Saturday, October 20, 2007

Minkowski diagrams

In special relativity we hear a lot about length contraction - objects such as spacecraft and the ubiquitous 'rigid rod' 'get shorter' as they approach light speed. People don't get this. Does the rod 'really' get shorter or not? It seems to depend on the observer, defying Aristotelian logic and common sense.

An excellent Wikipedia article on Minkowski diagrams clears the whole thing up. The problem is with our use of language. When we say 'spacecraft' or 'rod' we are making a spacial statement - the object considered at a 'now'. But that immediately invalidates special relativity.

Instead, we have to consider the space-time object extended in space and swept out over a defined time. This extended space-time object at a particular 'now' is viewed differently by differently-moving observers because their 'now's are differently-oriented. Specifically when you view an object travelling fast past you, the spatial slice you measure as being the length 'now' is rotated as compared with the 'now' length measured by an observer travelling with the rod. And the rotated view is shorter. It's made clear in the diagram below, with explanatory text pasted in from the article.

Notice, by the way, that this has nothing to do with the time taken for light to get to you from different parts of the object at your 'now'. Your length measurement has to correct for those effects, and after the correction you compute that the 'length' is shorter.

"Relativistic length contraction means that the length of an object moving relative to an observer is decreased and finally also the space itself is contracted in this system. The observer is assumed again to move along the ct-axis. The world lines of the endpoints of an object moving relative to him are assumed to move along the ct'-axis and the parallel line passing A and B respectively. For this observer the endpoints of the object at t=0 are O and A. For a second observer moving together with the object, so that for him the object is at rest, it has the length OB at t'=0. Due to OA being less than OB the object is contracted for the first observer."

* Excerpted from the Wikipedia article.

The point is made more clearly by the following thought experiment in the diagram above. Suppose, according to the 'blue' observer speeding by, that the rod flicked into existence 'all at once' for a millisecond along OB, then vanished again. The 'black' stationary observer would not see a rod at all, but would calculate a thin slice of material which sprang into existence close by and seemed to move away much faster than light (although it would visually appear to be slower than light due to light propagation delay from more distant parts of the rod). The 'spatial' rod would, in fact, have been rotated a little into time from the stationary observer's viewpoint.