Motion is relative, right? And most pop sci relativity explanations are somewhat incorrect?

Question

On page 20 of A Brief History of Time:

. . . all observers should measure the same speed of light, no matter how fast they are moving.

But in an observer's frame of reference, they're actually not moving at all. So the sentence "should" read, ". . . they are moving, relative to each other."

BTW, I do realize that this kind of writing is intended to be easy to read and thus omits the obligatory caveats to every claim. I'm just trying to deeply, intuitively understand the material.

From a UVA lecture, Conserving Momentum: the Relativistic Mass Increase:

Mass Really Does Increase with Speed . . . particles become heavier and heavier as the speed of light is approached . . .

And here, isn't ". . . heavier and heavier relative to an observer . . ." actually correct?

Answer 1

There are two concepts of mass in relativity. One is the rest mass which is the mass of an object measured in the frame of reference in which the object is at rest. Thus, rest mass by definition does not depend on the observer and hence is sometimes called the invariant mass. The other concept of mass is relativistic mass which is the mass measured in a frame of reference where the object is moving and hence does depend on the observer.

The latter can be seen from mass-energy equivalence which implies that relativistic mass of an object must depend on its kinetic energy. In relativity, kinetic energy of an object moving with velocity v is \begin{equation} E_k = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}-mc^2 = \frac{1}{2}mv^2 + ... \end{equation}

where m is the rest mass of the object. The rest energy of the object is

\begin{equation} E_0 = mc^2 \end{equation}

Hence, the total energy is

\begin{equation} E=E_0+E_k=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} \end{equation}

Now, since mass is energy divided by c², the observer in the frame of reference in which the object is at rest will find the mass of the object to be m. At the same time, the observer in the frame of reference relative to which the object is moving with velocity v will find the mass to be

\begin{equation} m'=\frac{m}{\sqrt{1-\frac{v^2}{c^2}}} \end{equation}

It can also be shown that the momentum of the object in the frame of reference where it is moving is, not unexpectedly

\begin{equation} p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} \end{equation}

Note that term "mass" is increasingly used to mean the rest mass and the term "energy" is used to refer to relativistic mass. This is justified by the fact that relativistic mass multiplied by c² become total energy of an object. This removes the ambiguity of the term "mass".

To answer your question directly: It is not always strictly necessary to mention dependence on observer's frame of reference. The mention that a quantity discussed depends on velocity implies the "relative to an observer's frame of reference" part which is then often omitted for brevity. This follows from the fact that to even define velocity one needs to refer to a frame of reference. Thus, the omission does not constitue an error on the author's part.

That said, you are right that in the quote referring to the increase in particles' mass adding "relative to an observer's frame of reference" would be correct since the author discusses relativistic mass (rather than the invariant mass).