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Golden Boar
science forum Guru
Joined: 17 May 2005
Posts: 651
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 Posted: Wed May 31, 2006 7:30 pm Post subject:
Re: Another look at the Lorentz factor.
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PD wrote:
| Quote: | Golden Boar wrote:
The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
I think it would benefit you to see the form of the Lorentz
transformations in terms of hyperbolic transcendental functions, where
gamma = cosh(rapidity),
about as simple as you can get.
Moreover, the rapidity variable is exceeding useful as well as being
more physically fundamental.
See, for example, http://www.everything2.com/index.pl?node_id=312518
The important thing to note is that, rather than reading up on things
that have already been done a long long time ago, you are spending your
time futzing with equations to stumble on the work yourself. This in
itself is not a bad thing. Thinking that you've done something original
or of interest to others --- that you'll find people have a lesser
opinion of.
PD
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Hey, I didn't say it was original, and as for it not being of interest
to others, I can say without a doubt in my mind, that somebody
somewhere will find it interesteing. |
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Golden Boar
science forum Guru
Joined: 17 May 2005
Posts: 651
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| Posted: Thu Jun 01, 2006 9:17 am Post subject:
Re: Another look at the Lorentz factor.
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PD wrote:
| Quote: | Golden Boar wrote:
The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
I think it would benefit you to see the form of the Lorentz
transformations in terms of hyperbolic transcendental functions, where
gamma = cosh(rapidity),
about as simple as you can get.
|
I don't know.
The rapidity is given by,
(c + v) / sqrt(c^2 - v^2)
So you are implying that,
cosh((c + v) / sqrt(c^2 - v^2))
is simpler than,
c / (sqrt(c^2 - v^2)
| Quote: | Moreover, the rapidity variable is exceeding useful as well as being
more physically fundamental.
See, for example, http://www.everything2.com/index.pl?node_id=312518
The important thing to note is that, rather than reading up on things
that have already been done a long long time ago, you are spending your
time futzing with equations to stumble on the work yourself. This in
itself is not a bad thing. Thinking that you've done something original
or of interest to others --- that you'll find people have a lesser
opinion of.
PD |
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PD
science forum Guru
Joined: 03 May 2005
Posts: 4363
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| Posted: Thu Jun 01, 2006 12:45 pm Post subject:
Re: Another look at the Lorentz factor.
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Golden Boar wrote:
| Quote: | PD wrote:
Golden Boar wrote:
The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
I think it would benefit you to see the form of the Lorentz
transformations in terms of hyperbolic transcendental functions, where
gamma = cosh(rapidity),
about as simple as you can get.
I don't know.
The rapidity is given by,
(c + v) / sqrt(c^2 - v^2)
So you are implying that,
cosh((c + v) / sqrt(c^2 - v^2))
is simpler than,
c / (sqrt(c^2 - v^2)
|
No, I'm implying that using rapidity is simpler than using c and v. Why
would you use two parameters for speed when rescaling will give you
one?
In a sensible system of units, you can either choose the maximal speed
to be 1 or infinity (rapidity).
PD
| Quote: |
Moreover, the rapidity variable is exceeding useful as well as being
more physically fundamental.
See, for example, http://www.everything2.com/index.pl?node_id=312518
The important thing to note is that, rather than reading up on things
that have already been done a long long time ago, you are spending your
time futzing with equations to stumble on the work yourself. This in
itself is not a bad thing. Thinking that you've done something original
or of interest to others --- that you'll find people have a lesser
opinion of.
PD |
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avergon@verizon.net
science forum Guru Wannabe
Joined: 30 Jan 2006
Posts: 282
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| Posted: Thu Jun 01, 2006 11:16 pm Post subject:
Re: Another look at the Lorentz factor.
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Golden Boar wrote:
| Quote: | Eric Gisse wrote:
Golden Boar wrote:
Eric Gisse wrote:
Golden Boar wrote:
The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
Excellent - you can do algebra. More than can be said for a fair bit of
folks here, though.
Now why are you doing this?
No reason really, it's just a neater more intuitive equation, which I
came up with, and you never know it might help someone.
...and what makes you think it is more intuitive? The form of the
equation as usually presented is the result of the derivation. Adding
another step of algebra doesn't add any insight.
Have a look at this triangle
http://en.wikipedia.org/wiki/Image:Trigonometry_triangle.svg
The speed of light is side h.
The velocity is side a.
So side b = sqrt(c^2 - a^2)
The Lorentz factor is then c / b = sec(A) = 1/cos(A)
Uhhh...no.
All you did, whether you realise it or not, is multiply it by a form of
1 while bringing the extra c term inside the square root.
That c was meant to be h.
I made a few mistakes in the above posts becuase at first I was using a
triangle with sides a,b and c. So I will satrt again.
Have a look at this triangle,
http://en.wikipedia.org/wiki/Image:Trigonometry_triangle.svg
The speed of light is side h.
The velocity is side a.
So side b = sqrt(h^2 - a^2)
The Lorentz factor is then h / b = sec(A) = 1/cos(A).
Also, gamma * beta is then a / b = tan(A) = sin(A)/cos(A).
|
VERGON
So what?
If you realize that the parameters of SR are trigonometric in character
(as is the MM experiment) then you should be able to figure out the
kinetic energy from the triangle abc.
Where velocity a is given where c =1
And b is sqrt(1- a^2) [[It is also the Lorentz transform]]
Ek = ( sqrt[ 1 + a^2/ {1-a^2} ] -c)mc^2
a^2
Ek = (sqrt 1 + ---------- - c ) mc^2
1-a^2 |
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avergon@verizon.net
science forum Guru Wannabe
Joined: 30 Jan 2006
Posts: 282
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| Posted: Fri Jun 02, 2006 12:18 am Post subject:
Re: Another look at the Lorentz factor.
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Vert wrote:
| Quote: | Golden Boar wrote:
Eric Gisse wrote:
Golden Boar wrote:
Eric Gisse wrote:
Golden Boar wrote:
The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
Excellent - you can do algebra. More than can be said for a fair bit of
folks here, though.
Now why are you doing this?
No reason really, it's just a neater more intuitive equation, which I
came up with, and you never know it might help someone.
...and what makes you think it is more intuitive? The form of the
equation as usually presented is the result of the derivation. Adding
another step of algebra doesn't add any insight.
Have a look at this triangle
http://en.wikipedia.org/wiki/Image:Trigonometry_triangle.svg
The speed of light is side h.
The velocity is side a.
So side b = sqrt(c^2 - a^2)
The Lorentz factor is then c / b = sec(A) = 1/cos(A)
Uhhh...no.
All you did, whether you realise it or not, is multiply it by a form of
1 while bringing the extra c term inside the square root.
That c was meant to be h.
I made a few mistakes in the above posts becuase at first I was using a
triangle with sides a,b and c. So I will satrt again.
Have a look at this triangle,
http://en.wikipedia.org/wiki/Image:Trigonometry_triangle.svg
The speed of light is side h.
The velocity is side a.
So side b = sqrt(h^2 - a^2)
The Lorentz factor is then h / b = sec(A) = 1/cos(A).
Also, gamma * beta is then a / b = tan(A) = sin(A)/cos(A).
VERGON
So what?
If you realize that the parameters of SR are trigonometric in character
(as is the MM experiment) then you should be able to figure out the
kinetic energy from the triangle abc.
Where velocity a is given where c =1
And b is sqrt(1- a^2) [[It is also the Lorentz transform]]
Ek = ( sqrt[ 1 + a^2/ {1-a^2} ] -c)mc^2
a^2
Ek = (sqrt 1 + ---------- - c ) mc^2
1-a^2
|
I MADE A TYPO
a^2
Ek =[ {sqrt (1 + --------------) } - c] mc^2
1-a^2 |
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xxein@bellsouth.net
science forum Guru Wannabe
Joined: 12 Sep 2005
Posts: 272
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| Posted: Fri Jun 02, 2006 1:48 am Post subject:
Re: Another look at the Lorentz factor.
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Golden Boar wrote:
| Quote: | The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2) |
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The Ghost In The Machine
science forum Guru
Joined: 25 Mar 2005
Posts: 1551
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| Posted: Sat Jun 03, 2006 5:00 am Post subject:
Re: Another look at the Lorentz factor.
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On Tue, 30 May 2006 20:38:08 -0700, Golden Boar wrote:
| Quote: | The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
|
An interesting if incomplete thought. One can also, of course, write it
as
gamma = c / (sqrt(c-v) * sqrt(c+v)) [*]
or
gamma = 1 / (sqrt(1-v/c) * sqrt(1+v/c) )
which can come in handy when deducing that the frequency ratio is
f/f0 = sqrt(c+v)/sqrt(c-v) = sqrt(1+v/c) / sqrt(1-v/c)
for an approaching light source (in one dimension, anyway; three
dimensions make things a little more complicated), and
f/f0 = sqrt(c-v)/sqrt(c+v) = sqrt(1-v/c) / sqrt(1+v/c)
for a receding one.
Note that in SR, the wavelength ratio for these are:
l/l0 = sqrt(c-v)/sqrt(c+v) = sqrt(1-v/c) / sqrt(1+v/c)
for an approaching light source, and
l/l0 = sqrt(c+v)/sqrt(c-v) = sqrt(1+v/c) / sqrt(1-v/c)
for a receding one.
By contrast, Newtonian theory computes l/l0 = 1, and
f/f0 = (c+v)/c for approaching, (c-v)/c for receding sources.
Of course s/s0 = (c+v)/c for approaching and s/s0 = (c-v)/c for receding
objects in Newtonian theory, where I've written s0=c.
This also means E/E0 = (c+v)^2/c^2 = (f/f0)^2 in Newtonian theory for
approaching light sources, but E/E0 = f/f0 in Einstein's theory;
presumably one can test this using Compton scattering from semiconductors.
(It's worth noting that the error between Newtonian and Einsteinian theory
is very small for v << c, as it should be.)
[*] c is a scalar here. Since this is a one dimensional problem v is a
signed scalar.
--
#191, ewill3@earthlink.net
It's still legal to go .sigless. |
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Spoonfed (www.spoonfedrel
science forum Guru Wannabe
Joined: 28 Apr 2006
Posts: 144
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| Posted: Sat Jun 03, 2006 6:48 pm Post subject:
Re: Another look at the Lorentz factor.
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Golden Boar wrote:
| Quote: | PD wrote:
Golden Boar wrote:
The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
I think it would benefit you to see the form of the Lorentz
transformations in terms of hyperbolic transcendental functions, where
gamma = cosh(rapidity),
about as simple as you can get.
I don't know.
The rapidity is given by,
(c + v) / sqrt(c^2 - v^2)
|
Red X. Minus 1.
| Quote: | So you are implying that,
cosh((c + v) / sqrt(c^2 - v^2))
is simpler than,
c / (sqrt(c^2 - v^2)
|
I've made this error before, too. Actually, the rapidity is
A=ln[sqrt((c+v)/(c-v))]
then Cosh(A) = (e^A+e^-A )/2
= (sqrt((c+v)/(c-v)) +sqrt((c-v)/(c+v)))/2
after a little algebra, this yields
(c+v) +(c-v)
---------------- *1/2
sqrt(c^2-v^2)
Cosh(A)= c/sqrt(c^2-v^2)
And to put it into your terms, PD means that
(e^ln[sqrt((c+v)/(c-v))]+e^-ln[sqrt((c+v)/(c-v))] )/2
is simpler than
(c + v) / sqrt(c^2 - v^2)
where, here, I have represented everything in terms of v and c, and
made efforts to make the first representation look as complicated as
possible.
If you write everything in terms of rapidity, of course, then PD's way
looks less complicated.
Also, another method of interest is to write the transformation thusly:
(ct') = /Cos[45] -Sin[45] \ / k 0 \ /Cos[45] Sin[45]\ /ct\
( x') = \Sin[45] Cos[45] / \ 0 1/k/ \-Sin[45] Cos[45]/ \ x/
where k=sqrt[(c+v)/(c-v)].
The rapidity, A=ln(k) is considered to be more fundamental than
velocity or v/c. because you can get it to look as simple as this:
/ct'\ = / Cosh[A] -Sin[A]\ /ct\
\ x'/ \-Sinh[A] Cos[A]/ \ x/
There may be other reasons as well, but I've not heard them.
| Quote: | Moreover, the rapidity variable is exceeding useful as well as being
more physically fundamental.
See, for example, http://www.everything2.com/index.pl?node_id=312518
The important thing to note is that, rather than reading up on things
that have already been done a long long time ago, you are spending your
time futzing with equations to stumble on the work yourself. This in
itself is not a bad thing. Thinking that you've done something original
or of interest to others --- that you'll find people have a lesser
opinion of.
PD
|
Between the boredom of people who have no idea what your talking about,
and the disdain of those who do, you will find many books and journal
articles, and you are expected to read them all from cover to cover
before posting here and humiliating yourself before the court of your
superiors.  |
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Bilge
science forum Guru
Joined: 30 Apr 2005
Posts: 2816
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| Posted: Sat Jun 03, 2006 10:04 pm Post subject:
Re: Another look at the Lorentz factor.
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Golden Boar:
| Quote: |
Bilge wrote:
Golden Boar:
The Lorentz factor is usually given by the equation:
gamma = 1/sqrt(1-v^2/c^2)
or by
beta = v/c
gamma = 1/sqrt(1-beta^2)
The equation can be written in a more intuitive (for me at least) way
as shown below:
gamma = c / sqrt(c^2 - v^2)
OK, your next assignment is to begin with an arbitrary lagrangian
density, L[q(x^u), d_v q(x^u)] and find the conserved currents and
charges if L is invariant under an infinitesimal spacetime displacement,
x^u -> x^u + a^u.
Nah, I don't think I will.
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Then you really have no basis for your rants against mass. |
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The Sorcerer
science forum Guru
Joined: 22 May 2006
Posts: 363
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| Posted: Sat Jun 03, 2006 10:49 pm Post subject:
Re: Another look at the Lorentz factor.
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|
"Spoonfed" <good4usoul@yahoo.com> wrote in message
news:1149360536.521651.286830@y43g2000cwc.googlegroups.com...
| Between the boredom of people who have no idea what your talking about,
| and the disdain of those who do, you will find many books and journal
| articles, and you are expected to read them all from cover to cover
| before posting here and humiliating yourself before the court of your
| superiors. ;-)
c = (dx-dx)/dt = 0/0.
http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
(With the utmost disdain for stupid spoofed inferior cunts with heads
up their arse who have no idea what they are babbling about.)
Androcles. |
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Spoonfed (www.spoonfedrel
science forum Guru Wannabe
Joined: 28 Apr 2006
Posts: 144
|
| Posted: Sun Jun 04, 2006 4:24 am Post subject:
Re: Another look at the Lorentz factor.
|
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The Sorcerer wrote:
| Quote: | "Spoonfed" <good4usoul@yahoo.com> wrote in message
news:1149360536.521651.286830@y43g2000cwc.googlegroups.com...
| Between the boredom of people who have no idea what your talking about,
| and the disdain of those who do, you will find many books and journal
| articles, and you are expected to read them all from cover to cover
| before posting here and humiliating yourself before the court of your
| superiors.
|
For the record, I do regret making the the comment Androcles quoted.
It was a completely sarcastic remark, (not out of character for me, I'm
afraid,) but meant only in fun, teasing PD's attitude toward "futzing
with equations." I probably should have asked what PD meant by
futzing. For instance
| Quote: |
c = (dx-dx)/dt = 0/0.
|
Represents futzing to what appears to be an inexcusable level...
(Are you trying to convince me that you're just stupid Androcles? It
won't work. You are an artist!)
I would actually give some credit to this level of futzing, perhaps try
to give it some analysys and figure out how the futzer had come upon
his idea if it weren't followed by:
....which is clearly incorrect, because if it were true, I would
suffocate.
Androcles generally doesn't believe anything he is writing. |
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The Sorcerer
science forum Guru
Joined: 22 May 2006
Posts: 363
|
| Posted: Sun Jun 04, 2006 11:24 am Post subject:
Re: Another look at the Lorentz factor.
|
|
|
"Spoonfed" <good4usoul@yahoo.com> wrote in message
news:1149395074.890296.114570@u72g2000cwu.googlegroups.com...
|
| The Sorcerer wrote:
| > "Spoonfed" <good4usoul@yahoo.com> wrote in message
| > news:1149360536.521651.286830@y43g2000cwc.googlegroups.com...
| > | Between the boredom of people who have no idea what your talking
about,
| > | and the disdain of those who do, you will find many books and journal
| > | articles, and you are expected to read them all from cover to cover
| > | before posting here and humiliating yourself before the court of your
| > | superiors.
|
| For the record, I do regret making the the comment Androcles quoted.
| It was a completely sarcastic remark, (not out of character for me, I'm
| afraid,) but meant only in fun, teasing PD's attitude toward "futzing
| with equations." I probably should have asked what PD meant by
| futzing. For instance
|
| >
| > c = (dx-dx)/dt = 0/0.
|
| Represents futzing to what appears to be an inexcusable level...
| (Are you trying to convince me that you're just stupid Androcles? It
| won't work. You are an artist!)
|
| I would actually give some credit to this level of futzing, perhaps try
| to give it some analysys and figure out how the futzer had come upon
| his idea if it weren't followed by:
|
| >
| > http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
| > (With the utmost disdain for stupid spoofed inferior cunts with heads
| > up their arse who have no idea what they are babbling about.)
| > Androcles.
|
| ...which is clearly incorrect, because if it were true, I would
| suffocate.
|
| Androcles generally doesn't believe anything he is writing.
http://www.fourmilab.ch/etexts/einstein/specrel/www/figures/img7.gif
Choking yet?
http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
(With the utmost disdain for stupid spoofed inferior cunts with heads
up their arse who have no idea what they are babbling about.)
Androcles.
| |
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Sue...
science forum Guru
Joined: 08 May 2005
Posts: 2684
|
| Posted: Sun Jun 04, 2006 11:49 am Post subject:
Re: Another look at the Lorentz factor.
|
|
|
The Sorcerer wrote:
| Quote: | "Spoonfed" <good4usoul@yahoo.com> wrote in message
news:1149395074.890296.114570@u72g2000cwu.googlegroups.com...
|
| The Sorcerer wrote:
| > "Spoonfed" <good4usoul@yahoo.com> wrote in message
| > news:1149360536.521651.286830@y43g2000cwc.googlegroups.com...
| > | Between the boredom of people who have no idea what your talking
about,
| > | and the disdain of those who do, you will find many books and journal
| > | articles, and you are expected to read them all from cover to cover
| > | before posting here and humiliating yourself before the court of your
| > | superiors.
|
| For the record, I do regret making the the comment Androcles quoted.
| It was a completely sarcastic remark, (not out of character for me, I'm
| afraid,) but meant only in fun, teasing PD's attitude toward "futzing
| with equations." I probably should have asked what PD meant by
| futzing. For instance
|
|
| > c = (dx-dx)/dt = 0/0.
|
| Represents futzing to what appears to be an inexcusable level...
| (Are you trying to convince me that you're just stupid Androcles? It
| won't work. You are an artist!)
|
| I would actually give some credit to this level of futzing, perhaps try
| to give it some analysys and figure out how the futzer had come upon
| his idea if it weren't followed by:
|
|
| > http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
| > (With the utmost disdain for stupid spoofed inferior cunts with heads
| > up their arse who have no idea what they are babbling about.)
| > Androcles.
|
| ...which is clearly incorrect, because if it were true, I would
| suffocate.
|
| Androcles generally doesn't believe anything he is writing.
http://www.fourmilab.ch/etexts/einstein/specrel/www/figures/img7.gif
Choking yet?
1905 ???
http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
(With the utmost disdain for stupid spoofed inferior cunts with heads
up their arse who have no idea what they are babbling about.)
Androcles.
|
The Catholic church was rather ham-handed in their treatment
of witches. Androcles would have extracted some vengence
from Mother Teresa before she died of natural causes but all
his navigation equipment assumed a particle model of light
propagation so he couldn't locate here.
<< It is to be found rather in the fact of his recognition
that the four-dimensional space-time continuum of the
theory of relativity, in its most essential formal properties,
shows a pronounced relationship to the three-dimensional
continuum of Euclidean geometrical space. 1 In order to
give due prominence to this relationship, however, we must
replace the usual time co-ordinate t by an imaginary magnitude
(sqrt -1)
ct proportional to it. Under these conditions, the natural
laws satisfying the demands of the (special) theory of
relativity assume mathematical forms, in which the time
co-ordinate plays exactly the same rôle as the three
space co-ordinates. Formally, these four co-ordinates
correspond exactly to the three space co-ordinates in
Euclidean geometry. >>
Albert Einstein (1879-1955). Relativity:
The Special and General Theory. ***1920***.
http://www.bartleby.com/173/17.html
giggle
Sue...
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The Sorcerer
science forum Guru
Joined: 22 May 2006
Posts: 363
|
| Posted: Sun Jun 04, 2006 2:16 pm Post subject:
Re: Another look at the Lorentz factor.
|
|
|
"Sue..." <suzysewnshow@yahoo.com.au> wrote in message
news:1149421785.177426.262600@c74g2000cwc.googlegroups.com...
The Sorcerer wrote:
| Quote: | "Spoonfed" <good4usoul@yahoo.com> wrote in message
news:1149395074.890296.114570@u72g2000cwu.googlegroups.com...
|
| The Sorcerer wrote:
| > "Spoonfed" <good4usoul@yahoo.com> wrote in message
| > news:1149360536.521651.286830@y43g2000cwc.googlegroups.com...
| > | Between the boredom of people who have no idea what your talking
about,
| > | and the disdain of those who do, you will find many books and
journal
| > | articles, and you are expected to read them all from cover to cover
| > | before posting here and humiliating yourself before the court of
your
| > | superiors.
|
| For the record, I do regret making the the comment Androcles quoted.
| It was a completely sarcastic remark, (not out of character for me, I'm
| afraid,) but meant only in fun, teasing PD's attitude toward "futzing
| with equations." I probably should have asked what PD meant by
| futzing. For instance
|
|
| > c = (dx-dx)/dt = 0/0.
|
| Represents futzing to what appears to be an inexcusable level...
| (Are you trying to convince me that you're just stupid Androcles? It
| won't work. You are an artist!)
|
| I would actually give some credit to this level of futzing, perhaps try
| to give it some analysys and figure out how the futzer had come upon
| his idea if it weren't followed by:
|
|
| > http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
| > (With the utmost disdain for stupid spoofed inferior cunts with heads
| > up their arse who have no idea what they are babbling about.)
| > Androcles.
|
| ...which is clearly incorrect, because if it were true, I would
| suffocate.
|
| Androcles generally doesn't believe anything he is writing.
http://www.fourmilab.ch/etexts/einstein/specrel/www/figures/img7.gif
Choking yet?
1905 ???
http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
(With the utmost disdain for stupid spoofed inferior cunts with heads
up their arse who have no idea what they are babbling about.)
Androcles.
|
The Catholic church was rather ham-handed in their treatment
of witches. Androcles would have extracted some vengence
from Mother Teresa before she died of natural causes but all
his navigation equipment assumed a particle model of light
propagation so he couldn't locate here.
I never have trouble locating 'here', it's 'there' that is the problem.
Androcles.
<< It is to be found rather in the fact of his recognition
that the four-dimensional space-time continuum of the
theory of relativity, in its most essential formal properties,
shows a pronounced relationship to the three-dimensional
continuum of Euclidean geometrical space. 1 In order to
give due prominence to this relationship, however, we must
replace the usual time co-ordinate t by an imaginary magnitude
(sqrt -1)
ct proportional to it. Under these conditions, the natural
laws satisfying the demands of the (special) theory of
relativity assume mathematical forms, in which the time
co-ordinate plays exactly the same rôle as the three
space co-ordinates. Formally, these four co-ordinates
correspond exactly to the three space co-ordinates in
Euclidean geometry. >>
Albert Einstein (1879-1955). Relativity:
The Special and General Theory. ***1920***.
http://www.bartleby.com/173/17.html
giggle
Sue...
|
|
| Back to top |
|
|
Spoonfed (www.spoonfedrel
science forum Guru Wannabe
Joined: 28 Apr 2006
Posts: 144
|
| Posted: Sun Jun 04, 2006 4:56 pm Post subject:
Re: Another look at the Lorentz factor.
|
|
|
Sue... wrote:
| Quote: | It is to be found rather in the fact of his recognition
that the four-dimensional space-time continuum of the
theory of relativity, in its most essential formal properties,
shows a pronounced relationship to the three-dimensional
continuum of Euclidean geometrical space. 1 In order to
give due prominence to this relationship, however, we must
replace the usual time co-ordinate t by an imaginary magnitude
(sqrt -1)
ct proportional to it. Under these conditions, the natural
laws satisfying the demands of the (special) theory of
relativity assume mathematical forms, in which the time
co-ordinate plays exactly the same rôle as the three
space co-ordinates. Formally, these four co-ordinates
correspond exactly to the three space co-ordinates in
Euclidean geometry.
Albert Einstein (1879-1955). Relativity:
The Special and General Theory. ***1920***.
http://www.bartleby.com/173/17.html
giggle
Sue...
|
Did he mean this in an exact manner?
A rapidity change in the x-t plane
/ct'\ = / Cosh[A] -Sinh[A]\ /ct\
\ x'/ \-Sinh[A] Cosh[A]/ \ x/
is equal to
/ct'\ = / Cos[i A] -Sin[i A]\ /ct\
\ x'/ \-Sin[i A] Cos[i A]/ \ x/
which is similar but not quite the same as a rotation in the x-y plane
with the real-valued angle replaced with an imaginary angle.
/ y'\ = / Cos[B] -Sin[B]\ / y\
\ x'/ \ Sin[B] Cos[B]/ \ x/
But here, I am referring to treating the rapidity as an imaginary
angle, whereas Einstein was talking in terms treating the value of t as
imaginary. So considerably more futzing is needed to show whether
Einstein's statement (replace c*t with i*c*t and it's exactly the same)
was correct. |
|
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