Can Planck Units Resolve the Incompatibility of Units for Space and Time?

[Count Iblis]

If we reject the notion of fundamental dimensionful constants and interpret the constants h-bar, c and G as rescaling constants, an artifact of declaring certain quantities to be incompatible and using inconsistent units (e.g. for space and time we use the meter and second and we assume that space and time are fundamentally incompatible quantities), we can reason as follows.

The Planck length is

[tex]l_p = \sqrt{\frac{\hbar G}{c^3}}=1.616 252 10^{-35}\text{ meter}[/tex]

But in Planck units hbar = c = G = 1, we have:

[tex]l_p = 1[/tex]

If we take this literally and assume that [tex]l_p[/tex] is a pure number, we have:

[tex]1.616 252 10^{-35}\text{ meter} = 1[/tex]

Or:

[tex]\text{meter }= 6.187746\times 10^{34} [/tex]

Now, this assignment of a pure number to the meter depends, of course, on the way we assign numbers to hbar, c and G when we define Planck units. Assuming that the "correct" assignment will only differ from putting hbar = c = G = 1 by factors of order unity, we can put:

[tex]\text{meter }= 6.187746\times 10^{34}\lambda [/tex]

where [tex]\lambda\approx 1[/tex]

If we substitute this in an expression involving only lengths that is dimensionally correct, the constant lambda will drop out exactly. But we can write down expressions that don't make any sense from a fundamentalistic dimensional point of view, in which the lambda dependence is nevertheless negligible. Consider e.g. Log(meter). We have:


[tex]\log(\text{meter})= 80.11 + \log(\lambda) [/tex]

We can, of course, express this in terms of meters by dividing this by 6.187746 10^(34) times lambda, but then the result will strongly depend on lambda.

 

[AndyW]

First of all, I want to clarify that the concept of fundamental dimensionful constants is not something that can be rejected or accepted. These constants, such as h-bar, c, and G, are fundamental in the sense that they are universal and do not depend on any arbitrary units or scales. They represent the underlying physical properties of our universe and are not simply rescaling factors.

The idea of Planck units, where h-bar, c, and G are set to 1, is a useful tool for theoretical calculations and can provide insight into the fundamental nature of our universe. However, it is important to remember that this is just a mathematical convenience and does not change the fundamental nature of these constants.

Furthermore, the Planck length, while it may have a value of 1 in Planck units, is still a physical length in our universe and cannot be reduced to a pure number. It represents the scale at which quantum effects become important and is not simply an artifact of incompatible units.

Lastly, the concept of lambda as a constant that can be arbitrarily chosen and dropped out of calculations is not valid. In reality, any change in the value of lambda would have significant consequences for physical phenomena and would not simply "drop out" of calculations. It is important to approach scientific concepts with a critical and analytical mindset and not make assumptions or oversimplifications.

 

[sir-pinski]

The content being discussed here is related to the concept of Planck units and how they can be used to interpret fundamental constants such as h-bar, c, and G as rescaling constants rather than fundamental quantities. This approach challenges the traditional notion of fundamental dimensionful constants and suggests that the incompatibility of units for space and time can be resolved by using Planck units.

The Planck length, denoted as l_p, is a fundamental length scale that is derived from the three constants mentioned above. In Planck units, where h-bar, c, and G are all equal to 1, the Planck length is simply 1. This leads to the idea that the meter, which is traditionally used to measure length, can be expressed as a pure number multiplied by a factor of lambda. The value of lambda is approximately equal to 1 and depends on the specific assignment of numbers to h-bar, c, and G in defining Planck units.

However, this approach can lead to nonsensical expressions when applied to quantities that involve lengths. For example, the logarithm of the meter, Log(meter), would have a value of 80.11 plus a term that is dependent on lambda. This term can be removed by dividing by the factor of 6.187746 10^(34) times lambda, but this would result in a value that strongly depends on lambda.

In conclusion, while the use of Planck units can provide a consistent way to interpret fundamental constants, it is important to consider the implications of assigning a pure number to a unit of measurement. This approach may work for certain quantities, but it can also lead to nonsensical results for others. Ultimately, the interpretation of fundamental constants and units is a complex and ongoing discussion in the field of physics.