2
$\begingroup$

Setup: Consider a scalar field $\phi(x)$ which classically satisfies the curved spacetime equation,

\begin{equation} g^{ab}\nabla_a\nabla_b \phi-m^2\phi =0, \end{equation}

where we assume that the background spacetime metric $g_{ab}$ is fixed in the sense that it does not depend on $\phi$ (i.e., the $\phi$-field has negligible contribution to the stress-energy).

Now suppose that the metric can be written as a small perturbation away from Minkowski spacetime,

\begin{equation} g_{ab} = \eta_{ab} + h_{ab}, \end{equation} where the components of $h_{ab}$ are much less than 1 in some global inertial coordinate system of $\eta_{ab}$ (see Wald section 4.4 for discussion). Importantly, further suppose that this perturbation $h_{ab}$ is also fixed (i.e., $\phi$ does not contribute to the stress energy tensor even at the order of $h_{ab}$).

That is, consider a scalar field $\phi(x)$ satisfying the classical equations of motion: [note $g^{ab}\equiv g_{ab}^{-1} = \eta^{ab}-h^{ab}$ so that $g^{-1}_{ac}g_{cb}=\delta^{a}_{\; b}+\mathcal{O}(h_{ab}^2)$]

\begin{align} 0&=g^{ab} \nabla_a\nabla_b \phi-m^2\phi\\ &=(\eta^{ab}-h^{ab})(\partial_a\partial_b-\Gamma^c_{\; ab}\partial_c)\phi-m^2\phi\\ &=(\eta^{ab}-h^{ab})\partial_a\partial_b\phi-\eta^{ab}\Gamma^c_{\; ab}\partial_c\phi-m^2\phi+\mathcal{O}(h_{ab}^2), \label{eom} \tag{*} \end{align}

where $h_{ab}$ is independent of $\phi$ and,

\begin{equation} \Gamma^c_{\; ab}=\frac{1}{2}\eta^{cd}(\partial_a h_{bd} + \partial_b h_{ad} - \partial_d h_{ab})+\mathcal{O}(h_{ab}^2). \end{equation}

Question: If $g_{ab}=\eta_{ab}$, then I might simply introduce global inertial coordinates $(t,\vec{x})$, use these coordinates to Fourier decompose the field into its momentum modes $\phi(x)\propto \int d^3k \hat{\phi}(k)\exp{(\vec{k}\cdot \vec{x})}$, plug this mode expansion into the equations of motion, and then proceed along the lines of Peskin and Schroeder section 2.3 to "quantize" the field. However, it's not clear to me how to proceed along these lines here since the metric is no longer precisely invariant under the Lorentz transformations.

Supposing that the spacetime is globally hyperbolic and static in the asymptotic future/past, then the quantum theory should be well-defined with a prefered vacuum state in the asymptotic future/past. Furthermore, it seems that one should, in principle, be able to obtain the two-point function of the theory, e.g., as some perturbative expansion in orders of $h_{ab}$. Nevertheless, I don't see how to get started in this direction.

$\endgroup$
4
  • $\begingroup$ The whole idea of linearization is you treat your $\eta$ part of the metric as the background metric, and $h$ as a dynamic variable, (in particular NOT the background metric). Then you just have a field theory in flat spacetime with some extra fields $h_{ab}$. $\endgroup$
    – zzz
    May 2, 2017 at 22:17
  • $\begingroup$ That is one use for linearization (and a common one), but I think the setup I have given is clear: I want to consider a scenario in which spacetime is nearly flat but not precisely and the entire metric is independent of the phi field. This would be of interest, e.g., if one is considering different renormalization prescriptions for the stress-energy tensor in curved spacetime and want to consider a metric that is not flat but is also not "too curved" either. $\endgroup$
    – user143410
    May 2, 2017 at 22:29
  • $\begingroup$ Just trying to understand the question here, are you saying you're linearizing around some $\eta$ that's not the $(1, 3)$ Minkovski metric? $\endgroup$
    – zzz
    May 2, 2017 at 22:40
  • $\begingroup$ As I say in the question, I am considering perturbations off of the usual Minkowski metric $\eta_{ab}$, where the perturbation components are "small" in the sense described by Wald section 4.4. $\endgroup$
    – user143410
    May 2, 2017 at 22:43

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.