What's the first appearance of ultrastatic spacetimes (that is, spacetimes with a metric of the form $ds^2 = -dt^2 + h$, with $h$ a Riemannian metric)? The oldest reference I can find on the topic is Fulling's 1977 paper "Alternative vacuum states in static space-times with horizons", but from the article it seems to not be the first appearance of the term, but I'm unable to find an older reference from there.

  • $\begingroup$ Isn't Minkowski spacetime "ultrastatic" on one of the signature conventions? $\endgroup$
    – Conifold
    Oct 22, 2017 at 22:14
  • $\begingroup$ Yes, but that's not quite the same thing as the term or the notion of ultrastatic being discussed $\endgroup$
    – Slereah
    Oct 22, 2017 at 22:56

1 Answer 1


Both MathSciNet and APS do not even find Fulling'77 (probably because "ultrastatic" is not used in the abstract). The earliest metadata occurrences are Fulling, S. A. Narcowich, F. J. Wald, Robert M. Singularity Structure of the Two-point Function in Quantum Field Theory in Curved Spacetime, II. Ann. Physics 136 (1981), no. 2, 243–272; and Page, Don N. Thermal Stress Tensors in Static Einstein Spaces, Phys. Rev. D (3) 25 (1982), no. 6, 1499–1509. 83C47 (81E20). This suggests that "ultrastatic" was not a term/keyword before 1980s.

On the other hand, Fulling'77 does not credit or reference anybody when it defines "ultrastatic" on p. 919, and on p. 948 it even states:

"This procedure and interpretation are presumably physically trustworthy when the orbits of the Killing vector are also the geodesics normal to each constant-time hypersurface, as in the class of models we have called ‘ultrastatic’". [emphasis mine]

Of course, this could just mean "we have called within this paper", but then it would be odd not to mention the source of definition. Moreover, according to Fulling, his inspiration was Unruh, specifically Unruh's Notes on Black-Hole Evaporation, Phys. Rev. D 14, 870 (1976):

"The key observation, due to Unruh (1976), is that the horizon can be embedded in a flat or ultrastatic space-time; equivalently, that each of the two null surfaces making up the horizon is invariant under translation in its affine parameter, V or U."

But Unruh's paper does not use the term "ultrastatic" despite deriving results about ultrastatic spacetimes. So it seems entirely plausible to me that Fulling came up with the term himself.


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