QFT on curved spacetime, uniqueness of spacelike hypersurface

Question

Consider the Lagrangian of a real, scalar field coupled to gravity via the metric $g_{\mu\nu}$ and covariant derivative $\nabla_\mu$

$$\mathcal{L} = \sqrt{-g} (-\frac{1}{2} g^{\mu\nu} \nabla_\mu \phi \nabla_\nu \phi -\frac{1}{2}m^2\phi^2). $$

After canonical quantization, we can expand the field operator in terms of positive-negative frequency modes as (putting the theory in a box) $$ \hat{\phi}(x) = \sum_k [\hat{a}_k u_k(x) + \hat{a}_k^\dagger u^*_k(x)]. $$

However, generally different observers wouldn’t agree on this split between the modes and identify particle states differently. So, this expansion is not unique and for each such expansion we need to find a complete set of orthonormal modes $\{v_k(x)\}$, where orthonormality is w.r.t. the Klein-Gordon inner product: $$ (\phi_1, \phi_2) := -i \int \sqrt{g_\Sigma} [\phi_1 \nabla_\mu \phi_2^* - \phi_2^* \nabla_\mu \phi_1] d\Sigma^\mu.$$ Here $\Sigma$ is a space-like hypersurface and $g_\Sigma$ the induced metric.

My question is about the uniqueness of this hypersurface. In particular, how arbitrary is this choice? For example, would choosing a coordinate chart or defining a worldline fix this surface? Also, if coordinates fix it, I’d expect isometries to result in equivalent definitions of vacuum. Is this correct?

Answer 1

how arbitrary is this choice?

It needs to be a Cauchy surface. In other words, it has to intersect every causal curve exactly once. If it missed some curve, you would be missing information on the spacetime. If it intersected some curve more than once, you'd have issues with causality in the spacetime and might be giving too much information to the equations of motion. The existence of such a surface is equivalent to requiring the spacetime to be globally hyperbolic, which is a common assumption within QFTCS.

Any choice of Cauchy surface is equivalent, as one can show from the conservation laws associated to the Klein–Gordon equation and fooling around with Stoke's theorem.

If I recall correctly, this is discussed on Chap. 14 of Wald's General Relativity textbook (in bold, because I do not mean the QFTCS book).

For example, would choosing a coordinate chart or defining a worldline fix this surface?

Not necessarily. For example, pick the Schwarzschild coordinates in the maximally extended Schwarzschild spacetime. They only cover a part of the spacetime, so they are not enough to specify a Cauchy surface. They might be enough to specify a Cauchy surface on a smaller portion of the spacetime, though (such as one of the outer regions of the black and white holes).

Defining a worldline won't help you fix such a surface, since you actually need a spacelike hypersurface cutting through all of spacetime. A single observer does not have access to that.

Also, if coordinates fix it, I’d expect isometries to result in equivalent definitions of vacuum. Is this correct?

A timelike Killing field often leads you to a preferred notion of vacuum, so, in this sense, yes. However, there are caveats.

1. Kerr spacetime, for example, does not have a Killing-invariant non-singular vacuum state. (Wald's QFTCS book mentions this when discussing the Unruh effect in curved spacetime, there's probably a detailed reference in there)
2. Different observers tangent to the same Killing field might have different notions of particles. For example, pick two accelerated observers with different accelerations in Minkowski spacetime. Both of them are moving tangentially to the boost Killing field, but they disagree on the temperature of the Unruh bath, since they have different accelerations.