The question is how to interpret the Lorentz transformation. In a previous post, *Lorentz generalized*, a modest generalization of the Lorentz transform was derived. Absolute reference speeds were combined with a relative actual speed.

Let’s step back and look at a map of space and time:

Interstate Drive Times & Distances Sample

This map of nodes and links on the U.S. interstate highway system displays travel distances and driving times between cities. If you look closer, you can see that it is based on a standard travel speed of about 50 mph (with some local variations). So each point on the network represents a travel distance and a travel time: in other words, space and time are in sync.

Now compare this map with some actual travel experience, say, one traveler going at 40 mph and another at 60 mph. If they start together, after one hour of travel they will have gone 40 and 60 miles respectively, compared to the standard of 50 miles. After one hour, the standard “map” distance is 50 miles but the actual distances are 40 and 60, so space and time are not in sync with these travelers.

The problem is space and time can no longer be mapped together: either the distance traveled or the travel time can be mapped but not both. At most all the distances for one travel time or all the travel times for one distance can be mapped.

A physicist approaching this situation might ask, is there some function of space and time that can still be mapped? Is there a quantity that is invariant no matter what the travel speed is? Can an alternate map be constructed?

The answer is yes and the key is the Lorentz transformation. Note that this is for an alternate map: if travel speeds equal the standard speed, no new map is needed. So we’re looking at speeds *u* and *u’* that differ from the standard speed, *c*.

The alternate map has one limitation: it’s from the point of view of one traveler. But an alternate map can be constructed for any traveler and the principles of its construction are the same for all travelers. That’s the best that can be done.

*Related*