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Point-particle explanations: the case of gravitational waves

  • S.I. : Infinite Idealizations in Science
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Abstract

This paper explores the role of physically impossible idealizations in model-based explanation. We do this by examining the explanation of gravitational waves from distant stellar objects using models that contain point-particle idealizations. Like infinite idealizations in thermodynamics, biology and economics, the point-particle idealization in general relativity is physically impossible. What makes this case interesting is that there are two very different kinds of models used for predicting the same gravitational wave phenomena, post-Newtonian models and effective field theory models. The paper contends that post-Newtonian models are explanatory while effective field theory models are not, because only in the former can we eliminate the physically impossible point-particle idealization. This suggests that, in some areas of science at least, models invoking ineliminable infinite idealizations cannot have explanatory power.

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Fig. 1

(Source: NASA)

Fig. 2
Fig. 3

(adapted from Goldberger 2007, p. 16)

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Acknowledgements

I am grateful to Peter Zimmerman for Fig. 2, for research support, and for many patient discussions about gravitational waves. Thanks also to Eric Poisson for helpful conversations, and to two anonymous referees for valuable comments. This work was supported financially by a grant from the Social Sciences and Humanities Research Council of Canada (Grant No. 430-2012-0555).

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Appendices

Appendix A: PN derivation of gravitational waves

This appendix provides an overview of post-Newtonian derivations of gravitational waves. The starting point for post-Newtonian models are the fundamental equations of motion in GR, the EFEs. When the harmonic gauge condition

$$\begin{aligned} \partial _\alpha p^{\alpha \beta }=0 \end{aligned}$$
(2)

has been imposed, the restricted (so-called “relaxed”) EFEs can be written

$$\begin{aligned} \square p^{\alpha \beta }=\frac{16\pi G}{c^{4}}\tau ^{\alpha \beta }, \end{aligned}$$
(3)

where \(\tau ^{\alpha \beta }\) is defined as the sum of matter sources \(T^{\alpha \beta }\) and sources that contain only the field \(\Lambda ^{\alpha \beta }\),

$$\begin{aligned} \tau ^{\alpha \beta }\equiv \left( {-g} \right) T^{\alpha \beta }+\frac{c^{4}}{16\pi G}{\Lambda }^{\alpha \beta }. \end{aligned}$$
(4)

The tensor field \(p^{\alpha \beta }\) is defined as

$$\begin{aligned} p^{\alpha \beta }\equiv \sqrt{-g}g^{\alpha \beta }-\eta ^{\alpha \beta }, \end{aligned}$$
(5)

where \(g^{\alpha \beta }\) is the full spacetime metric and \(\eta ^{\alpha \beta }\) is the Minkowski metric.

In order to write the relaxed EFEs in the form (3), an assumption was made in (4) that \(T^{\alpha \beta }\) has compact support (and is conserved), or in other words that the system has only a compact region of spacetime with non-zero stress-energy values. This assumption is physically realistic in a two-body inspiral system, because the two compact bodies are the only non-zero regions.

The \(\hbox {PM}_{\mathrm{far}}\) approximation aims to construct valid solutions of the relaxed EFEs (3) in the far zone. Here, spacetime is void of matter and (3) can be written

$$\begin{aligned} p^{\alpha \beta }={\Lambda }^{\alpha \beta }. \end{aligned}$$
(6)

The approximation assumes that we can expand the field in powers of a small parameter \({\upvarepsilon }\sim v^{2}/c^{2}\sim 1/c^{2}\), so that

$$\begin{aligned} p^{\alpha \beta }=\varepsilon p_{1}^{\alpha \beta } +\varepsilon ^{2}p_2^{\alpha \beta } +\cdots =\mathop \sum \nolimits _{m=1}^{\infty } \varepsilon ^{m}p_{m}^{\alpha \beta } . \end{aligned}$$
(7)

Because \({\Lambda }^{\alpha \beta }\) is a nonlinear operator,

$$\begin{aligned} {\Lambda }^{\alpha \beta }\left[ {\varepsilon p_{1} +\varepsilon ^{2}p_{2} } \right] \ne \varepsilon {\Lambda }^{\alpha \beta }\left[ {p_{1} } \right] +\varepsilon ^{2}{\Lambda }^{\alpha \beta }\left[ {p_{2} } \right] , \end{aligned}$$
(8)

its expansion in the \(m{\mathrm{th}}\) power of \(\upvarepsilon \) involves all orders less than m,

$$\begin{aligned} \square p_{m}^{\alpha \beta } ={\Lambda }_m^{\alpha \beta } \left[ {p_1 ,\cdots ,p_{m-1} } \right] . \end{aligned}$$
(9)

The solution can be obtained as a sum of multipole coefficients, and the details are complex. Qualitatively, the solution requires imposition of the harmonic gauge condition and appropriate boundary conditions at infinity. It also uses a retarded propagator, and this propagator requires \({\Lambda }_{n}^{\alpha \beta }\) be determined for all \(r > 0\). However, the solution is not valid in the near zone, and indeed (9) becomes singular as \(r\rightarrow 0\). The solution must be regularized to eliminate this divergence. The regularization procedure used, analytic continuation, is part of the standard mathematical toolkit of contemporary physics.

The \(\hbox {PN}_{\mathrm{near}}\) approximation aims to construct solutions of the relaxed EFEs (3) that are valid in the near zone. Again, the starting point is an expansion of the field and the matter source (since the near zone contains the compact bodies) in powers of a small parameter \(\upvarepsilon \). The expansion of the field is of the form

$$\begin{aligned} p^{\alpha \beta }=\mathop \sum \nolimits _{n=1}^{\infty } \varepsilon ^{n}p_{n}^{\alpha \beta } , \end{aligned}$$
(10)

and the expansion of the source is

$$\begin{aligned} \tau ^{\alpha \beta }=\mathop \sum \nolimits _{n=-1}^{\infty } \varepsilon ^{n}\tau _{n}^{\alpha \beta } , \end{aligned}$$
(11)

where \(\tau ^{\alpha \beta }\) is given by (4).

The first-order solution takes the field \({\Lambda }^{\alpha \beta }=0\) so that (3) becomes

$$\begin{aligned} \Delta p_{1}^{\alpha \beta } =\left( {-g} \right) T^{\alpha \beta }. \end{aligned}$$
(12)

From (10), (11) and (12) the recursion relation

$$\begin{aligned} {\Delta }p_{n+1}^{\alpha \beta } =16\pi G\tau _{n-1}^{\alpha \beta } +\partial _{0}^{2} p_{n}^{\alpha \beta } \end{aligned}$$
(13)

follows. As in the \(\hbox {PM}_{\mathrm{far}}\) approximation, the expansions of \(p^{\alpha \beta }\) are singular and the integrals required to solve (13) for any given n diverge. Again, regularization techniques have been successful at eliminating these divergences.

It is here that the point particle idealization has been incorporated into the \(\hbox {PN}_{\mathrm{near}}\) model. Recall that in order to write the relaxed EFEs in the form (3), the assumption is made in \(T^{\alpha \beta }\) has compact support. In order to construct the \(\hbox {PN}_{\mathrm{near}}\) model, however, a further assumption is made: that \(T^{\alpha \beta }\) in (4) is nonzero only at two points, corresponding to the centres of mass of the two compact bodies. This is expressed as

$$\begin{aligned} T^{\alpha \beta }=m_1 \cdot u_{1}^\alpha u_{1}^{\beta } \frac{\delta ^{4}\left( {x-z_1 \left( \tau \right) } \right) }{\sqrt{-g}}+m_2 \cdot u_{2}^{\alpha } u_{2}^{\beta } \frac{\delta ^{4}\left( {x-z_{2} \left( \tau \right) } \right) }{\sqrt{-g}}, \end{aligned}$$
(14)

where \(u_{1}^{\alpha } \) is the four-velocity of one compact object, \(u_{2}^{\alpha } \) is the four-velocity of the other, and \(z_{i} \left( \tau \right) \) is the worldline of each particle. It is important to keep in mind that the point-particle idealization (14) not only applies to the first-order solution (12); rather, it is used at all orders of the \(\hbox {PN}_{\mathrm{near}}\) approximation.

In order to construct a uniformly valid solution, the post-Newtonian model relies on the fact that the \(\hbox {PM}_{\mathrm{far}}\) and \(\hbox {PN}_{\mathrm{near}}\) approximations can be matched in the buffer zone. Matched asymptotic expansions connect the near zone and far zone models by using the multipole coefficients of the source, obtained from the near-zone \(\hbox {PN}_{\mathrm{near}}\) multipole expansion, to fix the multipole coefficients of the \(\hbox {PM}_{\mathrm{far}}\) field. A matched expansion of order n in the small parameter \(\upvarepsilon \sim v^{2}/c^{2}\) is called the \(n\hbox {PN}\) post-Newtonian model. For example, the 2.5PN model is of the order \({\upvarepsilon }^{2.5}\sim v^{5}/c^{5}\).

Appendix B: EFT derivation of gravitational waves

This appendix provides an overview of EFT derivations of gravitational waves. We start by assuming that the target contains a gravitational field and two point particles. The degrees of freedom are thus (i) the metric field \(g_{\alpha \beta } ( x)\), and (ii) the worldlines of the bodies \(\gamma :\lambda \mapsto x^{\mu }\left( \lambda \right) \), where \(\lambda \) is arbitrary affine parameter running along the trajectories. The symmetries are (i) coordinate invariance \(x^{\alpha }\rightarrow x^{\alpha }\left( x \right) \), and (ii) world line reparameterization invariance \(\lambda \rightarrow {\bar{\lambda }} \left( \lambda \right) \) (Goldberger 2007, p. 20). We assume the inspiralling bodies are not spinning and are uncharged. A nonzero spin or charge would imply additional degrees of freedom and different symmetry constraints in the model, but these additional complications are not relevant to the assessment of the explanatory merits of EFT models in gravitational physics. The next step in the construction of an EFT model is to write the fundamental equations of the global theory of GR in terms of a general effective action,

$$\begin{aligned} S_{eff} \left[ {x^{\alpha },g_{\alpha \beta } } \right] =S_{EH} \left[ {g_{\alpha \beta } } \right] +S_{pp} \left[ {x^{\alpha },g_{\alpha \beta } } \right] , \end{aligned}$$
(15)

where \(x^{\upalpha }\) is the particle coordinate, \(S_{EH}\) is the Einstein–Hilbert action, and \(S_{pp}\) is the point particle action. The Einstein–Hilbert action is given by

$$\begin{aligned} S_{EH} \left[ {g_{\alpha \beta } } \right] =\frac{c^{3}}{16\pi G}\smallint R\left( x \right) \sqrt{-g} d^{4}x, \end{aligned}$$
(16)

where R is the curvature scalar defined in terms of the Ricci tensor as \(g^{\alpha \beta }R_{\alpha \beta }\). The Einstein–Hilbert action, plus the harmonic gauge condition, encodes information from the relaxed EFEs. The point-particle action is given by

$$\begin{aligned} S_{pp} \left[ {x^{\alpha },g_{\alpha \beta } } \right] =-m\smallint d\tau +a_\mathcal{E} \smallint \mathcal{E}_{{\upalpha \upbeta }} \mathcal{E}^{{\upalpha \upbeta }}d{\uptau }+\hbox {a}_\mathcal{B} \smallint \mathcal{B}_{{\upalpha \upbeta }} \mathcal{B}^{{\upalpha \upbeta }}d{\uptau }+\cdots , \end{aligned}$$
(17)

where \(\uptau \) is the proper time along the world line given by \(d\uptau ^{2}=g_{\alpha \beta } dx^{\alpha }dx^{\beta }\). The first term \(-m\smallint d{\uptau }\) simply describes the geodesic motion of a point particle. The point particle action also needs to encode higher multipole moments that are supposed to model the effects of the fact that these are actually extended bodies. The \(\mathcal{E}\) and \(\mathcal{B}\) terms are constructed from the Riemann tensor and are the lowest-order terms inducing non-geodesic motion in the spacetime described by the Einstein–Hilbert action. These terms, along with an infinite number of higher-order terms, are supposed to model the effects of the internal structure of the body in the EFT. In short, \(\mathcal{S}_{pp}\) models finite-size effects of what is actually an extended body following a geodesic as a point particle undergoing non-geodesic motion. The next step is to take advantage of the separation of scales to construct the EFT model and predict observable phenomena of interest, namely gravitational waves.

Recall that the system has three distinct and well-defined scales: \(r \sim r_{s}\), the radius of a body; \(r \sim d\), the distance between the orbiting bodies; and \(r\gg \uplambda \), the wavelength of the emitted radiation (see Fig. 2). In a nutshell, the strategy is to construct an effective action for an effective field at each scale and match coefficients at the boundaries to integrate out the shorter-range physics. The process is illustrated schematically in Fig. 3.

The smallest scale is that of the radius of the body \(r_{s}\). The actual system consists of a gravitational field and two compact extended bodies with internal structure; the state and dynamics of this system is determined by the EFEs and the equation of motion of the compact bodies. At the compact body scale, we construct an idealized EFT model containing two point particles with multipole moments. The coupling coefficients \(\hbox {a}_\mathcal{E} \) and \(\hbox {a}_\mathcal{B} \) are used to encode information about the internal structure at the smallest scale (16). This is analogous to the UV cut-off for EFTs in particle physics. This EFT works all the way up to the scale of the orbital radius d using renormalization group (RG) rescaling techniques.

At the orbital scale d, a second EFT is constructed as follows. Assume that the gravitational field metric \(g_{\alpha \beta } \) can be described by a flat space-time metric \(\eta _{\alpha \beta } \) and a small metric perturbation \(h_{\alpha \beta }^{F} \), where \(\left| h \right| \ll 1\),

$$\begin{aligned} g_{\alpha \beta } =\eta _{\alpha \beta } +h_{\alpha \beta }^{F} . \end{aligned}$$
(18)

To integrate out the orbital scale physics, split the gravitational field metric,

$$\begin{aligned} h_{\alpha \beta }^{F}=H_{\alpha \beta } +h_{\alpha \beta } , \end{aligned}$$
(19)

where \(H_{\alpha \beta } \) represents short-wavelength potential modes that mediate the forces in the bound system (to a first approximation this is simply the Newtonian potential), and \(h_{\alpha \beta } \) represents long-wavelength radiation modes that propagate out as gravitational waves. The goal at the orbital scale is to integrate out the short wavelength H modes by computing the path integral

$$\begin{aligned} e^{iS_{NR} \left[ {h,x_a } \right] }=\int DH_{k,\alpha \beta } \left( {x^{0}} \right) e^{i\left( {S_{EH} \left[ {h\,+\,H} \right] \,+\,S_{pp} \left[ {h\,+\,H,x_a } \right] } \right) } \end{aligned}$$
(20)

What we are doing here is matching the long-distance EFT (the radiation modes) to the orbital scale particle worldlines to obtain an effective action that depends only on h and \(x_{a}\). Computing the path integral involves summing over all the diagrams with internal H lines to a given order in the expansion parameter v. In other words, what is being done here is approximating the full unknown effective action, the right-hand side in equation (20), as an effective action dependent only on the radiation modes h and a new point particle world line \(x_{a}\) that now represents the two-particle bound state.

Fig. 4
figure 4

(Goldberger 2007, p. 37)

A \(2\mathrm{nd}\) -order Feynman diagram in the EFT model

Figure 4 illustrates the Feynman diagram for the second-order terms in the EFT. The double line to the right of the horizontal arrow indicates a vertex at scale \(r \approx d\) or above, an EFT in which the two point particles are treated as a single effective bound state. Terms to the left of the arrow indicate vertices in the EFT at \(r \approx r_{s}\) scale (see Fig. 3). These include terms from the multipole expansion, point-particle coupling terms, and graviton couplings to the point particles.

Our ultimate goal is to predict gravitational wave observables in the far zone. To do this, write

$$\begin{aligned} e^{iS_{eff} \left[ {x_a } \right] }=\int Dh_{\alpha \beta } \left( x \right) e^{iS_{NR} \left[ {h,x_a } \right] } \end{aligned}$$
(21)

Next, compute diagrams with first-order radiative corrections (one-loop Feynman diagrams), using a prescription from QFT that the real part of \(S_{\textit{eff}} \left[ {x_a } \right] \) yields coupled equations of motion of the two bodies while the imaginary part yields the total power radiated by gravitational waves (Fig. 5).

Fig. 5
figure 5

(Goldberger 2007, p. 38)

One-loop Feynman diagram in the far zone \(\hbox {r}>> \lambda \)

The heuristic argument here is that, in analogy with particle physics, the imaginary part of the effective action corresponds to the rate of emission of graviton particles in the EFT model. The rate of change of the binding energy over time is equated with energy of the graviton emission, so the far-zone power can be calculated. Finally, the latter gives the observable time-dependent phase. This prescription is based on what we have seen to be a somewhat tenuous analogy with QFT. EFT models, although they make use of calculational machinery borrowed from QFT, are non-quantum models of purely classical point particles in GR (Section 4). EFT models are inconsistent with classical (causal) physics because they rely on the extracting the imaginary part from the instantaneous power. However, the prescription does work to yield observables we are interested in, such as the time evolution of the phase of gravitational waves detected by LIGO.

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Wayne, A. Point-particle explanations: the case of gravitational waves. Synthese 196, 1809–1829 (2019). https://doi.org/10.1007/s11229-017-1638-y

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