# Are inertial frames relative?

I am not very sure of what my doubt is and will lay down the questions along the lines of which I feel problems.

Does an absolute inertial frame exist? If yes how do we define it? How can we say that it's acceleration is zero...I mean zero with respect to what? If not then what are approximately inertial frames and why are they so called?

Most importantly, What are the implications of approximating frames like say our earth which are actually non-inertial as inertial frames?

I have tried to find answers in other questions present on the site but most of them seem to be above my current level of knowledge, involving cosmic background radiation and what not.

P.S.- I am a grade 12 student and do not know much about Theory of Relativity or Quantum stuff.

• Some nice bedtime reading: plato.stanford.edu/entries/spacetime-iframes (philosophical/historical account)! Note how difficult it was (and unfortunately still is for many textbooks) to set up classical mechanics in a non-tautological way. One good reference is Classical Mechanics: A Contemporary Approach by José; the only other textbook on classical mechanics you will probably ever need to read is by Goldstein. In both, the first few pages will be accessible to you and are very relevant. I'm aware this is not an answer, but these references should help improve your question. Mar 8, 2020 at 21:22

First, an inertial object is one which is not subject to active forces. Literally, inertial means not active, from Latin, the language in which Newton wrote the Principia. Newton considered that an active force required contact. Ideally, we would define inertial as meaning that there are no contact interactions with other matter (including interation with light, or photons). This is not strictly possible, but we can get arbitrarily close to it:

• An inertial object is one such that, in any reference frame, the effect on its motion due to contact interactions with other matter is negligible.

We can now restate Newton's first law as a local law:

• N1*: An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter.

Newton's first law was needed in Newtonian mechanics to determine "absolute space", but relativity replaces this with the idea of a local inertial reference frame, based on N1*:

• An inertial reference frame is one in which inertial bodies remain at rest or in uniform motion.

It is implicit in this definition that inertial reference frames are local. That is to say, an inertial reference frame describes only a finite region of spacetime in which deviation from rest or uniform motion is not measurable and can be neglected. The size of this region depends upon the accuracy of measurement, but it is worth noting that a second in time corresponds to a light-second in distance. In terms of normal timescales, local refers to quite short intervals of time.

Einstein used a different definition "the laws of physics in their simplest form" but Newton's first law is more concrete, and it is sufficient to define inertial reference frames.

The Earth can be regarded as an inertial body, and, when you remove rotation, it defines an inertial reference frame, but this only works for a short amount of time. For a full description of spacetime you have to paste together inertial reference frames. The result uses the same mathematics, differential geometry, as for curved surfaces which can be seen as near flat in small enough regions. This is how we form "curved" spacetime but it must be understood that this is a mathematical definition of curvature - it does not mean that spacetime is actually curved in the same sense as a curved surface.

"Inertial" means "orthogonal", which in turn means this: A frame {e1,e2,e3,e4} is orthogonal (or inertial) if e1.e1=-1 , e2.e2 = e3.e3 = e4.e4 = 1, and all the various other inner products you can form (like e2.e3 or e1.e4) are zero.

There is nothing relative about this criterion. The equations above either hold or they don't. So everyone can agree on which frames are inertial.

• If you wish to describe frames with basis vectors on spacetime the right approach IMO is the one by Sachs & Wu in "GR for Mathematicians". Then a reference frame is a unit future-directed timelike vector field $Z$. You can always add $e_1,e_2,e_3$ so that with $e_0 = Z$ you have $g(e_\mu,e_\nu)=\eta_{\mu\nu}$. This does not make the reference frame inertial. An inertial reference frame, mathematically, is one for which $\nabla Z =0$, (c.f. Sachs & Wu for details, section 2.3 and Exercise 2.3.12).