Month: July 2026

Inertial Motion in a Spherical or Hyperbolic Expanding Universe

Daniel Shields

The analysis of inertial motion in my previous post can be extended in a straightforward way to spherical and hyperbolic universes. In spherical and hyperbolic universes also, if a cosmological constant is present and the expansion never ending, then any inertially moving object will come to rest with respect to space on its own without any forces acting on it. (The space still expands, and so the object’s distance from an observer still increases.)

The mathematical analysis is simplified if one utilizes an alternative system of coordinates. In my previous analysis I was using the following standard form of an FLRW metric:

ds2=−dt2+a2[11−kr2dr2+r2(dθ2+sin2dϕ2)]ds^2=-dt^2 +a^2[\frac{1}{1-kr^2}dr^2+r^2(d\theta^2+sin^2d\phi^2)]

This is, of course, simplest to work with when the universe is flat (k = 0). When k = 1 the universe is spherical and finite in size. The variable r is the radial component from the origin. If it were possible to travel far enough in a single direction one would return to where one started. When r = 1 the position is one quarter of the way around the entire universe. In the opposite direction, r = 1 would be another quarter of the way around the universe. But r = 1 is also a coordinate singularity since one ends up dividing by zero. Another standard version of the FLRW metric avoids this problem and covers the entire spherical universe:

ds2=−dt2+a(t)2[dχ2+Σ(χ)2(dθ2+sin2dϕ2)]ds^2=-dt^2 +a(t)^2[d\chi^2+\Sigma(\chi)^2(d\theta^2+sin^2d\phi^2)]

When k = 0 (flat universe), sigma of chi = chi and these coordinates are identical to the previous coordinates. When k = 1 (spherical universe), sigma of chi equals the sine of chi. When k = -1 (hyperbolic universe), sigma of chi equals the hyperbolic sine of chi. In a spherical universe, when chi goes to pi one has gone half way around the universe. Chi going to pi in the opposite direction then covers the other half of the universe.

From this metric one can derive the connection coefficients. We are interested in the case of radial motion only, and so the following connection coefficients (for the k = 1 case) are the only ones relevant:

Γtχχ=Γχtχ=a˙a\Gamma^\chi_{t \chi}=\Gamma^\chi_{\chi t}=\frac{\dot{a}}{a}

The geodesic equation then gives:

d2χdt2=−2a˙adtdτdχdτ=−2a˙aγuχ\frac{d^2\chi}{dt^2}=-2\frac{\dot{a}}{a}\frac{dt}{d\tau}\frac{d\chi}{d\tau}=-2\frac{\dot{a}}{a}\gamma u_\chi

Presuming that the velocity is much less than the speed of light, gamma is approximately one. If the velocity is appreciable compared to light, gamma will be more than one and the deceleration will be greater.

The geodesic equation has given us a differential equation, which, under the simplifying assumption that gamma is a constant equal to one, is solved by the following equation:

vχ=vχ0a2v_\chi=\frac{v_{\chi 0}}{a^2}

The distance covered by the object’s proper velocity (as opposed to the comoving velocity of space) is given by:

l=∫t0∞g11vχdt=∫t0∞avχdt=∫t0∞vχ0adtl=\int^\infty_{t_0}\sqrt{g_{11}}v_\chi dt=\int^\infty_{t_0}av_\chi dt = \int^\infty_{t_0}\frac{v_{\chi 0}}{a} dt

If the universe has a cosmological constant and a density low enough not to recontract and end in a big crunch, then eventually the scale factor increases as

a(t)=CeHta(t)=Ce^{Ht}

Thus the total distance covered by an inertially moving object in infinite time is (H is the Hubble parameter which approaches a constant value determined by the cosmological constant):

l=vχ0C∫t0∞e−Htdt=0−−Hvχ0Ce−Ht0l=\frac{v_{\chi 0}}{C}\int^\infty_{t_0}e^{-Ht} dt = 0 -\frac{-H v_{\chi 0}}{C}e^{-Ht_0}

A nearly identical analysis applies to a hyperbolic universe (k = -1.) Thus inertially moving objects not only slow down and approach a state of rest, but approach a terminal point in space. This might be welcome news to Aristotelians.