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Zoltan Szekely: From Galilean to Lorentz transformation - and beyond

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Wednesday, 7. December 2022, 11:00 - 12:00
This talk explores some ideas in Special Relativity from an algebraic point of view
that might enhance students’ perception and understanding of the theory. We
restrict our attention to the kinematics of the special theory, involving inertial
motions, although some steps are taken in the direction of the general theory by an
elementary approach to speed change/ acceleration, without entering the subtlety
of General Relativity. We also take a minimalist approach by working with 2-
dimensional spacetime reference frames/ coordinate systems only.
Considering the usual spatial Galilean transformations of classical Newtonian
physics, we arrive to the idea of time shifts and temporal Galilean transformations.
By extending the concept of an inertial frame, we include into our arguments the
postulates of special relativity, the relativity principle and invariance of light speed
across inertial frames. By combining spatial and temporal Galilean transformations,
we obtain the well-known Lorentz transformation from an inertial frame K to K’ with
its formulas following the speed of light = c = 1 convention:
\[

\begin{bmatrix} t' \\ x' \end{bmatrix} =g \begin{bmatrix} 1 & -b \\ -b & 1 \end{bmatrix} \begin{bmatrix} t \\ x \end{bmatrix} \text{ or } t' =g(t-bx),\ x'=g(x-bt)
\]
where b is the relative speed of K’ from K and g is Lorentz-constant.
Touching the usual concepts of time dilation and length contraction, we look at time
contraction and length dilation, as well, that leads us toward mixed coordinates and
an introduction to standard trigonometric functions into the computation of
transformations.
As time allows, we talk about accelerating motions as approximated by equidistant
polygons. The corresponding simultaneity curves are considered using the concept
of light boxes, i.e. rectangles with sides of slope 1 or -1. Relying on a concept of
initialization and validation of moving clocks, we discover the spread of simultaneity
through the light boxes for finite and periodic motions. We also look at the motion
of extended objects/ sticks.
Assuming a virtual entity called time piece, we relate it to a moving object. While
the path/ world line of and object’s motion in the spatial domain draws the
temporal axis of a co-moving reference frame, the curve of the spread of the
associated time piece in the temporal domain provides the spatial axis of the frame.
By this association, we may introduce a special view of the correlated processes of
a moving object and a spreading of the associated time piece. These considerations
point to the direction of the general theory of relativity without entering it.
 
The lecture will be in the Riesz lecture room.
 

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