Local coordinates and motion of a test particle in the McVittie spacetime

This paper analyzes the orbital motion of a test particle in the McVittie spacetime using local coordinates and osculating elements to demonstrate that while cosmological expansion does not affect the orbit's size up to second order, it induces a time-dependent precession and alters the mean orbital frequency, with the precession direction determined by the Hubble and deceleration parameters.

The Gist

Imagine the universe as a giant, expanding balloon. As you blow more air into it, the surface stretches, and everything painted on it moves further apart. This is the Hubble expansion.

Now, imagine you draw a tiny, intricate spiral galaxy or a solar system right on that balloon. The big question physicists have been asking for decades is: Does the stretching of the balloon pull apart the tiny spiral? Does the universe's expansion rip planets away from their stars, or do they stay glued together?

This paper by Vishal Jayswal and Sergei Kopeikin tackles that question using a specific mathematical map called the McVittie metric. Here is the breakdown of their findings in simple terms.

1. The Problem: Two Different Maps

To study this, the authors had to switch between two different "maps" of reality:

• The Global Map (The Balloon): This is the view from far away, looking at the whole universe expanding. It's great for cosmology but terrible for looking at a single solar system because it makes everything look like it's drifting apart, even if it's not.
• The Local Map (The Observer): This is the view from someone standing right next to the black hole or star. They want to know: "Is my planet actually moving away from me?"

The authors realized that the "Global Map" creates optical illusions. It looks like the expansion is pulling things apart, but that's just a coordinate trick. To get the real physics, they had to mathematically "zoom in" and create a Local Coordinate System attached to the massive object (the black hole). This is like switching from a satellite view of the Earth to a street-level view of your neighborhood.

2. The Main Discovery: The "Rubber Band" Effect

Once they switched to the Local Map, they ran the numbers to see how the expansion of the universe affects a planet orbiting a black hole.

The Big Surprise:
They found that the size of the orbit does not change.

• Analogy: Imagine a rubber band holding a marble to a heavy stone. Even if the room you are in is slowly expanding, the rubber band doesn't stretch, and the marble doesn't drift away. The local gravity of the black hole is so strong that it "locks" the planet in place, completely ignoring the gentle tug of the expanding universe.
• The Result: To a very high level of precision (up to the second order of the expansion rate), the semi-major axis (the size of the orbit) and the eccentricity (how oval the orbit is) remain constant. The universe's expansion does not tear the solar system apart.

3. The Subtle Twist: The "Slow Dance"

While the size of the orbit doesn't change, the orientation of the orbit does.

The Precession:
Imagine the orbit isn't a perfect circle, but an oval (an ellipse). The point where the planet gets closest to the black hole is called the pericenter.

• The authors found that the expansion of the universe causes this "closest point" to slowly rotate or precess over time.
• The Metaphor: Think of a spinning top. As it spins, the axis wobbles. The universe's expansion acts like a very faint, invisible wind that makes the entire oval orbit slowly rotate around the black hole.

What determines the direction of this wobble?
It depends on how the universe is expanding:

• Accelerating Expansion (Dark Energy): If the universe is speeding up its expansion (which it is, right now), the orbit rotates in one direction.
• Decelerating Expansion: If the universe were slowing down, the orbit would rotate in the opposite direction.
• Constant Expansion: If the expansion rate were perfectly steady, this specific type of rotation would vanish.

4. Real-World Numbers: How Big is the Effect?

The authors did the math for real systems, like stars orbiting the supermassive black hole at the center of our Milky Way (Sagittarius A*) and the Sirius binary star system.

The Verdict:
The effect is incredibly tiny.

• The "wobble" caused by the universe's expansion is billions of times smaller than the wobble caused by the black hole's own gravity (a known effect called General Relativity precession).
• Analogy: If the black hole's gravity caused the orbit to rotate like a clock hand moving one full circle in a year, the universe's expansion would move that hand by less than the width of a single atom in that same year.

5. Why This Matters

This paper is important because it settles a debate. Some theories suggested that the expansion of the universe might slowly pull planets away from stars over billions of years.

The authors say: "No, not really."

• Local systems (solar systems, galaxies, black hole binaries) are decoupled from the cosmic expansion. They are like islands in a rising tide; the water rises around them, but the islands don't float away.
• The only thing the expansion does is cause a microscopic, almost unmeasurable "drift" in the orientation of the orbit, and even that is dwarfed by other gravitational effects.

Summary

• The Setup: Can the expanding universe pull a planet away from its star?
• The Method: The authors created a "local map" to remove the optical illusions of the expanding universe.
• The Result: The orbit's size stays the same. The universe cannot rip a solar system apart.
• The Catch: The orbit's orientation slowly rotates (precesses) due to the expansion, but this effect is so small it's practically invisible compared to the black hole's own gravity.

In short: The universe is expanding, but your solar system is safe. The cosmic stretching is too weak to break the strong gravitational bonds that hold our local neighborhood together.

Technical Summary

Here is a detailed technical summary of the paper "Local coordinates and motion of a test particle in the McVittie spacetime" by Vishal Jayswal and Sergei M. Kopeikin.

1. Problem Statement

The paper addresses the long-standing problem of determining the orbital motion of a test particle (e.g., a star or planet) gravitationally bound to a massive central body (such as a black hole) embedded within an expanding cosmological universe.

• The Conflict: Standard cosmological models (FLRW metric) describe an expanding universe, while local systems are described by the Schwarzschild metric. The McVittie metric provides an exact solution to Einstein's field equations that couples a central mass $m$ with a cosmological background (FLRW).
• The Challenge: The McVittie metric is naturally expressed in global isotropic coordinates $(t, r)$, which are comoving with the Hubble flow. These coordinates are unsuitable for interpreting local physical observations because they introduce non-physical, coordinate-dependent effects (such as apparent expansion of local orbits) that violate the Einstein Equivalence Principle (EEP) locally.
• Goal: To derive the equations of motion for a test particle in a local coordinate system $(T, R)$ attached to the central mass, eliminate spurious expansion effects, and calculate the secular perturbations of orbital elements (semi-major axis, eccentricity, pericenter precession) caused by cosmological expansion and spatial curvature up to the second order in the Hubble parameter ($H^2$).

2. Methodology

The authors employ a rigorous mathematical framework combining general relativity, perturbation theory, and celestial mechanics:

A. Coordinate Transformation (Global to Local)

The core of the methodology is transforming the McVittie metric from global coordinates $(t, r)$ to local coordinates $(T, R)$ where the metric approaches the Minkowski metric at the origin (satisfying the EEP).

1. Tetrad Formalism: The authors utilize the tetrad (vierbein) formalism to define a locally inertial frame. They express the metric in terms of auxiliary functions $f_1, g_1, g_2$.
2. Radial Transformation: They transform the radial coordinate $r$ to $R$ using a relation derived from the metric structure, involving the scale factor $a(t)$ and the mass function $\mu(t) = m/a(t)$.
3. Time Transformation: To eliminate the off-diagonal term ($g_{TR}$) in the metric, they introduce a time transformation $t = T - \chi(T, R)$.
4. Lagrange Inversion Theorem: Since the transformation equations involve a 6th-order algebraic polynomial that cannot be solved in radicals, the authors apply the Lagrange Inversion Theorem to express the global coordinate $r$ as a power series in the local coordinate $R$. This allows them to expand the metric coefficients.

B. Post-Friedmannian Approximation

The transformed metric is expanded in a series of small parameters:

• $HR/c \ll 1$ (Hubble expansion relative to local scale)
• $m/R \ll 1$ (Post-Newtonian limit)
• $m/a \ll 1$
  The expansion is carried out up to second-order terms in the Hubble parameter $H$ and its time derivative $\dot{H}$.

C. Equations of Motion and Perturbation Analysis

1. Geodesic Equation: The equation of motion is derived from the geodesic equation in the local coordinates, resulting in a Newtonian-like form: $\ddot{\vec{R}} = \vec{F}_N + \vec{F}_{pN} + \vec{F}_H + \vec{F}_K$.
   • $\vec{F}_N$: Newtonian gravity.
   • $\vec{F}_{pN}$: Post-Newtonian corrections.
   • $\vec{F}_H$: Perturbations due to Hubble expansion (dependent on $H$ and $\dot{H}$).
   • $\vec{F}_K$: Perturbations due to spatial curvature $k$.
2. Osculating Elements: The authors analyze the evolution of the six Keplerian orbital elements ($a, e, i, \Omega, \omega, M$) using the method of osculating elements.
3. Time-Averaging Technique: To isolate secular (long-term) changes from periodic oscillations, they apply a time-averaging technique over one orbital period. This involves calculating averages of functions involving the true anomaly $f$ using Hansen coefficients.

3. Key Contributions

1. Rigorous Local Coordinate Construction: The paper provides a generalized transformation of the McVittie metric to local coordinates for all three spatial curvature cases ($k = -1, 0, +1$), overcoming limitations of previous works that were restricted to flat universes ($k=0$).
2. Second-Order Hubble Expansion: The analysis includes terms up to $H^2$ and $\dot{H}$, providing a more complete picture than first-order approximations.
3. Deceleration Parameter Dependence: The authors explicitly link the direction of orbital precession to the deceleration parameter $q$ (where $\dot{H} = -H^2(1+q)$), showing that the sign of $q$ determines whether the cosmological precession adds to or subtracts from the relativistic precession.
4. Validation of Equivalence Principle: The study confirms that to the order of $H^2$, the average size ($a$) and shape ($e$) of the orbit are not affected by cosmological expansion, consistent with the Einstein Equivalence Principle.

4. Key Results

A. Orbital Elements Stability

• Semi-major axis ($a$) and Eccentricity ($e$): The secular rates of change $\langle \dot{a} \rangle$ and $\langle \dot{e} \rangle$ are zero up to the order of $H^2$. This implies that bound systems do not expand with the universe; their average size and shape remain constant.
• Inclination ($i$) and Node ($\Omega$): These remain constant due to the conservation of angular momentum (the perturbing forces have no component normal to the orbital plane).

B. Pericenter Precession ($\omega$)

The total rate of change of the argument of pericenter is the sum of Post-Newtonian, Hubble, and Curvature terms:
$$\langle \dot{\omega} \rangle = \langle \dot{\omega}_{pN} \rangle + \langle \dot{\omega}_{H} \rangle + \langle \dot{\omega}_{K} \rangle$$

• Hubble Contribution: The cosmological expansion causes a precession that depends on $q$.
  • If $q < 0$ (accelerating universe, e.g., dark energy dominance): The precession is positive, adding to the standard relativistic precession.
  • If $q > 0$ (decelerating universe): The precession is negative, opposing the relativistic precession.
  • If $q = 0$ (constant expansion rate): The $q$-dependent term vanishes, but a residual negative precession remains due to the $H^2$ term.
• Curvature Contribution: Spatial curvature ($k$) induces an additional precession term proportional to $\kappa = k/a^2$.

C. Mean Anomaly ($M$)

The mean anomaly changes over time, which corresponds to a shift in the frequency of the mean orbital motion ($n$).

• The frequency shift depends on $q$ and $k$.
• For an accelerating universe ($q < 0$), the mean orbital frequency decreases.

D. Numerical Estimates

The authors applied their formulas to specific astrophysical systems:

• S-stars (S2, S62) orbiting the supermassive black hole Sgr A* at the Milky Way center.
• Sirius binary system.
• Result: The cosmological contribution to the pericenter precession is extremely small (on the order of $10^{-9}$to$10^{-18}$microarcseconds per year) compared to the Post-Newtonian precession (which is$\sim 10^7$to$10^8$$\mu$as/yr for S-stars). While currently undetectable with existing technology, the theoretical framework is established for future precision tests.

5. Significance

• Theoretical Consistency: The work resolves ambiguities regarding the "expansion of local systems." It rigorously demonstrates that while the global universe expands, local bound systems (like solar systems or binary stars) do not expand in their average dimensions, preserving the validity of local physics.
• Generalization: By including spatial curvature ($k$) and the deceleration parameter ($q$), the results generalize previous studies (e.g., by Carrera & Giulini, Kerr, Arakida) which were often limited to flat, de Sitter-like universes.
• Future Observational Tests: Although the effects are currently too small to measure, the paper provides the necessary theoretical precision for future astrometry missions (like those targeting the Galactic Center) that may eventually detect deviations in orbital precession caused by the specific nature of dark energy ($q$) and spatial curvature.
• Asymptotic Analysis: Appendix C provides an exact asymptotic solution showing that orbital elements oscillate around their initial values and do not drift secularly over infinite time, reinforcing the stability of bound orbits in an expanding universe.

In summary, this paper establishes a robust local framework for analyzing orbital dynamics in a cosmological context, proving that while cosmological expansion does not tear apart bound systems, it does induce subtle, measurable (in principle) precessions in their orbits dependent on the universe's expansion history.