Free-Particle Dynamics from Spacetime Symmetry

This note compares massive free particles in Galilean and Minkowski spacetime. The logical order is

Passing to a curved spacetime requires an additional universal-free-fall postulate and a specified connection. It does not follow from coordinate covariance alone.

Scope and assumptions¶

The result below concerns a structureless massive point particle with no external fields.

Physical principles

1. Inertial frames are related by the Galilei group in the non-relativistic theory and by the proper orthochronous Poincare group in the relativistic theory.

2. Space and time translations and spatial rotations are physical symmetries.

3. The law of motion has the same form in every inertial frame.

Mathematical assumptions

1. The worldline is twice differentiable.

2. The law is local and second order: acceleration at an event depends only on the event and the velocity there. It has no dependence on higher derivatives, internal variables, or hidden background fields.

3. In the relativistic case the worldline is timelike and is parametrized by proper time.

These assumptions define the class in which uniqueness is claimed. Symmetry alone does not exclude higher-derivative laws, nonlocal laws, massless particles, or particles carrying additional internal data.

We use the spacetime structures derived in the metric note: Minkowski spacetime in the Poincare branch and the temporal metric plus spatial co-metric in the Galilei branch.

Inertial motion¶

Galilean spacetime¶

In an inertial Cartesian frame, write the most general law in the stated class as

Spatial and temporal translation invariance give

for every constant $\mathbf a$ and $s$. Hence

Use the Galilean boost convention

Covariance means that the primed observer uses the same function $\mathbf F$:

Describing the same worldline with the unprimed equation therefore gives

for every $\mathbf v$ and every boost velocity $\mathbf b$. Given any $\mathbf v_1,\mathbf v_2$, choose $\mathbf b=\mathbf v_1-\mathbf v_2$; then $\mathbf F(\mathbf v_2)=\mathbf F(\mathbf v_1)$. Thus $\mathbf F$ is a constant vector $\mathbf C$.

Rotation covariance now requires

for every $R\in SO(3)$. The only vector fixed by every rotation is zero, so

Boost covariance removes velocity dependence; isotropy removes the remaining constant acceleration.

Minkowski spacetime¶

Let

Translation invariance reduces the most general covariant law in the stated class to

Lorentz covariance is the equivariance condition

for every proper orthochronous Lorentz transformation $\Lambda$.

Choose the rest four-velocity $u_0=(c,\mathbf 0)$. Every spatial rotation $R$ fixes $u_0$. Equivariance then gives

Consequently the spatial part of $A(u_0)$ vanishes, so $A(u_0)$ is proportional to $u_0$. On the other hand, differentiating the fixed normalization of the four-velocity gives

Thus $A(u_0)$ is also orthogonal to $u_0$. Since $u_0$ is non-null, both conditions can hold only when $A(u_0)=0$.

Every future-directed timelike four-velocity is $\Lambda u_0$ for some Lorentz transformation. Therefore

and hence

This argument uses the full Lorentz orbit of timelike velocities, not one special boost. In an inertial coordinate system it is equivalent to constant three-velocity.

What has and has not been proved¶

Within the local second-order class above, inertial-frame symmetry uniquely selects straight, uniformly parametrized worldlines. No action or momentum formula was used.

This does not prove that the equation has a unique Lagrangian. Inverse variational problems generally admit inequivalent Lagrangians producing the same trajectories. To determine the familiar free-particle Lagrangians we must add more input.

Adding an action principle¶

Assume Hamilton’s principle for a local first-order action,

Two Lagrangians differing by

have the same Euler-Lagrange equations. Any uniqueness statement below is therefore only modulo such a total derivative.

We also identify canonical momentum,

with the mechanical momentum independently derived from inertial-frame symmetry, collision balance, extensivity, and regularity in the unified energy-momentum note. Those extra collision assumptions, not spacetime symmetry by itself, determine the mass normalization.

Galilean Lagrangian¶

The imported momentum law is

In a homogeneous gauge $L=L(\mathbf v)$, the canonical-mechanical identification gives

Integrating each component yields

The constant is dynamically irrelevant because $C=d(Ct)/dt$. Restoring the general total-derivative freedom,

The coefficient $m$ is the Bargmann central charge carried by the particle; the bare Galilei spacetime geometry does not fix its value.

Relativistic Lagrangian¶

For a structureless free particle, require the action to be local, additive along adjacent worldline segments, reparametrization invariant, and Lorentz invariant. Let $\lambda$ be an arbitrary increasing worldline parameter and write

Translation invariance removes explicit $x$-dependence. For a timelike worldline, Lorentz invariance permits dependence on $\dot x^\mu$ only through

so $\mathcal L(\dot x)=f(z)$. Under a constant reparametrization for which $\dot x\mapsto k\dot x$, invariance of $\mathcal L\,d\lambda$ requires positive homogeneity,

for every $k>0$. Taking $k=\sqrt{w/z}$ gives

so $f(z)=\beta\sqrt z$ for one constant $\beta$. Because $\sqrt z\,d\lambda=c\,d\tau$, the action has the form

with $\alpha=\beta c$, a constant having dimensions of energy. Thus locality, Lorentz invariance, and reparametrization invariance fix the form; they do not fix this particle-dependent normalization.

Using

we obtain

Its canonical momentum is

The independently derived relativistic momentum is

Equating the two fixes $\alpha=-mc^2$. Thus

This conclusion applies to massive timelike particles. A massless particle requires an auxiliary worldline field or an equivalent affine-parameter action and is a separate branch.

Equations of motion from the actions¶

For the Galilean action,

For the relativistic action,

For $m>0$ and $|\mathbf v|<c$, the map $\mathbf v\mapsto\gamma m\mathbf v$ is one-to-one, so conserved momentum implies constant $\mathbf v$ and hence $a^\mu=0$. The actions therefore reproduce, rather than supply, the previously derived inertial motion.

Noether charges¶

After the action principle has been introduced, spatial and temporal translations give canonical momentum and energy. Choosing the homogeneous representatives $\chi=0$ gives

For the relativistic particle,

It follows algebraically that

In Galilean mechanics an additive constant in the energy remains conventional. In the relativistic theory, $(E/c,\mathbf p)$ must transform as a four-vector, so a frame-independent additive shift of $E$ is not compatible with Lorentz covariance.

Curved spacetime and the geodesic postulate¶

Coordinate covariance is a requirement on how an already specified geometric law is represented; it is not a physical force law and does not determine a connection.

To pass from inertial motion to gravity, add:

1. universal free fall: all structureless test bodies couple to one affine connection, independently of mass and composition;

2. minimal coupling: in the absence of nongravitational forces, no additional curvature-dependent force term is present;

3. an affine parameter: proper time in Lorentzian spacetime, and an affine parameter compatible with absolute time in Newton-Cartan spacetime.

The invariant equation is then

or in coordinates,

In general relativity, imposing torsion freedom and metric compatibility makes $\Gamma$ the Levi-Civita connection of $g_{\mu\nu}$. In Newton-Cartan theory, compatibility with the temporal metric and spatial co-metric does not uniquely determine $\Gamma$; gravitational and Coriolis data remain in the connection, as discussed in the Newton-Cartan note.

At any event one can choose normal coordinates in which the symmetric part of the connection vanishes at that event, reducing the equation there to $d^2x^\lambda/d\lambda^2=0$. This is a pointwise statement. It does not make the connection vanish on a neighborhood and does not imply zero curvature.

Continuum conservation laws¶

For matter fields, conservation laws require a matter action and its field equations. Diffeomorphism invariance of that action gives an on-shell Noether identity.

In a Lorentzian theory with matter coupled only to the metric, the identity is

In Newton-Cartan theory the background data are degenerate and independent, so the corresponding Ward identities involve the mass current, momentum current, and energy current. They should not be compressed into the Lorentzian formula without specifying the Newton-Cartan matter couplings.

These are consequences of the matter action’s diffeomorphism invariance and the matter equations of motion. They are not consequences of inertial-frame symmetry alone, and deriving them does not require a gravitational field equation.

Dependency summary¶

The non-circular chain is therefore not “symmetry fixes everything.” Spacetime symmetry fixes inertial motion within a stated class; additional, explicit physical assumptions determine its variational representation, normalization, coupling to gravity, and continuum conservation laws.