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It is primarily a reflection of the fact that we humans established our systems of units long before we stumbled upon relativity theory.

As such, we did not realize that what we perceive as space and what we perceive as time are fundamentally manifestations of the same entity, spacetime. And stupidly, we ended up using different units to measure space vs. time.

Or maybe not so stupidly. It is, after all, convenient. Just as it is convenient, for instance, for airplane pilots to measure horizontal distance in nautical miles but vertical distance (altitude) in feet. So they can actually tell you, for instance, that they are ascending 100 feet for every nautical mile traveled. Now feet per nautical mile sounds like a rather silly unit of measurement (why don’t we just use the number, 6076.12?) but we use it nonetheless, because it is, as I said, convenient.

For the same reason, we also use units like meters per second or miles per hour. We could, in principle, use a natural system of units in which the fundamental conversion factor between the unit of length and unit of time is 1. Say, measure time in seconds and length in light-seconds. In these units, Einstein’s famous formula is just [math]E=m.[/math] No conversion factor needed; it really shows the true meaning of the equation, mass-energy equivalence.

But in our everyday reality, such natural units are rather inconvenient. A light-second is a huge distance. For instance, if I had to provide biometric information, say, when applying for a passport, I’d have to write down my height as 0.0000000061 light-seconds. So we continue using things like meters or feet or whatever, just as pilots continue using incompatible units for horizontal and vertical distance, for convenience.

Similarly, we usually (but not always!) find it convenient to use units of mass when we discuss inertia, and units of energy when we discuss, say, kinetic or potential energy, despite the fact that we know that mass and energy are really the same. But it is much easier to talk about an object weighing one kilogram as opposed to 89,875,517,873,681,764 joules, for instance. Usually. But not always. Particle physicists, for instance, usually express the mass of a particle using units of energy: e.g., they tell you that an electron’s mass is 511 keV (kiloelectronvolt; 1 eV = 0.000000000000000000160218 joules) as opposed to [math]9.11\times 10^{-31}[/math] kilograms.

As to the fact why it is speed of light squared (and not cubed, or first power, or whatever) that is present in the formula that relates our concept of mass and energy, it has to do with the fact that kinetic energy is proportional to an object’s mass and the square of its velocity: [math]K=\frac{1}{2}mv^2.[/math] This formula was developed long before the mass-energy equivalence relationship was realized. An alien species that discovered things differently might have chosen to write [math]K=\frac{1}{2}E\beta^2[/math] (where we recognize [math]\beta[/math] as the dimensionless velocity, i.e., the ratio [math]v/c[/math]) and not use the concept of mass at all. Einstein’s 1905 paper, after all, tells us that the inertia of a body is proportional to its energy-content; there is no need, or use, for the separate concept of mass.


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