# Generalized transformations

**Introduction**

Despite the continued success of special relativity (SR), there remains a core of theorists who are dissatisfied with the standard Lorentz transformations discussed in the previous chapter. This dissatisfaction is motivated in part by the loss of absolute simultaneity in the transition from Galilean relativity to SR. This has prompted some theorists to explore the properties of *generalized transformations, *a class of linear transformations between inertial frames that includes Galilean transformations (GTs) and Einstein-Lorentz transformations (ELTs) as special cases. In this chapter we discuss generalized transformations in terms of a toy model, a simplified version of space-time.

## Constraints

Our toy model includes the essential properties of generalized transformations and excludes properties that are regarded as superfluous. The model is based on the following assumptions:

- 1. All aspects of gravitation such as curvature of spacetime are excluded and the concept of an infinitely extended inertial frame is valid.
- 2. There is a preferred inertial frame F
_{0}in which the speed of light in all spatial directions is the same and denoted by*c.*This frame can be interpreted in several ways. - (a) It is the frame of Newton’s Absolute Space and Time.
- (b) It is the rest frame of the
*Aether,*the medium that nineteenth-century physicists believed was responsible for the transmission of electromagnetic waves. - (c) It is a local frame in which the dipole anisotropy of the cosmic microwave background radiation (CMBR) field is zero [WMAP, 2013].

*Images of Time.* First Edition. George Jaroszkiewicz.

© George Jaroszkiewicz 2016. Published in 2016 by Oxford University Press.

- 3.
**Each inertial frame consists of a chorus of observers each at rest relative to that frame and each carrying a standard clock with a rate common to that frame and determined by the physics of the***Universal Handbook**.*^{4} - 4.
**Given two relatively moving inertial frames, there are no length contraction effects transverse to the direction of motion. Therefore, any discussion involving two such frames can be restricted to one time dimension and one spatial dimension in the direction of relative motion.** - 5.
**Given point 4, the standard time and space coordinates of an event***A***in frames F**_{0}and*F**'***are denoted**

**respectively, where the symbol ^ means ***‘**is represented by’**.*

**6. Ignoring transverse coordinates, a generalized coordinate transformation from F _{0} to **

*F*

*'*

**can always be reduced to the form**

**where ***а**, **в**, **в***, and ***S* **are dimensionless parameters and ***в* **= v/c. Without loss of generality we take ***а >* **0, ***S >* **0. Here ***t* **and ***x* **are time and space coordinates of an event as observed by the F _{0} chorus whilst **

*t*

*'*

**and**

*X*

**are the time and space coordinates of the same event as observed by the**

*F*

*'*

**chorus. These time and space coordinates are considered**

*physically observable*

**(i.e., measurable) coordinate values and not just mathematical coordinate patch artefacts. Therefore, we must take care in defining the various protocols associated with various measurements.**

**7. It is implicit in all such discussions that ***after* **all observations and experiments have been concluded, chorus observations are reported back to some primary observer: the information so collected forms the basis of a reconstruction in the mind of that primary observer of past events.**

**The constant ***v* **in (18.2) is by inspection the velocity of the origin of ***F' ***coordinates as seen in the F _{0} frame. This is readily seen by setting **

*x*

**= 0 in (18.2).**

**At this stage, the generally accepted null result of the Michelson-Morley experiment has not been introduced. We shall come to it presently.**