Faster Than Light

We believe that we know what that phrase means, but we only have a Newtonian understanding of it. We really need a relativistic understanding of it if we want to consider claims of things that actually move faster than light. As usual, we will use imaginary experiments to guide our reasoning.

In terms of four-dimensional geometry, we want to consider events separated from each other by a spacelike interval rather than events separated from each other by a timelike interval. More specifically, we want to consider events that are attended by the same object and separate them in a way that obliges that object to move faster than light. To start we look at Hermann Minkowski’s metric equation for flat spacetime, the four-dimensional analogue of the Pythagorean theorem;

(Eq’n 1)

For simplifying convenience, as usual, we assume that all relevant motions occur only in the x-direction, so the y- and z-components zero out of that equation.

Consider two events that occur a distance x and an interval t apart. To calculate the four-distance between those events we have

(Eq’n 2)

If the temporal interval multiplied by the speed of light is greater than the spatial interval, then s is an imaginary number and we say that it represents a timelike interval. An object attending both events would move at a speed less than the speed of light (x/t<c). If the temporal interval multiplied by the speed of light is smaller than the spatial interval, then s is a real number and we say that it represents a spacelike interval. An object attending both events would move at a speed greater than the speed of light (x/t>c).

In the lore of Relativity we see a statement to the effect that nothing can fly faster than light, that nothing can attend two events separated by a spacelike interval. The dynamic reason behind that statement, physicists tell us, lies in the statement that an object becomes more massive as it gains velocity and that the object’s mass tends to infinity as the object’s speed approaches the speed of light. Experiments performed with cyclotrons and synchrotrons have proven and verified the existence of that mass-increase effect. But there are also more purely geometric reasons why nothing can move faster than light.

We know that, because time does not elapse on the boundary of the Universe, nothing can have the possibility of reaching that boundary. That statement necessitates that nothing can move faster than the speed, the speed of light, at which the boundary recedes from all objects. That fact necessitates in turn that spacetime consist of an infinite set of inertial frames of reference, all related to each other through the Lorentz Transformation, which comes from the four-dimensional analogue of the Pythagorean theorem. Objects don’t move through space; they occupy and mark inertial frames. Thus, trying to make an object move faster than light through direct acceleration in normal space has somewhat the character of trying to get a point off a circle by rotating the circle.

When those facts became clear, science-fiction writers conceived the idea of hyperspace, a realm that somehow stands outside of normal space and allows starships to move in a way that corresponds to flying faster than light in normal space. Of course, the fact that we can conceive the idea of something does not necessitate that the something actually exist. Nonetheless, a little fantasy can help us clarify our understanding of real things. We don’t need to concern ourselves with how ships get into and out of hyperspace; we need only assume that they can do so.

Let us then imagine sending a variety of objects from Starbase Asimov to Starbase Biggle. For the duration of our experiments the distance X=10 lightyears between the two starbases does not change. Let I.S.S. (InterStellar Ship) Altair travel from Starbase Asimov to Starbase Biggle by traveling through hyperspace, arriving at Starbase Biggle, by the starbase’s clocks, 36.5 days after leaving Starbase Asimov. The ship has made the journey in a manner equivalent to a normal-space speed one hundred times the speed of light (warp factor 4.64 if Altair uses Star Trek’s warp drive). By Minkowski’s equation we have the four-distance between Altair’s insertion into hyperspace and Altair’s emergence from hyperspace as the square root of 100-0.01=99.99 or 9.9995 lightyears. That’s a real number, so the four-distance is spacelike.

At the same time as Altair departs from Starbase Asimov, I.S.S. Tambora passes the same point at a speed designated as Lorentz factor 2, 86.6 percent of the speed of light relative to the starbase’s frame of reference, which will get Tambora to Starbase Biggle in 11.55 years as measured on the starbases’ clocks. But distances and durations look different as measured by Tambora’s crew. Actually, observers on board Tambora don’t have to make their own measurements: they can use the Lorentz Transformation to translate measurements made by observers at the starbases. In this case they would use

(Eq’ns 3)

in which v represents the relative velocity between Tambora’s inertial frame and that of the starbases (expressed as a fraction of the speed of light) and we calculate the Lorentz factor from the usual

(Eq’n 4)

Note that we express time in years and distance in lightyears. Thus, by Tambora’s measurements, that ship’s journey crosses 5 lightyears in 5.78 years.

The first of Equations 3 seems to give us a wrong answer. Using that equation, the observers aboard Tambora measure a distance of zero between their leaving Starbase Asimov and arriving at Starbase Biggle. But that calculation is, if fact, correct: as measured from Tambora the events of Tambora leaving Starbase Asimov and arriving at Starbase Biggle truly lie zero lightyears apart. If we want to know the instantaneous distance between the starbases as measured from Tambora, which is generally what we mean when we ask for the distance Tambora moved, we must apply the Lorentz-Fitzgerald contraction, dividing the distance between the starbases as measured in the starbase frame of reference by the Lorentz factor between that frame and Tambora’s frame to obtain x/L=5 lightyears.

What does Altair’s mission look like from Tambora? Equations 3 give us x’=19.8268 lightyears and t’=-17.12 years, which means that in Tambora’s frame Altair went from Starbase Biggle to Starbase Asimov at an effective speed of 1.158 times the speed of light. We can calculate that speed more directly if we divide the first of Equations 3 by the second. If we define the equivalent speed H of the starship by writing x=Ht, then we have

(Eq’n 5)

If Tambora’s crew had a God’s eye view of their region of space, they would see Altair emerge from hyperspace at Starbase Biggle and go about its business, albeit in dilated time, and then, a little over seventeen years later, they would see Altair leave Starbase Asimov and vanish into hyperspace.

That result appears to vex causality, with an effect (Altair emerging from hyperspace) occurring before its cause (Altair entering hyperspace). But a simple calculation indicates that a signal emitted from Altair upon emergence from hyperspace will not reach Starbase Asimov before Altair enters hyperspace, so the effect cannot alter the cause in any way. That statement will stand true to Reality as long as the magnitude of x’ is greater than t’; that is, it will be true as long as H’>1. Since H is always greater than one (that’s what hyperspace is for, after all), then that criterion will be met and causality will be preserved.

As long as our hyperspace meets that criterion, it has a possibility of existing. As for how to get ships into it and out of it and what happens between those events, we don’t know. We will have to gather more information on the nature of Reality before we can even hazard a guess.

eabf