On the Ontology of Particle Mass and Energy in Special Relativity
OOn the Ontology of Particle Mass and Energy inSpecial Relativity
Forthcoming in
Synthese
Kevin Coffey ∗ Department of PhilosophyNew York University Abu Dhabi
Abstract
Einstein claimed that the fundamental dynamical insight of special rel-ativity was the equivalence of mass and energy. I disagree. Not onlyare mass and energy not equivalent (whatever exactly that means) buttalk of such equivalence obscures the real dynamical insight of spe-cial relativity, which concerns the nature of 4-forces and interactionsmore generally. In this paper I present and defend a new ontology ofspecial relativistic particle dynamics that makes this insight perspicu-ous and I explain how alleged cases of mass–energy conversion can beaccommodated within that ontology.
Special Relativity is widely recognized as having transformed our under-standing of the relationship between energy and mass. Over a decade after ∗ Thanks to Gordon Belot, Carl Hoefer, Mario Hubert, Tim Maudlin, Trevor Teitel,and two anonymous referees from
Synthese for helpful comments and suggestions, and toaudiences in Dubrovnik, Helsinki, London, Prague, San Sebastian, and Winston-Salem.I am particularly indebted to Sam Fletcher and Chip Sebens for extensive discussionand feedback, and to the John Bell Institute for the Foundations of Physics in Hvar forproviding a serene and glorious environment for finishing the paper. In the words of one mathematician–historian, the energy–mass relationship broughtabout by special relativity constitutes “the most striking example of unification that hasbeen effected in the present century”. (Whittaker, 1958, p.96) a r X i v : . [ phy s i c s . h i s t - ph ] F e b he theory’s development, Einstein wrote that the special relativistic recon-ceptualization of the mass–energy relationship, as expressed in the equation E = mc , was the theory’s most important and lasting contribution to ourunderstanding of the physical world—not the relativity of simultaneity, timedilation, or length contraction. When one reflects on the role this equationplayed in the development of nuclear weapons and nuclear energy—processesin which mass is apparently converted into energy—it’s difficult to disagree.Mass and energy seem to bear a substantive physical connection to eachother in the relativistic context that is quite unlike their classical relation-ship.Although the mathematical expressions relating mass and energy arestraightforward, the precise nature of the physical reconceptualization re-mains poorly understood. Are mass and energy the same physical property,or are they instead distinct but related quantities? What does it mean to saythat mass is “converted” into energy (or energy into mass)? Does E = mc suggest some new, non-classical ontology of mass and energy?The slogans physicists use to characterize the mass–energy relationshipprovide little clarity. d’Inverno (1992) writes that [ E = mc ] is not just a mathematical relationship between two differentquantities, namely energy and mass, but rather states that energy andmass are equivalent concepts . (48) Faraoni (2013) puts the relationship in a seemingly different way: [ E = mc ] expresses the fact that a free particle possesses energy justbecause it has mass, and that a small mass can free up an enormousamount of energy because the factor c is large in ordinary units (as Einstein (1919, p.230). See also Rindler (1991, p.73) 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 metaphysi-cally. This view is echoed in the philosophical literature by, e.g., Butterfield (1984, p.104),Earman (1989, p.18), and Teller (1991, p.382). emonstrated in nuclear reactions and nuclear bombs): this is the equivalence of mass and energy . (144) Do these authors agree on the nature of the alleged equivalence? Strikingly,Bondi and Spurgin (1987) reject any apparent equivalence between massand energy:
Mass and energy are not interconvertible. They are entirely differentquantities and are no more interconvertible than are mass and volume,which also happen to be related by an equation, V = mρ − ...The bestway to appreciate Einstein’s conclusion is to realise that energy hasmass...[All] should be warned against believing erroneous statementsthat mass and energy are interconvertible, and they should be urgedto avoid such terminology as ‘the equivalence of mass and energy’.(62–63) Evidently, not all of these physicists can be right.This paper proposes a new ontology for the dynamics of special rela-tivistic particles that is motivated by an attempt to clarify the relationshipbetween energy and mass, as understanding that relationship is ultimatelygrounded in what the fundamental physical properties of particles are . Iargue that energy and mass are not equivalent: the appearance of inter-conversion is the product of an inadequate (if ubiquitous) view of the un-derlying dynamics. On the view developed here, the surprising and centraldynamical insight of special relativity lies not in any relationship betweenenergy and mass, as Einstein claimed, but rather in the nature of interac-tions between particles mediated by 4-forces in Minkowski spacetime. Helliwell (2010, pp.143–152) expresses a similar view, whereas Rindler (1991, pp.81–84) seems to combine both passages. Flores (2005) identifies six interpretations of Einstein’s equation represented in theliterature on special relativity, tentatively endorsing the view that mass and energy areinequivalent physical properties that can—but need not—be converted into each other. Ithink that both his positive argument and his grounds for rejecting several of the compet-ing interpretations rest on conceptual misunderstandings, but will confine my commentaryto footnotes. The view developed here is not among the six interpretations Flores can-vasses. This paper is restricted to the special relativistic dynamics of (spinless) particles. One dynamically (as opposed to kinematically )novel—an account on which there is no deep ontological connection betweenenergy and mass. might feel that a clear understanding of energy–mass ‘equivalence’ can’t be adequatelyaddressed 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 goodreasons to think the energy–mass relationship can be investigated in an illuminating wayin the limited context of special relativistic particle dynamics, and in fact that such arestricted context is the appropriate starting point for an inquiry into the relationshipbetween energy and mass. First, the original association of mass with energy, articulatedin Einstein (1905), draws solely upon special relativistic particle dynamics. There is thusa straightforward conceptual question about how such an equivalence is to be understoodthat predates any general relativistic or field-theoretic considerations. Einstein thoughtthe identification of mass and energy was already grounded in the comparatively simplerelativistic theory of point particle dynamics. Second, the philosophical challenges raisedto 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 withthe simpler case in the hopes that a clear understanding of special relativistic particledynamics might point the way towards understanding more elaborate contexts. Whetherthe interpretation developed here can be suitably extended to classical fields, includinggeneral relativity, is an open question. The Received View
Traditional presentations of energy and mass in special relativity generallyproceed from the definition of the relativistic energy of a free particle: E = γ u mc . Here m is the particle’s mass, c is the speed of light, u is the particle’s speed,and γ u is the Lorentz factor given by γ u = 1 (cid:112) − u /c . In the relativistic context, unlike the classical one, a particle is evidentlyrecognized as possessing energy simply in virtue of possessing mass, for ina frame in which a particle is stationary ( u = 0) the relativistic energyexpression reduces to E = mc , where E is called the rest energy . Thatrest energy stands in a fixed ratio to mass has suggested to many physiciststhat mass just is rest energy (sometimes suggestively called mass-energy ).For example, Taylor and Wheeler (1992) write that [ E = mc ] is the most famous equation in all physics. Historically, thefactor c captured the public imagination because it witnessed to thevast store of energy available in the conversion of even tiny amounts of Here ‘mass’ is being used in the modern sense of the property that determines how aparticle 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 ‘quantityof matter’. Jammer (1997) discusses the history of this conceptual transformation, whichhas its origins in Euler’s work in the 18th century. For a philosophical justification of thisreconceptualization, see Cartwright (1975) and Lange (2001). The quantity representedby 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 anintrinsic, frame-independent property of a particle. That Flores (2005) fails to appreciatethis point leads to a misunderstanding—and misplaced criticism—of Lange’s view. (Seenote 19 below.) Mass should be distinguished from a particle’s so-called ‘relativistic mass’,given by m R = γ u m , for which rest mass is the special case corresponding to u = 0. ass to heat and radiation. The units of mc are joules; the units of m are kilograms. However, we now recognize that joules and kilogramsare units different only because of historical accident. The conversionfactor c , like the factor of conversion from seconds to meters or milesto feet, can today be counted as a detail of convention rather than asa deep new principle. (pp.203–206; diagrams on pp.204–205) This understanding, which I take to be held by many physicists, holds thatmass and rest energy are “equivalent” in the sense that they are one and thesame physical property; their terms are coreferential . Given the intercon-vertability between different forms of energy, one might also say that massis equivalent to energy in general , and that mass and kinetic (or potential)energy are “two forms of the same thing”. Mass and energy are intercon-vertible in the same sense that different forms of energy are interconvertible.
But channeling Bondi and Spurgin (1987) in the passage above, we don’t identify photon energy ( E ) and frequency ( ν ), even though they, too, arerelated in fixed proportion as E = ¯ hν , nor do we treat energy and frequencyas “two forms of the same thing”. So why interpret E = mc differently?Why not say that a certain amount of energy ( E ) is ‘associated’ with anobject’s mass?The central reason is that there seem to be physical processes in whichmass and other forms of energy are interconverted . One oft-cited illustrationconcerns ‘mass defects’ associated with radioactive decay. When a tritium(or hydrogen-3) nucleus decays in a nuclear fission reaction into a helium-3nucleus, an electron, and a neutrino H → He + e − + ¯ ν e , Sternheim and Kane (1991, p.493). See also Moore (2013, pp.34–35). greater than the sum of the massesof the individual daughter bodies. Part of the mass of the tritium hasapparently been converted into the kinetic energies of the outgoing bodies,for the loss of mass corresponds to a loss of rest energy determined by∆ E = ∆ mc —precisely the amount needed to account for the increased kinetic energy.A second example concerns inelastic particle collisions. Consider a genericinelastic collision between two particles of equal mass, as viewed from theircenter-of-momentum frame:Prior to collision, the mass and energy of the system are given by m sys = m + m = 2 mE sys = E + E = 2 γ u mc . Given energy conservation, the final system energy is now also given by E sys = γ u (cid:48) M c = M c , where u (cid:48) = 0 and γ u (cid:48) = 1. Equating both expressions for system energy2 γ u mc = M c implies that M = 2 γ u m > m. gained in the collision process, and this gain isaccompanied by a corresponding loss in the system’s kinetic energy. Kineticenergy has apparently been converted into mass, for the loss in kinetic energycorresponds precisely to the gain in actual mass one would expect if massand rest energy were two forms of the same thing. This widespread understanding of the energy–mass relationship has beenchallenged by (Lange, 2001, 2002), who argues that the differing ontologicalstatuses of mass and energy in the theory undermines their alleged equiv-alence. Lange’s argument is grounded in two related and widely-acceptedfeatures of special relativity. First, that all inertial frames are physicallyequivalent in the sense that the dynamical laws take the same form in all in-ertial frames and, on their basis, do not permit the identification of any onesuch frame as physically privileged or distinguished. Second, and relatedly,that any numerical quantities representing primitive or basic physical prop-erties are Lorentz-invariant—that is, are not frame- or observer-dependent.(This is of course not to suggest that all fundamental physical propertiesmust be represented by scalar quantities.) Versions of both claims are alsogenerally taken to hold for classical dynamics, although in special relativity The kinetic energy ( T ) of a free particle in special relativity is given by T = ( γ v − mc so that E = γ v mc = E + T . In the inelastic collision, the kinetic energy lost is2( γ v − mc , which corresponds precisely to the rest energy (and hence mass) gained: Mc − mc = 2 γ v mc − mc = 2( γ v − mc . There are subtle issues concerning how to make precise sense of this notion of equiv-alence and of the role of symmetries in theory interpretation in general, but that is atopic for another paper. That the physical equivalence of all inertial frames lies at thefoundation of special relativity is explicit in Rindler (1991, pp.1,7,50), and Lange’s invo-cation of the notion of Lorentz-invariance is standard in both the physics and philosophyliteratures. Consider how these interpretive commitments get applied to discussionsof relativistic length and distance . As standardly understood, there is nofundamental fact in special relativity about how long a measuring rod is,as its length (as represented by a number) is not Lorentz-invariant. Itsvalue changes from one frame to the next in accordance with the Lorentztransformations. There are facts about the length of the rod in differentframes—there is a perfectly real and objective physical fact about its lengthin the inertial frame in which it’s at rest (its ‘rest length’), say, assumingsuch a frame exists—but there is no frame-independent fact about the lengthof the rod. Length is not an absolute property of the rod, and thus not acandidate for being fundamental. None of this is to suggest that we oughtnot to be interested in such derivative frame-dependent physical facts andproperties. Indeed, such features of the world are often the most readilyaccessible and convenient with which to work, given the modes of descriptionthat come naturally to us. But this is all the more reason why we must becautious in making judgments about the theory’s fundamental ontology. Failure to keep track of which individual quantities represent fundamen- Throughout this paper I make use of the distinction between a fundamental physicalproperty and a derivative or non-fundamental physical property. However those meta-physical notions are made precise, it is this author’s opinion that the distinction is asubstantive one, and moreover one that is implicit in physical practice. The interpretive constraint that the mathematical objects and equations characteriz-ing a theory’s fundamental ontology satisfy certain sorts of symmetry requirements—inthis case, requirements associated with Lorentz transformations—is not without its philo-sophical puzzles. See, e.g., Dasgupta (2016) for a critical discussion of how such symmetrydemands might be justified. However, this paper is not intended as an exploration of thisissue, and I will take the overall cogency of this interpretive constraint for granted in whatfollows. Were this interpretive principle to be jettisoned, it would have consequencesfor the physical content of special relativity that go far beyond the relationship betweenenergy and mass. m and is often interpreted as a fundamentalphysical property, whereas its kinetic energy ( γ v − mc and total free en-ergy γ v mc are manifestly not Lorentz-invariant (depending as they both doon frame-dependent speed) and thus not candidates for representing funda-mental physical properties. How, then, are we to understand the allegedphysical conversion between mass and kinetic energy—a central motivationfor the received view—if one quantity in that conversion is fundamental andthe other is not? Indeed, in what sense could mass and kinetic energy be“two forms of the same thing” if mass is a primitive physical property butenergy is not? On the face of it, the widely-publicized equivalence betweenmass and energy appears to be the product of a ubiquitous conceptual con-fusion. They simply cannot be equivalent (or interconvertible) because theydon’t share the same ontological category. One could simply insist that the quantity mc represents a distinct andfundamental type of energy, but it’s unclear what the motivations for sucha view would be or why the resulting property would warrant the label ‘en-ergy’. A defining feature of energy is its ability to be converted from oneform to another. What sense does it make to call mc ‘rest energy’ if it, butnot any other forms of energy, are fundamental physical properties—if we Of course something similar is also true in classical dynamics, as speed isn’t Galileaninvariant either, but in that context no one alleges an equivalence between mass andenergy. 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 aperfectly real physical property (just like speed). The issue is that it’s not fundamental. See Rindler (1991, p.74). Lange’s solution is to deny the identification of mass as a form of energy onthe grounds that there are no genuine physical conversions between massand energy. Mass is no more a type of energy than photon frequency. Theimpression of conversion that various physical processes elicit is, on thisview, an artifact of our choice of description—a feature of the perspectivethat we adopt. In this sense, we convert energy to mass (or mass to energy),not nature.The diagnosis here hinges on the observation that mass in special rela-tivity is non-additive : the mass of a system is generally not the sum of themasses of its parts. Consider a system of eight particles of equal mass m ,each moving radially outward with the same speed u from a common origin.This is depicted in the following diagram: Again, it’s worth emphasizing that Lange’s puzzle raises no concerns about the con-sistency of E = γmc (or E = mc ) within the mathematical formalism or about its usein empirically successful applications. Rather, the issue here is a conceptual one abouthow the theory’s ontology is to be understood. E sys = M c = E + . . . + E . But for each particle i , E i = γ u mc , and so we can write M c = 8 γ u mc , from which we conclude: M = 8 γ u m > m. Here of course there are no collisions or any other postulated interactions,and thus no pretense of converting energy into mass. Instead, the lesson isthat the mass of the system is more than the sum of its constituent masses.Re-expressing the system mass as M = ( m + . . . + m ) + 1 c ( T + . . . + T ) See footnote 9. u → u (cid:48) > u , the mass of the system would increase: E (cid:48) sys = M (cid:48) c = 8 γ u (cid:48) mc M (cid:48) = 8 γ u (cid:48) m > γ u m because γ u (cid:48) > γ u , and yet the fundamental particle masses would remainfixed.This example might reasonably make us suspicious regarding claims ofmass–energy interconversion. The special relativistic formalism tells us thatthe system mass goes up when the particles are uniformly boosted in theirrespective directions, and thus it appears energy is being converted intomass. Yet when we look at the particles themselves only their kinetic ener-gies are changing. Their masses remain fixed, and so it looks like whateverenergy is applied to the system in the boost is converted into the kineticenergies of the constituents. At no point does the mass of any electron inthis process change, for example, so in what sense is mass being gained orlost?To make sense of these and other apparent cases of mass–energy con-version, Lange distinguishes two ways we might characterize a system ofparticles. First, there is the ‘component level’ description. Here the col-lection is described in terms of its constituent particles and the forces andinteractions they experience. Second, there is the ‘system level’ description,where we think of the collection as a single unit interacting with an externalenvironment and ignore the internal dynamics. We shouldn’t be misled intothinking each description is on an ontological par: the component level isfundamental. While it is true that energy is apparently converted into mass13hen a ball of gas is heated (to use Lange’s example), this ‘conversion’ onlyoccurs when the system-level description is invoked. Described entirely interms of basic constituents, Lange argues, there is no conversion betweenmass and energy—just as there is no conversion in our initial example whenall eight particles are boosted. This is why he writes that mass–energyconversion is an artifact of our perspective and not a real physical process.Other cases involving particle systems mislead us into thinking thatenergy–mass conversion occurs because we inadvertently switch perspectivesmidway through our analysis. For instance, consider how Lange diagnosesthe typical textbook treatment of an inelastic collision. Recall from sec-tion 2.2 that the kinetic energies of the pre-collision bodies appeared to beconverted into mass upon impact, as outlined in the diagram below:But the total pre-collision mass is 2 m only if we adopt a component-leveldescription. As one can see in the following diagram, on a system-level de-scription (owing to non-additivity) the pre-collision mass is 2 γ v m —preciselythe mass of the system after collision: Bondi and Spurgin (1987) use this example to argue that energy has mass. Read in astraightforward way, this claim is confused. Mass is a property of inertial resistance: themass of an object is a measure of how much that object resists changes to inertial motionin light of impressed forces. So for energy to have mass, energy must be the sort of thingto which impressed forces can be applied, and energy simply isn’t that sort of thing. appearance of conversion from energy to mass in this instance arisesbecause textbooks switch descriptive levels halfway through the analysis.If one describes the process from a fixed perspective, there is no conver-sion. A similar diagnosis applies to the apparent mass defect involved intritium decay, although there the descriptive switch occurs in the oppositedirection: mass appears to be converted into energy because our pre-decaydescription is at the system-level, whereas our post-decay description is atthe component level. One worry with Lange’s account, raised by Flores (2005), is that the underlying onto-logical 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 toimplicitly privilege a particular frame—namely, the frame in which the object is (instan-taneously) at rest. Like length, then, it is a perfectly real physical property, but on thesurface 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 restmass and relativistic mass (Lange, 2001, pp.226–7) that rest mass is understood merelyas the u = 0 mathematical limit of relativistic mass. As previously noted (see note 7above) the quantity m itself represents an ontologically fundamental and frame-invariantparticle property for Lange, which also happens to be equal to the particle’s relativisticmass in the frame in which the particle is at rest. But that equality is not constitutive ofthe property. In this way, contra Flores (2005), Lange’s proposed ontology preserves thespecial relativistic maxim that all inertial frames are physically equivalent. Puzzles of Dynamical Interpretation
However, Lange’s diagnosis of alleged mass–energy conversions fails to con-vince. In this section I explain why. In the process I identify several puzzleswe must confront in light of adhering to Lorentz invariance as an interpre-tive constraint in understanding the fundamental ontology of special rela-tivistic particle dynamics. These puzzles then help to motivate the accountof what makes special relativity dynamically novel that I develop in the fi-nal sections—an account that makes no mention of any equivalence betweenmass and energy.
Assume for the moment that Lange has correctly diagnosed the alleged casesof mass–energy conversion discussed above and that such conversions are, infact, unphysical. An initial problem is that his analysis doesn’t generalize.There are a wide class of particle collisions where apparent energy–massconversion can’t be explained away as a perspectival shift. The most obviousinstances are electron–positron creation and annihilation: γ + γ → e − + e + (electron–positron creation) e − + e + → γ + γ (electron–positron annihilation)In the first case, two high-energy massless photons collide to create an elec-tron and a positron, both elementary particles of equal non-zero mass. Inthe second, the reverse happens. In both instances there is an apparent con-version between mass and energy that can’t be explained away by appealingto different perspectives or levels of description. The pre-collision situation16s characterized in terms of two distinct things, as is the post-collision situ-ation. But in the creation reaction the input bodies have no mass , whereasthe output bodies do. Where has the mass come from? Certainly not fromswitching our perspectives part way through the analysis. Some of the pho-tons’ energy really has, it would seem, been converted into mass. Indeed,the kinetic energy difference pre- and post-collision precisely matches theenergy associated, via ∆ E = ∆ mc , with the electron and positron masses.Contra Lange (especially Lange (2001, p.230)), there is no plausible case tobe made here that mass alone is conserved. It is total energy that’s con-served in this reaction, where that total energy would seem to include massas one particular form.These examples are not isolated or unique. A host of particle reactionscan’t be accommodated within Lange’s analysis: π → γ + γ (neutral pion decay) γ + p → π + p (pion photoproduction) p + p → p + p + p + ¯ p (proton–antiproton pair production) e − + e + → e − + e + + e − + e + (electron–positron pair production)In each, the apparent conversion between energy and mass occurs at thefundamental descriptive level, so no plausible story about shifting perspec-tives is available. These reactions really do seem to involve mass cominginto and out of existence.Unfortunately, the existence of these reactions does little to quell theoriginal sense of puzzlement. In the particle reactions it seems hard todeny that mass is created and destroyed, and mass–energy conversion in This terminology follows Freund (2008, p.247). E = ∆ mc is the natural explanation. But how is sucha story conceptually coherent if mass is a fundamental physical propertyand energy (being Lorentz- variant ) is not?Here we must be careful to manage expectations and to distinguish twoissues. There is, on the one hand, the physical account of what’s going onin these reactions and why they occur. What physical laws govern electron-positron annihilation and creation, for example? We should not expectspecial relativity (or at least special relativistic particle dynamics) to answerthis question. The collision reactions at issue are inherently quantum innature and our best understanding of them lies in quantum field theory, notin the non-quantum particle dynamics of special relativity.On the other hand, quantum field theory is designed to preserve cen-tral conceptual and dynamical features of special relativity, such as thephysical equivalence of all Lorentz frames, and the collision reactions notedin this section are often cited in support of the purely special relativisticidentification of mass with energy. Is there a way of understanding thefundamental ontology of special relativistic particle dynamics that remains consistent with —even if not explanatory of —the existence of these colli-sion reactions, without thereby either committing the conceptual error ofidentifying fundamental and derivative physical properties or giving up theprinciple that all inertial frames are physically equivalent? That seems likean entirely reasonable question. What Lange’s puzzle makes vivid is thatstandard textbook accounts, according to which there is simply a conversionbetween mass and energy, fail to provide this. 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 theexperimental evidence from particle physics for the basic principles of special relativisticparticle dynamics. .2 Composite Mass Let us back up a step: is Lange’s analysis adequate for the cases he considersexplicitly? Return to his diagnosis of the ball of gas and recall that, when thegas is heated, there is an apparent conversion of energy into mass. Lange’sclaim is that this conversion is unphysical and the product of a particularperspectival switch, namely, from that of treating the ball of gas as composedof distinct molecules to that of treating the ball of gas as a single unit. Whenthe gas is viewed from the more ontologically fundamental perspective—thatof the gas molecules themselves—the application of heat merely changesthe kinetic energies and not the molecular masses. On this basis, Langeconcludes that “this ‘conversion’ of energy into mass is not any kind of realphysical process taking place in nature. We ‘converted’ energy into masssimply by changing our perspectives on the gas: from treating it as manybodies to treating it as a single body” (Lange, 2001, p.235).This diagnosis only makes the situation of the gas more mysterious, how-ever, for there is a genuine and fundamental physical process occurring herein need of identification. If we imagine the gas (as a system) to be uniformlydistributed in an enclosing box or shell, that box will be harder to move—willexhibit more inertial resistance to impressed forces—after it’s been heated.This is a real empirical effect, not a magician’s conjuring trick! There isan undeniable physical change in the gas’ total mass after heating, and thatchange must be grounded in changes to the fundamental properties of thebasic entities constituting the gas. That is, there is a genuine physical factabout the inertial resistance the gas puts up—a fact that is grounded in See Bondi and Spurgin (1987). This point is also noted in Flores (2005). A similarphenomena occurs in a compressed spring, which has a larger mass than a relaxed springowing (it would seem) to its increased potential energy. On this point, see Dib (2013). Turn now to Lange’s analysis of inelastic collisions. The charge wasthat in the process of describing the collision we inadvertently switchedperspectives, and in the process ‘converted’ the kinetic energy of the initialbodies into the rest mass of the final body. If we think of the two pre-collisionbodies as a single system, the total system mass will equal the mass of thesingle body post-collision and the alleged conversion evaporates. Again (theclaim is) the conversion is revealed to be unphysical, the result of descriptivelegerdemain.But for this analysis to be compelling, Lange must do more than showthat the mass of the pre-collision two-body arrangement, when viewed as asingle system, is the same as the mass of the post-collision body. He must (in This puzzle—the puzzle of understanding the nature of composite mass—is not uniqueto macroscopic objects or objects whose constituents only interact via collisions (as onegenerally assumes for gases). For all but the most fundamental particles, mass seems tobe partly constituted by ‘internal’ energy, whether in the form of kinetic energy or someform of binding or potential energy. from the same perspective , andthat requires saying a good deal more about how exactly these ‘perspectivaldescriptions’ are to be understood. What justifies Lange’s claim that thetextbook analysis switches perspectives? That we view the pre-collisionarrangement as two bodies and the post-collision arrangement as one? AmI switching perspectives when I smash a rock and describe the result as acollection of shards?More importantly, though, Lange’s diagnosis fails in this case for thesame reason it did for the ball of gas. There is a genuine physical processoccurring here in which mass is changing—a process that is grounded inchanges to the fundamental properties of the particles that constitute thecolliding bodies. Imagine that two equally massive balls of putty collide inan inelastic collision. If I then cut the post-collision body in half, each willhave a greater mass than either pre-collision body. This is a real physicalchange and Lange’s talk of ‘switching perspectives’ obscures it. The obviousexplanation of this change is that the kinetic energies of the putty moleculeshave increased, but now we’re back at non-additivity and the ball of gaspuzzle.
The situation we now face is the following: there are compelling groundsto deny the inter-convertibility and equivalence of mass and energy, andyet such a denial seems hard to reconcile with important experimental phe-nomena. What should we make of this? I suggest that the difficultiesstem from an inadequate (or inadequately specified) ontological picture of21pecial relativity and from a hitherto overlooked feature of its dynamicalfoundations. Accordingly, this section develops a new account of the fun-damental ontology of special relativistic particle dynamics. This requiresidentifying the metaphysically basic entities and properties, indicating howthey interact and evolve dynamically, and specifying the mathematical ob-jects and structures and equations that encode that physical picture. Theontology is based on a generally covariant (or geometrical) formulation ofspecial relativity, briefly outlined in the first subsection. Several indepen-dent motivations for my ontological picture are offered, although the cen-tral claim—developed in sections 7 through 9—is that this interpretationprovides a satisfying resolution of the puzzles associated with mass–energy‘equivalence’ while clarifying the dynamical foundations of special relativityand the meaning of E = mc .It is now commonplace to formulate special relativity in 4-dimensionalterms, so one might not think this section has anything new to offer. But:(1) this formulation is rarely accompanied by any discussion of the funda-mental ontological picture associated with the dynamics; (2) there has beenno discussion in the literature of how the experimental phenomena associatedwith alleged energy-mass conversion ought to be understood in geometricalterms; and, most importantly for foundational concerns, (3) despite theubiquity of 4-dimensional presentations, it has yet to be recognized—as Ithink it ought to be—that the central dynamical insight of special relativityhas nothing to do with the relationship between energy and mass. Indeed,4-dimensional expositions of special relativity remain frustratingly elusive The approach here is in the spirit of Maudlin (2018). Standard philosophical references include Friedman (1983), Earman (1989), and Mala-ment (2012).
Geometrically, special relativistic theories are framed against the backdropof Minkowski spacetime, which is a space of events represented by a 4-dimensional (pseudo-Riemannian) manifold equipped with a flat metric η µν of (Lorentzian) signature ( − , + , + , +). The trajectory or worldline of amaterial particle is the collection of events that together constitute the his-tory of that body, and it is a basic postulate of special relativity that suchobjects are represented by timelike worldlines and light rays by null trajec-tories. Each point along a particle’s worldline is associated with a unique 4- Where there is little risk of confusion I will be lax about the distinction betweenmathematical representation and physical feature represented. Recall that the metric structure divides the spacetime at any point (call it the ‘originpoint’) into distinct regions, which can be characterized by the vectors (4-vectors) at thatpoint. The timelike region consists of those events whose displacement vectors from theorigin point have negative magnitude. All such events are said to be timelike separatedfrom the origin point and any 4-vector that points from the origin point to a timelikeseparated event is said to be a timelike 4-vector. The lightlike ( spacelike ) region is theset 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 manifoldis said to be timelike (null, spacelike) if the tangent 4-vector at each point along it istimelike (null, spacelike).Note that photons play a rather curious role in textbook presentations of special relativ-ity, with some authors smoothly sliding between initial talk of light rays or signals to latertalk of photons and other authors acknowledging that photons are quantum mechanicalin 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 wasoriginally understood: in the 1920s, well after the acceptance of special relativity, Bohrand 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 silenton 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 mentioneither of inertial frames or of light’s constitution and ‘speed’. P µ called the (or energy-momentum 4-vector ), whichis always tangent to the curve and is traditionally defined in textbooks as P µ = mU µ , where U µ = dx µ dτ is the 4-velocity along the worldline. τ is the proper time parameter alongthe particle’s worldline. In an arbitrary inertial frame the 4-momentum canbe expressed in coordinate form as: P µ = ( γ u mc, γ u m(cid:126)u ) = ( E/c, (cid:126)p ) , where (cid:126)u = d(cid:126)xdt is the spatial velocity in that frame, E = γ u mc is the relativis-tic energy of the particle, and (cid:126)p = γ u m(cid:126)u is its relativistic spatial momentum(or relativistic 3-momentum). These features are illustrated in the followingspacetime diagram: [Text] E/m [T p μ Some authors use bolded capital letters (e.g., (cid:126)A ) for 4-vectors and bolded lower-caseletters (e.g., (cid:126)a ) for spatial 3-vectors, indicating the components in an inertial frame S bywriting (cid:126)A S → ( A , A , A , A ) or (cid:126)a S → ( a , a , a ). I will occasionally use this notation, butmore often will refer to 4-vectors by writing things like A µ , understood to represent thecomponents of the 4-vector (cid:126)A in some arbitrary inertial frame. F µ a particle obeys what appears tobe a 4-dimensional relativistic analogue of Newton’s equation of motion: F µ = dP µ dτ . The components of the 4-force in an arbitrary inertial frame are given by F µ = γ u ( c ddt [ γ u m ] , d(cid:126)pdt ) = γ u ( 1 c dEdt , (cid:126)f ) . Here f is the relativistic 3-force (cid:126)f = d(cid:126)pdt = d ( γ u m(cid:126)u ) dt , which in the u/c → (cid:126)P tot , is always conserved: (cid:126)P tot = n (cid:88) i =1 (cid:126)P ( i ) = constant , where the (cid:126)P ( i ) are the 4-momenta of the incoming (or outgoing) particles.This is an independent dynamical principle and not something derived froma more basic law. Let us call it the . See Rindler (1991, pp.70–73, 90–92). For an isolated n -body system that interactsonly locally (i.e., effectively via collisions), a more general principle holds that the net4-momentum of the system remains constant, in the sense that the components of thetotal system 4-momentum do not change in any given inertial frame. Even though thespecific (cid:126)P ( i ) E = mc and suggest an alternative picture of whatmakes special relativity dynamically novel. What metaphysics ought to be associated with this formalism? Standardpresentations of classical dynamics encourage a particular ontological pic-ture, according to which there are (a) material particles possessing primitiveproperties of mass and instantaneous location and velocity in space or space-time, and (b) forces that mediate interactions between particles and generatechanges in their dynamical states and motions. Other quantities of physi-cal significance—e.g., “dynamical variables” like spatial momentum, angularmomentum, and energy—are ultimately understood as derivative properties summation is taken at an instant, in these circumstances Rindler (1991, pp.78–79) showsthat the total system 4-momentum (cid:126)P sys is a well-defined 4-vector. It follows from thismore general principle that relativistic energy and relativistic (spatial) momentum areboth independently conserved in these circumstances, even though the values of thosequantities are frame-dependent. For a more general discussion not restricted to inertialobservers, see Gourgoulhon (2010, pp.288–291).
When an n -body system can’t be treatedas isolated, such as when various sorts of field-theoretic considerations are included, thenthere isn’t generally a well-defined total 4-momentum associated with the system. How-ever, the puzzles developed above concern physical systems for which these field-theoreticconsiderations can be ignored. Recall (see footnote 6) that the guiding methodology hereis to tease out the significance of E = mc in the simplest dynamical cases and to leavemore complicated physical situations for subsequent work. There is a third dynamical principle—conservation of the angular momentum 4-tensoralong 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 fundamentalproperties relevant to the types of interactions it generates and the way in which it respondsto 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 partof its dynamical state—e.g., its position in space or spacetime. A non-interacting objectmoving inertially is constantly changing its position and yet its dynamical state remainsconstant. On the other hand, while typical presentations of special relativity of-ten contain extensive discussions of the radical ontological changes broughtabout by relativistic kinematics (e.g., the relativity of simultaneity, lengthcontraction, time dilation), they remain surprisingly quiet regarding anyontological changes demanded by the new dynamics —aside, as we’ve seen,from problematic claims about the equivalence of mass and energy. Indeed,the very manner in which the new dynamics is often introduced as “a mod-ification of the Newtonian scheme” and the similarity in terminology givesthe impression that much of the ontology standardly associated with classi-cal dynamics carries over, more or less, to the context of special relativisticparticle dynamics. We are told, for example, that classical momentum( (cid:126)p cl = m(cid:126)u ) must be ‘redefined’ in the relativistic context and that theequation (cid:126)f = d(cid:126)pdt is the ‘relativistic generalization’ of Newton’s second law.Many relativistic expressions appear as seemingly straightforward analoguesof Newtonian ones, thereby suggesting that the basic ontology underpinningthe dynamics remains essentially the same. See, e.g., Jos´e and Saletan (1998, pp.13–14). I do not mean to suggest that thispicture of classical ontology is uncontroversial. Among the issues raised in recent years,some philosophers have considered whether velocity ought to be taken as an ontologicallyprimitive property in its own right (see Arntzenius (2000), Carroll (2002), Meyer (2003),Smith (2003), Lange (2005), Easwaran (2014), McCoy (2018)), whereas others have won-dered whether attributions of mass ought to designate fundamental properties of particles(see Dasgupta (2013), Martens (forthcomingb), Martens (forthcominga)). And Butter-field (2006) has argued against understanding classical dynamics (or indeed any physicaltheory) in terms of properties defined at points of space or spacetime. There is also along tradition within the metaphysics of physics going back at least to Mach and Hertzpuzzling 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 theseliteratures will be relevant to the specific way I frame the ontological proposal sketchedbelow, but they do not affect the central argument and in the context of this paper I’vehad to set them aside. French (1968, p.4) See, e.g., French (1968, pp.21–23).
If we take the geometrical formulation of special relativity seriously andunderstand the dynamics as genuinely unfolding in Minkowski spacetime,the 4-momentum vector is the obvious candidate for encoding a particle’sinstantaneous dynamical state. There is a natural ontology associated withthis, according to which the 4-momentum encodes two distinct and fun-damental properties of a particle: one represented by the 4-momentum’s magnitude and the other represented by what I will call the 4-momentum’s orientation . (This is not to deny that 4-momentum components representobjectively real particle properties when indexed to a particular frame, butthose features are not fundamental.) For material bodies, whose worldlinesare timelike, these properties are physically independent. So the proposalis that, at any given location in spacetime at which a massive particle ex-ists, this pair collectively defines the instantaneous dynamical state of thatparticle at that location.As ontologically primitive features of particles, these properties are notsubject to reductive analysis or further explication, although one can still28onvey a feel for their physical content. The magnitude of a particle’s 4-momentum is in some sense a reflection of how much the particle resistschanges to its dynamical state on account of an applied 4-force. This quan-tity is often taken to be just another way of representing particle mass, animpression reinforced by the fact that (cid:107) P µ (cid:107) = (cid:112) − P µ P µ = mc holds as afixed relation for all material particles. I have no objection to this concep-tual gloss on the physical content of 4-momentum magnitude, but in keepingwith my emphasis on the covariant formalism I prefer to think of P µ as thecentral representational device: the customary definition of 4-momentum, P µ = mU µ , is, on this view, a definition of the quantity m . P µ directlyrepresents two basic dynamical properties of particles and is not to be under-stood as representing derivative features characterized by constellations ofmathematical objects that themselves represent more fundamental physical 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 substancehangs on this point: both (cid:107) P µ (cid:107) and m are understood as representing one and the sameprimitive 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 dynamicalequations governing derivative ontology. Since I think the focus on the frame-dependentequations for special relativistic particle dynamics has been the source of much ontologicalconfusion, I prefer to do without m and use (cid:107) P µ (cid:107) as the representation of particle mass.It’s worth emphasizing that the property represented by m (or (cid:107) P µ (cid:107) ) takes on a ratherdifferent character in relativistic dynamics than it does in classical dynamics. Withinan inertial frame a special relativistic particle, unlike a classical one, exhibits differentamounts of inertial resistance to impressed 3-forces in different directions. The resistanceto 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 directionperpendicular to its instantaneous spatial velocity—its ‘transverse mass’. Indeed, thespatial acceleration of a body in response to an impressed 3-force is generally not evenin the direction of the 3-force itself. The frame-dependent dynamical equation governingparticle motion is (cid:126)f = γ u m(cid:126)a + ddt [ γ u m ] (cid:126)u , and so the magnitude and direction of a particle’sacceleration 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). Therelationship (cid:126)f = m(cid:126)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 toa particle solely in the x-direction generates a velocity-dependent deceleration in the y-direction. orientation , reflects roughlyhow a particle is ‘directed in’ spacetime. Together, we can think of 4-momentum magnitude and orientation as reflecting how a particle is ‘mov-ing through’ spacetime. Although spacetime diagrams may suggest thatorientation is an artifact of one’s choice of coordinates, it is in fact only ourrepresentation that is coordinate-dependent. The orientation itself, in space-time , is entirely independent of any coordinate system. This is of coursenot the only respect in which spacetime diagrams of Minkowski spacetimecan mislead. The following two diagrams are equally adequate graphical rep-resentations of the same 4-momentum (same magnitude, same orientation),just drawn from the perspective of different Lorentz frames: ct x and Like (cid:107) P µ (cid:107) and m , P µ and U µ are equally good representations of a particle’s (4-momentum) orientation. This sort of discourse naturally suggests an underlying substantivalism of some formor other. I embrace this, but wish to remain agnostic here regarding whether such ametaphysical commitment is necessary for the ontology I propose. t ' x ' On the other hand, the two 4-momenta in the following diagram have thesame magnitude but different orientations, as they are being representedwith respect to the same Lorentz frame: ct x
We cannot specify the 4-momentum orientation of a particle except by in-voking indexical expressions—say, by specifying a coordinate system usingindexicals and then giving the 4-momentum components in that coordinatesystem—much as the proponent of substantival space can’t specify the ab-solute location of a particle except by saying (pointing) that it is here or there or providing its coordinates in some coordinate system. But that 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 in4-momentum orientations between particles, as (cid:126)P · (cid:126)P = (cid:107) (cid:126)P (cid:107)(cid:107) (cid:126)P (cid:107) γ ( v ) holds invari-antly (where v is the relative velocity between the two particles). See Rindler (1991,p.76). If (cid:126)P · (cid:126)P = (cid:107) (cid:126)P (cid:107)(cid:107) (cid:126)P (cid:107) then both 4-momenta have the same orientation, and if t as measured in a particularLorentz frame, subjected to a constant impressed 3-force in the direction ofits motion. During this interval the particle accelerates, and then returnsto inertial motion after the force is switched off. Intuitively, the dynami-cal state of the particle has changed: the particle’s state before the appliedforce is switched on is different from its state after the force is switched off.After all, if there were a second (force-free) particle originally at rest withrespect to the first, after ∆ t it would be in uniform relative motion withrespect to it (or in a different state of relative motion if the second particlewasn’t originally at rest). But what about the dynamical state of the parti-cle itself has changed? What fundamental physical properties does it havethat we can point to as having changed from one side of this interval to theother? All of the obvious candidates—kinetic energy, speed, 3-momentummagnitude—aren’t Lorentz-invariant and thus aren’t fundamental, and theorientation of its 3-momentum remains unchanged because the force is ap-plied in the direction of motion. The dynamical state of the particle haschanged, but when considered within a frame it’s not at all clear exactly which fundamental properties have changed. Other situations illustrate the same point. Consider a world consistingof two otherwise identical particles moving in the same direction at different (cid:126)P · (cid:126)P (cid:54) = (cid:107) (cid:126)P (cid:107)(cid:107) (cid:126)P (cid:107) then the extent to which (cid:126)P · (cid:126)P / (cid:107) (cid:126)P (cid:107)(cid:107) (cid:126)P (cid:107) > It’s of course true that the magnitude of the particle’s acceleration is non-zero duringthe interval ∆ t —and, being Lorentz-invariant, this points to a genuine physical differ-ence while the force is being applied—although, unlike Newtonian dynamics, in specialrelativity the non-zero magnitude of that acceleration is frame-dependent. The postulation of a fundamental property of 4-momentum orientationprovides an immediate and satisfying diagnosis of these situations. In thecase of a particle subjected to a 3-force for a finite interval, the orienta-tion of the particle’s 4-momentum does change. The orientations beforeand after the application of the force are distinct. Indeed, part of whatthe relativistic force law says is that the dynamical effect of an interactioncan be to change the 4-momentum orientation of a particle. Because I take4-momentum orientation to be a fundamental physical property that partlydefines a particle’s dynamical state, changes in orientation provide an imme-diate explanation of how the dynamical state of the particle changes across∆ t . In fact, it is the only fundamental dynamical difference. Similarly, In this instance we can see particularly clearly how the same issue arises for Newtoniandynamics, 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 thecontext of Newtonian theory (or don’t differ between the particles). The motivations devel-oped here thus apply equally to an analogous ontological picture of classical dynamics—although in that context there is no puzzle associated with energy and mass that theontology 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 the3-force component of a 4-force gives rise to frame-dependent changes in velocity and kineticenergy.
This is not the complete ontology. Turning to the dynamics of particle in-teractions, 4-forces themselves on this picture are fundamental, not 3-forces.Relativistic 3-forces are generally introduced as the analogues of Newtonianforces, so there’s a temptation to assign them the same ontological statuswithin each theory. But as has been emphasized throughout this paper, oneprinciple reflected in special relativity is that all inertial frames are physi-cally equivalent. A relativistic 3-force, as part of the spatial component ofa 4-force, simply does not transform in a Lorentz-covariant way. Rather, 3-forces transform on the model of ordinary spatial velocity. Letting S and S (cid:48) be two inertial frames in standard configuration, v the speed of S (cid:48) relativeto S (along the x-axis), (cid:126)u S → ( u , u , u ) = ( dxdt , dydt , dzdt ) the spatial velocityof a particle p in frame S , and (cid:126)f S → ( f , f , f ) the S-components of therelevant 3-force applied to p , the components of the 3-force (cid:126)f (cid:48) experiencedby p in S (cid:48) are given via the following transformations: f (cid:48) = f − v d ( γ u m ) dt (1 − u vc ) f (cid:48) = f γ v (1 − u vc ) See Rindler (1991, p.91) for discussion. (cid:48) = f γ v (1 − u vc ) . The transformation depends upon both the relative velocity between framesand (surprisingly) the velocity of the particle itself on which the force is act-ing. It is evident from these transformations that boosting from one frameto the next changes both the magnitude of the 3-force p experiences and theangular difference between it and p ’s 3-velocity (cid:126)u . That is, if p experiencesa particular 3-force applied in a specific direction in a given frame S , theapplied 3-force in a boosted frame S (cid:48) will generally have both a differentmagnitude and a different direction. There is no frame-independent factabout what 3-force is acting on a particle at any given event, and thus 3-forces are not the sorts of posits that ought to be taken as ontologicallyfundamental. (At least, one cannot take the 3-force as ontologically fun-damental without privileging some particular inertial frame.) Indeed, thereare situations in which a particle experiences no impressed 3-force in itsrest frame but a non-zero 3-force in all other frames, each parallel to thedirection in which the frame is boosted relative to the rest frame. Instead, I propose that we understand interactions between particlesby taking 4-forces as (or as representing something) fundamental. Thisis a natural proposal given the formal connection between 4-forces and 4-momenta changes and the fact that 4-momenta are here taken to encodethe instantaneous dynamical states of particles. As with spatial forces inNewton’s physics, 4-forces on this view are the fundamental things mediatinginteractions between particles and changing their dynamical states.Like 4-momenta, 4-forces possess magnitudes and orientations in space- Only when the force is ‘pure’ (discussed below) and the boost is in the direction of (cid:126)u will it be the case that (cid:126)f = (cid:126)f (cid:48) . See Rindler (1991, p.91) for discussion. Rindler (1991, p.92) calls such forces ‘heatlike’. through spacetime .Whereas the push or pull of a Newtonian force tries to change the waya body is moving through space, a 4-force tries to change the way a bodyis moving through spacetime. That spacetime isn’t isotropic means thatthe character of the push or pull exhibits itself in different ways dependingon: the orientation of the 4-force; the orientation of the 4-momentum of theparticle to which it’s applied; and the frame in which it’s being considered.For example, a 4-force applied orthogonally to a particle’s 4-momentum andconsidered in the rest frame of the particle will manifest itself as a purelyspatial push. In other frames it will also involve a ‘temporal push’, in eachcase rotating the orientation of the particle’s 4-momentum.In summary, then, the fundamental ontology I postulate for special rel-ativistic particle dynamics is one according to which there are two typesof basic entities, particles and 4-forces, each of which possess two distinctfundamental properties—one represented by a scalar magnitude and one as-sociated with an orientation in spacetime. (This is in addition to Minkowskispacetime and any properties that might come along with that.) Many of theentities and properties we typically associate with the dynamics of particlesin special relativity are, on the proposal here, not fundamental.
Returning to the puzzle of composite mass, recall that the issue there wasthat there seemed to be a variety of cases in which a physical process wasoccurring in a system even though none of the fundamental properties ofthe system’s constituents seemed to be changing. I agree with Lange that This view of Newtonian forces as pushes and pulls is emphasized in Wilson (2007).
36o physical conversion occurs between mass and energy when, say, a gasis heated, and yet a real physical process is occurring that is changing thesystem’s mass. If the gas were enclosed in a box, it would be harder tomove after being heated. What we would like is an explanation of how themass of the system is constituted by fundamental physical properties of itsunderlying constituents—properties that are changing as the gas is heated.The trick is to recognize that velocity-dependent quantities are not theonly features of the molecules that are changing in this process. Even ifthe magnitudes of the 4-momenta remain constant, the orientations of the4-momenta are changing in response to whatever 4-forces are transmittingthe heat. On the ontology advocated here, these are fundamental physicalchanges to the dynamical states of the constituent molecules. Recognizingthat the 4-momentum of the total system is just the ordinary vector sumin Minkowski spacetime of its constituent 4-momenta, it follows that thesechanges in 4-momenta orientations at the constituent level give rise to thechanging mass of the gas as a whole. To see this, let’s start with thediagram below for the (much simplified) case of a 2-body system: ct x
For any gas, of course, there are vastly more component 4-momentum vec- As noted in footnote 29, the situation is more complicated for systems whose physicaldescriptions require field-theoretical considerations, for in those cases there isn’t generallya well-defined total 4-momentum that can be associated with the system. and the orientations of the constituent’s4-momenta.Consider now what happens as the system is heated, as the orientationsof the constituent 4-momenta change but not their magnitudes (masses).Due to the geometry of Minkowski spacetime all such equal-magnitude 4-vectors have endpoints that lie along the hyperbola shown in the followingdiagram: ct x
The hyperbola itself is Lorentz invariant, although the inclinations at whichthe 4-vectors are drawn is frame-dependent. Here, the new 4-momenta (thesolid vectors) are drawn in the original rest frame of the system. Noticethat the 4-momentum of the system does not remain constant during theheating process. It retains its orientation in spacetime as the orientations ofits constituent 4-momenta change, but—unlike its constituents—its magni-tude also changes. As the constituent particles move faster in the originalframe—as their 4-momenta rotate their orientations—the magnitude of thesystem’s 4-momentum gets larger. So not only can the composite mass ofthe system be understood as constituted by fundamental physical proper-ties of its molecular constituents (namely, the physical properties encoded in38he constituents’ 4-momentum vectors) but we can understand what genuinephysical changes are occurring at the molecular level when the gas is heatedthat result in changing the mass of the gas as a whole. Unlike changes invelocity and kinetic energy, these are fundamental physical changes, and assuch provide a solution to the original puzzle of composite mass.Although Lange is right that the mass of an n -body system can be ex-pressed as M = ( m + . . . + m n ) + 1 c ( T + . . . + T n ) , the dependence of system mass on constituent energies is a coordinate-dependent manifestation of the more fundamental dependence of systemmass on 4-momenta magnitudes and orientations. Lange’s expression is anartifact of how the magnitude of the system’s total 4-momentum decomposesinto its constituents’ 4-momentum components in the center-of-momentumframe. It does not reflect the genuine physical properties that actually de-termine the mass of the gas. Those are the constituent 4-momentum mag-nitudes and orientations, a fact which (again) is obscured unless one adoptsthe 4-dimensional ontology of particle dynamics proposed here. Turning to inelastic collisions, the 4-Momentum Collision Principle tells ushow the total 4-momentum of an isolated system constrains the way in whichthe 4-momenta magnitudes and orientations of its constituents change whenthey undergo collision. Initially (i.e., pre-collision), the net 4-momenta ofeach ball are oriented in different directions, and that’s why the two collid-ing bodies have a total mass that is more than the sum of the two bodiesthemselves. (That is, the magnitude of the total 4-momentum vector is39reater than the sum of the magnitudes of the two component 4-momentumvectors, as was the case with the gas.) When the two bodies collide inelas-tically, each changes its 4-momentum orientation such that both come tobe oriented in the same direction. Because the 2-body system is isolated,however, the total 4-momentum before and after collision must remain thesame. This means that when the colliding bodies change their 4-momentaupon impact, their new 4-momentum vectors must each be oriented in thesame direction as the total system 4-momentum prior to impact and mustpreserve the magnitude of the total system 4-momentum prior to impact.But this can only happen if, when the orientations change upon impact,their magnitudes increase as well. This means that the mass of each bodygoes up upon impact, which is why if the resulting body post-collision werecut in half, each half would exhibit more inertial resistance than it did pre-collision. Of course, this consequence is an explanation at the ‘principle’level—it doesn’t tell us anything about the underlying dynamical mecha-nism in virtue of which this change might be brought about. That pointwill be addressed in the next section and lies at the heart of what is trulynovel about special relativistic particle dynamics. E = mc Let us (finally) address perhaps the most puzzling case of all—apparent casesof genuine conversion between mass and energy in the context of elementaryparticle collisions. What might really be going on in these reactions, atleast when viewed through the lens of special relativistic particle dynamics?Recall that in electron and positron creation collisions the incoming photonsare massless but the outgoing particles have mass. How do we reconcile40hat’s occurring with the principles of special relativity, without appealingto kinetic energy as a fundamental physical property?If we consider the system of colliding photons that initiate the collisionreaction, their 4-momenta (in green) are represented in the following center-of-momentum frame diagram: ct x
The 2-photon system as a whole clearly possesses a non-zero 4-momentummagnitude (and hence a non-zero mass), even though neither constituentdoes. According to the 4-Momentum Collision Principle this total 4-momentumgets preserved in the interaction, as in the following diagram: ct x
As with the inelastic collision case, we can straightforwardly deduce fromthe 4-Momentum Collision Principle that the output particles will exhibitdifferent inertial resistances than the inputs. For the inertial resistances ofthe output particles are determined by their 4-momentum magnitudes, and41hose in turn are determined by the 4-momentum magnitudes and orienta-tions of the input particles in accordance with the 4-Momentum CollisionPrinciple. There is a genuine transformation of objective physical propertiesin this reaction, but it’s not a conversion between mass and energy. Whatchanges in the course of the reaction are simply the magnitudes and orien-tations of the constituent 4-momenta. Mass is not created out of somethingelse , as the original puzzle seemed to suggest. The transformation in thisreaction is from one set of 4-momenta magnitudes and orientations to an-other. Mass (as 4-momentum magnitude) simply arises in virtue of thesechanges.Still, what we want—what is needed to complete the picture in a satisfy-ing way—is some account of the dynamics that underpins this transforma-tion. Can a dynamical story be given, in terms of 4-momentum magnitudesand orientations, according to which constituent masses ‘emerge’ in thisreaction? This is something that my interpretation of special relativisticparticle ontology ought to provide. (Again, we shouldn’t expect special rel-ativity to offer an explanation of this specific collision reaction, which isquantum in nature.)The answer, I think, turns on the nature and reality of 4-forces. Whenthe incoming constituents collide, they experience 4-forces that change their4-momenta. Most 4-forces are such that they change only the orientationbut not the magnitude of a particle’s 4-momentum, as in the case of theheated gas. A 4-force of this sort has (in a frame) the following components: F µ = γ u ( cm dγ u dt , (cid:126)f ) , where the associated relativistic 3-force f = m dγ u dt (cid:126)u + γ u m(cid:126)a
42s called a ‘pure’ force. The electromagnetic force on a charged body isthe canonical example of a pure 3-force. As Rindler (1991, p.92) shows, anecessary and sufficient condition for a 3-force to be pure is that its asso-ciated 4-force F always satisfies U · F = 0. As a consequence, if a 4-forcealways acts orthogonally in Minkowski spacetime to a particles 4-velocity(4-momentum), then the magnitude of the 4-momentum will be preserved.However, a second possibility is also built into the dynamical structure of F = d P dτ – namely, that a 4-force might act in such a way as to change the magnitude of a particle’s 4-momentum. Such forces can be expressed in aframe as F µ = γ u ( cm dγ u dt + cγ u dmdt , (cid:126)f ) , and their associated 3-forces f = ( m dγ u dt + γ u dmdt ) (cid:126)u + γ u m(cid:126)a are said to be ‘impure’. By their dynamical action alone, 4-forces of thissort are capable of changing the mass of a particle. Naturally, a necessaryand sufficient condition for a 3-force to be impure is that U · F (cid:54) = 0 hold forthe associated 4-force. Rindler (1991, p.92) shows using this condition thatany 4-force derivable from a 4-scalar potential Φ via an equation of the form F µ = ∂ Φ ∂x µ , such as the scalar meson theory of the nucleus, must be impure.Return now to the electron-positron creation reaction and to a diagramof possible timelike oriented outputs: A special case of such forces are what Rindler (1991, p.92) dubs ‘heatlike’ forces, theaction of which only changes a particles 4-momentum magnitude and not its orientation. t x As one can see from the diagram, if the outputs are not null-oriented the4-forces applied must be such that U · F (cid:54) = 0. (Recall that for a null 4-vector U in Minkowski spacetime, U · F = 0 iff F is parallel to U .) So if the 4-forcesexerted are such that they change the orientations of null 4-momenta, theymust also change their magnitudes . This holds because the applied forces inthe reaction (whatever they are exactly) must be impure, and it drops rightout of the dynamical equations. Mass, then, as the property of inertial resistance, can be created anddestroyed in special relativistic dynamics, and it is changes in 4-momentamagnitudes brought about by impure 4-forces that provide the dynamicalmechanisms for these changes. This is what is truly dynamically novelabout special relativistic particle mechanics: mass can be created and de-stroyed by 4-forces. Now, as it happens, the application of a 4-force (ingeneral) also changes the energy of a particle, although the rate of thatchange—just like the magnitude and the spatial direction of the 3-force—isframe-dependent. This misleads us into thinking that in some cases there A similar story can be given for the case of inelastic collisions, as discussed in thepreceding section. Evidently the 4-forces at work in such a collision must be impure so asto change the magnitudes of the incoming particles’ 4-momenta. Many textbooks are blithely indifferent to the possibility of impure forces. See, e.g.,French (1968, p.215) and Freund (2008, p.192), where dmdt = 0 is assumed without anycomment whatsoever.
44s a conversion between energy and mass. But there is no such conversion.Instead, what happens is that impure 4-forces change the 4-momentum mag-nitudes of the particles on which they act while also changing the energiesof those particles.
What E = mc tells us is how changes in particle massbrought about by 4-forces are correlated with changes in energy brought aboutby those same 4-forces. But the change in mass is a fundamental physicalchange, whereas the change in energy is a change in a non-fundamental orderivative property. That there is a precise and fixed correspondence be-tween these two changes is an important empirical fact, arising from theaction of 4-forces and facts about their frame-dependent decompositions. Inthis sense E = mc does encode a profound and novel dynamical discovery,but that discovery can’t be read off of the equation itself and it holds invirtue of the nature and existence of 4-forces. E = mc does not meanwhat it is often taken to mean, or what Einstein took it to mean.
10 Conclusion
Although often taken to be the central insight of special relativistic particledynamics, the alleged equivalence between energy and mass expressed inEinstein’s famous equation E = mc remains both controversial amongstphysicists and conceptually problematic, as it contravenes a central princi-ple regarding the content and interpretation of special relativity—namely,that all inertial frames are physically equivalent. Nevertheless, the apparentinter-convertability of energy and mass seems to have prima facie exper- In this sense, one really ought to stick to writing the equation as ∆ E = ∆ mc , whichis 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 the4-momentum magnitudes changing on account of the applied (impure) 4-forces. eferences
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