The Way To Derive The Lorentz Transformation In The Best Approach?

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06 June 2022

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Bondi's "k-calculus" (which is an algebra-primarily based technique) is the best method for introducing particular relativity, with the emphasis stored on the relativity principle, the invariance of the pace of gentle, and spacetime geometry (on a place-vs-time graph)... motivated by operational definitions.

The same old Lorentz Transformation components are seen as a secondary consequence of the strategy. (It's because Bondi works in the eigenbasis of the Lorentz boost transformation. One has to re-write his equations when it comes to rectangular coordinates to the extra recognizable system.) Right here is my Insight on this method (offering extra particulars that what Bondi presents to a normal viewers) https://www.physicsforums.com/insights/relativity-utilizing-bondi-ok-calculus/ , which underlies my Relativity on Rotated Graph Paper approach.

The correct-occasions are related by \startalign \tau_OE& =ok \tau_OC\\ \tau_ON& =ok\tau_OE=k^2\tau_OC \finishalign the place the relativity precept implies the identical worth of ##okay##. [##ok## began out as only a proportionality constant for every observer, however now are equal in response to relativity ...it's extra-familiar bodily interpretation is uncovered later.] These shall be known as the "radar coordinates of E (with origin at O)" ##u=\tau_ON## and ##v=\tau_OC##. (##v## will not be a velocity.)

If you are just fascinated in the Lorentz Transformation, you'll be able to skip some of these starting subsections (meant for physical interpretation and connection with standard textbook formulation). - [To interpret by way of rectangular coordinates....] The lab frame makes use of the "radar-technique" to assign rectangular coordinates, the place ##\tau_OC## is the clock-studying when the lab body sends a mild-signal to event E and ##\tau_ON## is the clock-studying when the lab frame receives a mild-signal from event E \startalign \Delta t_OE&=\frac12(\tau_ON + \tau_OC)=\frac12(okay^2T + T)\\ \Delta x_OE/c&=\frac12(\tau_ON - \tau_OC)=\frac12(okay^2T - T) \endalign [This is arguably more-bodily and extra-sensible for astronomical observations. (No lengthy rulers into area are needed. No distant clocks at rest with respect to the observer are needed.)] (The first of this pair displays "time-dilation" when compared to ##\tau_OE=kT##.)

- [To interpret ##okay## phrases of the relative velocity ##V##....] By division, we get a relation between the relative-velocity ##V## and ##k## (which turns out to be the Doppler system) \beginalign V_OE&=\frac\Delta x_OE\Delta t_OE=\frac\frac12c(ok^2T - T)\frac12(ok^2T + T)=\frack^2-1okay^2+1c \finishalign

- [To recognize the sq.-interval in rectangular form...] As a substitute, by addition and subtraction, we get \beginalign \tau_ON &=\Delta t_OE+\Delta x_OE/c=okay^2T\\ \tau_OC &=\Delta t_OE-\Delta x_OE/c=T \finishalign so we see that their product is invariant (to be known as the "squared-interval of OE") and is equal to the sq.-of-the-correct-time along OE \startalign \tau_ON \tau_OC &=\Delta t_OE^2-(\Delta x_OE/c)^2=(kT)^2=\tau_OE^2 \finishalign

- Again, with these "radar coordinates" ##u=\tau_ON## and ##v=\tau_OC##, we will find event E. (Be aware: ##v## isn't a velocity.)

To acquire the Lorentz-Transformation and the Velocity-Transformation, introduce another observer (Brian) and examine the right-times alongside Brian's worldline with what was obtained along the Lab Frame (Alfred): ##\tau_ON## and ##\tau_OC##.

We are able to see that Alfred and Brian's radar-coordinates for occasion E are associated by:

$$ \beginalign u &= ok_AB u'\\ v' &= ok_BA v. \finishalign $$ By the relativity-precept, ##okay_AB=okay_BA##. So, name it ##K_rel##.

Rewriting as $$ \startalign u' &= \frac1K_rel \ u\\ v' &= K_rel \ v, \finishalign $$ we've got the Lorentz Transformation in radar coordinates (i.e. within the eigenbasis). (Clearly, ##u'v'=uv##.... displaying アインシュタインの2大教義 終焉 of the interval along OE.) No time-dilation factor ##\gamma=\frac1\sqrt1-(V/c)^2## or velocity ##V## is needed ... just the Doppler factor $Okay$.

To obtain the Lorentz transformation in rectangular coordinates...: do addition and subtraction (and dropping the ##_rel## subscript), $$ \startalign u' +v' &= ( \frac1K \ u) + (Ok \ v ) \\ u' - v' &= ( \frac1K \ u) + (Ok \ v ) \endalign $$

Then, introducing the rectangular coordinates (dropping the ##\Delta##s) we've: \startalign 2 t' &= ( \frac1K \ (t+x/c) ) + (Okay \ (t-x/c) ) = (Okay+\frac1K)t - (Okay-\frac1K)x/c \\ 2 x'/c &= ( \frac1K \ (t+x/c) ) + (Ok \ (t-x/c) ) = -(Okay-\frac1K )t + (Ok+\frac1K)x/c \endalign

Some algebra shows that the time-dilation issue ##\gamma=(Ok+\frac1K)/2## and ##\gamma V=(K-\frac1K)/2##. This is less complicated if one writes ##K=e^\theta## and observes that ##V=c\tanh\theta## and ##\gamma=\cosh\theta##.

The Lorentz Transformation in radar-coordinates includes the Doppler Issue and is mathematically easier (because the equations for its coordinates are uncoupled) in comparison with the Lorentz Transformation in rectangular-coordinates, which involves the time-dilation factor and the velocity.

Physically, ##Okay## is less complicated to measure. Assuming these zero their clocks at their meeting.... As a gentle-signal is shipped, ship the picture of the sender's clock. When a sign is obtained, compare the sender's transmitted picture of his clock at sending with the receiver's clock there at receiving. The ratio of reception to emission is ##Okay##.

But they do not have to satisfy or zero their clocks. Just send two alerts...
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