# The Koranic Speed of Light

A reader sent me a really wonderfully wacko link. It's a fundamentalist islamic site, which tries to use relativity to argue for the divinity of the Koran. It's remarkably silly. (I also recently got a link to something similar, but from a Jewish perspective - claiming that the Torah disproves relativity. Alas, I screwed up and lost the link; if whoever sent me that link could re-send it, I'd really appreciate it!)

The claim that relativity proves the Koran is true. See, they claim that the Koran tells you what the speed of light was, and that the real absolute speed of light as described by relativity is stated in the Koran, and that it's tied to the muslim Koranic lunar calendar.

Now, most of us think that the speed of light is an absolute constant - roughly 3×108m/s. But according to our new wacko pals,
that's not true! That speed of light isn't really constant.

Their argument is based on the idea that the speed of light can vary in
certain accelerated reference frames - and that, therefore, the speed of light
is not a fixed quantity based on the speed of light in a vacuum - because the
speed of light in a vacuum can vary! But the Koran says that angels travel in
one day the same distance that the moon travels in 100 lunar years. Since in
the Islamic calendar, a lunar year consists of 12 lunar months, where each
month is a lunar orbit, that's 12,000 lunar orbits per day. And since
obviously, angels move at the speed of light, and 12,000 lunar orbits
per day works out to roughly 3×108m/s, the real absolute fixed, unvarying speed of light is 12,000 lunar orbits per day.

Let's start by checking their figures. The moons orbit is pretty close to circular - it's got a very low eccentricity of around 1/20th. According to Wikipedia, its average orbital radius is about 384,403 km; and its average orbital speed is 1.022 km/s.

Now, we come to our first problem. The moons orbital period around the earth is about 27.3 days long - 2358720 seconds. (This is the orbital lunar month). But because of the earths orbit around the sun, the apparent orbital period is 29.5 days - or 2548800 seconds. (This is the synodic lunar month.). In the month where the moon orbits the earth, the earth moves in its orbit. The direction that the the earth is facing at noon has shifted because of the earths motion in its orbit. So the moon has to go a bit further to appear to have completed a circle.

So we'll compute two different versions of the speed of light according to our new friends - the orbital month, and the synodic month. Using our two different months, we get slightly different answers for how far the moon moves in a month.

In the synodic month, the moon travels about 2,605,000 kilometers. In the orbital month, the moon travels about 2,411,000 kilometers. So, by our friends count, that means that the speed of light predicted by the synodic month is about 28,927,342,080 kilometers/day - or 334,800 kilometers/second.
In an orbital month, the moon travels about 2,411,000 kilometers - for
a predicted light speed of 28,932,000,000 kilometers/day - or about 335,000 km/sec. The actual speed of light is around 299,800 km/sec. So the "Koranic" speed of light is close, but not very close - assuming the best possibly estimate (the synodic), it's off by around 10%. That's actually pretty damned poor - an experimental apparatus built in 1849, using very mediocre measurements of distance, came closer than that.

But that's not the half of the problems with our new friends!

See, the reason that they care about this is because they don't understand relativity. It's like I always say: the worst math is no math. These guys have heard about how relativity says that time slows down under certain circumstances, and from that they conclude that the speed of light must change. But the Koran fixes all that! Because no matter what,
from earth, the speed of light will always be the same: 12,000 lunar orbits/day.

n observer outside gravitational fields measures the speed of light locally (in his location) at 299792.458 km/s but when he looks towards a black hole he sees the speed of light there to be as slow as a few meters/s. At the same time an observer freefalling into that black hole measures the speed of light locally (in his location) at 299792.458 km/s; when he looks towards the black hole he sees the speed of light there much slower; when he looks away from the black hole he sees the speed of light there much faster. When he looks towards outside gravitational fields he sees the speed of light there a zillion km/s. In 1915 (10 years after Special Relativity) Einstein developed another theory called General Relativity that deals with gravitational fields and according to this latest theory the speed of light appears to vary with the intensity of the gravitational field.

This is what happens when you try to use words instead of math. The whole point of relativity and time dilation is that it makes the speed of light be the same everywhere. If I'm moving towards you, the light coming from me towards you is moving at the same speed as the light from a stationary object
the same distance away. No matter what viewpoint you consider, the speed of light is always the same. The speed of light doesn't slow down when someone is falling into a black hole. It stays exactly the same; what relativity does is explain how it can be the same.

What our friends are doing is playing with a poor understanding
of general relativity. It's true that Einstein, when he formulated
light changing in a non-inertial frame of reference is consistent with - but not necessary to - general relativity. Modern interpretations of
relativity, based on the continuing research and extensive experimentation by physicists, conclude that it doesn't.

Where does this confusion come from? There's a nice explanation of it href="http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html">here.
The basic idea is that velocity can only be measured relative to something
that provides a system of coordinates. Velocity is measured in terms of
coordinates applied to space-time. But the coordinates are dependent on the
shape of space-time. So when we're trying to measure coordinates in an area
where space-time is heavily bent by a gravity well, measuring coordinates
becomes tricky - in fact, it becomes ambiguous, because there are
multiple different ways of applying coordinates that are, arguably, correct.
In some of those coordinate systems, used within a reference frame where there is a significant gravitational gradient, you can observe varying speeds of
light.

What we end up with when we work through the math is that for any
freely falling frame of reference, the speed of light in that frame
of reference is the well-known, unchanging speed of light. But if you use a
frame of reference which is in a fixed position relative to a gravity well,
then you can create the appearance of a varying speed of light. But that is
because of the fact that you've created a very artificial frame of reference:
a non-accelerated frame that isn't free-falling into a gravity well. So the
frame of reference is subject to the time-dilation effect of the gravity well,
but not to the acceleration. It's ignoring the warp of space-time
while simultaneously being affected by it.

What does this mean? Well - from the perspective of someone in the
kind of reference frame that creates the appearance of varying lightspeed - that is a fixed frame in a large gravity well - the speed of the moon
revolving around the earth also appears to vary. In other words,
the supposedly corrected Koranic absolute speed of light doesn't
fix the problem that it's supposed to correct. Because, you see, they didn't
do the math.

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Mark Chu-Carroll: 1, Koran: 0

Be fair. It's "Stupid muslim guy: 0, Mark Chu-Carroll: 1". The Koran doesn't make this stupid argument; it's one stupid idiot.

The way he makes up for the discrepancy is by multiplying a factor of cos(phi) where phi is the number of degrees passed by Earth's orbit within a sidereal moon orbit. Supposedly, this factor appears when you remove the earth-moon system from the sun, and you lose the energy from "gravitational twist" (whatever that is). It's terribly unclear to me why one would do this, or why the math should come out to cos(phi).

In any case, it's all a meaningless exercise in numerology.

I wonder though, the moon is slowly getting farther away from the earth, thus it's period of orbit is getting larger. Would this possibly account for the 10% error. I'm not sure how long ago the koran was written, or how fast the moon is moving away from us, but it might be a little weird if they sync up. I'm not defending the koran or anything as I know nothing about it, but it would be weird if it was right, because that could be the earliest prediction of the speed of light.

By Jivemasta (not verified) on 12 Jun 2008 #permalink

as a muslim i am offended by this blasphemy. inshallah your servers are in canada as i feel you are violating my civil rights....

First, a quibble: the moon is also orbiting the sun, that distance should be factored in, not to mention the distance the entire solar system is moving.

But my major objection is to your comment:

"This is what happens when you try to use words instead of math."

For shame! Math is not just symbolic manipulation! Math is about ideas and reasoning. It can be done just as well in words as in symbols. After all, words are just symbols too.

I had a math prof who insisted that all proofs be in complete sentences, with as few symbols as possible. In fact, in the history of mathematical thought, and even in modern research papers, there are a lot more words than symbols.

I wonder though, the moon is slowly getting farther away from the earth, thus it's period of orbit is getting larger. Would this possibly account for the 10% error.

The Koran dates from the 7th century CE. Lunar months didn't get 10% longer in 1300 years.

Just brilliant. I especially loved the part where they identify Dark Matter as Jinns.

revami is probably right saying that you can do words-only-math. However you should only do so if you have a thorough understanding of the topic, which the author obviously didn't. 'Proper' math with equations and stuff might be meant to keep those wackos away from physics ;)

Well, I did drop out of university the first time around before I finished my math minor, but isn't 100 lunar years 1,200 lunar orbits, not 12,000?

Revami wrote:

Math is not just symbolic manipulation! Math is about ideas and reasoning. It can be done just as well in words as in symbols. After all, words are just symbols too.

Yes, but there's a caveat. The symbols we use in maths are chosen to convey ideas as succinctly as possible. We could write "two hundred and ninety-nine million, seven hundred and ninety-two thousand, four hundred and fifty-eight metres per second" out in full ..... but it would ultimately lead to single equations taking up several pages. That way also lie mistakes (if you can't remember the beginning of a sentence by the time you have reached the end .....)

Now, don't get me wrong: redundancy in language can be a wonderful property when dealing with poor-quality communication media. And the fact that a concept can be expressed in more than one way instantly creates an artform. ("My girlfriend smells nice" is certainly not going to win any poetry awards.) But within the limited context of the exact sciences, such redundancy can be more of a hindrance: it is better to have a one-to-one mapping between abstract concepts and the ways of expressing them.

There is a minor movement in the Muslim world - lead by "scholars" with no scientific background and mainly financed by rich Gulf countries - that is trying to fabricate scientific explanations for many things that are mentioned in Koran. I personally do not recall any verse from Koran that states the speed of light. These "scholars" are irritating: instead of focusing on more important issues, they're waisting everyone's time on ridiculous items. Seeking knowledge in an honest and ethical fashion is very encouraged in Islam (according to Koran) ... but this group of scholars is the first one to violate this noble principle.

By AnArabMuslim (not verified) on 13 Jun 2008 #permalink

revami:

Math can be written as words, or as symbols. That's not the distinction that I'm trying to make. You can be completely non-symbolic, and yet be talking math. But math is a highly structured formal language - and that's what I'm trying to get at.

If you want to meaningfully talk about relativity, you can't just say "time gets slower around a black hole", and draw conclusions from that. "Time gets slower around a black hole" is a very imprecise statement. It's absolutely non-mathematical: it omits all of the necessary precision that allows you to meaningfully talk about what relativity says.

If you talk about relativity in just words - that is, the kind of informal discussion that doesn't have any mathematical rigor or precision - then you can't really meaningfully infer anything about what relativity implies. You can, however, also *talk* about relativity - using nothing but english - and do it in a precise way that is mathematically correct and precise. It tends to be very awkward to do that - and that's the reason why we invent symbolic notations and jargon: to make it easier to make the kinds of precise statements that are required by math.

Googling will find many Theophysics web pages on how fast Satan fell from heaven, clearly a branch of Angelology Kinetics, including this, from the Missionary Baptist Resource Center:

==========

The Chief of the Fallen Angels

"Lucifer" is sometimes known as the "light bearer"---very possibly the very high angel who had stewardship over light, energy and power in the old creation. As the "anointed guardian Cherub" in the garden of God (a place in the heavenlies corresponding with the garden in Eden on earth), he may well have had enormous power and influence over the entire angelic host. In the New Testament, Paul warns that he sometimes comes among men disguised as an angel of light (through false teachers and also as a giver of deceptive visions). In fact, he has now become the Prince of Darkness. Lucifer may have been originally God's appointed watch guard over the forces of nature; for example, he could have been in charge of such physical phenomena as the propagation of light. I believe it is quite possible that the observed decrease in velocity of light over time might have begun when Satan fell. {See separate files on the velocity of light]

==========

To add to the woo, that page wanders into thermodynamics:

The universe we now live in has been damaged by the fall of man and the angels---it is now a dying old creation, subject to the degenerative principle known in physics as the "Second Law of Thermodynamics." Biblically speaking this is referred to in Romans,

"I consider that the sufferings of this present time are not worth comparing with the glory that is to be revealed to us. For the creation waits with eager longing for the revealing of the sons of God; for the creation was subjected to futility, not of its own will but by the will of him who subjected it in hope; because the creation itself will be set free from its bondage to decay and obtain the glorious liberty of the children of God. We know that the whole creation has been groaning in travail together until now; and not only the creation, but we ourselves, who have the first fruits of the Spirit, groan inwardly as we wait for adoption as sons, the redemption of our bodies." (8:18-23)

You can see some examples of math done using language from before mathematical symbology was invented in John Derbyshire's Unknown Quantity: A Real and Imaginary History of Algebra. It's incredibly akward and confusing, at least to me. Actually, that's the book that led to me finding this blog, so I'm glad I finally have an excuse to recommend it.

Comparing an experiment from the 19th century to knowledge from the 7th is a little unfair. If they actually did have the speed of light down to within 10%, I'd be pretty darned impressed. But I can't imagine how they could have calculated the Moon's path with any great accuracy, much less timed flashes of light, so I wouldn't take this guy's word for it. I might expect knowledge reveled by an omnipotent deity to be a little more accurate though (if not a whole lot more with the language barrier and all).

Pakistani physicist Pervez Hoodbhoy* wrote a very nice book, "Islam and Science: Religious Orthodoxy and the Battle for Rationality" that talked about just such 'science.' A very nice book. From what I recall, such science was sponsored by the state during the time of Zia-ul-Haq%

*http://en.wikipedia.org/wiki/Pervez_Hoodbhoy
%Zia-ul-Haq: most recently seen in Tom Hank's movie about the cold war

Math is about ideas and reasoning. It can be done just as well in words as in symbols. After all, words are just symbols too.

This might be a math urban myth, but at a topology course the lecturer mentioned an unnamed prominent researcher that evidently claims he can't picture structures worth damn. Presumably he can work with only language and math symbolism.

It was supposed to comfort some of us. :-/

isn't 100 lunar years 1,200 lunar orbits, not 12,000?

Typo, the linked page has it as a 1000 years. Huge number. :-P

By TorbjÃ¶rn Lars… (not verified) on 13 Jun 2008 #permalink

D'oh! "I meant "urban math myth"", he said punsively.

By TorbjÃ¶rn Lars… (not verified) on 13 Jun 2008 #permalink

"urban math myth": math can and should be taught in Ebonics and other ghetto argot so as not to disrespect the culture of the urban student.

I fought that almost every day teaching Math in urban public schools.

I refer you to the wonderful book:

Twice as Less: Black English and the Performance of Black Students in Mathematics and Science, by Eleanor Wilson Orr, Norton, W. W. & Company, Inc., October 1997
# ISBN-13: 9780393317411
# 256pp
# Edition Description: REISSUE

Synopsis

Can Black English stand between black students and success in math and science? In this groundbreaking study, Eleanor Wilson Orr argues that the performance of black students in math and science is crippled not by lack of intelligence or diligence but by linguistic interference. Using student work from an experimental program she helped establish in the District of Columbia, Orr traces specific ways that nonstandard English usage can lead to misunderstanding and misrepresentation in the classroom. This controversial book challenges classroom teachers, school administrators, and citizens in general to rethink their views on how to improve the performance of minority youth in American schools. In a new introduction for this 1997 edition, Orr takes on the latest widespread debate over "Ebonics" and the role Ebonics-based programs might play in American education.

Publishers Weekly:

A veteran high school teacher of science and mathematics offers an unusual approach to the problem of underachievement among minority students. Founder with her husband of the Hawthorne School in the District of Columbia, Orr here describes the results of the school's experimental linguistic program from which her theory is developed: ``Differences between black English vernacular (BEV) and standard English can affect a BEV speaker's concept of certain quantitative relations.'' Observing the functional role of prepositions, conjunctions and relative pronouns in the identification of quantitative ideas, Orr pinpoints misunderstandings that beset students whose first language is nonstandard English. Her belief that BEV is rule-governed and not merely ``bad'' English is supported by data from her students who, for example, confuse ``twice'' and ``half'' or combine ``as'' and ``than'' in their partitive comparisons. The inquiry and explanations are complex, but Orr is generous with illustrations and invites compelling speculation on how the Hawthorne experiment might be replicated by educators seeking to unleash the scientific potential of disadvantaged black students. (August 19)

BTW, the Koran existed before there was a moon or a universe. Muslims believe that the Koran is eternal and has always existed, just as Allah has always existed.

There was a caliph in the 9th century named al-Ma'mun that followed the Mu'tazili branch of Islam, which combined traditional Muslim beliefs with Greek beliefs in rationalism. One of the major tenets of the Mu'tazili was that the Koran was created and was not eternal. They reached this belief using reason and logic, and although the Mu'tazili branch of Islam didn't last very long it coincided with what is now known as the "Golden Age of Islam".

Caliph al-Ma'mun created a great library called the "House of Wisdom" that saved a lot of Greek knowledge (such as Ptolemy's Almagest), and the West learned of both Algebra and Arabic/Hindu numbers from al-KhwÄrizmÄ«, who was one of the scholars in the House of Wisdom. The House of Wisdom had a ton of mathematicians as scholars (they tended to practice astrology). Unfortunately, within a couple generations of the death of al-Ma'mun the orthodox Muslims took over the religion. One of the biggest influences on orthodox thoughts was a Muslim named al-GhazzÄlÄ«, who wrote The Incoherence of the Philosophers, which flat out condemned the rationalism of the Greeks and claimed that Greek science and mathematics would only lead to atheism. The Muslim faith has never been the same since.

You write:
"The claim that relativity proves the Koran is true"
Um, I think you meant:
"They claim that relativity proves the Koran is true"

By Joshua Zucker (not verified) on 13 Jun 2008 #permalink

Interesting... I'm reading a book called Einstein and Religion by Max Jammer (http://www.amazon.com/Einstein-Religion-Max-Jammer/dp/0691006997), a physicist who apparently knew him.
The second part of the book deals with attempts by theologists to use relativity to prove or disprove either the existence of God or various properties of his.
I imagine you could criticize these attempts the same way you could criticize the attempts to prove the validity of the Koran using relativity.

By Nathan Fellman (not verified) on 14 Jun 2008 #permalink

"attempts by theologists to use relativity to prove or disprove either the existence of God or various properties of his." -- Egad. Plus, with the auctioning off of a recent letter written by Einstein, in which he is fairly condescending toward things religious, it's going to be harder for the theological camp (myself included) to "use" Einstein -- for those of us who do such things (myself not included) -- to buttress any position of theology as being true. Moral: try not to lean *too much* on any one person to form one's positions re: theism.

Nice counter-argument to the Koranic-speed-of-light thingy, Marc. Pithy.

Wasn't there some Brazilian physicist who posited a variable speed of light, way back @ the beginning of the universe? I forget his name, now, but have his book on my shelf somewheres.

Skeptic:

Well, that would be interesting had they actually responded to anything that I said. But they didn't. I specifically addressed the problem with their "variable speed of light" nonsense, but they ignored that. The variable speed of light is an apparent phenomenon in an accelerated reference frame viewed as if it weren't accelerated.

I'm not saying relativity is wrong, which is what their "response" claims. I'm saying that one particular interpretation of relativity in a peculiar frame of reference produces strange results until you work the details of that frame of reference into it. But those peculiar results aren't a refutation of the fundamental fact that the speed of light is a constant.

And it doesn't change the fact that their "calculations" of the speed of light from their Koranic citation don't arrive at the correct speed of light. And it doesn't change the fact that a poetic line in the Koran about how angels experience time is a really stupid basis for claiming that the Koran tells you the precise speed of light.

Ah, well both of you seem to be stating facts that contradict each other.
But I shall drop it since I presently know nothing about relativity.

In round numbers, the Moon is a quarter million miles from the earth, and orbits in a Moonth, which, in round numbers is 12 times a year. This distance is 12 * 1/4 * 2 * pi (which is 3 in the Bible) = 18 million miles in a year.

In round numbers, the Earth (and the Moon) orbit 100 million miles from the Sun in a year. So the distance is 100 * 2 * pi (still 3, because it's a constant) = 600 million miles.

So the ratio of speed of the Moon around the Sun to the speed around the Earth is 600 / 18. We're too lazy to get out a calculator, so this is 600 / 20 = 30x. This isn't 10 percent even in very round numbers.

Clearly, the angels are going much faster than the speed of light. But that's OK. It's always to place limits on what God can do. You're a limited mortal - it's not your job. So, if you can accept on faith (which is belief in something without evidence - a synonym is "delusion") that God made herself, then her angels can do anything she wishes. And she can make a rock so large that not even she can lift it. For example, what, exactly does it mean to "lift the Universe"? Nothing. What would she stand on? So granted that she made the Universe, it's something she can't lift. And BTW, the egg came first, well before birds, much less chickens. So there!

This math also explains why one side of the Moon you see doesn't seem to have more craters than the other side. Even at New Moon, when the speed of the Moon around the Earth is more or less subtracted from the speed of the Moon around the Sun, the Moon is moving more or less the same speed around the Sun. It's going backwards. And so, it plows into solar system debris on all sides more or less at equal speeds. This isn't true for moons around Saturn, but you can do that math, right?

and orbits in a Moonth, which, in round numbers is 12 times a year.

in "round numbers" it is closer to 13 times a year.

Even at New Moon, when the speed of the Moon around the Earth is more or less subtracted from the speed of the Moon around the Sun, the Moon is moving more or less the same speed around the Sun. It's going backwards.

If you actually plot the moon's motion around the sun, you will find that nowhere does it have any retrograde motion.

And this does not explain why there are fewer craters on the nearside of the moon. It has something to do with that big old obstruction in the way called the Earth.

Einstein theorized that the speed of light in a gravitational field is actually not a constant, but rather a variable depending upon the reference frame of the observer.

computer science guy wrote:
"What our friends are doing is playing with a poor understanding of general relativity. It's true that Einstein, when he formulated general relativity, was uncertain about this. The idea of the speed of light changing in a non-inertial frame of reference is consistent with - but not necessary to - general relativity. Modern interpretations of relativity, based on the continuing research and extensive experimentation by physicists, conclude that it doesn't"

False. All parts of General Relativity were individually verified. You can find the tests for General Relativity in the same sources that I provided. Actually modern cosmology is based on General Relativity. Today we are sure that an observer outside gravitational fields measures the speed of light locally (in his location) at 299792.458 km/s but when he looks towards a black hole he sees the speed of light there to be as slow as a few meters/s. At the same time an observer freefalling into that black hole measures the speed of light locally (in his location) at 299792.458 km/s but when he looks towards outside gravitational fields he sees the speed of light there a zillion km/s. This is confirmed; End of Story.

As for the 10% difference, we can use classical orbital mechanics to calculate the speed of light with zero% difference.

Supposing you are right with your reasoning, isn't it still a big mystery how an illiterate man in a desert even got close to that number 1400 years ago?

Nauman : "isn't it still a big mystery how an illiterate man in a desert even got close to that number 1400 years ago?"

No.

Furthermore, the world was filled with nicely dressed highly literate people 1400 years ago who could calculate much more precise numbers. So, which tradition do YOU want to follow?

Not that I have a problem with tachyonic angels. I have published stories about Djinns, from the pre-Koranic traditions of the Arabian subcontinent) and about Relativity applied to the afterlife. But I can always distinguish between when I'm writing Science/Math or writing fiction/poetry.

Forget about exact numbers for lunar orbits and so forth. It's much easier than that.

-

[Quran 70.4] The angels and the Spirit ascend to Him in a day, the measure of which is fifty thousand years.

That's supposedly an accurate prediction of the speed of light by "an illiterate man who lived 1400 years ago".

Proof:

v = c*sqrt (1-1/(50000*12*27.321661)^2) = .9999999999999981*c

Hmm. The big "miracle" is that the Quran says 50'000 and the result of the formula is very close to the speed of light.

If you enter a bigger number than 50k, you get a more accurate prediction.

If you enter a smaller number than 50k, you STILL get a very exact prediction:

v = c*sqrt (1-1/(1*12*27.321661)^2) = .999995348*c

!

That's still a pretty damn good approximation of c, isn't it? - Regardless of the difference of 49'999 days in the prediction!

How come?

Well... It's the mathematics of the formula that leads to the end result that creates the illusion of a proof.

=> This so called "accurate prediction of the Quran" is a scam. The way this "proof" is set up, any number >1 works to win over the gullible.

red25,

from your
"v = c*sqrt (1-1/(50000*12*27.321661)^2) = .999995348*c"

why is that 50000*12*27.321661 is taken as the lorentz factor?

having put any numbers as a lorentz factor as long as it's bigger or equal 3 for the lorentz factor would get c value to be > 90% from it's original value.

what is it that the lorentz factor has anything to do with the lunar/orbit framework proposed/argued for by the guy in the first place?

By Anonymous (not verified) on 31 Mar 2010 #permalink

red25,

http://en.wikipedia.org/wiki/Lorentz_factor

to be exact, as long as the factor is bigger than or equal to 2.294, (refer to the link above) u can always get 90% of the original c values.

perhaps u wanna explain further how u disprove the setup as a scam.

By Anonymous (not verified) on 31 Mar 2010 #permalink