Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?

9 questions, answered

For the teacher

1. 1

   Which result from his earlier paper does Einstein take as his starting point, and where is it found?

   The formula for the energy l* of a packet of plane light waves measured from a coordinate system moving with speed v, taken from § 8 of the June paper (“l. c. § 8”). Sources: [1, 3]

2. 2

   Describe the thought experiment: what does the body do, and in which system is it at rest?

   A body at rest in the system (x, y, z) sends out two equal packets of light, each of energy L/2, in opposite directions. It stays at rest in (x, y, z) while the system (ξ, η, ζ) moves along the x axis with speed v. Sources: [1]

3. 3

   Show that the two light packets together carry L/√(1 − (v/V)²) in the moving system.

   Add the two terms of the second equation: the terms in (v/V)cos φ have opposite signs and cancel, leaving (L/2 + L/2)/√(1 − (v/V)²) = L/√(1 − (v/V)²). Sources: [1]

4. 4

   Why can Einstein say that H − E differs from the kinetic energy K only by a constant?

   H and E are the energies of the same body measured from two systems, and the body is at rest in one of them, so their difference is its energy of motion in the other; energies are fixed only up to an arbitrary added constant, so a constant C remains, and it does not change during the emission. Sources: [1]

5. 5

   Using 1/√(1 − x²) ≈ 1 + x²/2 for small x, derive K₀ − K₁ ≈ (L/V²)(v²/2) and explain what it says about the mass.

   K₀ − K₁ = L(1/√(1 − (v/V)²) − 1) ≈ L(v/V)²/2 = (L/V²)(v²/2). This has the form ½mv² with m replaced by L/V². The speed v is the same before and after, so the kinetic energy can fall only if the mass falls, by L/V². Sources: [1, 8]

6. 6

   Where does the number 9 · 10^20 come from? How much mass does a body lose when it gives off 1 kilowatt hour?

   It is V² in cm²/s², with V about 3 × 10^10 cm/s. 1 kWh = 3.6 × 10^6 J = 3.6 × 10^13 erg, so the loss is 3.6 × 10^13 / 9 × 10^20 = 4 × 10^-8 gram, or 4 × 10^-11 kg with c² = 9 × 10^16 m²/s². Sources: [1, 7, 6]

7. 7

   Why did Einstein suggest radium salts as a test, and why did the test not work at first?

   Its energy content changes a great deal (“in hohem Maße veränderlich”) as it gives off radiation, so its mass should change the most. Because c² is so large, the loss of mass proved much too small to observe with the methods of the time. Sources: [1, 4]

8. 8

   Einstein ends with “Wenn die Theorie den Tatsachen entspricht” (if the theory matches the facts). How was the relation tested later?

   In 1932 Cockcroft and Walton compared the masses of lithium and a proton with the two alpha particles they produced and their energy; in 2005 researchers at MIT, NIST and the Institut Laue-Langevin compared the energy of gamma rays emitted by silicon and sulfur with measured atomic masses, and found E and mc² equal to within 4 parts in 10 million. Sources: [4, 5]

9. 9

   The paper says it does not matter that the energy leaves as radiation. Why is this generalisation the heart of the result?

   Einstein concludes that the mass of any body measures its energy content, whatever form the energy takes; earlier ideas of electromagnetic mass concerned only particular kinds of energy, whereas relativity makes the relation hold for any body, whatever its internal make-up. Sources: [1, 4]