SR as Geometrical Constructions

[bcrowell]

This is just a random thought, may be totally wrong.

Euclidean geometry was originally described as a constructive theory in which the axioms state the existence (and implied uniqueness) of certain geometrical figures. These constructions are the ones that can be done with two concrete tools: a compass and straightedge.

In a more modern presentation, we might describe (plane) Euclidean geometry as the geometry implied by the metric ##ds^2=dx^2+dy^2##.

Now SR (in 1+1 dimensions for simplicity) can be described as the geometry implied by the metric ##ds^2=dt^2-dx^2##. Could we describe this in terms of concrete tools, and if so, what would the be? Light beams and a clock? If we were to write axioms for SR in the Euclidean style ("Given a ..., to construct a ..."), how would they look?

 

[PeterDonis]

Interesting question! I think Einstein would have said light beams and a clock, yes--more specifically clocks, a "fleet" of inertially moving clocks, all at rest relative to each other, synchronized using the Einstein convention, and measuring distance between them using round-trip light travel times. But Einstein didn't really explain how you get from this to "constructing" arbitrary intervals.

I think you could pick one reference clock and send out projectiles from it at different relative velocities (measured using Doppler shift), and record the times at which they pass other clocks. Constructing an arbitrary interval would amount to choosing the right relative velocity for a given projectile, and the right clock to use to record when the projectile passes; the ##dt## for the interval would be the difference in clock readings, and the ##dx## for the interval would be the distance between the clocks. So, knowing ##dt## and ##dx##, you would first choose the clock at distance ##dx## from the reference clock, and then compute the velocity the projectile would have to have to take time ##dt## to travel that distance.

Of course this only works for timelike intervals. Null intervals should be constructible by a similar method, since you can just use a light beam as the projectile; the only choice to make is which clock is the target clock. Spacelike intervals should be constructible using standard Euclidean methods, provided the fleet of clocks is chosen to be at rest in the appropriate inertial frame.

 

[SlowThinker]

I was trying to think about something like that to solve the Gravity on Einstein thread.
I think you need a pulsed light source, a "radioactive comparator", and mirrors.
The pulsed light source sends a pulse in 2 directions. Each pulse pair should have a different color.
The radioactive comparator measures the time between 2 received pulses of the same color.
A mirror is an omnidirectional point mirror, also can act as a beam splitter. Would love to have one of those
Then you have a construction commands
- Put a comparator in a place where the difference between pulses from source S is T
- Put a mirror in a place where the difference between pulses from source S to comparator R is T
- Put a source in a place where the difference between pulses in detector R is T
- Put anything in a place where something else is already, or put a source anywhere (to get started)
Not very precise but you get the idea. (I hope )
The comparator not only sets scale, but is also needed to measure acceleration.

Edit: each pulse pair should have a different color
Edit: mirror can also be used to split a beam in 2, one continuing the previous path

 

[vanhees71]

I'd not develop the theory geometrically but algebraically and analytically. There is first of all physics in this geometry, implied by the choice of the nondegenerate indefinite fundamental form ("pseudometric"), which is the possibility to build a causality structure for pseudo-Euclidean spaces with a pseudo metric of signature ##(1,d)## (west-coast convention) or ##(d,1)##. Then you can postulate equations of motion for fields (or fields and classical point particles as "strangers in the theory" as Sommerfeld put it in his Lectures), using the action principle with a strong dose of Lie-Noether-group theory.

 

[martinbn]

I'd say that the analogs to stright edge and compas should be: a tool to construct geodecis, which happen to be striaght lines as well, so you would need light signals, inertial observers for the timelike geodesics and a stight edge for the space-like geodesics (which is ok since the space-time is static and I can measure thing that I call space distances). For the compas you need a tool that constructs hyperbolas (in space-time), at leaast that seems to be the analog of a circle.

 

[stevendaryl]

It seems that the Schild's Ladder approach to defining parallel transport is very much a geometric construction.