Re: virus: Science
Fri, 20 Sep 1996 23:44:28 -0500 (CDT)

On Fri, 20 Sep 1996, KMO prime wrote:

> On Thu, 19 Sep 1996 23:27:29 -0500 (CDT) writes:
> Okay Kenneth, you got me. I think you'll need to turn the laconic
> paragraph below into three our four pedantic ones before I'll know what
> to make of it. Heck, I knew I should have finished that Tipler book. (I
> really don't want to start a Frank Tipler is the man/an idiot thread.)
> >However, there are STILL objective measurements possible. For
> >instance,
> >[given a system of units], the 'proper time' a particle [or person,
> >objectively rather than subjectively] experiences is objective--it
> >doesn't
> >matter how badly your measurements are distorted by frame shifts
> >[rotating
> >frames, etc.], the proper time is invariant [up to quantum-mechanical
> >uncertainity]. [If its square is negative, that means the events are
> >space-like, and there is no sublight causal relation possible between
> >them. Until we have decent theoretical/experimental evidence, I'm not
> >going to consider 'no preferred space-time factorization for FTL
> >phenomena' as obvious. For sublight and lightspeed phenomena, there
> >is
> >rather extreme support.]
> >
> >//////////////////////////////////////////////////////////////////////////
> >/ Kenneth Boyd
> >//////////////////////////////////////////////////////////////////////////

Fair enough. I don't think this requires Tipler--I got most of it out of
chapter 2 ofJohn Wheeler's "Gravitation" text. I recommend familarity
with college algebra and matrices for that text--otherwise, it will be
extremely painful reading. This is the only text on General Relativity
I know of that will teach how to do the calculations in a fashion
sensible to any branch of mathematicians.

This will be more like 23 paragraphs than 4, KMO prime. I'm presuming
nothing beyond a vague recollection of college algebra.

First of all, General Relativity is a computational nightmare, because
coordinate systems as per College Algebra are impossible over 'too large
a scale'. The space curves out from them, until they start to create
noticeable problems. A 2-dimensional illustration is a globe. Look at
the longitude and latitude lines. Notice that the north and south poles
do NOT have decent coordinates--you can specify them by latitude alone,
and the longitude is ill-defined.

This is used to simulate the effects of 'the force of gravity' as follows:
An object is at rest if no forces act upon it; it then falls freely. I,
sitting in my chair, am NOT at rest--the Earth, via the chair, 'uses'
electromagnetic repulsion, combined with quantum-mechanics [all electrons
must be in distinct states], to accelerate me continuously at 1 g upwards.
Since our perceptual map is linked to our bodies, we perceive ourselves
as at rest, in a gravity field of 1 g.

We can use any coordinate charts we like [coordinates work over
reasonable scales]. However, physical invariants will look invariant
only when converted to coordinate charts 'at rest' i.e. freely falling.
Let's call these Lorentz frames. [I'm going to be dry--another popular
name for these is 'Einstein elevator'!]

Now, it is convenient [at large scales, for some purposes], to describe
positions in space-time locally by numbers t [for time], and x,y,z [for
space]. As long as we're at the Earth's surface, let's make z go
straight up. This is quite arbitrary--indeed, the entire idea of
coordinate charts is highly arbitrary, a computational device that isn't
really natural to General Relativity. IMO, this is key to why Quantum
Mechanics and General Relativity are hard to unify. IMO, this is also
key to why the Feynman diagrams give 100% agreement with measurements.
[I'm talking about Mathematics, not Science!]

Of course, from our perspective, these Lorentz frames are falling past
us, accelerating at 1 g downwards. There's some conversions involved.
Let us say that an 'area' of space-time is 'flat enough for a single
coordinate chart' if it's so small that we can't measure the effects of
gravity. Of course, this guarantees nothing about whether things are at
REST relative to this chart. These are the 'areas' that can be covered
by a single Lorentz frame.

Now, the proper time between two events A and B is defined as follows:
I happen, for theoretical reasons, to have two events A, B in
space-time. I also have a series of coordinate patches that will get me
from A to B, [Let's take say, someone sitting in a chair
for one second according to the clock on the computer they're using.]
Very loosely speaking, one computes the proper time between two
points as follows:
First, build a series of Lorentz frames that bridge the two events.
That is, build a series of coordinate patches that are, by themselves, so
small that we can't observe gravity in any one of them. We CAN observe
gravity by "the way they're spliced together".
Next, we consider the Lorentz metric--not really a metric in the
Euclidean sense, since it can be negative. Within each Lorentz frame, we
define tau^2 = t^2 - (x^2 + y^2 + z^2) for any two points in the patch. This
tau^2 is the measure of how the events are related within that patch.
Now, our path between events A and B is going to go through a lot of
these Lorentz frames.
Now [being careful to avoid double-counting, etc.], we can measure
the length of the path we have between A and B, in terms of the summed
tau^2. [After a certain point, calculus is more convenient. In
particular, we can use a single 'integral' across ALL of these Lorentz
frames, with suitable higher math. It is probable that lack of
mathematics is restricting developments in current physics.]
We then 'straighten out the path between A and B' to minimize the
absolute value of the tau^2 on the path, possibly changing the Lorentz
frames we use. Think of this as taking out the slack in a string.
The resulting minimal tau^2 is independent of whatever initial
measurements one uses. [We forced a conversion to a series of
coordinate charts that produce indistinguishable results.] An Earth-bound
physicist will get the same number as the proverbial spacecraft flying
by at one-half speed of light, or 99 percent speed of light--subject to
numerical imprecision, of course. This procedure IS computationally
difficult. We call this the proper time. If it is positive [our case],
this is the square of the time t separating the two events by a particle that
thinks it is ALWAYS in free fall, and at the fictitious 'absolute rest'.]
We say that A and B are time-like related, in this case.
A zero tau^2 means that the two events are connectable by a light
ray/photon/whatever [assuming matter is transparent! If neutrinos are/were
massless, they would be a better connector.] We say A and B are
light-like related, in this case.
A negative tau^2 means that -(tau^2) is the square of the distance
that the path between A and B would span in space under forced
simultaneity. [we can always choose to measure events so that any two
space-like events are simultaneous. Since the order of the two events
for purposes of conventional causality is observer-dependent, there are
not allowed to be conventionally causally related. Qualitative
conclusions are MUCH more important than numerical conclusions.] We say
that A and B are space-like related, in this case.
Also note that tau^2 doesn't depend on whether you go from B to A or
A to B--although, in our example, the latter direction is the one that is
forward in time.
The "time dilation" phenomenon is a result of the the coordinate
transformations that keep tau^2 constant.

My comments on the inobviousness of "no preferred factorization of
space-time for FTL events" is a result of the following conclusions [and
preferring to write almost-hard sci-fi rather than hard sci-fi. I want
FTL, and I need competent explanations as to how to generalize physics to
allow it!]
We must avoid the "grandfather paradox", i.e. being one's own
grandparent. [Yes, the idiom's chauvinist. Sorry.] Also, using the
'proper time' mentioned above, it is a mild algebra and Calc I exercise
[with the correct equations] to show that an object moving at a FINITE
FTL speed travels with all four coordinates purely imaginary [I'm using
complex numbers: a+b*i, i^2=-1]. It is convenient to regard this as
perpendicular to space-time as 'embedded locally into four complex
space-time coordinates'. In other words, the reason we don't observe
tachyons is that we can't bring our instruments to bear on them for an
instant--the uncertainity principle prevents us from looking at that
instant. An FTL drive based on this would be a highly unreliable
hyperdrive [cf. Star Wars and the Millenium Falcon for ideas.
The equations break down for light-speed and infinite speed, i.e.
teleportation with respect to a specific reference frame. If there are
no preferred frames for FTL events, we simply change reference frames to
where the 'travel' looks like it is a finite FTL speed, upon which the
comment above goes off. Thus, teleportation in the FTL sense is
vaguely possible if and only if there is a preferred reference frame.
Interestingly enough, allowing teleportation with respect to a unique
preferred global factorization of space-time into space and time STILL
"grandfather paradox" is completely ruled out. In any particular
reference system, the apparent time travel caused by the
now-apparently-finite-speed FTL is canceled out enough so that the
apparent arrival time between two conventionally-timelike events is
always forward. This, however is best with a preferred global
factorization of space into space and time. A local preferred
factorization still allows the "grandfather paradox". If the preferred
frame is nonunique, we also have the "grandfather paradox".
This loophole is also used (implicitly) in the description of all of
the warpdrives I have seen described under Transhumanist page/Space-time
technology. One can visualize this reasonably on paper as follows:
Pretend space is 1-d, and time is 1-d. Draw your space-axis as
horizontal, and your time-axis as vertical. Your light-cone should be an
X intersecting your origin, bisecting the angles between the space-time
axes. [Equations: draw y=x, and y=-x. Geometrically, recall the small
angles from a 45-45-90 triangle as in geometry or trigonometry.]
We can convert reference frames by drawing a second time axis, then
reflecting it around the light-cone hal/f it is 'leaning towards' to get
the corresponding space-axis. Now we can graphically compute various
methods of allowing teleportation [transit parallel to a space axis], and
verify directly (for large frame shifts) that allowing two different space
axes for apparent teleportation results in the grandfather paradox. My
claim about the unique frame is also graphically verifiable then.
There's some 800-calculus which says that the above manipulation is
qualitatively correct. Just equate the apparent space-like distance in 3-d
with the apparent space-like distance in 1-d.

/ Kenneth Boyd