Special Relativity - SR - Lorentz transformations

[Jan Slowak]

In the derivation of LT in [7], three special cases are used to determine the constants A, B, C, D.
SC1:

SC1:

LEx': x' = Ax + Bt
LEt': t' = Cx + Dt
SC1: x' = 0, x = vt
→ B = -Av

SC2:

SC2:

LEx': x' = Ax + Bt
LEt': t' = Cx + Dt
SC2: x' = -vt', x = 0
→ B = -Dv

But I have not seen a figure describing SC3. Can anyone help me with such a figure?

SC3:
LEx': x' = Ax + Bt
LEt': t' = Cx + Dt
SC3: x' = ct', x = ct

Thank you!

[Ghideon]

8 minutes ago, Jan Slowak said:

But I have not seen a figure describing SC3. Can anyone help me with such a figure?

Can you define every axis that are used? I get the x-axis but I’m not sure if S axis and the unlabeled axis are significant.

 

[Jan Slowak]

It is only about the x-axis and the x'-axis that coincide with each other.
[7] Modern Physics; Second edition; Randy Harris; Chapter 2; Special Relativity; 2008
I enclose pages 14-15 from [7].
You can draw a similar image as Fig. SC1 or Fig. SC2.

 

Modern Physics Randy Harris 2008 pages 14-15-v1.pdf

[Ghideon]

1 hour ago, Jan Slowak said:

It is only about the x-axis and the x'-axis that coincide with each other.

Something like this? One flash of light originating at the origin of two frames of reference moving relative to each other. At time t=0 the two frames of reference are in the same place. The x- and x'-axis overlap so I draw them separated by a dashed line instead of stacked on top of each other. The flash of light is moving with velocity c in both frames of reference. 

Note: this is a common diagram, I reduced it to one axis as requested.

https://thecuriousastronomer.wordpress.com/2013/03/10/derivation-of-the-lorentz-transformations-from-first-principles/

 

Edited June 3, 2019 by Ghideon
wrong picture

[Jan Slowak]

23 hours ago, Ghideon said:

Something like this? One flash of light originating at the origin of two frames of reference moving relative to each other. At time t=0 the two frames of reference are in the same place. The x- and x'-axis overlap so I draw them separated by a dashed line instead of stacked on top of each other. The flash of light is moving with velocity c in both frames of reference. 

Note: this is a common diagram, I reduced it to one axis as requested.

https://thecuriousastronomer.wordpress.com/2013/03/10/derivation-of-the-lorentz-transformations-from-first-principles/

 

We have two inertial reference systems S and S'. These move relative to each other at constant speed v. We say that S' moves to the right on our pictures. For the sake of simplicity, we treat a simplified case when the x-axis and the x'-axis coincide on the same line.

Then we have an event that is considered by the two reference systems. We denote the event with E = (x, t) for S and E' = (x', t') for S'. Note that physically it is about the same event!

The thought experiments dealt with in different derivations of LT begin with the moment when S and S' are in the same point. At this moment, the clocks in S and S' are synchronized. All this means that in the beginning we have
E = (x, t) = (0, 0)
 E' = (x', t') = (0, 0)
See the following image: Fig: t = 0, t' = 0

Do we agree on this?

I'm talking about the derivation of LT from [7]. You refer to the link above which we can designate with eg [DLTf1P].

The picture you have drawn is not enough to see exactly what is happening and what relationships we have between the coordinates and what conditions you set on the equation system that is formed.

One flash of light originating at the origin of two frames of reference moving relative to each other. (your words).

I now draw a picture where we see how things stand out after the time t > 0, t' > 0. Note that the image is not drawn to scale. S and S' consider the event located on the wave front on the x-axis (x'-axis). See picture Fig. t > 0, t' > 0.

Do we agree on this?

If we agree then I would ask you to help me write down the values of x, t, x', t' and the relationships between them and the value of the distance between S and S'. So as this picture shows. Thanks!

[Ghideon]

On 6/3/2019 at 6:58 PM, Jan Slowak said:

It is only about the x-axis and the x'-axis that coincide with each other.

I tried to do that and you respond with:

56 minutes ago, Jan Slowak said:

I now draw a picture where we see how things stand out after the time t > 0, t' > 0. Note that the image is not drawn to scale. S and S' consider the event located on the wave front on the x-axis (x'-axis). See picture Fig. t > 0, t' > 0.

 

The above looks like a flash of light extending in a circular fashion in two dimensions, or a sphere in three dimensions.Can you clarify what number of dimensions you wish to include? One, two or three? Can you put labels on all axes you wish to include; y, y' and z, z' for instance. (Or remove them if they are irrelevant)

 

59 minutes ago, Jan Slowak said:

Do we agree on this?

It depends. In which frame of reference does the circle belong?

 

[Ghideon]

20 hours ago, Jan Slowak said:

We have two inertial reference systems S and S'. These move relative to each other at constant speed v. We say that S' moves to the right on our pictures. For the sake of simplicity, we treat a simplified case when the x-axis and the x'-axis coincide on the same line.

Then we have an event that is considered by the two reference systems. We denote the event with E = (x, t) for S and E' = (x', t') for S'. Note that physically it is about the same event!

The thought experiments dealt with in different derivations of LT begin with the moment when S and S' are in the same point. At this moment, the clocks in S and S' are synchronized. All this means that in the beginning we have
E = (x, t) = (0, 0)
 E' = (x', t') = (0, 0)
See the following image: Fig: t = 0, t' = 0

Do we agree on this?

I revisited this after my response above and have some additional details. First I have my version of your picture above. I post this as a reference and starting point for other pictures.

I use three dimensions as in your picture but I prefer x, y, z instead of x, S, unlabeled. This is at time t=0, t'=0 as in your image. The flash is just starting to propagate; for visibility reasons it is a small circle instead of a dot in point 0,0. Note: there is only one flash of light, visible to an observer in S and an observer in S'

S and S' are moving relative to each other with speed v.

 

20 hours ago, Jan Slowak said:

I'm talking about the derivation of LT from [7]. You refer to the link above which we can designate with eg [DLTf1P].

The picture you have drawn is not enough to see exactly what is happening and what relationships we have between the coordinates and what conditions you set on the equation system that is formed.

One flash of light originating at the origin of two frames of reference moving relative to each other. (your words).

I now draw a picture where we see how things stand out after the time t > 0, t' > 0. Note that the image is not drawn to scale. S and S' consider the event located on the wave front on the x-axis (x'-axis). See picture Fig. t > 0, t' > 0.

Do we agree on this?

Here is a revised version of my hand drawn picture of SC3. It is intended to be exactly the same as my hand drawn one but colored, overlapping correctly and three dimensional.

This is again the situation at some time t>0 and t'>0 as seen from both S and S', using mainstream physics. Note that there are two frames of reference so two images of the same physical flash of light is drawn; there is no time component and we have not yet derived Lorentz transform. An observer in S or S' would measure speed of light = c in every direction and consider themselves in the center of an expanding sphere of light. Note again, the expanding spheres of light is the same physical event as seen from the two frames of reference.

Finally, to make sure I have drawn the right conclusions and discuss the correct situation, here is a reference. 

On 6/3/2019 at 6:23 PM, Jan Slowak said:

But I have not seen a figure describing SC3. Can anyone help me with such a figure?

SC3:
LEx': x' = Ax + Bt
LEt': t' = Cx + Dt
SC3: x' = ct', x = ct

In your referenced book* I find "3", this is the situation i have drawn and explained above.

Does this answer your question?

 

*) Modern Physics Randy Harris 2008 pages 14-15-v1.pdf

 

Edited June 5, 2019 by Ghideon
clarified one sentence

[Jan Slowak]

19 hours ago, Ghideon said:

I tried to do that and you respond with:

The above looks like a flash of light extending in a circular fashion in two dimensions, or a sphere in three dimensions.Can you clarify what number of dimensions you wish to include? One, two or three? Can you put labels on all axes you wish to include; y, y' and z, z' for instance. (Or remove them if they are irrelevant)

  

It depends. In which frame of reference does the circle belong?

 

We only talk about (x, t) and (x', t'). The image contains three axes for S and S' to see that it is a reference system i (x, y, z). The circle representing the wavefront is drawn in a plane but we only deal with the event, a point, where the circle crosses the x-axis.

Hope this helps understand the picture.

Then you should state the expression of the event's coordinates (x, t) for S and the event's coordinates (x', t') for S'.

 

14 minutes ago, Ghideon said:

I revisited this after my response above and have some additional details. First I have my version of your picture above. I post this as a reference and starting point for other pictures.

I use three dimensions as in your picture but I prefer x, y, z instead of x, S, unlabeled. This is at time t=0, t'=0 as in your image. The flash is just starting to propagate; for visibility reasons it is a small circle instead of a dot in point 0,0. Note: there is only one flash of light, visible to an observer in S and an observer in S'

S and S' are moving relative to each other with speed v.

 

Here is a revised version of my hand drawn picture of SC3. It is intended to be exactly the same as my hand drawn one but colored, overlapping correctly and three dimensional.

This is again the situation at some time t>0 and t'>0 as seen from both S and S', using mainstream physics. Note that there are two frames of reference so two images of the same physical flash of light is drawn; there is no time component and we have not yet derived Lorentz transform. An observer in S or S' would measure speed of light = c in every direction and consider themselves in the center of an expanding sphere of light. Note again, the expanding spheres of light is the same physical event as seen from the two frames of reference.

Finally, to make sure I have drawn the right conclusions and discuss the correct situation, here is a reference. 

In your referenced book* I find "3", this is the situation i have drawn and explained above.

Does this answer your question?

 

*) Modern Physics Randy Harris 2008 pages 14-15-v1.pdf

 

It would be good if you number pictures so that you can refer to them.

One thing is not right in your last picture:
There should be only the red circle, the wave front that occurs at the beginning of the thought experiment, then when the two reference systems coincide in the same point.

The wave front does not follow any reference system!

[swansont]

46 minutes ago, Jan Slowak said:

 The wave front does not follow any reference system!

The wave front follows all reference systems 

[Jan Slowak]

3 minutes ago, swansont said:

The wave front follows all reference systems 

Never! 

 

7 minutes ago, swansont said:

The wave front follows all reference systems 

Never!
The light speed and its direction are independent of the source and the observer's movements!

 

[swansont]

3 hours ago, Jan Slowak said:

Never! 

 

Never!
The light speed and its direction are independent of the source and the observer's movements!

 

Light speed, yes. Which is why it follows all reference frames. It’s an invariant.

[Jan Slowak]

15 minutes ago, swansont said:

Which is why it follows all reference frames. It’s an invariant.

I'll draw a picture.
A light signal, a pulse, occurs on the x-axis and moves to the right with speed c.
At the same time, two rockets start, one moves to the right with speed v and the other to the left with speed -v.

In what way does the light signal follow the two reference systems?

[Strange]

31 minutes ago, Jan Slowak said:

I'll draw a picture.
A light signal, a pulse, occurs on the x-axis and moves to the right with speed c.
At the same time, two rockets start, one moves to the right with speed v and the other to the left with speed -v.

Your diagram only shows one frame of reference.

Quote

In what way does the light signal follow the two reference systems?

In the way shown in Ghideon's diagram.

[swansont]

Strange beat me to it: you have only shown one frame. There are two rockets moving with respect to that frame.

But for each rocket, they would see a light pulse traveling at c. In any frame n, they will all agree that light has traveled a distance x_n = ct_n

That's what I mean by "it follows for all reference frame"

[Jan Slowak]

1 hour ago, swansont said:

Strange beat me to it: you have only shown one frame. There are two rockets moving with respect to that frame.

But for each rocket, they would see a light pulse traveling at c. In any frame n, they will all agree that light has traveled a distance x_n = ct_n

That's what I mean by "it follows for all reference frame"

I wrote: In what way does the light signal follow the two reference systems?
Two rockets = two reference systems.
Note the following:
- a light signal
- one or more reference systems
Each of these objects / phenomena is independent of each other.

The light speed and its direction are independent of the source and the observer's movements!

[Ghideon]

6 hours ago, Jan Slowak said:

One thing is not right in your last picture:
There should be only the red circle, the wave front that occurs at the beginning of the thought experiment, then when the two reference systems coincide in the same point.

Of course there are two wave fronts as several members have pointed out, since you asked for a picture with two frames of reference in relative motion. The key is in the specification of the case 3 you asked for; "Einstein's second postulate". 

 

28 minutes ago, Jan Slowak said:

The light speed and its direction are independent of the source and the observer's movements!

The observers in the frames S and S' will be in the centre of the expanding sphere of light in their own frame of reference. Another way to state the consequence shown in my picture: Both observers are in the centre of the light sphere in their frame of reference and both will claim that they see that the other observer is not in the centre of the light sphere.

If you wish to have a picture with one frame of reference that can be done. But it would not be the SC3 picture that you asked for.

 

[Strange]

1 hour ago, Jan Slowak said:

Two rockets = two reference systems.

But you have drawn a diagram from only one frame of reference (which is neither of the two rockets).

1 hour ago, Jan Slowak said:

The light speed and its direction are independent of the source and the observer's movements!

Yes. Which is why, when you draw multiple frames of reference, you get the diagram shown by Ghideon.

Here is an attempt to extend your diagram to add the frames of reference of each of the two rockets. It shows how far the rockets have gone in each frame of reference (upper arrows) and how far the light would have travelled in each frame of reference (lower arrows). Note that in each of the rockets' frames of reference, the distance between the rocket and the wavefront of the light is different.

 

Understanding the invariance of c is absolutely fundamental to understanding SR (and therefore the Lorentz transform).

40 minutes ago, Strange said:

Here is an attempt to extend your diagram to add the frames of reference of each of the two rockets. It shows how far the rockets have gone in each frame of reference (upper arrows) and how far the light would have travelled in each frame of reference (lower arrows). Note that in each of the rockets' frames of reference, the distance between the rocket and the wavefront of the light is different.

Here is another version that moves the origins of the frames of reference, so it is more like Ghideon's diagram.

 

Edited June 5, 2019 by Strange
Updated diagrams for clarity (hopefully)

[swansont]

1 hour ago, Jan Slowak said:

I wrote: In what way does the light signal follow the two reference systems?
Two rockets = two reference systems.
Note the following:
- a light signal
- one or more reference systems
Each of these objects / phenomena is independent of each other.

The light speed and its direction are independent of the source and the observer's movements!

Yes. c is invariant. It has the same value in every reference frame. In any reference frame, the light will have traveled a distance ct, where t is the time elapsed in that frame. It doesn’t matter what frame you are in. It’s true in all frames.

16 minutes ago, Strange said:

But you have drawn a diagram from only one frame of reference (which is neither of the two rockets).

Yes. Which is why, when you draw multiple frames of reference, you get the diagram shown by Ghideon.

Here is an attempt to extend your diagram to add the frames of reference of each of the two rockets. It shows how far the rockets have gone in each frame of reference (upper arrows) and how far the light would have travelled in each frame of reference (lower arrows). Note that in each of the rockets' frames of reference, the distance between the rocket and the wavefront of the light is different.

 

Not 2v, though, since velocities don’t add linearly.

[Strange]

10 hours ago, swansont said:

Not 2v, though, since velocities don’t add linearly.

True. But then again, the diagram doesn't represent length contraction or time dilation. 

I might see if I can label it better, though... (Done)

[Jan Slowak]

We start almost from the beginning.
Fig 5.1:
A reference system S is on the x-axis in the point O. Clock shows t = 0. The reference system S is at rest against the point O. We mark two points on the x-axis. Points V at distance v from O. Point C at distance c from O. Three distances are formed. OV, OC and VC. 
length (OV) = v, length (OC) = c, length (VC) = c - v.

Fig 5.2:
The reference system S is at rest against the point O. A light signal occurs at point O. This signal moves at the speed of light c. We only consider the light beam coinciding with the x-axis. When clock shows t = 1, wave-front reaches point C. The following still applies:
length (OV) = v, length (OC) = c, length (VC) = c – v.

Fig 5.3:
The reference system S moves from left to right with speed v < c. At the moment when S is in O a light signal occurs in O. We only consider the light beam coinciding with the x-axis. When clock shows t = 1, reference system S reaches point V and wave-front reaches point C. The following still applies:
length (OV) = v, length (OC) = c, length (VC) = c - v.

These three pictures show what happens in reality. They also show that the wave front movement is independent of any reference systems that would happen to be there or elsewhere.

Then I ask the question to all of you:
Is it true that the lengths of the three distances in Fig. 5.3 are the following:
length (OV) = v, length (OC) = c, length (VC) = c - v?

 

 

 

 

[Strange]

1 hour ago, Jan Slowak said:

We start almost from the beginning.

Sadly, that seems to be true. You really should have understood all this before launching an attack on SR. (Although if you understood all this, presumably you wouldn't need to attack SR.)

1 hour ago, Jan Slowak said:

The reference system S moves from left to right with speed v < c.

Here is a typical problem of the beginner: not being clear about frames of reference.

It might be better to refer to this new reference frame as S' to distinguish it from the stationary frame S. Because, for example, when you say that S' is moving with velocity v, you need to say what that velocity is relative to. I assume it is relative to the stationary frame S.

And the rest of your argument falls into the same trap: you are drawing and measuring everything as seen from S, not as seen in S'.

For example, in your third diagram, you show the distance ravelled by light as measured in the stationary frame S, not the frame S'. That is why the distance OC has not changed.

As measured in the moving frame S', the distance light travels in unit time will be a distance OC from S'; in other words, the light travels the same distance as measured in S' as it does in S.I have shown this in the diagram below. The distance light travels as seen in frame S' is O'C' (red).

[swansont]

Why are you plotting speeds on the x axis, and treating them like they are distances?

[Ghideon]

I’ll try to compose a picture “3” with the requirements stated by OP. It may not be useful as an explanation of SR in the general case but might help to clarify misconceptions in this case. Since there are several requirements stated by different sources here is a list; requirements from Jan Slowak: JS, Einstein postulates: EP, The referenced book: [7]. Some requirements may have been stated in more than one location.

JS: There must be one sphere of light
JS: The picture should show the situation at some time t>0 as in SC3
EP: Speed of light is invariant (stated in [7])
[7]: There are two frames of reference moving relative one another in situation 3
[7]: Clocks are synchronised
[7]: Observes are located in origo in each frame of reference

Other notes: We are attempting to derive Lorentz transform, picture may not rely on transformation between coordinate systems.
We still use a flash of light occuring at time t=0, t’=0 in origo when both frames S and S’ are at the same position.

The picture may look unintuitive so lets build it in steps and cover all requirements in the final step.

JS: There must be one sphere of light
JS: The picture should show the situation at some time t>0 as in SC3

Consequences: one circle radius r>0

 

EP: Speed of light is invariant (stated in [7])
[7]: There are two frames of reference moving relative one another in situation 3
[7]: Clocks are synchronised
[7]: Observes are located in origo in each frame of reference

Consequences: Two coordinate systems, both having origo in the center of the expanding sphere of light. Note: The Observers O and O’ in respective reference frame is not necessarily located at the same point in space, they just are drawn at the same place in the picture*.  

Final requirement:

JS: The picture should show the situation at some time t>0 as in SC3

Consequences: Since we have two different frames of reference in relative motion we have to draw each observers’ view. Wee need to show what does an observer see from their point of view, in their frame of reference and do that twice.
Observer O will claim that they see observer O’ located at some distance. This is drawn in reference frame S as a red italic O’ to the left in the picture. 
Observer O’ will claim they see observer O at the same distance. This is displayed using a green italic O in reference frame S’ to the right in the picture.

So The result is that by using one sphere of light and still under the requirement to show two frames of reference and stay in mainframe science and use no transforms we must draw each observer in more than one location.

Hopefully this helps clarify some misunderstandings.

The relative motion and consequences of Einsteins postulates shows that there are properties that differs for observers in different frames of reference. Which is why I started with my question "In which frame of reference does the circle belong?". The requirements talk about two frames of reference but OP seems to want to draw only one.

 

*) side note, maybe separate thread: does this kind of "overlapping coordinate systems & frames of reference" have a name? The relative movement is not shown even if time is elapsing. I know of comoving distances and conformal time,  is this somehow a related thing?

[Jan Slowak]

Answer to Strange, swansont, Ghideon:

I asked you to confirm that in Fig 5.1, 5.2 and 5.3 the lengths of the three distances are as follows:
length (OV) = v, length (OC) = c, length (VC) = c - v

Nothing else! You should not draw any other pictures. We talk about pictures I have drawn. You should not deviate from the subject. You should not ask any further questions.
If you are a physicist or mathematician with college education then you just need to answer my question. The answer should be Yes or No. If you answer No, you can argue briefly. Thanks!

[Ghideon]

8 minutes ago, Jan Slowak said:

Answer to Strange, swansont, Ghideon:

I asked you to confirm that in Fig 5.1, 5.2 and 5.3 the lengths of the three distances are as follows:
length (OV) = v, length (OC) = c, length (VC) = c - v

Nothing else! You should not draw any other pictures. We talk about pictures I have drawn. You should not deviate from the subject. You should not ask any further questions.
If you are a physicist or mathematician with college education then you just need to answer my question. The answer should be Yes or No. If you answer No, you can argue briefly. Thanks!

No