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On Mar 4, 8:35 pm, "Dirk Van de moortel" dirkvandemoor...@ThankS-NO-
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On Mar 4, 7:21 pm, "Dirk Van de moortel" dirkvandemoor...@ThankS-NO-
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On Mar 4, 10:52 am, "Dirk Van de moortel" dirkvandemoor...@ThankS-NO-
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On Mar 4, 6:08 am, "
wrote:
I'm not sure I know exactly how to use the Minkowski diagram. For
example, let's say you have a stationary observer and a moving
observer, and an event somewhere, for which the coordinates are x and
t in the stationary frame and x' and t' in the moving frame. How do
you get t', for example, from the Minkowski diagram?

Ram.

No, you can't.

Of course you can.

OK, you can if you are VERY careful. For example, note the effect of
the a boost, which is called a "rotation" in Minkowski space. However,
this is not your garden-variety rotation, in that both axes get
rotated, say, clockwise, to do the transformation. In fact, the x-axis
rotates one way and the t-axis rotates the other way.

All sorts of mapping issues result from this shift from Euclidean to
hyperbolic geometry. For example, trig relations get replaced by
hyperbolic trig relations (sinh, cosh, tanh, rather than sin, cos,
tan). All of these are trackable if you are careful and understand
what is fundamentally different about the Minkowski diagram from the
usual pair of axes on a flat piece of paper.

Since in Euclidean terms we don't allow imaginary angles, we
shouldn't really call this a "real" rotation ("real" like in The Real
Numbers).
I made this thing some time ago. It just took some straightforward
standard high school level analytic geometry with lines, slopes,
and intersections, and as you can see, even if you're not careful,
once you have that scale factor, one can easilily "read" the
transformed coordinates from the Minkovski diagram :-)

Dirk Vdm

Dirk -- You are a godsend. That Java thing rocks. I tried some values,
and I checked it and it gives the same result as the Lorentz
Transformation. However, I tried to get x' and t' analytically from
the graph, but it didn't come out like the x' and t' that the app
said. Do I have a computation mistake, or did I not understand how x'
and t' are retirieved? I'll tell you how I retrieved them: For x', for
example I took the point x' on the graph and calculated its distance
from O. Is that what I'm supposed to do?

See my last message where I calculated the scale factor
for X' when X and T are known.
I'll leave the calculation of T' as an exercise. Let me know
if you get stuck :-)

Dirk Vdm

Ahoi sailor, it worked!
So my conclusion is, use Minkowski diagram but multiply it by the
magic factor sqrt(1-v^2)/sqrt(1+v^2). So that means that Minkowski
diagrams are only good for telling you which events happened on the
same time, but it's pretty hard to read from them what exactly that
time was.

I don't agree. It is pretty easy to read. It is just a scaling factor.
You can put the x'=1 and t'=1 marks and then estimate by merely
looking.


Does this mean that this page:
http://www.physics.usyd.edu.au/super...i_Diagrams.pdf
Is rubbish? Because it doesn't mention the magic factor.

That's what I meant.
You will find that the factor (0.686 for v=0.6) is present in the
points (x',t') = (1,0) and (0,1).
Not rubbish at all - just not explicitly mentioned.

Is that
factor mentioned in books about SR?

No idea, I have never seen it in books.
I calculated it a while ago when I was playing with Geometer's
Skechpad.
Here's another screenshot I made a few days earlier:
http://users.telenet.be/vdmoortel/di...sformation.png
To bad that saving as html/java is very limited. Compare the
quality difference between my previous Lorentz.htm and
Lorentz.png :-)

Cheers,
Dirk Vdm