Newton s Second Law is Valid in Relativity for Proper Time

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1 Newton s Second Law is Valid in Relativity for Proper Time Steven Kenneth Kauffmann Abstract In Newtonian particle dynamics, time is invariant under inertial transformations, and speed has no upper bound. In special relativity, it is the observed particle s proper time, rather than the arbitrary observer s time, which is inertial-transformation invariant, and it is the particle s proper-time speed which has no upper bound. It is thus perhaps not surprising the Newton s Second Law is also valid in relativistic particle dynamics presented in terms of the particle s proper time. This follows from the special-relativistic dynamical principle that the time derivative of special-relativistic momentum is equal to the applied force, provided proper force is defined to have an additional factor of gamma. The four-vector fully Lorentz-covariant completion of proper force is obliged to have zero contraction with the particle s proper four-velocity. A special-relativistic particle in either a scalar or a four-vector electromagnetic potential is shown to adhere to the proper-time Newton s Second Law, and when it is consistently taken into account that a metric gravitational potential modifies the rate of change of a particle s proper time with observer time, that adherence is shown to hold for metric potentials as well. Observed versus proper-length constant velocity of a special relativistic object An object s constant velocity can be calculated by dividing the vector segment of its trajectory which it instantaneously intersects by the time it requires to traverse that segment, but special-relativistic observed length contraction of that trajectory segment by the factor γ 1 [1], where, 1 def γ = 1 ṙ/c , 1a implies that the object s special-relativistic observed constant velocity ṙ equals γ 1 times its proper-length constant velocity, i.e., its proper-length constant velocity equals γṙ, whose magnitude γṙ = ṙ 2 / 1 ṙ/c is unbounded, notwithstanding that ṙ < c. Moreover, γṙ is equal to dr/dτ, the object s velocity calculated using its Lorentz-invariant differential proper time dτ, which of course is, dτ = dt 2 dr/c b That γṙ is equal to dr/dτ is a general fact because, γ = 1/ 1 ṙ/c = dt 2 / dt 2 dr/c = dt 2 /dτ = dt/dτ γṙ = dt/dτṙ = dt/dτdr/dt = dr/dτ. 2 An object s Lorentz-transformation invariant differential proper time dτ is somewhat analogous to the Galilean-transformation invariant time of Newtonian physics and, since an object s proper-time speed dr/dτ = γṙ is unbounded, that speed is somewhat analogous to the unbounded speed of Newtonian physics. The Lorentz-covariant proper-time extension of Newton s Second Law The usual presentation of single-particle special-relativistic dynamics is, dp/dt = f, 3a where f is the force and the relativistic single-particle momentum p is given by, p = mγṙ, 3b where m is the particle s rest mass. From Eq. 2 we see that Eq. 3b can be rewritten, so Eq. 3a becomes, p = mdr/dτ, mddr/dτ/dt = f. 3c 3d Retired, American Physical Society Senior Life Member, SKKauffmann@gmail.com 1

2 We now multiply the left side of Eq. 3d by dt/dτ and its right side by γ, as per Eq. 2, which yields, mddr/dτ/dtdt/dτ = γf. 3e We simplify the left side of Eq. 3e and denote γf on its right side as the proper force F to obtain, m d 2 r/dτ 2 = F, 3f the relativistic extension of Newton s Second Law via proper time. An example of Eq. 3f is the proper force exerted by an electromagnetic field on a particle of charge e, namely, F = eγe + ṙ/c B. 3g The fully Lorentz-covariant four-vector completion of Eq. 3f must of course read, m d 2 x µ /dτ 2 = F µ, 3h but the nature of proper time ensures that only three of the four components of the proper force F µ can be mutually independent. We begin the demonstration of this fact by using Eq. 2 to show that, dx µ /dτ dx µ /dτ = ẋ µ ẋ µ dt/dτ 2 = ẋ µ ẋ µ γ 2 = c 2 ṙ 2 / 1 ṙ/c 2 = c 2, 3i which furthermore implies that, d 2 x µ /dτ 2 dx µ /dτ = 2ddx 1 µ /dτdx µ /dτ/dτ = 2 1 d c 2 /dτ = 0. 3j Eq. 3h together with Eq. 3j implies that, F µ dx µ /dτ = m d 2 x µ /dτ 2 dx µ /dτ = 0. 3k Therefore only three of the four components of the proper force F µ can be mutually independent. In greater detail, Eq. 3k together with Eq. 2 yields that, 0 = F µ dx µ /dτ = F µ ẋ µ dt/dτ = F 0 c F ṙ / 1 ṙ/c 2 1 2, which implies that F 0 = F ṙ/c. 3l We thus see that F 0 vanishes altogether in the nonrelativistic limit ṙ/c 0, for which it is also true that dt/dτ = γ = 1 ṙ/c , so in the nonrelativistic limit Eq. 3h reduces to Newton s m r = f. Eq. 3f shows that the concept of inertial mass, which is the same as rest mass, is just as relevant to relativistic physics as it is to Newtonian physics. Indeed, the development of Higgs field physics [2] has elaborated the inertial mass concept. An intriguing inertia issue is the existence of particles, e.g., photons, which have zero inertial mass these are asserted to not couple at all to the Higgs field. According to Eq. 3c, a zero-inertial-mass particle which has nonzero momentum p > 0 has infinite proper-time speed because dr/dτ = lim m 0 p /m =. We now show that this corresponds to observed speed ṙ being c by inverting the Eq. 2 relation of proper-time velocity dr/dτ to observed velocity ṙ, namely, dr/dτ = γṙ = 1 ṙ/c 2 12 ṙ. The inverse of this relation is, which has the asymptotic form, ṙ = dr/dτ 1 + dr/dτ/c 2 1 2, 3m ṙ c dr/dτ/ dr/dτ as dr/dτ/c. 3n This result shows that zero-inertial-mass particles of nonzero momentum p > 0, which therefore have infinite proper-time speed dr/dτ = lim m 0 p /m =, consequently have observed speed ṙ equal to c. We next work out the specific proper force F µ exerted on a relativistic particle of mass m by 1 a scalar potential φx α, 2 a four-vector electromagnetic potential A µ x α and 3 a dimensionless second-rank symmetric-tensor metric gravitational potential g µν x α. To obtain the needed three equations of motion, we first must construct the three relativistic single-particle Lagrangians for these three potentials. 2

3 A rest-frame approach to constructing special-relativistic particle Lagrangians A mass m, ṙ = 0 special-relativistic particle has energy mc 2 plus its ṙ = 0 rest potential energy V rest, H rest = mc 2 + V rest, 4a so since the usual Lagrangian term ṙ p vanishes entirely when ṙ = 0, the special-relativistic particle s rest Lagrangian L rest and consequent rest action S rest are, L rest = H rest = mc 2 + V rest Srest = mc 2 + V rest dt. 4b The special-relativistic extension of S rest is required to be Lorentz invariant, and therefore is of the form, S inv = mc 2 + V inv dτ = mc 2 + V inv dτ/dtdt, 4c where dτ is the particle s Lorentz-invariant differential proper time, and V inv is its Lorentz-invariant potential energy, which must reduce to V rest in the limit ṙ 0. Extending V rest to the Lorentz-invariant V inv is dealt with case-by-case. Eq. 4c implies that the full special-relativistic Lagrangian L rel is given by, L rel = mc 2 + V inv dτ/dt. 4d The proper force exerted by a scalar potential A relativistic particle of mass m which couples to a scalar potential φx α with dimensionless coupling strength k has both V φ rest and V φ inv equal to kφ, so from Eqs. 4d and 2, L φ = mc 2 + kφ dτ/dt = mc 2 + kφ γ 1 = mc 2 + kφ 1 ṙ/c a Since the generic equation of motion implied by any single-particle Lagrangian L is, d L/ ẋ i /dt = L/ x i, where i = 1, 2, 3, 5b it is very useful in the case of the Lagrangian L φ of Eq. 5a to note that, γ 1 / ẋ i = c 2 γẋ i = c 2 dt/dτẋ i = c 2 dx i /dτ. 5c From Eqs. 5a 5c we obtain that, d m + kφ/c 2 dx i /dτ /dt = k φ/ x i dτ/dt. 5d Upon multiplying both sides of Eq. 5d by dt/dτ and noting that x i = x i, it becomes, d m + kφ/c 2 dx i /dτ /dτ = k φ/ x i, 5e whose fully Lorentz-covariant four-vector completion clearly is, d m + kφ/c 2 dx µ /dτ /dτ = k φ/ x µ. 5f Noting that dφ/dτ = φ/ x ν dx ν /dτ, we carry out the outer d/dτ differentiation on the left side of Eq. 5f and then move all terms except m d 2 x µ /dτ 2 to its right side to obtain, m d 2 x µ /dτ 2 = k [ φ/ x µ 1/c 2[ dx µ /dτ φ/ x ν dx ν /dτ + φ d 2 x µ /dτ 2]] = F µ, 5g where F µ is the proper force φx α exerts on a mass m particle of coupling strength k. By applying the identities given by Eqs. 3i and 3j, namely that, dx µ /dτdx µ /dτ = c 2 and d 2 x µ /dτ 2 dx µ /dτ = 0, 5h we verify that the Eq. 5g proper force F µ satisfies the requirement F µ dx µ /dτ = 0 of Eq. 3k. 3

4 We also note that if the scalar potential φx α is constant in x α, Eq. 5g implies that, m + kφ/c 2 d 2 x µ /dτ 2 = 0, 5i i.e., the particle s mass m is effectively modified by the addition to it of the constant term kφ/c 2. The Higgs field is thought of as such a constant scalar potential which is able to give an effective mass to otherwise zero-mass particles if they have nonzero dimensionless coupling strength k with that scalar potential [2]. The proper force exerted by a four-vector electromagnetic potential A particle of mass m and charge e at rest in a four-vector electromagnetic potential A µ x α has potential energy Vrest Aµ = ea 0, with Lorentz-invariant extension Vinv Aµ = e/cdx ν/dτa ν. Thus from Eqs. 4d and 2, L A µ = mc 2 + e/cdx ν /dτa ν dτ/dt = mc 2 γ 1 e/cẋ ν A ν = mc 2 γ 1 ea 0 + e/cṙ A. 6a Applying Eqs. 5b and 5c to the Eq. 6a Lagrangian L A µ yields, d mdx i /dτ + e/ca i /dt = e/cẋ ν A ν / x i. 6b Multiplying both sides of Eq. 6b by dt/dτ and noting that x i = x i produces, d mdx i /dτ + e/ca i /dτ = e/cdx ν /dτ A ν / x i, 6c which we reexpress as, m d 2 x i /dτ 2 = e/c [ dx ν /dτ A ν / x i da i /dτ ]. 6d Since da i /dτ = A i / x ν dxν /dτ, we can rewrite Eq. 6d as, m d 2 x i /dτ 2 = e/cdx ν /dτ [ A ν / x i A i / x ν ], 6e whose fully Lorentz-covariant four-vector completion clearly is, m d 2 x µ /dτ 2 = e/cdx ν /dτ[ A ν / x µ A µ / x ν ]= F µ, 6f where F µ is the proper force A µ x α exerts on a mass m particle of charge e. The proper force F µ satisfies the requirement F µ dx µ /dτ = 0 of Eq. 3k because dx ν /dτdx µ /dτ is symmetric under interchange of ν and µ, whereas [ A ν / x µ A µ / x ν ] is antisymmetric under that interchange. Eq. 6f also implies Eq. 3g, since for µ = i = 1, 2, or 3, F i = e/cγẋ ν [ A ν / x i ] A i / x ν = eγ [ A 0 / x i 1/c A i] 3 ẋj + e/cγ [ A j / x i A i / x j] = eγ r A 0 1/cȦ i + e/cγ r ṙ A ṙ r A i = j=1 6g eγ E + ṙ/c r A i = eγ E + ṙ/c B i. The proper force exerted by a metric gravitational potential A sufficiently simple special-relativistic dynamical system is coupled to a dimensionless symmetric-tensor metric potential g µν x α by substituting g µν x α for occurrences of the Minkowski metric tensor η µν in the system s Lorentz-invariant action. To study a single particle s interaction with g µν x α only, the Lorentzinvariant action in which occurrences of η µν are replaced by g µν x α must be that of the free particle, i.e., S free = mc 2 dτ, 7a whose Lorentz-invariant proper differential time dτ is given by Eq. 1b, dτ = dt 2 dr/c = dx µ dx µ 1 2/c = dx µ η µν dx ν 1 2/c. 7b 4

5 Having expressed dτ in terms of η µν, we replace η µν by g µν x α, which changes the Eq. 7a free-particle action S free to the following action S gµν for the interaction of the particle with the metric potential g µν x α, S gµν = mcdx µ g µν dx ν 1 2 = mcẋ µ g µν ẋ ν 1 2 dt, from which the Lagrangian L gµν for the interaction of the particle with g µν x α follows, L gµν = mcẋ µ g µν ẋ ν 2 1 = mc ẋ 0 g 00 ẋ 0 + 2ẋ 0 3 j=1 g 0jẋ j j=1 k=1 ẋj g jk ẋ k 2, where ẋ 0 = c. 7c Since L gµν is given in the observer s time t, presenting its equation of motion in the particle s proper time τ requires dτ/dt we saw above that the particle s coupling to g µν x α changed dτ from dx µ η µν dx ν 1 2/c to dx µ g µν dx ν 2/c, 1 which implies that that coupling correspondingly changed dτ/dt from ẋ µ η µν ẋ ν 2/c 1 to, dτ/dt gµν def = ẋ µ g µν ẋ ν 1 2/c. 7d A crucial physics-related restriction on g µν x α is that for all x α, its four matrix eigenvalues are required have the same signs as the matrix eigenvalues of η µν, namely {+,,, } [3]. Therefore for all x α, g µν x α has a matrix inverse, which is conventionally denoted g λκ x α. Thus, for example, g λκ x α g κν x α = δ λ ν. 7e Before we work out the equation of motion implied by the Eq. 7c Lagrangian L gµν, we note the generalizations of the proper-velocity and proper-acceleration identities given by Eqs. 3i and 3j which ensue in the presence of a metric potential g µν x α. The Eq. 3i identity s generalization is easy to surmise, i.e., dx µ /dτg µν dx ν /dτ = ẋ µ g µν ẋ ν /dτ/dt gµν 2 = c 2, 7f where the last equality follows from Eq. 7d. The Eq. 3j identity s generalization is then obtained via differentiation with respect to τ of the Eq. 7f identity, 0 = d c 2 /dτ = ddx µ /dτg µν dx ν /dτ/dτ = 2 d 2 x µ /dτ 2 g µν dx ν /dτ + dx µ /dτ g µν / x κ dx κ /dτdx ν /dτ, 7g which implies the following generalization of the Eq. 3j identity, d 2 x λ /dτ 2 g λγ dx γ /dτ = 2dx 1 µ /dτ g µν / x κ dx κ /dτdx ν /dτ. 7h Given this identity, a purported proper force F λ on a particle of mass m that is claimed to adhere to, m d 2 x λ /dτ 2 = F λ, 7i must be such that it satisfies the consistency requirement, F λ g λγ dx γ /dτ = 1 2mdx µ /dτ g µν / x κ dx κ /dτdx ν /dτ. 7j We now work out the equation of motion implied by the Eq. 7c Lagrangian L gµν = mcẋ µ g µν ẋ ν 1 2, L gµν / ẋ i = 1 2mc 2g i0 ẋ j=1 g ijẋ /ẋ j µ g µν ẋ ν 2 1 = mcg iν ẋ ν /cdτ/dt gµν = mg iν dx ν /dτ, 7k and, L gµν / x i = 1 2mcẋ µ g µν / x i ẋ ν /ẋ µ g µν ẋ ν 2 1 = 1 2mcẋ µ g µν / x i ẋ ν /cdτ/dt gµν = 1 2mẋ µ g µν / x i dx ν /dτ. Inserting the Eq. 7k and 7l results into the generic Eq. 5b equation of motion produces, mdg iν dx ν /dτ/dt = 1 2mẋ µ g µν / x i dx ν /dτ. 7l 7m 5

6 After dividing both sides of Eq. 7m by dτ/dt gµν, this equation of motion can be reexpressed as, mdg iν dx ν /dτ/dτ = 1 2mdx µ /dτ g µν / x i dx ν /dτ. 7n The four-vector completion of Eq. 7n clearly is, mdg κν dx ν /dτ/dτ = 1 2mdx µ /dτ g µν / x κ dx ν /dτ. 7o Carrying out the outer d/dτ differentiation on the left side of Eq. 7o yields two terms, mg κν d 2 x ν /dτ 2 mdx µ /dτ g κν / x µ dx ν /dτ = 1 2mdx µ /dτ g µν / x κ dx ν /dτ. 7p Because dx µ /dτdx ν /dτ is symmetric under interchange of µ and ν, Eq. 7p can be rewritten as, mg κν d 2 x ν /dτ 2 = 1 2mdx µ /dτ[ g κν / x µ + g κµ / x ν g µν / x κ ]dx ν /dτ. 7q Making use of Eq. 7e, we multiply both sides of Eq. 7q by g λκ and sum over the index κ to obtain, m d 2 x λ /dτ 2 = 1 2mg λκ dx µ /dτ[ g κν / x µ + g κµ / x ν g µν / x κ ]dx ν /dτ = F λ, 7r where F λ is the proper force g µν x α exerts on a mass m particle. To check that F λ satisfies the consistency requirement of Eq. 7j, we use g λκ = g κλ to rewrite F λ as, F λ = 1 2mdx µ /dτ[ g κν / x µ + g κµ / x ν g µν / x κ ]dx ν /dτg κλ, 7s which, since g κλ g λγ = δ κ γ, yields that, F λ g λγ dx γ /dτ = 1 2mdx µ /dτ[ g κν / x µ + g κµ / x ν g µν / x κ ]dx ν /dτdx κ /dτ = 1 2mdx µ /dτ g µν / x κ dx κ /dτdx ν /dτ, 7t as required by Eq. 7j, where the last equality ensues after appropriately renaming contracted indices. It is to be noted that Eq. 7r is conventionally written using the Christoffel symbol Γ λ µν, i.e. [4], d 2 x λ /dτ 2 + dx µ /dτγ λ µνdx ν /dτ = 0 where, Γ λ µν = 1 2g λκ [ g κν / x µ + g κµ / x ν g µν / x κ ]. 7u References [1] Length contraction Wikipedia, contraction. [2] Higgs field Simple Wikipedia, field. [3] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity John Wiley & Sons, New York, 1972, Section 3.6, pp [4] S. Weinberg, op. cit., Eq , p. 71 and Eq , p