5.5 The lorentz transformation  (Page 4/12)

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$\text{Δ}r{\prime }^{2}={\left(\text{Δ}x\prime \right)}^{2}+{\left(\text{Δ}y\prime \right)}^{2}+{\left(\text{Δ}z\prime \right)}^{2}.$

That has the same value that $\text{Δ}{r}^{2}$ had. Something similar happens with the Lorentz transformation in space-time.

Define the separation between two events, each given by a set of x , y , and ct along a four-dimensional Cartesian system of axes in space-time, as

$\left(\text{Δ}x,\text{Δ}y,\text{Δ}z,c\text{Δ}t\right)=\left({x}_{2}-{x}_{1},{y}_{2}-{y}_{1},{z}_{2}-{z}_{1},c\left({t}_{2}-{t}_{1}\right)\right).$

Also define the space-time interval $\text{Δ}s$ between the two events as

$\text{Δ}{s}^{2}={\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}.$

If the two events have the same value of ct in the frame of reference considered, $\text{Δ}s$ would correspond to the distance $\text{Δ}r$ between points in space.

The path of a particle through space-time consists of the events ( x , y , z¸ ct ) specifying a location at each time of its motion. The path through space-time is called the world line    of the particle. The world line of a particle that remains at rest at the same location is a straight line that is parallel to the time axis. If the particle moves at constant velocity parallel to the x -axis, its world line would be a sloped line $x=vt,$ corresponding to a simple displacement vs. time graph. If the particle accelerates, its world line is curved. The increment of s along the world line of the particle is given in differential form as

$d{s}^{2}={\left(dx\right)}^{2}+{\left(dy\right)}^{2}+{\left(dz\right)}^{2}-{c}^{2}{\left(dt\right)}^{2}.$

Just as the distance $\text{Δ}r$ is invariant under rotation of the space axes, the space-time interval:

$\text{Δ}{s}^{2}={\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}.$

is invariant under the Lorentz transformation. This follows from the postulates of relativity, and can be seen also by substitution of the previous Lorentz transformation equations into the expression for the space-time interval:

$\begin{array}{cc}\hfill \text{Δ}{s}^{2}& ={\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}\hfill \\ & ={\left(\frac{\text{Δ}x\prime +v\text{Δ}t\prime }{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\right)}^{2}+{\left(\text{Δ}y\prime \right)}^{2}+{\left(\text{Δ}z\prime \right)}^{2}-{\left(c\frac{\text{Δ}t\prime +\frac{v\text{Δ}x\prime }{{c}^{2}}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\right)}^{2}\hfill \\ & ={\left(\text{Δ}x\prime \right)}^{2}+{\left(\text{Δ}y\prime \right)}^{2}+{\left(\text{Δ}z\prime \right)}^{2}-{\left(c\text{Δ}t\prime \right)}^{2}\hfill \\ & =\text{Δ}s{\prime }^{2}.\hfill \end{array}$

In addition, the Lorentz transformation changes the coordinates of an event in time and space similarly to how a three-dimensional rotation changes old coordinates into new coordinates:

$\begin{array}{ccc}\mathbf{\text{Lorentz transformation}}\hfill & & \mathbf{\text{Axis}}\phantom{\rule{0.2em}{0ex}}–\phantom{\rule{0.2em}{0ex}}\mathbf{\text{rotation around}}\phantom{\rule{0.2em}{0ex}}\mathbit{\text{z}}\mathbf{\text{-axis}}\hfill \\ \left(x,t\phantom{\rule{0.2em}{0ex}}\text{coordinates):}\hfill & & \left(x,y\phantom{\rule{0.2em}{0ex}}\text{coordinates):}\hfill \\ x\prime \phantom{\rule{0.3em}{0ex}}=\left(\gamma \right)x+\left(\text{−}\beta \gamma \right)ct\hfill & & x\prime =\left(\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)x+\left(\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)y\hfill \\ ct\prime =\left(-\beta \gamma \right)x+\left(\gamma \right)ct\hfill & & y\prime =\left(-\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)x+\left(\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)y\hfill \end{array}$

where $\gamma =\frac{1}{\sqrt{1-{\beta }^{2}}};\phantom{\rule{0.5em}{0ex}}\beta =v\text{/}c.$

Lorentz transformations can be regarded as generalizations of spatial rotations to space-time. However, there are some differences between a three-dimensional axis rotation and a Lorentz transformation involving the time axis, because of differences in how the metric, or rule for measuring the displacements $\text{Δ}r$ and $\text{Δ}s,$ differ. Although $\text{Δ}r$ is invariant under spatial rotations and $\text{Δ}s$ is invariant also under Lorentz transformation, the Lorentz transformation involving the time axis does not preserve some features, such as the axes remaining perpendicular or the length scale along each axis remaining the same.

Note that the quantity $\text{Δ}{s}^{2}$ can have either sign, depending on the coordinates of the space-time events involved. For pairs of events that give it a negative sign, it is useful to define $\text{Δ}{\tau }^{2}$ as $-\text{Δ}{s}^{2}.$ The significance of $\text{Δ}\tau$ as just defined follows by noting that in a frame of reference where the two events occur at the same location, we have $\text{Δ}x=\text{Δ}y=\text{Δ}z=0$ and therefore (from the equation for $\text{Δ}{s}^{2}=-\text{Δ}{\tau }^{2}\right)\text{:}$

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