Dimensional analysis

Dimensional analysis is a powerful tool based on a simple idea: look at the kinds of quantity involved in a system, as reflected by their units of measurement, and make sure that your equations and analysis are consisistent about this simple information about the quantities. The kind of a quantity is also called its dimension; but this leads to confusion with dimension in the sense involved in linear spaces, so I'll call it a kind here.

There are two primary uses for dimensional (kindly ?) analysis: sanity-checking some formula one has derived (are the quantities to be added together, or declared equal to one another, of the same kind ?) and working out what formulae could be relevant – which generally comes down to determining the ways one can combine the known data of the system to yield a dimensionless quantity – that is, one of the same kind as pure numbers; a ratio of quantities of like kind. Another variant on the second approach is to identify combinations of data of the system that are of some simple kinds, that let us reduce our analysis to these combinations rather than dealing independently with all the data they combine. For example, the system may offer us a natural charge-to-mass ratio that we can use to eliminate the difference between charges and masses in the system – or, even to eliminate charges and masses altogether (see the second example below).

It is usual to express the kinds of quantities in terms of four primitive kinds:

I'll write ~ for is of kind, so any mass ~ M, speed ~ L/T, and so on. Thus energy ~ L.L.M/T/T; electrostatic potential (voltage) is energy per charge so ~ L.L.M/Q/T/T; and electrical impedence is potential over current so ~ L.L.M/T/Q/Q.

The simple pendulum

To illustrate the approach, let's consider the case of a simple pendulum. This comprises a massive body attached to one end of a rod (typically light enough that we can ignore its mass relative to the main one) whose other end is fixed but free to turn, with negligible friction, so that the rod and mass can swing back and forth. It is observed that such a pendulum swings to and fro, repeating its earlier motion cyclically with a definite period which doesn't seem to depend much on the size of the movement, as long as this is tiny compared to the length of the rod. The known data of our system are:

m = mass
How heavy is the object on the end of the rod ? Of kind M.
h = length
How far is this mass from the pivot point ? Of kind L.
r = range
How far apart are the extremes of the object's movement ? Of kind L.
t = period
How long does the pendulum take to complete one cycle of its movement ? Of kind T.
g = gravitational field strength
How rapidly do unsupported massive objects accellerate downwards ? Of kind L/T/T.

Now, we're given that t doesn't seem to depend much on r, at least as long as r/h is small, so let's ignore r for now. Only m has M in its kind, so we can immediately see that it can't play any part in the relationship among the other quantities: we can predict that the period of a pendulum doesn't depend on its mass (as long as this is large enough to dwarf the mass of its supporting rod, and large enough that its weight dwarfs the forces we're ignoring, such as air resistance). We can make this physical prediction based only on looking at the kinds of quantity involved !

Furthermore, when we look at h, t and g we can see that the only dimensionless quantity we can form from them is g.t.t/h. We can expect any such dimensionless quantity to be a constant determined by the abstract geometry (without reference to absolute scale) of our system; it may have stray factors of π and other pure numbers in it, but our system is simple enough that we can expect it to have little more. We can thus predict that the period, t, should be proportional to √(h/g).

Now, of course, such predictions should always be tested experimentally: there may be some other quantities involved which we haven't noticed, which might be of kinds that mix up the quantities we were taking into account, thereby complicating the story. There might be some universal constant (or parameter of our context, c.f. gravitational field strength) which we didn't take into account, that isn't dimensionless. However, it commonly happens that one does identify the correct quantities and, from a knowledge of which forces are involved, which universal constants are in play. From these, dimensional analysis yields the most obvious hypotheses to test; and these often are correct, at least to a good approximation. This can save a lot of effort on the way to discovering a full and accurate description of the system.

Planck's Units

While I'll leave the main discussion of this elsewhere, Planck's units are an example of dimensional analysis. In 1904, Max Planck noticed that certain theoretical complications in the relationships between electrodynamics and thermodynamics could be resolved by assuming that light's interaction with matter only happens in discrete chunks, transferring an amount of energy proportional to the frequency of the light involved. The constant of proportionality, h, is known as Planck's constant; and h ~ energy / frequency = L.L.M/T. This only involves length mass and time, but Planck noticed that it couldn't be expressed in terms of the kinds of the other known universal constants: the speed of light, c ~ L/T, and Newton's constant, G ~ L.L.L/M/T/T. (The electromagnetic constants μ0, ε0 and Z0 can all be determined, given c, from any one; and each involves charge; so could not be relevant.) This meant that c, h and G are three independent kinds expressible purely in terms of the three base kinds M, L and T; from which he could infer that it must be possible (because there are three of each) to invert the system and express M, L and T in terms of c, h and G. The interested reader is encouraged to work out how to do that, before reading on.

We can use c to eliminate either L or T from the others: G/c/c ~ L/M and h/c ~ L.M. Multiplying these together we get L.L while dividing the second by the first yields M.M; the square roots of these are then √(h.G/c)/c ~ L and √(c.h/G) ~ M. Applying c to the first we get √(h.G/c)/c/c ~ T. When we work out the values of these, we find that the length and time are spectacularly tiny (even when compared to, say, the diameter of an atomic nucleus, although Planck didn't know about those at the time); while the mass is merely small – a droplet of water with this mass is about a third of a millimetre across and may have fairly complex life-forms doing interesting things inside it. It is common to use these units in analysing problems in theoretical physics, so as to make c, h and G all take the value 1 and thus save the need to keep track of them: I, however, prefer to keep track of our arguably spurious units, since doing so leaves me able to use dimensional analysis to check that I haven't made stupid mistakes (or, often enough, makes it easier to find my stupid mistake when I have made one).

General Relativity

When it comes to looking at a smooth manifold, we can distinguish displacements within the manifold from lengths and times: the metric (and the speed of light) perform conversions between these, but displacements themselves are not lengths or times. So add a kind S to describe displacements. At the same time, relativity definitely tells us to treat length and time as equivalent; so I'll suppress T in favour of L but keep track of the factors of c (the speed of light) that this implies.

Basic structure

The metric of space-time, g (not to be confused with the simple pendulum's g), is combined multiplicatively with two displacements to yield the square of a length (or time, but I'll stick with length). So g ~ L.L/S/S. Now, g is a linear map from tangents, intrinsically of kind S, to gradients, intrinsically of kind 1/S; but it's throwing in a dimensional factor of L.L in the process. In practice, we can't measure displacements, per se, only lengths and the like: so we can't directly examine any tensor rank unless its intrinsic factors of S all cancel. Thus we can look at linear maps from tangents to tangents (because their tensor rank is a tangent as output times a gradient that consumes the input, so the implicit factors of S cancel) but we can't look at tangents themselves. Equally, we can't measure g; but we can use it to convert tensor ranks to ones we can measure; doing so amounts to converting every factor of S to a factor of L.

The differential operator on space-time (with respect to which g is constant) maps any function to one which, when multiplied by a displacement, yields a change in the original function; so the differential operator's kind is 1/S. The Riemann tensor is a linear map from gradient fields to tensor fields of third rank, all factors in which are gradient fields; it determines the result of antisymmetrising the second derivative of a gradient field in the two factors of gradient field brought into play by the two uses of the differential operator. Thus the Riemann tensor's kind is just that of the square of the differential operator, 1/S/S. The Ricci tensor is a trace of the Riemann tensor which contracts out the tensor factors associated with the input gradient field, so is consequently of the same kind.

Sanity checks

We have a gravitational field equation, linking the Ricci tensor to the energy-momentum-stress tensor, an electromagnetic source equation linking the relativistic electromagnetic field (which unifies the electric and magnetic fields) to the 4-current and various equations of consistency. These last all equate some single expression to zero – tacitly the zero of whatever kind the relevant thing is – so don't give us any scope for dimensional analysis (although they would if they equated some sum of expressions to zero). The Bianchi identities, constraining the Ricci tensor, are of this kind; as are the conservation-law for current, d^(μ(J)) = zero, and the consistency equation of the electromagnetic field, d^F = zero (see below for what μ, J and F are).

The electromagnetic field tensor, F, encodes the electric and magnetic fields; it's of the same kind as E or B.c, so it's a force per unit charge. However, it's also a tensor field of rank gradient times gradient, so the physical-ranked thing derived from it that really has this kind is F/g; so F/g ~ M.L/Q/T/T = c.c.M/Q/L and F ~ c.c.M.L/Q/S/S. The 4-current density, J, is a tangent quantity so needs a factor of S/L times its intrinsic current per unit area or speed times charge per unit volume, c.Q/L/L/L, yielding J ~ c.Q.S/L/L/L/L. It's linked to F by a field equation

wherein Z0 is the impedance of free space (a universal constant), g\F/g = Alt(2, g, F) is a tangent times tangent quantity obtained from F by applying g's inverse to each (gradient) tensor rank of F and μ×μ = −det(g) defines (up to a constant sign) μ as an alternating form, of kind Alt(dim, {gradients}) when the manifold has dimension dim, which can be used as a linear map which contracts away tangent ranks of a tensor (antisymmetrically); when it contracts away n of them, it contributes dim−n remaining gradient ranks to the result. Thus μ's rank is L/S raised to power dim; but it appears on both sides of the equation, so we can ignore it in our dimensional analysis. We can now sanity-check our equation:

g\F/g
~ (L.L/S/S) \ c.c.M/Q/L
= c.c.S.S.M/Q/L/L/L;
J.Z0
~ (c.Q.S/L/L/L/L).(L.L.M/T/Q/Q)
= c.S.M/Q/T/L/L
= c.c.S.M/Q/L/L/L,

agreeing with what's left of the equation's other side after d^ (the alternating differential operator) has knocked out one factor of S.

The remaining contribution of F to the (dimensionally non-fatuous) field equations is via its contribution to the energy-momentum-stress tensor, which is (F/g\F −g.trace(F/g\F/g)/dim)/c/Z0 in which the second term trivially has the same dimensions as the first (it divides by a factor of g and multiplies by a factor of g) so I'll ignore it. Fitting together the parts we have yields F/g\F/c/Z0 ~ c.c.M/L/S/S, in which c.c.M is indeed an energy; the tensor is of rank gradient times gradient, the same as g, so dividing it by g would turn the /S/S into /L/L and give us an energy per unit volume, or energy density, as we should expect.

The field equation for gravity multiplies the energy-momentum-stress tensor by κ = 8.π.G/c/c/c/c and identifies it with Ricci −g.(trace(Ricci/g)/2 +Λ); as before, the g.trace term is transparently of the same kind as the first, 1/S/S, so I'll ignore it, aside from noting that the cosmological constant, Λ, has the same kind as Ricci/g, so Λ ~ 1/L/L is an inverse area. Now, G is Newton's constant, so ~ force.L.L/M/M = c.c.L/M, so κ ~ L/M/c/c and κ.F/g\F/c/Z0 ~ 1/S/S matches Ricci.

Reformulation

In the last, we scaled F/g\F, which has two factors of F in it, by κ/c/Z0 = 8.π.G/Z0/c/c/c/c/c ~ Q.Q/M/M/c/c/c/c, which is the square of a Q/M/c/c or charge-to-energy ratio. If we scale F ~ c.c.M.L/Q/S/S by a Q/M/c/c, the result would be a simple L/S/S, whose physical expression is obtained by dividing by g to obtain a 1/L. The natural charge-to-energy ratio derived from the universal constants of the field equations is ρ = √(4.π.G/Z0/c)/c/c – in which the factor of 4.π comes from G's rôle in Newton's equation of gravitation being equivalent to that of 1/(4.π.ε0) = c.Z0/4/π in Coulomb's equivalent force law for electrostatics – with a value of about 0.9587e-27 Coulomb / Joule. So let f = ρ.F ~ L/S/S and the equation of gravitation becomes

wherein W is the energy-momentum-stress contribution from everything but the electromagnetic field (which is presumably zero in the free-field case). We can re-arrange this as

in which the left side is expressed in terms of fields which only involve length and displacement as kinds. Naturally, we must now replace F with f throughout our other field equations; d^F = zero simply becomes d^f = zero, which is easy enough. The interesting one is then

naturally encouraging us to introduce j = J.ρ.Z0 ~ S/L/L/L, a displacement density, whose physical manifestation would be an inverse area. Its conservation law would simply be d^(μ(j)) = zero as for J. The Lorentz force law would then give the (proper time) rate of transfer, per unit volume, of 4-momentum between a charged medium carrying current density j and the electromagnetic field as

4-force density
= g\F·J/c

in which the g\ is present to turn the covector (gradient-valued) F·J into a vector (tangent-valued) quantity and the /c is present to turn the current density J into a charge density to be acted on by the force per unit charge, F.

= g\f·j/ρ/ρ/Z0/c

The constant ρ.ρ.Z0.c = 4.π.G/c/c/c/c ~ L/M/c/c, so its inverse (by which we're multiplying g\f·j) has dimensions of energy over length, i.e. force; while g\f·j ~ S/L/L/L/L, an S/L per unit volume, so our 4-force density's physical manifestation would indeed have dimensions of force per unit volume. Note that the force constant, a kind of Planck force, is the only physical constant remaining in the free-field theory (i.e. when W is zero); and its value is

which, when divided by Earth's surface gravity to get the mass whose weight it would equal, comes out at about 1.6 times the mass of our Milky Way galaxy.


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