Where did Frames go in the geometric formulation of Newton's laws? (Schuller)

Question

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1. What the question is really asking

The student has learned Newton’s laws in the usual textbook way:

“A particle that is not acted on by a net force moves with constant velocity.”
The law is written in a particular inertial reference frame as

[ \mathbf F_{\text{net}} = m\frac{d^{2}\mathbf x}{dt^{2}}\;, \qquad \mathbf F_{AB}= -\,\mathbf F_{BA}. ]

In the geometric (coordinate‑free) formulation used by Schuller the laws are written as

[ \nabla_{\mathbf u}\mathbf u = \mathbf F,\qquad \mathbf F_{AB}= -\,\mathbf F_{BA}, ]

where (\mathbf u) is the particle’s 4‑velocity and (\nabla) is a covariant derivative on Newtonian (Galilean) space‑time.
There is no explicit mention of “inertial frames”.

Question: Where did the notion of reference frames (especially inertial frames) go in this geometric picture? How do we recover the usual textbook statements?

2. Step‑by‑step answer

2.1 Newtonian (Galilean) space‑time as a geometric object

Manifold (\mathcal M): a 4‑dimensional smooth manifold whose points are events ((t,\mathbf x)).
Absolute time: a closed 1‑form (\tau) (often written (dt)) that assigns a time value to each event. Its kernel (\ker\tau) consists of the three‑dimensional spatial directions.
Spatial metric (h): a symmetric, positive‑definite bilinear form defined on (\ker\tau); it measures spatial distances but does not raise a time component.
Connection (\nabla): a torsion‑free affine connection satisfying

[ \nabla \tau =0,\qquad \nabla h =0. ]

These two compatibility conditions encode the idea that time is the same for all observers and that spatial distances are preserved by parallel transport.

The quadruple ((\mathcal M,\tau,h,\nabla)) is called Galilean (Newtonian) space‑time.

2.2 World‑lines, velocity and acceleration

A particle’s world‑line is a curve (\gamma : \mathbb R \to \mathcal M) with (\tau(\dot\gamma)=1); the parameter is the absolute time (t).
The 4‑velocity is the tangent vector (\mathbf u = \dot\gamma).
The covariant derivative along the curve gives the (coordinate‑free) acceleration:

[ \mathbf a \equiv \nabla_{\mathbf u}\mathbf u . ]

In any coordinate system ((t,x^i)) that respects the structure ((\tau = dt)), the components read

[ \bigl(\nabla_{\mathbf u}\mathbf u\bigr)^i = \frac{d^{2}x^{i}}{dt^{2}} + \Gamma^{i}{}{jk}\frac{dx^{j}}{dt}\frac{dx^{k}}{dt} + 2\Gamma^{i}{}{0j}\frac{dx^{j}}{dt} + \Gamma^{i}{}_{00}, ]

where indices (0) refer to the time coordinate.

2.3 Free particle ⇒ geodesic

The geometric version of Newton’s first law is

[ \nabla_{\mathbf u}\mathbf u = 0 . \tag{1} ]

Equation (1) says the acceleration vanishes with respect to the connection (\nabla); i.e. the world‑line is a geodesic of (\nabla).

2.4 Where are inertial frames?

An inertial frame is simply a coordinate chart ((t,x^i)) that makes the connection coefficients vanish:

[ \Gamma^{\mu}{}_{\nu\rho}=0 . \tag{2} ]

Because (\nabla\tau=0) and (\nabla h=0), the condition (2) can be satisfied globally on a flat Galilean space‑time (the one used in Newtonian physics). The coordinates satisfying (2) are called affine (Galilean) coordinates.

In such a chart, (1) reduces to

[ \frac{d^{2}x^{i}}{dt^{2}} = 0 , ]

which is exactly the textbook statement “a free particle moves with constant velocity”.
Any other chart related to an affine chart by a Galilean transformation

[ t’ = t + t_{0},\qquad \mathbf x’ = \mathbf R\mathbf x + \mathbf v\,t + \mathbf a, ]

also has (\Gamma^{\mu}{}_{\nu\rho}=0); thus all Galilean transformations map inertial frames to inertial frames.

Consequently, the notion of an inertial frame has not disappeared; it is encoded in the choice of coordinates for which the affine connection looks trivial. In the coordinate‑free language we simply do not have to mention the frame because the law is written directly in terms of geometric objects that are independent of any particular chart.

2.5 Adding forces

When external forces act, the geometric law becomes

[ \nabla_{\mathbf u}\mathbf u = \frac{1}{m}\,\mathbf F . \tag{3} ]

(\mathbf F) is a spatial vector field (i.e. (\tau(\mathbf F)=0)) defined along the world‑line.
In an inertial chart where (\Gamma^{\mu}{}_{\nu\rho}=0), (3) reads

[ m\frac{d^{2}x^{i}}{dt^{2}} = F^{i}, ]

which is the familiar Newton’s second law.

The action‑reaction principle remains untouched because it is a statement about the force vectors themselves:

[ \mathbf F_{AB} = -\,\mathbf F_{BA}, ]

which is a coordinate‑free equality and therefore holds in any frame.

2.6 Summary of the correspondence

Textbook formulation	Geometric formulation	How frames appear
Inertial frame → coordinates with no “fictitious” terms	Covariant derivative (\nabla) (torsion‑free, compatible)	Inertial frames are precisely the affine charts where (\Gamma^{\mu}{}_{\nu\rho}=0)
(\displaystyle \frac{d^{2}\mathbf x}{dt^{2}} = 0) for a free particle	(\displaystyle \nabla_{\mathbf u}\mathbf u = 0)	No explicit frame needed; the equation is valid in any chart; in inertial charts it reduces to the textbook form
(m\mathbf a = \mathbf F)	(\displaystyle \nabla_{\mathbf u}\mathbf u = \frac{1}{m}\mathbf F)	Same reduction as above
(\mathbf F_{AB} = -\mathbf F_{BA})	Same equality of spatial vectors	Frame‑independent, so unchanged

3. Final answer

In the geometric (coordinate‑free) formulation the reference‑frame language is hidden inside the affine connection (\nabla). An inertial frame is any coordinate system that makes the connection coefficients vanish; in such a frame the covariant‑derivative equation (\nabla_{\mathbf u}\mathbf u = \mathbf F/m) reduces exactly to the textbook Newtonian equations. Thus frames have not disappeared; they are simply encoded by the choice of coordinates that trivialise the connection.

4. Common mistakes when dealing with this topic

Mistake	Why it’s wrong	How to avoid it
Thinking that “no frames appear → frames are unnecessary.”	The geometric law is frame‑independent, but to recover the familiar (d^{2}\mathbf x/dt^{2}) you must choose an inertial chart.	Remember that any tensor equation holds in all frames; only when you write components do you need a frame.
Assuming any coordinate system is inertial.	Only those for which the Galilean connection coefficients vanish are inertial. Accelerating or rotating coordinates introduce extra terms (Coriolis, centrifugal).	Check the connection coefficients (\Gamma^{\mu}{}_{\nu\rho}); if they are non‑zero, the frame is non‑inertial.
Confusing the Newtonian connection with the Levi‑Civita connection of a Lorentzian metric.	Newtonian space‑time has a degenerate spatial metric and a separate absolute‑time 1‑form; its compatible connection is not the same as the relativistic one.	Keep the structure ((\tau,h,\nabla)) in mind; the compatibility conditions are (\nabla\tau=0,\ \nabla h=0).
Treating (\mathbf F) as a 4‑vector with a time component.	In Galilean space‑time forces are spatial; the time component of (\mathbf F) is forced to be zero by (\tau(\mathbf F)=0).	Explicitly project forces onto (\ker\tau) or state “(\mathbf F) is spatial”.
Dropping the torsion‑free condition on (\nabla).	Torsion would introduce extra “force‑like” terms even for free particles, violating Newton’s first law.	Always require (\nabla) to be torsion‑free when modelling Newtonian mechanics.

Original question: Where did Frames go in the geometric formulation of Newton's laws? (Schuller) on Physics Stack Exchange, licensed CC BY-SA.

1. What the question is really asking

2. Step‑by‑step answer