Field Quantisations in Schwarzschild Spacetime: Theory versus Low-Energy Experiments

TL;DR

The paper probes how quantum dynamics and particle concepts translate from quantum mechanics to quantum field theory in the curved spacetime around Earth, focusing on Schwarzschild geometry. It derives the non-relativistic limit with Newtonian gravity, constructs covariant modes, and analyzes both n- and h-type propagators, including Hawking quanta, in the far-horizon region. A key finding is that the Hawking-particle propagator ⟨h(x)|h(X)⟩ differs from the Minkowski/path-integral expectation, while the n-mode propagator ⟨n(x)|n(X)⟩ reproduces the standard quantum-mechanical propagator in the appropriate limit; this reveals a tension between traditional particle notions in curved spacetime and observable quantum phenomena. The work implies that Hawking particles cannot be straightforwardly mapped onto conventional quantum-mechanical particles, with potential implications for their experimental detection and for interpreting gravity's role in quantum dynamics.

Abstract

Non-relativistic quantum particles in the Earth's gravitational field are successfully described by the Schrödinger equation with Newton's gravitational potential. Particularly, quantum mechanics is in agreement with such experiments as free fall and quantum interference induced by gravity. However, quantum mechanics is a low-energy approximation to quantum field theory. The latter is successful by the description of high-energy experiments. Gravity is embedded in quantum field theory through the general-covariance principle. This framework is known in the literature as quantum field theory in curved spacetime, where the concept of a quantum particle is, though, ambiguous. In this article, we study in this framework how a Hawking particle moves in the far-horizon region of Schwarzschild spacetime by computing its propagator. We find this propagator differs from that which follows from the path-integral formalism -- the formalism which adequately describes both free fall and quantum interference induced by gravity.

Figures (6)

• Figure 1: Numerical computations of $(\omega/k)\,|B_{l}(\omega,M)|^2$ as a function of $l$ for various values of $\omega R_S$ and $MR_S$. We compute $\mathcal{A}_l(\omega,M)$ by evaluating the confluent Heun functions entering $\mathcal{H}_{\omega l}(r_s)$ and their derivatives with respect to $r_s$ at $r_s = 10^3 R_S$. We next confirm that the values of $\mathcal{A}_l(\omega,M)$ are essentially independent of $r_s$ by computing some of those also at $r_s = 10^4 R_S$. Our numerical results shown here and below are however based on the computations of $\mathcal{A}_l(\omega,M)$ at $r_s = 10^3 R_S$, because it requires less computational resources. Left: We first assume the relativistic regime, i.e. $k \gg M$. Our numerics agree with the DeWitt approximation, see (144) in DeWitt-1975. Right: We next consider the non-relativistic regime, i.e. $M \gg k$. In this case, $|B_{l}(\omega,M)|^2$ approximately equals $(k/\omega)\,\theta(l_\text{max} - l)$ with $l_\text{max} \equiv (3\sqrt{2}/2)\,MR_S$.
• Figure 2: Numerical computations of the gray-body factor $\Gamma(\omega,M)$ as a function of $kR_S$ for various values of $MR_S$. We compute the transmission probability $|B_l(\omega,M)|^2$ by using the method outlined in the caption of Fig. \ref{['fig:1']}. Left: Numerical results for $\Gamma(\omega,M)$ with $MR_S = 0$, in accord with DeWitt's approximation DeWitt-1975. Right: Numerical results for $\Gamma(\omega,M)$ with $MR_S \in \{5,10\}$, suggesting $\Gamma(\omega,M)$ increases with growing $MR_S$ and vanishes if $kR_S \to 0$, which agrees with \ref{['eq:a1']} as $kR_S \to 0$ if $c \to \infty$.
• Figure 3: Left column: Numerical computations of $\mathcal{N}_{\omega}(\boldsymbol{x},\boldsymbol{x})$ for various values of $\omega$ and $M$, while the Schwarzschild radial coordinate $r_s \in [2,100]{\times}R_S$. Our numerics support \ref{['eq:g_nk_asymp']} at $r_s \to \infty$. Shortly we shall derive this spatial-infinity asymptotic of $\mathcal{N}_{\omega}(\boldsymbol{x},\boldsymbol{x})$ by use of $\mathcal{N}_{\omega l}^{(1)}(r)$ given in \ref{['eq:app-radial-mode-solutions-n']}. It will also reveal the functional origin of the oscillations shown in the subplots. Right column: Numerical computations of $\mathcal{H}_{\omega}(\boldsymbol{x},\boldsymbol{x})$ for the same $\omega$ and $M$, while $r_s \in [1.0001,1.01]{\times}R_S$. Our numerics verify \ref{['eq:g_hk_asymp']} at $r_s \to R_S$. This asymptotic of $\mathcal{H}_{\omega}(\boldsymbol{x},\boldsymbol{x})$ at $r_s \to R_S$ also agrees with Candelas for $M = 0$.
• Figure 4: Left column: Numerical computations of $\mathcal{N}_{\omega}(\boldsymbol{x},\boldsymbol{x})$ for the same $\omega$ and $M$ as in Fig. \ref{['fig:3']}, while $r_s \in [1.0001,1.01]{\times}R_S$. Our numerics support \ref{['eq:g_nk_asymp']} at $r_s \to R_S$. Right column: Numerical results for $\mathcal{H}_{\omega}(\boldsymbol{x},\boldsymbol{x})$ with $r_s \in [2,100]{\times}R_S$, which confirm \ref{['eq:g_hk_asymp']} at $r_s \to \infty$. In contrast to $\mathcal{N}_{\omega}(\boldsymbol{x},\boldsymbol{x})$, which approaches unity at $r_s \to \infty$ as shown in Fig. \ref{['fig:3']}, $\mathcal{H}_{\omega}(\boldsymbol{x},\boldsymbol{x})$ vanishes as $(1/r_s)^{2}$ at spatial infinity. This circumstance particularly implies that the radial modes $N_{\omega lm}(x)$ and $H_{\omega lm}(x)$ differently behave in the regime in which Newton's mechanics successfully works. This has impact on how the one-particle states $|n(x)\rangle$ and $|h(x)\rangle$ propagate at $|\boldsymbol{x}| \gg R_S$, as will be shown in the subsequent sections.
• Figure 5: Numerical computations of $\mathcal{N}_\omega(\boldsymbol{x},\boldsymbol{X})$ for various values of $\omega$ and $M$, whereas $\boldsymbol{x} = \Delta\boldsymbol{x} + \boldsymbol{X}$ with $\boldsymbol{X} = (0,0,Z)$ and $\Delta\boldsymbol{x}$ being either $(\Delta{x},0,0)$ or $(0,0,\Delta{z})$. For a given value of $Z$ in units of $R_S$, we thus compute how $\mathcal{N}_\omega(\boldsymbol{x},\boldsymbol{X})$ changes by varying $\Delta\boldsymbol{x}$ either perpendicularly or parallelly to $\boldsymbol{X}$. Left column: $\Re\Delta_\perp\mathcal{N}_\omega(\boldsymbol{x},\boldsymbol{X})$ with $\Delta\boldsymbol{x} = (\Delta{x},0,0)$ and $Z_s/R_S \in \{50,100,150\}$, while the subplots show $\Im\mathcal{N}_\omega(\boldsymbol{x},\boldsymbol{X})$. Right column: $\Re\Delta_\parallel\,\mathcal{N}_\omega(\boldsymbol{x},\boldsymbol{X})$ with $\Delta\boldsymbol{x} = (0,0,\Delta{z})$ and the same values of $Z_s$ and the subplots display $\Im\Delta_\parallel\,\mathcal{N}_\omega(\boldsymbol{x},\boldsymbol{X})$.