# Relativity Using the Bondi K-calculus

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Although Special Relativity was formulated by Einstein (1905), and given a spacetime interpretation by Minkowski (1908) [which helped make special relativity more accessible and acceptable], it could be argued that these approaches are still too abstract and too mathematical for most students.

In the early 1960s, Hermann Bondi advocated a presentation of special relativity (called the k-calculus [which involves no calculus–just simple algebra]) that is advertised to be a “simple logical extension of well-known Newtonian ideas, without any of its mathematical trappings.” (I think what this means is that Bondi will de-emphasize and postpone the derivation of the Lorentz Transformation.)

His book “Relativity and Common Sense” (1962, 1964) is available at archive.org/details/RelativityCommonSense . In addition, Bondi presented “E=mc2: Thinking Relativity Through”, a series of ten lectures on BBC TV running from Oct 5 to Dec 7, 1963. [Are these accessible online?] There is an accompanying pamphlet “E=mc2: An Introduction to Relativity” (

Unfortunately, Bondi’s simple approach is not well-known.
I haven’t seen it in any introductory-physics or modern-physics textbook. Physics majors might encounter it in intermediate-level introductions to relativity (e.g. Woodhouse, d’Inverno, Ellis&Williams, which–maybe not surprisingly–are authored by relativists who studied at British universities). My earlier Insight ( Relativity on Rotated Graph Paper ) is, in part, my attempt to replace some of the algebra of the k-calculus by counting and arithmetic.

The radar-diagram from Bondi’s pamphlet [corrected version].

In this Insight, I wish to introduce the Bondi k-calculus, plus some of my geometrical insights.
(When my audience are students in an introductory course, I would simplify the presentation given below.)
[Admittedly, if you want a quick presentation, the Wikipedia entry is pretty good https://en.wikipedia.org/wiki/Bondi_k-calculus .]

To explain Bondi’s diagram above, let me first describe a more general situation, with a little extra notation [for clarity].

#### the k-factor

In the above diagram, OA and OB are along the worldlines of inertial observers, Alice and Bob, who met momentarily at event O. Henceforth, we assume that the speed of light is the same for all inertial observers… and we have chosen units so that all light-signals are drawn at 45-degrees.

Later, Alice broadcasts a 1-hour TV show starting at event A and ending at event C on her worldline–thus, ##\Delta T_{AC}^{\cal Alice}=1## hour. Due to the finite speed of light, there is a delay before Bob starts receiving the broadcast. Bob receives the transmitted show starting at event B and ending at event E on his worldline. Since, after event O, Bob and Alice are steadily separating, it takes longer for the transmission at event C to arrive at event E, compared to the transmission at event A to arrive at event B. In other words, it takes longer than 1 hour for Bob to view Alice’s 1-hour show–Bob views Alice’s 1-hour show in slow motion.

Call this slow-down factor for Bob watching Alice ##k^{\cal Bob}_{\cal Alice}##. If Bob, at some distance away, were at rest with respect to Alice, there would still be a delay for Bob, but the slow-down factor ##k^{\cal Bob}_{\cal Alice}## would be equal to 1 [no slow down]. The greater the rate of separation, the larger the slow-down factor. This factor does not depend on when Alice began broadcasting after event O or how long her broadcast lasted. This factor depends only on the relative-speed between Alice and Bob. That is, ##k^{\cal Bob}_{\cal Alice}## is a proportionality constant. (Mathematically, think “scaling”… or “similar triangles”. Physically, think “ratio of the period of reception by the receiver to the period of transmission of the source”… “Doppler factor”.)

Analogously, when Bob broadcasts a show to Alice, she views it in slow motion with slow-down factor ##k^{\cal Alice}_{\cal Bob}##. By the Principle of Relativity, these factors must be equal. So, let us refer to these factors as simply ##k## (and henceforth refer to it as the ##k##-factor relating these observers).

So, how does one measure k?

If we knew the period between the tickmarks of Bob’s wristwatch along Bob’s worldline, we could determine ##k##. Either we ask Bob how long it took to watch Alice’s 1-hour show, or we ask Bob to broadcast a 1-hour show and ask Alice how long it took for her to watch it. However, often the situation is that we don’t know the period between Bob’s ticks–one has to calibrate Bob’s wristwatch.

We demonstrate a radar method that Alice can use, which uses her wristwatch and radar signals she sends to and receives from Bob. Alice broadcasts her 1-hour show to Bob, who immediately rebroadcasts [or passively reflects] it back to Alice. So, Alice watches her originally broadcasted 1-hour show in super-slow-motion… with factor ##k^2##.

So, the ratio of reflected-reception period to transmission period is ##{\displaystyle\frac{\Delta T^{\cal Alice}_{MN}}{\Delta T^{\cal Alice}_{AC}}}={\displaystyle\frac{k^2\Delta T^{\cal Alice}_{AC}}{\Delta T^{\cal Alice}_{AC}}}=k^2##, which is independent of ##\Delta T^{\cal Alice}_{AC}## and event ##A##. (Note the similarity of ##\bigtriangleup ABM## and ##\bigtriangleup CEN##.)

Now, let’s simplify to Bondi’s diagram by starting the broadcast at event O and ending at event C [effectively moving events A, B, and M to coincide with event O].

#### Interpreting k and Calculating with k

To interpret ##k## in more familiar terms, Alice measures vector ##\vec {OE}## using a radar method to assign (t,x)-coordinates to distant event E on Bob’s worldline. [The set of radar events (##C## and ##N##) for Alice to measure event E is determined by the intersection of Alice’s inertial worldline and the light-cone of event E. Alice chooses an origin event ##O## on her worldline, which is often the separation event for convenience.]

We provide the calculations here. (For details, consult https://en.wikipedia.org/wiki/Bondi_k-calculus#Radar_measurements_and_velocity or my Insight Relativity on Rotated Graph Paper )

Alice assigns an elapsed time ##\Delta t_{OE}## as average of her radar times (half the sum of) ##\Delta t_{ON}## and ##\Delta t_{OC}.##
(This defines an event P on Alice’s worldline that Alice regards as simultaneous with distant event E. That is, for Alice, ##\Delta t_{OP}=\Delta t_{OE}##. Geometrically, P is the midpoint of segment CN—although this wasn’t drawn accurately in Bondi’s diagram.)
$$\Delta t_{OE}=\frac{1}{2}(k^2T + T)$$
The spatial distance ##\Delta x_{OE}## is half of the round-trip time (half the difference), times the speed of light ##c##,
$$\Delta x_{OE}=\frac{1}{2}c(k^2T – T)$$
(For Alice, this means that ##\Delta x_{OE}=\Delta x_{PE}## and thus, ##\vec {OE}=\vec {OP}+\vec {PE}##,
where ##\vec {OP}## and ##\vec {PE}## are along Alice’s time- and space-axes [and are thus Minkowski-perpendicular]. )

The [constant] velocity is therefore
$$V=\frac{\Delta x_{OE}}{\Delta t_{OE}}=\frac{\frac{1}{2}c(k^2T – T)}{\frac{1}{2}(k^2T + T)}=\frac{k^2-1}{k^2+1}c$$
Rather than solve the above velocity equation for ##k##, let us add and subtract the expressions for elapsed time and spatial distance to get ##k##:

$$\Delta t+\frac{\Delta x}{c}=k^2 T$$

$$\Delta t-\frac{\Delta x}{c}=T,$$
which expresses the radar times on Alice’s worldline to measure ##\vec {OE}##, in terms of the elapsed time and spatial distance (that is, the time- and space-components of ##\vec {OE}## according to Alice).

These are related to “light-cone coordinates“, the coordinates in the eigenbasis of the Lorentz Transformation. In my convention, ##\Delta u=\Delta t+\frac{\Delta x}{c}## and ##\Delta v=\Delta t-\frac{\Delta x}{c}##. So, these equations be rewritten as

$$\Delta u=k^2 T$$

$$\Delta v=T,$$
The ##k##-factor is an eigenvalue of the Lorentz Transformation. [See my earlier insight for details: Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity.]

• By division [or by eliminating ##T##],
$$k^2 =\frac{\Delta u}{\Delta v}=\frac{\Delta t+\frac{\Delta x}{c}}{\Delta t-\frac{\Delta x}{c}}=\frac{1+\frac{\Delta x}{c\Delta t}}{1-\frac{\Delta x}{c\Delta t}}=\frac{1+(V/c)}{1-(V/c)},$$we recognize that the ##k##-factor is the Doppler factor $$k=\sqrt{\frac{1+(V/c)}{1-(V/c)}}.$$
• By multiplication, we obtain an interesting expression of the square-magnitude of ##\vec {OE}## [on Bob’s worldline] in terms of a Pythagorean-like combination involving ##k## and Alice’s ##\Delta t## and ##(\Delta x/c)## and her ##T## (which we have made explicit):
$${\Delta u}{\Delta v}=\left(\Delta t^{\cal Alice}_{OE}\right)^2-\left(\frac{\Delta x^{\cal Alice}_{OE}}{c}\right)^2=(k_{Alice,Bob} T^{\cal Alice})^2,$$
Using ##\Delta t^{\cal Bob}_{OE}=k_{Alice,Bob} T^{\cal Alice}##, we have
$${\Delta u}{\Delta v}=\left(\Delta t^{\cal Alice}_{OE}\right)^2-\left(\frac{\Delta x^{\cal Alice}_{OE}}{c}\right)^2=(\Delta t^{\cal Bob}_{OE})^2,$$
where the expression on the right depends on Bob’s measurements alone [not Alice’s].
This suggests that if another observer, (say) Carol, measures ##\vec {OE}##, her components would satisfy an expression of the same form. That is,
$${\Delta u^{\cal Carol}}{\Delta v^{\cal Carol}}=\left(\Delta t^{\cal Carol}_{OE}\right)^2-\left(\frac{\Delta x^{\cal Carol}_{OE}}{c}\right)^2=(\Delta t^{\cal Bob}_{OE})^2.$$
In other words, for any observer measuring ##\vec{OE}##, the expression ##{\Delta u^{\cal obs}_{OE}}{\Delta v^{\cal obs}_{OE}}## or equivalently ##\left(\Delta t^{\cal obs}_{OE}\right)^2-\left(\frac{\Delta x^{\cal obs}_{OE}}{c}\right)^2## is an invariant, independent of observer. This is called the “squared-interval of ##\vec{OE}##”, denoted by ##\Delta s^2_{OE}##.
• It was observed by N.D Mermin (“Space-time intervals as light rectangles,” N. D. Mermin, Am. J. Phys. 66, 1077–1080 (1998); http://dx.doi.org/10.1119/1.19047 ) that the Doppler factor ##k## and the square-interval ##\Delta s^2_{OE}## describe the aspect-ratio ##\frac{\Delta u}{\Delta v}## and the area ##{\Delta u}{\Delta v}##of the “causal diamond” of OE. This was the inspiration of my AJP article (“Relativity on rotated graph paper,” Roberto B. Salgado, Am. J. Phys. 84, 344-359 (2016); http://dx.doi.org/10.1119/1.4943251 ). See also my Relativity of Rotated Graph Paper Insight for details.

#### formulas for Velocity Composition and the Lorentz Transformation

Bondi, in his Relativity and Common Sense (archive.org/details/RelativityCommonSense), derives the Velocity Composition and Lorentz Transformation formulas. We’ll switch to Bondi’s names: Alfred, Brian, and Edgar.

#### velocity composition

From Bondi’s diagram above, if Alfred broadcasts a ##T=1##-hour show, Brian watches that in ##k^{\cal Brian}_{\cal Alfred}T## hours (called ##kT## in the diagram). If Edgar watches Brian’s instant rebroadcast, it takes Edgar ##k^{\cal Edgar}_{\cal Brian}(k^{\cal Brian}_{\cal Alfred} T)  ## hours (called ##k’kT## in the diagram), which could be thought of as Edgar watching Alfred’s original ##T=1##-hour show in ##k^{\cal Edgar}_{\cal Alfred}T## hours. Thus, ##k^{\cal Edgar}_{\cal Alfred}T=k^{\cal Edgar}_{\cal Brian}(k^{\cal Brian}_{\cal Alfred} T)##, which implies this multiplicative relation of k-factors
$$k^{\cal Edgar}_{\cal Alfred}=k^{\cal Edgar}_{\cal Brian}k^{\cal Brian}_{\cal Alfred}.$$

Since ##\displaystyle V^{\cal Edgar}_{\cal Alfred}=\frac{(k^{\cal Edgar}_{\cal Alfred})^2-1}{(k^{\cal Edgar}_{\cal Alfred})^2+1}c=\frac{(k^{\cal Edgar}_{\cal Brian} k^{\cal Brian}_{\cal Alfred})^2-1}{(k^{\cal Edgar}_{\cal Brian}k^{\cal Brian}_{\cal Alfred})^2+1}c##
and
##\displaystyle (k^{\cal Edgar}_{\cal Brian})^2={\frac{1+V^{\cal Edgar}_{\cal Brian}/c}{1-V^{\cal Edgar}_{\cal Brian}/c}}## and ##\displaystyle (k^{\cal Brian}_{\cal Alfred})^2={\frac{1+V^{\cal Brian}_{\cal Alfred}/c}{1-V^{\cal Brian}_{\cal Alfred}/c}}##,
we obtain
$$V^{\cal Edgar}_{\cal Alfred}=\frac{V^{\cal Edgar}_{\cal Brian}+V^{\cal Brian}_{\cal Alfred}}{1+V^{\cal Edgar}_{\cal Brian}V^{\cal Brian}_{\cal Alfred}/c^2}.$$
(Sorry, if it looks cluttered… but, in this case, I think it’s better than  ‘, ”, and unprimed. In a calculation by hand, I used initials instead of full names.)

#### Lorentz Transformation

From Bondi’s diagram above, we have two inertial observers doing a radar experiment to assign coordinates to a distant event. Note the intersections of their worldlines with the lightcone of the distant event. Bondi has absorbed the factor of ##c## into the units of distance.

Restoring the factors of ##c##, we have in rectangular and light-cone coordinates:
$$(t+\frac{x}{c})=k(t’+\frac{x’}{c})\qquad u=ku’$$

$$(t’-\frac{x’}{c})=k(t-\frac{x}{c})\qquad v’=kv$$

Putting the primed quantities (Brian’s measurements) on the left, we have

$$(t’+\frac{x’}{c})=\frac{1}{k}(t+\frac{x}{c})\qquad u’=\frac{1}{k}u$$

$$(t’-\frac{x’}{c})=k(t-\frac{x}{c})\qquad v’=kv$$

By multiplication, the ##k##-factors cancel to display the invariance of the square-interval
$$t’^2-\left(\frac{x’}{c}\right)^2=t^2-\left(\frac{x}{c}\right)^2\qquad u’v’=uv.$$

By addition and subtraction, we can solve the system of equations for ##t’## and ##x’##:
##
\begin{align*}
(t’+\frac{x’}{c})+(t’-\frac{x’}{c})
&=\frac{1}{k}(t+\frac{x}{c})+k(t-\frac{x}{c})\\
2t’&=(k^{-1}+k)t+(k^{-1}-k)\frac{x}{c}\\
t’&=\left(\frac{k+k^{-1}}{2}\right)t-\left(\frac{k-k^{-1}}{2}\right)\frac{x}{c}=(\gamma) t – (\gamma V) \frac{x}{c}
\end{align*}
##

##
\begin{align*}
(t’+\frac{x’}{c})\color{red}{-}(t’-\frac{x’}{c})
&=\frac{1}{k}(t+\frac{x}{c})\color{red}{-}k(t-\frac{x}{c})\\
2\frac{x’}{c}&=(k^{-1}\color{red}{-}k)t+(k^{-1}\color{red}{+}k)\frac{x}{c}\\
\frac{x’}{c}&=\left(\frac{k\color{red}{-}k^{-1}}{2}\right)t-\left(\frac{k\color{red}{+}k^{-1}}{2}\right)\frac{x}{c}=(\gamma V)  t – (\gamma) \frac{x}{c}
\end{align*}
##
where ##k=\sqrt{\frac{1+\frac{V}{c}}{1-\frac{V}{c}}}## and ##\gamma=\frac{1}{\sqrt{1-\left(\frac{V}{c}\right)^2}}##.

These relationships between ##k##, ##V##, and ##\gamma## are possibly less obscure when one recognizes that ##k=e^\theta##, ##V=c\tanh\theta##, and ##\gamma=\frac{e^\theta+e^{-\theta}}{2}=\cosh\theta##, where ##\theta## is the rapidity[-angle].  [See my earlier insight for details: Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity.]

So, in Bondi’s method, special relativity [in (1+1)-dimensions] can be developed, with physical interpretation and relatively-simple arithmetic, from ##k## (the Doppler factor). Note that the square-roots may appear at the end of a calculation if one wishes to express things in terms of ##V## instead of ##k##.

“Relativity and Common Sense” Hermann Bondi (Dover, 1962);
available at archive.org/details/RelativityCommonSense .

“E=mc2: Thinking Relativity Through”, a series of ten lectures by Hermann Bondi on BBC TV running from Oct 5 to Dec 7, 1963. [Are these accessible online?]
http://genome.ch.bbc.co.uk/schedules/bbctv/1963-10-05

“E=mc2: An Introduction to Relativity”, Hermann Bondi
[a pamphlet that accompanies the BBC broadcast]
(

https://en.wikipedia.org/wiki/Bondi_k-calculus

“Space-time intervals as light rectangles,” N. D. Mermin,
Am. J. Phys. 66, 1077–1080 (1998); http://dx.doi.org/10.1119/1.19047

“Relativity on rotated graph paper,” Roberto B. Salgado,
Am. J. Phys. 84, 344-359 (2016); http://dx.doi.org/10.1119/1.4943251