There exists a Positive Energy theorem for General Relativity in Anti-de Sitter and asymptotically flat spacetimes, but there is no equivalent theorem for de Sitter spacetimes

Question: Is there a lower bound theorem on negative mass-energy density on de Sitter spacetimes?

The intuition says that the absence of a positive energy theorem in dS has to do with the fact that for small enough positive energy densities, the cosmological expansion beats the gravitational attraction, which means that positive energy densities need to exceed a threshold in order to behave attractively from far away. Is this intuition correct?

  • $\begingroup$ Not an answer, but I found this recent paper $\endgroup$
    – Trimok
    Jul 28, 2014 at 15:13
  • $\begingroup$ thanks, that paper was the one that brought the question in my head in the first place :-) $\endgroup$
    – lurscher
    Jul 28, 2014 at 15:26

1 Answer 1


The positive energy theorem talks about the lower bound on the total energy/mass, like the ADM mass.

To be able to define such a concept of the total energy/mass in general relativity, one needs some asymptotic region respecting a time-translational symmetry. That's the region where the gravitational potential (something like the deviation of $g_{00}$ from the vacuum value) goes like $GM/r$.

Minkowski and anti de Sitter space have this global time-like Killing vector and the required asymptotic region where the ADM-like mass may be measured. However, de Sitter space doesn't have one.

So not only there is no positive energy theorem in de Sitter space. There is even no well-defined definition of a conserved mass in that spacetime background! To understand all these things, one has to see why there is no nontrivial conserved energy/mass in cosmology or general backgrounds of general relativity, see e.g.


  • $\begingroup$ thanks. What would you say that is the reason we haven't seen/detected negative energy/mass yet? Is it forbidden by some quantum/stringy symmetry? It is just a symmetry-breaking thing that could change if temperature gets to inflationary-epoch levels? Or maybe negative mass is composed of sterile/dark energy that doesn't interact with our sector? $\endgroup$
    – lurscher
    Jul 28, 2014 at 15:29
  • $\begingroup$ Negative energy is really forbidden by relativity plus the stability of space. If regions with negative energy could be created, they could be created along with ordinary regions with positive energy - without violating any conservation laws - and the vacuum could decay in this way. We clearly don't observe that. Also, negative-energy particles are/may be tachyons. I am ready to bet anything that we won't see negative-energy objects in our lifetime. ;-) It's not about "yet". Negative energy density is impossible, that's why things that forbid it are called "theorems" and not "hypotheses". $\endgroup$ Jul 28, 2014 at 16:35
  • $\begingroup$ but I thought that was what the lack of a positivity theorem was implying: since our universe is de-Sitter, there is no such thing as a 'theorem' that forbids them. At least not a known one yet. Or is there some complementary non-negativeness theorem beyond the one being discussed here? You mention that it is forbidden because the vacuum "obviously" doesn't collapse and we are here :-) so, are you saying that the reason negative energy is not seen is purely anthropic? $\endgroup$
    – lurscher
    Jul 28, 2014 at 19:17
  • $\begingroup$ Well, the total conserved energy can't be well-defined in our (nearly) de Sitter space, but in every region of the Universe that is nearly flat (whose curvature radius is much longer than the typical length scale of the objects) and much smaller than the visible Universe, one may still use the rules for the Minkowski space. Inflation produced energy out of nothing, but every solar-system-sized region it produced is effectively flat and the Minkowski positivity and conservation laws apply there. $\endgroup$ Jul 28, 2014 at 19:23
  • $\begingroup$ The total energy density (incl. the gravitational terms) is really an ill-defined concept in GR even in spaces that are asymptotically Minkowski or anti de Sitter - only the total energy becomes more well-defined over there. One may still talk about the energy density of matter fields, ignoring any gravitational contribution, in the case that the gravitational curvature is small (because the curvature is small), and this energy density still obeys energy conditions (some positivity) and this is equally true whether the universe is globally flat, de Sitter, or anti de Sitter! $\endgroup$ Jul 28, 2014 at 19:25

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