Highlighting the back-action contribution of matter to quantum sensor network performance in multi-messenger astronomy

arising from C. Dailey et al. Nature Astronomy https://doi.org/10.1038/s41550-020-01242-7 (2021)

In their study, Dailey et al.¹ claimed that networks of quantum sensors can be used as sensitive multi-messenger probes of astrophysical phenomena that produce intense bursts of relativistic bosonic waves that interact non-gravitationally with ordinary matter. The most promising possibility considered in ref. ¹ involved clock-based searches for quadratic scalar-type interactions, with greatly diminished reach in the case of magnetometer-based searches for derivative-pseudoscalar-type interactions and clock-based searches for linear scalar-type interactions. In this Matters Arising, we show that the aforementioned work overlooked the ‘back action’ of ordinary matter on scalar waves with quadratic interactions, which can greatly affect the detection prospects of clock networks. In particular, back action can cause strong screening of scalar waves near Earth’s surface and by the apparatus itself, potentially rendering clock experiments insensitive to extraterrestrial sources of relativistic scalar waves. Back-action effects can also slow the propagation of scalar waves through the interstellar and intergalactic media, which can in turn substantially delay the arrival of scalar waves at Earth, compared with their gravitational-wave counterparts, and prevent multi-messenger astronomy on timescales comparable to the average human life expectancy. Our findings and conclusions apply to quadratic scalar-type interactions with the particular signs chosen in ref. ¹.

A real scalar field ϕ may interact non-gravitationally with standard-model fields via the following ϕ² interactions:

$${{{\mathcal{L}}}}=\frac{{\phi }^{2}}{{({{{\varLambda }}}_{\gamma }^{{\prime} })}^{2}}\frac{{F}_{\mu \nu }{F}^{\,{\mu \nu }}}{4}-\mathop{\sum}\limits_{\psi }\frac{{\phi }^{2}}{{({{{\varLambda }}}_{\psi }^{{\prime} })}^{2}}{m}_{\psi }\bar{\psi }\psi \,,$$

(1)

where the first term represents the interaction of the scalar field with the electromagnetic field tensor F, while the second term represents the interaction of the scalar field with the standard-model fermion fields ψ, m_ψ is the ‘standard’ mass of the fermion and $\bar{\psi }={\psi }^{{\dagger} }{\gamma }^{0}$ the Dirac adjoint. The indices μ and ν denote the space-time coordinates and run over 0,1,2,3; the superscript † denotes the conjugate transpose; 𝛾 denotes a Dirac matrix. Note that we employ the natural system of units ℏ = c = 1 throughout, where ℏ is the reduced Planck constant and c is the velocity of light in a vacuum. The interaction parameters Λ′_γ,ψ have dimensions of energy. Note that the signs in equation (1) are identical to those used in ref. ¹. In the presence of ordinary matter, the effective potential V_eff experienced by the scalar field differs from the bare potential V(ϕ) = m²ϕ²/2 assumed in ref. ¹, where m is the bare scalar mass, and instead reads:

$${V}_{{{{\rm{eff}}}}}(\phi )=\frac{{m}^{2}{\phi }^{2}}{2}+\mathop{\sum}\limits_{X=\gamma ,{\mathrm{e,N}}}\frac{{\rho }_{X}{\phi }^{2}}{{\left({{{\varLambda }}}_{X}^{{\prime} }\right)}^{2}}\,,$$

(2)

where we have assumed that the ordinary-matter fields are associated with non-relativistic atoms. Here, ρ_γ = ∣E∣²/2 ≈ −F_μνF^μν/4 is the Coulomb energy density of a non-relativistic nucleus, where E is the electric field, and ρ_e and ρ_N are the electron and nucleon mass energy densities, respectively. In the presence of ordinary matter, therefore, the effective scalar mass m_eff depends on the local ordinary-matter density ρ:

$${m}_{{{{\rm{eff}}}}}^{2}(\,\rho )={m}^{2}+\mathop{\sum}\limits_{X=\gamma ,{\mathrm{e,N}}}\frac{2{\rho }_{X}}{{\left({{{\varLambda }}}_{X}^{{\prime} }\right)}^{2}}\,.$$

(3)

We thus see that, in the presence of the ϕ² interactions (1), ordinary matter has a back action on ϕ. Analogous back-action effects in various models of scalar fields with ϕ² interactions have previously been explored in different contexts—see refs. ^2,3,4,5,6 and references therein.

The increase in the effective scalar mass in the presence of matter, according to equation (3), causes the scalar field to tend to be expelled from dense regions of matter, in a similar manner to the expulsion of magnetic fields from the interior of a superconductor due to the generation of an effective photon mass inside the superconductor⁷. Consider a relativistic scalar wave with particle energy ε propagating through a vacuum and incident onto a dense spherical body of radius R. Strong screening of the scalar wave near the surface of and inside the dense body occurs when ${m}_{{{{\rm{eff}}}}}^{2} > {\varepsilon }^{2}$ inside the dense body and qR ≳ 1, where $q=\sqrt{{m}_{{{{\rm{eff}}}}}^{2}-{\varepsilon }^{2}}$. In Fig. 1 we show the regions of parameter space for the scalar–photon interaction in equation (1) where a scalar wave is strongly screened near the surface of and inside Earth, by Earth’s atmosphere, and by a typical apparatus or satellite (see ‘Elemental compositions and mass–energy contributions of matter’ in the Methods for details of the assumed elemental compositions of these systems). The analysis presented in ref. ¹ focused on the limiting case of small wave dispersion, corresponding to a coherent burst. We note that there is an inconsistency between the analysis presented in the text of ref. ¹ and its implementation in fig. 3 of ref. ¹. The analysis in ref. ¹ explicitly assumed a small spread in the scalar particle energy, Δε ≪ ε, corresponding to a coherent burst. However, the burst duration of τ = 100 s assumed in fig. 3 of ref. ¹ implies, by the time–energy uncertainty relation, a minimum particle energy spread of Δε ≈ 1/τ ≈ 10⁻¹⁶ eV. This in turn implies that Δε ≳ ε, which corresponds to an incoherent burst, for the vast majority of scalar particle energies in fig. 3 of ref. ¹. Here we restrict ourselves to the simpler case of a coherent burst. The relevant scattering problem can be solved analytically in the simple limiting case of a monochromatic continuous burst (see ‘Scattering of a relativistic scalar wave by a dense spherical body’ in the Methods). In the low-energy limit pR ≪ 1, where $p=\sqrt{{\varepsilon }^{2}-{m}^{2}}$ is the scalar particle momentum in a vacuum, and when qR ≫ 1, the scalar-wave amplitude at height h ≪ R above the surface of the dense body is suppressed by the factor ≈ h/R + 1/(qR); in the limit q → ∞, one recovers the well-known result for hard-sphere scattering⁸. The leading anisotropic correction to the scalar-wave amplitude near the surface of the dense body is dipolar in nature and is suppressed compared with the dominant monopolar term by a factor of ${{{\mathcal{O}}}}(pR)$. Inside the dense body (or a dense spherical shell), the scalar-wave amplitude becomes exponentially suppressed at depths d ≳ 1/q. In Fig. 1 we show how the sensitivity of ground-based clocks to the scalar–photon interaction is degraded when the effect of screening of a scalar wave by Earth’s non-gaseous matter is taken into account, assuming a typical clock height of h ≈ 1 m above Earth’s surface. It is apparent that, when the effects of screening of a scalar wave by Earth’s non-gaseous matter and Earth’s atmosphere are collectively taken into account, ground-based clocks become insensitive to extraterrestrial scalar waves when ε ≳ 5 × 10⁻²¹ eV or Λ′_γ ≲ 10⁷–10⁸GeV, whereas space-based clocks become insensitive to scalar waves due to screening by the apparatus and satellite when Λ′_γ ≲ 10⁴–10⁵ GeV. We thus see that space-based clock experiments can provide a more powerful probe of relativistic scalar waves than ground-based clock experiments, even when the space-based clocks are intrinsically less precise. Note the sharp transition from the weak screening regime to the strong screening regime for ground-based clocks (solid cyan line) at around ε ≈ 10⁻²¹ eV in Fig. 1.

Fig. 1: Regions of parameter space for the quadratic interaction of ϕ with the photon.

The scalar field is strongly screened near the surface of and inside Earth (light grey region), by Earth’s atmosphere (dark grey region) and by a typical apparatus or satellite (black region). The solid and dashed cyan lines denote the estimated sensitivities of ground-based experiments using state-of-the-art optical clocks with and without screening by Earth’s non-gaseous matter taken into account, respectively. The dashed blue line denotes the estimated sensitivity of space-based experiments using microwave clocks (GPS, Global Positioning System). For the clock-based experiments, we have chosen burst parameters that yield a coherent burst over the entire relevant range of ε, with the dashed cyan and blue lines matched to their counterparts in fig. 3 of ref. ¹. The region above the solid red line denotes the region of parameter space where the time delay (δt) of a scalar wave relative to its gravitational-wave counterpart exceeds 100 yr due to the back-action effects of the interstellar and intergalactic media on the scalar wave. See the main text and Methods for more details.

The increase in the effective scalar mass in the presence of matter, according to equation (3), also slows the propagation of a scalar wave through matter. In the limiting case when the scalar wave remains relativistic, the speed of the scalar wave is given by:

$$v\approx 1-\frac{{m}_{{{{\rm{eff}}}}}^{2}}{2{\varepsilon }^{2}}\,,$$

(4)

and the time delay between the arrival of a relativistic scalar wave and its gravitational-wave counterpart (assumed to be travelling at the velocity of light) reads:

$$\updelta t\approx \frac{{m}_{{{{\rm{eff}}}}}^{2}L}{2{\varepsilon }^{2}}\,,$$

(5)

where L is the distance from the source to the detector, and we have assumed that the intervening medium is homogeneous. In Fig. 1 we show the region of parameter space for the scalar–photon interaction in equation (1) where the time delay exceeds 100 yr due to the back-action effects of the interstellar and intergalactic media on the scalar wave. We have assumed the favourable scenario where ${m}_{{{{\rm{eff}}}}}^{2}\gg {m}^{2}$; however, if ${m}_{{{{\rm{eff}}}}}^{2}\approx {m}^{2}$, then δt > 100 yr is possible even in the limit Λ′_γ → ∞ (that is, when the time delay occurs entirely due to the bare scalar mass). We have also assumed that the source is located outside our Galaxy and that the scalar wave favourably enters our Galaxy parallel to the normal of the thin Galactic Disk; in this case, the interstellar medium within our Galaxy provides the larger contribution to the time-delay effect, with a comparable contribution from the intergalactic medium only for sources located in the farthest reaches of the observable Universe. If the scalar wave instead propagates along a substantial fraction of the plane of the Galactic Disk (for example, if the source is located near the Galactic Centre), then the time-delay effect increases by at least an order of magnitude. We refer the reader to ‘Elemental compositions and mass–energy contributions of matter’ in the Methods for further details. We thus see that δt > 100 yr can occur in large regions of parameter space, which would prevent multi-messenger astronomy with clock-based experiments on human timescales and greatly exceed the value δt ≲ 10 h given in ref. ¹ for the extragalactic source GW170608. We also note that, depending on the mechanism responsible for producing relativistic scalar waves, the generation of an effective scalar mass in dense environments may strongly suppress the production of scalar waves in burst-type phenomena if ${m}_{{{{\rm{eff}}}}}^{2} > {\varepsilon }^{2}$ at the source.

In this Matters Arising we have highlighted how back-action effects can drastically affect the detection prospects of clock networks in searches for relativistic scalar waves with the quadratic interactions in equation (1). For ϕ² interactions of the type in equation (1) but with the signs reversed, the scalar field will tend to be anti-screened by dense regions of matter when $0 < {m}_{{{{\rm{eff}}}}}^{2} < {m}^{2}$. However, for the clock-based sensitivity curve parameters in fig. 3 of ref. ¹, ${m}_{{{{\rm{eff}}}}}^{2} < 0$ for the matter densities encountered in Earth’s interior and atmosphere, as well as apparatus and satellite components; in this case, the effective potential has an unstable maximum at ϕ = 0 and the classical scalar-field equation of motion admits a tachyonic solution, at least when higher-order terms in the bare potential, higher-dimensional operators and finite spatial extent of the dense region are neglected. For further discussion of such types of ‘opposite-sign’ interactions, including discussions of possible non-perturbative effects, we refer the reader to refs. ^{5,6,9,10,11,12}. In the case of linear scalar-type interactions, the interaction generates a source term in the classical equation of motion for the scalar field and hence does not give rise to back-action effects for a harmonic bare potential, when the classical scalar-field equation of motion is linear; however, back-action effects can arise for more complicated potentials that give rise to nonlinear term(s) in the scalar-field equation of motion (see, for example, refs. ^4,6). The back-action effects due to spin-unpolarized matter considered here do not apply to derivative-pseudoscalar-type interactions at leading order. Possible back-action effects due to spin-polarized matter, such as spin-polarized electrons in certain types of magnetic shielding and spin-polarized geoelectrons inside Earth, warrant separate investigation.

Methods

Elemental compositions and mass–energy contributions of matter

The main mass–energy contributions in an electrically neutral atom containing A nucleons and Z electrons are as follows:

$${M}_{{{{\rm{atom}}}}}\approx A{m}_{\mathrm{N}}+Z{m}_{\mathrm{e}}+\frac{{a}_{\mathrm{C}}Z(Z-1)}{{A}^{1/3}}+Z{a}_{\mathrm{p}}+(A-Z\,){a}_{\mathrm{n}}\,.$$

(6)

The first two terms in equation (6) correspond to the nucleon and electron mass energies, respectively. The third term corresponds to the energy associated with the electrostatic repulsion between protons in a spherical nucleus of uniform electric charge density with the coefficient a_C ≈ 3α/(5r₀) ≈ 0.7 MeV, where α ≈ 1/137 is the electromagnetic fine-structure constant and r₀ ≈ 1.2 fm is the internucleon separation parameter that is determined chiefly by the strong nuclear force. The final two terms in equation (6) correspond to the electromagnetic energies of the proton and neutron, respectively, with the coefficients a_p ≈ +0.63 MeV and a_n ≈ −0.13 MeV derived from the application of the Cottingham formula¹³ to electron–proton scattering¹⁴. We applied equation (6) to determine the fractional mass–energy contributions due to the electromagnetic, electron mass and nucleon mass components in various systems listed below, assuming the following elemental compositions for these systems. Note that the fractional mass–energy contribution due to the nucleon mass component for all discussed systems is K_N = 1.0.

Earth’s non-gaseous interior

We assumed that the elemental composition of Earth’s non-gaseous interior is a 1: 1: 1 ratio of ²⁴Mg¹⁶O, ²⁸Si¹⁶O₂ and ⁵⁶Fe by number. In this case, the fractional mass–energy contribution due to the electromagnetic component is K_γ = 1.9 × 10⁻³ and the contribution due to the electron mass component is K_e = 2.4 × 10⁻⁴. We treated Earth as a uniform sphere of radius R_⊕ ≈ 6,400 km with a density of ρ_⊕ ≈ 5.5 g cm⁻³.

Earth’s atmosphere

We treated Earth’s atmosphere as a 4: 1 ratio of ¹⁴N₂ and ¹⁶O₂ by number, with a constant density of ρ ≈ 10⁻³ g cm⁻³ extending out from Earth’s surface to an altitude of h ≈ 10 km. In this case, K_γ = 9.5 × 10⁻⁴ and K_e = 2.7 × 10⁻⁴.

Interstellar and intergalactic media

We assumed that the elemental composition of the interstellar and intergalactic media is a 3: 1 ratio of ¹H and ⁴He by mass, and neglected the effects of stellar nucleosynthesis. In this case, K_γ = 6.3 × 10⁻⁴ and K_e = 4.4 × 10⁻⁴. We treated the interstellar and intergalactic media as homogeneous media, with particle densities of ~10 cm⁻³ and ~10⁻⁷ cm⁻³, respectively. Our Galactic Disk has a radius of ≈ 5 × 10⁴ light yr and a thickness of ~1 × 10³ light yr. The distance from the Sun to the Galactic Centre is ≈ 25 × 10³ light yr. The present-day size of the observable Universe is ≈ 1 × 10¹¹ light yr, while the distances from Earth to the recently observed extragalactic sources GW170608 and GW170817 are ≈ 1 × 10⁹ light yr and ≈ 1 × 10⁸ light yr, respectively.

Apparatus and satellites

The details of the apparatus (including the sizes, materials and geometries of the apparatus components, as well as the details of the surrounding shielding and the laboratory or satellite environment) vary between individual experiments. Atomic clocks consist of low-density atomic vapours or individual ions contained within spherical or cylindrical vacuum chambers. Common materials for vacuum chamber walls include aluminium (density ≈ 3 g cm⁻³) and stainless steel (density ≈ 8 g cm⁻³). Ground-based clocks in laboratories are surrounded by reinforcing structures of buildings that are commonly made of reinforced concrete (density ≈ 2–3 g cm⁻³). For simplicity, we modelled a typical apparatus or satellite as a uniform sphere of radius R ≈ 10 cm with a density comparable to Earth’s average density and assumed the same elemental composition quoted above for Earth’s non-gaseous interior.

Scattering of a relativistic scalar wave by a dense spherical body

Here we solve the problem of a relativistic scalar wave propagating through vacuum and incident onto a dense spherical body of radius R when the scalar field has the effective potential (2). We found it convenient to work with the complex scalar field ${{\varPhi }}=(\phi +i\eta )/\sqrt{2}$, where ϕ and η are two real scalar fields, and then take the real part of Φ at the end of the calculation. In this case, the differential equation for the complex scalar field reads:

$$\frac{{\partial }^{2}{{\varPhi }}}{\partial {t}^{2}}-{{{{\boldsymbol{\nabla }}}}}^{2}{{{{\Phi}}}}+\left[{m}^{2}+\frac{2{\rho }_{X}(r)}{{\left({{{\varLambda }}}_{X}^{{\prime} }\right)}^{2}}\right]{{\varPhi }}(t,r,\theta ,\varphi )=0\,,$$

(7)

where θ is the polar angle and φ is the azimuthal angle; with an analogous equation for Φ^*. In equation (7) and below, summation over the mass–energy components X = γ, e, N is implicit. The radial density profile reads:

$${\rho }_{X}(r)=\left\{\begin{array}{ll}{\rho }_{X}\,,&0\le r < R\,;\\ 0\,,&r > R\,.\end{array}\right.$$

(8)

In the limit of a dispersionless scalar wave, Δε → 0, corresponding to a monochromatic continuous source that produces scalar particles with energy ε in a non-expanding Universe, the solutions of the differential equation (7) are stationary and can be factorized as follows:

$${{\varPhi }}(t,r,\theta ,\varphi )=\mathop{\sum }\limits_{l=0}^{\infty }\mathop{\sum }\limits_{n=-l}^{+l}{\chi }_{l}(r){Y}_{l}^{\,n}(\theta ,\varphi )\exp (-i\varepsilon t)\,,$$

(9)

where ${Y}_{l}^{\,n}$ are the spherical harmonic functions of degree l and order n, and the radial functions χ_l are solutions of the following ordinary differential equation:

$$\frac{{d}^{\,2}{\chi }_{l}}{d{r}^{2}}+\frac{2}{r}\frac{d{\chi }_{l}}{dr}+\left[{m}_{{{{\rm{eff}}}}}^{2}[\,\rho (r)]-{\varepsilon }^{2}-\frac{l(l+1)}{{r}^{2}}\right]{\chi }_{l}(r)=0\,.$$

(10)

Far from the source and far from the dense spherical body of interest, the generated scalar wave can be approximated by a plane wave, which we took to be travelling along the negative z axis towards the dense spherical body, centred at the origin of the spatial coordinates. As the dense body is spherically symmetric, the resulting scalar-field wave function must be independent of φ, and so only spherical harmonics with n = 0 can contribute to the scalar-field wave function in equation (9). The external solution is the sum of the incident plane wave and reflected spherical waves. With the aid of the plane-wave expansion, the external solution that remains finite as r → ∞ can be written in the following form:

$$\begin{array}{l}{{{\varPhi }}}_{{{{\rm{out}}}}}(t,r,\theta )=A\mathop{\sum }\limits_{l=0}^{\infty }(2l+1){i}^{l}{j}_{l}(\,pr){P}_{l}[\cos (\theta )]\exp (-i\varepsilon t)\\\qquad\qquad\quad\;+\,\mathop{\sum }\limits_{l=0}^{\infty }{B}_{l}{h}_{l}^{(1)}(\,pr){P}_{l}[\cos (\theta )]\exp (-i\varepsilon t)\,\end{array},$$

(11)

where A and B_l are parameters with dimensions of energy, P_l is the Legendre polynomial of order l, j_l is the spherical Bessel function of the first kind and order l, ${h}_{l}^{(1)}$ is the spherical Hankel function of the first kind and order l and $p=\sqrt{{\varepsilon }^{2}-{m}^{2}}$ is the scalar particle momentum in a vacuum. Strong screening of the scalar wave near the surface of and inside the dense body occurs when ${m}_{{{{\rm{eff}}}}}^{2} > {\varepsilon }^{2}$ inside the dense body and qR ≳ 1, where $q=\sqrt{{m}_{{{{\rm{eff}}}}}^{2}-{\varepsilon }^{2}}$. In this case, the internal solution, which remains finite as r → 0, reads as follows:

$${{{\varPhi }}}_{{{{\rm{in}}}}}(t,r,\theta )=\mathop{\sum }\limits_{l=0}^{\infty }{C}_{l}\,{y}_{-l-1}(-iqr){P}_{l}[\cos (\theta )]\exp (-i\varepsilon t)\,,$$

(12)

where C_l are parameters with dimensions of energy and y_l is the spherical Bessel function of the second kind and order l. The requirement for continuity of Φ and dΦ/dr at r = R, together with the orthogonality of Legendre polynomials with different values of l, fixes the coefficients B_l and C_l in equations (11) and (12) in terms of A:

$${B}_{l}=\frac{\begin{array}{l}A{i}^{l}(2l+1)\left\{\right.p{y}_{-l-1}(-iqR)[\,{j}_{l+1}(\,pR)-{j}_{l-1}(\,pR)]\\-iq\,{j}_{l}(\,pR)[\,{y}_{-l-2}(-iqR)-{y}_{-l}(-iqR)]\left.\right\}\end{array}}{\begin{array}{l}iq{h}_{l}^{(1)}(\,pR)[\,{y}_{-l-2}(-iqR)-{y}_{-l}(-iqR)]\\+p{y}_{-l-1}(-iqR)[{h}_{l-1}^{(1)}(pR)-{h}_{l+1}^{(1)}(\,pR)]\end{array}},$$

(13)

$${C}_{l}=\frac{2A{i}^{l}(2l+1)}{\begin{array}{l} p{R}^{2}\left\{q{h}_{l}^{(1)}(\,pR)[\,{y}_{-l-2}(-iqR)-{y}_{-l}(-iqR)]\right. \\ \left.-ip{y}_{-l-1}(-iqR)[{h}_{l-1}^{(1)}(\,pR)-{h}_{l+1}^{(1)}(\,pR)]\right\} \end{array}}.$$

(14)

In the low-energy limit pR ≪ 1, and when qR ≫ 1, the dominant contribution to the scalar-field wave function at small heights h ≪ R above the surface of the dense body is monopolar in nature (l = 0) and reads:

$${{{\varPhi }}}_{{{{\rm{out}}}}}(t,r)\approx A\left(\frac{h}{r}+\frac{1}{qr}\right)\exp (-i\varepsilon t)\,,$$

(15)

while the leading anisotropic correction to the scalar-field wave function at small heights is dipolar in nature (l = 1) and reads:

$$\updelta {{{\varPhi }}}_{{{{\rm{out}}}}}(t,r,\theta )\approx 3iApR\left(\frac{hR}{{r}^{2}}+\frac{R}{q{r}^{2}}\right)\cos (\theta )\exp (-i\varepsilon t)\,.$$

(16)

In the limit q → ∞, we recover the well-known result for hard-sphere scattering⁸. Note that Klein’s paradox for a complex scalar field does not apply in our present problem. On the one hand, a strongly repulsive electrostatic potential V causes the scalar particle momentum to be real inside the potential barrier when V > ε + m. On the other hand, when the effective scalar particle mass inside a dense body satisfies ${m}_{{{{\rm{eff}}}}}^{2} > {\varepsilon }^{2}$, the scalar particle momentum becomes imaginary, resulting in the attenuation of the scalar-wave amplitude inside the dense body. Meanwhile, the internal solution reads as follows, at leading order:

$${{{\varPhi }}}_{{{{\rm{in}}}}}(t,r)\approx 2\,A\exp (-qR)\exp (-i\varepsilon t)\,\,{{{\rm{for}}}}\,qr\ll 1\,,$$

(17)

$${{{\varPhi }}}_{{{{\rm{in}}}}}(t,r)\approx \frac{A\exp [-q(R-r)]}{qr}\exp (-i\varepsilon t)\,\,{{{\rm{for}}}}\,qr\gg 1\,.$$

(18)

We see that at depths d ≳ 1/q inside the dense body, the scalar-wave amplitude becomes exponentially suppressed. Likewise, the scalar-wave amplitude is also exponentially suppressed at depths d ≳ 1/q inside a dense spherical shell (such as Earth’s atmosphere). The exponential suppression (rather than abrupt vanishing) of the scalar-wave amplitude inside dense bodies when ${m}_{{{{\rm{eff}}}}}^{2} > {\varepsilon }^{2}$ is a consequence of the position–momentum uncertainty relation.