Bondis k calculus

Bondis k- calculus is an approach to the special theory of relativity that is particularly widespread in the Anglo-Saxon region . It was made famous in the 1960s by Hermann Bondi and can also be found in many (mainly English-language) books on the theory of relativity.

Many introductions to relativity begin with the concept of speed and the derivation of the Lorentz transformation . Other concepts such as the so-called time dilation , the so-called length contraction , the relativity of simultaneity and the resolution of the so-called twin paradox as well as the relativistic Doppler effect are then derived from this, all as functions of speed.

Bondi reversed this order in his book Relativity and Common Sense (1964). He starts with what he calls 'fundamental ratio' and denotes k (which turns out to be the longitudinal Doppler factor).

On this basis he explains the twin paradox and the relativity of simultaneity, the so-called time dilation and length contraction, all expressed by k . Only later does he establish a connection between k and the speed. The Lorentz transformation does not appear until the end of the book.
JL Martin also uses the k calculus in his book General Relativity .

prehistory

As early as 1935, the calculus was used by Edward Arthur Milne , who designated a constant Doppler factor with s . But he dealt with the more general case of non-inertial motion (and thus also a variable Doppler factor). Instead, Bondi used the letter k and simplified Milne's view by limiting himself to constant Doppler factors. He also introduced the term " k- calculus".

Bondis k factor and the inverse k factor

Space-time diagram for the motivation of the k -factor including the reciprocal             World line from Anja              Björn's world line              World line of david              Light signal

Two inertial observers Anja and Björn (mostly Alice and Bob in English) are modeled here. Both move away from each other at a constant relative speed. Anja sends Björn light signals at the same time intervals T (according to her own clock), which of course reach Björn with a delay.
As they move away from each other, the delay increases, and thus the 'period duration' T, in which Björn receives the signals, is stretched by a constant factor that depends exclusively on the relative speed of both observers and with it

${\displaystyle k_{AB}>1}$

referred to as. The frequency is also reduced by this factor ("redshift"), because vibrations are not added while the signal is traveling. Therefore the factor k can also be interpreted as a Doppler factor.

In Björn's direction of movement there is a third inertial observer, David (Dave). His distance to Anja and thus the delay between sending by Anja and receiving by David is constant, so

${\displaystyle k_{AD}=1}$

is. Björn amplifies the signals received from Anja and sends them on immediately. There

${\displaystyle k_{AB}\cdot k_{BD}=k_{AD}=1\Leftrightarrow k_{BD}={\frac {1}{k_{AB}}}.}$

This generally applies:

1. The k factor depends exclusively on the relative speed.
2. It is greater than 1 as the distance increases and smaller as the distance decreases.
3. The k -factors for equally fast distance and approach are reciprocal values ​​of each other.

Watch comparison

Spacetime diagram for the clock paradox              World line from Anja              Björn's world line              World line from Carla              World line of david              Light signal

A fourth inertial observer Carla (English: Carol) is now moving from David to Anja just as quickly as Björn in the opposite direction, and in such a way that she passes David at the same time as Björn. The times measured by Anja, Björn and Carla are referred to as . ${\displaystyle t_{A},t_{B},t_{C}}$

When they meet, Anja and Björn synchronize their watches

${\displaystyle t_{A}=t_{B}=0.}$

At the point in time , Anja Björn sends a light signal afterwards, which Björn according to the definition of the k -factor at the point in time ${\displaystyle t_{A}=T}$

${\displaystyle t_{B}=k_{AB}T=:kT}$

receives by his own clock when he reaches David and meets Carla. Carla synchronizes her watch with Björns

${\displaystyle t_{C}=t_{B}=kT.}$

When they meet, Björn and Carla send light signals to Anja at the same time. Björn's signal reaches Anja at the time

${\displaystyle t_{A}=k^{2}T.}$

since the k -factor from Anja to Björn and that from Björn to Anja must be identical, as Galileo's principle of relativity requires. For reasons of symmetry, the same must apply to Carla's signal and also take her way from David to Anja according to her own clock . When you arrive your watch must tell the time ${\displaystyle kT}$

${\displaystyle t_{C}=2kT}$

Show. The k factor for this part of the journey must be the reciprocal factor . Therefore, for signals sent from Carla to Anja, a sending interval corresponds to a receiving interval . The time for Anja's watch when Carla arrives ${\displaystyle 1/k}$ ${\displaystyle kT}$ ${\displaystyle T}$

${\displaystyle t_{A}=(k^{2}+1)T>t_{C}=2kT}$

indicates. That t _A > t _C follows from

${\displaystyle t_{A}-t_{C}=(k^{2}-2k+1)T=(k-1)^{2}T>0,}$

for and , as it was assumed. According to Newton, t _A = t _C would have been expected. ${\displaystyle k\neq 1}$ ${\displaystyle T>0}$

Radar measurements and speed

Spacetime diagram for radar measurement              World line from Anja              Björn's world line              World line of david              Radar pulse

In the k- calculus, distances are measured with radar . Radio waves propagate like visible light with c . An observer sends a radar pulse to a target at time T ₁ and receives an echo at time T ₂ , each measured with his own watch. The distance to the target - at the point in time t _A to be determined - is therefore

${\displaystyle x_{A}={\tfrac {1}{2}}c(T_{2}-T_{1}).}$

With the prerequisite that the way there and back are the same length and take the same amount of time

${\displaystyle t_{A}={\tfrac {1}{2}}(T_{2}+T_{1}).}$

In this particular case, Anja is the observer and Björn (who is just passing David) the target. The k calculus delivers

${\displaystyle T_{2}=k^{2}T_{1},}$

and so is

${\displaystyle x_{A}={\tfrac {1}{2}}c(k^{2}-1)T_{1}}$

${\displaystyle t_{A}={\tfrac {1}{2}}(k^{2}+1)T_{1}.}$

Since Björn passed Anja at , Björn's speed is through relative to Anja ${\displaystyle t_{A}=0,x_{A}=0}$

${\displaystyle v={\frac {x_{A}}{t_{A}}}={\frac {{\tfrac {1}{2}}c(k^{2}-1)T_{1}}{{\tfrac {1}{2}}(k^{2}+1)T_{1}}}=c{\frac {k^{2}-1}{k^{2}+1}}=c{\frac {k-k^{-1}}{k+k^{-1}}}}$

given. This equation expresses the speed as a function of Bondis k factor. You can also by k resolve to k as a function of v express:

${\displaystyle k={\sqrt {\frac {c+v}{c-v}}}={\sqrt {\frac {1+v/c}{1-v/c}}}.}$

Addition theorem for velocities

Space-time diagram for the combination of k -factors             Anja              Bjorn              Erwin              Light signal

A fifth inertial observer Erwin (English: Ed) moves in the same direction as Björn, but with a different speed v _AE . In order to distinguish Björn's speed relative to Anja, it is denoted here with v _AB .

Anja sends light signals in time segments from T (according to his own clock), which Björn receives according to his watch in time segments k _AB T and Erwin according to his watch in time segments k _AE T.

Björn always sends his own in the same direction at the moment he receives Anja's signals, according to his own clock, i.e. also at time intervals of k _AB T.

Björn's signals reach Erwin according to his clock at time intervals of k _BE ( k _AB T). Since Anja's and Björn's signals must leave Björn at the same time and at the same speed

${\displaystyle k_{BE}(k_{AB}T)=k_{AE}T}$

be. Thus, k factors simply have to be multiplied.

Finally, the substitution results

${\displaystyle k_{AB}={\sqrt {\frac {1+v_{AB}/c}{1-v_{AB}/c}}},\,k_{BE}={\sqrt {\frac {1+v_{BE}/c}{1-v_{BE}/c}}},\,v_{AE}=c{\frac {k_{AE}^{2}-1}{k_{AE}^{2}+1}}}$

the relativistic addition theorem for speeds

${\displaystyle v_{AE}={\frac {v_{AB}+v_{BE}}{1+v_{AB}v_{BE}/c^{2}}}.}$

The absolute distance

Spacetime diagram for the derivation of the absolute distance and the Lorentz transformation              Anja              Bjorn              Radar pulse

With the radar method described above, Anja assigns the coordinates ( t _A , x _A ) to an event by specifying the point in time

${\displaystyle t_{A}-x_{A}/c}$

emits a radar pulse and the echo at the time

${\displaystyle t_{A}+x_{A}/c}$

receives, of course according to their watch.

Similarly, Björn assigns the coordinates ( t _B , x _B ) to the same event by assigning the point in time

${\displaystyle t_{B}-x_{B}/c}$

emits a radar pulse and the echo at the time

${\displaystyle t_{B}+x_{B}/c}$

receives, according to his watch. Instead of sending his own signal, he can also use Anja's signal.

The application of the k- calculus to the signal from Anja zu Björn gives

${\displaystyle k={\frac {t_{B}-x_{B}/c}{t_{A}-x_{A}/c}},}$

its application to the signal propagating in the opposite direction results

${\displaystyle k={\frac {t_{A}+x_{A}/c}{t_{B}+x_{B}/c}}.}$

Equation and transformation supplies

${\displaystyle c^{2}t_{A}^{2}-x_{A}^{2}=c^{2}t_{B}^{2}-x_{B}^{2}.}$

This shows that the size

${\displaystyle c^{2}t^{2}-x^{2}}$

is an invariant: It has the same value in all inertial systems.

The Lorentz Transformation

The two equations for k in the previous section can also be added and subtracted to find expressions for converting ( t _A , x _A ) to ( t _B , x _B ):

${\displaystyle ct_{B}={\tfrac {1}{2}}(k+k^{-1})ct_{A}-{\tfrac {1}{2}}(k-k^{-1})x_{A}}$

${\displaystyle x_{B}={\tfrac {1}{2}}(k+k^{-1})x_{A}-{\tfrac {1}{2}}(k-k^{-1})ct_{A}}$

This is the Lorentz transformation , expressed by Bondi's k factor instead of speed. Through the substitution

${\displaystyle k={\frac {1+v/c}{1-v/c}}}$

can be the "more traditional" form

${\displaystyle t_{B}={\frac {t_{A}-vx_{A}/c^{2}}{\sqrt {1-v^{2}/c^{2}}}};\,x_{B}={\frac {x_{A}-vt_{A}}{\sqrt {1-v^{2}/c^{2}}}}.}$

receive.

Rapidity

When combining (collinear) speeds, k- factors are multiplied. With rapidity

${\displaystyle \varsigma =\ln(k),\,k=e^{\varsigma }}$

is there an additive size, because

${\displaystyle k_{AE}=k_{AB}\cdot k_{BE}}$

${\displaystyle \Leftrightarrow \varsigma _{AE}=\varsigma _{AB}+\varsigma _{BE}.}$

The k- factor version of the Lorentz transform becomes like this

${\displaystyle ct_{B}=ct_{A}\cosh \varsigma -x_{A}\sinh \varsigma }$

${\displaystyle x_{B}=x_{A}\cosh \varsigma -ct_{A}\sinh \varsigma ,}$

which mathematically corresponds to a rotation through an angle.

The speed turns out to be the hyperbolic tangent of the rapidity up to a factor c :

${\displaystyle v=c{\frac {k-k^{-1}}{k+k^{-1}}}=c\tanh \varsigma }$

Individual evidence

1. ↑ z. B. Woodhouse, NMJ (2003), Special Relativity , Springer, ISBN 1-85233-426-6 , pp.58-65
2. ↑ z. B. Ray d'Inverno: Introducing Einstein's Relativity . Clarendon Press, 1992, ISBN 0-19-859686-3 , Chapter 2: The k -calculus.
3. ^ Hermann Bondi: Relativity and Common Sense . Doubleday & Company, New York 1964 ( archive.org - Also from Heinemann 1965 in Great Britain, reissued 1980 by Dover).
4. ↑ John Martin Martin: General Relativity . Prentice Hall, Hertfordshire 1996, ISBN 0-13-291196-5 (2nd edition; first published by E. Horwood Limited in 1988).
5. ↑ Bondi (1964), pp. 80-90
6. ↑ ^{a b} Bondi (1964), p.103
7. ↑ Woodhouse (2003), p.64
8. ↑ Woodhouse (2003), p.65
9. ↑ ^{a b} Bondi (1964) p.105
10. ↑ ^{a b c} Bondi (1964), p.118
11. ↑ ^{a b} Woodhouse (2003), p.67
12. ↑ Woodhouse (2003), p.71