Newton's laws in (3+1)D
In order to track interactions (particularly of conserved quantities like energy E = γmc2 and momentum p = mw) between multiple traveling objects, we now must move beyond those traveler-point variables, which sadly are good for describing the motion of only one traveler at a time. This will lead to a work-energy theorem (δW = Σξ•δx = mα•δx) familiar at any speed, and a map-based force-momentum theorem, namely Σf ≡ δp/δt, where each map-based force f equals ξ||w + ξ⊥w/γ in terms of the corresponding frame-invariant felt-force ξ. The latter, in turn, obey the frame-invariant felt force-acceleration relation Σξ = mα.
These relations, along with (3+1)D versions of the equations summarized on our (1+1)D dynamics page, are listed for reference in the table below.
A good worked example of this is our use of frame-invariant felt force ξ to show how the magnetic force on a moving charge from current in a neutral wire is, from the moving charge's perspective, simply an electrostatic force from a charged wire. Put another way, magnetic forces are a direct and everyday consequence of these "anyspeed" dynamical laws, when they are applied to the electrostatic (Coulomb's law) force f between charged particles.
The force-momentum and force-acceleration relations reduce to Newton's 2nd law (Σf ≈ mα ≈ ma) in terms of coordinate-acceleration a ≡ δ2r/δt2 at low speeds i.e. when γ ≈ 1. For unidirectional motion, map-based forces f and frame-invariant felt forces ξ are equal, so that Σf = δp/δt = Σξ = mα works at any speed. Momentum-conservation in the map-frame, of course, also gives rise to a general version of Newton's 3rd action-reaction law in flat spacetime, i.e. fAB = -fBA.
For the special cases of curved spacetimes (like that we live in here on earth) and for accelerated frames, this may also be a good place to discuss the utility of pretending that non-proper geometric forces (like gravity and centrifugal), which are invisible to our cell-phone accelerometers, are also proper forces for the purpose of predicting local trajectories. This is of course a tradition begun by Isaac Newton himself, in his treatment of gravity as a proper force.
On the topic of pre-20th century insights, it also turns out that proper forces can always be broken down into static and kinetic components (as shown in the table above). The latter components vanish in the frame of objects feeling the force. This breakdown is especially useful when oppositely-signed force-carriers exist. In that context an elegant notation was developed in the mid-1800's for the treatment of electromagnetic forces. This notation was "relativity-smart" a half-century before the connection between space and time was uncovered, and remains fundamental to new developments in electromagnetic communications even today.
Worked examples of these include the link between differential-aging and potential energies associated with (i) Schwarzschild gravity, (ii) centrifugal "gravity", and (iii) a linearly accelerating spaceship. What else?
Three-vector addition of relative proper-velocities may also be useful to mention here. Folks might enjoy this even less than they do the relative-velocity section in a standard Newtonian text, although it has the advantage that it works at any speed using a very similar construction. It also makes it pretty obvious why a collider is a way better investment than an accelerator, if you are trying to break the relative land speed record (in proper-velocity units, of course) for an electron, a proton, or a uranium nucleus.
Worked examples of this include (i) the unidirectional relation where coordinate-velocities add while the aging-factors multiply to yield a relative proper velocity, and (ii) the more general velocity addition problems e.g. involving an enemy spaceship dropping out of hyperspace in the neighborhood of a starfleet ship orbiting a ringworld. What else?