The statement, "The speed of light from Lₚ / Tₚ (may) or may not be valid. Since Planck values for length and time were derived using light's speed (a circular reasoning?)", rests on an incomplete understanding of the origin and intent of Planck units. It is also not as an example of circular reasoning.
Planck introduced his natural unit system in 1899, well before Einstein’s special and general relativity theories were published (in 1905 and 1916, respectively). At that time, the speed of light (c) was not interpreted through a relativistic lens. Instead, Planck's goal was to construct a unit system derived entirely from fundamental constants of nature.
The Planck scale is defined using three constants:
- the speed of light c,
- the gravitational constant G, and
- the reduced Planck constant h-bar.
These constants were not chosen to define each other, but to provide universal units of length, time, mass, etc., that remain invariant across physical contexts. The expressions for Planck length Lp and Planck time Tp use c, G and h-bar, but not in a way that implies circular logic. Rather, c is treated as a known constant—just as G and h-bar are—serving as part of a dimensional bridge between quantum mechanics and gravitation.
Thus, there is no logical fallacy involved. The Planck units are internally consistent and reflect natural scales set by the interplay of fundamental constants. Their construction is not dependent on any single one being derived from the others, and especially not on a derived dependence of c from Lp / Tp.
Therefore, no circular reasoning arises in this context, and the usage of c in defining Planck units is a matter of dimensional coherence, not definitional dependence.
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