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The earliest text I've been able to find that explain the GR energy conditions is "The large scale structure of space-time" (1973) by Hawking and Ellis.
However in Barcelo and Visser's paper "Twilight for the energy conditions?" they speak of the trace energy condition which was popular in the 60s and has since now been abandoned. This is obviously before 1973 so who actually created the energy conditions?
I emailed Matt Visser to ask for help, his reply is reproduced below with his permission:
From: Matt Visser
Sent: Saturday, March 21, 2015 11:49 PMI think the name "trace energy condition" was actually invented by Carlos Barcelo and myself; but the condition (without the explicit name) was used extensively --- for instance, in the neutron star community.... see for instance, see also. A google search on "Zeldovich trace stress energy" will pick up these references and link back to even older stuff...
Following up on Matt's links one finds that the "trace energy condition" goes back further than 1960-s. In a 1939 paper On Massive Neutron Cores Oppenheimer and Volkoff obtained the limiting mass of $0.71$ solar masses for a neutron star, only half of the corresponding Chandrasekhar's limit for white dwarves. They used the stress energy tensor of a free Fermi gas of neutrons in their calculations, but remarked that their conclusions "would not be appreciably affected by any modification of these equations at supranuclear densities that was physically reasonable in the sense that the trace of the stress-energy tensor did not become negative". This relied on an unpublished result of von Neumann mentioned by Chandrasekhar in a 1935 paper.
However, the free model discarded the effects of nuclear interaction, and while von Neumann's result gave a perfectly fine asymptotic, treating it as a bound proved to be an overreach. In 1959 Skyrme constructed a model of nuclear interaction, which now bears his name, and Cameron applied it to model a gas of interacting neutrons. The Skyrme model was not relativistic however. Finally, in 1961 Zeldovich constructed an interacting gas model compatible with special relativity, but with a stress-energy tensor whose trace wasn't always non-negative. Correspondingly, the mass limit went up to $2.0$, and eventually to $3.0$ solar masses, which is much more consistent with modern observations.
