A thrown stone arcs through the air and falls. A planet orbiting a star is the same stone, but it falls forever, looping around the star in a closed curve. In Newton's world, that curve is a perfect ellipse, and the planet returns to the same point each orbit. In reality, the orbit slowly rotates: its closest point to the star, the perihelion, shifts.
In the new work, the authors didn't just calculate the shift for nearly circular orbits: they found a general solution for deviations in both theories. In the Newtonian case, spacetime curvature plays no role, and the orbit remains closed only for certain forms of the potential (interaction energy). In general relativity, things are more complicated: the geometry of spacetime itself forces the orbit to precess — to slowly rotate. The authors derived an explicit formula for this shift in a static, spherically symmetric field — for example, around a non-rotating star or a black hole.
Of particular interest is the influence of the cosmological constant (the very quantity associated with dark energy). Even a tiny addition to the metric (the description of spacetime geometry) can change the perihelion shift. The researchers estimated this effect: it is negligible within the Solar System but could become noticeable on galactic scales. And at the end of the article, there's a surprising result: they found a metric 'recipe' for which the shift vanishes entirely. Such a geometry is a theoretical rarity, but it reminds us that nature doesn't have to follow our expectations.
The perihelion shift of Mercury was noticed as early as the 19th century, but only Einstein could explain it, and that became the first triumph of general relativity.
In Orson Scott Card's novel 'Ender's Game,' characters manipulate gravity to alter asteroid orbits, such precision would require knowledge of all shifts, including the effects described in this article.