Monday, September 22, 2014

dimensional analysis - How can the speed of light be a dimensionless constant?


This is a quote from the book A first course in general relativity by Schutz:



What we shall now do is adopt a new unit for time, the meter. One meter of time is the time it takes light to travel one meter. The speed of light in these units is $$\begin{align*}\end{align*}$$


$$\begin{align*} c &= \frac{ \text{distance light travels in any given time interval}}{\text{the given time interval}}\\ &= \frac{ \text{1m}}{\text{the time it takes light to travel one meter}}\\ &= \frac{1m}{1m} = 1\\ \end{align*}$$


So if we consistently measure time in meters, then c is not merely 1, it is also dimensionless!



Either Schutz was on crack when he wrote this, or I'm a dope (highly likely) 'cos I can't get my head around this:


The space-time interval between different events at the same location measures time, and between different events at the same time measures space. So they're two completely different physial measurents: One is a time measurement using a clock, the other a space measurement using a ruler. In which case the units of $c$ should be $ms^{-1}$



Does Schutz correctly show how $c$ can be a dimensioness constant?




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