Sorry for the delay.

Jorrie wrote:I was referring to these diagrams … for your radial time presentation. They have a decidedly Epsteinian 'look and feel'. ;) … Spacetime diagrams (including Epstein's) have high utility in depicting spacetime events and spacetime world-lines of objects in relative motion

That 'look and feel' is intuitive Euclidean geometry available to both Epstein diagrams and my interval-time diagrams. Epstein achieves this by rotating one coordinate frame relative to another. Interval-time coordinates don't require this because all frames have

invariant interval coordinates in common, so one simply compares local worldlines.

Interval-time coordinates derive naturally from

curved-space, radial-time (below) at any event (

p). The rest frame of the cosmos is "special" as it is the oldest accessible reference frame and also exhibits perfect 3-spherical symmetry (equal recessional red shift in all directions).

From Left: A temporal field radiating outward in 4D from the Big Bang (BB) similar to electric charge. A 2D slice at rest with respect to the cosmos reveals interval-time coordinates corresponding to rest (v_{0}) and natural speed limit c (v_{max}) enforced by unidirectional time. A rest frame in relative motion (v_{1}) sees the cosmos shorter in the direction of motion. In moving "frame c" light emission & future absorption are a single interval contact event as simultaneity (t_{2}) collapses flatly on radial time.A resulting

finite,

universal,

constant and

invariant speed limit is itself

more than enough to justify a

curved-space, radial-time model.

c is the tangent limit because greater inclination violates unidirectional time. Interval coordinates correspond to this. There,

c is an

absolute speed limit because contact is an

absolute proximity limit. Similarly,

c is an

absolute speed limit because nothing is slower than

absolute rest (i.e. interval rest:

∆d/∆t = 0). Unlike spacetime, interval-time maintains "frame

c", even as space and time collapse. That's

utility! Explaining

c in spacetime is

futility.

Jorrie wrote:[altered] I fail to see how [interval-time] coordinates can show… the all-powerful light-cone.

That's because for any event (

p) in

curved-space, radial-time there are an infinite number of incoming light cones and just one outgoing "light disk". This is best depicted from the perspective of a real (i.e. subluminal) observer, here conveniently in the rest frame of the cosmos. For incoming light cones, consider a paper cone resting tangent to a small balloon at its circle of contact. The cone keeps this relation, widening for expanded balloons. Those are like incoming light cones for a fixed vertex event

p.

Like incoming light, outgoing light occurs tangent to the simultaneity of the

emitter. From any event

p, this describes a tangent disk.

Left: Incoming light cones (shown in cross section) widen with concentric, expanded, past cosmic simultaneities (3-spheres enclosing the Big Bang). Tracing their emission events yields a valentine shape. Right: Outgoing light occurs on a "light disk" tangent to the spatial simultaneity of the emitter (p) and aiming futureward but independent of aging (i.e. the emitter's timeline).Jorrie wrote:…how would you show a light ray being reflected back from an object (with the point of reflection as an event)…

As you know, that varies with reference frame. Since light transmission occurs by

interval contact, emission is depicted with a mere dot

p in frame

c . Reflection is a future dot

q. In both cases, future space (in the direction of transmission) is collapsed onto the emitter's timeline.

"

…time [aging] would stop if you could travel at precisely the speed of light. You would abolish distance entirely, so that your point of departure and your destination seem to be at the same place."

Calder p.96

In the cosmic rest frame, light maintains its direction tangent to the simultaneity of the emitter, but the path now appears to have length. This is consistent with the

defining notion that a wormhole (here, a "pinhole") has different internal and external spans. To be consistently non-aging the light paths from emitter and reflector are each normal to their respective timelines.

Left: In frame c, incident light originates and terminates at the same event p despite different spacetime coordinates. Reflected light does the same from a future event q. Right: In the cosmic rest frame, light transmits without aging from event p and incident upon spatially-remote future event q. From there, light is reflected tangent to space at the reflector.Jorrie wrote: how would you show … [the worldline of] the "away-twin" in the old "twin paradox"?

The same as light in the cosmic frame above but with subluminal worldlines less inclined away from radial time. With interval-time coordinates, mutual age contraction is given by the simple Euclidean relation

t' = tcosθ. Velocity as a fraction of

c is given by

v = csinθ, the degree to which a worldline is lightlike. This relates to the LFTs by the identity: cos² = 1 - sin², so cosθ = √(1-v²).

Inertial clocks (and aging) run mutually slow as depicted with interval-time coordinates. t' is the age each sees of the other at their own age t. In the rest frame of the "moving" object, its timeline would be vertical while the other tilts away at the same angle but opposite direction. Neither age is locked in relative to the other unless some force puts both clocks into the same rest frame.A round trip combines two such paths. Let the trip be sufficiently local to neglect the curvature of space. Specify a

forced instant path reversal at a point indicated on a vertical line. Now we know why

F = ma! All force is lightlike, imparting a lightlike component to an object's worldline, thus changing its inclination. This is always a non-aging component in the rest frame of the object experiencing the force.

A round trip (green) of a massive object requires a lightlike force to accelerate its return. This locks in a non-aging component compared to a relatively stationary object (i.e. one not experiencing force) which ages t. The accelerated object ages t'.(Implied forces f to start and end travel have been neglected.)Nothing ages faster than the cosmos, which is always at rest with respect to itself. All forces act within.