Our unit of length, the meter, is not an arbitrary construct. It's deeply intertwined with the speed of light. In fact, we can understand the meter as representing a specific fraction of light-time: 1/c seconds. When we state "1 meter," we are, in essence, describing the distance that light travels in 1/c seconds. This isn't just a mathematical trick or a convenient conversion; it's a fundamental aspect of our measurement system. Our meter already embodies a relationship between space and time.
The Unity at c=1: A View of Fundamental Relationships
When we envision a system where the speed of light (c) is equal to one, the consequences are profound. Setting c=1 (by changing the definition of our unit of length/time so that a light second is one of those unit lengths) unveils a unity between mass, energy, and momentum that is usually obscured. The familiar equations:
at c=1, reduce to:
This tells us that the properties that we normally see as distinct quantities are actually expressions of the same underlying mass. This implies that our physical constants, at their fundamental levels, are describing the interactions of mass with spacetime.
Scaling Away from Unity: The Divergence of Properties
The process of moving away from this c=1 unity, where all photon properties are numerically equal to the mass value, is essentially the same as scaling the meter. Instead of using a light-second as our basic unit of length, we start to break that down into smaller distances, using fractions of that large value for convenient measurement for day to day work. Our decision to use a smaller unit of distance, like fractions of how far light goes in a second, 1/c, has the following implications:
Segmentation of Space: As we define the meter (smaller than a light second), we are effectively segmenting space into fractions of light-time. This segmentation is critical to our ability to practically measure and work with the universe.
Divergence of Energy and Momentum: The decision to use a meter and making C no longer =1 results in a separation between energy, momentum and mass, all which were unified at c=1. This is because, while mass is fixed (we are measuring that same mass), the distance it is measured over changes based on our meter definition.
Energy and Momentum Spread: The energy and momentum increase until they match the mass at c=1. We have a tendency to express distance in units of time by saying 10 meters is really 10/c seconds. As we make the length unit smaller and smaller we are increasing the energy and momentum of the mass over that small distance, and they spread apart from each other, as well as mass. We see them separate as the meter becomes an ever smaller fraction of the speed of light. That spacing is seen by mc = p and pc = E
A Natural Unit System
Our system, though it may seem arbitrary, is actually a natural way of expressing these relationships.
If we were to try to work at c=1 we would be trying to describe distances in seconds of light travel and would have to start to break that down until we eventually landed at something comparable to a meter. Because of that, our current system is not arbitrary, it has essentially segmented light-time into manageable parts that allow us to easily express the relationships between mass, energy, and momentum.
In short, we are already working with fractions of light-time when measuring distances. We could state a distance as 10/c seconds and simply give it another name. In the same way, if we were to try to work at c=1, we would eventually find ourselves having to work with fractions of the light-second for manageable units of space.
This perspective reveals something fundamental: the way we segment and measure space is intimately linked to the way mass expresses itself through energy and momentum. The relationships that we see at c=1 are ever-present, but our choice of segmentation (our unit system) can make them either more or less obvious.
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