${}$ What exactly is a dimension?

Question

Intuitively, I understand that if something is measured in meters, it has the dimension of length, which we denote as $L$. For example, if we take length and time as fundamental dimensions, then velocity has the dimensions of length divided by time, or $LT^{-1}$.

More formally, we can define dimensions in terms of scaling. Suppose we have one system of units and another system of units. The dimensions of a quantity describe how its numerical value changes when we change the units. If we reduce the unit of length by a factor of $L$ and the unit of time by a factor of $T$, then the unit of velocity scales by $LT^{-1}$.

It is clear that we have some freedom in choosing the base units within our system of units. However, what puzzles me is the following:

Are the terms "dimension of time, space,..." just some conventions which is used within out standard system of units, or there exists a more fundemental reason for such notion?

Answer 1

Many words have more than one meaning, including “dimensions”. The usage here

if something is measured in meters, it has the dimension of length

refers to a way to determine if measurements are of the same kind. In this sense, measurements with the same dimension can be compared, added, and subtracted.

However, the usage here

dimension of time, space

refers to the idea of a space, like a vector space, which requires a certain number of basis vectors to span the space. In this sense, dimensions describe the space itself, rather than specific measurements.

In the first sense, the dimensionality of a measured quantity is a matter of convention. Different units will use different conventions, which may not always be compatible. In geometrized units length has the same dimensionality as time. In SI units current has its own dimensionality, but in cgs units the dimension of current is derived from purely mechanical units.

In the second sense, dimensions are usually associated with a meaningful metric on the space. For example, in relativity physical space and time form a four dimensional space called spacetime. It is meaningful because there is a physical notion of “distance” that involves both clocks and rulers.

Answer 2

Are the terms "dimension of time, space,..." just some conventions which is used within out standard system of units, or there exists a more fundemental reason for such notion?

To be clear you are talking about dimensional analysis. This depends on the physical framework that we use. In Newtonian physics, time and space are fundamental dimensions because time cannot be converted into space and vice-versa. Thus here the dimension of speed, is as you say, L/T.

However, in the framework of special relativity space is convertible into time and vice-versa. Thus L = T. This the dimension of speed is L/T = L/L = 1. Thus it is dimensionless. We can see this by expressing speed as a fraction of $c$, the speed of light. The latter is a dimensionless constant. Thus we see speed, as a fraction of this, is dimensionless.

Since special relativity is the more fundamental theory, should we then take speed to be dimensionless? This essentially depends on 'speed' scale. If relativistic speeds appear in physics you are probing then yes, you should. But if the physics you are probing has low speeds (compared to $c$) then you can take the classical approximation and then space and time are not interconvertible, so the dimension of speed here will be taken to be L/T.

What exactly is a dimension, ie mass?

The dimensions of a theory is the set of scales in a theory which can't be interconverted with each other. For example, in the Newtonian framework there is a scale of length (L), time (T) & mass (M) and these can't be interconverted with each other. Note that although space is 3d, we don't have three dimensions of length, since a length along the x-axis can be converted to a length along the z-axis by rotation. Thus there is simply one dimension of space here which we standardly call length.

Answer 3

A system of units starts with a set of standards. So SI starts with a standard length, standard mass, standard time. In the old days, the standard mass was literally a chunk of material kept somewhere in France, but now the standard is defined in terms of physical properties so in principle anyone could reproduce it. Then, any other length, mass, and time is specified relative to these standards.

The reason we allow quantities to have dimensions is so that we know how to rescale the quantities if the standards change. If we change from a unit system with meter as the standard length to foot as the standard length, then keeping track of dimensions let us convert all lengths so they are correct relative to the new standard. Since physics can't depend on the choice of standard, any physically correct equation must be valid in any choice of standard, leading to dimensional analysis.

There is no rule that the standards must be length, time, and mass. There are systems of units with additional standards. For example, in some systems of units, temperature is a separate unit from energy, and in others it is not. In aviation in the US, it is common to specify horizontal distances in miles and vertical distances (aka the altitude of the plane) in feet -- so in that language, there is a different standard unit for vertical vs horizontal lengths. And, there are systems of units with fewer standards. In Planck units, by definition constants $\hbar=G=c=1$, and that completely fixes how all physical quantities are measured. In that system, "length" is understood to be "number of Planck lengths" (which as you can guess is not going to be a very useful system for everyday physics!)