What is the definition of mass?

Question

From what I have seen so far, there seem to be two fundamental criteria for considering something to have mass:

• First is that it must have inertia and therefore momentum,
• Second is that it interacts gravitationally i.e. it curves spacetime.

Even when we measure the mass of something, we can't do it directly, we calculate it through measuring its inertia or through its weight i.e. gravitational interaction.
But then this seems to create a contradiction in considering photons to be massless, as photons fulfill both¹ these² criteria. So then the definition of mass has to be different, and if so what is the actual definition of mass? If the definition is as I said, then why are photons considered massless?

For more reference to the questions that this question was built on:

If photons have no mass, how can they have momentum?
How is light affected by gravity?

Answer 1

The mass $m$ of an object is defined in terms of its energy $E$ and momentum $\mathbf p$ by the equation

$$(mc^2)^2=E^2-(\mathbf pc)^2$$

or

$$m^2=E^2-\mathbf{p}^2$$

in units where $c=1$.

This equation has a geometrical interpretation in terms of Minkowski spacetime: the mass is the Lorentz-invariant “length” of the energy-momentum four-vector $(E, \mathbf p)$.

Energy and momentum are important quantities because they are conserved. But observers in different inertial reference frames disagree on their numerical values: energy and momentum are frame-dependent. By contrast, inertial observers all agree on the mass; it is a frame-independent quantity and thus an intrinsic property of the object.

This should not be surprising. For a three-vector in Euclidean space, observers in various frames rotated with respect to one another disagree on the components of the vector but agree on its length.

Photons have both energy and momentum. But these are related by $E=|\mathbf p|c$, so $m=0$ for photons.

An object with nonzero mass moving with velocity $\mathbf v$ has energy

$$E=\frac{mc^2}{\sqrt{1-\frac{\mathbf{v}^2}{c^2}}}$$

and momentum

$$\mathbf p=\frac{m\mathbf v}{\sqrt{1-\frac{\mathbf{v}^2}{c^2}}}.$$

$E$ and $|\mathbf p|$ both become infinite as $v\to c$, which explains why a massive object cannot move at the speed of light.

These formulas are not useful for photons, since they give the indeterminate ratio 0/0. However, you can argue that if $m=0$ then $v$ must equal $c$, otherwise $E$ and $|\mathbf p|$ would both be zero.

Answer 2

The distinction between inertial and gravitational mass makes little sense. Mass is (strictly: was) defined in terms of the standard $1$ kg platinum-iridium block kept in Paris .

It is not defined as "inertial mass". Nor as "gravitational mass". But simply as "mass".

The fundamental "MKS" mechanical units are mass (kg), length (m) and time (s) . Force not being one of these, it has to be defined in terms of them, for instance via Newton's second law. If a force applied to the standard $1$ kg mass in Paris results in an acceleration of $1 \frac{\text m}{\text s ^2}$, then the value of that force is by definition $1$ N.

This allows the masses of other objects to be determined. If a force applied to a body gives an acceleration a; and the same force applied to the standard $1$ kg mass in Paris produces an acceleration $a_1$; then the mass $M$ of that body is by definition $M=a_1/a$.

In possession of operational procedures for measuring quantitatively force and mass, Newton's gravitational constant $G$ can be determined experimentally. And that's it. No separate inertial and gravitational masses. Simply mass.