Historical research guide · 0693 · 2022
The Return of the Singularities: Applications of the Smeared Null Energy Condition
Ben Freivogel · Eleni-Alexandra Kontou · Dimitrios Krommydas
In one minute
Ben Freivogel, Eleni-Alexandra Kontou and Dimitrios Krommydas ask the question that decides which geometries the universe will permit: how much negative energy can quantum fields actually deliver, and over how long a stretch? General relativity on its own allows wormholes and warp bubbles, because you may draw any geometry you like and then read off the matter it would need. The classic singularity theorems ruled those geometries out by assuming energy is never negative anywhere — an assumption every quantum field theory violates. The authors work instead with a replacement they call the smeared null energy condition: the null energy averaged along a finite stretch of a light ray, weighted by a smooth window function, cannot fall below a floor set by Newton’s constant. They motivate the bound, show it survives the standard counter-example once the cutoff is imposed in a Lorentz-invariant way, and test it against an evaporating black hole and against the Maldacena–Milekhin–Popov traversable wormhole. Then they prove a Penrose-style singularity theorem from it.
Connection
Chapter 4 turns on a single accounting question — how much energy may sit below the ambient vacuum level, for how long, in how thin a shell — and this paper writes that budget down as a theorem rather than a hope. It is also the cleanest statement of what the energy conditions do and do not settle: they never forbid negative energy, which quantum field theory guarantees exists, they only price it, and the price is the design constraint every wormhole and warp geometry in chapter 4 has to meet.
From the source
The classical pointwise energy conditions are not a law of nature. They are bounds on the stress tensor at a single point, and they were proven to be violated in all quantum field theories — every quantum field theory contains states in which the null-contracted stress tensor has a negative expectation value at a point.Section 1, Introduction, paragraphs 1 and 2
The smeared null energy condition replaces the pointwise rule with a budget: the null energy averaged along an achronal segment of a null geodesic, weighted by a smooth smearing function, is bounded below by four times a constant B divided by Newton’s constant, times the integral of the squared derivative of that smearing function. Because Newton’s constant carries the number of matter species with it, the bound is exactly as strong in a theory with many fields as in a theory with few.Section 1, equations (2) and (3)
The Fewster–Roman counter-example does not defeat a bound of this form. Boosting the two-particle states so their momenta run nearly parallel to the light ray does drive the smeared null energy arbitrarily negative in free field theory, but imposing the ultraviolet cutoff on the transverse momentum — which is the Lorentz-invariant way to impose it — leaves a divergence going as one over the cutoff length, milder than the bound’s own inverse-square behaviour.Section 2.3, equations (22) to (28)
The traversable wormhole of Maldacena, Milekhin and Popov obeys the four-dimensional smeared bound without saturating it, provided the number of field species is large; the same wormhole does saturate the two-dimensional quantum energy inequality, whose curved-spacetime version the authors prove here. A long wormhole is therefore not ruled out by this accounting — it sits inside the budget.Sections 3.3.1 and 3.3.2
A Penrose-type singularity theorem follows from the smeared bound alone, with no pointwise assumption. For the value of the constant derived from induced gravity on a brane, any spherically symmetric surface inside a Schwarzschild-like horizon whose radius is at most two-thirds of the Schwarzschild radius guarantees an incomplete null geodesic — so black-hole spacetimes still contain singularities even where the null energy condition fails.Section 4.2, equations (92) and (93)
The bound is not yet proven for interacting fields in general curved spacetimes, no example is known that saturates it in four dimensions, and it is an open question whether curvature correction terms are needed — so a stronger and more restrictive bound may exist, and finding a saturating example is the measurement that would settle it.Section 5, Conclusion, paragraphs 2 and 5
The way in
https://doi.org/10.21468/SciPostPhys.13.1.001How to cite it
Ben Freivogel, Eleni-Alexandra Kontou, Dimitrios Krommydas (2022) The Return of the Singularities: Applications of the Smeared Null Energy Condition doi:10.21468/SciPostPhys.13.1.001