The Proceedings of the Eighth International Conference on Creationism (2018)

and (2) a misapplication of the relativity principle in uncritically applying the Lorentz transformation after 0 t = , rather than deriving a consistent set of coordinates based on the radar-ranging method and based on the first principle of special relativity, viz., the constancy of the speed of light independent of the observer’s speed. Recall that the Lorentz transformation was derived on the assumption of two different inertial observers in uniform relative motion; therefore, any uncritical application of the transformation to accelerated observers is invalid. The following derivation of continuous time coordinates based on “radar-ranging” is not essential to the main purpose of this paper but is provided as an example of a consistent and operational construction of coordinates based on the fundamental physics of the principle of relativity. In this regard, it is the same theoretical procedure that Einstein used to construct the coordinates in the case of globally inertial observers; and illustrates how to avoid the pitfalls of naively applying Lorentz transformations to cases in which they are inapplicable. A correct method is to use radar-ranging. Refer to Figure 13. In the radar-ranging method, a light pulse is transmitted by the observer in frame S ′ at time A t , subsequently reflected from an object at ( ) , x t and then received at time B t . (It should be noted that the artificial definition of simultaneity based on the LT results in the assignment of time coordinates to events that have not yet been observed at time 0 t = . It is thus at odds with the very operational construction of SR itself. We note that when using the radar- ranging method – a fully operationalist construction – coordinates are always assigned to events that lie in the observer’s past; and thus, have already occurred. It is necessarily compatible with the reality of time and its flow. This method is thus fully causal and results in causally consistent coordinates that can be used for computing physical quantities.) So, we consider an observer who undergoes an instantaneous velocity increase at time t = 0 . (See Figure 12) Before t = 0 , S ′ is an inertial frame coincident with S . After t=0, S ′ is in an inertial frame with velocity v in the x-direction relative to S . The radar-ranging method consists of the observer emitting a light pulse at time A t ′ to a target at a location a distance B t ′ and measuring the time of observation of the reflected signal at time B t ′ . Since the speed of light is invariant we can compute the time of reflection based on the clock readings , B A t t ′ ′ in frame S ′ as: ( ) 1 2 B A t t t ′ ′ ′ = + The range of the object measured by S ′ is then given by: ( ) 1 2 B A r t t ′ ′ ′ = − We divide Minkowski space into four coordinate charts, labeled , , , I II III IV M M M M . These are selected based on the speed of S ′ at the time of transmission and reception. In I M , transmission and reception occur when S is at rest relative to S . Thus, for all events in I M , S and S ′ agree as to assigned coordinates: t t x x ′ = ′ = In IV M , transmission and reception occur when S ′ is moving at speed v + relative to S . Thus in IV M , since S ′ is moving at constant speed, the full Lorentz transformation applies for all events in S ′ : 2 2 1 1 t vx t v x vt x v − ′ = − − ′ = − In II M and III M , however, transmission occurs when S ′ is at rest relative to S , and reception occurs when S ′ is moving at speed S relative to S . Thus in III M and III M the Lorentz transform does not apply and the transformation between the reference frames must be derived from first principles. Finally, note that, in II M and III M the x coordinate is positive or negative, respectively, for the radar method. In II M the x coordinate is the same as range, while in III M the x-coordinate is the negative of the range. We now derive the coordinate charts for II M and II M , using the constancy of the speed of light and the time dilation of S ′ clock in region IV M . In region II M , writing the kinematic equations for the null rays, gives: B B A t t x vt t t x − = − − = or 1 B A x t t v t t x + = + = − The radar range equations yield: Dennis ◀ Young earth relativistic cosmology ▶ 2018 ICC 32 Figure 13. Radar-ranging method for the frame S ′ . The observer is not globally inertial, but is inertial before and after t=0 . Coordinates are assigned by sending a light signal to a distant event and receiving the returned signal. Using the constancy of the speed of light, consistent coordinates can be assigned that are physically based on the fundamental principle of Einstein’s principle of relativity.