body subject to a central force potential will always lie in a plane. Figs. 4 and Fig. 5 show two types of orbiting two-body systems. The motion restricted to the plane is valid in both Newtonian and Einstein gravity theory. Hence, in our analysis, it will be sufficient to consider only geodesics that lie in that plane.

                                                  

Figure 4: Two bodies with the same mass orbiting a common barycenter external to both bodies (which occur for example binary stars)(source).

                                         

Figure 5: Two bodies with different masses orbiting a common barycenter (source).

A third condition for the validity of this analysis is that the radial coordinate r must be much larger than the radius of the Sun. This is not a problem since the Schwarzschild radius of the Sun is much smaller than the Sun’s radius. More specifically, the Schwarzschild radius of the Sun approximately 2.95×10³ m, while the radius of the Sun is close to 6.96×10⁸ m.

 

Figure 6: The German physicist and astronomer Karl Schwarzschild (source)

         

Symmetries in a given spacetime are associated with conserved quantities for particles and photons moving in it. Since the metric g of the Schwarzschild solution is both time-independent (or time translation invariant) and spherically symmetric, the energy of massive particles and the energy of the photon are both conserved. We can see that mathematically as follows.

In a spacetime with metric g, a free-falling material particle or photon obeys the geodesic equation (the generalization of a “straight line” to curved spacetime) associated with that spacetime, which is given by (see Schutz):

                                                                    

Equation 2: The geodesic equation, obeyed by freely falling material particles or photons.

                                                            

Note that since photons will also be considered, the parameter λ cannot be the proper time τ. The geodesic equation can also be written as:

                                                      

                Equation 3: The geodesic equation, written in an alternative form.

Now note that:

                                               

         Equation 4: Constant components of the metric in time and in the coordinate ϕ.

      

Eqs. 3 and 4 imply that:

                                     

            Equation 5: Constants of motion on the geodesic.

We then make the following definitions:

                                       

Equation 6: The energy per unit mass of a massive particle and the energy of the photon.

The ~ on top of the massive particle’s energy is used (see Schutz) to indicate that this energy is per unit of mass. In the same vein, as a consequence of the independence of g of φ, angular momentum is conserved. We define:

                                                                

Equation 7: The angular momentum per unit mass for a massive particle and the angular momentum for the photon.

where the left term is the angular momentum per unit mass of a massive particle and the right term is the angular momentum of the photon. We now need the equations for the orbit. The three components of the momentum of the massive particle are:

                                                          

      Equation 8: The three momentum components of the massive particle.

The momenta for the photon are:

                              

Equation 9: The three components of the momentum of the photon.

We now use the momentum components we just derived, substitute them in the equations |p|=-m² for the particle and for the photon and solve for dr/dλ. The equations for dr/dλ then read:

                                                    

                                 Equation 10: The equations for dr/dλ squared.

          

Now intuition tells us to rewrite these equations using effective potentials, namely:

                                                        

 Equation 11: The definitions of the effective potentials for the massive particle and the photon.

The potentials are plotted in Fig. 7. Note that, since the left-hand side of both equations is positive, the effective potential must be smaller than the energy. Fig. 7 shows the effective potential (note that E and V in Fig. 7 indicate the same quantities with the tilde ~ above it) for massive and massless particles. The figure also indicates the turning points where dr/dλ=0, forbidden regions (where E < V) and circular orbits (both stable and unstable), where dV²/dr=0.

 

                                                 

Figure 7: Effective potentials for massive and massless particles (photons). The figure indicates the turning points, forbidden regions, and circular orbits, both stable (minima) and unstable (maxima).

 

Precession of the Perihelion of Mercury

From now on, let us consider only the motion of massive objects since our goal is to calculate the precession of the perihelion of Mercury.

A stable circular orbit occurs in the minimum of the effective potential. Let M be the mass of the Sun. Differentiating the effective potential, setting the result to zero and solving for r we obtain the radius of the stable circular orbit:

                                                       

Equation 12: Radius of the stable circular orbit of a massive particle oscillating around the Sun.

In Newtonian mechanics, a complete orbit of a planet in a circular orbit returns to its initial φ. Now using the fact that circular orbits have E²=V², and using the expressions derived until now, we obtain the time it takes for a planet to have Δφ=2π, which is the period P:

 

                                                       

 Equation 13: Newtonian result for the period of a planet rotating around the Sun. 

              

Now, in general relativity, a rotating planet does not return to its initial point. If the relativistic effects are small we should have an ellipse that slowly rotates around its center. What we can do (see Schutz) is to examine the motion of the perihelion of the orbit. To do that we perform three quick calculations (see Schutz):

·        Derive an expression for dφ/dλ in terms of angular momentum

·        Derive expressions for dt/dλ in terms of energy per unit mass

·        Define the new variable u ≡ 1/r

Substituting them in Eq. 10 we get:

                                              

                           Equation 14: Relativistic expression for du/dφ.

We now define y, the deviation for circularity, as follows:

                                                 

            Equation 15: Definition of the variable y, the deviation for circularity. 

 

For a Newtonian orbit, y=0. To obtain the relativistic expression we substitute Eq. 15 into Eq. 14 and drop terms of the order of y³. We get the following equation for an almost circular orbit:

                                                 

Equation 16: Relativistic expression for dy/dφ.

The solution reads:                             

                                              Equation 17: Solution y(φ) of Eq. 16.

 whee B depends on initial conditions. From the argument of the cosine, we conclude that the orbit returns to the same radius when Δ(kφ) = 2π. The presence of k different from 1 is what differentiates it from the Newtonian result! If the relativistic effects are small we can make a few other simple approximations to obtain:

Equation 18: The perihelion advance from consecutive orbits.

For Mercury specifically one obtains a shift of 0.43" per year which as mentioned in the beginning of the article was the value confirmed experimentally.

It seems that even Einstein was stunned by the result. After finding the result of the calculation, he could not work for days. In his own words, he became “beside himself with joy.”

References:-

1. Theories of Relativity- By Albert Einstein.

2. Gravitation & Cosmology- By Steven Weinberg.