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The authors propose a new way to handle space-time singularities in general relativity. Instead of fixing the metric or fields at the singular boundary, they treat the singularity as a free boundary where variations are unconstrained. Requiring the gravitational action to be stationary under such free variations produces on-shell boundary conditions that the geometry must satisfy at the singularity.
To make this precise they work with the usual gravitational action plus the standard boundary term (the Gibbons–Hawking–York term) and use the tetrad (coframe) formalism near a spacelike singularity. Varying the coframe while keeping the bulk equations of motion gives a simple condition in the limit toward the singularity: a combination of the extrinsic curvature (which measures how spatial slices bend) and the spatial volume element must go to zero. In other words, the way space “squeezes” to zero volume at the bang is restricted by the action principle.
When applied to cosmological solutions, this boundary condition rules out a large class of anisotropic singular behaviors known as Kasner-like or Belinski–Khalatnikov–Lifshitz (BKL) space-times. By contrast it admits conformally regular space-times, including the standard Friedmann–Lemaître–Robertson–Walker (FLRW) cosmologies, provided the matter is a perfect fluid with pressure P and energy density ρ satisfying 0 ≤ P < ρ (equivalently an equation-of-state parameter w between 0 and 1). As an example, radiation-dominated universes (P = ρ/3) meet the condition.
The paper also studies small (linear) perturbations around nearly isotropic FLRW backgrounds. For those cases the boundary condition is equivalent to demanding conformal regularity at the bang together with reflecting boundary conditions on perturbations. Concretely, certain time-derivatives of the scalar metric potential and of tensor (gravitational-wave) modes vanish at the initial time. Vector perturbations are not allowed and, in the special cases of dust or radiation, the perturbations are analytic and symmetric under time reflection about the bang. The authors note that these reflecting conditions are compatible with large-scale cosmological observations.
There are important caveats. The argument relies on a particular variational set-up: the chosen gravitational action with its boundary term, the use of the tetrad formalism, and the assumption that the matter Lagrangian does not itself produce boundary terms (an assumption that holds for perfect fluids). The detailed results are proved for linear perturbations around homogeneous, isotropic closed spatial slices and for perfect-fluid equations of state with 0 ≤ w < 1; the stiff case w = 1 gives a divergent on-shell action and is excluded. Finally, the excerpt is not the full paper, so other technical conditions or extensions may appear in the complete work.