Wormhole theory

[Jonathan Kirwan]

Cool. Finally something I have the math for.

Something to keep in mind about calculus is that it is really VERY
simple, in basic concept. The older books I used to learn from made
it lots more complex that it needed to be. I think there may be some
newer ones based on something that I discovered entirely for myself,
but then discovered that others had already discovered it -- namely
"non-standard analysis." (Abraham Robinson)

But the very simple way to view things in calculus is that it is just
like algebra, except that you get a new kind of special variable type
that adds to what's already in algebra. This special kind of
variable, unlike the usual ones, is ONLY ALLOWED to hold infinitely
small values (which aren't exactly 0, but are smaller than any finite
value.) In other words, in algebra, variables can only hold finite
values. In calculus, you get to keep that type of variable and add a
new one that can only hold infinitesimal values.

Take a look at a falling body under gravitation, assuming that at the
start of time (t=0) the velocity is zero (v=0.) Now you release the
object and it begins to fall. Under normal algebra, you often find
the simple formula that looks like:

D = v * t

So that the distance some object travels is the speed it is going
times the time it went at that speed. But this, of course, assumes
that the speed is the same for the entire time or that the speed given
is a useful average, at least.

In the case of a falling object, though, the velocity is constantly
changing. So what do we use for V? It's never the same twice.

Now the cheating answer is to simply say that under constant
acceleration, that the average velocity must be 1/2 of the final
velocity and we can compute that final velocity as g*t. So by that,
we can say:

D = [(1/2)*g*t]*t = g*t^2/2

Which is correct.

But if you want to think about things from a calculus perspective (a
better one, in general), then what you do instead is ask yourself:

"Well, since the velocity is constantly changing all the time, exactly
when is the velocity a precise value and not some average thing?"

In other words, looking at D=v*t, ask yourself for how long an exact
velocity is still exact. If you think about that, you have to answer,
"It is a particular finite value only for an instant of time."

Once you say this to yourself, now ask.. "Well, if the velocity is
exact only for an instant, how far will it travel in that instant?"

And to this, you answer yourself, "For a very small distance, namely
only an infinitely small distance." Kind of, well, an instant of
distance. So you write:

dD = v * dt

Which is to say that the object travels an infinitely small fraction
of the total distance, computed by multiplying the instantaneous and
finite velocity at some moment by the duration of that moment, and
that is itself infinitely small.

It makes sense that when you multiply a finite value by an infinitely
small one, that you should also get another infinitely small result.
Right?

Now remember, that dD isn't two things, but one thing. It's just a
variable. We could have called it Q, if you prefer. But we need some
way to keep track of the fact that it can ONLY HOLD infinitely small
values. So if we just remember the little-d in front as kind of
modifying the variable name so that it helps us remember this, then we
are fine. So dD is a variable and dt is a variable. They are
different variables, too.

In any case, keep in mind that these tiny, infinitesimally small
values are NOT necessarily the same as each other. Just as two
different variables in algebra are not necessarily the same as each
other, despite both representing some finite value. In like fashion,
the variables dD and dt, although each are infinitely small, they can
be different from each other and yet both infinitely small.

It's perfectly possible to relate one infinitely small variable to
another, like this:

dD/dt

In this case, all we are doing is dividing one infinitely small value
by another. And this often produces a finite value. For example, it
could be the case that:

dD/dt = 1

In this case, the two variables track each other in a 1:1
relationship. They are equally infinitely small. But we could have
written:

dD/dt = 2

In this case, dD is twice as big as dt. They are both infinitely
small, but it is still true that dD is twice as big as dt.

Now, as you already know about velocity, it is:

v = D/t

Well, an even better definition that works for objects that don't
always travel at uniform speed, is this:

V(t) = dD/dt

In this case, we are just saying that the speed at some moment (t) is
best computed as the instantaneous distance it traveled divided by the
instant of time it traveled. This is always true, even for things
that aren't always moving around at the same speed. So in that sense,
it is a more powerful and better way to see speed defined. It is a
more universal definition. The old one uses averages. The new one
uses instantaneous precision.

Of course, that's all just useless unless at some point you can get
back to finite values for real measurements.

How many infinitely small distances do you need to add up to make a
finite distance? Well, an infinite number of them, of course. So
calculus invents a new symbol, the integral sign, to indicate this
particular kind of infinite sum.

If you had an infinite number of dD pieces, what does it add up to?
Well, D of course! What else?

And like in algebra, what you do to one side of an equation you do to
another side. So if you infinite sum one side, you infinite sum the
other side, too. This is called "integration."

Going back to:

dD = v * dt

To clear it up and get rid of the infinitesimal variables, we just sum
both sides:

SUM(dD) = SUM(v * dt)

Since we also know that v=g*t, we can substitute to get:

SUM(dD) = SUM(g * t * dt)

Now comes the fun part. We already know what the SUM(dD) is, so:

D = SUM(g * t * dt)

That was easy. But what about the rest?

Well, since 'g' is a constant (as given), then we already know that
any common factor found in terms of a long sum can be extracted, just
as in (a*b+a*c+a*d) = a*(b+c+d). So our sum can be adjusted so that:

D = g*SUM(t * dt)

Okay, now we are stuck. 't' itself is NOT a constant, so we cannot
just pull it out. Similarly, dt, although a constant is infinitely
small and needs to be treated by the SUM(). So both stay in there.

Any time you see a product, think of areas. In this case, imagine
that you have an area that is 't' high and 'dt' wide. That inner part
could be seen that way, just fine. So we are summing up an infinite
number of tiny areas. What would that look like?

Well, imagine laying each tiny rectangle side by side, so that the
width is 'dt' and they are sorted according to their heights, 't'. As
in:

|
| |
| | |
.. | | | |

And so on. Each is very slim, dt wide. But when you stack them side
by side like that, they all add up to something that is 't' wide.
Similarly, how high are they? Well, as we go from 0 to t, they grow
from 0 high to 't' high. So in the end, we have a triangle that is
't' wide and 't' high. What is the total area of all that? Well, 1/2
of t^2, of course. So:

D = g*t^2/2

Just like before.

You can also cancel out various variables you see on different sides
of the equation.

Getting this back to electronics, which I suppose I should do, you are
told that the charge on a capacitor is:

Q = C * V

But if the V is changing, then the Q is changing. We could rewrite it
more precisely as:

dQ = C * dV

Just like we did earlier for speed, time, and distance relationships.

And if we want to, we could compare both sides at each moment of time,
so that:

dQ/dt = C * dV/dt

Dividing both sides by the same variable is just fine. It looks
funny, in a way, but it is just another variable and we can divide
both sides by this infinitesimal variable if we want to. Or we cancel
them back out by multiplying both sides by dt and get back what we
started with.

Just like in algebra.

By the way, dQ/dt is just I (current) so:

I = C * dV/dt

A useful thing to remember.

Anyway, calculus is just algebra with a new type of variable that can
only hold infinitely small values. That forces you, at some point
later on, to use an infinite sum if you want to see finite values.
But you can delay that while you think and cancel things just as in
regular algebra and, if you are lucky, there won't be that much left
on which to have to use an infinite sum.

Jon

 

[Robert]

Something to keep in mind about calculus is that it is really VERY
simple, in basic concept. The older books I used to learn from made
it lots more complex that it needed to be. I think there may be some
newer ones based on something that I discovered entirely for myself,
but then discovered that others had already discovered it -- namely
"non-standard analysis." (Abraham Robinson)

[bunch of math cut]

This was the basic problem that tossed these out after Newton.

How can two numbers both be infinitely small but not the same?

One would be smaller than the other but they are both infinitely small which
is as small as you get.

Infinite + 1 (or any finite number) is the same infinite number.

1/[infinite] is not going to be smaller than 1/[infinite+1] though you say
it is.

Georg Cantor went nuts thinking about these kinds of questions, stop before
it's too late for you! <g>

I do want to see how Robinson has done it one of these days. I suspect it's
like the square root of -1. You just declare it as a different type of
number and deal in multiples of it as you do with j or i. But mixing it in
with regular numbers is the path of insanity.

Robert

 

[Kevin Aylward]

That is obvious. I was just talking about time dilation.
Oh.


It's an optical illusion that can slow the decay of particles.

There is a terminology issue here. My phrase "conventional time
dilation" is *only* referring to the reciprocal effect of each observer
both seeing the others clocks run slow. It is not referring to the real
life time increase of relativly moving particles, which is *not* an
optical illusion effect.


Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Robert Baer]

(please note I don't understand the math of the Lorentz transformations

or

GR yet...)

The math behind the Lorentz transformation is dead easy. Simple
algebra.

First, try very hard to mentally assume away any idea of an absolute
(or 'preferred') reference frame and ask yourself how you might start
out to describe the motion of a photon particle traveling at the speed
of light from two different non-preferred perspective frames:

x = c*t coordinate system K

and,

x' = c*t' coordinate system K'

In other words, we make no presumptions about the two reference frames
to each other, __except__ for the fact that they will each observe the
same velocity for the photon. It is as though they are completely
independent to each other.

To make this point come to home deeply and seriously, imagine that the
photon is simultaneously observable in two completely different
universes that have nothing in common to each other except that they
both interact with shared photons.

The above two equations do express the fact that x and x' have nothing
to do with each other except for the speed of light, and that t and t'
similarly have nothing to do with each other except again for the
speed of light. The only thing in common in those two equations is c,
itself. Both independent frames will see that as the same value,
whatever it turns out to be. Other than that, the two frames are
completely without any other shared referents.

Now we write these same two, but with the photon traveling in the
opposite direction, just to get all the possibilities:

x = -c*t

and,

x' = -c*t'


Then we can write:

x - c*t = 0, and also from the last case, x + c*t = 0

and,

x' - c*t' = 0, and also from the last case, x' + c*t' = 0

Of course. Nothing special about that.

But since these are zero, we can now make some statement that combines
both frames:

x - c*t = A*(x' - c*t'), and, x + c*t = B*(x' + c*t')

We don't yet know what A and B are likely to be, but that's the
problem to solve.

We can now add and subtract both of these from each other:

x - c*t = A*(x' - c*t') x - c*t = A*(x' - c*t')
+ [x + c*t = B*(x' + c*t')] - [x + c*t = B*(x' + c*t')]
----------------------------- -------------------------------
2*x = A*x'+B*x'-A*c*t'+B*c*t' 2*c*t = A*x'-B*x'-A*c*t'-B*c*t'

or,

2*x = (A+B)*x' - (A-B)*c*t' 2*c*t = (A-B)*x'-(A+B)*c*t'

All this is pretty basic algebra, accessible to most anyone I think.

At this point, to get rid of the factor 2 (the constants A and B are
just arbitrary ones, so we can create any new ones we want based on
them for convenience and use those instead), we can define the
following new derived constants:

R = (A+B)/2, and S = (A-B)/2

From that, we can re-write the above as:

x = R*x' - S*c*t' and c*t = S*x'-R*c*t'

Remember these two, we'll refer to them below.

Well, that's a start. Of course, we still haven't figured out what R
and S are. So let's do so.

At the origin of the one of the frames, we have x = 0, so

x = R*x' - S*c*t' = 0

or,

x' = t' * [S*c/R]

This appears to be in a familiar x'=v'*t' form, if we defined the
velocity v':

v' = [S*c/R]

If so, it would be the velocity for which the origin (or really, any
point) in K is moving with respect to K'.

But in perhaps a little better form, we can write that the
instantaneous change in x' is:

dx' = dt' * [S*c/R]
dx'/dt' = S*c/R
v' = dx'/dt' = S*c/R

Same thing. And v' is the relative velocity of the two systems.

Now, we also know that length of a rest-object in system K observed
from system K' must be the same as the length of a rest-object in
system K' observed from system K (due to the assumption of relativity
-- no preferred point of view.) So let's take a picture, so to speak,
at time t'=0, then

x = R*x'

Let's now look at two points in system K (x-axis) separated by a
distance of exactly 1:

x1 = R*x1'
x2 = R*x2'

and we can also say that, by definition:

x2 = x1 + 1
x2 - x1 = 1
dx = x2 - x1 = 1

But also,

R*x2' = R*x1' + 1
R*x2' - R*x1' = 1
R*(x2' - x1') = 1
x2' - x1' = 1/R
dx' = x2' - x1' = 1/R

Therefore,

dx'/dx = 1/R.

But now, if we look at the picture taken from system K at t=0 and if
we then work to eliminate t' from:

x = R*x' - S*c*t' and c*t = S*x'-R*c*t'

then we find:

c*t = S*x'-R*c*t' @ t=0
c*0 = S*x'-R*c*t'
0 = S*x'-R*c*t'
R*c*t' = S*x'
t' = S*x'/[R*c]

substituting into:

x = R*x' - S*c*t'
x = R*x' - S*c*{S*x'/[R*c]}
x = R*x' - S^2/R*x'
x = (R - S^2/R)*x'
x = (R/R)*(R - S^2/R)*x'
x = R*(1 - S^2/R^2)*x'

Now, recall that:

v' = S*c/R

So that it should be clear that:

v'^2 = S^2*c^2/R^2
v'^2/c^2 = S^2/R^2

Substituting,

x = R*(1 - S^2/R^2)*x'
x = R*(1 - v'^2/c^2)*x'
dx = R*(1 - v'^2/c^2)*dx'
dx/dx' = R*(1 - v'^2/c^2)

But we also know that, from elsewhere above:

dx'/dx = 1/R

And from the assumed relativity principle, that:

dx/dx' = dx'/dx

Therefore,

R*(1 - v'^2/c^2) = 1/R
R^2*(1 - v'^2/c^2) = 1
R^2 = 1/(1 - v'^2/c^2)
R = SQRT[1/(1 - v'^2/c^2)]
R = 1/SQRT(1 - v'^2/c^2)

From:

v' = S*c/R

we find:

S = v'*R/c
S = v'*[1/SQRT(1 - v'^2/c^2)]/c
S = (v'/c)*[1/SQRT(1 - v'^2/c^2)]
S = [1/(c/v')]*[1/SQRT(1 - v'^2/c^2)]/c
S = 1/[(c/v')*SQRT(1 - v'^2/c^2)]
S = 1/SQRT(c^2/v'^2 - 1)

So, we've solved for both R and S. Remembering,

x = R*x' - S*c*t' and c*t = S*x'-R*c*t'

And with R and S in hand, we are set to find:

x = (x' - v'*t') / SQRT(1 - v'^2/c^2)
t = (t' - x'*v'/c^2) / SQRT(1 - v'^2/c^2)

And there you have it all.

All this is based on translation and not upon systems K that rotate
with respect to K' or are under some acceleration, gravitational or
otherwise. That is left for the general theory of relativity to deal
with, where it makes the assertion that there is no difference between
acceleration and gravity, from the point of view of the observer, and
thus that inertial mass (which relates to acceleration) must be the
same as gravitational mass (which, of course, relates to gravity) in a
1:1 relationship. With that assumption in hand, all the fun begins.

Jon

I'd get that if you explained the apostrophe. I know it has something to do
with derivatives, but I don't know how to take derivatives quite yet. I'm
getting up to the level at which I can learn to though.

Fairly simple..
Start with a straight line at some slope Y=m*X+b; the slope "m" si
the derivitave and a constant.
Step up to a parabola.
Make it simple with the parabola pointing "up" (to catch water??) and
not too skinny and not too flat, with it symmetrical about the Y axis.
Draw a tangent at the bottom (minimum Y); it will be horizontal ("m"=0).
Draw another at some point at the right side; positive slope.
Draw another on the left side, displaced vertically exactly the same
as the previous one; negative slope exactly the same.
Now draw a line, using three points: (a) first one X=same as
left-side tangent, Y=the slope found, (b) X=zero, Y=zero (the bottom
line slope), (c) X=same as right-side tangent, Y=the slope found.
This will be a straight line that accurately describes the slope or
derivitave of that parabola.
**
You probably had a good idea of this anyway, and now are ahead of
this mundane explaination (if it merits such a label).
Have fun; i have complete confidence in you.
Do yourself the courtesy of having confidence in yourself.




and at some arbritrary point along it, draw a tangent: hmmm... looks
like you drew Y*m*X+b there.

 

[Jonathan Kirwan]

Something to keep in mind about calculus is that it is really VERY
simple, in basic concept. The older books I used to learn from made
it lots more complex that it needed to be. I think there may be some
newer ones based on something that I discovered entirely for myself,
but then discovered that others had already discovered it -- namely
"non-standard analysis." (Abraham Robinson)

[bunch of math cut]

This was the basic problem that tossed these out after Newton.

How can two numbers both be infinitely small but not the same?

One would be smaller than the other but they are both infinitely small which
is as small as you get.

Infinite + 1 (or any finite number) is the same infinite number.

1/[infinite] is not going to be smaller than 1/[infinite+1] though you say
it is.

Georg Cantor went nuts thinking about these kinds of questions, stop before
it's too late for you! <g>

You know, I had discovered this idea entirely for myself. I'd been
taking calculus classes and "doing it the traditional way" looking at
these funny dx, dy, and dz things as a kind of "special notation" of
which I somehow needed to master the use. I treated them as a kind of
"paste on" thing to keep track, but that's about it. And some aspect
of being completely at ease and facile always seemed to be just out of
reach.

Then in an insight, it all came crashing in. And I got really angry,
after a fashion. I was angry at the books and at the teachers for
withholding the near trivial insight from me that made all this so
very much easier. And I felt cheated that no one had bothered to tell
me the simple truth and had kept it hidden from me.

I'm no mathematician and was learning calculus for my physics classes.
But decades later, when I was talking with a mathematician friend I
know about this, he just said, "Oh, what you are talking about was put
onto a rigorous basis in the 1960's."

I was learning calc starting about 1973 and it wasn't until a few
years later that it 'hit me.' I'd never read anything suggesting the
idea (that I'm aware of, anyway) and since then hadn't dug much deeper
into that insight. It was innate and natural and 'felt right' after
some exploration, but it was not the least bit well thought out by me.
But it sure made sense and made things MUCH easier to follow.

I've no idea why it took so long for professional mathematicians to
put it onto a rigorous foundation. I'd have thought they would have
been all over this thing (like teenage boys on a cheap prostitute) and
that if I'd ever mentioned this to them they'd have just said, "Oh,
that's dead obvious. Anyone can see that. Who thinks otherwise?"

But I guess I was glad to hear that it wasn't just an idiocy of my own
that couldn't hold up.

The way I was forced to learn calculus was exactly out of the 19th
century, the same stuff from when calculus was made rigorous by
Dedekind and Weirstrauss, I learned everything from the concept of
the "limit," using the epsilon and delta formalism. What a damned
pain it all is to me, looking back at it.

I guess I never had to wrestle with accepting the idea of different
infinitesimals. I'm not a mathematician so accepting them didn't
bother me in the least. And since it was my own internal "discovery"
that made all this so much easier to think about for me, it never has
crossed my mind to worry much about it, afterwards. It feels natural
to me, as it must, since my mind just naturally tumbled to the idea
many years ago. I'm comfortable with it, like an old shoe.

I do want to see how Robinson has done it one of these days. I suspect it's
like the square root of -1. You just declare it as a different type of
number and deal in multiples of it as you do with j or i. But mixing it in
with regular numbers is the path of insanity.

I've only just yesterday for the very first time tried to search the
web on this subject of "non-standard analysis." I've never read a
single paper on the subject, and only knew about it from that
conversation I had about 6 months ago with my math friend over dinner.
I didn't even know what it was called, because he didn't tell me more.
But when others here brought up the name on a different thread, I
instantly *knew* that it had to be what he was talking about (the
dates matched), so I started looking it up.

I will be getting some of the calculus books based on this idea to see
what they say, now. I think it would be interesting to see this from
a more thorough and well-considered point of view. For me, it's still
in my gut and instinctive and "works well." But I'd like to formalize
it some.

And no, I don't think it will be defined quite as you say. More like
some kind of distinct, continuous number line.

The way I kind of think about it is that there are smaller and larger
infinitesimals along some continuum leading down towards zero itself.
And as you go there, you can also take (1/that-infinitesimal) to get a
similar continuous line of infinite values. And that this is an easy
short cut to seeing why division by zero is uniquely undefined and why
division by an infinitesimal is reasonable. If you think of the
dx=infinitesimals and their 1/dx infinities, you can see the line of
infinities disappearing away into the distance as your dx gets closer
to zero, but the connecting line between these two number lines
becomes "parallel" at division by zero and thus never intersects the
other line -- which is "why" it is undefined.

Jon

 

[Kevin Aylward]

On Tue, 29 Mar 2005 00:21:46 GMT, "~~SciGirl~~"

Cool. Finally something I have the math for.

Something to keep in mind about calculus is that it is really VERY
simple, in basic concept. The older books I used to learn from made
it lots more complex that it needed to be. I think there may be
some newer ones based on something that I discovered entirely for
myself, but then discovered that others had already discovered it
-- namely "non-standard analysis." (Abraham Robinson)

[bunch of math cut]

This was the basic problem that tossed these out after Newton.

How can two numbers both be infinitely small but not the same?

One would be smaller than the other but they are both infinitely
small which is as small as you get.

Infinite + 1 (or any finite number) is the same infinite number.

1/[infinite] is not going to be smaller than 1/[infinite+1] though
you say it is.

Georg Cantor went nuts thinking about these kinds of questions, stop
before it's too late for you! <g>

You know, I had discovered this idea entirely for myself. I'd been
taking calculus classes and "doing it the traditional way" looking at
these funny dx, dy, and dz things as a kind of "special notation" of
which I somehow needed to master the use. I treated them as a kind of
"paste on" thing to keep track, but that's about it. And some aspect
of being completely at ease and facile always seemed to be just out of
reach.

Then in an insight, it all came crashing in. And I got really angry,
after a fashion. I was angry at the books and at the teachers for
withholding the near trivial insight from me that made all this so
very much easier. And I felt cheated that no one had bothered to tell
me the simple truth and had kept it hidden from me.

And what truth was this?

I'm no mathematician and was learning calculus for my physics classes.
But decades later, when I was talking with a mathematician friend I
know about this, he just said, "Oh, what you are talking about was put
onto a rigorous basis in the 1960's."
Oh?


I was learning calc starting about 1973 and it wasn't until a few
years later that it 'hit me.' I'd never read anything suggesting the
idea (that I'm aware of, anyway) and since then hadn't dug much deeper
into that insight. It was innate and natural and 'felt right' after
some exploration, but it was not the least bit well thought out by me.
But it sure made sense and made things MUCH easier to follow.

I've no idea why it took so long for professional mathematicians to
put it onto a rigorous foundation.
Indeed.

I'd have thought they would have
been all over this thing (like teenage boys on a cheap prostitute) and
that if I'd ever mentioned this to them they'd have just said, "Oh,
that's dead obvious. Anyone can see that. Who thinks otherwise?"

What your suggesting here is that all those other mathematicians were so
stupid for not seeing the "obvious". This is very, very naive. You can
bet your boots that this has been looked at extensively. The issue is
that these "obvious" ideas, when actually rigorously investigated,
usually don't work. There is not a chance in hell that these "obvious"
ideas not investigated in excruciating detail, and found lacking in
various ways. It is only much later that mathematics became refined
enough to go back and see what went wrong when those ideas were *first*
rejected.

Indeed, the very first link I got on "non-standand analysis" was
http://members.tripod.com/PhilipApps/nonstandard.html

"Newton and Leibniz used infinitesimal methods in their development of
the calculus, but were unable to make them precise, and Weierstrass
eventually provided the formal epsilon-delta idea of limits"

Some of tye problems are identified
http://everything2.com/index.pl?node_id=901233, 2nd paragraph

So, with all due respect, I doubt you really understand the details as
to why such "obvious" ideas were subsequently rejected when first
thought of. NSA is non-trivial. If this were not the case, it *would*
indeed have been done prior. Things only become "obvious" *after* the
fact. There are millions of "obvious facts" that turned out to be
completely wrong.


Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Jonathan Kirwan]

<snip>
What your suggesting here is that all those other mathematicians were so
stupid for not seeing the "obvious".

No, I'm not, Kevin.

I have no such sense at all and it must have been the poor way I wrote
what I did, if that came across. I had assumed that all this would
have been dealt with on a rigorous basis a long time ago -- yes. And
I frankly have absolutely NO IDEA at all why it took so long -- yes.

But I also know that my experience is terribly limited in this regard
and I assumed then and still assume know (in other words, I believe)
that there are VERY GOOD reasons why it did take that time and that I
just haven't yet been exposed to them.

I just don't know what they are, that's all.

This is very, very naive.

No, it's not. Because that isn't what I was thinking, despite how it
may have come across to you. Sorry about that.

You can bet your boots that this has been looked at extensively.

Agreed. I just had no idea about "why until 1964?" Kevin. But I'm
sure there are good reasons for this.

The issue is
that these "obvious" ideas, when actually rigorously investigated,
usually don't work. There is not a chance in hell that these "obvious"
ideas not investigated in excruciating detail, and found lacking in
various ways. It is only much later that mathematics became refined
enough to go back and see what went wrong when those ideas were *first*
rejected.

Well, as I already pointed out, Kevin, I've not yet (but will soon)
read up on it. I'll be ordering some textbooks this next week to add
to my library when they arrive and I'm sure I'll probably be much
better for having done so. I still don't know why it wasn't put on a
rigorous basis until 1964 or so, but I'm sure it will become clearer
to me over time.

My own individual awakening was a personal one and I didn't mean it to
reflect on anyone else, Kevin. It was just very nice when it happened
and I was surprised a bit that my teachers and their textbooks seemed
to studiously avoid saying anything like it. But my exposure is
limited. So that may all there is to say about it.

I apologize if I left the wrong impression.

Jon

 

[Jonathan Kirwan]

assume know

^^^^
now

 

[Kevin Aylward]

No, I'm not, Kevin.
Ok.


I have no such sense at all and it must have been the poor way I wrote
what I did, if that came across. I had assumed that all this would
have been dealt with on a rigorous basis a long time ago -- yes. And
I frankly have absolutely NO IDEA at all why it took so long -- yes.

ok.

I read into this that there was a criticism being presented, when there
wasn't.

But I also know that my experience is terribly limited in this regard
and I assumed then and still assume know (in other words, I believe)
that there are VERY GOOD reasons why it did take that time and that I
just haven't yet been exposed to them.

I just don't know what they are, that's all.


No, it's not. Because that isn't what I was thinking, despite how it
may have come across to you. Sorry about that.


Agreed. I just had no idea about "why until 1964?" Kevin. But I'm
sure there are good reasons for this.

And, apparantly, there are reasons why this "new" 1964 approach, is not
all that its craked up to be.

This link, http://encyclopedia.laborlawtalk.com/Non-standard_analysis,
at the criticism section is interesting.

Well, as I already pointed out, Kevin, I've not yet (but will soon)
read up on it. I'll be ordering some textbooks this next week to add
to my library when they arrive and I'm sure I'll probably be much
better for having done so. I still don't know why it wasn't put on a
rigorous basis until 1964 or so, but I'm sure it will become clearer
to me over time.

Maybe never. I can't say I'm convinced that this later approach adds
anything of real value. The issue is that major breakthroughs taht give
people grand prizes don't happen very often, e.g. Shrodinger equation,
General Relativity, tellytubbies etc, so people simply invent some
"wonderful" thing, that isn't.

My own individual awakening was a personal one and I didn't mean it to
reflect on anyone else, Kevin. It was just very nice when it happened
and I was surprised a bit that my teachers and their textbooks seemed
to studiously avoid saying anything like it. But my exposure is
limited. So that may all there is to say about it.

I apologize if I left the wrong impression.

I probably jumped the gun a bit here Jon. Its hard to read between the
lines as to how people are really thinking.

What I have found is that there are very few things that are "simple"
that have not been already looked at in great depth. There's always a
catch.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[John Woodgate]

I read in sci.electronics.design that Jonathan Kirwan
<[email protected]>) about 'Wormhole theory',

The way I kind of think about it is that there are smaller and larger
infinitesimals along some continuum leading down towards zero itself.

That concept is a lot older than the 1960s. You can see it in
l'Hopital's method for evaluating functions that have a limit of the
form 0/0 for some value of the independent variable.

Incidentally, I learned differential calculus the 'old' way. I'm no
mathematician, either, but I didn't find it difficult. I had a *good*
teacher. That makes a lot of difference, maybe all the difference.

You may need the 'limit' concept for the geometrical applications of
calculus, staring with the gradient of the tangent.

 

[Robert Monsen]

There is a terminology issue here. My phrase "conventional time
dilation" is *only* referring to the reciprocal effect of each observer
both seeing the others clocks run slow. It is not referring to the real
life time increase of relativly moving particles, which is *not* an
optical illusion effect.

How fast does the particle think our clocks are going? Or put another
way, if you were zipping past the earth at 0.99c, and you happened to
look at a clock on the earth, would it be going slower than yours?

The point is that your perception of time is a function of relative
velocity. The particle decays after a certain interval, but that
interval is perceived by stationary observers to be longer than what is
experienced by the particle itself. There isn't a problem with this,
because your *perception* of time in objects is affected by their
relative motion to you; if you consider a motion in some other reference
frame, that motion generates an interval, the difference of two
spacetime points. Your motion through 'time' also generates an interval.
The projection of the interval created by their motion onto the interval
created by your non-motion is the perception you have of their velocity
and elapsed time, at least according to SR.

A rendezvous between two particles at two separate points in spacetime
is, as you (and JW) say, a different matter, requiring accelerations,
which orient the trajectories towards one another again. (Can these
accelerations be dealt with properly using SR? Even in x and t?)
However, thinking about it as a path through spacetime makes it much
more believable that the elapsed time experienced on a particular path
would be different for different trajectories.

Now, on a separate but related subject, I wonder why there should be a
difference between inertial paths and accelerated paths. What is an
'accelerated path' anyway? If I'm orbiting the earth, I don't feel any
acceleration, but I'm apparently aging a bit more slowly. Your math
pages link points out the idea that two orbits, one circular, and one
eliptical, will experience different 'proper time'. Neither can tell
they are accelerated unless they look out the window. Thus, by comparing
notes when they cross paths, they could violate the principle of
relativity, and figure out how fast they were going in relation to one
another. I guess GR covers this.

Also, when one thinks about the spacetime momentum+energy vector, and
sees that mass is just momentum that we happen to be travelling along
with, it all gets a bit confusing. To a photon, YOU are the photon, and
any photons it happens to be travelling with in the same direction are
mass, perhaps making up photon planets and galaxies. To it, your time is
standing still, just like to us, a photon experiences no time. An
infinite number of parallel realities, corresponding to the infinite
directions a photon can move...

Nyuck Nyuck Nyuck.

--
Regards,
Robert Monsen

"I'm tryin' ta think, but nuttin's happenin!"
Curly

 

[~~SciGirl~~]

How fast does the particle think our clocks are going? Or put another

way, if you were zipping past the earth at 0.99c, and you happened to
look at a clock on the earth, would it be going slower than yours?

The particle can't think

 

[Robert]

The way I kind of think about it is that there are smaller and larger
infinitesimals along some continuum leading down towards zero itself.
And as you go there, you can also take (1/that-infinitesimal) to get a
similar continuous line of infinite values. And that this is an easy
short cut to seeing why division by zero is uniquely undefined and why
division by an infinitesimal is reasonable. If you think of the
dx=infinitesimals and their 1/dx infinities, you can see the line of
infinities disappearing away into the distance as your dx gets closer
to zero, but the connecting line between these two number lines
becomes "parallel" at division by zero and thus never intersects the
other line -- which is "why" it is undefined.

Jon

They were using your mental picture of infinitesimals from Newton up to at
least the Bernoulli brothers for calculus. IIRC, they finally went to the
limit concept of Weierstrass, Dedekind, and others because of problems with
the infinitesimals. You could use the infinitesimal to derive nonsense. If
you are interested in the details take a look at a very readable explanation
in Edna Kramer's " The Nature and Growth of Modern Mathematics"

Amazon shows used copies from about $5.00
http://www.amazon.com/exec/obidos/ASIN/0691023727/103-1113358-7469469

Robert

 

[Jim Thompson]

The particle can't think

"Looking" screws up the data. See Heisenberg's Uncertainty Principle.

...Jim Thompson

 

[Terry Given]

I read in sci.electronics.design that Jonathan Kirwan
<[email protected]>) about 'Wormhole theory',



That concept is a lot older than the 1960s. You can see it in
l'Hopital's method for evaluating functions that have a limit of the
form 0/0 for some value of the independent variable.

Incidentally, I learned differential calculus the 'old' way. I'm no
mathematician, either, but I didn't find it difficult. I had a *good*
teacher. That makes a lot of difference, maybe all the difference.

You may need the 'limit' concept for the geometrical applications of
calculus, staring with the gradient of the tangent.

Heaviside's operational calculus.....

Cheers
Terry

 

[Terry Given]

Heaviside's operational calculus.....

Cheers
Terry

Oh yeah, I was astounded too, when I realised calculus can be replaced
with algebra. And wondered why nobody ever mentioned it to me in that way.

Cheers
Terry

 

[Joel Kolstad]

Terry,

Oh yeah, I was astounded too, when I realised calculus can be replaced
with algebra. And wondered why nobody ever mentioned it to me in that way.

Your teachers were probably either brilliant and therefore figured it was
'obvious' and not worthy of mention, or else they weren't so savvy and
didn't recognize it themselves.

 

[Robert Monsen]

"Looking" screws up the data. See Heisenberg's Uncertainty Principle.

...Jim Thompson

This doesn't ring true. The times can be computed exactly. There is no
uncertainty. Both see the other's clocks as going slow.

Feynman* actually has a great explanation of this whole thing. He
describes a light clock, where a light beam is going back and forth
between two mirrors. The mirrors are oriented so the surfaces are
orthogonal to the direction of relative motion. Each time it hits the
mirror, a 'click' happens. The guy in the rocket carrying the clock sees
it going back and forth, ticking at a certain rate, defined by c and the
distance between the mirrors. The guy on the ground hears the ticking at
a slower rate, because to him, the beam must be going in a zigzag
direction, because the rocket and mirrors are moving with some velocity
u. If you work out the distance with the pythagorean theorem, and assume
the light is moving at c, the guy on the ground hears ticks at a rate
which is smaller by a factor of sqrt(1 - u^2/c^2), which is what is
predicted by those Lorentz transforms.

However, it's obvious that the same thing will occur if a spaceship guy
looks at one of these light clocks on the earth as he passes. He'll see
it as ticking more slowly than it should be ticking.

An attempt to explain this is given by Feynman*, where somebody
mistakenly believes that he can measure width and depth of an object by
the angle different surfaces make with his eyes (the surfaces' "apparent
size"). This is clearly false, since you can rotate the object and get
different apparent sizes for the different surfaces. Same thing, except
for spacetime, it's relative velocity and not position that is doing the
rotating.

(* Lectures on Physics, Vol I)

--
Regards,
Robert Monsen

"Your Highness, I have no need of this hypothesis."
- Pierre Laplace (1749-1827), to Napoleon,
on why his works on celestial mechanics make no mention of God.

 

[Jonathan Kirwan]

They were using your mental picture of infinitesimals from Newton up to at
least the Bernoulli brothers for calculus. IIRC, they finally went to the
limit concept of Weierstrass, Dedekind, and others because of problems with
the infinitesimals. You could use the infinitesimal to derive nonsense. If
you are interested in the details take a look at a very readable explanation
in Edna Kramer's " The Nature and Growth of Modern Mathematics"

Amazon shows used copies from about $5.00
http://www.amazon.com/exec/obidos/ASIN/0691023727/103-1113358-7469469

Thanks, I'll look.

From what little reading I've done in the last couple of days, the
gist is that Abraham Robinson has put the physicist-type insightful
methods of Newton (and what I tumbled to, when studying physics) on as
solid a mathematical footing as what Dedekind and Weirstrauss did in
the 19th century.

Personally, I much much prefer this geometrical thinking mode and it
has NOT let me down, in terms of producing the same results others
produce when using the traditional techniques, and enables faster and
clearer insights. I'm not likely to let loose of it. It's worked too
well.

But I need to dig into non-standard analysis and see. I'll get the
book you mention above and also some of the newer textbooks teaching
1st year calculus using non-standard analysis. But I doubt that
mathematicians would write these textbooks unless they had satisfied
themselves that it was a solid idea. Time will tell me, though.

Thanks, again.

Jon

 

[Rich Grise]

This doesn't ring true. The times can be computed exactly. There is no
uncertainty. Both see the other's clocks as going slow.

Feynman* actually has a great explanation of this whole thing. He
describes a light clock, where a light beam is going back and forth
between two mirrors. The mirrors are oriented so the surfaces are
orthogonal to the direction of relative motion. Each time it hits the
mirror, a 'click' happens.

I saw this very demonstration in animation on teevee about fifty
years ago. It was a Disney/Bell Labs thing, I think. And it didn't
go "click", it went "boink." ;-) The one time I did get to go to
Disneyland, Tomorrowland was closed for repairs or some such. )-;
(in retrospect, given the timeframe, they were probably transforming
it to punkland or gothland or something.)

Thanks!
Rich