The Planck length, denoted as "ℓP," is derived from fundamental physical constants, namely the speed of light (c), the gravitational constant (G), and Planck's constant (h). It represents a scale at which our current understanding of physics breaks down, and it is thought to be the smallest meaningful length scale in our universe. Here are a few reasons why it is considered the smallest measurable length:
Quantum Gravity Scale: The Planck length arises when attempting to reconcile the principles of quantum mechanics and general relativity, known as quantum gravity. At distances smaller than the Planck length, it is believed that the fabric of spacetime becomes highly curved and fluctuates in a manner not described by our current theories. It is an energy regime where both quantum effects and gravitational effects become significant, making our conventional understanding of physics inadequate.
Uncertainty Principle: The Heisenberg uncertainty principle, a fundamental principle of quantum mechanics, states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. This principle implies that there is a fundamental uncertainty associated with measuring distances. The Planck length is related to the uncertainty in position measurements and represents the scale at which this uncertainty becomes significant.
Black Hole Physics: The Planck length is also associated with black hole physics. According to general relativity, if the mass of an object is compressed within a region smaller than the Schwarzschild radius (a characteristic radius associated with the mass), it will form a black hole. For a black hole of Planck mass, its Schwarzschild radius would be roughly equal to the Planck length, indicating that it is the smallest meaningful scale for the formation of a black hole.
Lack of Experimental Evidence: While the Planck length is derived from theoretical considerations, there is currently no experimental evidence to confirm or refute its existence or properties. Our current experimental techniques and technologies are far from being able to directly probe distances on the order of the Planck length.
It's important to note that our understanding of the universe is constantly evolving, and future scientific advancements may shed new light on the nature of spacetime at extremely small scales. However, based on our current understanding, the Planck length represents a fundamental limit to the measurability of lengths in our universe.