Physics 1905

Does the Inertia of a Body Depend Upon Its Energy Content?

Albert Einstein

Mass and energy are one and the same thing, joined by the speed of light squared.

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In depth · the introduction

Mass and energy are the same thing wearing different clothes — and the exchange rate between them is staggeringly large.

The big idea

For three centuries physics treated mass and energy as separate things, each conserved on its own. Einstein noticed they were never really separate. When an object gives off energy — light, heat, radiation — it also becomes very slightly lighter. The mass it loses equals the energy released divided by the speed of light squared.

That divisor is what makes the relationship so dramatic. The speed of light is about 300 million metres per second; squared, it is a 9 followed by sixteen zeros. So a tiny pinch of mass corresponds to an enormous amount of energy — turn a single gram of matter entirely into energy and you release roughly the blast of twenty thousand tonnes of TNT.

How it came about

This was the last and shortest of the four papers Einstein published in 1905, his “miracle year,” while working as a clerk in the Swiss patent office in Bern. An earlier 1905 paper had already laid out special relativity; this three-page sequel asked a simple follow-up — does losing energy change a body's mass? — and answered it with a short calculation. Einstein didn't even write “E = mc²” in the modern way; the now-iconic formula is just the compact form of his result.

Why it mattered

The equation revealed that ordinary matter is a vast, locked storehouse of energy. It explained how the Sun and stars could shine for billions of years, predicted the colossal power inside the atomic nucleus, and tied together two laws that had stood apart since Newton's day. Few sentences in science have done so much with so little.

A way to picture it

Think of mass as frozen energy and the speed of light as the exchange rate at the bank. The rate is so absurdly high that you only ever cash in a sliver: even the blazing Sun trades just a tiny fraction of its mass for all its light. Convert a whole gram and the payout is a city's worth of power. Use the converter below to turn a speck of matter into its hidden energy.

Where it sits

Einstein's relativity rebuilt the stage on which Newton's mechanics had played for two hundred years, showing that space and time themselves bend with motion and gravity. E = mc² is the part of that revolution that reached furthest into everyday life — through nuclear power, medical imaging, and our understanding of the stars.

The original document

Original source text

A. Einstein · Annalen der Physik 18 (1905): 639–641 · trans. Perrett & Jeffery (1923)

The results of the previous investigation lead to a very interesting conclusion, which is here to be deduced.

I based that investigation on the Maxwell-Hertz equations for empty space, together with the Maxwellian expression for the electromagnetic energy of space, and in addition the principle that:— The laws by which the states of physical systems alter are independent of the alternative, to which of two systems of co-ordinates, in uniform motion of parallel translation relatively to each other, these alterations of state are referred (principle of relativity).

With these principles as my basis I deduced inter alia the following result. Let a system of plane waves of light, referred to the system of co-ordinates (x, y, z), possess the energy l; let the direction of the ray (the wave-normal) make an angle φ with the axis of x of the system. If we introduce a new system of co-ordinates (ξ, η, ζ) moving in uniform parallel translation with respect to the system (x, y, z), and having its origin of co-ordinates in motion along the axis of x with the velocity v, then this quantity of light — measured in the system (ξ, η, ζ) — possesses the energy l* = l · (1 − (v/c) cos φ) / √(1 − v²/c²), where c denotes the velocity of light. We shall make use of this result in what follows.

The energy balance

Let there be a stationary body in the system (x, y, z), and let its energy — referred to the system (x, y, z) — be E₀. Let the energy of the body relative to the system (ξ, η, ζ), moving as above with the velocity v, be H₀.

Let this body send out, in a direction making an angle φ with the axis of x, plane waves of light of energy L/2 measured relatively to (x, y, z), and simultaneously an equal quantity of light in the opposite direction. Meanwhile the body remains at rest with respect to the system (x, y, z). The principle of the conservation of energy must apply to this process, and in fact with respect to both systems of co-ordinates.

Calling the energy of the body after the emission of light E₁ and H₁ respectively, measured relatively to the systems (x, y, z) and (ξ, η, ζ), we obtain, using the relation given above, E₀ = E₁ + L and H₀ = H₁ + (L/2)·[ (1 − (v/c) cos φ) + (1 + (v/c) cos φ) ] / √(1 − v²/c²) = H₁ + L / √(1 − v²/c²).

By subtraction we obtain from these equations (H₀ − E₀) − (H₁ − E₁) = L · { 1 / √(1 − v²/c²) − 1 }.

The two differences of the form H − E occurring in this expression have simple physical significations. H and E are energy values of the same body referred to two systems of co-ordinates which are in motion relatively to each other, the body being at rest in one of the two systems. Thus the difference H − E can differ from the kinetic energy K of the body, with respect to the other system, only by an additive constant C, which depends on the choice of the arbitrary additive constants of the energies H and E. Since C does not change during the emission of light, we may set H₀ − E₀ = K₀ + C and H₁ − E₁ = K₁ + C, so that K₀ − K₁ = L · { 1 / √(1 − v²/c²) − 1 }.

The conclusion

The kinetic energy of the body with respect to (ξ, η, ζ) diminishes as a result of the emission of light, and the amount of diminution is independent of the properties of the body. Neglecting magnitudes of fourth and higher orders we may set K₀ − K₁ = ½ (L/c²) v².

From this equation it directly follows that:— If a body gives off the energy L in the form of radiation, its mass diminishes by L/c². The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that —

The mass of a body is a measure of its energy-content; if the energy changes by L, the mass changes in the same sense by L/9 × 10²⁰, the energy being measured in ergs, and the mass in grammes.

It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test. If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies.

Bern, September 1905.