UNIVERSAL REALITY
UNIVERSAL REALITY
The ultimate shortcoming of General Relativity as a geometric theory is not being general enough to cover space-time at any range; this irony has its root in a misapprehension of Special Relativity. The attempt to generalise what we will show to be an incomplete account of the theory was, hence, doomed. Accordingly we will revisit Special Relativity to expose where current understanding is deficient.
The theory of Special Relativity demonstrates that an observer in a Lorentz reference frame will measure the length of an object traveling at a speed close to that of light to be shorter (contracted) in the direction of motion (i.e. in the line-of-sight) than if the same object were at rest in the observer's reference frame. Yet another observer who moves with the object will measure the same length for the object as if it were at rest in the first observer's reference frame. A clock attached to an object whose speed is close to that of light will similarly appear to run slower (its time is dilated) when compared with the passing of time in a reference frame that is stationary with respect to the observer. Relativity shows that an object, or event concerning an object, is not defined by just its spatial extension but inseparably forms a hyperbolic function with some defining duration of its existence or persistence. When we observe an object or event in any number of inertial Lorentz reference frames, the difference of the squares of the spatial and temporal increments obtained in each will compute the same value:
δt2 - δs2 = (δt')2 - (δs')2 [2]
where δt denotes time and δs space coordinate increments in the observer’s reference frame and the primed variables denote the corresponding increments in a second reference frame
The square root of this difference (the hyperbolic function) is consequently invariant and called the (Einstein or Minkowski) interval. The fact that the interval is common to all inertial Lorentz frames (and that the coordinates of different reference frames are covariant) actually signifies that it is feasible to conceive one reference frame to which all inertial reference frames are related, i.e. a universal reference frame in which a defining attribute concerning any event is unique rather than invariant (conceptually there can be only one reference frame that is universal). In other words, invariance, like relative speed, is but a vestige of Lorentz geometry and instead actually provides evidence that the concept of the universal reference frame, not the notion of local Lorentz space-time reference frames, is correct. (If the differential speed between any two Lorentz reference frames is known, Relativity's Lorentz transformations will convert space-time variables measured in one frame to a corresponding set that would be measured in the other Lorentz frame).
The hyperbolic (Einstein) interval defines a space-time dimension of an object or event that is invariable regardless of one's vantage point in any Lorentz reference frame. Although contrary to natural Euclidean inclinations, the interval's invariance can also be interpreted to have an additional meaning: the observed length and time increments in Lorentz reference frames are both geometric projections of the Einstein interval. Projections in the microscopic world of elementary phenomenon are more than the illusory vision of a real object in the macroscopic world but have a reality no less than that what is being projected. Think of such a projection in the same sense as a force acting at an angle being smaller than if the angle with direction of action were zero, but it is still one and the same real force: the term projection herein is meant in the sense of component, as in component vector. It is in the nature of projection that different objects or aspects of an object can result in the same projection (e.g. diagonals whose projection is a fundamental vector of two intersecting rectangular surfaces), while other objects or aspects of an object may not have a projection other than one without spatial extension such as an infinitesimal. It is important to realise that a projection and how it is discerned are both the product of the observer's orientation, i.e. the orientation provides a reference for the direction and so indirectly a measurement for the magnitude of an object's defining radials- composite vectors (the progenitors of the component vectors) that are free to rotate with respect to our reference. Consequently, a change in reference orientation or to another geometry will precipitate a change to a projected component vector and its relation to its radial vector. Much of the discussion that follows will elaborate on the geometrical implications of projections in different vector spaces.