I've enjoyed frequent discussions of relativity. Occasionally, the concept of 4-velocity and 4-momentum arise, which I will quickly summarize.

Every particle has a "world line", which describes its path through 4D spacetime. A constant progression, for example, appears in a typical spacetime diagram as a straight line. This is equally true for conventional velocity (e.g. meters/sec.) and for a particle "at rest", which has a vertical world line also known as its timeline. Typically, space is assigned a horizontal axis and time a vertical one. Because they have different units, a proportionality constant is employed.

It is foundational to relativity that there is a universal speed limit, c. As such, a dimensionless* fraction (between 0 and 1) corresponds to every conventional velocity. Dimensionless expressions would be especially useful in communicating with intelligent aliens who never heard of a "meter" or a "second". c is the proportionality constant note above.

A consequence of dimensionless velocities is that both momentum (p = mv) and energy (e.g. KE = ½mv²) are found to have the same units (mass). These are combined to express a "4-momentum"[1] (a.k.a. "momenergy"[2]), where the energy component (E) is ascribed to the time axis and the momentum components (p) remain along the spatial axes. This allows the expression of an invariant** mass (mi) where mi² = E² - p², from which (when p = 0) the more famous equation E = mc² is readily derived (recall, c = 1).

Given 4-momentum, I must assert the existence of a 4-ANGULAR momentum. Such a notion exists, indeed even "rotational invariance" [3]. But here's the problem. While angular momentum components about the spatial axes are recognized, the analogous component, which MUST exist about the temporal axis (my "chronaxial spin"), is disgracefully avoided. To wit:

"However, there is another type of angular momentum, called spin angular momentum (more often shortened to spin)... Almost all elementary particles have spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: Spin is an intrinsic property of a particle, unrelated to any sort of motion in space. All elementary particles have a characteristic spin, for example electrons always have 'spin½'..."[3,4]

Of course, quantum spin is "unrelated to any sort of motion in space". We're talking about spaceTIME! What a ridiculous cop out! Or, am I misreading this?

*Dividing every velocity by c, yields each as a fraction of "lightspeed" having no units.

**Relativistically "invariant" means viewed the same by all inertial (constant velocity) observers. Speed c is invariant (c = c'). So are the intervals (d) separating events in space (x) and time (t) by an analogous equation d² = x² - t².

1] http://en.wikipedia.org/wiki/Four-momentu

2] http://physics.bu.edu/~duffy/ns547_spri ... energy.pdf

3] http://en.wikipedia.org/wiki/Angular_mome

4] http://en.wikipedia.org/wiki/Angular_mome

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