The Biggest Ideas in the Universe: Space, Time, and Motion

by Sean Carroll

Energy isn’t a kind of substance, like water or dirt. It’s a property that things have, depending on what they are and what kind of situation they’re in.

Mass, on the other hand, is an intrinsic property; roughly speaking, mass is the resistance that an object has to being accelerated.

Speed is a number, a certain number of meters per second. Whereas velocity is a vector—a quantity with both a magnitude and a direction.

These days the momentum vector is usually denoted . (The letter m is reserved for mass, so we take the symbol from the Latin word for momentum, petere.)

Classical mechanics says that the world is made of things with definite, measurable values, obeying deterministic equations of motion; it stands in contrast with quantum mechanics. Newtonian mechanics adds specific ideas about absolute space and time. It stands in contrast with “relativistic” mechanics, which is classical but not Newtonian, and in which space and time become unified.

Noether’s theorem states that every smooth, continuous symmetry transformation of a system is associated with the conservation of some quantity.

Invariance under spatial shifts leads to conservation of momentum, and invariance under temporal shifts leads to conservation of energy.

The view that only the present moment is real is called presentism, sensibly enough. It can be contrasted with eternalism, the view that all moments of time are equally real, which we implicitly assumed in some of the discussion above. Eternalism is also called the block universe view, since it takes the real world to be the whole four-dimensional block of spacetime.

a third tempting possibility is possibilism, or the “growing present” view. There, we count both the past and the present as real while denying that status to the future.

Anything that moves at the speed of light, including light itself, doesn’t experience the passage of time. It’s tempting to ask yourself what it would be like to travel at that speed. (Apparently Einstein wondered about this when he was in high school.) The short answer is that there is nothing “it would be like” to travel at the speed of light; if somehow you could do it, you wouldn’t experience the passage of time, so you wouldn’t have any perceptions or conscious thoughts at all.

Paths in spacetime that move at the speed of light are called null trajectories (because the proper time is zero) or lightlike trajectories (for obvious reasons). When we choose units where c = 1, null trajectories are drawn as diagonal lines that tilt at 45 degrees on a spacetime diagram, traversing equal amounts of space and time. Anything that is moving slower than the speed of light will traverse more time than space, so such paths are known as timelike trajectories, moving upward in the spacetime diagram. We can also consider spacelike trajectories, which move sideways and traverse more space than time.