Of Particular Significance

Chapter 8, Endnote 7

• Quote: The lesson of the relativity formula is that intransigence, the stubbornness that resists changes in motion, comes from energy. It’s the energy stored inside an object that makes it harder to throw or catch.

• Endnote: A proof that the intransigence of a stationary object is proportional to its internal energy requires using Einstein’s formulas for relativity, which show that to change an object’s speed from zero to v requires adding motion energy that’s proportional to the object’s internal energy. (More specifically, the required motion energy equals the object’s internal energy times a simple function of speed, namely, [1 − (v/c)2 ]-1/2 − 1 .)

In Einstein’s generalization of Newton’s laws of motion, the energy of an stationary object of mass m is its internal energy

• $E_0 = m c^2 \ .$

The energy of the same object moving at speed v is

• $E_v = m c^2/ \sqrt{1 - v^2/c^2} \ .$

(For a derivation of this formula, see the discussion for endnote 9 of this chapter).

For small v, as we encounter for ordinary objects in ordinary life, we can Taylor-expand the square root, giving approximately

• $E_v \approx m c^2 + \frac{1}{2} mv^2 \ = E_0 + \frac{1}{2} mv^2$

which is the object’s internal energy plus the motion energy that Newtonian physics would have expected the object to have. In other words, for Newton, the motion energy is the difference between the total energy Ev and the internal energy, which is the same as E0 :

• ${\rm motion \ energy} = E_v - E_0 \approx \frac{1}{2} mv^2 \ .$

However, for larger v, we need Einstein’s formulas, which calculate a different amount of motion energy for the moving object:

• ${\rm motion \ energy} = E_v - E_0 = m c^2 \left[\frac{1}{\sqrt{1 - v^2/c^2}}-1\right] = E_0 \left[\frac{1}{\sqrt{1 - v^2/c^2}}-1\right] \ .$

This more complicated formula is the one that appears in the endnote. Thus motion energy Ev – E0 is indeed proportional to internal energy E0.

This confirms that intransigence arises, fundamentally, from energy. The amount of energy that we have to put into an object, in order to accelerate it from rest to a speed v, is proportional to E0 ; an object with double the internal energy requires twice as much energy input to bring it up to speed.

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