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Freixas

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@PeterDonis has been a big influence on how I try to approach S.R. What follows is a summary of some basic rules he gave me (any errors in paraphrasing are mine, of course):

- If something is an invariant, it reflects actual physics.
- Things that are not invariant are artifacts of a choice of coordinates, but can still be useful in calculating invariants.
- Any analysis that
*is*coordinate dependent is going to tell you*some*things that have nothing to do with the actual physics, but are artifacts of your choice of coordinates.

I’ve found definitions similar to this:

*Relativistic invariance (or Lorentz invariance) means "the same regardless of frame of reference".*

I think observables are invariants. The definition I’ve found for an observable is “a physical quantity that can be measured.”

Are events invariants? Some definitions are that an event is a “point in spacetime”. Others say an event matches a physical occurrence with a point in spacetime. A physical occurrence sounds like an observable, although some physical occurrences might not be measurable. Time and space, however, are coordinates that are generally not thought of as invariants.

Bob and Alice meet. This sounds like an event and even sounds observable, but I’m not clear what we would measure.

I came up with an invariant-qualification scheme that will lead to some further questions.

- Type I invariants have no qualifiers. I would like to include events in this class. A less problematic example would be
*c*, the speed of light (the two-way speed, anyway). - Type II variants are qualified only by their frame of reference. The only examples I have are proper time and proper location—the term “proper” identifies that these are measured relative to the inertial frame in which the observer and any relevant objects are at rest.
- Type III invariants require a frame of reference and a simultaneity convention.

*not*invariant. (Does anyone say “variant” instead of “not invariant”?)

Let’s consider a simple problem: Two observers meet. One is at rest; the other is moving at 80%c relative to the first. How long (by the first observer’s clock) after their meeting will the second observer be 4 LY away (a proper length relative to the first)?

We can use a 3D lattice as per Taylor/Wheeler, with observation equipment and clocks at each lattice junction. When a lattice point 4 LY from the first observer notes the presence of the second, it marks the time. This information can later be downloaded.

The meeting of the second observer with this lattice point is an event; let’s call it A. The lattice will also contain other events whose time matches this event’s time. One of those events will match a lattice point with the first observer; let’s call it B.

The clock time for event A is an invariant, but to say B occurs

*at the same time as*A seems like an artifact of the coordinate system. But both things depend on the same (arbitrary) simultaneity convention used to initialize the lattice’s clocks. Also, that the two events occurred

*at the same time*is literally what the lattice tells us.

In this example, a single time value is associated with event B. We only get in trouble if we try to match this time to a time occurring elsewhere. There are some type III invariants, such as lengths, velocities, and clock rate comparisons, which always require matching the time for two points separated by some distance. But we could imagine, for example, a ship passing a planet and then having the observed length of the ship posted on a huge sign. This seems to transform a length into an observable and thus an invariant.

I am clearly hazy as to whether what I’ve called type III invariants are actually invariants at all or only under certain conditions.

**P.S.**I'm sure there are other types of invariants. I just focused on the basic ones I'm familiar with.