How the Whole Universe May Slow Your Clock: New Study Tally’s Cosmic Time Dilation

Every clock on Earth is running slow, and not just because of the planet’s own gravity. According to a new open-access study in Astrophysics and Space Science by Richard Petarius III and Robert J. Nemiroff of Michigan Technological University, every bit of mass-energy in the cosmos, from the nearest galaxy to the most distant quasar, contributes a tiny gravitational time dilation to clocks here on Earth. The question the authors set out to answer is deceptively simple: when you add up all of those contributions across the entire observable universe, does the total settle down to a finite number, or does it grow without bound? The answer, they find, depends in surprising ways on the kind of universe we live in, and the result has implications for some of the most precise tests of Einstein’s equivalence principle ever attempted.

The starting point is a fact that has been measured directly both on Earth and across the Solar System: matter dilates time in its surroundings. The canonical description comes from the Schwarzschild metric, the general relativistic solution for the spacetime around an uncharged, non-rotating mass. In that solution, the rate of a clock at radial distance r from a mass is slowed relative to a clock infinitely far away by a factor involving the mass’s Schwarzschild radius, which is proportional to the mass itself. Pound and Rebka measured such dilation in a laboratory stairwell in 1959, and Shapiro delay measurements with the Cassini spacecraft confirmed the effect to exquisite precision. What Petarius and Nemiroff add is a cosmological twist: mass-energy anywhere, even at distances of billions of light years, contributes to the dilation experienced by an observer on Earth.

A crucial subtlety is that gravitational time dilation is a scalar quantity and does not sum the way gravitational forces do. The gravitational pull inside a spherical shell of matter is exactly zero, a classic Newtonian result, yet the time dilation produced by that shell at its center is not zero. This follows from Birkhoff’s theorem, which states that the interior of an empty spherically symmetric shell is flat, described by the Minkowski metric, while the exterior is Schwarzschild. Because the metric is flat inside the shell, there is no gradient of gravitational potential and hence no relative time dilation between two interior points. But compared with an external reference clock, the entire interior sits at a constant, non-zero dilation. It is this global, comparative effect, not the local redshift, that the new work tallies.

The authors build up the calculation with a striking thought experiment. Place the Earth one Schwarzschild radius from a black hole’s center, and time at the Earth effectively freezes compared with a distant observer, an infinite dilation. Now split the black hole in two and place the halves on opposite sides of the Earth. Each fragment has half the mass, so the Earth sits two Schwarzschild radii from each. In the weak-field approximation where the individual potentials simply add, the total dilation is found by multiplying the two factors, and the result is that an external observer would see Earth clocks ticking at exactly half speed, even though the total mass is unchanged. Keep splitting the mass into ever more fragments distributed around the Earth, and the dilation weakens: the more fragmented and distributed the surrounding mass, the smaller the effect. In the limit of a thin spherical shell built from infinitely many pieces, the cumulative factor converges to an exponential form, exp of minus the shell’s effective Schwarzschild radius over twice its distance, which is precisely the expression that arises independently in the Newtonian gauge of cosmological perturbation theory. The authors are careful to note this agreement is a consistency check on their multiplicative approximation, not a derivation of it.

With that machinery in place, the team first considers a purely Newtonian universe filled uniformly with mass. There, the amount of mass enclosed grows with the cube of distance while the potential contribution of each shell falls only as the inverse of distance, so the cumulative gravitational potential, and with it the cumulative time dilation, diverges as the considered radius grows to infinity. This divergence is the mathematical heart of the paper’s central puzzle. The authors point out a famous coincidence that emerges if the divergence is capped at the radius of the observable universe: using the present critical density and a comoving radius of about 46.5 billion light years, the enclosed mass has a Schwarzschild radius roughly equal to that radius, a coincidence long noted in discussions of whether our universe could be the interior of a black hole. Taken at face value in that capped scenario, the cumulative dilation factor would be on the order of ten to the minus four, meaning cosmic mass-energy alone would slow Earth clocks by roughly that fraction relative to a hypothetical empty reference.

The relativistic treatment is more subtle. In Friedmann-Lemaître-Robertson-Walker cosmologies, the authors work in the Newtonian gauge, where the metric component g00 takes the form that yields a clock rate of approximately one plus the gravitational potential over c squared, matching the weak-field expansion of the Schwarzschild result. They model the universe as a series of concentric comoving shells centered on the Earth and compute the dilation of each shell relative to the same FLRW universe with that shell emptied of mass-energy, then multiply the factors, or equivalently add their logarithms, across all shells out to the particle horizon. Because every location in a homogeneous universe sees the same total dilation, the calculation is inherently comparative: there is no external clock at infinity to reference, so the filled-versus-empty-shell construction is what isolates the gravitational contribution. The authors stress that this is exploratory basic science rather than a measurement, and that general relativity’s non-linearity means the summation is not exact.

The headline result is that the fate of the cumulative dilation depends dramatically on what the universe is made of. When plotted as a function of redshift, the cumulative time dilation converges to finite values in universes dominated by matter, by radiation, or by the concordance mixture of roughly 68.5 percent dark energy, 31.5 percent matter, and a trace of radiation that matches current Planck satellite parameters. The reason is geometric: in those cosmologies, the comoving distance to a shell flattens out at high redshift, so integrating over all redshifts only ever includes a finite amount of mass-energy. The lone exception is a universe continually dominated by dark energy alone, where the comoving distance itself diverges with redshift, allowing shells from arbitrarily far away to enter the sum and driving the cumulative dilation to infinity. Meanwhile, as a function of comoving distance rather than redshift, the dilation generally diverges in several flat cosmologies, echoing the Newtonian result.

The authors also extend the framework to inhomogeneous universes by redefining the baseline. In a perfectly uniform universe, any overdensity slows time relative to the mean while any underdensity, a void, accelerates it. They show that dividing each spherical shell into n pieces, concentrating all its mass into one dense clump and leaving the rest empty, produces a net relative time dilation that grows as the universe becomes more concentrated, with the curves for different concentration levels converging toward a common behavior at high redshift. This connects their work to the classic Rees-Sciama Swiss cheese picture, in which masses surrounded by voids produce their own characteristic redshift signatures, though the present calculation focuses on the global dilation at the Earth rather than redshifts along specific lines of sight.

Why does any of this matter beyond conceptual housekeeping? The motivation traces back to tests of the weak equivalence principle using photons from distant gamma-ray bursts and fast radio bursts. Those tests compare the arrival times of photons of different energies that have crossed billions of light years of gravitational potential, and they are limited by how much of the universe’s mass distribution is included in the calculation. Many previous analyses used only the potential of the Milky Way, or extended to the Laniakea supercluster, precisely because including more distant mass seemed to invite divergences that researchers worked around rather than confronted. Petarius and Nemiroff argue that facing those divergences head-on is a necessary stepping stone: cumulative time dilation, unlike gravitational force, accumulates monotonically with enclosed mass-energy and is unusually sensitive to the distant universe. The work also distinguishes observable gravitational time dilation from a kind of raw potential that persists even when redshift effects between two locations cancel, raising the possibility that if different forms of matter or radiation respond to that raw potential differently, the large-scale universe itself could become a laboratory for testing the equivalence principle. The authors caution that their infinities may not be physical, since they are sensitive to conditions in the early universe where the underlying assumptions of FLRW cosmology, and perhaps general relativity itself, may break down. But by mapping exactly where the divergences appear and where they vanish, the study turns a long-ignored embarrassment into a tractable question, one that future measurements with cosmic transients may finally be able to answer.

Subject of Research: Cumulative gravitational time dilation from cosmological mass-energy distributions in Newtonian and FLRW universe models

Article Title: Cumulative gravitational time dilations across the universe

Article References: Petarius, R., III, & Nemiroff, R. J. (2026). Cumulative gravitational time dilations across the universe. Astrophysics and Space Science, 371(10), Article 113. https://doi.org/10.1007/s10509-026-04645-6

Image Credits: AI Generated

DOI: 10.1007/s10509-026-04645-6

Keywords: gravitational time dilation, general relativity, Schwarzschild metric, FLRW cosmology, Newtonian gauge, cosmological redshift, dark energy, equivalence principle, gamma-ray bursts, fast radio bursts, comoving distance, cosmology