A problem with a basic thought experiment in Special Relativity

This is for a class on special relativity I am to give to some school children.

Moe is moving at 0.9c. At the instant he passed Joe he emits a flash of light. One microsecond later, Joe(at rest) sees that Moe has traveled 270m and the light flash has traveled 300m. That is, the distance between Moe and the light flash is 30m. All perfectly good so far.

But Moe, who does not know he is moving, waits one microsecond and observes that the light has traveled 300m in front of him. Also perfectly good.

The reason that Moe sees that the light has traveled 300m and not 30m is that his time has been dilated, so that Moe's microsecond is longer than Joe's microsecond. Also, when he measures how far ahead the light flash is, he is using a shortened ruler.

All very good. Except that if you use the Lorenz transformation equations, you find that his time has dilated to 2.29 microseconds and his ruler is shorter by 1/2.29. Those are not enough to explain a 300m measurement for Moe: It only provides for the light to be 157m ahead, not 300m ahead!

What am I missing here?

• I quoted the description of the thought experiment to make it clearer - I hope I got it right! If not, DavidM, please feel free to fix it with your own edit. – David Z Feb 10 '14 at 1:35

As always, a spacetime diagram is crucial for insight. However, this can be dispensed with without one.

What am I missing here?

The event that Moe's wristwatch reads $1\mu s$ has coordinates, in Joe's frame, of $(2.29\mu s, 618m)$. In other words, according to Joe, Moe's wristwatch reads $1\mu s$ when Joe's clock reads $2.29\mu s$.

According to Joe, Moe has travelled $618m$ and the light has travelled $687m$ so, according to Joe, there is a distance of $68.7m$ separating Moe and the light pulse at that time, not $30m$.

Further, the two events - Moe's wristwatch reading $1\mu s$ and the light pulse is $300m$ ahead of Moe according to Moe's rulers - while simultaneous according to Moe - are not simultaneous according to Joe.

Thus, the number $(2.29)^2 \times 30m = 157m$ is not a meaningful calculation.

According to the Lorentz transformations, the events $(1\mu s, 0), (1\mu s, 300m)$ in Moe's frame have coordinates, in Joe's frame, of $(2.294\mu s, 619.4m),(4.359\mu s, 1308m)$ (these are the events that Moe's wristwatch reads $1\mu s$ and the light pulse is $300m$ distant according to Moe).

The spatial separation of these two events, according to Joe, is about $688m$ which is as it should be.

According to Moe, the spatial separation between the events is $300m$ and, according to Moe, Joe's rulers are shorter by a factor of about 2.294 and thus, Joe should measure the distance to be about $688m$