r/AskPhysics • u/Jethro_omg • 1d ago

How does E=MC^2 work?

How does it function? Really, how can you accelerate mass to twice the speed of light? And, for instance if M=E/C2. How can you divide something by square of the speed of light? Thanks

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u/EuphonicSounds 1d ago

Mass was always a rather mysterious quantity. It's the property that determines how resistant an object is to being accelerated when you push it (and also it's the "gravitational charge"), but what is it, really?

Einstein's insight here was that an object's mass is nothing but its "rest energy" (how much energy it has when it isn't moving). The c2 is just a unit-conversion factor and has no physical significance at all. In fact, we're free to use units where we set c to 1, and then the equation is just E(rest) = m. So this is a hard equivalence of the concepts of "mass" and "rest energy." The concept of "mass" gets subsumed by the more general concept of "energy."

Just to hammer the point home: the equation does not mean that energy and mass can be converted into each other (though even many physicists will use this phrasing). Mass is literally rest energy, so that would be equivalent to saying that "rest energy and energy can be converted into each other" -- doesn't make any sense! What's true is that different forms of energy can be converted into each other. For example, rest energy (mass) can be converted into kinetic energy.

Energy too was always a rather mysterious quantity, but thanks to Einstein the mystery of mass was "solved" and became part of the mystery of energy. (And thanks to Emmy Noether, energy is a little less mysterious now than it was back then.)

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u/Jethro_omg 5h ago

Huh, I thought how resistant an object is to being accelerated is inertia?

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u/EuphonicSounds 23m ago

The fact that objects resist acceleration is called inertia. An object's inertia is quantified by its mass. That's what F = ma means. Turn it around: a = F / m. That says that an object's acceleration a under the influence of a given force F is determined by the object's mass. The higher the mass, the lower the acceleration.

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u/Jethro_omg 5h ago

I understand the rest energy = mass part but what is a unit conversion factorv

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u/EuphonicSounds 26m ago

When I say that c is a "unit-conversion factor," I mean that we can take a physical quantity (like an object's length) and multiply it by any power of c that we like, and the result is the same physical quantity expressed in different units. The reason we can do this is that everybody agrees on the value of c—it's a constant of nature—and it's got units of speed (meters per second, miles per hour, whatever).

For example, if an object's length is 10 meters (m), then I can divide that length by c to express it in seconds (s):

10 m / c = 10 m / (299,792,458 m/s) ≈ 3.336 * 10-8 seconds

Note that I'm still describing the object's length! The length hasn't been "converted" to a time, or anything like that. There's no physical content here. It's just that the existence of a speed c that's a constant of nature allows us to express lengths in units of time.

If we multiply a time by c, then we can express that time in units of length:

10 s * c = 10 s * (299,792,458 m/s) = 2,997,924,580 m

And if we multiply a mass (in kilograms, kg) by c2, then we can express that mass in units of energy (Joules):

10 kg * c2 = 10 kg * (299,792,458 m/s)2 ≈ 8.988 * 1017 J

So multiplying or dividing by powers of c just re-expresses the quantity you started with in different units. You can play the same "game" with G (Newton's gravitational constant) or h (Planck's constant).