Mass Unit in General Relativity

In summary, James Clerk Maxwell deduced that the unit of mass has the dimensions of (L^3)(T^-2) in his discussion of a universal system of units. This is similar to the use of geometric units in general relativity, where mass, length, and time all have the same units. However, in quantum field theory, mass cannot have the same units as length, and the value of Newton's gravitational constant (G) in these units is not equal to 1. This highlights the limitations of using a single unit of mass in different theories.
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Hornbein
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TL;DR Summary
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?
 
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Hornbein said:
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.

Hornbein said:
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.
 
  • #3
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.
 
  • #4
Hornbein said:
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.

Hornbein said:
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.
 
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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.
 
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FAQ: Mass Unit in General Relativity

1. What is the mass unit used in General Relativity?

The mass unit used in General Relativity is the kilogram (kg). This is the standard unit of mass in the International System of Units (SI) and is used to measure the mass of objects in both classical and relativistic physics.

2. How does General Relativity define mass?

In General Relativity, mass is defined as the amount of matter an object contains. This is different from the classical definition of mass as a measure of an object's inertia. In General Relativity, mass is also a form of energy and is related to the curvature of spacetime.

3. Is mass conserved in General Relativity?

Yes, mass is conserved in General Relativity. This means that the total mass of a closed system remains constant over time, even though it may change form or location. This is a fundamental principle in physics and is supported by observations and experiments.

4. How does General Relativity account for the mass-energy equivalence?

In General Relativity, mass and energy are considered to be equivalent, as described by the famous equation E=mc^2. This means that mass can be converted into energy and vice versa. This concept is crucial in understanding the behavior of massive objects, such as black holes, in the theory of General Relativity.

5. Can mass affect the curvature of spacetime in General Relativity?

Yes, mass can affect the curvature of spacetime in General Relativity. This is one of the main principles of the theory, known as the Einstein Field Equations. The more massive an object is, the more it will curve the surrounding spacetime, leading to effects such as gravitational lensing and time dilation.

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