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$CPT$ symmetry is a well-known symmetry that holds for all observed systems, and experimental efforts have been unable to find a breakdown of $CPT$. The proof relies on Lorentz symmetry, so in principle it applies to flat spacetimes. Okay, so far so good. However, when I think of a general scenario where a curved spacetime is given, I do no longer understand this symmetry. For example, in a simple model of FRW spacetime with flat spatial sections but an overall scale factor, $$ds^2=-dt^2+a^2(t)d\vec{x}^2,$$ time reversal symmetry is explicitly broken (to see this, compute for example the action of a scalar field in this background to find that the scale factor renormalizes dynamically some of the parameters of the QFT). Therefore, the combination of charge conjugation $C$ and parity $P$ must also be broken. So the reasoning that confuses me is

  • Take $a(t)=1$. We have time-reversal symmetry, and so the combination $PC$ must not be broken, in order for $CPT$ to hold.
  • Now, allow $a(t)$ to have some explicit dependence on time. This breaks $T$, but in principle there is nothing that breaks also $C$ or $P$, since we are only changing the temporal part of the metric.

So my confusion is on how this interpolation between the flat spacetime and the curved spacetime case work. Is it that $CPT$ must not be a symmetry in a general spacetime, or is it that actually a time-dependent $a(t)$ breaks either parity $P$ or charge conjugation $C$, which compensates the explicit breakdown of time-reversal symmetry, so that $CPT$ holds?

TopoLynch
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1 Answers1

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For reference, please read The Cosmological CPT Theorem. The metric you are referring to is equation (3.31) which is conformally transformed to the Poincar'e patch in de Sitter since it is easier to discuss then.

The primary observation is that CPT invariance in (non-time evolving) Minkowski space is replaced with CRT where the $R$ stands for reflections. It is shown that parity conservation on the Poincar'e patch in de Sitter is equivalent to parity transformations. Something important to remember is that in cosmology, observations are made with correlators, which are on a surface (such as the inflation surface, last heating surface, CMB...) and therefore there is a conformal symmetry (that in real scenarios is, if I can recall correctly, is badly broken), and so the Lorentz symmetry is upgraded to the Lorentz group plus conformal transformations (dilatations, special conformal transformations, etc).

The way the correlators actually transform under CRT is in section 3.4. The statement of the cosmological CPT theorem is made at the beginning of section 7.

MathZilla
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