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?
Best Answer
The infinitesimal length interval between two events in spacetime $ds$ is defined by
$$ds^2=c^2 dt^2 - dx^2 - dy^2 - dz^2$$
The creature is dimensionally consistent, because time is multiplied with a speed. You can think of $(t,x,y,z)$ as the four coordinates of spacetime $(x^0,x^1,x^2,x^3)$ and $c$ appears naturally in the equations. However, the usual approach is setting $x^0=ct$ and so all the four coordinates have units of length. The definition of infinitesimal interval is then
$$ds^2=(dx^0)^2-(dx^1)^2-(dx^2)^2-(dx^3)^2$$
By doing that, $c$ vanishes from equations. For convenience, $x^0$ is named $t$ and, since it is actually proportional to time, it is called time or the time coordinate, but it is not really time. It is a distance. The original idea of Minkowski was even more bizarre, since he originally did $x^0=ict$, an imaginary (complex) distance.
The quantity that matters is a distance. It is proportional to time, but it is nevertheless a distance. The whole conceptual business is quickly (but not very enlightening) explained by saying "we work in units such that $c=1$" and the beginner is lost...
So, when you see $t$ you must remember that you are calling "time" to a quantity that is actually a distance. This trick affects also other physical quantities, so that you will be calling energy to something that is NOT energy, but energy divided by $c^2$. You will call speed to an adimensional quantity that must be multiplied by c to recover the "real" speed... It is not so unfamiliar to you: a supersonic plane may fly at "match 2.5", which really means 2.5 times the speed of sound.
Look at this list you may derive and check by yourself as an exercise:
Things that happen when you call "time" to the distance $ct$:
What you call LENGTH is still a length.
What you call TIME is a distance.
What you call MASS is still a mass.
What you call SPEED is something adimensional.
What you call ACCELERATION is an acceleration divided by $c^2$
What you call MOMENTUM is a momentum divided by $c$.
What you call ENERGY is energy divided by $c^2$
What you call ELECTRIC CURRENT is an electric current divided by $c$
By looking at this list, you now know how to "undo" the definitions so that, for instance, you have to multiply by $c^2$ when you want to recover the energy from the E that appears in the equations (e.g. the famous Einstein equation relating mass and energy is written $E=m$).
Don't worry, you will get used very soon. If you think this is bizarre, then look at what particle physicists do: They rename things so that not only the speed of light vanishes from the equations, but also the Planck constant and the electron charge, all at once. Don't ask me how to "undo" that...
EDIT: Some users keep asking in the comments so, for the ease of beguinners, here is how to reproduce that list:
Decompose every physical quantity into the basic dimensions length, mass, time. For instance:
Now, take every [time] factor you see and change it into [length]:
That is, by pretending that time is a length, we are pretending too that Energy is mass. And so on with the other quantities.
SECOND EDIT, About the metric signature:
One user was puzzled about the sign of the interval $ds^2$. This edit attempts to make this point clear:
Einstein defined in the 1921 Princeton lectures the interval as:
$$ds^2= + c^2 dt^2 - dx^2 - dy^2 - dz^2 $$ this sign convention is referred to as $(+---)$ as a shorthand notation. It is used by many authors, both in the field of Relativity and Particle Physics, like Weinberg, Peskin & Schröder or Zee.
However, there are too very good, canonical books in both fields with the opposite convention $(-+++)$, that is $$ds^2= - c^2 dt^2 + dx^2 + dy^2 + dz^2 $$
For instance Schutz himself, Wald (except in the spinors chapter), MTW or Srednicki. It is also the convention used in Wikipedia.
The first thing you have to look at, when consulting a book, is which convention is the author using, because usually there are sign differences in the same equation when based upon one convention or the other.