How to derive the Lorentz transformation in the simplest way?

[alan123hk]

Is there the simplest, direct, and easy-to-understand method that only needs to apply the most basic algebra and logic to completely and strictly derive the Lorentz transformation?

Thanks for your help.

 

[PeroK]

Is there the simplest, direct, and easy-to-understand method that only needs to apply the most basic algebra and logic to completely and strictly derive the Lorentz transformation?

Thanks for your help.

It depends on your starting point. There are two main options:

1) Starting from only the homogeneity and isotropy of space and time:

http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

2) Starting from the postulates of SR ("speed of light" etc.)

This can be done by first deriving time dilation, length contraction and the relativity of simultaneity (leading clocks lag rule), then using these to derive the Lorentz transformation.

Or, more directly, as here for example:

https://oyc.yale.edu/sites/default/files/notes_relativity_3.pdf

 

[alan123hk]

To be honest, although I have seen many explanations of the theory of relativity in the media (YouTube etc.), I have never studied it seriously and systematically. But recently I have free time, so I am interested in learning from the basics.

The articles in the two links you provided are valuable, but the article below seems simpler and I will take the time to read it.
https://oyc.yale.edu/sites/default/files/notes_relativity_3.pdf

Thank you for your help

 

[PeroK]

To be honest, although I have seen many explanations of the theory of relativity in the media (YouTube etc.), I have never studied it seriously and systematically. But recently I have free time, so I am interested in learning from the basics.

The articles in the two links you provided are valuable, but the article below seems simpler and I will take the time to read it.
https://oyc.yale.edu/sites/default/files/notes_relativity_3.pdf

Thank you for your help

If you are serious about systematic study, I would recommend Helliwell's book:

https://www.goodreads.com/book/show/6453378-special-relativity

Alternatively, the first chapter of Morin's book is free to get you started:

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf

And you can see how you go. My recommendation is to use a proper textbook, rather than ad hoc pieces (however good they may be).

 

[Nugatory]

I have never studied it seriously and systematically. But recently I have free time, so I am interested in learning from the basics.

The Taylor and Wheeler's "Spacetime Physics" is available online at https://www.eftaylor.com/spacetimephysics/ and

 

[alan123hk]

The Taylor and Wheeler's "Spacetime Physics" is available online at https://www.eftaylor.com/spacetimephysics/ and

This is a very good book. For example, it uses simple algebra and logical reasoning to derive the Lorentz transformation, which is exactly what I want. This book has more than 300 pages and detailed explanations. It uses only simple algebra as a whole, and does not involve advanced mathematics such as tensor analysis. It is suitable for beginners. Thanks for sharing the link, I downloaded the full version.

Additional comment : However, it seems that this book does not even use calculus, which is a bit strange and unexpected.

 

[Nugatory]

Additional comment : However, it seems that this book does not even use calculus, which is a bit strange and unexpected.

Calculus is a tool to be used on problems that require it, which is to say pretty much anything in which speeds, forces, accelerations are changing. The essential concepts of special relativity can be communicated using only examples with constant speeds so no need for calculus. That 's what behind the subtitle of Einstein's (classic and excellent) popularization "Relativity - the Special and General Theory" which is something about "using only high school algebra..."

However once you have the basic concepts down it's useful to be able to take on some of the more mathematically challenging problems. You will often find people incorrectly asserting that special relativity cannot handle problems involving acceleration; they have been misled into thinking that SR can't handle these problems because their intro courses didn't touch on them. What's really going on is that tthese problems are appreciably more mathematically demanding while contributing no new deep insight so they're not introduced in the intro presentations.

So... learn special relativity with or without calculus

 

[paralleltransport]

The lorentz transformation come from a symmetry of space-time, therefore the most direct and straightforward derivation uses the definition directly.

Basically, the lorentz transformation are transformation that preserve the invariant "length". From there you realize this is the same as a rotation in imaginary time and all the cosines and sines turn into cosh and sinh (reference: landau vol. 2)

s2=(ct)2−x2s2=(ct)2−x2​

 

[rayj]

Here is a pair of presentations that roll the equations into simple pictures:
https://www.relativity.li/en/epstein2/read/c0_en/c8_en
which points to:
http://www.physastromath.ch/uploads/myPdfs/Relativ/Relativ_06_en.pdf

 

[robphy]

Here is a pair of presentations that roll the equations into simple pictures:
https://www.relativity.li/en/epstein2/read/c0_en/c8_en
which points to:
http://www.physastromath.ch/uploads/myPdfs/Relativ/Relativ_06_en.pdf

Epstein-type diagrams have limitations. (Similar comments [for different reasons] apply to Loedel diagrams and Brehme diagrams.)
These attempts try to hold on too much to Euclidean geometry.
Special relativity is fundamentally associated with Minkowskian spacetime geometry.

(If all one wants are "equations" or
one is only interested in very simple problems in special relativity,
then Epstein, Loedel, or Brehme are probably okay.
But for more complicated problems, or progressing to general relativity,
these approaches will hinder you more than help you.)

In these Epstein-diagrams, where are the light-cones?
(I don't think Epstein diagrams can display all of the events that a Minkowski spacetime diagram can.)

Possibly useful ancient threads:
https://www.physicsforums.com/threads/wheres-the-catch.151075/
https://www.physicsforums.com/threa...se-of-the-minkowski-spacetime-diagram.203490/

 

[Sagittarius A-Star]

Is there the simplest, direct, and easy-to-understand method that only needs to apply the most basic algebra and logic to completely and strictly derive the Lorentz transformation?

1. From the light clock scenatio, derive time dilation factor $1/\gamma$ and calculate $\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ with the Pythagorean theorem.
2. Use this result and the standard twin paradox scenario to argue, that the length contraction factor must be $1/\gamma$.
3. Derive the Lorentz transformation from length contraction by using the following diagram.

Solve the shown equation for $x'$:
$$x'=\gamma(x-vt)$$
With a symmetry argument, you get the inverse transformation:
$$x=\gamma(x'+vt')$$
Eliminate $x'$ between the two previous equations and then solve for $t'$ to get the time transformation:
$$t'=\gamma(t-\frac{v}{c^2}x)$$
With a symmetry argument, you get the inverse transformation:
$$t=\gamma(t'+\frac{v}{c^2}x')$$

 

[robphy]

From what I have seen,
Bondi's "k-calculus" (which is an algebra-based method) is the simplest approach for introducing special relativity, with the emphasis kept on the relativity principle, the invariance of the speed of light, and spacetime geometry (on a position-vs-time graph)... motivated by operational definitions.

The usual Lorentz Transformation formula are seen as a secondary consequence of the approach.
(This is because Bondi works in the eigenbasis of the Lorentz boost transformation.
One has to re-write his equations in terms of rectangular coordinates to the more recognizable formula.)
Here is my Insight on this approach (providing more details that what Bondi presents to a general audience)
https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/ ,
which underlies my Relativity on Rotated Graph Paper approach.

The proper-times are related by
\begin{align}
\tau_{OE}& =k \tau_{OC}\\
\tau_{ON}& =k\tau_{OE}=k^2\tau_{OC}
\end{align}
where the relativity principle implies the same value of $k$.
[$k$ started out as just a proportionality constant for each observer,
but now are equal according to relativity
...it's more-familiar physical interpretation is uncovered later.]
These will be called the "radar coordinates of E (with origin at O)" $u=\tau_{ON}$ and $v=\tau_{OC}$. ($v$ is not a velocity.)

If you are just interested in the Lorentz Transformation,
you can skip some of these beginning subsections (intended for physical interpretation and connection with standard textbook formulas).

• [To interpret in terms of rectangular coordinates...]
  The lab frame uses the "radar-method" to assign rectangular coordinates, where $\tau_{OC}$ is the clock-reading when the lab frame sends a light-signal to event E and $\tau_{ON}$ is the clock-reading when the lab frame receives a light-signal from event E \begin{align}
  \Delta t_{OE}&=\frac{1}{2}(\tau_{ON} + \tau_{OC})=\frac{1}{2}(k^2T + T)\\
  \Delta x_{OE}/c&=\frac{1}{2}(\tau_{ON} - \tau_{OC})=\frac{1}{2}(k^2T - T)
  \end{align}
  [This is arguably more-physical and more-practical for astronomical observations.
  (No long rulers into space 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 $k$ terms 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 formula)
  \begin{align}
  V_{OE}&=\frac{\Delta x_{OE}}{\Delta t_{OE}}=\frac{\frac{1}{2}c(k^2T – T)}{\frac{1}{2}(k^2T + T)}=\frac{k^2-1}{k^2+1}c
  \end{align}
• [To recognize the square-interval in rectangular form...]
  Instead, by addition and subtraction,
  we get
  \begin{align}
  \tau_{ON} &=\Delta t_{OE}+\Delta x_{OE}/c=k^2T\\
  \tau_{OC} &=\Delta t_{OE}-\Delta x_{OE}/c=T
  \end{align}
  so we see that their product is invariant (to be called the "squared-interval of OE") and is equal to the square-of-the-proper-time along OE
  \begin{align}
  \tau_{ON} \tau_{OC} &=\Delta t_{OE}^2-(\Delta x_{OE}/c)^2=(kT)^2=\tau_{OE}^2
  \end{align}
• Again, with these "radar coordinates" $u=\tau_{ON}$ and $v=\tau_{OC}$, we can locate event E. (Note: $v$ is not a velocity.)
  
  To obtain the Lorentz-Transformation and the Velocity-Transformation,
  introduce another observer (Brian) and compare the proper-times along Brian's worldline
  with what was obtained along the Lab Frame (Alfred): $\tau_{ON}$ and $\tau_{OC}$.
  
  
  
  We can see that Alfred and Brian's radar-coordinates for event E are related by:
  
  $$
  \begin{align}
  u &= k_{AB} u'\\
  v' &= k_{BA} v.
  \end{align}
  $$
  By the relativity-principle, $k_{AB}=k_{BA}$. So, call it $K_{rel}$.
  
  Rewriting as
  $$
  \begin{align}
  u' &= \frac{1}{K_{rel}} \ u\\
  v' &= K_{rel} \ v,
  \end{align}
  $$
  we have the Lorentz Transformation in radar coordinates (i.e. in the eigenbasis).
  (Clearly, $u'v'=uv$... displaying the invariance of the interval along OE.)
  No time-dilation factor $\gamma=\frac{1}{\sqrt{1-(V/c)^2}}$ or velocity $V$ is needed
  ... just the Doppler factor $K$.
• To obtain the Lorentz transformation in rectangular coordinates...:
  do addition and subtraction (and dropping the ${}_{rel}$ subscript),
  $$
  \begin{align}
  u' +v' &= ( \frac{1}{K} \ u) + (K \ v ) \\
  u' - v' &= ( \frac{1}{K} \ u) + (K \ v )
  \end{align}
  $$
  
  Then, introducing the rectangular coordinates (dropping the $\Delta$s) we have:
  \begin{align}
  2 t' &= ( \frac{1}{K} \ (t+x/c) ) + (K \ (t-x/c) ) = (K+\frac{1}{K})t - (K-\frac{1}{K})x/c \\
  2 x'/c &= ( \frac{1}{K} \ (t+x/c) ) + (K \ (t-x/c) ) = -(K-\frac{1}{K} )t + (K+\frac{1}{K})x/c
  \end{align}
  
  Some algebra shows that the time-dilation factor $\gamma=(K+\frac{1}{K})/2$
  and $\gamma V=(K-\frac{1}{K})/2$. 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 involves the Doppler Factor and is mathematically simpler (since the equations for its coordinates are uncoupled)
compared to
the Lorentz Transformation in rectangular-coordinates, which involves the time-dilation factor and the velocity.

Physically, $K$ is simpler to measure.
Assuming these zero their clocks at their meeting...
As a light-signal is sent, send the image of the sender's clock.
When a signal is received, compare the sender's transmitted image of his clock at sending
with the receiver's clock there at receiving. The ratio of reception to emission is $K$.

But they don't have to meet or zero their clocks.
Just send two signals... and work with differences.

 

[vanhees71]

Here is a pair of presentations that roll the equations into simple pictures:
https://www.relativity.li/en/epstein2/read/c0_en/c8_en
which points to:
http://www.physastromath.ch/uploads/myPdfs/Relativ/Relativ_06_en.pdf

I think the Epstein approach is more confusing than helpful. The best geometric representation still is the good old Minkowski diagram or the equivalent approach by Bondi, using light-cone coordinates. See, e.g.,

https://www.physicsforums.com/insights/relativity-rotated-graph-paper/
https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/
https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/

 

[Sagittarius A-Star]

Here you find Still the "World’s Fastest Derivation of the Lorentz Transformation" from Alan Macdonald:
https://arxiv.org/abs/physics/0606046v1

via German Wikipedia:
https://de.wikipedia.org/wiki/Lorentz-Transformation#Herleitung_aus_der_Zeitdilatation

 

[PeroK]

Here you find Still the "World’s Fastest Derivation of the Lorentz Transformation" from Alan Macdonald:
https://arxiv.org/abs/physics/0606046v1

via German Wikipedia:
https://de.wikipedia.org/wiki/Lorentz-Transformation#Herleitung_aus_der_Zeitdilatation

Can he derive the Lorentz Transformation faster than the speed of light? That's the question!

 

[robphy]

Here you find Still the "World’s Fastest Derivation of the Lorentz Transformation" from Alan Macdonald:
https://arxiv.org/abs/physics/0606046v1

via German Wikipedia:
https://de.wikipedia.org/wiki/Lorentz-Transformation#Herleitung_aus_der_Zeitdilatation

Macdonald sets up a radar measurement and then presents this first numbered set of equations
\begin{eqnarray}
T+X =\gamma(1+v)(T'+X')\\
T-X =\gamma(1-v)(T'-X')
\end{eqnarray}
(with $X'=0$, $X=vT$, and $T=\gamma T'$).
This is essentially Bondi's starting point,
"Eq 9 and 10" in my post above (which is a re-write of "Eq 1 and 2" in my post above).

$T+X$ and $T-X$ are light-cone coordinates, which can be re-written as radar-measurement times,
and
$\gamma(1+v)$ and $\gamma(1-v)$ are Doppler-Bondi factors $k=\gamma(1+v)$ and $k^{-1}=\gamma(1-v)$.

The pair of equations already contains the Lorentz Transformation (in light-cone coordinates).
By addition and subtraction, one gets the transformation in rectangular coordinates.

 

[Sagittarius A-Star]

$T+X$ and $T-X$ are light-cone coordinates

This can be shown by rotating the coordinate system by 45°:
https://www.physicslog.com/blog/2019/03/lightcone-coordinates/

 

[robphy]

This can be shown by rotating the coordinate system by 45°:
https://www.physicslog.com/blog/2019/03/lightcone-coordinates/

Yes, this is why I use "rotated graph paper".
https://www.physicsforums.com/insights/relativity-rotated-graph-paper/

From a spacetime viewpoint,
the light-cone coordinates U and V are equivalent to radar time measurements according to an observer.
The observer's T and X coordinates of an event are defined by these light-cone coordinates (radar times).
Then one can go back and write U and V in terms of T and X.

U and V are arguably more primitive than T and X
since U and V are along the eigenvectors of the boost (the basis in which the boost takes the simplest form).