Question

In accordance with special relativity, if we take into account length contraction then ideally the Planck length which is the smallest possible length possible should be frame dependent. Well we know that the number of particles are invariant in different frames. So, a frame travelling with some relative velocity will see the same number of particles but their length contracted by a factor. According to this logic, the Planck length should change with different frames. Yet it is defined by using the fundamental constants G,c and h which obviously do not change with frame. This could only imply two things :-

1. One of the fundamental constants G,c,h are frame dependent (which I do not believe could be since they are fundamental constants)
2. The shortest possible length cannot be written only in terms of fundamental constants
The second conclusion seems more apt but is it right or is there something more to this?

Answer 1

The Planck length, defined by fundamental constants such as the speed of light, gravitational constant, and Planck's constant, appears frame-independent and does not undergo length contraction per special relativity, hinting at the need for a complete theory such as quantum gravity to understand it fully.

Step-by-step explanation:

Making Connections: Length Contraction and Planck Length

According to Einstein's theory of special relativity, an object in motion relative to an observer experiences length contraction. This means that an object, such as a stick, will appear shorter when it is moving at a velocity close to the speed of light compared to when it is at rest. This rest length is known as the proper length Lo. However, when we consider the Planck length, which is often cited as the smallest possible length, we encounter a conceptual challenge.

The Planck length is indeed defined using the fundamental constants of nature: the gravitational constant G, the speed of light c, and Planck's constant h. These constants are deemed frame-independent, meaning they do not change with different inertial frames of reference. This fact suggests that the Planck length should also be frame-independent, which seems to contradict the idea that lengths should contract in frames moving at a relative velocity.

This apparent paradox leads to the idea that the Planck length may represent a minimum scale at which the classical concepts of space and special relativity cease to be applicable. Instead, it is likely that a theory of quantum gravity, which would combine the principles of quantum mechanics and general relativity, would provide a better understanding of the Planck length.

Thus, while special relativity predicts that observed lengths contract in moving frames, this does not apply to the Planck length, suggesting that we may need a more complete theoretical framework to fully understand its nature.