Cosmology · Gravity and an expanding Universe

Why bound objects need not expand with the Universe

Compare a pair following the cosmic flow with a system governed by local binding. A changing background scale does not require every physical size to track it, but deciding what a real system does still requires dynamics.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Distinguish a comoving trajectory from a locally bound system.
Calculate the coordinate change corresponding to a stipulated fixed physical length.
Explain why bound does not mean perfectly static or exactly immune to every cosmological influence.

Does an expanding background require an expanding ruler?

The last two lessons followed ideal markers with fixed comoving separations. If the scale factor doubled, their proper separations doubled. A tempting next step is to apply that rule to a ruler, an atom or an orbit. That step silently assumes that the parts of each system follow the same comoving trajectories.

A bound system has interactions that constrain its relative motion. Gravity can bind an orbital system; electromagnetic interactions are central to atomic and material structure. Their equations and physical states help determine sizes and motions. A cosmological background does not remove those local interactions or prescribe fixed comoving coordinates for every constituent.

The real question is dynamical: how does a particular system respond to a specified environment and cosmic history? It is not answered by stretching every length in a drawing. The ideal smooth background is an approximation for large-scale behavior, not a complete local solution inside every concentration of matter.

Specify what is being held fixed

Use a flat-background coordinate comparison with dimensionless a and length-valued comoving separation χ. A comoving pair with χ = 2 u has proper separation D = 2a u, where u is an invented length unit. At a = 1, 1.5 and 2, its proper separations are 2, 3 and 4 u.

For comparison, stipulate a second separation with constant physical length L = 2 u over that interval. Writing the same relation L = aχ requires χ = L/a. Its comoving separations are 2, 4/3 and 1 u at the three epochs. Constant physical length and constant comoving separation are different prescriptions.

The second trace is a chosen kinematic comparison, not a solved galaxy, orbit or quantum atom. It shows that fixed physical length is compatible with a changing coordinate scale; it does not establish that a particular binding mechanism produces exactly that trace. A physical solution would have to justify the motion.

In the same flat-background description, the separation-rate decomposition is Ḋ = HD + v_pec, where v_pec is the relative peculiar contribution along the pair's line in this example. For a stipulated constant D, that contribution is −HD. The minus sign cancels the background-coordinate growth term; it is not evidence of a hidden force or inward wind of space.

Stipulated trajectories in a coordinate model; u is an invented length unit
Scale factor a	Comoving pair: proper D/u	Fixed-length comparison: proper L/u	Fixed-length comparison: comoving χ/u
1	2	2	2
1.5	3	2	4/3
2	4	2	1

Two kinematic prescriptions are plotted against dimensionless a from 1 to 2 on linear axes. A solid-circle fixed physical length L = 2 u has constant proper length 2 u and comoving separation (2 u)/a. A dashed-square comoving pair with χ = 2 u has proper separation 2a u and constant comoving separation 2 u. At a = 1, 1.5 and 2, the fixed-length comoving values are 2 u, 4/3 u and 1 u; the pair proper values are 2 u, 3 u and 4 u. These are stipulated comparisons, not calculated galaxy, orbital or quantum-atom solutions. The horizontal axis is scale factor rather than time. Original Ark teaching schematic.

Bounded motion can still change

An orbit can remain bound while its separation oscillates. A galaxy can be gravitationally bound while it rearranges, merges or responds to its surroundings. A material can change size when heated. Bound therefore does not mean motionless, unchanging or described by one universal radius.

Nor should we declare that the cosmological environment has exactly zero influence on all local systems. The answer can depend on the background history, the binding strength, the initial state and the precision of interest. A useful approximation can neglect an effect that is small for its question without asserting a theorem that the effect cannot exist.

The relevant comparison comes from the actual equations and their time scales. It is not enough to calculate HD across an object and call that an outward force. HD has speed units. A force or acceleration requires additional dynamics, and a coordinate separation rate is not itself a local stretching interaction.

An expanding scale need not supply an outward acceleration

Price and Romano's pedagogical study examines a specified classical system with an attractive inverse-square binding force in an expanding spacetime model. It distinguishes physical radius from cosmological coordinates and checks its nonrelativistic description against a relativistic construction. The classical model is not a quantum calculation of a real atom or a universal model of a galaxy.

In that controlled system, the physical-radius equation contains a background term (ä/a)r, where r is physical radius. The factor ä/a has units of inverse time squared, so multiplying it by r produces acceleration units. Its sign depends on the history's acceleration, not just on whether a is increasing.

For the earlier synthetic linear history a = 1 + t/T, a increases but ä is zero. This particular background acceleration term is therefore zero, even though H is positive. In the synthetic quadratic history a = (1 + t/T)², it is positive. These substitutions illustrate the mathematical distinction; they do not solve the bound-system equation or give a disruption threshold.

The study's bounded solutions can respond or oscillate rather than undergoing continuous Hubble-like growth. The lesson is to calculate a response within a specified model. Extending that result to self-gravitating structures, quantum systems or a different background requires its own analysis. There is no single size boundary derived here that divides everything that expands from everything that never changes.

Keep local behavior and cosmic inference distinct

A useful local investigation follows relative positions or motions over time, models the binding and external influences, and compares predicted changes with measurement uncertainty. A change due to local mass loss or interactions should not automatically be labelled cosmological expansion merely because the large-scale Universe expands.

Conversely, measuring a stable orbit does not independently determine the large-scale expansion history. That history is inferred from suitable cosmic observations with their own distance definitions and models. Different questions can require different approximations without making the descriptions inconsistent.

This also clarifies the pictures used in popular explanations. A model can display fixed-size markers whose separations grow. It should say whether those markers merely label positions or represent bound objects under an approximation. The next lesson examines how balloon and raisin analogies can help—and how their literal details can mislead.

Worked reasoning

Check a fixed-length prescription

A stipulated system has physical separation L = 6 u at a = 0.5 and at a = 1. What are its comoving separations in the convention L = aχ? Does their change establish a measured local contraction?

At a = 0.5, χ = L/a = 6/0.5 = 12 u.
At a = 1, χ = 6/1 = 6 u.
The physical separation was stipulated to remain 6 u. The changing coordinate length follows from the coordinate scale; it is not a decrease in that physical separation.

χ changes from 12 to 6 u while L stays 6 u. This is a coordinate comparison, not an observed contraction or a derived bound-object solution.

Try the reasoning

Can a system remain bound while its internal separation changes repeatedly?
Show the answer and reasoning
Yes.
A bound orbit can oscillate in radius. Binding constrains dynamics; it does not require every relative position to stay fixed.
Does a positive HD across a chosen length, by itself, specify an outward force on that object?
Show the answer and reasoning
No.
HD has units of speed, and its interpretation refers to the background separation rate. The local response needs an equation of motion and the relevant interactions.
Does the statement that a bound system need not follow the Hubble flow prove that every cosmological influence on it is exactly zero?
Show the answer and reasoning
No.
The size of any influence depends on the system and background model. A negligible effect in one approximation is not a universal zero-effect theorem.

What to carry forward

Fixed comoving coordinates and local binding describe different trajectory conditions.
A physical length can stay bounded while its comoving coordinate changes.
Bound systems can evolve; the cosmological response must be assessed in a specified dynamical model.
A background expansion rate is not an extra universal outward force applied to every object.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.