Four-Dimensional Formulations of Newtonian Mechanics
First we reproduce section 2 from Michael Friedman’s “Simultaneity in Newtonian Mechanics and Special Relativity” in Foundations of Space-Time Theories (ed. Earman et al., UMinn, 1977), p.405-407. Then we provide the dual.
According to the spatio-temporal point of view, the basic object of both our theories is a four-dimensional manifold. I shall use R4, the set of quadruples of real numbers, to represent the spatio-temporal manifold. Both theories agree that there is a natural system of straight lines defined on this manifold. If (a0, a1, a2, a3), (b0, b1, b2, b3) are two fixed points in R4, then a straight line is a subset of R4 consisting of elements (x0, x1, x2, x3) of the form
(1) x0 = a0r + b0
x1 = a1r + b1
x2 = a2r + b2
x3 = a3r + b3
where r ranges through the real numbers. A curve on R4 is a (suitably continuous and differentiable) map σ: R → R4. Such a curve σ(u) is a geodesic if and only if it satisfies
(2) x0 = a0u + b0
x1 = a1u + b1
x2 = a2u + b2
x3 = a3u + b3
where (x0, x1, x2, x3) = σ(u) and the ai and bi are constants. So if a curve is a geodesic its range is a straight line. Note that the geodesies are just the curves that satisfy
(3) d2xi/du2 = 0 i = 0, 1, 2, 3.
The importance of straight lines and geodesies is due to the fact that both theories agree that the trajectories of free particles are spatio-temporal straight lines. So we can represent such trajectories as geodesies in R4.
A coordinate system is a one-one (suitably continuous and differentiable) map φ: R4 → R4. A coordinate system is affine if and only if it is a linear transformation of R4, i.e., it satisfies
(4) yi = Σ aij xj + bi i, j = 0, 1, 2, 3
where the aij and bi are constants and (y0, y1. y2, y3) = φ(x0, x1, x2, x3). Affine coordinate systems are precisely those that preserve the condition
(5) d2yi/du2 = 0 i = 0, 1, 2, 3.
for geodesies. As we shall see, such coordinate systems are a natural representation of the physicist’s frames of reference.
So far, Newtonian mechanics and special relativity agree on the spatio-temporal structure. But the two theories differ over what further structures exist on the spatio-temporal manifold, and, in particular, over the individual natures of space and time. …
(a) Newtonian Mechanics
The central object that Newtonian kinematics postulates on the spatio-temporal manifold is an absolute time: a real-valued function t: R4 → R defined by t(x0, x1, x2, x3) = x0. Think of it as assigning a time to each spatio-temporal point (event). The hypersurfaces t = constant are called planes of absolute simultaneity. Two points in R4 are simultaneous if and only if they lie on the same t = constant hypersurface. Furthermore, on each plane of absolute simultaneity Newtonian kinematics postulates a Euclidean metric, h, defined by
(6) h((t, xl, x2, x3), (t, x′l, x′2, x′3))2 = (x1 − x′1)2 + (x2 − x′2)2 + (x3 − x′3)2.
Now any geodesic curve σ(u) satisfies x0 = a0u + b0, so if we use x0 = t as a parameter for σ it remains a geodesic: i.e., σ(t) satisfies
(7) d2xi/dt2 = 0 i = 0, 1, 2, 3.
This is just Newton’s law of inertia.
An inertial coordinate system is an affine coordinate system which is generated by a Galilean transformation; i.e., y0, y1. y2, y3 is inertial if and only if
(8) y0 = x0 = t
yi = Σ aij xj + bi i, j = 1, 2, 3
where the aij, i, j = 1, 2, 3 form an orthogonal matrix: Σ aij akj = δik = 1 if i = k, 0 if i ≠ k. Inertial coordinate systems are just those that preserve the above form of the law of inertia and the above form of the spatial metric h. I shall say that an inertial coordinate system y0, y1. y2, y3 is adapted to a trajectory σ(t) if and only if σ(t) satisfies the equations y0 = t, yi = 0, i = 1, 2, 3. Thus one can think of σ as representing a particle at rest at the origin of y0, y1. y2, y3. There exists an inertial coordinate system adapted to σ if and only if σ is a geodesic. So if σ is a geodesic and φ is an inertial coordinate system adapted to σ, I shall call the pair (σ, φ) an inertial frame. In inertial frames free particles satisfy Newton’s first law.
According to the temporo-spatial point of view, the basic object of both our theories is a four-dimensional manifold. I shall use R4, the set of quadruples of real numbers, to represent the temporo-spatial manifold. Both theories agree that there is a natural system of straight lines defined on this manifold. If (a0, a1, a2, a3), (b0, b1, b2, b3) are two fixed points in R4, then a straight line is a subset of R4 consisting of elements (t0, t1, t2, t3) of the form
(1) t0 = a0r + b0
t1 = a1r + b1
t2 = a2r + b2
t3 = a3r + b3
where r ranges through the real numbers. A curve on R4 is a (suitably continuous and differentiable) map σ: R → R4. Such a curve σ(u) is a geodesic if and only if it satisfies
(2) t0 = a0u + b0
t1 = a1u + b1
t2 = a2u + b2
t3 = a3u + b3
where (t0, t1, t2, t3) = σ(u) and the ai and bi are constants. So if a curve is a geodesic its range is a straight line. Note that the geodesies are just the curves that satisfy
(3) d2ti/du2 = 0 i = 0, 1, 2, 3.
The importance of straight lines and geodesies is due to the fact that both theories agree that the trajectories of free particles are temporo-spatial straight lines. So we can represent such trajectories as geodesies in R4.
A coordinate system is a one-one (suitably continuous and differentiable) map φ: R4 → R4. A coordinate system is affine if and only if it is a linear transformation of R4, i.e., it satisfies
(4) si = Σ aij tj + bi i, j = 0, 1, 2, 3
where the aij and bi are constants and (s0, s1. s2, s3) = φ(t0, t1, t2, t3). Affine coordinate systems are precisely those that preserve the condition
(5) d2si/du2 = 0 i = 0, 1, 2, 3.
for geodesies. As we shall see, such coordinate systems are a natural representation of the physicist’s frames of reference.
So far, Newtonian mechanics and special relativity agree on the temporo-spatial structure. But the two theories differ over what further structures exist on the temporo-spatial manifold, and, in particular, over the individual natures of time and space. …
(a) Newtonian Mechanics
The central object that Newtonian kinematics postulates on the temporo-spatial manifold is an absolute distance: a real-valued function x: R4 → R defined by x(t0, t1, t2, t3) = t0. Think of it as assigning a distance to each temporo-spatial point (event). The hypersurfaces x = constant are called planes of absolute simuldistance. Two points in R4 are simuldistant if and only if they lie on the same x = constant hypersurface. Furthermore, on each plane of absolute simuldistance Newtonian kinematics postulates a Euclidean metric, h, defined by
(6) h((x, t1, t2, t3), (x, t′1, t′2, t′3))2 = (t1 − t′1)2 + (t2 − t′2)2 + (t3 − t′3)2.
Now any geodesic curve σ(u) satisfies t0 = a0u + b0, so if we use t0 = x as a parameter for σ it remains a geodesic: i.e., σ(x) satisfies
(7) d2ti/dx2 = 0 i = 0, 1, 2, 3.
This is just Newton’s dual law of facilia.
A facilial coordinate system is an affine coordinate system which is generated by a Galilean transformation; i.e., s0, s1. s2, s3 is facilial if and only if
(8) s0 = t0 = x
si = Σ aij tj + bi i, j = 1, 2, 3
where the aij, i, j = 1, 2, 3 form an orthogonal matrix: Σ aij akj = δik = 1 if i = k, 0 if i ≠ k. Facilial coordinate systems are just those that preserve the above form of the law of facilia and the above form of the temporal metric h. I shall say that a facilial coordinate system s0, s1. s2, s3 is adapted to a trajectory σ(s) if and only if σ(s) satisfies the equations s0 = x, si = 0, i = 1, 2, 3. Thus one can think of σ as representing a particle at rest at the origin of s0, s1. s2, s3. There exists a facilial coordinate system adapted to σ if and only if σ is a geodesic. So if σ is a geodesic and φ is a facilial coordinate system adapted to σ, I shall call the pair (σ, φ) an facilial frame. In facilial frames free particles satisfy Newton’s first law.