Links between the Lorentz equations and SR/GR

[John SpaceY]

TL;DR Summary

Links between the Lorentz equations and the Theories of SR and GR

I have a question linked to the equations of Lorentz:

The Theory of the Special Relativity (SR) of Albert Einstein comes from the equation of Lorentz and we have the following equation on the time t’ in the moving frame:

The space time (x’, y’, z’, t’) is moving at a speed v measured from the space time (x, y, z, t)

t’ = g . (t – (v.x)/c^2)

If we write x = v.t we have the following equation:

t’ = g . (t – (v^2t) /c^2) = g . t (1 – (v^2)/c^2)

with g = 1 / (√(1 – v^2/c^2) )

And so we find: t’ = (√(1 – v^2/c^2) ) . t

And so t = (1 / (√(1 – v^2/c^2) ) ). t’ = g . t’

And the equation t = g . t’ is the equation of the Theory of the Special Relativity and proves that the SR comes from the equations of Lorentz.

And this result is due to the fact that we have for Lorentz x = v.t (see above), or v.x = v^2.t

And according to the SR we have also:
v = x’/t = x/t’
and also x = x’/g and t = g . t’

But, in the equation of Lorentz, we should replace x by v.t’ and not by v.t in order to be OK with the SR (see the above equations). And if we replace x by v.t’ we will have v.x = v. v.t’ = v^2.t’ and it is not v^2.t

Or we can see also that it will work if we replace v.x by v.x’ in the equation of Lorentz in the sense that:
v.x’ = v. (v.t) = v^2.t because v = x’/t according to the SR

And so it seems to have some errors somewhere to explain the equations of the SR starting from the equation of Lorentz.

Could you explain me what happen in this reasoning ? am I right or not ?

 

[Ibix]

Um... SR is the Lorentz transforms. I've not followed through whatever it is you are doing with them, but you do rather seem to have missed the point.

 

[Ibix]

Summary:: Links between the Lorentz equations and the Theories of SR and GR

The space time (x’, y’, z’, t’) is moving at a speed v measured from the space time (x, y, z, t)

No. There's only one spacetime. One frame may be moving with respect to another, but there's only one spacetime and it's incoherent to speak of it moving.

And the equation t = g . t’ is the equation of the Theory of the Special Relativity and proves that the SR comes from the equations of Lorentz.

No, that's the time dilation equation, which is just a special case of the Lorentz transforms for a clock that is at rest in the primed frame.

Summary:: Links between the Lorentz equations and the Theories of SR and GR

And according to the SR we have also:
v = x’/t = x/t’

No. You seem to be considering an object moving at $v$ in the unprimed frame. Thus $v=x/t$, but in the primed frame the clock is stationary, so $x'=0$ for all times. I'm not sure why you think dividing primed and unprimed quantities is going to give you anything meaningful - it won't.

The rest of the post just seems to perpetuate this confusion.

Could you explain me what happen in this reasoning ? am I right or not ?

Fundamentally, you seem to be very confused about what you are actually doing. You seem to start off by doing something with a single clock in inertial motion, but then you just seem to get lost.

Try starting by telling us what physical situation you are trying to describe. I think it is a single clock moving inertially, described in one unprimed frame where it has speed $v$ and one primed frame where it has speed 0. Is this correct?

 

[John SpaceY]

Thanks Ibix for your comments
I am French and I think I am not using the good names
Just to be sure:
My frame at rest is your unprimed frame : is this OK ?
The frame moving at v in the unprimed frame is your prime frame : is this OK ?
For me the distance seen from the unprimed frame is x and if we see this distance from the primed frame, this distance should be higher, according to the SR and this distance is x' : from the unprimed frame we should see a contracted distance x.
And in the prime frame we should see a contracted time t' (lower than t in the unprimed frame)
And I thought that starting by the Lorentz equations I should find again the equation of the SR and so
x = x’/g and t = g . t’

 

[Ibix]

For me the distance seen from the unprimed frame is x

What distance? What are you measuring?

x and if we see this distance from the primed frame, this distance should be higher, according to the SR and this distance is x' : from the unprimed frame we should see a contracted distance x.

It depends what you are measuring. The length of an object that is stationary in one frame will be shorter in frames in which it's moving, but you haven't specified what you are measuring so we don't know which frame (if any) is its rest frame. Furthermore, you are going to confuse yourself if you use $x$ for the length of something when it is also the usual symbol for a coordinate. Use a different symbol, such as $l$.

And in the prime frame we should see a contracted time t' (lower than t in the unprimed frame)

Again, the time measured by what clock? And again, don't use the symbol $t$, the time coordinate, for a particular measured duration. Use $\Delta t$ or something. I suspect that failing to distinguish between coordinates and lengths/durations is part of why you are confused.

And I thought that starting by the Lorentz equations I should find again the equation of the SR

The Lorentz transforms are the equations of SR. If you are hoping to derive the length contraction and time dilation formulae that can be done, but you need to state what physical system (e.g. a clock at rest in one frame, or a ruler at rest in one frame) you are measuring. Just announcing "a length" isn't enough.

 

[John SpaceY]

Thanks for your comments
I will try to be more clear
Suppose that D is a distance between 2 planets : our Earth (the unprimed frame) and another planet
And I am inside a spacecraft , near the second planet and this spacecraft is moving a v in the unprimed frame which is the Earth. The spacecraft is the primed frame
The distance seen from the spacecraft between our Earth and the spacecraft should be D and the same distance seen from our Earth should be d with D = g . d (d is lower than D and this is the contracted distance).
I the same way the time to make the travel seen from our Earth will be T1 (Delta T in fact) and the time for the travel seen from inside the spacecraft should be t2 (Delta t in fact) and we should have T1 = g . t2
T1 is higher than t2 (t2 is the contracted time)

And by starting of the Lorentz equation
t’ = g . (t – (v.x)/c^2)
I would like to find again T1 = g . t2
T1 is a delta t
t2 is a delta t'

 

[Ibix]

Suppose that D is a distance between 2 planets : our Earth (the unprimed frame) and another planet

The distance measured in which frame? From your later comment that $d<D$ I think you mean measured in the Earth's rest frame.

Also note that the Earth is not a frame. I think you mean the inertial frame in which the Earth is at rest.

The spacecraft is the primed frame

Again, a spacecraft is not a frame. Here I think you mean the frame in which the spacecraft is at rest, which is moving at speed $v$ with respect to the Earth's rest frame.

The distance seen from the spacecraft between our Earth and the spacecraft should be D and the same distance seen from our Earth should be d with D = g . d (d is lower than D and this is the contracted distance).

No. Wrong way round. The longest distance will be measured in the frame where the object is at rest - so the longer distance is measured in the Earth's rest frame.

T1 is higher than t2 (t2 is the contracted time)

Yes.

And by starting of the Lorentz equation
t’ = g . (t – (v.x)/c^2)
I would like to find again T1 = g . t2
T1 is a delta t
t2 is a delta t'

You would also need the $x'$ transform. This is fairly simple. Have a clock at rest in the primed frame. It measures some time interval between $t'=0$ and $t'=\Delta t'$. What is the time difference between these events in the unprimed frame?

Similarly, have a ruler of rest length $l$ at rest in the primed frame. What are the $x'$ coordinates of its end points at some arbitrary time $t'=T$? Where and when are those events in the unprimed frame? What has to be true for the difference between $x$ coordinates to be a length? Can you write down the length in the unprimed frame?

 

[Sagittarius A-Star]

I the same way the time to make the travel seen from our Earth will be T1 (Delta T in fact) and the time for the travel seen from inside the spacecraft should be t2 (Delta t in fact) and we should have T1 = g . t2
T1 is higher than t2 (t2 is the contracted time)

And by starting of the Lorentz equation
t’ = g . (t – (v.x)/c^2)
I would like to find again T1 = g . t2
T1 is a delta t
t2 is a delta t'

First I recommend, that you use Latex for formulas, see the "LaTeX Guide" below.

Lorentz transformation:

$\require{color} \Delta x' = \gamma (\Delta x-\color{red} v\Delta t \color{black})$
$\Delta t' = \gamma (\Delta t-v\Delta x/c^2)$

The traveler is at rest in the primed frame. For his clock ticks is valid:
$\Delta x' = 0$
=> $\require{color} \Delta x = \color{red}v\Delta t$
Put this into the above Lorentz transformation for time:
$\require{color} \Delta t' = \gamma (\Delta t-v*\color{red}v\Delta t\color{black}/c^2)$
$\Delta t' / \Delta t= \gamma (1-v^2/c^2)$
$\Delta t' / \Delta t= 1/\gamma$.

 

[John SpaceY]

Thanks Sagittarius A and Ibix
I think I have understood now
Regards
John

 

[robphy]

(Pet peeve: primed and unprimed is (in my opinion) poor bookkeeping unless accompanied by a diagram.
I prefer Alice and Bob... and, with that, I still prefer a diagram. )

I think it is useful to consider an ordinary Euclidean rotation of axes to study
properties of, e.g., a vector or maybe a pair of parallel lines
in the original axes and in the rotated axes.
Interpret the meaning of various operations...
is the ratio of two particular quantities a slope? or a cosine?
or something else meaningful? or something not-meaningful?

Then consider the Lorentz Transformation equations to study
properties of, e.g., a vector or maybe a pair of parallel lines
...

My $0.02.

 

[Sagittarius A-Star]

Thanks Sagittarius A and Ibix
I think I have understood now
Regards
John

A easier solution would be the inverse Lorentz transformation for time:
$\Delta t = \gamma (\Delta t' + v\Delta x'/c^2)$
with $\Delta x' = 0$ follows:
$\Delta t = \gamma \Delta t'$.

 

[Dale]

Pet peeve: primed and unprimed is (in my opinion) poor bookkeeping unless accompanied by a diagram.
I prefer Alice and Bob... and, with that, I still prefer a diagram.

Interesting. I actually prefer primed and unprimed over Alice and Bob. I don’t like the constant “observer” stuff. It makes new students think that SR is about optical illusions seen by people.

 

[robphy]

Interesting. I actually prefer primed and unprimed over Alice and Bob. I don’t like the constant “observer” stuff. It makes new students think that SR is about optical illusions seen by people.

My usual question is "which is prime and which is unprimed" (especially when there is no diagram)?
Is "prime" more special than "unprimed"?
For steak, yes. For inertial astronauts in special relativity, no!

With Alice and Bob,
I can say "Bob [radar] measures... " and "Bob sees..." and talk about "Bob's meterstick", and similarly for Alice... (With "prime" and the similar sounding "unprimed", the analogous sentences will sound like a tongue twister.)

The above can be shortened to "B [radar] measures..." and "B sees..." and similarly for A.

Well, I guess prime and unprimed can be shorted as well..
' measures... and ' sees...
and
measures... and sees... (there was a leading space there for unprimed's nickname ... Quote this post and see.)Bob can more easily be non-inertial than "prime guy".

I'd rather have Carol as a third astronaut than "double-prime".

It's easier to explain $v_{AC}=\frac{v_{AB}+v_{BC}}{1+v_{AB}v_{BC}}$
and rephrase as $v_{AB}=\frac{v_{AC}-v_{BC}}{1-v_{AC}v_{BC}}$.
I think a novice could learn the pattern above more easily than a prime-unprimed version.

There are no optical illusions if one emphasizes the verbs and provides an operational definition for measurements.

To me, human names make the subject matter more inviting than prime and unprimed.
Telling a story about astronauts and their adventures in spacetime sounds more natural with human names.

My $0.03.

 

[PeroK]

I prefer Alice and Bob... and, with that, I still prefer a diagram. )

Like this? With Carol and Ted thrown in.

 

[Ibix]

I generally prefer named frames ("the Earth's inertial rest frame", or "Earth's frame" for short), a policy I adopted about three seconds after Peter resolved an issue I had with understanding Kruskal diagrams by pointing out that regions III and IV are inconsistently labelled in the literature. However, this still bumps into the issue of how to notate coordinates in those frames. One or other system needs the primes, so we end up making one or other frame "the primed frame" by implication. Or else we need non-standard notation like $x_E,t_E$ for the Earth frame and $x_s,t_s$ for the ship frame. And those look more like constants than coordinates to me.

 

[Sagittarius A-Star]

Like this? With Carol and Ted thrown in.

Eventually, they need to replace Alice and Bob. Reason:

My approach would be to throw Alice and Bob to the crocodiles

 

[vanhees71]

Sometimes you need three guys. E.g., in quantum cryptography there are Alice and Bob who want to securely communicate and Eve who wants to spy on them ;-)).

Of course, in relativity it's just important to define a frame of reference by the physics being discussed. There are standard choices for different tasks. E.g., in continuum mechanics you define everything related with intrinsic properties of the matter (mass, charge densities, temperature, etc.) in the local rest frame of the "fluid cells". Then you express everything in a manifest covariant form in this frame, and then it's valid in any frame.

 

[robphy]

Then you express everything in a manifest covariant form in this frame, and then it's valid in any frame.

Yes… so with Alice-and-Bob I can suggest that a quantity is invariant by dropping subscripts like A and B.

With prime-and-unprimed, I think I have to introduce a new variable (maybe a variant that is capitalized, or in a different font or color or boldness) for fear of confusing unprimed-quantities with invariant-quantities.