Mercury's Precession: Geodesic Equations in Action

Hello! Welcome back to our exploration of General Relativity.

In our last lesson, we analyzed the motion of particles and light in the Schwarzschild spacetime, deriving the fundamental radial equation of motion and discovering key features like the Innermost Stable Circular Orbit (ISCO) and the photon sphere. We saw that the dynamics are governed by an effective potential which includes a purely relativistic term, proportional to $1/ r^{3}$ .

Today, we will see the direct, observable consequence of this term. Our learning outcome is to calculate the perihelion precession of Mercury using the derived geodesic equations. This calculation was one of the first great triumphs of General Relativity, explaining a long-standing puzzle in celestial mechanics and providing powerful evidence for Einstein's new theory of gravity.

1. From Newtonian Ellipses to Precessing Orbits

In Newtonian gravity, a planet orbiting the Sun follows a perfect, closed ellipse. The point of closest approach, the perihelion, remains fixed in space, orbit after orbit. However, for decades before Einstein, astronomers knew this wasn't quite right for Mercury. Its perihelion was observed to be advancing, or precessing, by a tiny amount.

This precession is what General Relativity predicts. The $1/ r^{3}$ term in the effective potential, though small, is enough to perturb the orbit so that it no longer closes perfectly. After one complete radial oscillation (from perihelion to aphelion and back), the planet has traveled slightly more than $36 0^{\circ}$ , causing the perihelion to shift forward.

Exaggerated Perihelion Precession of Mercury — This diagram illustrates the concept of perihelion precession. The elliptical orbit of a planet like Mercury does not perfectly close. Instead, the point of closest approach to the Sun (the perihelion, labeled p1, p2, etc.) advances slightly with each orbit. The effect shown here is enormously exaggerated for clarity.

Our goal is to calculate the magnitude of this advance.

2. The Equation of the Orbit

Let's begin with the tools we developed in the last lesson. For a massive particle ( $ϵ = 1$ ), we have the radial equation and the angular momentum equation:

r ˙^{2} = E ~^{2} - (1 - \frac{2 M}{r}) (1 + \frac{L ~ ^{2}}{r ^{2}}) and ϕ ˙ = \frac{L ~}{r ^{2}}

Here, $E ~$ is the conserved energy and $L ~$ is the conserved angular momentum, both per unit mass. The dot denotes differentiation with respect to proper time $τ$ .

To find the shape of the orbit, $r (ϕ)$ , we need to eliminate the proper time. We can use the chain rule:

r ˙ = \frac{d r}{d ϕ} ϕ ˙ = \frac{d r}{d ϕ} \frac{L ~}{r ^{2}}

Substituting this into the radial equation gives a differential equation for $r (ϕ)$ . However, this equation becomes much simpler with a change of variables to $u (ϕ) = 1/ r$ . With this change, we have:

\frac{d r}{d ϕ} = \frac{d}{d ϕ} (\frac{1}{u}) = - \frac{1}{u ^{2}} \frac{d u}{d ϕ}

So, $r ˙$ becomes:

r ˙ = (- \frac{1}{u ^{2}} \frac{d u}{d ϕ}) (L ~ u^{2}) = - L ~ \frac{d u}{d ϕ}

Now substitute this and $r = 1/ u$ into the radial equation:

(- L ~ \frac{d u}{d ϕ})^{2} = E ~^{2} - (1 - 2 M u) (1 + L ~^{2} u^{2})

L ~^{2} (\frac{d u}{d ϕ})^{2} = E ~^{2} - 1 - L ~^{2} u^{2} + 2 M u + 2 M L ~^{2} u^{3}

This is a first-order differential equation for $u (ϕ)$ . To get to a more familiar form, we can differentiate the entire equation with respect to $ϕ$ .

L ~^{2} (2 \frac{d u}{d ϕ} \frac{d ^{2} u}{d ϕ ^{2}}) = - 2 L ~^{2} u \frac{d u}{d ϕ} + 2 M \frac{d u}{d ϕ} + 6 M L ~^{2} u^{2} \frac{d u}{d ϕ}

Assuming the orbit is not circular ( $\frac{d u}{d ϕ} \neq = 0$ ), we can divide through by $2 \frac{d u}{d ϕ}$ :

L ~^{2} \frac{d ^{2} u}{d ϕ ^{2}} = - L ~^{2} u + M + 3 M L ~^{2} u^{2}

Dividing by $L ~^{2}$ , we arrive at the relativistic orbit equation:

\frac{d ^{2} u}{d ϕ ^{2}} + u = \frac{M}{L ~ ^{2}} + 3 M u^{2}

This is a beautiful result. Let's examine it:

The left-hand side, $\frac{d ^{2} u}{d ϕ ^{2}} + u$ , is the equation for a simple harmonic oscillator.
The first term on the right, $M / L ~^{2}$ , is a constant driving force. In Newtonian gravity, this is the only term on the right, and the solution is $u (ϕ) = \frac{M}{L ~ ^{2}} (1 + e cos ϕ)$ , the equation for a conic section (an ellipse for eccentricity $e < 1$ ).
The second term, $3 M u^{2}$ , is the General Relativistic correction. It is this non-linear term, arising from the $1/ r^{3}$ part of the effective potential, that causes the precession.

3. Solving the Orbit Equation Perturbatively

For planetary orbits in our solar system, the relativistic correction is very small. This makes it ideal for a perturbative solution. We will treat the $3 M u^{2}$ term as a small perturbation to the Newtonian orbit. The following reading provides a clear, step-by-step derivation.

The Schwarzschild Solution and Precession of Perihelion

The following notes from a GR course provide a clear derivation of the orbit equation and its solution using perturbation theory. This is a standard and elegant method for this problem.

Please read section 4, 'Precession of Perihelion of planets'. Follow the steps starting from the geodesic equations (which we've just reviewed). Pay close attention to how the change of variables to u = 1/r is used to derive the orbital differential equation (9). Then, see how this equation is solved perturbatively by treating the final term as a small correction to the Newtonian case. The final result for the precession angle per revolution is the key takeaway.

As derived in the text, the precession angle $Δ ϕ_{P}$ per revolution is:

Δ ϕ_{P} = 6 π (\frac{GM}{L ~})^{2}

(Note that the resource uses $h$ for angular momentum per unit mass, which is our $L ~$ ). This is the excess angle the planet sweeps out by the time it returns to its perihelion.

To connect this theoretical formula to observable astronomical quantities, we need to express the angular momentum $L ~$ in terms of the properties of the orbit. For a Newtonian ellipse with semi-major axis $a$ and eccentricity $e$ , the angular momentum is given by:

L ~^{2} = GM a (1 - e^{2})

Substituting this into our precession formula gives the final result, where we have restored the speed of light $c$ for the calculation:

Δ ϕ_{P} = \frac{6 π GM}{a ( 1 - e ^{2} ) c ^{2}}

Test your understanding!

Looking at the final formula for precession, which orbital parameters make the effect larger? Does this match your intuition for where relativistic effects should be strongest?

Show answer

The precession $Δ ϕ_{P}$ is proportional to $1/ a$ and $1/ (1 - e^{2})$ . This means the effect is largest for planets with:

Small semi-major axis ( $a$ ): The planet gets closer to the central mass, where the gravitational field is stronger.
Large eccentricity ( $e$ ): A more eccentric orbit means the planet's closest approach (perihelion) is even closer for a given semi-major axis ( $r_{p er i} = a (1 - e)$ ), where it experiences the strongest gravitational effects.

This matches our intuition perfectly: the most relativistic effects occur for planets that are closest to the star and have the most elliptical orbits. Mercury fits this description best in our solar system.

4. Comparison with Observation

The final step is to see if this prediction matches reality. The video below provides a slightly different but conceptually similar derivation based on direct integration and then discusses the historical context and the stunning success of the result.

14. Orbital Motion in the Schwarzschild Spacetime (General Relativity)

The calculation we just performed is quite abstract. To see it carried out in a slightly different way and to appreciate its historical significance, please watch the following video.

Watch from 33:40 to the end. The first part (33:40 - 57:40) shows an alternative derivation by directly integrating the orbital equation. You don't need to follow every algebraic step, but appreciate the strategy of using series expansions. The final part (57:40 - end) puts the formula into context, plugging in the numbers for Mercury and explaining why this was a monumental success for General Relativity.

As the video explains, when we plug in the values for Mercury:

$G = 6.674 \times 1 0^{- 11} N m^{2} / kg^{2}$
$M_{S u n} = 1.989 \times 1 0^{30} kg$
$a = 5.79 \times 1 0^{10} m$
$e = 0.2056$
$c = 3 \times 1 0^{8} m/s$

The formula yields a precession of about 43 arcseconds per century. This value precisely matched the anomalous precession that had been observed by astronomers since the 1850s, after all known Newtonian effects (like the gravitational pull of other planets) had been accounted for. It was a spectacular confirmation of General Relativity.

Conclusion

In this lesson, we have taken the formal machinery of geodesics and applied it to a crucial real-world test. We saw how the subtle relativistic correction to the Newtonian laws of gravity leads to a measurable effect: the precession of planetary orbits.

Key Takeaways:

The equation for a particle's orbit in Schwarzschild spacetime can be written as $\frac{d ^{2} u}{d ϕ ^{2}} + u = \frac{M}{L ~ ^{2}} + 3 M u^{2}$ , where $u = 1/ r$ .
The $3 M u^{2}$ term is a small relativistic correction that causes orbits to precess.
By solving this equation perturbatively, we derived the formula for the precession per orbit: $Δ ϕ_{P} = \frac{6 π GM}{a ( 1 - e ^{2} ) c ^{2}}$ .
This formula successfully explained the observed 43 arcsecond per century anomalous precession of Mercury's perihelion, providing one of the first and most important pieces of experimental evidence for General Relativity.

Preview of Next Lesson:

Having successfully tested our theory against the motion of massive particles, we will next turn our attention to massless ones. In the upcoming lesson, we will calculate the deflection of light by a massive object, another of the classic tests of GR. This will involve analyzing the null geodesics we first encountered in the previous lesson.

Recommended resources

1. From Newtonian Ellipses to Precessing Orbits

2. The Equation of the Orbit

3. Solving the Orbit Equation Perturbatively

4. Comparison with Observation

Conclusion