From Lorentz transformatios to Bondi's approach

[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?
Thanks in advance for your answers.

 

[DrGreg]

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?

Thanks in advance for your answers.

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).

 

[bernhard.rothenstein]

lorentz and bondi

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

 

[bernhard.rothenstein]

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?
Thanks for your answer.

 

[DrGreg]

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?

Thanks for your answer.

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.

 

[bernhard.rothenstein]

lorentz bondi

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.

 

[DrGreg]

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?

 

[bernhard.rothenstein]

Lorentz Bondi

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