The Historical Development of the Warp Drive

via GIPHY

Preface

This is another of my "Covid-Projects," consequent to my "Star Trek History of the Twenty-First Century" project (which is drafted but still needs a lot of work). Basically, I wanted to work out an overall rationalization of the development of the Warp Drive that best fits what has been shown on screen and explains certain oddities that manifested themselves along the way. It varies considerably from a lot of fan-created Treknological assumptions through the years as well as semi-official resources published  and statements made along the way, even by producers and technical advisers to the various series and films. 

In the process of writing the "History of the Twenty-First Century," this kind of went on the back burner until I came upon an article on the blog Occasional Enthusiast entitled "Alternative Star Trek Warp Speed Scale and Related Equations" (2016) [LINK] which takes a very different approach but has some really cool ideas. My comment on that post led to a very helpful exchange with the author (at least, helpful to me) and inspired me to pull this back out, polish it a bit, and go ahead and post it.

Take it for what it is -- one fan's interpretation. And remember that, "Dammit, Jim, I'm an historian, not a mathematician!"

The Facts As We Know Them From the Episodes and Movies

Although other spacefaring peoples of the Star Trek universe, such as the Vulcans, had previously possessed the Warp Drive (or something closely enough akin to it that no distinction has ever been made on screen), Earth’s first Warp Drive was invented by Zefram Cochrane and his associate Lily Sloane in the mid twenty-first century. Launching in a converted ICBM which he had dubbed Phoenix from an abandoned missile silo near Bozeman, Montana, on Thursday 05 April 2063, Cochrane and a support crew of two cleared Earth’s atmosphere, engaged the prototype warp engine, “broke the warp barrier,” and dropped back into normal space approximately fifty-five seconds and somewhat more than that many light-seconds away from Earth. Whatever exactly “broke the warp barrier” means in this specific context (and that will be discussed later), for now simply understand that Phoenix had exceeded the speed of light even if only for a short time. Cochrane and his crew returned safely to their point of origin and were there several hours later when an alien spacecraft descended on Bozeman. The Vulcans had long been observing Humankind from above but had thus far adhered to their policy of not initiating contact with any primitive people, even spacefaring ones such as Humanity had been for a century, until that people had achieved faster-than-light travel. The Vulcan survey vessel T’Plana-Hoth was in the solar system when it detected Phoenix surpassing light-speed and thus tracked her flight to her landing. And so Humanity experienced the first of many “First Contacts” to come.[1]

What then occurred has not been definitively revealed on screen. The broad strokes are that over the next few decades, under a sometimes-irksome oversight by the oh-so-superior Vulcans, Humanity slowly moved out into the local interstellar neighborhood. By the early twenty-second century (2119) there had been founded near Bozeman, Montana, a “Warp Five Research Institute” (popularly known simply as “the Warp Five Complex”) under the direction of Dr. Henry Archer. According to the “Warp Factor scale” initially formulated by Cochrane himself, Warp Factor 1 was the speed of light, while Warp Factor 2 was eight times the speed of light. Even that latter velocity not having been attained as of 2119, Warp Factor 5 – 125 times the speed of light – seemed a very long way off. Although the Vulcans clearly had the capability, they refused to share it, deeming humanity not “mature” enough as a species to be encouraged access to the wider universe unfettered by relatively slow lower warp speeds. But Warp 2 – or the “Warp 2 Barrier” – was “broken” in 2143 by test pilot Cmdr. A. G. Robinson flying the first NX-prototype test vehicle. In one minute and twenty-seven seconds of increasingly rough acceleration past Warp 2 from warp insertion near Earth before experiencing a catastrophic warp field collapse at Warp 2.2. The NX-Alpha broke up near Jupiter.[2] Robinson survived and became the first Human to eject in an escape pod at faster-than-light velocity. The ever-helpful Vulcans promptly urged Earth’s Starfleet to put its warp trials on hold indefinitely – but a few days later test pilot Cmdr. Jonathan Archer (long-deceased Henry Archer’s son) joined Robinson in an illicit test flight of Starfleet’s only other NX-prototype, the NX-Beta, employing a new intermix formulation overseen by engineer Lt. Charles “Trip” Tucker III – and attained a steady Warp 2.5 before returning safely to Earth. After only a year and a half grounding and review of the program, test flights resumed with new and improved NX-prototypes. Test pilot Cmdr. Jacques Duvall attained Warp 3 in 2145 in the NX-Delta.[3] It is unknown exactly when Warp 4 was broken (although the first Earth starship capable of that velocity was Franklin[4]), but Earth’s first Warp-Five Starship, Enterprise (NX-01), launched on its first trip outside the Solar System under the command of Capt. Jonathan Archer on Friday, 16 April 2151, three weeks ahead of schedule.[5] At that time Enterprise had not yet made Warp 5 a reality – tests inside the Solar System had maxed at Warp 4.5, but continued improvements “on the fly” by Chief Engineer Commander “Trip” Tucker allowed the ship to break the Warp 5 barrier for a short time on Friday 09 February 2152.[6] Although speeds in excess of Warp 5 would remain a rarity, she ultimately set her own record at Warp 5.2 (141c) in November 2154.[7]

We have no further specifics on when Warp Factors 6, 7, and 8 were attained. However, we know that a century later, in 2254, Lt. José Tyler of the United Federation of Planets Starship USS Enterprise (NCC-1701), enthused to supposed survivors of the crashed scientific survey ship SS Columbia – stranded on the planet Talos IV for the past eighteen years – that “You won’t believe how fast you can get back – the time barrier’s been broken!” During that same period, the term “hyperdrive” is used for Enterprise’s translight propulsion system, along with the term “Time-Warp Factor” – specifically “Time-Warp Factor 7” is mentioned.[8] Columbia was lost in 2236; Enterprise was commissioned in 2245 after her shakedown cruise, and was probably being constructed through the early 2240s. Whatever advance Tyler means had to occur between 2236 and 2245, probably no later than 2240; let’s just say “circa 2240.”

By a decade later, that same USS Enterprise (NCC-1701) had a cruising velocity of Warp 6 and an emergency speed of Warp 8, with brief bursts up to Warp 9. Extraordinary circumstances on a couple of occasions during Capt. James T. Kirk’s first Five-Year Mission (2266-2270, inclusive) saw incredible speeds of up to Warp 14.1 attained.[9]

By the 2280s, experiments with a new “Transwarp Drive” had proven successful enough in trials to warrant the prototype of the next generation starship, USS Excelsior (NX-2000), to be constructed as a test bed. Much to the evident chagrin of her captain, Lawrence Edward Styles, Excelsior failed her first test when USS Enterprise was stolen from the immediately adjacent berth in Spacedock by her own former officers in 2285. Having just, earlier in the day, exulted to Cmdr. Montgomery Scott, formerly Chief Engineer of Enterprise, that they would be breaking all of Enterprise’s speed records on the morrow,[10] Styles warned Kirk and company that they stood no chance of outrunning Excelsior and that if they did not stand down their careers in Starfleet would be ended – then he then had to watch Enterprise disappear into warp as his own ship ground to a halt.[11] Later investigation revealed the truth – that the Transwarp failure was due to sabotage on the part of Scott before he went to join his fellow officers on Enterprise. Nevertheless, Transwarp would never be mentioned on-screen again,[12] and when Excelsior was next seen, she bore the hull number NCC-2000 and flew under the command of Capt. Hikaru Sulu.[13] The terminology remained simply “Warp Drive” and “Warp” or “Warp Factor” x.

By three-quarters of a century later, in the 2360s and beyond, while terminology continued to reflect “Warp Drive,” Warp Factors beyond 10 were rare to non-existent. Eventually (and I’m uncertain exactly when, but believe it was in an episode of Star Trek: Voyager) Warp 10 was asserted on-screen to be a theoretically unattainable “infinite velocity,” at which an object would occupy every point in the universe simultaneously – although the plot of the episode concerned one of the characters doing exactly that in a modified shuttle-pod.[14] Despite several other instances when supposed travel “faster than Warp 10” was reported in other episodes of the “Next Generation Era,” such as when USS Enterprise (NCC-1701-D) jumped almost instantaneously from our own galaxy to one of the neighboring clusters of the M33 Triangulum Galaxy, 2.7 million light-years distant[15]  -- and the depiction of one future timeline alternate to 2370 in which  Capt. Jean-Luc Picard witnessed a dramatically[16] upgraded Enterprise D capable of Warp 13![17] – it was clear that something radical had changed and that the Warp Factor Scale was now meant to go from zero to ten.

These are the main data of which I am aware based on the various episodes and movies of all the Star Trek series to date. Notice that I am not considering the recent Star Trek: Discovery and its “Spore Drive.” That is not Warp Drive.[18] But it does raise one point which is worthy of consideration. “Warp Drive” is not the only means of faster-than-light travel in the Star Trek Universe. Various other technologies mentioned as allowing the “universal speed limit” to be flouted include tachyon-based Bajoran lightships[19] and Borg “transwarp conduits,”[20] graviton-based propulsion,[21] wormholes such as that in the Bajoran System,[22] and many other forms of “spacetime manipulation.”[23] And those do not include such apparently instantaneous means of traveling through space and time as the Guardian of Forever.[24] I am not concerned with any of those here, however – just the plain old Warp Drive, regarding which I think it obvious that the history has not been fully explained – or at least not to my satisfaction. The purpose of this essay is to do just that, and propose a somewhat different understanding of what and how the “Warp Scale” alluded to earlier as having been devised by Zefram Cochrane himself really works and how it relates to the progression of Warp Drive technology through at least two major advances following its invention in the mid-twenty-first century.

The Problem As I See It

Examining the facts as we know them as narrated above, I think the first of those propositions is clear. Summarizing it into a brief timeline will perhaps highlight the two major advances that have occurred:

• 2063: Phoenix attains Warp 1, e.g., the “warp barrier.”
• 2143: NX-Alpha exceeds Warp 2 briefly and breaks up at Warp 2.2. A few days later, using an adjusted intermix formula, NX-Beta sustains Warp 2.5 for a time and returns safely to Earth.
• 2145: NX-Delta attains Warp 3.
• Ca. 2150?: Franklin attains Warp 4.
• 2151: Enterprise (NX-01) sustains Warp 4.5.
• 2152: Enterprise (NX-01) briefly attains Warp 5.
• 2154: Enterprise (NX-01) sets her own record at Warp 5.2.
• Ca. 2240: Something called “the time barrier” is broken.
• 2254: USS Enterprise (NCC-1701) travels faster than light employing a “hyperdrive” or “time-warp drive” at an attainable “Time Warp Factor” of at least 7.
• Ca. 2255: An odd visual and audio effect associated with the “time-warp drive” is corrected.
• 2260s: Enterprise (NCC-1701) (and her sister ships) cruise at Warp 6, can endure Warps 8 and 9, and under extraordinary circumstances attain up to Warp 14.1.
• 2280s: Experiments in Transwarp Drive with results unknown – except that Starfleet would not have built a totally new, bigger starship as a test bed ready for launch in 2285 had there not been plenty of successful tests and it being essentially a proven technology. Perhaps there were still bugs to work out, but the single spectacular failure that is the only thing we truly know about it must be remembered to have resulted from sabotage, not failure of concept, design, or construction.
• 2290s ff.: Terminology has reverted to “Warp Drive” and “Warp Factor” or simple “Warp” as an expression of speed.
• 2360s ff.: Although the terminology remains the same – “Warp Drive,” Warp [Factor]” – suddenly Warp 10 is an unattainable (or at least rarely attainable) “infinite velocity.”

I think it is immediately apparent that there are at least three general periods in the history of the Warp Drive from its invention in 2063 to its latest canonical presentation, in the 2020 series, Star Trek: Picard, set in 2399. But there are other considerations that must be examined before discussing those.

What Do We Mean By “Warp Drive”?, and What is the “Warp Scale”?

It will be easier to step more explicitly out of the Star Trek universe, I think.

First off, “Warp Drive” is a term introduced during the early days of the original Star Trek series. The common understanding that eventually developed was that it depended on “warping” space in a controlled fashion to provide propulsion without the relativistic effects of simply attempting to accelerate to and beyond the speed of light – relativistic effects that would, in fact, render “beyond the speed of light” impossible because of time dilation, dimensional contraction, and energy and mass exponentiation as any physical object approaches the speed of light. The theory depends on the idea that space itself is not subject to those concerns, and that if one can therefore warp space in such a way that a given spatial domain moves relatively faster than light as compared to another, one can ride along with it and effectively be moving faster than light oneself. It is not as out there as it sounds. Real-world physicists have made advances toward working out the science involved, most famously Miguel Alcubierre of the Max Planck Institute for Gravitational Physics at the National Autonomous University of Mexico and NASA engineer Harold "Sonny" White. Experiments proceed, although it is cautioned that the science and engineering are both in their infancy. [LINK]

Regarding the “Warp Scale,” as far back as I can remember, a simple cubic exponentiation was proposed to explain how fast the Starship Enterprise might be traveling in terms of the speed of light (c):

Table 1: The Original Warp Scale

Warp Factor	Multiple of the Speed of Light
1	13 = 1 c
2	23 = 8 c
3	33 = 27 c
4	43 = 64 c
5	53 = 125 c
6	63 = 216 c
7	73 = 343 c
8	83 = 512 c

I do not know if this scale was ever confirmed as such on-screen, but in noted science fiction author James Blish’s prose adaptation of the episode “Tomorrow is Yesterday,” Warp 4 is stated to be 64c. “The Enterprise, on a rare trip back toward the Sol sector and Earth, hit the black star traveling at warp factor four – sixty-four times the speed of light.”[25] It tracks, and shows that Blish was aware of the scale. It might seem incredibly fast, but there does not seem to be a fundamental problem with Warp 14.1 being 2,803 times the speed of light. There were other instances of warp speeds beyond Warp Factor 10, even reference in one of The Animated Series episodes to a ship clocking in at a truly incredible Warp 36 (or 46,656c)![26] Spock is avowedly incredulous: “This object is traveling at a rate theoretically impossible for matter to achieve. More specifically […] it is moving at a speed on the order of warp-thirty-six.”[27]

But why is it “theoretically impossible”? – It is simple math.

Nevertheless, by the late 1980s when The Next Generation premiered, viewers were suddenly presented with a practical “warp speed limit” of Warp 10. It was immediately apparent even in the first episode, “Encounter at Farpoint” (1987), when Enterprise (NCC-1701-D) tried to outrun the mysterious ball of energy that apparently was the godlike trans-dimensional being calling itself “Q.” Was any other “old-Trek” fan wondering why Enterprise was going so slow? Warp 9, Warp 9.2, Warp 9.3…. This is a century after Kirk and company; surely the speeds the “original” Enterprise attained in extraordinary circumstances (Warps 10 and 11, even 12) would be ordinary by now! But noooo … this new Enterprise crawled to a maximum of 9.7 in that chase. Say what? … Eventually (as I understand it, in Star Trek: Voyager, specifically in “Threshold”) Warp 10 was stated to be an unattainable “infinite velocity.” Say what?

The “what” is that by the late 1980s Star Trek creator and executive producer Gene Roddenberry wanted to prevent the rampant “warp factor inflation” that had occurred late in the Original Series by arbitrarily imposing a Warp 10 speed limit – for dramatic purposes. That is a perfectly fine “real world” explanation, even if I do not happen to agree with it. Really – Is constantly adding nines to the decimal (Warp 9.9, 9.99, 9.999, …) truly any better than adding integral numbers (Warp 9, 10, 11, 12, …)? I would say it really diminishes the sense of radically increasing speed. Was there ever an “in-universe” explanation? I do not recall there ever being one, at least on screen. I don’t think there was even a “quasi-in-universe” explanation beyond reference to “Gene [Roddenberry]’s recalibration” and “Eugene’s Limit” (Warp 10) in the Star Trek: The Next Generation Technical Manual by Rick Sternbach and Michael Okuda (1991) which (pp. 54 ff.) gives all kind of gobbledegook exposition about Zefram Cochrane’s “continuum distortion propulsion” (CDP, only informally known as the “Warp Drive”) engine, which in its first iteration allowed only a “straddling” of exactly the speed of light for alternating instants no longer than Planck time, 1.3 x 10^-43 seconds and consequently a four-year voyage even to the nearest star, with the subsequent development of “asymmetrical peristaltic field manipulation” (APFM) allowing “layering” of field layers imparting velocities beyond light speed dependent on the number of layers per unit time … and so forth. You get the picture.[28] As meaningless as all that is, it could work as an explanation for Original Series Warp Drive as well as Next Gen, no problem.[29] But when Sternbach and Okuda progressed to “Warp Power Measurement” they set forth a scale that is different from that previously published in Making and well-known among fans:

Table 2: The Original and Next Generation Warp Scales

Warp Factor	Original Series Scale (extended) Multiple of the Speed of Light	Next Generation Scale Multiple of the Speed of Light
1	13 = 1 x the speed of light (c)	1 c
2	23 = 8 c	10 c
3	33 = 27 c	39 c
4	43 = 64 c	102 c
5	53 = 125 c	214 c
6	63 = 216 c	392 c
7	73 = 343 c	656 c
8	83 = 512 c	1024 c
9	93 = 729 c	1516 c
10	103 = 1,000 c	∞

Okay… What is the new “Warp Equation”? Hint – there really is not one that fits Warp Factors from 1 to 10 inclusive. Actually, there is a modification of the old formula that will work for 1 to 9. Instead of the exponent being a factor of 3, it is the endlessly repeating decimal resulting from the fraction 10/3 = 3.3333….

Table 3: The Exponential 10/3 Warp Scale

W Warp Factor	X = W(10/3)
1	1 c
2	10.08 c
3	38.94 c
4	101.59 c
5	213.75 c
6	392.50 c
7	656.14 c
8	1024 c
9	1516.38 c
10	2154.43 c

--- rounded to the nearest integer. But that scale does not yield infinity for Warp 10, which seems mathematically inconsistent – because it is.

I don’t like mathematical inconsistency.

There is an acknowledgment of the problem in a commentary paragraph on TNG Tech Manual p. 56. Reconciling two imperatives was “pretty complicated” – Roddenberry’s new dictate of a Warp 10 limit conflicted with the reasonable expectations of a viewership that the twenty-fourth-century Enterprise-D be significantly faster than her twenty-third-century namesake that had attained as high as Warp 14.1. Sternbach and Okuda’s “solution was to redraw the warp curve so that the exponent of the warp factor increases gradually, then sharply as you approach Warp 10. At Warp 10, the exponent (and the speed) would be infinite, so you could never reach this value. (Mike [Okuda] used an Excel spreadsheet to calculate the speeds and times.)” Essentially, however, he freehanded it past Warp 9.

(As to an acknowledgment of the difference from the Original Series, that would only come in a somewhat exasperated commentary much later in the chapter of the TNG Tech Manual, p. 62: “Oh, very well: The current warp values are presumably much faster than those achieved by the original Enterprise in the first series, but the “old” and the “new” Warp 1 are the same, the speed of light. The ‘old’ Warp 6 is about Warp 5 on the new scale. The (then) amazing speed of Warp 14.1, achieved by the first Enterprise under extreme duress in “Is There in Truth No Beauty?” [sic – it was actually in “That Which Survives”] now works out to around Warp 9.7, which the new ship achieved while fleeing Q during “Encounter at Farpoint.” That they had presumably been bombarded with questions about this shows in that initial “Oh, very well”….)

Nevertheless, the long and short is that there is now no way to express the full continuum of Warp Factors from zero to maximum velocity by a consistent mathematical formulation – and that has always bothered me. Ultimately, “Eugene’s Limit” created a dramatic quandary for the creators that, to my mind, took away from the overall effect. Moving at Warp 9.96, Capt. Picard commands, “Increase speed to Warp 9.97 – Engage!” And I yawn…. – Moving at Warp 13, he commands, “Increase speed to Warp 14 – Engage!” – now that has a greater impact.

One problem Sternbach and Okuda admitted, however, was that while one could devise an appropriate mathematical equation approaching asymptotically to infinity as you near Warp 10, the result would skew lower warp speeds to incredibly greater velocities compared to their Original Series counterparts. As they envisioned it, the previously fast but not incredible Warp 9 would now make the opposite end of the galaxy attainable and most previously visited planets reachable in a matter of days if not hours.

Of course, if they had owned up to the change on-screen by simply admitting that this was a new generation of “Warp Drive” making the old warp scale obsolete, those viewers who cared would have accepted it, and the creators could have continued using numbers like Warp 7 and 8 to mean “really, really fast” just like in the old days. Moreover, there is a way to construct a warp scale curve keeping up to Warp 9 in a reasonable range even with 10 being infinity, as long as the viewer is periodically reminded that, e.g., Warp 7 is the maximum sustainable velocity, Warp 8 is sustainable only for a matter of hours, and Warp 9 will quickly burn out the warp coils and leave you stranded in deep space.

But no, they didn’t do that, and left it to schmucks like me to try to come up with a workable theory and history of the Warp Drive that makes consistent sense in all eras:

• One that does not depend on arbitrarily “recalibrating” the Warp Scale between the Original Series era and the Next Generation era without good reason. There must have been some kind of fundamental change in warp technology to warrant such.
• One that provides some kind of predictable, calculable system or systems to figure effective velocities based on stated Warp Factors.
• One that accounts for occasional shifts in terminology where those shifts have meaning and yet reasonably revert to the older terminology while implicitly meaning something slightly different in the minds of the characters.

My Propositions

Returning to the timeline summarized in the section entitled “The Problem As I See It,” I propose that there are three distinct eras in human warp technology, three different generations of warp drive based on fundamentally similar and yet substantially different theories:

1. First Generation: The Warp Drive, 2063-ca. 2240. The fundamental technology was that devised by Zefram Cochrane and Dr. Lily Sloane and refined by a series of scientists and engineers such as Dr. Henry Archer and Cmdr. Trip Tucker as well as many others whose names are unknown to us. The propulsion system was called the “Warp Drive,” and velocities were expressed in units called “Warp Factors,” or simply “Warp” for short.
2. Second-Generation: The Time-Warp Hyperdrive, ca. 2240-ca. 2285. Some advance allowed something called the “time barrier” to be broken and resulted in speeds that would have been amazing only a few years before its advent. After a brief period when the terms “Hyperdrive” and “Time-Warp Factor” were employed, the older terms fell back into use among any but the most pedantic, being simpler, widely understood, and describing technology and effects that were, for most people, indistinguishable from what had prevailed before.
3. Third-generation: The Transwarp Drive, ca. 2285 ff. Yet another advance resulted in substantially faster velocities and even a “recalibration” of the open-ended warp scale from one beginning at zero and progressing steadily with no apparent ceiling to one beginning at zero but divided into ten units where the tenth exponentiates to infinity. And yet, after a short time, the prevailing terminology fell back to that of the previous generation – again, it was simpler, widely understood, and described technology and effects that were, for most people, indistinguishable from what had prevailed before. (Yes, I know that in response to fans’ speculation that this was exactly the case, the producers of The Next Generation rejected that notion, proclaiming that the Transwarp Drive was a failed technology, as shown in Star Trek III: The Search For Spock. But that “failure” was, as I covered previously, due to an act of sabotage. Pouring water in a gas tank might ruin a particular internal combustion engine, but it does not invalidate the technology itself. And such a refitting from Second-Generation to Third-Generation Warp Drives in the late twenty-third century is a better explanation in-universe for the Warp Scale Recalibration than “The Great Bird of the Galaxy Says It Must Be So.”)

Corollary Questions and Propositions

One question is begged in just about any discussion of the Warp Drive and Warp Factor system to describe velocity. There are others that are just as implicit:

1. Why? – Why are those terms used? Why not just use a more straightforward expression of how “times the speed of light” or how many “Lights” the ship is travelling? Especially if the latter shorthand is used, in what way is “Warp 2” a better expression of speed than “eight Lights”?
2. Another such question: Why are Warp Factors in any age generally (although not exclusively) expressed as integers? There is a definite preponderance of whole number Warp Factors – “Warp Factor 3, Mr. Sulu!” – not “Warp Factor three point six one five, Mr. Sulu!” Why do starships customarily travel at those integral Warp Factors when there is ample evidence that the scale is in fact continuous, e.g., Warp 2, Warp 2.1, Warp 2.2, Warp 2.3, etc.
3. Why, in addition to the easily explicable “Warp [1] barrier” of the speed of light – a truly unattainable velocity according to Einstein’s Theory of Relativity (even if there seems to be no mathematical reason why something could not travel faster than the speed of light,[30] getting to such a velocity would require passing through the speed of light, which really does appear to be an impassable barrier. You cannot get there from here) – is there a “Warp 2 Barrier,” a “Warp 3 Barrier,” a “Warp 5 Barrier,” and presumably other such barriers at the integral Warp Factors?
4. What exactly was the “Time Barrier” something to be “broken,” apparently ca. 2240?

The original answer to the first question is basically that there is a virtue in vagueness from a storytelling standpoint. I’d say that Roddenberry perceived correctly, even though he had no inkling of the magnitude of the cultural phenomenon he was creating, and even in the days before the Internet allowed virtually instantaneous feedback, that there were those among the audience that were going to be soaking in every nugget of information provided and point out every little inconsistency. “Eight times the speed of light” is exact; “Warp 2,” at least in the beginning, was not. “Hey, wait a minute, there’s no way Enterprise can travel from A to B in C time at just eight times the speed of light! It’s going to take 28 weeks, three days, seventeen hours, 52 minutes and 48 seconds!” (That’s from Earth to Alpha Centauri, by the way.) One might also consider how cumbersome expressing higher velocities could get in terms of “Lights” – “Keptin, enemy wessel is moving at four-hundred-tree Lights!” – “Bring us to four-hundred-fifty Lights, Mister Chekov! Don’t let them get away!” (Remember, we speak in “word numbers,” not “symbol-numbers”.) So, “Warp” is both simpler and more concise, in addition to giving a sense that this they are based on some as-yet-undeveloped technology.

The next two are really related, and depend on what the “Warp Barriers” are – which is actually fairly easily explainable based on something established in the Star Trek: The Next Generation Technical Manual for what I am considering the “Third Generation Warp Drive” but which works as well for the others as well. There is something intrinsic in the “Warp Factors.” As laid out on pages 55 and 56, including Figure 5.1.1, “Warp speed/power graph” (which, incidentally, appeared on-screen in an episode of Enterprise[31] and so was, indeed, applicable to that era as well). Quoting from the top of p. 56:

The amount of power required to maintain a given warp factor is a function of the cochrane value of the warp field. However, the energy required to initially establish the field is much greater, and is called the peak transitional threshold. Once that threshold has been crossed, the amount of power required to maintain a given warp factor is lessened. ….

Warp fields [i.e., speeds] exceeding a given warp factor, but lacking the energy to cross the threshold to the next higher level, are called fractional warp factors. Travel at a given fractional warp factor can be significantly faster than travel at the next lower integral warp, but for extended travel, it is often more energy-efficient to simply increase to the next higher integral warp factor.

Putting it another way, it takes more energy to approach and surpass each integral Warp Factor that it does to maintain a lower fractional Warp Factor beyond it – up to a point. Then it becomes more energy efficient to “jump the threshold” past the next Warp Factor and end up traveling substantially faster for the same energy consumption. In other words, energy usage increases as you approach Warp 1, spikes,[32] then dramatically falls off past Warp 1 before starting to increase as you approach Warp 2, spikes, then falls off, rinse and repeat for each Warp Factor. Since Warp 1 is c, the speed of light, that makes sense (in fact, theoretically it would take infinite energy to attain it, hence the “continuum distortion propulsion” theory of oscillating presence straddling precisely the speed of light that I made fun of above), such a “spike” (of infinite magnitude) makes sense – and also that the integral Warp Factors would bear some relationship to the speed of light.

Figure: Warp Power Consumption Scale 

Note that although this graph goes up to Warp 14.5, the highest Warp Factor known to have been attained and survived by a starship is Warp 14.1. [Okuda's original: LINK]

Recall that in the Enterprise episode “First Flight,” the key to successfully surpassing Warp 2 and settling into sustained flight at a “fractional warp factor” beyond that was Cmdr. Tucker’s revised “intermix formulation” – perhaps a more efficient fusion of matter and antimatter to provide the necessary energy to the outboard warp nacelles to establish and maintain the required warp field strength. Other factors were undoubtedly involved, but that one was explicitly mentioned, and I see no reason to doubt that similar refinements were involved in “breaking” each successive warp barrier.

I think that basically answers questions two and three – and suggests an answer to the fourth.

What is the relationship of each integral Warp Factor to the speed of light? I think the original answer provided is fundamental and makes sense. We live in a three-dimensional universe. Breaking free of that universe into faster-than-light travel does not, to my mind, invalidate that, although one effect of the warp drive is to compress space in the direction of travel – to warp it in that direction. By what factor? – the “Warp Factor” – and, given the fundamental three-dimensions, it just makes sense that the number three would lie at the heart of the progressive warping of space as successive multiples of the speed of light were attained. So, from one perspective, a ship at Warp 2 is moving at (theoretically) twice the speed of light – but from another, it is moving at two-cubed times the speed of light. Hence, the cubic scale referenced above. It is a perfectly elegant solution, one that (in the real universe) prevailed in everyone’s mind very quickly – and in (my imagined) Star Trek universe explains the peak transitional thresholds as well. I also think it served from the time Zefram Cochrane invented the “space-warp” until a fourth dimension was added to the equation almost two hundred years later.

The “Time-Warp Drive”

The “Time Barrier” – bearing in mind that it was initially introduced only minutes into the first ever episode of Star Trek, thus in context it could have both more and less meaning than it does to us now, considering it in a much expanded context – seemed to refer to a recent advance that revolutionized space flight. It was an advance of such magnitude that:

1. Tyler enthused that the supposed Columbia survivors would be “amazed” at the speed; and
2. In the short term it was on everyone’s mind to the point that the “Time-Warp Factor” terminology noted above dominated; but
3. The shift was not so fundamental that the distinction in terminology between “Time-Warp” and simple “Warp” prevailed in the long run. By a decade or so later (2254-2265) speech had been shortened back simply to “Warp.”

How to account for a substantial enough revision without rewriting the warp scale entirely?

Since time is often considered the “fourth dimension,” I would propose the following – simply replace the cubic exponentiation with a quartic exponentiation – to the fourth power. Yes, I am saying that during the Original Series era, Enterprise was moving substantially faster than the cubic scale would seem to indicate – and I know that this suggestion will meet resistance from Star Trek fandom based on the long-standing authority of the original cubic equation (as well as its modification by Sternbach and Okuda to 10/3 rather than simply 3), but I think it provides a workable solution to the conundrum of the “Time-Warp.” Remember, moreover, that the cubic equation (as far as I know) was never “canonized” on-screen. The new quartic warp-scale would look like (and compare to the cubic scale as follows):

Table 4: The Cubic and Quartic Warp Scales

Warp Factor	The Cubic “Warp Scale” Multiple of the Speed of Light	The Quartic “Time-Warp Scale” Multiple of the Speed of Light
1	13 = 1 x the speed of light (c)	14 = 1 x the speed of light (c)
2	23 = 8c	24 = 16c
3	33 = 27c	34 = 81c
4	43 = 64c	44 = 256c
5	53 = 125c	54 = 625c
6	63 = 216c	64 = 1,296c
7	73 = 343c	74 = 2,401c
8	83 = 512c	84 = 4,096c
9	93 = 729c	94 = 6,561c
10	103 = 1,000c	104 = 10,000c

Tyler is therefore telling the supposed Columbia survivors that, at Warp 6, they will get home six times as fast as they would have expected based on the warp scale they were familiar with when they crashed in 2236. I imagine they would be “amazed.” The Quartic Scale provides a pretty substantial boost to velocity. And yet, conceptually, it is similar enough to the old Cubic Scale that the old “Warp” terminology ultimately prevailed.

How was that extra “time-dimension” added to the Warp Drive? I have no idea. Perhaps by some kind of nesting of warp-fields – or even adding a hyperspatial component to the previous warping of space into the subspatial realm ([33]), but I’m not terribly concerned with that, just the effects and determining a “realistic”[34] way to calculate them. As far as the term, “Time Barrier,” I imagine that to have been just a euphemism based on the previous experience of the “Warp Barriers.”

Following on this theme, if the first great advance in Warp Technology after Cochrane was effected by adding another dimension to the equation, what then was the second great advance that came less than half a century later? What was the “Transwarp”?

The “Transwarp Drive”

Similarly to the “Time-Warp” advance of ca. 2240, the “Transwarp” advance of ca. 2285:

• Must have been superior enough to Enterprise’s Warp Drive in Star Trek III that Styles gloated that Excelsior’s Transwarp Drive was about to break every speed record the Enterprise had ever set; and yet:
• Although “Warp 6,” etc., continued to be the terminology, meaning that, again, it was not so fundamental a shift; although: 
• It did compel the “recalibration” of the Warp Scale from the simple exponential scales to the “Zero-to-Infinity-in-Ten” scale described above.

How and why? I think it had to do with the “time-warp” nature of the Second-Generation Warp Drive. In late 2266, while Enterprise was observing the breakup of planetoid Psi 2000, a unique set of circumstances compelled her to initiate warp drive deep in an unstable gravitational field. The effect was a reversal of time. In approximately three minutes of shipboard time the ship traveled seventy-one hours back in objective time. During the process, the helmsman reported the speed of the ship to be “off the scale,” traveling faster than theoretically possible in our universe. It is unclear what he meant by that, but this event was followed by a similar occurrence only a few months later.

In March 2267: Enterprise was headed toward Earth at Warp 4 when she was caught in the intense gravitational field of an uncharted “black star.” In breaking loose from its grip, the ship experienced a slingshot effect and was thrown all the way to Earth – 297 years in the past. Drawing on the experience at Psi 2000 as well as sensor data obtained from their most recent accidental experience, science officer Lt. Cmdr. Spock theorized a method of duplicating the effect in reverse by using the gravity well of Earth’s own sun. It worked – first proceeding a few days further back into the past to rectify an unfortunate temporal contamination resulting from their unplanned appearance in the past, then proceeding into the future, returning to the same time from whence they had left.

In the wake of these two incidents, Starfleet initiated an intensive research program in an attempt to understand, duplicate, control, and exploit the technique. A little over a year later, in April 2268, as an experiment, Enterprise repeated her experience yet again, in a planned slingshot emerging almost exactly three hundred years into the past. Making historical observations from orbit, Enterprise found herself once again involved in current events. The danger of temporal contamination of the past being recognized, the secret of the “Light-Speed Breakaway” method of Time Travel was henceforth classified Above Top Secret.[35]

The second development resulted from the observation that during the Light-Speed Breakaway process, the ship’s effective velocity did not follow the predicted Second-Generation Warp Factor Scale. Instead, as noticed above, the velocity exponentiated “off the scale,” approaching infinity as the Warp Factor approached 10 (at which point the Time Warp effect was seen). This was found to result from the gravitational gradient in the presence of such high intensity gravity fields such as generated by the imploding planet Psi 2000, the uncharted “black star,” and deep in a star’s gravity well. But once that effect was recognized, the Transwarp Development Program was born, tasked with duplicating the increased effective velocities at lower Warp Factors without initiating a Time Warp. It took almost two decades, but by the mid-2280s the Transwarp Drive was tested, proven, and deployed with very little fanfare, most likely for security purposes, not to broadcast to potential adversaries that the Federation possessed a strategically advantageous faster method of interstellar travel. As always, such a secret could not be maintained forever, of course, and by the early twenty-fourth century all the major powers of local space possessed Transwarp or comparable technology

Regarding that short-term, however, I must digress. The “official” chronology has Star Trek III: The Search for Spock occurring sometime in 2285. The venerable old Starship Enterprise (NCC-1701), commissioned in 2245, was scuttled to keep her from being captured by the Klingons. The “official” chronology places Star Trek IV: The Voyage Home’s beginning sometime in 2286 – about three months after Kirk and company, now fugitives guilty of “grand theft starship,” fly a captured Klingon light destroyer to Vulcan and take refuge while Vulcan mystics restore Spock’s katra to his regenerated body. I have some minor quibbles with that specific dating, but they do not matter here. What is important is that in 2285 the Transwarp Drive is a new technology undergoing final tests on the Excelsior – tests that were undoubtedly set back a bit by Scotty’s sabotage. At the end of Star Trek IV, Kirk, having been court-martialed (again) and convicted of disobeying direct orders (again) and yet having just saved Earth (again) is demoted and assigned command of a new USS Enterprise (NCC-1701-A). Of the various suggestions, I like this “new” ship being the newly refit USS Yorktown (NCC-1704) as Gene Roddenberry himself suggested, although that doesn’t seem consistent with the Yorktown having been in space and disabled at the beginning of that same movie. On the other hand, I don’t know how long it was between that event, the return of Kirk and company from their adventures in 1987, Kirk’s court-martial, and their being assigned to Enterprise-A. The next movie, Star Trek V: The Final Frontier, is set in 2287, at which point Enterprise-A is still undergoing final testing, so it could be several months. Which might all seem tangential except it brings up the question: Did the refit Yorktown have Warp or Transwarp Drive? Personally, I think it was still Second-Generation Warp Drive, and that this plus her already advanced age (being initially built as part of the same construction contract series as the original Enterprise, in the 2240s) probably is the reason that she is to be decommissioned a mere six years later, in 2293, at the end of Star Trek VI: The Undiscovered Country. She is all but fifty years old at that point – and by then, I think the Transwarp Drive has been proven, and USS Excelsior now has a growing family of Transwarp sister-ships – enough that even Excelsior’s Captain Sulu used the traditional terminology of “Warp.”

Such a progression of technologies would, I believe, neatly explain what was shown on screen better than the idea of a “failed” technology (that Starfleet nevertheless had enough confidence in to build a new top-of-the-line starship for) followed by continuance of the same old technology with an otherwise unexplained “recalibration” of the Warp Factor Scale. I suggest that the Transwarp Advance somehow redefined the relationship between multiples of c in one perspective (the Warp Factors) and the effective velocity (V_eff) in another perspective, such that a simple exponentiating equation based on n dimensions no longer prevails. Now the exponentiating equation depends on some function of x (f(x)) where x is itself a function of the Warp Factor (W). To write it quasi-mathematically, no longer is the equation:

V_eff = Wⁿc

… but rather something akin to:

V_eff = W^f(x)c

where x = f(W)

Maintaining the Time-Warp Quartic dimension, because I propose this as an advance not a fundamentally new technology, I have come up with the following[36]:

V_eff = W⁴(3/√(10-W))c

Three divided by the inverse-square-root of the Warp 10 differential is (I would say) a (three-dimensional/fourth-dimensional) space/time displacement correction factor (proving I can do gibberish too).  In any case, comparing this proposed scale with the the original cubic scale and the Okuda “modified 10/3” scale: 

WF	Original Cubic Warp Scale	Okuda’s Modified (10/3) Warp Scale	The Professor's Proposed Transwarp Scale
1	1c	1c	1.00c
2	8c	10c	16.97c
3	27c	38c	91.85c
4	64c	102c	313.53c
5	125c	214c	838.53c
6	216c	393c	1,944.00c
7	343c	656c	4,158.65c
8	512c	1,024c	8,688.93c
9	729c	1,516c	19,683.93c
9.5	857c	2,500 (*) c	34,556.57c
9.75	927c	3,000 (*) c	54,221.27c
9.9	970c	4,000 (*) c	91,130.14c
9.95	985c	20,000 (*) c	131,500.9c
10	1000c	∞	∞

Note: (*) Okuda’s modified figures above Warp 9 are eyeballed from his chart.

Yes, it’s considerably faster, but even comparing it with the Original Series cubic scale, it’s in the same ballpark at the lower Warp Factors. At Warp 6 it approaches a full order of magnitude faster, but it does not really start exponentiating out of sight the higher Warp Factors. Even then, it is consistent. Of course, Warp 8 is screaming. At Warp 9, Scotty would be screaming! At Warp 10, no one can hear you scream!

But it is consistent and calculable. It’s not based on a fudged freehand curve.

The Three Generations of Warp Drive

So, to sum up, I propose that internal evidence from the episodes and movies themselves suggest that there are three generations of Warp Drive between its invention in 2063 and ca. 2400, our latest data, and that these can be correlated with events depicted in the series:

1. 2063-ca. 2240: The First Generation, the “Warp Drive.” Invented by Zefram Cochrane and Dr. Lily Sloane, refined by a series of scientists and engineers including Dr. Henry Archer, Cmdr. Trip Tucker, and many others. Velocity was expressed in units called “Warp Factors” according to a three-dimensional cubic scale based on the speed of light, the units being commonly shortened to “Warp.”
2. Ca. 2240-ca. 2285: The Second Generation, the “Time-Warp Hyperdrive.” Some advance (perhaps related to insights derived during the development of Subspace Radio?) allowed an exponentiation of the pre-existing Warp Drive into a fourth-dimension (commonly associated with “time”) yielding a new quartic “Time-Warp Factor” scale whereby the multiples of the speed of light were expressed in powers of four. Fundamentally, where a localized relative velocity of some multiple of c had yielded an effective velocity of c to the third power, now that same localized relative velocity yielded an effective velocity of c to the fourth power. After a relatively short time (less than a decade), the terminology reverted to the preexisting “Warp Drive” and “Warp Factors” or simply “Warp.”
3. Ca. 2085 ff.: The Third Generation, the “Transwarp Drive.” Yet another advance consequent to time-travel phenomena experienced by NCC-1701 Enterprise brought a new relation between the localized relative velocities in terms of c and a much faster effective velocity, such that now a localized relative velocity of 10c (previously “Warp Factor 10,” now “Transwarp Factor 10”) was unattainable because it yielded infinite velocity. Initially, the terminology reflected this fact – “Transwarp Drive,” “Transwarp Factor” (although it is unattested, it stands to reason) – but again, within a short time terminology reverted to the familiar “Warp Drive,” “Warp Factors,” and “Warp.”

Now, comparing the three Warp Factor Scales:

Table 5: The Three Generations of Warp Drives Compared

WF	First-Generation Warp Drive Cubic Scale (2063-ca. 2240)	Second-Generation Time-Warp Hyperdrive Quartic Scale (ca. 2240-ca. 2285)	Third-Generation Transwarp Drive “Transquartic Scale” (ca. 2285 ff.)
1	1c	1c	1.00c
2	8c	16c	16.71c
3	27c	81c	89.93c
4	64c	256c	304.44c
5	125c	625c	808.57c
6	216c	1,296c	1,882.43c
7	343c	2,401c	4,158.65c
8	512c	4,096c	8,688.93c
9	729c	6,561c	19,683.93c
9.5	857c	8,145c	34,556.57c
9.75	927c	9,037c	54,221.27c
9.9	970c	9,606c	91,130.14c
9.95	985c	9,801c	131,500.9c
10	1000c	10,000c	∞
11	1,331c	14,641c
12	1,728c	20,736c
13	2,197c	28, 561c
14	2,744c	38,416c
14.1	2,803c	39,525c

Addenda:

The Cochrane Factor and Sloane’s Constant

A common perception among fans, with (I believe) some warrant from televised information, although I do not have references, is that actual warp velocities are also affected by local gravimetric fields, etc. The first time I ever encountered this idea was in the Introduction to Navigation booklet included with the Star Trek Maps (ca. 1980), and, although I do not think the effects are as pronounced as proposed in that work, I think the basic idea stands to reason -- so I incorporated it into "my" vision of the development of the Warp Drive. I ask that the reader remember, once again, that I am not a mathematician; these equations in particular were derived by trial and error to create come up with the effects I wanted, as well as to make the units cancel out: 

In the “History of the Twenty-First Century” (of the Star Trek universe) which I am currently writing, under the year 2074 there is an explanation for another facet of space-warp physics – the Cochrane Factor. I quote from my current draft:

The astute reader may have noticed a discrepancy in the warp factor attained by Enterprise [Earth’s first starship, which looks amazingly like the IXS-110 Enterprise designed by NASA engineer Sonny White in 2013 [LINK]], her real-space equivalent velocity, and the time taken to travel the 4.395 light years from Earth to Alpha Centauri in 2072-2073. The discrepancy was definitely noticed by the captain and crew even during their passage before they arrived at in the outer Centauri system a full six months earlier than they had expected when they left Earth. They had no way to report the conundrum back to Earth – and no way to know that by the time they returned the mystery would be explained if not actually solved and that the insights derived from the investigation would produce the next significant breakthrough in evading the light-speed limit prevailing in normal space [what will eventually be called "Subspace Radio].

Dr. Lily Sloane had noticed immediately in studying the telemetry from Cochrane’s first flight that Phoenix had not ended up where fifty-five seconds at Warp 1.01 should have placed it. Several more flights gathered sufficient data for her to solve the mystery – as far as anyone has – within a decade.  At the time Enterprise left Earth on its first voyage, she had not, but even in Sloane’s absence – she was not aboard – Enterprise confirmed the solution she was only in the meantime discovering. Oversimplifying it considerably, although Cochrane himself had perceived that the warping of three dimensional space would result in the cubic equation that would be the most fundamental fact of space warp propulsion known by every school child today – effective real-space velocity equals the cube of the warp factor times the speed of light, or v_eff = W³c, there is another factor multiplying the effect even further. Sloane called it “The Cochrane Factor.”  It depends on a complex interaction of forces including but not limited to the masses and separation of the two stars between which a warp-ship is traveling (if so), the total mass of the galaxy and one’s position within it relative to the galactic core, the local density of the interstellar medium, as well as the local masses and distances of other nearby stars and planets – most of which are not (and cannot be) known exactly even today and thus can only be estimated. Sloane ultimately derived a complex equation to predict X modifying the cubic equation such that v_eff = XW³c is correct within half a percent … most of the time. In order to do so, she found it necessary to apply a constant, K, which is always rendered today as K_S, Sloane’s Constant. It was a solution she herself found inelegant, and many theses and dissertations have resulted from almost a century’s efforts to derive K_S. She herself came to believe, although she could never prove, that the solution was to be found in the very Heim theory that Cochrane worked on as a graduate student.

The best approximation of the overall effect may be calculated by the following formula:

X=1+K_SG_cc^-2𝚲^-1[(m₁+m₂)m_or₁₂²+0.5m_om_H(m₁+m₂)m_Pr_H²+m_Gr_G²]

Where:

·        K_S = Sloane’s constant, empirically derived as 5.03 x 10^-31

·        G_c = the gravitational constant, 6.674 x 10^-14 m³g^-1s^-2

·        c = the speed of light, 2.998 x 10⁸ ms^-1

·        𝚲= the cosmological constant, 1.056 x 10^-52 m^-1

·        m₁ and m₂ = respectively, the mass of the star(s) at departure and the mass of the star(s) at destination, expressed in terms of solar mass m_o

·        m_o = one solar mass, the mass of the star Sol, 1.989 x 10³³ g

·        r₁₂ = the distance between stars 1 and 2 expressed in meters, where one light year = 9.46 x 10¹⁵ m

·        m_H = the mass of one hydrogen molecule (H₂), 3.346 x 10^-24 g

·        m_p = the mass of the payload being conveyed – i.e., the ship, expressed in grams.

·        r_H = the mean separation between hydrogen molecules in the local interstellar medium, estimated at 1.416 x 10^-2 m

·        m_G = the mass of the Galaxy, 2.984 x 10⁴⁵ g

·        r_G = the distance from the center of the Galaxy, expressed in meters.

And, just to show I do not know when to stop....

Let’s Do the Time Warp Again – The “Slingshot Effect”

Using similar concepts, it is possible to calculate the Time Warp via the “Light-Speed Breakaway” (more properly, “Warp-Speed Breakaway” in high-gravitic-gradient situations.

First, “high gravitic gradient” is a relative term. Empirical evidence based on the Psi 2000 and Black Star incidents described above indicate that the gradient threshold required to initiate Time Warp is very low, on the order of 1x10^-7m^-1s^-1, calculated as follows:

dg = mG_c(r^-2-(r²-2r+1)^-1)TK_T 

Where:

·        dg = the gravitic gradient per meter

·        m = the mass of the gravity source

·        G_c = the gravitational constant, defined above

·        r = the distance from the center-point of the gravity source

Sudden initiation of a warp field of (Second-Generation) Warp Factor 10 or higher in the presence of such a gravitic differential shears the space-time continuum and creates a time-warp. Depending on whether one is traveling toward the gravity source or away from it, one will be displaced either backward (toward) or forward (away) in time to a degree dependent on several factors. The simplified formula for estimating such temporal displacement is as follows – with the caveat that the conditions are fluid during the process because each of the following factors will be constantly changing in a moving ship:

T_W = V_eff(W₃x)TmG_c(r^-2-(r²-2r+1)^-1)K_T(V_eff(W₂10)^-1r^-2

Where;

·        T_W = Time Warp displacement

·        V_eff(W₃x) = the effective velocity multiplier of 3^rd-generation Warp x

·        T = subjective time at Time Warp velocity

·        K_T = Time-Warp Constant (empirically-derived, -1.462 x 10²⁰)

·        V_eff(W₂10) = the effective velocity multiplier of 2^nd-generation Warp 10 (=10,000)

The divisor of V_eff(W₂10) is included rather than the number 10⁴ to emphasize the crucial fact that, whereas normal, predictable 2^nd-generation Warp Physics seems to apply up to that point, when (unavoidably confusing considering the association of the Second-Generation Warp Drive advance being characterized as “breaking the time barrier”) the ship … breaks the time barrier. At that point, spatial velocity becomes meaningless, and further “acceleration” (i.e., increased power input to the warp field according to what would become known as the Third-Generation Warp Scale, starting at the Second-Generation Warp 10 equivalent of Third -Generation Warp 6.193) manifests as increased displacement through time.

---

[1] The foregoing story was told in the movie, Star Trek: First Contact (1996).

[2] Calling it an average of Warp 2.1, NX-Alpha would have covered 241,550,958.6 km in 87 seconds. The minimum distance from Earth to Jupiter is about 4 AU, where 1 AU (astronomical unit = the distance from the Earth to the sun) is 149,597,870 km. Four AU would be about 600,000,000 km. In other words, the NX-Alpha should not have been anywhere near Jupiter. But see way below under “The Cochrane Factor and Sloane’s Constant.”

[3] The foregoing narrative is synthesized largely from “First Flight” (Enterprise, Episode 2:24, 2003) although aspects of it permeate the series, especially the first episode, “Broken Bow” (2001).

[4] Star Trek Beyond (2016).

[5] “Broken Bow.”

[6] Enterprise, Episode 1:23, “Fallen Hero” (2002).

[7] Star Trek: Enterprise, Episode 4: 15, “Affliction” (2005),

[8] All of this is from the original Star Trek pilot episode, “The Cage” (1964). When announcing the diversion to Talos IV, Capt. Pike uses the term “time-warp factor” to designate the speed: “Our destination is the Talos Star Group. Our time-warp … factor seven.” Interestingly, as they accelerate the ship and crew waver translucent and the stars can clearly be seen rushing by – through – them, and as Pike leans over Tyler’s shoulder, Tyler uses hand-signals to indicate “7,” implying that sounds cannot be heard. This effect is never seen again. Obviously, it was corrected sometime between 2254 and the next time, chronologically, we see ships of this 2250s era. If one accepts Star Trek: Discovery (2017 ff.), that would be 2256, I believe; if one does not, then it would be 2265. In any case, although I am not discussing it further in the main text, I would suggest that the introduction of the “time-warp drive” resulted in a brief “reality lag” or “hyperdimensional flux” – that perhaps manifested only during acceleration or deceleration – that was corrected ca. 2255.

Later in the episode, when Spock, as acting captain, for the safety of the ship and crew, orders Enterprise to break orbit, he adds the term, “hyperdrive”: “All decks prepare for hyperdrive. Time warp factor – ” – then he is cut off by Tyler’s report that the ships controls have gone dead.

[9] Star Trek, Episode 3: 14, “That Which Survives” (1969).

[10] Even Warp 14.1?

[11] Star Trek III: The Search for Spock (1984)

[12] This is not entirely true. The term was used quite a bit, but obviously meaning something different, especially in Star Trek: Voyager.

[13] Star Trek VI: The Undiscovered Country (1991)

[14] Star Trek: Voyager, Episode 2: 15, “Threshold” (1996).

[15] Star Trek: The Next Generation, Episode 1: 06, “Where No One Has Gone Before” (1987).

[16] By which I mean, “hideously.”

[17] Star Trek: The Next Generation, Episode 7: 25/26: “All Good Things.…” (1994).

[18] And, frankly, one of the more ludicrous aspects of that series.

[19] Star Trek: Deep Space 9, Episode 3: 22, “Explorers” (1995)

[20] Star Trek: Voyager, Episode 4: 03, “Day of Honor” (1997)

[21] Star Trek, Episode 2: 18, “Obsession” (1967); Star Trek: Voyager, Episode 6: 09, “The Voyager Conspiracy” (1999); etc.

[22] Station Deep Space 9’s raison d’être.

[23] Too many references to enumerate.

[24] Star Trek, Episode 1: 28, “City on the Edge of Forever” (1967)

[25] Quoted from Star Trek 2 (1968), as reprinted in Star Trek: The Classic Episodes (2016 Sterling Books Edition), p. 183.

[26] Star Trek: The Animated Series, Episode 2: 06, “The Counter-Clock Incident” (1974).

[27] Quoted from Alan Dean Foster’s prose adaptation in Star Trek The Animated Series Log Seven (1976), as printed in Star Trek The Animated Series Logs Seven and Eight (2006), p.22.

[28] Yes, I’m having a little fun with it. I really shouldn’t. They were attempting to create a seemingly plausible explanation for technologies so far beyond anything considered possible today that anything they might come up with, as impressive as it might look, was going to be largely gobbledegook. They succeeded. But it’s still gobbledegook … that did not stand in its details more than four years. There’s nothing about “continuum distortion propulsion” or “CDP” or “straddling” light-speed in Star Trek: First Contact (1996) – just attaining warp speed.

[29] Until First Contact….

[30] There are issues of causality, however – but discussion of faster-than-light travel in Star Trek has generally ignored those issues for the last fifty-five years, and so will I.

[31] “First Flight”

[32] As shown in the associated figure here, which is inspired by but does not reproduce the aforementioned Figure 5.1.1 in the TNG Technical Manual, I think it would be most logical for the spike in power consumption at each integral warp factor to be mathematically infinity. One of the great mysteries of the Warp Drive would then be exactly what happens at that phase transition from just below the integral Warp Factor to just above it in, as Okuda and Sternbach suggest, Planck Time. Obviously, infinite power cannot in reality be consumed. A disquieting theory is that the mass and very being of the starship and all its payload – including the crew – are momentarily annihilated at each warp barrier, ascending and descending, for that instant of Planck Time. Given McCoy’s aversion to the transporter based on a similar effect, perhaps it would be better if he did not know this…..

[33] Gobbledegook!

[34] -- ?! – Bwahahahahah!

[35] Nevertheless, almost two decades later, Enterprise’s crew – now flying an appropriated Klingon light destroyer – used the technique yet again: Early 2286, during the Whalesong Crisis (Star Trek IV: The Voyage Home, 1986).

[36] With help from blogger occen of the Occasional Enthusiast blog in private correspondence after I outlined my basic ideas in a comment on the post, “Alternative Star Trek Warp Speed Scale and Related Equations” (28 Nov 1016) http://www.occasionalenthusiast.com/alternative-star-trek-warp-speed-scale-and-related-equations/