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Warp Technology Development 
Warp Technology DevelopmentIn the 2290s, transwarp drive was expected to be the propulsion breakthrough which would lead to significantly increased warp speeds. Prior to transwarp, the highest speed attained by a starship in service was approximately 1750c while the projected transwarp velocity was on the order of 2800c. In trials, the Excelsior obtained a speed of only 1900c which was a tremendous disappointment and a major embarrassment for Star Fleet given the resources expended in transwarp development. What was not known at the time is that the Excelsior and other ships were experiencing the phenomenon of 'quantum compressibility'.
The 'failure' of transwarp drive caused Federation scientists and engineers to begin a major reevaluation of the warp theory set down by Cochrane and others in the 21st century. Classical warp theory held that the warp field effect caused the starship to travel on a cubic integrated spacetime curve where the speed in multiples of lightspeed is the warp factor cubed – speed = w^{E} where E=3 (Warp factor 5 would be equal to 125c). Dr. Katherine Ballentine showed in 2310 that an optimum warp field would cause the ship to move on a polynomial spacetime curve where the warp factor exponent becomes 10/3 (3.333...) rather than 3 – speed = w^{E} where E=10/3 (Warp 5 would be 213.75c). Early single lobe warp field engines could not bring ships to this optimum while later multilobe warp field engines of the late 23rd century could indeed approach the optimum.
Classical warp theory also held that there was no upper limit to the speeds obtainable given available warp field energy densities. An associate of Dr. Ballentine, Dr. Phillip Eugene, showed that quantum effects due to Planck time considerations, later popularly called 'quantum compressibility', were present at all warp speeds. As a ship increases warp speed and the cycle time of warp field approaches the Planck time, the expanding warp field layers begin to compress and coalesce into a single wave front with a reduction in the subspace shearing forces. The required energy to maintain the warp field increases dramatically thereby limiting the attainable warp speed.
Dr. Eugene's extensions to warp theory result in a correction to the warp factor exponent which account for quantum compressibility considerations – speed = w^{E} where E=10/3+(1/10w)*10)^{5/3} (Warp 5 would be equal to 214.25c). The difference in predicted speed for the uncorrected Ballentine equation and corrected Eugene equation is minimal at lower warp speeds, reaching only 4% at warp 9. However under the Eugene equation, above warp 9 the apparent speed and the required warp energy increase dramatically until reaching infinity at warp 10  hence a mathematical limit speed of warp 10. This limit is now known as Eugene's Law.
In 2321, Star Fleet redefined the standard warp factor curve to match the extension of warp theory by Dr. Eugene. Subsequent Federation starships would penetrate farther into the speed range beyond warp 9 by application of higher warp power densities and serious attention to warpdynamic shaping to optimize the ship's SearsHack mass distribution.
(Theory by Steve Justice)

