Observer and frame of reference are confusing me, here..
If x is positive and v is negative, the observer will never know of the event anyway.
At least, that appears to be the case. V is a vector, but it always points at the observer, no?
(This is why I stick with Newtonian approximations at v<<C)
The magnitude of a vector has to be nonnegative real number. However, th vector itself is characterized by as an ordered set of real numbers that can be positive, negative or zero. Each real number is called a component of the vector.
The English language fortunately distinguishes between velocity and speed. Velocity is the vector and speed is its magnitude. So speed is nonnegative, properly speaking in English. However, a one dimensional vector has only one component. T
A one dimensional vector can be characterized by one real number that can be positive, negative or zero. So if v is a magnitude (e.g., speed), then it has to be either zero or a positive real number. However, the one dimensional number is a component. It can be positive, negative, or zero.
In the case of a zero dimensional vector, the absolute value of the real number is the magnitude of the vector. So the magnitude is always nonnegative because the absolute value is nonnegative. However, the actual velocity is the same as the component in the one dimensional case.
In problems with only one degree of freedom, a vector is characterized by a real number. The magnitude
The Lorentz transform uses the velocity, which is a vector. The Lorentz transform is really not determined by the speed of a reference frame. The Lorentz transform is determined by the velocity of the reference frames. very often, the coordinate system is chosen so the only one degree of freedom is important. Therefore, the velocity is merely a real number that can be positive, negative or zero. There is no speed in the Lorentz transform, even in the one-dimensional cases usually shown.
Presenting a one dimensional velocity to lay people (Maartenns word, not mine) can be confusing especially when the path of the observer is changing. The direction of a vector could be changing while the magnitude remains the same. In rotational motion, the speed of the observer often does not change while the direction changes.
The important thing to note is that the velocity vector of a body can not change unless a mechanical force is applied. This third law of motion is valid both in the 'absolute space' of Newton and the inertial frame of Einstein. Hence, the velocity of an observer can not change unless a mechanical force is applied to the observer.
So in a lot of problems, the velocity in the Lorentz transform changes due to a mechanical force. In the so called twin paradox, the sign of the velocity changes due to a mechanical force that is the thrust of the rocket. The speed of the observer does not change due to the fact that the thrust is an infinitesimal impulse. However, the velocity changes a great deal. So understanding the twin paradox relies on accepting the fact that the mechanical force is not negligible.
This occurs when the mechanical force (also a vector) on a body is perpendicular to the velocity (a vector). If the force on a body is constant and perpendicular to the velocity, the body will keep the same speed at all times. The body will move on a circular path where the velocity is parallel to the path. So to simplify the problem, the velocity is approximated as a one-dimensional vector lying on the circular path.
Suppose that you look at a quantity with units of speed that I designate to be v. How do you decide whether it is a magnitude (speed) or a vector (one dimensional velocity).
When 'v' is capable of changing sign, it is likely to be a one dimensional vector. In that case, 'v' can be positive or negative. It may be part of a three dimensional component that is changing direction. However, the direction of a one dimensional vector has only two choices: positive or negative.
I think this confusion about v being a speed or v being a one dimensional vector is really important in understanding the Hefele-Keating experiment. There, the velocity of the earths surface is a one dimensional vector.
Understanding the relativistic paradoxes without mathematics requires one to visualize the mechanical forces. The conundrums of relativity often involve determining what mechanical forces are acting on the observer. The observer can be 'conscious' of the mechanical force acting on him.
The hidden assumption is that the mechanical force on an observer is locally objective. Space and time are subjective, in that they vary with the mechanical force acting on an observer. However, the reference frame of an observer is determined by the mechanical force on the observer.
Here, I think the phenological philosophy parts way with relativity. Popper and Husseri appear to think that mechanical force is locally subjective.