How to plot correctly spacetime diagrams

[RiccardoVen]

Similarly, I ask you, would it make sense to use the yellow markings along the t1 axis to measure the x1 distance between the bottom two events? Again, they are the same physical distance on the drawing.

English is not my preferred main language, even I use it every day for my job.
So I cannot understand the "mood" of your sentence just above. It seems a bit
you are doing some irony, but I'm not sure ( notice there's not any polemic in my words, I'm just trying
to figure out if you are confirming what I stated or you are trying to disprove it, somehow ).
I guess you are suggesting to use t1 marking to measure x'1 just because the x' inclined axis
has not any mark on it. But, looking better to your last plot, you can count the grid lines parallel
to t' and you get the red wordline intersecting the x' axis pretty after 5 of them. So you can lead
to 5 as it was in your orthogonal version. So, adding some marks on x' axis as well, would fix this ambiguity, I think.

So the answer is: yes. That logic can be applied both for times and for space axis, of course, since the symmetry of Lorentz transformations are actually "stretching" the unit of measure in the same way( this mainly because the Lorentz transformation IS linear ). Drawing the grids really helps a lot in explaining better what I meant.
And if Minkowski has seen it as well, this is a good point for sure :-).

I was just trying to explain this is not adding any new physics to it. It's straightforward, when you notice a distance in inclined x't' frame is 6, and this is also 6 when the x't' is orthogonal at rest, you are stating nothing really new. You are just noticing a distance is the same when measured in the same frame ( primed in this case ). Of course, if you use euclidean metric to measure that segment when the x' is inclined, you'd get a longer segment. But this, of course, doesn't mean that distance is measured as stretched from xt frame. that distance makes sense only at the same t' time, and cannot be interpreted as a distance in xt, since that would be no more simultaneous in xt frame.
So that's why I'm stating the unit of measure is stretched when we transform it form orthogonal to inclined.

If you look to this article on Loedel diagrams, for instance:

http://www.einsteins-theory-of-relativity-4engineers.com/loedel-diagrams.html

you can read:

One of the benefits of the diagram is that unlike in Minkowski diagrams, the scales of both axes of both frames are identical. In effect, the worldlines of both observers progress along their respective time axes at the same rate. Hence, the Loedel diagram does not give apparent preference to one of the inertial frames

so "different scaling" is also another way to me to tell "units are scaled ( so contracted or stretched, according to one you are referring to ).


Anyhow, it seems ( I hope ) you got my point, so yes using grid my point ismore noticable. So, coming back to my latest plot, I was just saying the DC segment is actually the klingon's ship length since it lasts from the extremes on klingon's wordlines.
So, if you had an hypothetical ruler owning the stretched units of primed inclined frame, you would be able to measure ( on the plot I mean ) the DC length and you got the length expressed as that distance was still in the dual orthogonal x't' frame.
If you don't have that ruler ( of course is difficult to own one of that kind :-) ), you could store how the unit segment in orthogonal x't' primed is stretched inclining it in xt frame. Let's say you are jumping from 1 to 1.45.
So, for example, if you measure by a ruler the DC segment would be 14.5, you could state for sure the klingon's proper length is 10. So you could use that proper length, and use it for choosing the A point: since the Enterprise and the Klingon's ship share the same length when both at rest, you could use that to choose an A point which x coordinate is 10 units far from C.x ( this means of course forcing the AC segment to be 10 units in length ).

So, theoretically, you would be able to sketch and gather the Enterprise length's without the need to transform and jump between 2 frames. This was an explanation on what I meant as "more information".

Of course this is not a trick can be used "really" to sketch wordlines, this is because is really to measure distances by a ruler and use that reading to add new segments based on that measurement, without loosing in precision. Your way is of course by far more reliable and precise, but for a "rough" analysis would be enough, I think.

Ricky

 

[ghwellsjr]

English is not my preferred main language, even I use it every day for my job.
So I cannot understand the "mood" of your sentence just above. It seems a bit
you are doing some irony, but I'm not sure ( notice there's not any polemic in my words, I'm just trying
to figure out if you are confirming what I stated or you are trying to disprove it, somehow ).
I guess you are suggesting to use t1 marking to measure x'1 just because the x' inclined axis
has not any mark on it. But, looking better to your last plot, you can count the grid lines parallel
to t' and you get the red wordline intersecting the x' axis pretty after 5 of them. So you can lead
to 5 as it was in your orthogonal version. So, adding some marks on x' axis as well, would fix this ambiguity, I think.

I was trying to get you to agree that if you would not use the t spacings to measure x distances in the normal frame, then you shouldn't use the t' spacings to measure the x' distances in the slanted frame, even if it works. It makes more sense to put in the grid lines and markings in both frames and use the t or t' markings to measure time intervals and the x or x' markings to measure the distance spacings. One applies exclusively to time and the other applies exclusively to distances. I'm not denying that your trick works, just that it's not a good idea to use it.

So the answer is: yes. That logic can be applied both for times and for space axis, of course, since the symmetry of Lorentz transformations are actually "stretching" the unit of measure in the same way( this mainly because the Lorentz transformation IS linear ). Drawing the grids really helps a lot in explaining better what I meant.
And if Minkowski has seen it as well, this is a good point for sure :-).

I was just trying to explain this is not adding any new physics to it. It's straightforward, when you notice a distance in inclined x't' frame is 6, and this is also 6 when the x't' is orthogonal at rest, you are stating nothing really new. You are just noticing a distance is the same when measured in the same frame ( primed in this case ). Of course, if you use euclidean metric to measure that segment when the x' is inclined, you'd get a longer segment. But this, of course, doesn't mean that distance is measured as stretched from xt frame. that distance makes sense only at the same t' time, and cannot be interpreted as a distance in xt, since that would be no more simultaneous in xt frame.
So that's why I'm stating the unit of measure is stretched when we transform it form orthogonal to inclined.

There's no stretching going on. Let me try another argument. Here is a diagram from my previous post:

Now here's another diagram that depicts exactly the same scenario:

Now would you say this second diagram depicts a stretched scenario compared to the first one? I hope not and I hope you understand that it is the grid lines and the markings that establish that the two diagrams are exactly the same. This is why the slanted frame is not stretched from the normal frame.

If you look to this article on Loedel diagrams, for instance:

http://www.einsteins-theory-of-relat...-diagrams.html

you can read:

One of the benefits of the diagram is that unlike in Minkowski diagrams, the scales of both axes of both frames are identical. In effect, the worldlines of both observers progress along their respective time axes at the same rate. Hence, the Loedel diagram does not give apparent preference to one of the inertial frames

so "different scaling" is also another way to me to tell "units are scaled ( so contracted or stretched, according to one you are referring to ).

I looked at the article and there's no mention of any stretching (or contracting) going on.

If you read carefully, they are saying that the Loedel diagram has three frames, the two that are drawn and the reference frame which is "imaginary", meaning it is not drawn. But this is the same thing that you would get if you had two of the Minkowski diagrams with one skewed one way and the other skewed the other way and then you eliminated the axes, grid lines and markings for the normal frame. There's nothing special about the Loedel diagram that other diagrams can't also show.

By the way, do you understand Fig. 2?

Anyhow, it seems ( I hope ) you got my point, so yes using grid my point ismore noticable. So, coming back to my latest plot, I was just saying the DC segment is actually the klingon's ship length since it lasts from the extremes on klingon's wordlines.
So, if you had an hypothetical ruler owning the stretched units of primed inclined frame, you would be able to measure ( on the plot I mean ) the DC length and you got the length expressed as that distance was still in the dual orthogonal x't' frame.
If you don't have that ruler ( of course is difficult to own one of that kind :-) ), you could store how the unit segment in orthogonal x't' primed is stretched inclining it in xt frame. Let's say you are jumping from 1 to 1.45.
So, for example, if you measure by a ruler the DC segment would be 14.5, you could state for sure the klingon's proper length is 10. So you could use that proper length, and use it for choosing the A point: since the Enterprise and the Klingon's ship share the same length when both at rest, you could use that to choose an A point which x coordinate is 10 units far from C.x ( this means of course forcing the AC segment to be 10 units in length ).

I got your point, I just don't think it's a good one. And I don't understand this paragraph. I don't know why you can't just follow the normal processes using the Lorentz Transformation and draw your diagram based on the results you get from it.

So, theoretically, you would be able to sketch and gather the Enterprise length's without the need to transform and jump between 2 frames. This was an explanation on what I meant as "more information".

But you are still jumping between 2 frames if you are using a diagram like the one in your link:

Do you see how they specify two frames in the drawing? They call the normal one the Enterprise Frame and they call the slanted one the Klingon Frame. They actually show the t and x axes for both frames. Unfortunately, they don't show any grid lines or markings for either frame so it isn't quite a obvious that there are two frames and two sets of implied coordinates for each event.

Furthermore, you should not think of the Enterprise Frame as containing only the Enterprise spaceship. It also contains the Klingon spaceship. Same with the Klingon frame--it also contains the Enterprise space ship. All frames contain all objects.

I have redrawn the Minkowski diagram to move the markings for the normal frame to the left and bottom edge and I have added markings for the slanted frame at the top and right:

Note that I have put in three events.

The coordinates for these three events in the normal frame and the primed frame are:

Blue   t=-3, x=-4   t'=-1, x'=-3
Red    t=4,  x=1    t'=4,  x'=-1
Green  t=-3, x=3    t'=-5, x'=5

Note that I can read these coordinates right off the diagram without having to do any Lorentz Transformation calculations. If you look up Minkowski diagram in wikipedia, you'll see that that is the point of a Minkowski diagram--you don't have to do any calculation to perform the Lorentz Transformation between two frames.

But now that we have computers that can draw the transformed diagrams for us, I don't know why we want to continue using the graphical techniques, especially sense they tend to be rather confusing to people who don't realize what they are actually depicting.

So here is a diagram for the three events in the normal frame:

And here is a diagram for the three events in the slanted (primed) frame:

Do you understand that just because I drew two separated diagrams for these two frames, the one Minkowski diagram also has two frames on the one diagram?

 

[RiccardoVen]

I'm not denying that your trick works, just that it's not a good idea to use it.

This is not a good point to me. you are expressing a personal opinion, which is
not actually really meaningful to me. If my idea is working ( and I can assure it's working )
that's all no opinion is needed.

There's no stretching going on.

Now would you say this second diagram depicts a stretched scenario compared to the first one?

Of course not. you are comparing two diagrams related to the same rest frame. I'm comparing
the frame at rest with the SAME frame slanted. Here there's a changing in scale for sure.

I looked at the article and there's no mention of any stretching (or contracting) going on.

This is not true. the article tells there's a different scaling between rest frame and slanted ones
on "normal" overlapped Minkowsky diagram. Probably you lost that part, and you forgot to quote it as well, since I posted it in my above post, i.e.:

One of the benefits of the diagram is that unlike in Minkowski diagrams, the scales of both axes of both frames are identical. In effect, the worldlines of both observers progress along their respective time axes at the same rate. Hence, the Loedel diagram does not give apparent preference to one of the inertial frames

This is just taken from that article. I've fully understood Loedel diagrams and they are using an intermediate "fake" inertial frame in order to make the 2 in analysis to be symmetric. And you can see from the above sentence that this remove the different scaling between rest and moving frame. I know my English is poor, but different means to me not equal, which again leads to dfferent units.
this is written in the quoted part above.

I got your point, I just don't think it's a good one. And I don't understand this paragraph.

I think this is the real problem. Since you don't understand it, you don't find a good one
( I was wondering how you can state to having got my point but at the same time stating
you didn't understand it ).
I cannot find any other words to explain you better. Sorry.

I don't know why you can't just follow the normal processes using the Lorentz Transformation and draw your diagram based on the results you get from it.

this is the other problem. It seems you are traying to tell me not to use my method, almost just
because you didn't understood it. And this is clear since you are telling: "using Lorentz transformation". I'm actually using them. Your way is not just the unique way to use.
You started to express personal opinion. I've used your method and I will use it for sure.
So, why are you trying to convince myself mine is wrong?
I think trying to disprove it, as you did it from the very beginning, it's a not a good
way to help understand it you better.

Do you see how they specify two frames in the drawing? They call the normal one the Enterprise Frame and they call the slanted one the Klingon Frame. They actually show the t and x axes for both frames. Unfortunately, they don't show any grid lines or markings for either frame so it isn't quite a obvious that there are two frames and two sets of implied coordinates for each event.

Furthermore, you should not think of the Enterprise Frame as containing only the Enterprise spaceship. It also contains the Klingon spaceship. Same with the Klingon frame--it also contains the Enterprise space ship. All frames contain all objects.

All these things are straightforward to me. When I say "jumping between 2 frames" I really meant "your" method
about building the wordlines in primed rest frame and then "press" your app button to get them in slanted version.
So "jumping" meant this to me.
So, fir the last time I hope, I was just stating you can use the info the slanted klingon's ship width in Enterprise rest frame ( let's say 6 ) could be used to deduce the Enterprise width, as explained in the paragraph it seems you don't understand.
We are using different approaches: you started plotting the 4 wordlines, while as explained as above, I started putting the C event point and to choose the rear klingon's wl to be coincident with the slanted t' ( which is
the choice used by the author as well ).
Doing that, you got directly the contracted klingon's width, so I was looking for a method to deduce the Enterprise's width WITHOUT jumping as you did. This led to my "not good" approach ( at least to you, of course ).

That said, just to let egos in peace, I will use your method for sure ( without any computer, of course). Mine was just a trick to understand better diagrams, but, AS STATED, it cannot be used regularly.

But now that we have computers that can draw the transformed diagrams for us, I don't know why we want to continue using the graphical techniques, especially sense they tend to be rather confusing to people who don't realize what they are actually depicting.

I totally disagree with you. Using computers for doing for us things it's like to use calculators for doing divisions. This is actually making minds sleeping.
I do them manually and I will do my plots as well for sure. So agan, since you are expressing personal
opinions which actually don't concern at all with physics, let's try to avoid them/

that said, sorry, but I have not too much time to convince you I'm correct and to explain you better
my point, and I'm not interested in convince you neither, indeed.

Thanks for your support, Regards

Ricky

 

[ghwellsjr]

I understand your trick. Let me express it. You want to use the distance of the Proper Time ticks of a moving object on a diagram to measure the distance of the Proper Length for the moving object on the same diagram.

But that trick only works because we usually draw our spacetime diagrams with the spacing of the horizontal and vertical coordinates so that it will work. I had to do a lot of work on my computer program to make that happen. It's not trivial. You should try to write you own program to make it work and you'll see what I mean. But there is nothing wrong with having the horizontal and vertical scaling be arbitrary, as a computer program will normally do without extra work. For example, here's a repeat of one of the diagrams from my first post:

Now here is another version of the same diagram but with the horizontal and vertical scaling changed:

This is a perfectly legitimate diagram conveying exactly the same information as the previous one. But your trick won't work, will it?

If you always use the time coordinates to establish time information and you use the spatial coordinates to establish length information, and you use the Lorentz Transformation equations to draw a new diagram with any coordinates you choose, then you'll never get tricked. This always works, even in situations where you trick would also work.

 

[RiccardoVen]

I understand your trick. Let me express it. You want to use the distance of the Proper Time ticks of a moving object on a diagram to measure the distance of the Proper Length for the moving object on the same diagram.

Not true. I'm using proper distance to measure the proper length. time is not involved at all.
Try to tell it better: look to my diagram I've already posted:

I'm talking about measuring the DC segment and using it to apply it to the AC one.
The DC segment is actually the proper distance of the klingon's ship evaluated at the
same t' in klingon' slanted frame. We can say it's a proper distance since it's measured
at the same t' and it's the segment between the klingon's front and end wls. So it's
actually the klingon's proper length.
Now, we can try to count the streched dots pierced by this DC distance ( which represent to me
the stretched unit in primed slanted system ). Let's say we count them to be 8 ( just for sake of simplicity ).
Then, since this is the proper distance and it's the same of the Enterprise's proper distance,
we can use it to fix the A point: this will be 8 units horizontally from the left of C, but evaluated using the x unit of measure belonging to the Enterprise's frame ( i.e. the distance between 2 dots
on X axis, which are unstretched ).
And all of this can be done on the same plot, without the need to put temporarily the x't' frame
in orthogonal view ( i.e. putting x't' at rest ).

So, actually the time is involved at all and there's not ambiguity with any different scaling
between t' and x' axes.

DC is actually representing a proper length, and not a proper time at all.

I can see this approach is unconventional, but it works.

 

[ghwellsjr]

Not true. I'm using proper distance to measure the proper length. time is not involved at all.
Try to tell it better: look to my diagram I've already posted:

View attachment 61899

I'm talking abot measuring the DC segment and using it to apply it to the AC one.
The DC segment is actually the proper distance of the klingon's ship evaluated at the
same t' in klingon' slanted frame. We can say it's a proper distance since it's measured
at the same t' and it's the segment between the klingon's front and end wls. So it's
actually the klingon's proper length.

If you had put scaling marks along the x' axis then I would agree but without those scaling marks or something equivalent like one inch represents 100 feet, then I don't know what you mean by "measuring the DC segment".

Now, we can try to count the streched dots pierced by this DC distance ( which represent to me
the stretched unit in primed slanted system ). Let's say we count them to be 8 ( just for sake of simplicity ).

What dots? I don't see any dots. Are you saying that you will provide the scaling if you had drawn in the dots correctly?

Then, since this is the proper distance and it's the same of the Enterprise's proper distance,
we can use it to fix the A point: this will be 8 units horizontally from the left of C, but evaluated using the x unit of measure belonging to the Enterprise's frame ( i.e. the distance between 2 dots
on X axis, which are unstretched ).

Then why didn't you just use the x unit of measure belonging to the Enterprise's frame (if there were one drawn it)?

So, actually the time is involved at all and there's not ambiguity with any different scaling
between t' and x' axes.

DC is actually representing a proper length, and not a proper time at all.

I can see this approach is unconventional, but it works.

I don't see how it works in this explanation. I thought I understood it when you used one of my drawings that had dots in it, but I'm totally lost now.

 

[RiccardoVen]

Try to draw with your app something owning a regular spacing of 1 on xt and then to add it those trasformed dots
on slanted x't'. So, yes this means put scaling on X. I did not add to my figure, but it was straightforward, since
any equal subdivision on X axis is actually transformed equally subdivided, but stretched, on X' axis.

Are you saying that you will provide the scaling if you had drawn in the dots correctly

Yes.

I will eventually try to redo a plot correctly spaced with dots added ( but I'm sure you would be faster
than me with your app ). This is basically as we'd add a slanted grid as well, of course.

Anyhow, yes, any explanation I did using your dotted plots, still rules. I'm just using dots to represent
the units on both frames. So, after you have correctly got the dots on the inclined x' axis, you could
use the same spacing on DC segment ( which means to me: use stretched units on x' to "measure" DC
segment on the plot ).

the resulting point is the one you already noticed: "if you read 8 "stretched" dots for DC, you can use that
value for the Enterprise as well, using 8 units of xt frame.

Which is the same as "proper distances of the 2 ships are the same for both frames"

 

[RiccardoVen]

OK,
let refer to this extended version on my handmade plot, in which I've added some suitable dots:

Let's remember I've done this handmade plot using your trick of drawing klingon's wl in x't' orthogonal version and then to slant it back into xt at rest. So wls are actually coherent with
beta = v/c = 0.6 as we have already stated in out previous posts.

That said, let's assume the x unit of measure on x-axis is actually 1 square ( which is actually
5mm in length, but we can keep "A Square" ( cit. from Flatland ) as the x unit of measure.
You can see I've plot 2 dots on the x axis. One of them is actually delimiting the OZ segment, which represents the segmen x unit of measure.

Using Lorentz transformations, we are able to compute OZ' length, i.e. about 1.42 square. You can get it easily using Lorentz transformations, i.e.:

x = (x' + beta*t')*gamma ( where gamma = 1.25, starting from 0.6 )
t = (t' + beta*x')*gamma

starting from x'=1, t'=0 you get:

x = 1.25
t = 0.75

So, you can get the OZ' length as sqrt(x^2+t^2) = 1.42...

That said, I'm pointing my attention to the PQ segment. This is actually the Klingon's
proper length evaluated at t'=0. So, just for sake of clearity, I've started in plotting the
dots on x' from the P points. You can also use a regular spacing on x' from O, leading
to some points not exactly piercing PQ. But I'm sure you got the point.
That said, you can see the PQ segment is actually piercing almost exactly 7 dots, i.e. 6
units segment between those 7 "stretched" dots.
So, JUST looking at this PQ segment, we could state the klingon's ship at rest is 6 unit,
where we are assuming unit are referred to x' ( i.e. 6 of OZ' segments ).

As you can see, this is EXACTLY the width of the AC segment, i.e. 6 units, where now
we are assuming the x "square" unit as reference.

This is my point. As you can see, I've used proper distances and not time at all, so there's
not any mixing in proper times with proper distances.

I really see this is exactly as overimposing two grids, one slanted and one "orthogonal".
Again this doesn't add any physics to it, it's just a math trick.
I hope you got the point this time and to be a bit more clear the before.

 

[ghwellsjr]

I really see this is exactly as overimposing two grids, one slanted and one "orthogonal".
Again this doesn't add any physics to it, it's just a math trick.
I hope you got the point this time and to be a bit more clear the before.

This I totally agree with, thanks for clarifying.

I just wouldn't characterize the dots as being "stretched", they are merely marking some of the coordinates for x'. If you had drawn them all in, you would see that they correspond exactly with the correct markings along the x' axis. But even if they hadn't lined up exactly, what you are doing is providing a distance measure along the x' axis. You are doing the correct math to achieve this. As such, I don't see why you would call this a math trick. It's the normal math you have to do to achieve the correct distance along the slanted axis. I also don't see why you call what I did a trick, it's also just the usual way to apply the Lorentz Transformations.

 

[RiccardoVen]

This sounds good to me. Probably I had to plot directly what I really meant the very first time
I talked about it. I know you don't agree completely to my description about the unit stretching
and I see your point. that's why I was calling a math trick. Of course, the distances are not really
stretched, I mean there isn't any physics in it. It's just a consequence of the transformation.
Of course, the only physical event we can measure about distances is length contraction, but not
any space dilation indeed.
I'm a bit bipolar, so sometimes I'm more addict to maths, some other to physics. In this situation, when I saw your dots the first time, I realized they could be interpreted as unit of length measurement on all the 2 axes. I see this is a bit misleading, I mean this could lead noob people ( at least more noob than me ) in identifying them as "real" stretched rods, or so. This was not my intention and, of course, the solution is to keep them as they are, i.e. merely space subdivision ( all in all, when we use meters or feet we are introducing a "fake" space subdivision for our purposes, exactly as we introduce suitable frames just to describe easier the physics in them ). I see here and there someone else which is interpreting them as units, so I really think it's a matter of talking the same language.
Nevertheless, I'm glad we finally land to the same page.
That said, as already told you, my core business was to be able to draw from scratch space-time diagrams, and you fed me with a real valid and robust method, and I thank you for this. Incidentally, working on it, I found to use your dots to apply directly measures on the plot. But of course the information content is the same in both cases.

Thanks, regards

Ricky