Mass Unit in General Relativity

[Hornbein]

TL;DR

In his 1873 A Treatise on Electricity and Magnetism James Clerk Maxwell suggests that the monochromatic light emitted by an atom be used to define both the meter and the second. These two quantities could then be used to define the unit of mass, as it has the dimensions of (L^3)(T^-2). But this conclusion was deduced under the assumption that Newton's equation for gravity was correct. What would be the result under general relativity?

James Clerk Maxwell deduced that the unit of mass has the dimensions of (L^3)(T^-2). But he assumed Newton's Law. What would it be under general relativity?

 

[PeterDonis]

James Clerk Maxwell deduced that the unit of mass has the dimensions of (L^3)(T^-2). But he assumed Newton's Law.

I don't see how he could have deduced any such thing, since in Newtonian physics mass is an independent unit and cannot be expressed in terms of the units of length and time.

I suspect you are misreading or misinterpreting something.

What would it be under general relativity?

In GR, "geometric" units are often used, in which mass and length and time all have the same units (i.e., the speed of light $c = 1$ and Newton's gravitational constant $G = 1$). However, these units are not really "fundamental", since they do not take into account quantum mechanics. "Natural" units in quantum field theory have $c = \hbar = 1$, so length and time have the same units, and mass and energy have units of inverse length/time; in these units Newton's gravitational constant $G$ has units of length squared, or inverse mass squared.

 

[Hornbein]

You may read Prof. Maxwell's writings for yourself at https://ia600209.us.archive.org/28/items/electricandmagne01maxwrich/electricandmagne01maxwrich.pdf on page 3. In Adobe Acrobat it is page 42.

The relevant passage is

In descriptive astronomy the mass of the sun or that of the
earth is sometimes taken as a unit, but in the dynamical theory
of astronomy the unit of mass is deduced from the units of time
and length, combined with the fact of universal gravitation. The
astronomical unit of mass is that mass which attracts another
body placed at the unit of distance so as to produce in that body
the unit of acceleration.

In framing a universal system of units we may either deduce
the unit of mass in this way from those of length and time
already defined, and this we can do to a rough approximation in
the present state of science ; or, if we expect soon to be able to
determine the mass of a single molecule of a standard substance,
we may wait for this determination before fixing a universal
standard of mass.

We shall denote the concrete unit of mass by the symbol M
in treating of the dimensions of other units. The unit of mass
will be taken as one of the three fundamental units. When, as
in the French system, a particular substance, water, is taken as
a standard of density, then the unit of mass is no longer independent, but varies as the unit of volume, or as L^3.

If, as in the astronomical system, the unit of mass is defined
with respect to its attractive power, the dimensions of M are
(L^3*T^-2).

For the acceleration due to the attraction of a mass m at a
distance r is by the Newtonian Law m/r^2 . Suppose this attraction
to act for a very small time t on a body originally at rest, and to
cause it to describe a space s, then by the formula of Galileo, s = mt^2/2r^2 whence m = 2r^2s/t^2. Since r and s are both lengths, and t is a time, this equation cannot be true unless the dimensions of m are(L^3*T^-2). The same can be shewn from any astronomical equation in which the mass of a body appears in some but not in all of the terms f.

 

[PeterDonis]

The relevant passage is

In this passage, Maxwell is not saying mass is not an independent fundamental unit. (In an earlier passage, he already said it was, so his position on that is clear). What he is doing is much the same as what I described GR as doing: he is picking a convenient unit of mass for use in the kind of physics he wants to discuss. See below.

For the acceleration due to the attraction of a mass m at a
distance r is by the Newtonian Law m/r^2 .

But that is not the Newtonian Law. The Newtonian Law is that the acceleration is $G m / r^2$. The $G$ can't just be handwaved away. It's there for a reason: because $m / r^2$ by itself, in Newtonian physics, does not have the units of acceleration. So there has to be a physical constant in there to make the units balance.

What Maxwell is doing is basically picking a system of units similar to the "geometric units" I described, where we set $G = 1$ for convenience. Maxwell (perhaps suprisingly, given his discussion of how to determine the units of length and time just previously) does not take the additional step that relativity takes of setting $c = 1$ so that length and time have the same units. If he had done that, then his units for mass of [L^3 T^-2] would just end up as [L], exactly as in GR.

But, as I noted in my previous post, setting $G = 1$ can't be viewed as really "fundamental" in view of quantum field theory, because in QFT, mass can't have the same units as length; in "natural" QFT units ($c = \hbar = 1$) it has the units of inverse length. And in those units, you cannot have $G = 1$; $G$, in fact, is the inverse Planck mass squared in these units.

 

[Hornbein]

Gosh! That helps tremendously. In a month or year or whatever maybe I'll have learned enough to truly understand it. But for now it is very helpful to be shown the correct direction.