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The Right Way To Derive The Lorentz Transformation In The Only Way?
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Bondi's "k-calculus" (which is an algebra-based mostly method) is the best strategy for introducing special relativity, with the emphasis kept on the relativity principle, the invariance of the speed of mild, and spacetime geometry (on a position-vs-time graph)... motivated by operational definitions.
The usual Lorentz Transformation method are seen as a secondary consequence of the approach. (It is because Bondi works within the eigenbasis of the Lorentz boost transformation. One has to re-write his equations when it comes to rectangular coordinates to the extra recognizable method.) Right here is my Insight on this approach (providing more particulars that what Bondi presents to a common viewers) https://www.physicsforums.com/insights/relativity-utilizing-bondi-okay-calculus/ , which underlies my Relativity on Rotated Graph Paper method.
The correct-times are associated by \startalign \tau_OE& =k \tau_OC\\ \tau_ON& =k\tau_OE=k^2\tau_OC \endalign where the relativity precept implies the same worth of ##k##. [##okay## began out as just a proportionality constant for every observer, but now are equal in response to relativity ...it is more-familiar bodily interpretation is uncovered later.] These will be known as the "radar coordinates of E (with origin at O)" ##u=\tau_ON## and ##v=\tau_OC##. (##v## just isn't a velocity.)
If you are just interested within the Lorentz Transformation, you possibly can skip a few of these starting subsections (supposed for physical interpretation and connection with normal textbook formulation). - [To interpret by way of rectangular coordinates....] The lab body makes use of the "radar-method" to assign rectangular coordinates, the place ##\tau_OC## is the clock-studying when the lab body sends a gentle-sign to occasion E and ##\tau_ON## is the clock-reading when the lab body receives a light-sign from event E \beginalign \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) \finishalign [This is arguably extra-bodily and more-practical for astronomical observations. (No lengthy rulers into house are wanted. No distant clocks at relaxation with respect to the observer are needed.)] ( アインシュタインの２大教義 終焉 of this pair shows "time-dilation" when compared to ##\tau_OE=kT##.)
- [To interpret ##okay## terms of the relative velocity ##V##....] By division, we get a relation between the relative-velocity ##V## and ##okay## (which turns out to be the Doppler formula) \beginalign V_OE&=\frac\Delta x_OE\Delta t_OE=\frac\frac12c(ok^2T - T)\frac12(k^2T + T)=\frack^2-1k^2+1c \endalign
- [To acknowledge 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-proper-time alongside OE \beginalign \tau_ON \tau_OC &=\Delta t_OE^2-(\Delta x_OE/c)^2=(kT)^2=\tau_OE^2 \endalign
- Again, with these "radar coordinates" ##u=\tau_ON## and ##v=\tau_OC##, we are able to locate event E. (Note: ##v## isn't a velocity.)
To obtain the Lorentz-Transformation and the Velocity-Transformation, introduce another observer (Brian) and evaluate the correct-instances alongside Brian's worldline with what was obtained along the Lab Body (Alfred): ##\tau_ON## and ##\tau_OC##.
We are able to see that Alfred and Brian's radar-coordinates for occasion E are related by:
$$ \beginalign u &= okay_AB u'\\ v' &= k_BA v. \endalign $$ By the relativity-precept, ##k_AB=okay_BA##. So, name it ##K_rel##.
Rewriting as $$ \startalign u' &= \frac1K_rel \ u\\ v' &= Ok_rel \ v, \finishalign $$ we've the Lorentz Transformation in radar coordinates (i.e. in the eigenbasis). (Clearly, ##u'v'=uv##.... displaying the invariance of the interval alongside OE.) No time-dilation issue ##\gamma=\frac1\sqrt1-(V/c)^2## or velocity ##V## is required ... simply the Doppler factor $K$.
To acquire the Lorentz transformation in rectangular coordinates...: do addition and subtraction (and dropping the ##_rel## subscript), $$ \startalign u' +v' &= ( \frac1K \ u) + (Ok \ v ) \\ u' - v' &= ( \frac1K \ u) + (K \ v ) \finishalign $$
Then, introducing the rectangular coordinates (dropping the ##\Delta##s) we now have: \beginalign 2 t' &= ( \frac1K \ (t+x/c) ) + (Ok \ (t-x/c) ) = (K+\frac1K)t - (Okay-\frac1K)x/c \\ 2 x'/c &= ( \frac1K \ (t+x/c) ) + (Okay \ (t-x/c) ) = -(Okay-\frac1K )t + (Okay+\frac1K)x/c \endalign
Some algebra reveals that the time-dilation factor ##\gamma=(K+\frac1K)/2## and ##\gamma V=(Ok-\frac1K)/2##. This is easier 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) compared to the Lorentz Transformation in rectangular-coordinates, which involves the time-dilation issue and the velocity.
Physically, ##K## is less complicated to measure. Assuming these zero their clocks at their meeting.... As a mild-signal is sent, ship the image of the sender's clock. When a signal is acquired, examine 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 don't have to satisfy or zero their clocks. Simply send two alerts...
The usual Lorentz Transformation method are seen as a secondary consequence of the approach. (It is because Bondi works within the eigenbasis of the Lorentz boost transformation. One has to re-write his equations when it comes to rectangular coordinates to the extra recognizable method.) Right here is my Insight on this approach (providing more particulars that what Bondi presents to a common viewers) https://www.physicsforums.com/insights/relativity-utilizing-bondi-okay-calculus/ , which underlies my Relativity on Rotated Graph Paper method.
The correct-times are associated by \startalign \tau_OE& =k \tau_OC\\ \tau_ON& =k\tau_OE=k^2\tau_OC \endalign where the relativity precept implies the same worth of ##k##. [##okay## began out as just a proportionality constant for every observer, but now are equal in response to relativity ...it is more-familiar bodily interpretation is uncovered later.] These will be known as the "radar coordinates of E (with origin at O)" ##u=\tau_ON## and ##v=\tau_OC##. (##v## just isn't a velocity.)
If you are just interested within the Lorentz Transformation, you possibly can skip a few of these starting subsections (supposed for physical interpretation and connection with normal textbook formulation). - [To interpret by way of rectangular coordinates....] The lab body makes use of the "radar-method" to assign rectangular coordinates, the place ##\tau_OC## is the clock-studying when the lab body sends a gentle-sign to occasion E and ##\tau_ON## is the clock-reading when the lab body receives a light-sign from event E \beginalign \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) \finishalign [This is arguably extra-bodily and more-practical for astronomical observations. (No lengthy rulers into house are wanted. No distant clocks at relaxation with respect to the observer are needed.)] ( アインシュタインの２大教義 終焉 of this pair shows "time-dilation" when compared to ##\tau_OE=kT##.)
- [To interpret ##okay## terms of the relative velocity ##V##....] By division, we get a relation between the relative-velocity ##V## and ##okay## (which turns out to be the Doppler formula) \beginalign V_OE&=\frac\Delta x_OE\Delta t_OE=\frac\frac12c(ok^2T - T)\frac12(k^2T + T)=\frack^2-1k^2+1c \endalign
- [To acknowledge 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-proper-time alongside OE \beginalign \tau_ON \tau_OC &=\Delta t_OE^2-(\Delta x_OE/c)^2=(kT)^2=\tau_OE^2 \endalign
- Again, with these "radar coordinates" ##u=\tau_ON## and ##v=\tau_OC##, we are able to locate event E. (Note: ##v## isn't a velocity.)
To obtain the Lorentz-Transformation and the Velocity-Transformation, introduce another observer (Brian) and evaluate the correct-instances alongside Brian's worldline with what was obtained along the Lab Body (Alfred): ##\tau_ON## and ##\tau_OC##.
We are able to see that Alfred and Brian's radar-coordinates for occasion E are related by:
$$ \beginalign u &= okay_AB u'\\ v' &= k_BA v. \endalign $$ By the relativity-precept, ##k_AB=okay_BA##. So, name it ##K_rel##.
Rewriting as $$ \startalign u' &= \frac1K_rel \ u\\ v' &= Ok_rel \ v, \finishalign $$ we've the Lorentz Transformation in radar coordinates (i.e. in the eigenbasis). (Clearly, ##u'v'=uv##.... displaying the invariance of the interval alongside OE.) No time-dilation issue ##\gamma=\frac1\sqrt1-(V/c)^2## or velocity ##V## is required ... simply the Doppler factor $K$.
To acquire the Lorentz transformation in rectangular coordinates...: do addition and subtraction (and dropping the ##_rel## subscript), $$ \startalign u' +v' &= ( \frac1K \ u) + (Ok \ v ) \\ u' - v' &= ( \frac1K \ u) + (K \ v ) \finishalign $$
Then, introducing the rectangular coordinates (dropping the ##\Delta##s) we now have: \beginalign 2 t' &= ( \frac1K \ (t+x/c) ) + (Ok \ (t-x/c) ) = (K+\frac1K)t - (Okay-\frac1K)x/c \\ 2 x'/c &= ( \frac1K \ (t+x/c) ) + (Okay \ (t-x/c) ) = -(Okay-\frac1K )t + (Okay+\frac1K)x/c \endalign
Some algebra reveals that the time-dilation factor ##\gamma=(K+\frac1K)/2## and ##\gamma V=(Ok-\frac1K)/2##. This is easier 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) compared to the Lorentz Transformation in rectangular-coordinates, which involves the time-dilation issue and the velocity.
Physically, ##K## is less complicated to measure. Assuming these zero their clocks at their meeting.... As a mild-signal is sent, ship the image of the sender's clock. When a signal is acquired, examine 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 don't have to satisfy or zero their clocks. Simply send two alerts...
