[ExI] The Bell Effect

[The Avantguardian]

--- Lee Corbin <lcorbin at rawbw.com> wrote:
> John Clark writes
> 
> > It occurs to me that this entire spaceship and string deal
> > would make an absolutely first rate MythBusters episode.
> 
> I assume that you read Stuart's excellent analysis. So
> you must be suggesting that even with a current 
> expenditure running into the billions, it is *John Bell's*
> and *Special Relativity's* so-called "myths" that
> would be busted.

Well this is an interesting philosophical question. When in the realm of
theoretical gendankens filled with unobtainium spaceships and magical 100%
matter conversion drives, what is myth and what is fact? I believe that John
Bell's intent with the paradox was to demonstrate to people that the two
equally accelerated spaceships would drift apart as they continued to
accelerate. His paradox does demonstrate that but only by over-simplifying a
rather tricky engineering problem.

If the string in his example were made out of a material that had a tensile
strength of nearly zero then yes, John Bell would have been exactly right and
the string would always break at reasonable relativistic speeds.

But when real strings with real material properties are used, then breaking the
string using Lorentz contractions alone is much harder as my previous example
demonstrated. This is not to say that it is *impossible* to break the string
using relativity. Just that it is certainly not as simple as the official
explanations lead one to believe.

> However, Stuart or someone could certainly have the
> absolutely last word on this paradox by calculating a set
> of (acceleration, mass, string-length, time) quadruples under
> which the string does break, as theoretically but correctly noticed by
> John Bell, by the first two pages of the Matsuda and Kinoshita article
> http://www.aapps.org/archive/bulletin/vol14/14_1/14_1_p03p07.pdf .

Actually if you read the last couple of pages of the Matsuda and Kinoshita
paper, the discussion section after the math talks about the Bell's Spaceship
Paradox but in the context of a very weak spring instead of a string. Although
anybody familar with Hooke's Law will realize that for all practical purposes
strings are weak springs. From the paper:

"In order to understand this, let us imagine two spaceships
separated by L and joined by a spring, which is not very strong.
If two spaceships accelerate in the same way, the distance between
them increases at first viewed from the spaceships, as
was discussed before. However, the distance begins to shrink
because of the spring and settles to the original distance L
viewed from the spaceships. In this way, the distance between
two spaceships contracts following Eq. (1) viewed on S."

This sounds very similar to what I said in my example. The string develops
tension and then pulls the spaceships back together.

However, I have come to the conclusion that I was mistaken, in that the paradox
does hold true a lot of the time. There are a set of parameters that will allow
you to break any string using the Lorentz contraction. I took Lee's advice and
figured them out. 

Here are the equations I derived from considerations of structural engineering,
conservation of total energy in the accelerating starships, and the Lorentz
transform:

Parameters
-----------------
R:= relativistic tension in string {Newtons}
I:= inertial tension in string {Newtons}
B:= breaking tension of string {Newtons}  
m:= rest mass of one ship {kilograms}
g:= acceleration {meters/second^2}
P:= initial distance between centers of mass {meters}
L:= initial length of string {meters}
X:= cross-sectional area of string {meters^2}
d:= density of string {kilograms/meters^3}
c:= speed of light {299,792,458 meters/second}
S:= ultimate tensile strength {Pascals}
v:= velocity of ships relative to observer at starting point {meters/second}
t:= time {seconds}

1. Time-independent Relativistic Tension Equation:

R = 2*m*P*c^2*g^2/(c^2-v^2)^2

2. Time-dependent Relativistic Tension Equation:

R = 2*m*P*c^2*g^2/(c^2-g^2*t^2)^2

3. Inertial Tension Equation

I = X*L*d*g

4. Breaking Tension Equation

B = S*X

5. Breakage Condition

If R+I > B then string breaks.
 
The breaking tension of the string (B) is a property of the string that is
determined by the composition and cross-sectional area of the string. It is
found by looking up the tensile strength of the material of which the string is
made and multiplying by its cross-section. Wikipedia has a good list of tensile
strengths:
http://en.wikipedia.org/wiki/Tensile_strength

Part of the difficulty of this problem is that the string cannot distinguish
between inertial and relativistic forces. They simply add together to put
stress on the string. Breakage occurs whenever R + I > B or the combined
relativistic and inertial tensions exceed the breakage tension. In order to
break the string using relativity alone as per the paradox, one has to minimize
the inertial tension while maximizing the relativistic tension. 

Another way of writing the breakage condition is R > B-I. Substituting in
equations 1,3, and 4 above gives us:

2*m*P*g^2*c^2/(c^2-v^2)^2 > X*S-X*L*g*d

Some information that simple inspection of this inequality gives us right away
is that for any given string there is a maximum possible acceleration beyond
which the string will fail due to inertial forces alone. This gmax can be
calculated by setting the right-hand side of the inequality to zero and then
solving for g.

X*S-X*L*g*d = 0 

6. gmax = S/(L*d)    

One can think of gmax as the maximum acceleration at which the string can still
support its own weight. As one can see, this is determined by solely by
physical properties of the string: the tensile strength, density, and length.

A curious fact to note at this point is that properties of the rope only affect
the inertial terms and breakage condition. That is to say that the relativistic
forces that governs the Bell Effect are independent of the length, composition,
or other properties of the string itself. This is important if we want to try
to break a string using the Bell Effect or conversely want to keep it one piece
at relativistic speeds.

Now if you want to break the string any old way, one has merely to have the
ships accelerate at greater than gmax. This will instantly break the string at
any speed, no relativity required. But if they accelerate below gmax, a curious
thing happens. A maximum relativistic speed, and corresponding time and
distance travelled, becomes associated with the each string that depends on how
far below gmax your acceleration is.

If your g is just a smidgeon below gmax, then your string will only last a few
seconds and will break at a fairly low speed. If however your acceleration is
quite a lot less than gmax, g << gmax, then you will probably run out of fuel
before you snap the string because the necessary velocity will be vanishingly
close to the speed of light.

For example if the string in my last post, with the two 100 ton spaceships, had
been a single spider silk fiber that was 3.5 microns in diameter and 100 meters
long, then its gmax would be 10,939 meters/second^2 or an astonishing 1116
times earth's surface gravity. Taking inertial forces into account and a
comfortable acceleration of 1 g, the spider silk thread would remain intact up
to a velocity of .9991 light speed or for 353.7 days if you are a
clock-watcher. At 250 g, it would fail at .6815 light speed or about 23 hours
and 10 minutes. A shorter thread would last a lot longer, because its gmax
would be higher.        

As one can see that the limit of the relativistic tension term
2*m*P*g^2*c^2/(c^2-v^2)^2 approaches infinity as v approaches c. Therefore if
one had unlimited fuel and really made an effort, one could theoretically break
*any* string no matter how thick or strong using the Bell Effect if you were
willing to wait long enough. This would include structural items like steel
beams, bulkheads, and welds. 

So in the final analysis, the Bell Spaceship Paradox sometimes works and
sometimes doesn't depending on a bunch of parameters. But the Bell Effect is
real and in a non-resource constrained mathematical sense, it will always work
given that one can get arbitrarily close to the speed of light.


Stuart LaForge

"A portion of mankind take pride in their vices and pursue their purpose; many more waver between doing what is right and complying with what is wrong." - Horace