# From Lorentz transformatios to Bondi's approach

• bernhard.rothenstein
In summary, the conversation discusses the use of Bondi's approach to derive the Lorentz transformations via radar detection and the use of the Doppler factor. It also considers the use of Einstein synchronization and the k-calculus method to obtain the same transformations. There is a question about the synchronization of clocks and the possibility of starting with the Lorentz transformation and replacing the relative velocity with the Doppler factor.
bernhard.rothenstein
As we know, Bondi derives the Lorentz transformations (LT) via the radar detection of the space-time coordinates of a distant event.
Once derived from other starting points the LT should account for the results obtained by Bondi expressing them as a function of the Doppler factor
D=sqrt[(1+b)/(1-b)] which can be derived without using the LT (b=V/c). The outgoing radar signal generates the event E(x,y=0,t=x/c) when detected from I and E'(x'=ct',y'=0,t'=x'/c). In accordance with the LT we have
x=(x'+cbt')/sqrt(1-b^2). (1)
But
b=(D^2-1)/1+D^2
sqrt(1-b^2)=2D/(1+D^2)
with which (1) becomes
x=[x'(1+D^2)+ct'(D^2-1)]/2D=Dx'=Dct' (2)
In a simillar way we obtain
t=[(1+D^2)x'+(D^2-1)x'/c]/2D=Dt'=Dx'/c. (3)
If the event is generated by a tardyon moving with speed u(u') the generated events are
E(x=ut,y=0,t=x/u) and E'(x'=u't',y'=0,t'=x'/u') and the corresponding LT could be espressed again as a function of D.
Do you find flows above? Do you consider that it could be a good exercise for the beginner teaching him to handle the LT?

bernhard.rothenstein said:
As we know, Bondi derives the Lorentz transformations (LT) via the radar detection of the space-time coordinates of a distant event.

Once derived from other starting points the LT should account for the results obtained by Bondi expressing them as a function of the Doppler factor D=sqrt[(1+b)/(1-b)] which can be derived without using the LT (b=V/c). The outgoing radar signal generates the event E(x,y=0,t=x/c) when detected from I and E'(x'=ct',y'=0,t'=x'/c). In accordance with the LT we have

x=(x'+cbt')/sqrt(1-b^2). (1)

But

b=(D^2-1)/1+D^2

sqrt(1-b^2)=2D/(1+D^2)

with which (1) becomes

x=[x'(1+D^2)+ct'(D^2-1)]/2D=Dx'=Dct' (2)

In a simillar way we obtain

t=[(1+D^2)x'+(D^2-1)x'/c]/2D=Dt'=Dx'/c. (3)

If the event is generated by a tardyon moving with speed u(u') the generated events are
E(x=ut,y=0,t=x/u) and E'(x'=u't',y'=0,t'=x'/u') and the corresponding LT could be espressed again as a function of D.

Do you find flows above? Do you consider that it could be a good exercise for the beginner teaching him to handle the LT?

I think you need to explain more carefully the situation and the symbols you use. What exactly is E? And what are you trying to prove?

You seem to be using the same symbol x with two different meanings when you say x=0 and t=x/c, because otherwise that would imply t=0.

By the way, it would make your posts easier to read if you left a blank line between paragraphs (as I have done to your quoted message above).

lorentz and bondi

DrGreg said:
I think you need to explain more carefully the situation and the symbols you use. What exactly is E? And what are you trying to prove?

You seem to be using the same symbol x with two different meanings when you say x=0 and t=x/c, because otherwise that would imply t=0.

By the way, it would make your posts easier to read if you left a blank line between paragraphs (as I have done to your quoted message above).
Thanks for your answer. E(x,y,t) stands "event E characterized in I by its space-time coordinates x,y,t. The approach I propose is in a single space dimension. When I say x=0 I mean an event that takes place at the origin O. E(x,0,t=x/c) characterizes an event generated by a light signal emitted from O at a point M(x,0).
My purpose is to show that the Lorentz transformations for the space-time coordinates of the same event are derived without using Bondi's approach we can recover them.
In short I think that the Lorentz transformations account for all scenarios compatible with physics.
Regards

Lorentz and Bondi

I rephrase my initial thread as follows:
1. Are the times, in the transformation equations obtained by Bondy displayed by clock synchronized a la Einstein?
2. If so in order to obtain them we could start with the Lorentz transformation replacing there the relative velocity V with the Doppler factor from the Doppler shift formula D=sqrt[(1+V/c)/(1-V/c)] which can be derived without using the Lorentz-Einstein transformations?

bernhard.rothenstein said:
I rephrase my initial thread as follows:

1. Are the times, in the transformation equations obtained by Bondy displayed by clock synchronized a la Einstein?

2. If so in order to obtain them we could start with the Lorentz transformation replacing there the relative velocity V with the Doppler factor from the Doppler shift formula D=sqrt[(1+V/c)/(1-V/c)] which can be derived without using the Lorentz-Einstein transformations?

I am not 100% sure exactly which equations you mean by "the transformation equations obtained by Bondi".

Take a look at this post, where I derive the Lorentz Transform using Bondi's k-calculus method, directly from Einstein's postulates and using Einstein synchronization to define a time coordinate t.

If you can rephrase your questions in terms of the equations and concepts in that post, then I may understand what you are asking. The k in my post is the same as the D in your post.

There was a small error in my original post. The equation v = (k-k-1)/(k+k-1) is dimensionally incorrect; it is missing a c. It should really be

v = c(k-k-1)/(k+k-1)

In general terms, there are lots of different results in relativity which depend on each other. So, for example, if you assume results A and B you can prove result C, but equally if you assume results A and C you can prove result B. And so on.

lorentz bondi

DrGreg said:
I am not 100% sure exactly which equations you mean by "the transformation equations obtained by Bondi".

Take a look at this post, where I derive the Lorentz Transform using Bondi's k-calculus method, directly from Einstein's postulates and using Einstein synchronization to define a time coordinate t.

If you can rephrase your questions in terms of the equations and concepts in that post, then I may understand what you are asking. The k in my post is the same as the D in your post.

There was a small error in my original post. The equation v = (k-k-1)/(k+k-1) is dimensionally incorrect; it is missing a c. It should really be

v = c(k-k-1)/(k+k-1)

In general terms, there are lots of different results in relativity which depend on each other. So, for example, if you assume results A and B you can prove result C, but equally if you assume results A and C you can prove result B. And so on.

Thanks. I have seen your derivation and I will use it.
I rephrase my question:
The Bondi derivation is based on clock synchronization a la Einstein. So it should be equivalent with
x=g(x'-Vt') (1)
t=g(t'-Vx'/cc)
where g stands for gamma.
The Doppler factor D=k enable us to express V/c via the Doppler shift formula that can be derived without using the Lorentz transformations.

bernhard.rothenstein said:
Thanks. I have seen your derivation and I will use it.
I rephrase my question:
The Bondi derivation is based on clock synchronization a la Einstein. So it should be equivalent with
x=g(x'-Vt') (1)
t=g(t'-Vx'/cc)
where g stands for gamma.
The Doppler factor D=k enable us to express V/c via the Doppler shift formula that can be derived without using the Lorentz transformations.
Sorry, I'm still not sure what you are asking.

You can use the Bondi method to prove the Lorentz transformation, as you quoted, for Einstein-synced coords, and to find Einstein-velocity v as a function of Doppler factor k. (As I did in my previous post.)

Or you can start with the Lorentz transformation and find the Doppler factor k as a function of Einstein-velocity v, a standard result in most textbooks, e.g. Relativistic Doppler effect. Actually, you don't need the full Lorentz transformation, you need only time-dilation, but how do you prove that?

Lorentz Bondi

DrGreg said:
Sorry, I'm still not sure what you are asking.

You can use the Bondi method to prove the Lorentz transformation, as you quoted, for Einstein-synced coords, and to find Einstein-velocity v as a function of Doppler factor k. (As I did in my previous post.)

Or you can start with the Lorentz transformation and find the Doppler factor k as a function of Einstein-velocity v, a standard result in most textbooks, e.g. Relativistic Doppler effect. Actually, you don't need the full Lorentz transformation, you need only time-dilation, but how do you prove that?
Thanks again. I think our discussion could be more easier in front of a blackboard.
The classical Doppler shift formula performed in I and involving say a stationary laser which emits successive short light signals and a moving target relates the proper emission period t(e) and the coordinate (nonproper) reception period t(r). An observer attached to the target, measures using his wrist watch a proper reception time interval t'(r) related to t(r) by the time dilation formula which can be derived without using the LT with which the classical Doppler shift formula becomes the relativistic one.
I learned much from Peres' "Relativistic telemetry"*in what concerns how formulas that account for relativistic effects could be derived without using the LT.
Such discussions are of big help for me. Thanks again.
*Asher Peres, "Relativistic telemetry," Am.J.Phys.55, 516-519 1987

## 1) What are Lorentz transformations?

Lorentz transformations are mathematical equations that describe the relationship between space and time in special relativity. They were developed by Hendrik Lorentz in the late 19th century and are used to explain the effects of relative motion on measurements of space and time.

## 2) How are Lorentz transformations related to Einstein's theory of relativity?

Lorentz transformations are a key component of Einstein's theory of special relativity. In this theory, the laws of physics are the same for all observers in uniform motion, and Lorentz transformations help to explain how measurements of space and time can appear different for different observers.

## 3) What is Bondi's approach?

Bondi's approach is a mathematical framework for understanding the effects of general relativity on the behavior of light and other electromagnetic radiation. It was developed by Hermann Bondi in the late 20th century and is used to explain phenomena such as gravitational lensing and the redshift of light in a gravitational field.

## 4) How do Lorentz transformations and Bondi's approach differ?

Lorentz transformations and Bondi's approach both deal with the effects of relativity on space and time, but they approach the subject from different perspectives. Lorentz transformations focus on the relationship between objects in motion, while Bondi's approach focuses on the behavior of electromagnetic radiation in the presence of gravity.

## 5) What real-world applications do Lorentz transformations and Bondi's approach have?

Lorentz transformations and Bondi's approach have numerous applications in modern physics and engineering. For example, they are used in the design of GPS systems, particle accelerators, and spacecraft navigation. They also have implications for our understanding of the universe and its origins.

• Special and General Relativity
Replies
1
Views
2K
• Special and General Relativity
Replies
22
Views
2K
• Special and General Relativity
Replies
9
Views
1K
• Special and General Relativity
Replies
10
Views
762
• Special and General Relativity
Replies
7
Views
775
• Special and General Relativity
Replies
5
Views
485
• Special and General Relativity
Replies
1
Views
901
• Special and General Relativity
Replies
17
Views
1K
• Special and General Relativity
Replies
9
Views
2K
• Special and General Relativity
Replies
5
Views
1K