Month: June 2026

In my book Nature and Nature’s God I claim that inertia is compatible with Thomistic natural philosophy. The fact that inertial motion continues on its own without an impelling force is not a real problem, as I show that for Aquinas motion in general does not require a continuously conjoined mover. The greater difficulty is the fact that inertia seems endless both in the sense of continuing forever and in the sense of having no goal. For Aquinas, every motion has its terminus.

I argue in the book that inertial motion, even though in principle infinite, has an effective terminus: complete isolation of a body from all interaction with other bodies. Not every Thomist I have communicated with has been comfortable with this solution. But I have since learned something very interesting about force-free motion in general relativity (GR). (Note that gravity does not count as a force in GR.)

In an expanding spacetime an object traveling force-free (no friction or any other retarding forces) in an empty expanding space decreases its proper velocity over time. In the simplest realistic case, every object asymptotically approaches a velocity of zero and also asymptotically approaches a limiting position in space. Its distance, however, from an observer will still increase because the space itself is expanding. The object, however, has no motion of its own (it “comoves” but has no “peculiar velocity.”) The simple but realistic case I have in mind is that of a flat, matter filled universe with a cosmological constant of sufficiently large value to prevent collapse and cause eventual runaway expansion. This is more or less the standard model of cosmology right now; the geometry is at least very close to flat, and dark energy is frequently interpreted as the cosmological constant, although this is currently under debate. The conclusion of a limiting position for a force free object may also hold in more complicated cases (e.g., positive spatial curvature, non-exponential expansion) but I have not been able to work through the more complicated mathematics.

An intuitive way to see why this phenomenon would occur so is to consider cosmological redshift. As the universe expands waves traveling through it are stretched thus undergoing redshift. This redshift applies not only to light but to the DeBroglie waves of massive objects. The DeBroglie wavelength of an object is inversely proportional to its momentum. Thus as the DeBroglie wave of a massive object traveling through space is redshifted and its wavelength increased, the momentum of the object decreases. Thus its proper velocity also decreases.

The more direct GR approach is to start with an FLRW metric. The metric determines a set of connection coefficients ($\Gamma_{\mu\nu}^{\alpha}$). If one plugs these connection coefficients into the geodesic equation ($\frac{d^2x^{\alpha}}{d\tau^2}=-\Gamma_{\mu\nu}^{\alpha}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}$), and makes the simplifying assumption that the motion is entirely in the radial direction ($d\theta=d\phi=0$) one obtains the differential equation $\frac{d^2r}{d\tau^2}=-\left(\frac{kr}{1-kr^2}+2\frac{\dot{a}}{a}v^r\gamma\right)$. If one assumes that the object’s speed is non-relativistic ($v \ll c$) so that the gamma factor can be neglected and that the universe is flat (k = 0) then the equation is easily solved to yield that $v = v_0 a^{-2}$, where a is the scale factor and $v_0$ is the object’s initial speed. If the scale factor increases exponentially with time, as the cosmological constant eventually causes to happen, then the distance covered by the object’s proper velocity is finite, that is $\int_{t_0}^\infin av dt$ is finite. A significant gamma factor value only increases the rate of deceleration, so this result should also hold for relativisitic speeds.

This means that a Thomist uncomfortable with infinite inertial motion may be able to avoid it without denying contemporary physics. Nevertheless, such a conclusion would require a universe whose expansion has no terminus. Thus the individual locomotion of objects in the universe might have a terminus, but not the growth motion of the universe itself. Whether this is helpful or not for a Thomist I leave to further reflection.