No, the fact that Planck units of length and time are dimensionless does not imply that you can measure time in length or vice versa.
The Planck units, named after the physicist Max Planck, are derived from fundamental constants of nature, such as the speed of light, the gravitational constant, and the reduced Planck constant. These units represent the scales at which the effects of quantum mechanics and general relativity become significant.
In the Planck system of units, the Planck length (L_P) and the Planck time (t_P) are defined as follows:
Planck length (L_P): It is the square root of the ratio of the gravitational constant (G) to the speed of light in a vacuum (c) cubed, multiplied by the reduced Planck constant (ħ). Mathematically, L_P = sqrt(Għ/c^3).
Planck time (t_P): It is the square root of the ratio of the gravitational constant (G) to the reduced Planck constant (ħ), cubed. Mathematically, t_P = sqrt(Għ/c^5).
Since these units are dimensionless, they do not carry any units of length or time. However, this does not imply that time and length are interchangeable. Time and length are distinct physical quantities, and they are measured and understood differently.
The dimensionless nature of Planck units arises from the choice of natural units, which is a system of units where certain fundamental constants are set to unity (e.g., c = 1, G = 1, ħ = 1). In this system, all physical quantities are expressed in terms of these fundamental constants, and their relative values become evident. However, it is important to note that in everyday life and most scientific contexts, we typically use units with dimensions to measure time and length separately.