Deriving the Planck Length, Time, and Mass

In summary, the Planck length, Planck time, and Planck mass are derived from the SI or cgs systems by starting with arbitrary definitions of unit length, mass, and time. From these definitions, units of velocity, momentum, force, and energy are derived. Physicists then measured universal constants such as the speed of E&M propagation, Planck's constant, and the universal gravitational constant. The Planck units are a conceptual representation of these constants, and can be considered a standard if measured in terms of temporary standards.
  • #1
Saketh
261
2
How are the Planck length, Planck time, and Planck mass "derived"?

I know their values, but I don't understand where we get them from.
 
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  • #2
it's similar to how the "Newton" was chosen to be exactly the force that would accelerate 1 kg of mass at a rate of 1 m/s2. in the SI or cgs systems, they start with a sort of arbitrary (from a universal POV) definition of unit length, unit mass, and unit time, and given those definitions, they naturally derive a unit of velocity, unit momentum, unit force, unit energy, etc.

then, given that set of units (pre-1960) physicists went out and measured a bunch of stuff including 3 universal constants intrinsic to free-space, that is the speed of E&M propagation [itex]c[/itex], Planck's constant [itex]\hbar[/itex], and the universal gravitational constant [itex]G[/itex].

remember that these constants take on the numbers that they do because of the 3 arbitrary base units we came up with to measure them. so if we chose to, we could choose the 3 base units (and adjust the consequential derived units) so that those three constants [itex]c[/itex], [itex]\hbar[/itex], and [itex]G[/itex] all take on the value "1" in terms of those units. that is what Planck units are.

check it out in the Wikipedia. at least the hard-core POV pushers haven't f*cked that one up too much.
 
  • #3
I once saw a problem which asked to "derive" the Planck units. Your explanation makes sense -- that the units are just a standard -- but are the units a logical outgrowth of a previous scientific result? Or are they nothing more than a standard?
 
  • #4
Saketh said:
I once saw a problem which asked to "derive" the Planck units. Your explanation makes sense -- that the units are just a standard -- but are the units a logical outgrowth of a previous scientific result? Or are they nothing more than a standard?

the units are more "conceptual" than a standard, although i suppose one could treat them as a standard if they want to have a Cavendish-like apparatus for measuring [itex]G[/itex], a Watt-balance for measuring [itex]\hbar[/itex] and a Michaelson-like thing (including a mirror on a mountain some distance away) for measuring [itex]c[/itex] in terms of existing temporary standards like a cesium-clock, platinum-iridium prototype meter bar, and prototype mass. if we measure these constants in terms of the temporary standards, then we can say how those temp standards stack up in relation to the Planck units.
 

1. What is the Planck length, and how is it derived?

The Planck length is a fundamental constant in physics that represents the smallest possible length scale in the universe. It is derived from three other fundamental constants: the speed of light, the gravitational constant, and Planck's constant. The formula for the Planck length is given by LP = √(ħG/c3), where ħ is Planck's constant, G is the gravitational constant, and c is the speed of light.

2. How is the Planck time calculated?

The Planck time is another fundamental constant that represents the smallest possible unit of time. It is derived from the same three fundamental constants as the Planck length: the speed of light, the gravitational constant, and Planck's constant. The formula for the Planck time is given by tP = √(ħG/c5), where ħ is Planck's constant, G is the gravitational constant, and c is the speed of light.

3. What is the significance of the Planck length and time?

The Planck length and time are significant because they represent the limits of our current understanding of the universe. They are the smallest possible units of length and time that can be measured, and they are associated with the fundamental forces of nature: gravity and electromagnetism. The Planck length and time are also used in theories such as string theory and loop quantum gravity to describe the structure of spacetime at a microscopic level.

4. Is there a formula for the Planck mass?

Yes, there is a formula for the Planck mass, which is another fundamental constant derived from the same three fundamental constants as the Planck length and time. The formula for the Planck mass is given by mP = √(ħc/G), where ħ is Planck's constant, c is the speed of light, and G is the gravitational constant.

5. How do the Planck length, time, and mass relate to each other?

The Planck length, time, and mass are all derived from the same three fundamental constants and are all related to each other through the laws of physics. For example, the Planck mass can be calculated by multiplying the Planck length by the speed of light and dividing by the Planck time. Additionally, the Planck length and time are related through the speed of light, as seen in their respective formulas. These fundamental constants provide a way to understand the universe at a very small scale and can help us develop a more comprehensive theory of everything.

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