iSoul In the beginning is reality.

# Relativity of orientation

The Principle of Relativity states that the laws of physics are the same in all inertial frames of reference (IRF). Since a frame of reference includes an orientation, that is, a convention as to which rectilinear semi-axes are positive (and so which are negative), and since there is no preferred frame of reference, each frame can have its own orientation. Galilean relativity have what is called “body-fixed” orientations.

A frame of reference is called “body-fixed” if it is conceptually attached to a rigid body, such as a vehicle, watercraft, aircraft, or spacecraft. Body-fixed frames are inertial frames if the body to which the frame is affixed is in inertial motion. The body is usually referenced in anthropomorphic terms, such as its left, right, face, or back, although some craft have their own terms, notably, ships with port, starboard, fore, and aft.

Consider the following scenario of cars in five lanes, oriented so that their forward direction is positive, with their unsigned speeds shown relative to the two parked cars in the middle lane:Compare the direction of cars B, C1, C2, and D according to the frames attached to the five cars:

 From Frame Frame B Frame C1 Frame C2 Frame D A ‒v ‒2v ‒2v ‒3v B 0 ‒v ‒v ‒2v C1 +v 0 0 ‒v C2 ‒v 0 0 +v D ‒2v ‒v ‒v 0 E ‒3v ‒2v ‒2v ‒v

If we compare (1) frames B and C1 or C2 and D, they show opposite signs. But if we compare (2) frames B and C2 or C1 and D, they show the same signs. What is the difference? In the former case (1), the frames are oriented in the same direction, but in the latter case (2) the frames are oriented in opposite directions. Which one is correct?

The problem is that a frame cannot observe the orientation of another frame: the relative orientation of frames is a matter of convention. In case (1) the convention is that frames are oriented in the same direction and corresponding velocities are in opposite directions, which is called the Reciprocity Principle. In the other case (2) the convention is that frames are oriented in opposite directions and corresponding velocities are in the same direction, which could be called the Kinematic Equivalence Principle.

The advantage of convention (1) is that there is no implicit rotation between inertial frames. However, it assumes an implicit third frame traveling at the “in-between” speed (see Rindler, Essential Relativity, p.31). The advantage of convention (2) is that the frames are directed relative to one another, rather than to a third frame, and the Principle of Relativity is respected by having the corresponding velocities equal.

Berzi and Gorini derived the Reciprocity Principle from the homogeneity and isotropy of space but “for simplicity” look only at “the case when the space axes of the two observers have the same orientation” [Reciprocity Principle and the Lorentz Transformation, Journal of Math. Physics, V.10, No.8 (Aug. 1969) p.1519].  That is, they excluded another orientation convention such as (2).

A reciprocity principle that covers both conventions is this: the direct and reciprocal relative velocities between the resting and moving frame have either (a) the same orientation with equal and opposite relative velocities, or (b) opposite orientations with equal relative velocities. I suggest calling (a) three-way or positional relativity and (b) two-way or relational relativity.