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I am a math student with some interests in physics. I picked up a book called "A First Course in General Relativity", and I am confused on the second page. I am assuming by notation or convention.

The chapter is on special relativity, and at this point they are just talking about how measurements of velocity are invariant by a constant. That is $v'(t) = v(t) - V$, where $v(t)$ is a measurement by one observer and $v'(t)$ is a measurement by another observer whose relative velocity to the original is $V$.

Then it says Newton's second law is unaffected by this replacement. It offers as an explanation, $$a' - dv'/dt = d(v - V)/dt = dv/dt = a.$$

I am confused in how this explains anything. Also as I read the notation, shouldn't $a' = dv'/dt$? So $a' - dv'/dt = 0$. Also the first equality is confusing, since $v' = v - V$, I thought, so using that replacement I get $$a' - dv'/dt = a' - d(v - V)/dt.$$ And I don't see where they go from there.

I assume I am just misunderstanding what is meant by a particular variable. Can anyone help shed some light on my confusion on this point? Thanks.

EDIT

I think I am being confused by a typo, replacing the $=$ with $-$ seems to make things make sense. I probably should have seen that.

$$a' = dv'/dt = d(v - V)/dt = dv/dt = a.$$

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2 Answers 2

up vote 0 down vote accepted

I think you are correct that it's simply a typo. Replacing the "-" with an "=" does indeed make the equation make sense.

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What you describe is a numerical invariance of the second law in the classical mechanics: not only the Newton equation has the same form $m\vec{a}=\vec{F}$, but also the left-hand and the right-hand side parts are numerically the same in different reference frames. I mean the Galilean transformations. Rotations, for example, preserve the form of equations, but change the numerical values of vector projections to the new axes.

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