A didactic-mathematical proposal for using the Planck scale: from photons to black holes

Abstract In this paper we present a theoretical study of the physical limits associated with a fundamental particle, the photon, in the Planck scale. In addition, we calculated the temperature of a black hole with Planck dimensions. From the dimensional analysis, we verified five units of the Planck scale (mass, time, length, energy, and temperature). First, we analyze a photon with dimensions of Planck length; Then we identify the energy of this photon with the wavelength of the order of the Schwarzschild ray ( r s ). Finally, we calculate the temperature of a black hole with Planck's dimensions (in this case, with its Schwarzschild radius equal to the Planck's length). The “photon ray” appears as a multiple of Planck's length, demonstrating the inaccessibility of scales smaller than Planck's due to the Uncertainty Principle. Among the results obtained, we can highlight three possible paths to the Planck scale, as well as the smallest possible photon measurement according to the laws of physics, which is characterized by maximum frequency and energy.