J. Rogers, SE Ohio
A small thought experiment.
This is a thought experiment in redefining our system of measurement. We will not change anything about the universe, nor will we discard the concept of physical dimensions like length, time, and mass. We will simply adjust the arbitrary scale of our base units—the meter, the kilogram, the Kelvin, and even the second—to build a more coherent system from the ground up.
This is explicitly not the system commonly called "Planck Units," which often involves the sloppy habit of treating fundamental constants as dimensionless numbers. Here, the constants retain their vital physical dimensions; we are merely rescaling our base units so that their value is 1, while retaining their units.
The Starting Point: An Arbitrary System
Our standard SI system is based on historical artifacts. The second is based on oscillations of a specific atom. The meter was once tied to the Earth's circumference. The kilogram was a lump of metal in a vault. As a result, the fundamental constants of nature have seemingly random, messy values that we must measure experimentally.
Our goal is to eliminate this arbitrariness, one step at a time, using the universe's own constants as our guide.
Step 1: Unifying Time and Space
We begin with the second as our anchor. However, we untether the meter from its historical definition. We create a new "meter" by rescaling it in such a way that the speed of light, c, to be exactly 1 unit of our new length per 1 unit of time. We are not defining a new speed of light, we just rescaled the meter, this results in a new c:
Definition: c = 1 new_meter / 1 second
All this does is define our new_meter as the distance light travels in one second (a "light-second"). The physical constant c is no longer a speed to be measured; it is now the perfect conversion factor. Its job is to "rotate" a quantity of time into a quantity of length. If you have 10 seconds of time, c tells you this is equivalent to 10 new_meters of length.
Step 2: Connecting Mass, Energy, and Quantum Action
Next, we untether the kilogram from its platinum-iridium cylinder. We rescale the meter, and we watch what is happening to h, we stop when:
Definition: h = 1 new_kilogram ⋅ new_meter² / 1 second
Since our new_meter is already defined in terms of the second, this definition fixes the scale of our new_kilogram based entirely on our choice of the second. The constant h now serves as the perfect conversion operator between the kinematic world (of motion and frequency) and the dynamic world (of mass and energy). In this new system, the famous equations E = mc² and E = hf reveal a simple numerical identity: Mass = Energy = Frequency.
Step 3: Unifying Temperature and Energy
We now redefine the Kelvin. Instead of tying it to the properties of water, we watch what is left of k_B after the other rescaling until:
Definition: k_B = 1 Joule / 1 new_Kelvin (where 1 Joule is 1 new_kg ⋅ new_meter² / s²)
This sets our new_Kelvin to be a direct measure of energy per degree of freedom. k_B is no longer a messy conversion factor between microscopic energy and macroscopic temperature; it is the identity operator that performs this conversion. Numerically, Temperature = Energy.
Step 4: The Final Rescaling - Anchoring to the Universe Itself
At this point, our entire system is coherent, but it is still anchored to our arbitrary, human-defined second. The gravitational constant, G, still has a seemingly random numerical value.
In the final step, we relinquish our last human artifact. We insist that G also have a value of 1 in our system's units. This is a little tricker because this is rescaling all the base units with the change in time, almost like time is the primary thing happening:
Definition: G = 1 new_meter³ ⋅ new_kilogram⁻¹ ⋅ new_second⁻²
With this final definition, the system is fully determined. We no longer have the freedom to choose the length of a second. We have four defining equations (c=1, h=1, k_B=1, G=1) for our four base units. We must now solve for the second.
This act defines a new_second based purely on the fabric of reality. This new_second then defines the new_meter, the new_kilogram, and the new_Kelvin. We have created a complete, interlocking system of measurement with no human artifacts.
So What?
We still have constants, none of the formulas has changed. The key to to see that the constants are now just present to preserve dimensional consistency. That is the only role they serve in this new way to measure things.
But this is not a new role, this change in the way we measured doesn't change the role of constants, measurement, or the way the universe works. The only role the constants ever served was to preserve dimensional consistency when the axis of measurement were scaled independently of each other.
Conclusion: The True Role of the Constants
Through this logical rescaling, we have created a set of base units that are inherent to the universe itself. In this system, the dimensional consistency of physics is perfectly preserved. The fundamental constants have not vanished. Their role has simply been clarified.
We're just adjusting our rulers and clocks to match the natural ratios that already exist in the universe.
That's it. No "profound insight about dimensional structure" or "revealing hidden truths."
The universe has certain simple ratios built into it—how the base units relate to each other. The J and N and E are built from those base measurements. We're simply choosing to measure things using units that align with those ratios, so the conversion factors come out to 1.
Constants are not mysterious numbers. They are the fundamental operators that translate, or "rotate," physical quantities between the differently scaled axes of measurement.
c rotates time into space.
h rotates frequency into mass-energy.
k_B rotates energy into temperature.
G rotates mass-energy into spacetime curvature.
That is all the constants ever do. Our conventional measurement system obscures this simple, elegant truth with arbitrary scales. This rescaled system reveals their true nature.
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