Endnotes
Note 5: Energy conservation and Einstein’s gravity
- Quote: This is a brilliant explanation because it reflects the way in which physics energy resembles money. It’s precisely measurable—we can figure out exactly how much an isolated object or set of objects has—and amounts flow from one “account” to another, though to follow the flow requires close attention.
- Endnote: Keeping track of energy and its conservation can become ambiguous in contexts when Einstein’s view of gravity is essential; these include black holes, the Big Bang, and the universe as a whole.
- Discussion: In short, conservation of energy (and momentum) is true in very small regions, i.e. locally, but when one looks at large regions, i.e. globally, defining the total energy and momentum in that region simply isn’t unique. The more complicated and strong the gravitational effects, i.e. the geometry of spacetime, the more ambiguous the definition of total energy and momentum becomes, at which point it’s meaningless to speak of whether they are conserved or not. (For advanced readers comfortable with physics and math, you can gain some additional insight from this link, which has a set of references at the end.)
Note 6: Conservation of total energy
- Quote: The total energy of an isolated object (or of an isolated set of objects) is conserved, while this is generally false for either motion energy or internal energy separately.
- Endnote: It’s perhaps surprising that a perspective-dependent form of energy can be conserved; you might expect some observers to see it as conserved and others to see it differently. But nature is clever. Although steadily moving observers, looking at an isolated object or set of objects, will disagree about how much total energy it has, they will all agree that the total energy is constant over time. That this all works out consistently is remarkable.
- Discussion: I previously gave an example of how conservation of energy and momentum works in this article — it is long, but the first half or so is mostly covered in the book’s Chapter 8, except for some discussion of momentum. If you’re already comfortable with the idea of momentum, you can jump to the section marked “Where We Are So Far“, in which the conservation of energy and momentum is discussed from several observers’ perspectives.
Note 7: Intransigence comes from energy
- 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 .)
- Discussion (coming soon)
Note 9: E = m[c2] in particle physics
- Quote: Ironically, most particle physicists don’t use either of these two simple interpretations of the relativity formula. We use a third one!
- Endnote: Taking E to be total energy and m to be rest mass, we view E = m[c2] as true only for stationary objects. Otherwise E > m[c2]; in words, the total energy of a moving object always exceeds its internal energy (by an amount that can be easily expressed in terms of the quantity called momentum.)
- Discussion: We write E2 = ( p c )2 + ( m c2 )2 where p is momentum. In terms of the object’s speed v, we don’t use Newton’s formula for momentum, p = mv, but instead we use an Einsteinian version, p = (E / c2) v . Because ( p c )2 is always positive if v isn’t zero, E = [ ( p c )2 + ( m c2 )2 ]1/2 > m[c2] for a moving object; if v=0, then p=0 and then E = m[c2]. [See also this article in which I discuss and give intuition as to how Einstein’s formulas were a sensible generalization of Newton’s.]
