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 space-time point of view, the basic object of both our theories is a four-dimensional manifold. I shall use R^{4}, the set of quadruples of real numbers, to represent the space-time manifold. Both theories agree that there is a natural system of straight lines defined on this manifold. If (*a*_{0}, *a*_{1}, *a*_{2}, *a*_{3}), (*b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}) are two fixed points in R^{4}, then a straight line is a subset of R^{4} consisting of elements (*x*_{0}, *x*_{1}, *x*_{2}, *x*_{3}) of the form

(1) *x _{0} = a_{0}r + b_{0}*

*x*

_{1}= a_{1}r + b_{1}*x*

_{2}= a_{2}r + b_{2}*x*

_{3}= a_{3}r + b_{3}where *r* ranges through the real numbers. A curve on R^{4} is a (suitably continuous and differentiable) map σ: R → R^{4}. Such a curve σ(*u*) is a geodesic if and only if it satisfies

(2) *x _{0} = a_{0}u + b_{0}*

*x*

_{1}= a_{1}u + b_{1}*x*

_{2}= a_{2}u + b_{2}*x*

_{3}= a_{3}u + b_{3}where (*x _{0}, x_{1}, x_{2}, x_{3}*) = σ(

*u*) and the

*a*

_{i}and

*b*

_{i}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) *d ^{2}x_{i}*/

*du*= 0

^{2}*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 straight lines in space-time. So we can represent such trajectories as geodesies in R^{4}.

A *coordinate system* is a one-one (suitably continuous and differentiable) map φ: R^{4} → R^{4}. A coordinate system is affine if and only if it is a linear transformation of R^{4}, i.e., it satisfies

(4)* y*_{i} = Σ *a*_{ij} *x*_{j} + *b*_{i} *i, j* = 0, 1, 2, 3

where the *a*_{ij} and *b*_{i} are constants and (*y _{0}, y_{1}. y_{2}, y*

_{3}) = φ(

*x*). Affine coordinate systems are precisely those that preserve the condition

_{0}, x_{1}, x_{2}, x_{3}(5)* d ^{2}y_{i}*/

*du*= 0

^{2}*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 structure of space-time. But the two theories differ over what further structures exist on the space-time manifold, and, in particular, over the individual natures of space and time. …

(a) Newtonian Mechanics

The central object that Newtonian kinematics postulates on the space-time manifold is an absolute time: a real-valued function *t*: R^{4} → R defined by *t*(*x _{0}, x_{1}, x_{2}, x_{3}*) =

*x*

_{0}. Think of it as assigning a time to each point (event) in space-time. The hypersurfaces

*t*= constant are called

*planes of absolute simultaneity*. Two points in R

^{4}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*, *x*_{l}, *x*_{2}, *x*_{3}), (*t*, *x′*_{l}, *x′*_{2}, *x′*_{3}))^{2} = (*x _{1} − x′_{1}*)

^{2}+ (

*x*)

_{2}− x′_{2}^{2}+ (

*x*)

_{3}− x′_{3}^{2}.

Now any geodesic curve σ(*u*) satisfies *x _{0} = a_{0}u + b_{0}*, so if we use

*x*

_{0}=

*t*as a parameter for σ it remains a geodesic: i.e., σ(

*t*) satisfies

(7)* d ^{2}x_{i}*/

*dt*= 0

^{2}*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., *y _{0}, y_{1}. y_{2}, y*

_{3}is inertial if and only if

(8) *y _{0} = x_{0} = t*

*y*

_{i}= Σ

*a*

_{ij}

*x*

_{j}+

*b*

_{i}

*i, j*= 1, 2, 3

where the *a _{ij}, i, j* = 1, 2, 3 form an orthogonal matrix: Σ

*a*= 1 if

_{ij }a_{kj}= δ_{ik}*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

*y*

_{0}, y_{1}. y_{2}, y_{3}is

*adapted*to a trajectory σ(t) if and only if σ(t) satisfies the equations

*y*,

_{0}= t*y*

_{i}= 0,

*i*= 1, 2, 3. Thus one can think of σ as representing a particle at rest at the origin of

*y*

_{0}, y_{1}. y_{2}, y_{3}. 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 time-space point of view, the basic object of both our theories is a four-dimensional manifold. I shall use R^{4}, the set of quadruples of real numbers, to represent the time-space manifold. Both theories agree that there is a natural system of straight lines defined on this manifold. If (*a*_{0}, *a*_{1}, *a*_{2}, *a*_{3}), (*b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}) are two fixed points in R^{4}, then a straight line is a subset of R^{4} consisting of elements (*t*_{0}, *t*_{1}, *t*_{2}, *t*_{3}) of the form

(1) *t _{0} = a_{0}r + b_{0}*

*t*

_{1}= a_{1}r + b_{1}*t*

_{2}= a_{2}r + b_{2}*t*

_{3}= a_{3}r + b_{3}where *r* ranges through the real numbers. A curve on R^{4} is a (suitably continuous and differentiable) map σ: R → R^{4}. Such a curve σ(*u*) is a geodesic if and only if it satisfies

(2) *t _{0} = a_{0}u + b_{0}*

*t*

_{1}= a_{1}u + b_{1}*t*

_{2}= a_{2}u + b_{2}*t*

_{3}= a_{3}u + b_{3}where (*t _{0}, t_{1}, t_{2}, t_{3}*) = σ(

*u*) and the

*a*

_{i}and

*b*

_{i}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)* d ^{2}t_{i}*/

*du*= 0

^{2}*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 straight lines in time-space. So we can represent such trajectories as geodesies in R^{4}.

A *coordinate system* is a one-one (suitably continuous and differentiable) map φ: R^{4} → R^{4}. A coordinate system is affine if and only if it is a linear transformation of R^{4}, i.e., it satisfies

(4)* s*_{i} = Σ *a*_{ij} *t*_{j} + *b*_{i} *i, j* = 0, 1, 2, 3

where the *a*_{ij} and *b*_{i} are constants and (*s _{0}, s_{1}. s_{2}, s*

_{3}) = φ(

*t*). Affine coordinate systems are precisely those that preserve the condition

_{0}, t_{1}, t_{2}, t_{3}(5)* d ^{2}s_{i}*/

*du*= 0

^{2}*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 structure of time-space. But the two theories differ over what further structures exist on the time-space manifold, and, in particular, over the individual natures of time and space. …

(a) Newtonian Mechanics

The central object that Newtonian kinematics postulates on the time-space manifold is an absolute stance: a real-valued function *x*: R^{4} → R defined by *x*(*t*_{0}, *t*_{1}, *t*_{2}, *t*_{3}) = *t*_{0}. Think of it as assigning a stance to each point (event) in time-space. The hypersurfaces *x* = constant are called *planes of absolute simulstanceity*. Two points in R^{4} are *simulstanteous* if and only if they lie on the same *x* = constant hypersurface. Furthermore, on each plane of absolute simulstanceity Newtonian kinematics postulates a Euclidean metric, *h*, defined by

(6)* h*((*x*, *t*_{1}, *t*_{2}, *t*_{3}), (*x*, *t′*_{1}, *t′*_{2}, *t′*_{3}))^{2} = (*t _{1} − t′_{1}*)

^{2}+ (

*t*)

_{2}− t′_{2}^{2}+ (

*t*)

_{3}− t′_{3}^{2}.

Now any geodesic curve σ(*u*) satisfies *t _{0} = a_{0}u + b_{0}*, so if we use

*t*

_{0}=

*x*as a parameter for σ it remains a geodesic: i.e., σ(

*x*) satisfies

(7)* d ^{2}t_{i}*/

*dx*= 0

^{2}*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., *s _{0}, s_{1}. s_{2}, s*

_{3}is facilial if and only if

(8) *s _{0} = t_{0} = x*

*s*

_{i}= Σ

*a*

_{ij}

*t*

_{j}+

*b*

_{i}

*i, j*= 1, 2, 3

where the *a _{ij}, i, j* = 1, 2, 3 form an orthogonal matrix: Σ

*a*= 1 if

_{ij }a_{kj}= δ_{ik}*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

*s*

_{0}, s_{1}. s_{2}, s_{3}is

*adapted*to a trajectory σ(s) if and only if σ(s) satisfies the equations

*s*,

_{0}= x*s*

_{i}= 0,

*i*= 1, 2, 3. Thus one can think of σ as representing a particle at rest at the origin of

*s*

_{0}, s_{1}. s_{2}, s_{3}. 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.