$$\require{cancel}$$

# 3.8: Faster-than-light frames of reference?

• • Contributed by Benjamin Crowell
• Professor (Physics) at Fullerton College

Skills to Develop

• Explain the phenomenon of faster-than-light motion in relativity

We recall from Section 3.4 that special relativity doesn’t permit the existence of observers who move at $$c$$. This is because if two observers diﬀer in velocity by $$c$$, then the Lorentz transformation between them is not a one-to-one map, which is physically unacceptable.

But what about a superluminal observer, one who moves faster than $$c$$? With charming naivete, the special-eﬀects technicians for Star Trek attempted to show the frame of reference of such an observer in scenes where a ﬁeld of stars rushed past the Enterprise. (Never mind that the stars, which pass in front of and behind the spaceship, should actually be a million times larger than it.) Actually such an observer would consider her own world-line, which we call spacelike, to be timelike, while the world-line of a star such as our sun, which we consider timelike, would be spacelike in her opinion. Our sun’s world-line might, for example, be orthogonal to hers, in which case the sun would not appear to her as an object in motion but rather as a line stretching across space, which would wink into existence and then wink back out. A typical transformation between our frame and the frame of such an observer would be the map $$S$$ deﬁned by $$(t',x') = (x,t)$$, simply swapping the time and space coordinates. The “swap” transformation $$S$$ is one-to-one, and therefore not subject to the objection raised previously to frames moving at $$c$$. $$S$$ happens to be a boost by an inﬁnite velocity, but we can also obtain boosts for velocities $$c < v < ∞$$ and $$-∞ < v < -c$$ by combining $$S$$ with a (subluminal) Lorentz transformation; given a superluminal world-line $$l$$, we ﬁrst transform into a frame in which $$l$$ is a line of simultaneity, and the we apply $$S$$.

But this was all in $$1+1$$ dimensions. In $$3+1$$ dimensions, what is the equivalent of $$S$$? One possibility is something like $$(t',x',y',z') = (x,t,t,t)$$, but this isn’t one-to-one. We can’t squish three dimensions to one or expand one to three without merging points or splitting one point into many.

Another possibility would be a one-to-one transformation such as $$(t',x',y',z') = (x,t,y,z)$$. The trouble with this version is that it violates the isotropy of spacetime (section 2.3). For example, consider the vector $$(1,0,1,0)$$ in the unprimed coordinates. This lies on the light cone, and could point along the world-line of a ray of light. After the transformation to the primed coordinates, this vector becomes $$(0,1,1,0)$$, which points along a line of simultaneity. The primed observer says that the speed of light in this direction is inﬁnite, and yet there are other directions in which it has a ﬁnite value. This clearly violates isotropy.

A surprisingly large number of papers, going all the way back to the birth of relativity, have been written by people trying to ﬁnd a way to extend the Lorentz transformations to superluminal speeds, and these have all turned out to be failures. In fact, there are no-go theorems showing that there can be no such thing as a superluminal observer in our $$3+1$$-dimensional universe.1,2

The nonexistence of FTL frames does not immediately rule out the possibility of FTL motion. (After all, we do have motion at $$c$$, but no frames moving at $$c$$.) For more about faster-than-light motion in relativity, see section 4.7.

# References

1 Gorini, “Linear Kinematical Groups,” Commun. Math. Phys. 21 (1971) 150. Open access via Project Euclid at projecteuclid.org/DPubS?service= UI&version=1.0&verb=Display&handle=euclid.cmp/1103857292.

2 Andreka et al., “A logic road from special relativity to general relativity,” arxiv.org/abs/1005.0960, theorem 2.1.