Abstract
The aim of this paper is to address, from a fresh perspective, the question of whether Newtonian mechanics can legitimately be regarded as a limiting case of the special theory of relativity (STR), or whether the two theories should be deemed so radically different as to be incommensurable in the sense of Feyerabend and Kuhn. Firstly, it is argued that focusing on the concept of mass and its transformation across the two varieties of mechanics is bound to leave the issue unsettled. On the one hand, the idea of a speeddependent ‘relativistic mass’, which has been invoked in support of incommensurability claims, results from a particular, often innocuous but unnecessary and inappropriate reading of certain basic formulae. On the other hand, the existence of an invariant rest mass in STR does not warrant its identification with the Newtonian mass, be it in a suitable limit. This invariant notwithstanding, those who follow Feyerabend and Kuhn can still uphold their views with regard to the two theories. It is shown, however, that the two mechanics embody relativistic frameworks that are direct consequences of the same set of assumptions. As a result, if Newton’s mechanics cannot simply be regarded as a limiting case of STR, the possibility of ‘recovering’ from the latter some elements of the former can be traced to a common source, belying claims of logical disconnection between the two theories.
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Notes
 1.
Feynman’s emphasis on m as a nonclassical notion of mass also appears to be at odds with his diagrammatic method for the calculation of terms in QED, since that method hinges on the concept of a Lorentzinvariant mass, i.e., on \(m_0 \), rather than any framedependent mass.
 2.
For the sake of consistency, the symbols m and \(m_0\) have been substituted in the quoted passage wherever Einstein uses M and m. Notational ambiguity is pervasive in this matter and a hindrance to clarification.
 3.
Inertia only reduces to mass (\(m_{0 } )\) in the rest frame of the body, i.e as \(p=0\) in the fundamental relation \(E^{2}=p^{2}c^{2}+m_{0} ^{2}c^{4}\). The case of light is special in that \(E=pc\) : there is no such rest frame and \(m_0 =0)\). Einstein (1935) includes rare emphasis on the energymass relation being \(E_0 =m_0 c^{2}\), where \(E_0 \) is the rest energy, and not \(E=mc^{2}\), where \(m=\gamma m_0 \). In contrast, Bergmann’s (1942) book, endorsed by Einstein, makes explicit reference to ‘relativistic mass’ (besides ‘rest mass’), but without discussing its physical significance.
 4.
See e.g., Sandin (1991) for a rare defence, in recent years, of \(\gamma m_0 \) as the proper concept of mass in STR.
 5.
Flores (2005), who discusses various positions regarding the significance of massenergy equivalence and the idea of conversion between one and the other, appears to regard \(E=mc^{2}\) as a correct expression of such equivalence.
 6.
As emphasized e.g., by Torretti (1990, Sect. 2.6.5).
 7.
Excluding photons, for which there is no rest frame and correspondingly no PIM (see footnote 3).
 8.
 9.
Uniform translational (rectilinear) motion means that, given a uniform method for measuring time and uniformly laid out spatial markers, distance covered is proportional to the time of coverage.
 10.
Homogeneity should not be regarded as reflecting intrinsic properties of space or time, but as a sensible and convenient demand made on certain transformations. Thus, in the theory of system processing and control, homogeneity comes down to the requirement that a change in the amplitude of the input signal should give rise to a proportional change in the output signal’s amplitude. The resulting transformation is linear.
 11.
The abovesketched derivation, like those footnote 8 refers to, hinges on the convenient reduction to 1 of the number of relevant spatial dimensions. As Feigenbaum (2008) points out, this oversimplifies the matter by reducing isotropy to parity. However, the outcome of Feigenbaum’s threedimensional treatment mainly differs from more ‘pedestrian’ approaches in that isotropy and homogeneity (or rather UTUM) jointly suffice to determine a group law, which does not need to be assumed from the outset. If the LorentzEinstein transformation is identical with \(T_{R\rightarrow R^{\prime }} \) upon identification of \(\kappa \) with c, this is essentially because, since speed is a scalar, basing a RF on the constancy of the speed (not the velocity) of anything, be it light, implicitly assumes rotational invariance. This assumption goes a long way towards determining the form of the transformation. Isotropy is not enough, however, to uniquely select the Lorentz group (rather than the conformal group). Nonlinearity, in particular, remains possible unless the homogeneity/UTUM assumption is brought in.
 12.
The limitations of this approach are those of any mathematical idealization. We do not live in a frictionless and gravityfree world—if there could be such a world. Uniform rectilinear motion is an abstraction, and so are true inertial frames. However, Einstein’s ‘geometrodynamics’ (Misner et al. 1973)—which is not a more general theory of relativity—gives way to STR dynamics whenever curvature effects induced by gravity are deemed negligible, so that in effect the geometry becomes indistinguishable from that of Minkowski ‘flat’ spacetime.
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Appendix: derivation of relativistic transformations
Appendix: derivation of relativistic transformations
Given two inertial frames \(R\equiv (O;x,y,z)\) and \(R^{\prime }\equiv (O^{\prime };x^{\prime }, y^{\prime },z^{\prime })\), we seek a general transformation \(T_{R\rightarrow {R}^{\prime }}\) such that the expression P(x, y, z, t) of a physical quantity, relative to frame R, systematically transforms into \(Q\left( {x^{\prime },y^{\prime },z^{\prime },t^{\prime }} \right) \) relative to \(R^{\prime }\), where t and \(t^{\prime }\) refer to clock readings in R and \(R^{\prime }\) respectively. Isotropy warrants choosing the direction of the relative rectilinear uniform motion of R and \(R^{\prime }\) as that of an arbitrary axis (Ox) with respect to R, and that of (\(O^{\prime }x^{\prime })\) with respect to \(R^{\prime }\), such that (Ox) and (\(O^{\prime }x^{\prime })\) are parallel. Since directions orthogonal to (\(Ox)/(O^{\prime }x^{\prime })\) do not contribute anything nontrivial to the changeofframe transformation \(T_{R\rightarrow {R}^{\prime }} \), and assuming that the origins and three pairs of parallel axes (\(Ox)/(O^{\prime }x^{\prime })\), (\(Oy)/(O^{\prime }y^{\prime })\) and (\(Oz)/(O^{\prime }z^{\prime })\) coincide when clocks \(C_{R}\) in R and \(C_{{R}^{\prime }}\) in \(R^{\prime }\) both read 0, then \(y=y^{\prime }\) and \(z=z^{\prime }\) for all readings of the two clocks. Deriving \(T_{R\rightarrow {R}^{\prime }}\) then reduces to working out f and g such that \(x^{\prime }=f\left( {x,t} \right) \) and \(t^{\prime }=g\left( {x,t} \right) \).
Spatial homogeneity requires that the difference \(f\left( {x_2 ,t} \right) f\left( {x_1 ,t} \right) \) (‘rod length’ in the direction of motion) should not be affected by an arbitrary shift in the x coordinate i.e.
Dividing by \(\varepsilon \) and taking the \(\varepsilon \rightarrow 0\) limit, this condition amounts to \(\left. {\frac{\partial f}{\partial x}} \right _{x_1 } =\left. {\frac{\partial f}{\partial x}} \right _{x_2 } \): given the arbitrariness of \(x_1 \) and \(x_2 \), f is a linear function of x, and so is g. Similar considerations with respect to the endpoints of any time interval lead to the conclusion that f and g are linear functions of time. Alternatively, since \(dx^{\prime } = ~\frac{{\partial f}}{{\partial x}}dx + ~\frac{{\partial f}}{{\partial t}}dt\) and \(dt^{\prime } = ~\frac{{\partial g}}{{\partial x}}dx + ~\frac{{\partial g}}{{\partial t}}dt\) , that uniform motion in R appears as uniform in \(R^{\prime }\) (UTUM) requires all partial derivatives to be independent of x and t. \(T_{R\rightarrow {R}^{\prime }}\) thus takes the form \(\left\{ {{\begin{array}{l} {x^{\prime }=\gamma x+\delta t} \\ {t^{\prime }=\alpha x+\beta t} \\ \end{array} }} \right. \), where the coefficients \(\alpha , \beta \), \(\gamma \), \(\delta \) depend only on the relative velocity of the two frames.
As seen from R, the origin \(O^{\prime }\) of \(R^{\prime }\) (\(x^{\prime }=0)\) is such that \(\gamma x=\delta t\): it is observed to be moving with the velocity \(v=\frac{\delta }{\gamma }\). From the viewpoint of \(R^{\prime }\), the origin O of R (\(x=0)\) moves in the opposite direction with velocity \(v=\frac{\delta }{\beta }\) . Therefore, \(\beta =\gamma \) and since \(\delta =\gamma v\), this reduces the task to working out the form of two unknown coefficients \(\alpha \) and \(\gamma \), functions of v only, such that \(T_v \equiv T_{R\rightarrow {R}^{\prime }} : \left\{ {{\begin{array}{l} {x^{\prime }=\gamma \left( v \right) x\gamma \left( v \right) vt} \\ {t^{\prime }=\alpha \left( v \right) x+\gamma \left( v \right) t} \\ \end{array} }} \right. \).
\(T_v \) should be invariant under a reflection in the parallel planes \((yz)/(y'z')\). If \(\tilde{R}\) is the image of R under such reflection, and \(\tilde{R^{\prime }}\) that of \(R^{\prime }\), then \(\tilde{x} = x, \widetilde{x^{\prime }} =x^{\prime }, {\tilde{t}} =t, \tilde{t^{\prime }} =t^{\prime }\). As seen from \(\tilde{R}, \tilde{R^{\prime }}\) is in uniform rectilinear motion with velocity \(v\) i.e. \(T_{\tilde{R}\rightarrow \tilde{R^{\prime }}}\, \equiv \, T_{v} \) . It follows that \(\gamma \) is an even function of \(v: \gamma \left( {v} \right) =\gamma \left( v \right) \), and that \(\alpha \) is odd: \(\alpha \left( {v} \right) =\alpha \left( v \right) \) (the same conclusions obtain upon considering time reversal instead of a mirror image).
A third inertial frame \(R^{{\prime }{\prime }}\) is introduced, which is in rectilinear uniform motion relative to \(R^{\prime }\), with velocity \(v^{\prime }\), in the (\(O^{{\prime }{\prime }}x^{{\prime }{\prime }})\) direction parallel to (Ox). A clock \(C_{{R}^{{\prime }{\prime }}}\) at rest in \(R^{{\prime }{\prime }}\) shows time \(t^{{\prime }{\prime }}\). The origins and axes of all three frames are most conveniently chosen so as to coincide when all of their respective clocks read 0. Requiring that \(T_{v^{\prime }} T_v =T_{v^{\prime \prime }} \), where \(v^{{\prime }{\prime }}\) is the velocity of \(R^{{\prime }{\prime }}\) relative to R, is equivalent to the set of simultaneous equations:
where \(\alpha ^{\prime }\equiv \alpha (v^{\prime }),\gamma ^{{\prime }{\prime }}\equiv \alpha \left( {{v}^{{\prime }{\prime }}} \right) )\) etc.
From \(\left( a \right) \) and \(\left( d \right) : \frac{\alpha ^{\prime }}{{{\gamma }^{{\prime }}{v^{\prime }}}}=\frac{\alpha }{\gamma v}\) and since \(\frac{\alpha }{\gamma v}\) depends on v only, and \(\frac{\alpha ^{\prime }}{{{\gamma }^{{\prime }}{v^{\prime }}}}\) on \(v^{\prime }\), equality implies that those ratios are independent of the relative velocity of the frames, hence equal to a ‘universal’ constant \(\lambda \). Given \(\alpha =\lambda \gamma v\), dividing \(\left( b \right) \) by \(\left( a \right) \) then yields the composition law for velocities: \(v^{{\prime }{\prime }}=\frac{v+v^{\prime }}{1\lambda vv^{\prime }}\) .
From \(\left( a \right) \) with \({v}^{\prime }= v\) and \(v^{{\prime }{\prime }}=0~: \gamma \left( v \right) \gamma \left( {v} \right) =\frac{\gamma \left( 0 \right) }{1+\lambda v^{2}}\), and letting \(v=0\, (\gamma \ne 0)\) implies \(\gamma \left( 0 \right) =1\). Since \(\gamma \) is an even function of v, \(\gamma \left( v \right) \gamma \left( {v} \right) =\gamma \left( v \right) ^{2}\), hence \(\gamma \left( v \right) =\frac{1}{\sqrt{1+\lambda v^{2}}}\) . This completes the derivation of \(T_v \) : \(\left\{ {{\begin{array}{l} {x^{\prime }=\frac{xvt}{\sqrt{1+\lambda v^{2}}}} \\ {t^{\prime }=\frac{t+\lambda vx}{\sqrt{1+\lambda v^{2}}}} \\ \end{array} }} \right. \)

\(\lambda =0\) yields the Galilei transformation: \(\left\{ {{\begin{array}{l} {x^{\prime }=xvt} \\ {t^{\prime }=t} \\ \end{array} }} \right. \).

If \(\lambda < 0\), then with \(\lambda =\frac{1}{\kappa ^{2}}, T_{R\rightarrow {R}^{\prime }} \equiv T_{v,\kappa } :\left\{ {{\begin{array}{l} {x^{\prime }=\frac{xvt}{\sqrt{1\frac{v^{2}}{\kappa ^{2}}}}} \\ {t^{\prime }=\frac{t\frac{v}{\kappa ^{2}}x}{\sqrt{1\frac{v^{2}}{\kappa ^{2}}}}} \\ \end{array} }} \right. \),
which is such that \(x^{{\prime }{2}}\kappa ^{2}t^{{\prime }{2}}=x^{2}\kappa ^{2}t^{2}\), a nonEuclidean metric invariant that suggests formulating the (S)RF in terms of a fourdimensional spacetime, in contrast with the ‘3+1’ formulation of GalileiNewtonian mechanics. \(T_{v,c}\) is the Lorentz transformation given identification of \(\kappa \) with the speed of light c.
As developed in Sect. 3, dynamics in the \(\lambda < 0\) case is subject to an energymomentum changeofframe transformation, the form of which is a transposition of \(T_{v,\kappa } \):
The corresponding invariant \(E^{{\prime }2}p^{{\prime }{2}}\kappa ^{2}=E^{2}p^{2}\kappa ^{2}\) is \(E_0 ^{2}=\mu \kappa ^{2}\), where \(\mu \) can be interpreted as a Proper Inertial Mass (PIM) and \(E_0 \) is the total energy in the rest i.e. ‘zeromomentum’ frame of the object.
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Delhôtel, J. Relativistic frameworks and the case for (or against) incommensurability. Synthese 195, 1569–1585 (2018). https://doi.org/10.1007/s112290161283x
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Keywords
 Special theory of relativity
 Relativistic mass
 Covariance
 Incommensurability
 Theory change