Abstract
Einstein claimed that the fundamental dynamical insight of special relativity was the equivalence of mass and energy. I disagree. Not only are mass and energy not equivalent (whatever exactly that means) but talk of such equivalence obscures the real dynamical insight of special relativity, which concerns the nature of 4-forces and interactions more generally. In this paper I present and defend a new ontology of special relativistic particle dynamics that makes this insight perspicuous and I explain how alleged cases of mass–energy conversion can be accommodated within that ontology.
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Notes
In the words of one mathematician–historian, the energy–mass relationship brought about by special relativity constitutes “the most striking example of unification that has been effected in the present century”. Whittaker 1958, p. 96.
See also French (1968, pp. 16–20), Sternheim and Kane (1991, p. 493), and Torretti (1983, pp. 306–307, n. 13), who equate energy and mass either conceptually or metaphysically. This view is echoed in the philosophical literature by, e.g., Butterfield (1984, p. 104), Earman (1989, p. 18), and Teller (1991, p. 382).
Flores (2005) identifies six interpretations of Einstein’s equation represented in the literature on special relativity, tentatively endorsing the view that mass and energy are inequivalent physical properties that can—but need not—be converted into each other. I think that both his positive argument and his grounds for rejecting several of the competing interpretations rest on conceptual misunderstandings, but will confine my commentary to footnotes. The view developed here is not among the six interpretations Flores canvasses.
This paper is restricted to the special relativistic dynamics of (spinless) particles. One might feel that a clear understanding of energy–mass ‘equivalence’ can’t be adequately addressed independently of general relativity or broader field-theoretic considerations. See Lehmkuhl (2011, p. 454, n. 1) for an expression of this attitude. But there are good reasons to think the energy–mass relationship can be investigated in an illuminating way in the limited context of special relativistic particle dynamics, and in fact that such a restricted context is the appropriate starting point for an inquiry into the relationship between energy and mass. First, the original association of mass with energy, articulated in Einstein (1905), draws solely upon special relativistic particle dynamics. There is thus a straightforward conceptual question about how such an equivalence is to be understood that predates any general relativistic or field-theoretic considerations. Einstein thought the identification of mass and energy was already grounded in the comparatively simple relativistic theory of point particle dynamics. Second, the philosophical challenges raised to the received view discussed below resurface in the broader context of general relativity. As noted by Hoefer (2000), the conceptual status of energy and mass is, if anything, more problematic in that context. It is thus good philosophical methodology to start with the simpler case in the hopes that a clear understanding of special relativistic particle dynamics might point the way towards understanding more elaborate contexts. Whether the interpretation developed here can be suitably extended to classical fields, including general relativity, is an open question.
Here ‘mass’ is being used in the modern sense of the property that determines how a particle resists changes to its state of motion (see, e.g., Moore (2013, p. 2)). Newton (1999) famously thought of mass differently—as in some sense a measure of a body’s ‘quantity of matter’. Jammer (1997) discusses the history of this conceptual transformation, which has its origins in Euler’s work in the eighteenth century. For a philosophical justification of this reconceptualization, see Cartwright (1975) and Lange (2001). The quantity represented by m is sometimes misleadingly called a particle’s ‘rest mass’, although I follow Lange (2001, (2002), Moore (2013), Rindler (1991), Wald (1984), and others in treating it as an intrinsic, frame-independent property of a particle. That Flores (2005) fails to appreciate this point leads to a misunderstanding—and misplaced criticism—of Lange’s view. (See footnote 19 below.) Mass should be distinguished from a particle’s so-called ‘relativistic mass’, given by \(m_{\!_{{R}}}=\gamma _um\), for which rest mass is the special case corresponding to \(u=0\).
The kinetic energy (T) of a free particle in special relativity is given by \(T=(\gamma _u - 1)mc^2\) so that \(E=\gamma _umc^2=E_0 + T\). In the inelastic collision, the kinetic energy lost is \(2(\gamma _u - 1)mc^2\), which corresponds precisely to the rest energy (and hence mass) gained: \(Mc^2 - 2mc^2 = 2\gamma _umc^2 - 2mc^2 = 2(\gamma _u - 1)mc^2\).
There are subtle issues concerning how to make precise sense of this notion of equivalence and of the role of symmetries in theory interpretation in general, but that is a topic for another paper. That the physical equivalence of all inertial frames lies at the foundation of special relativity is explicit in Rindler (1991, pp. 1, 7, 50), and Lange’s invocation of the notion of Lorentz-invariance is standard in both the physics and philosophy literatures.
Throughout this paper I make use of the distinction between a fundamental physical property and a derivative or non-fundamental physical property. However those metaphysical notions are made precise, it is this author’s opinion that the distinction is a substantive one, and moreover one that is implicit in physical practice.
The interpretive constraint that the mathematical objects and equations characterizing a theory’s fundamental ontology satisfy certain sorts of symmetry requirements—in this case, requirements associated with Lorentz transformations—is not without its philosophical puzzles. See, e.g., Dasgupta (2016) for a critical discussion of how such symmetry demands might be justified. However, this paper is not intended as an exploration of this issue, and I will take the overall cogency of this interpretive constraint for granted in what follows. Were this interpretive principle to be jettisoned, it would have consequences for the physical content of special relativity that go far beyond the relationship between energy and mass.
Something similar is also true in classical dynamics, as speed isn’t Galilean invariant either, but in that context no one alleges an equivalence between mass and energy.
I interpret Lange’s characterization of mass as a “real property” to be that it is a fundamental property. See Lange (2001, p. 227). When indexed to a frame, energy is a perfectly real physical property (just like speed). The issue is that it’s not fundamental.
See Rindler (1991, p. 74).
Again, it’s worth emphasizing that Lange’s puzzle raises no concerns about the consistency of \(E=\gamma mc^2\) (or \(E_0=mc^2\)) within the mathematical formalism or about its use in empirically successful applications. Rather, the issue here is a conceptual one about how the theory’s ontology is to be understood.
See footnote 9.
Bondi and Spurgin (1987) use this example to argue that energy has mass. Read in a straightforward way, this claim is confused. Mass is a property of inertial resistance: the mass of an object is a measure of how much that object resists changes to inertial motion in light of impressed forces. So for energy to have mass, energy must be the sort of thing to which impressed forces can be applied, and energy simply isn’t that sort of thing.
One worry with Lange’s account, raised by Flores (2005), is that the underlying ontological picture arises from inconsistent application of the relevant interpretive principles. Lange’s use of ‘mass’ to designate an object’s rest mass (Lange 2001, p. 225) appears to implicitly privilege a particular frame—namely, the frame in which the object is (instantaneously) at rest. Like length, then, it is a perfectly real physical property, but on the surface ought to be no more fundamental than particle mass in any other inertial frame—which is to say, not fundamental. However, it is clear from Lange’s discussion of rest mass and relativistic mass (Lange 2001, pp. 226–7) that rest mass is understood merely as the \(u = 0\) mathematical limit of relativistic mass. As previously noted (see footnote 7 above) the quantity m itself represents an ontologically fundamental and frame-invariant particle property for Lange, which also happens to be equal to the particle’s relativistic mass in the frame in which the particle is at rest. But that equality is not constitutive of the property. In this way, contra Flores (2005), Lange’s proposed ontology preserves the special relativistic maxim that all inertial frames are physically equivalent.
This terminology follows Freund (2008, p. 247).
Many of the core classical principles of relativistic particle dynamics are strikingly well-confirmed by the very reactions at issue. See French (1968) for a discussion of some of the experimental evidence from particle physics for the basic principles of special relativistic particle dynamics.
This puzzle—the puzzle of understanding the nature of composite mass—is not unique to macroscopic objects or objects whose constituents only interact via collisions (as one generally assumes for gases). For all but the most fundamental particles, mass seems to be partly constituted by ‘internal’ energy, whether in the form of kinetic energy or some form of binding or potential energy.
The approach here is in the spirit of Maudlin (2018).
Where there is little risk of confusion I will be lax about the distinction between mathematical representation and physical feature represented.
Recall that the metric structure divides the spacetime at any point (call it the ‘origin point’) into distinct regions, which can be characterized by the vectors (4-vectors) at that point. The timelike region consists of those events whose displacement vectors from the origin point have negative magnitude. All such events are said to be timelike separated from the origin point and any 4-vector that points from the origin point to a timelike separated event is said to be a timelike 4-vector. The lightlike (spacelike) region is the set of events whose displacement 4-vectors from the origin point are of null (positive) magnitude. This definition extends to 4-vectors as above. A curve through the manifold is said to be timelike (null, spacelike) if the tangent 4-vector at each point along it is timelike (null, spacelike). Note additionally that photons play a rather curious role in textbook presentations of special relativity, with some authors smoothly sliding between initial talk of light rays or signals to later talk of photons and other authors acknowledging that photons are quantum mechanical in nature and thus not a part of special relativistic dynamics proper. (Compare Rindler (1991, pp. 84–86) and Faraoni (2013, p. 173).) The latter view is how special relativity was originally understood: in the 1920s, well after the acceptance of special relativity, Bohr and others continued to express doubts about the existence of photons. See, e.g., Pais (1991, pp. 230ff) and Murdoch (1990, pp. 19ff). The presentation here is deliberately silent on the behavior of photons in Minkowski spacetime. Indeed, as Rindler (1991, p. 8) notes, light itself is not essential to the spacetime structure of special relativity. Maudlin (2012, ch. 4) makes a related claim, developing the kinematics in a way that makes no mention either of inertial frames or of light’s constitution and ‘speed’.
Some authors use bolded capital letters (e.g., \(\mathbf {A}\)) for 4-vectors and bolded lower-case letters (e.g., \(\mathbf {a}\)) for spatial 3-vectors, indicating the components in an inertial frame S by writing \(\mathbf {A} {\mathop {\rightarrow }\limits ^{S}} (A^0,A^1,A^2,A^3)\) or \(\mathbf {a} {\mathop {\rightarrow }\limits ^{S}} (a_1,a_2,a_3)\). I will occasionally use this notation, but more often will refer to 4-vectors by writing things like \(A^\mu \), understood to represent the components of the 4-vector \(\mathbf {A}\) in some arbitrary inertial frame.
See Rindler (1991, pp. 70–73, 90–92). For an isolated n-body system that interacts only locally (i.e., effectively via collisions), a more general principle holds that the net 4-momentum of the system remains constant, in the sense that the components of the total system 4-momentum do not change in any given inertial frame. Even though the specific \(\mathbf {P}_{(i)}\) 4-vectors that are elements of this sum are frame-dependent because the summation is taken at an instant, in these circumstances Rindler (1991, pp. 78–79) shows that the total system 4-momentum \(\mathbf {P}_\text {sys}\) is a well-defined 4-vector. It follows from this more general principle that relativistic energy and relativistic (spatial) momentum are both independently conserved in these circumstances, even though the values of those quantities are frame-dependent. For a more general discussion not restricted to inertial observers, see Gourgoulhon (2010, pp. 288–291). When an n-body system can’t be treated as isolated, such as when various sorts of field-theoretic considerations are included, then there isn’t generally a well-defined total 4-momentum associated with the system. However, the puzzles developed above concern physical systems for which these field-theoretic considerations can be ignored. Recall (see footnote 6) that the guiding methodology here is to tease out the significance of \(E_0=mc^2\) in the simplest dynamical cases and to leave more complicated physical situations for subsequent work.
There is a third dynamical principle—conservation of the angular momentum 4-tensor along any particle’s world line—but it won’t play a role in the argument that follows.
Roughly speaking, a dynamical state of a particle is the collection of fundamental properties relevant to the types of interactions it generates and the way in which it responds to different types of interactions. A particle may possess properties that are relevant to whether it experiences a particular impressed force or interaction, but which are not part of its dynamical state—e.g., its position in space or spacetime. A non-interacting object moving inertially is constantly changing its position and yet its dynamical state remains constant.
See, e.g., José and Saletan (1998, pp. 13–14). I do not mean to suggest that this picture of classical ontology is uncontroversial. Among the issues raised in recent years, some philosophers have considered whether velocity ought to be taken as an ontologically primitive property in its own right (see Arntzenius (2000); Carroll (2002); Meyer (2003); Smith (2003); Lange (2005); Easwaran (2014); McCoy (2018)), whereas others have wondered whether attributions of mass ought to designate fundamental properties of particles (see Dasgupta (2013); Martens (2019a, (2019b)). And Butterfield (2006) has argued against understanding classical dynamics (or indeed any physical theory) in terms of properties defined at points of space or spacetime. There is also a long tradition within the metaphysics of physics going back at least to Mach and Hertz puzzling over just what sort of thing a force really is. For more recent discussion, see, e.g., Ellis (1976), Bigelow et al. (1988), and Wilson (2007). Some of the issues raised in these literatures will be relevant to the specific way I frame the ontological proposal sketched below, but they do not affect the central argument and in the context of this paper I’ve had to set them aside.
French (1968, p. 4).
See, e.g., French (1968, pp. 21–23).
There is some precedent in the physics literature for thinking of m this way. See, e.g., Gourgoulhon (2010, p. 272). I do not think anything of deep philosophical substance hangs on this point: both \(\Vert P^\mu \Vert \) and m are understood as representing one and the same primitive physical property. However, within the special relativistic formalism the quantity m is most salient in frame-dependent contexts, where it plays a central role in dynamical equations governing derivative ontology. Since I think the focus on the frame-dependent equations for special relativistic particle dynamics has been the source of much ontological confusion, I prefer to do without m and use \(\Vert P^\mu \Vert \) as the representation of particle mass. It’s worth emphasizing that the property represented by m (or \(\Vert P^\mu \Vert \)) takes on a rather different character in relativistic dynamics than it does in classical dynamics. Within an inertial frame a special relativistic particle, unlike a classical one, exhibits different amounts of inertial resistance to impressed 3-forces in different directions. The resistance to being accelerated in a direction parallel to a particle’s instantaneous spatial velocity—its ‘longitudinal mass’—is different from its resistance to being accelerated in a direction perpendicular to its instantaneous spatial velocity—its ‘transverse mass’. Indeed, the spatial acceleration of a body in response to an impressed 3-force is generally not even in the direction of the 3-force itself. The frame-dependent dynamical equation governing particle motion is \(\mathbf {f} = \gamma _um\mathbf {a} + \frac{{d}}{{ d}t}[\gamma _um]\mathbf {u}\), and so the magnitude and direction of a particle’s acceleration in response to an impressed force depends on properties other than just m and the direction in which the force is applied (e.g., its spatial velocity in a frame). The relationship \(\mathbf {f}=m\mathbf {a}\) holds only in the instantaneous rest frame of the particle. Freund (2008, pp. 195–198) describes a very simple example where a constant 3-force applied to a particle solely in the x-direction generates a velocity-dependent deceleration in the y-direction.
Like \(\Vert P^\mu \Vert \) and m, \(P^\mu \) and \(U^\mu \) are equally good representations of a particle’s (4-momentum) orientation.
This sort of discourse naturally suggests an underlying substantivalism of some form or other. I embrace this, but wish to remain agnostic here regarding whether such a metaphysical commitment is necessary for the ontology I propose.
For more on the role of indexicals in this sort of context, see Maudlin (1993, pp. 189–191). There are, of course, easily expressible facts about relative differences in 4-momentum orientations between particles, as \(\mathbf {P}_1\cdot \mathbf {P}_2 = \Vert \mathbf {P}_1 \Vert \Vert \mathbf {P}_2 \Vert \gamma (v)\) holds invariantly (where v is the relative velocity between the two particles). See Rindler (1991, p. 76). If \(\mathbf {P}_1\cdot \mathbf {P}_2 = \Vert \mathbf {P}_1 \Vert \Vert \mathbf {P}_2 \Vert \) then both 4-momenta have the same orientation, and if \(\mathbf {P}_1\cdot \mathbf {P}_2 \ne \Vert \mathbf {P}_1 \Vert \Vert \mathbf {P}_2 \Vert \) then the extent to which \(\mathbf {P}_1\cdot \mathbf {P}_2 / \Vert \mathbf {P}_1 \Vert \Vert \mathbf {P}_2 \Vert > 1\) provides a measure of how their orientations differ.
It’s of course true that the magnitude of the particle’s acceleration is non-zero during the interval \(\varDelta t\)—and, being Lorentz-invariant, this points to a genuine physical difference while the force is being applied—although, unlike Newtonian dynamics, in special relativity the non-zero magnitude of that acceleration is frame-dependent.
In this instance we can see particularly clearly how the same issue arises for Newtonian dynamics, as the standard properties one might be inclined to appeal to aren’t Galilean-invariant either and thus aren’t candidates for being fundamental properties within the context of Newtonian theory (or don’t differ between the particles). The motivations developed here apply equally to an analogous ontological picture of classical dynamics—although in that context there is no puzzle associated with energy and mass that the ontology helps to resolve.
Clearly, such changes in orientation are also associated with several coordinate-dependent effects, including changes in momentum and kinetic energy. This is how the 3-force component of a 4-force gives rise to frame-dependent changes in velocity and kinetic energy.
See Rindler (1991, p. 91) for discussion.
Only when the force is ‘pure’ (discussed below) and the boost is in the direction of \(\mathbf {u}\) will it be the case that \(\mathbf {f} = \mathbf {f^\prime }\). See Rindler (1991, p. 91) for discussion.
Rindler (1991, p. 92) calls such forces ‘heatlike’.
This view of Newtonian forces as pushes and pulls is emphasized in Wilson (2007).
As noted in footnote 29, the situation is more complicated for systems whose physical descriptions require field-theoretical considerations, for in those cases there isn’t generally a well-defined total 4-momentum that can be associated with the system.
A special case of such forces are what Rindler (1991, p. 92) dubs ‘heatlike’ forces, the action of which only changes a particle’s 4-momentum magnitude and not its orientation.
A similar story can be given for the case of inelastic collisions, as discussed in the preceding section. Evidently the 4-forces at work in such a collision must be impure so as to change the magnitudes of the incoming particles’ 4-momenta.
In this sense, one really ought to stick to writing the equation as \(\varDelta E_0 = \varDelta mc^2\), which is the form that actually gets used in physical practice.
The ontology proposed here deals with the apparent reality of energy released in, say, a nuclear explosion by saying that the energy released is a frame-dependent effect of the 4-momentum magnitudes changing on account of the applied (impure) 4-forces.
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Acknowledgements
Thanks to Gordon Belot, Carl Hoefer, Mario Hubert, Tim Maudlin, Trevor Teitel, and two anonymous referees from Synthese for helpful comments and suggestions, and to audiences in Dubrovnik, Helsinki, London, Prague, San Sebastian, and Winston-Salem. I am particularly indebted to Sam Fletcher and Chip Sebens for extensive discussion and feedback, and to the John Bell Institute for the Foundations of Physics in Hvar for providing a serene and glorious environment for finishing the paper.
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Coffey, K. On the ontology of particle mass and energy in special relativity. Synthese 198, 10817–10846 (2021). https://doi.org/10.1007/s11229-020-02754-5
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DOI: https://doi.org/10.1007/s11229-020-02754-5