How to plot correctly spacetime diagrams

[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