The topic comes up, as it sometimes does, of the mass-energy equivalence from relativity. There are different tangents to this — what does the equivalence really mean, can you really turn energy into mass, does a photon have rest mass, what is the difference between relativistic mass and rest mass, and is the use of relativistic mass grounds for justifiable homicide, or is one compelled to stop at maiming?

E = mc² is the equation everyone knows, but what many don’t know is that the equation already assumes one is at rest. The actual equation is E² = p²c² + m²c⁴, which reduces to the more familiar form when the object is at rest. The implications of this are that photons have no mass, the mass term for massive particles doesn’t change when you move — that energy is in the kinetic term, (which renders relativistic mass moot) and also that the mass will increase if you add energy that does not appear in the kinetic term, i.e. extra energy in the center-of-momentum frame appears as mass.

The last concept showed up at Cosmic Variance recently, in the context of the mass of a spinning top

The spinning gyroscope has more energy than the non-spinning one. As a test, we can imagine extracting work from the spinning gyroscope — for example, by hooking it up to a generator — in ways that we couldn’t extract work from the stationary gyroscope. And since it has more energy, it has more mass. And the weight is just the acceleration due to gravity times the mass — so, as long as we weigh our spinning and non-spinning gyroscopes in the same gravitational field, the spinning one will indeed weigh more.

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