# Absolute (as opposed to relative) concept of inertial frame

In mechanics there is a relative concept of "inertial frame": frame A is inertial with respect to frame B if A moves uniformly with respect B. That concept is easy to understand.

There also seems to be an absolute concept of "inertial frame". I keep reading things like "A is an inertial frame", without specifying with respect to which other frame B. Every time I read that kind of statements I get stuck. I cannot see how A can be "inertial". I can only see how it can be "inertial with respect to B".

Related to this, I keep reading things like "the solar system is accelerating" or "an object is moving" (for example here and here). Those statements I simply can't understand, unless they specify with respect to what frame (or object) that movement is defined.

I suspect my inability to understand the absolute concept of inertial frame is related to my inability to understand the statement "an object is moving". I only keep wondering: "with respect to what?".

So, my question is: can you really say "A is inertial" or "B is moving" in an absolute sense? (i.e without adding "with respect to C"). If so, how is that interpreted?

You really have two questions here

1. How do we identify inertial frames?
2. How is it that acceleration is not relative when position and velocity are?

The first one is harder than the second.

## Identifying inertial frame

We define an inertial frame as one in which the laws of physics take on their usual (simple) form, and identify non-inertial frame by the need to impose (pseudo-)forces like the Coriolis Force that depend explicitly on place or direction to explain the behavior of objects.

The definition feels a little circular, but it does give rise to a simple method of finding additional inertial frames once you have identified a single one: any frame moving at constant velocity relative an inertial frame is also inertial.

## Acceleration not relative?

Allow that in some inertial frame $S$ we have identified all the forces acting on a body, applied newton's law and computed the acceleration $a = F_{net}/m$ of the body. The equation of motion of the body is (using one dimension only to simplify the mathematical expressions) $$x(t) = x_0 + v_0 t + \frac{1}{2} a t^2 \,.$$

Now we observe the same action from a second frame $S'$ moving at velocity $u$ as seen from frame $S$. For simplicity (again) we'll assume that at $t = t' = 0$ the origins of $S$ and $S'$ coincided. That means that we can compute positions as measured in either frame from the position as measured in the other by \begin{align*} x' &= x - ut & x &= x' + ut \,. \end{align*} In particular it means that $x_0 = x'_0$ We also know that the initial velocity of our object in frame $S'$ was $v'_0 = v_0 - u$.

Taking these facts together we write the equation of motion in frame $S'$ by transforming that in \begin{align*} x' &= x - u t \\ &= (x_0 + v_0 t + \frac{1}{2} a t^2) - u t \\ &= x'_0 + (v_0 - u) t + \frac{1}{2} a t^2\\ &= x'_0 + v'_0 t + \frac{1}{2} a t^2 \,. \end{align*} Two things are immediately apparent: (1) that the algebraic form of the equation of motion is identical in the primed frame as that in the unprimed frame and (2) that the acceleration is also the same.

If you are of a particularly skeptical bent you may not quite believe this, but try choosing a set of values for $x_0$, $v_0$ and $a$; tabulating the motion for a while, converting the tabulation to the primed frame and computing the acceleration in the primed frame from the table.

• Thanks! You say "it does give rise to a simple method of finding additional inertial frames once you have identified a single one". But that's my relative concept of inertial frame, which I have no problem with. The absolute concept elludes me, though. – Luis Mendo Jan 25 '15 at 17:20
• @LuisMendo Well, that is the harder problem. The best I've got for you is "If the laws of physics take on their simplest form (i.e. without pseudo-forces) in your current reference frame, then you are in an inertial frame." That's what John is getting at. Only he's trying to boot the discussion all the way to GR. I stuck with the Newtonian concept here, because I find it easier to start there and move up. – dmckee --- ex-moderator kitten Jan 25 '15 at 17:23
• But how do you know if a force is pseudo- or not? Say you experience a force/pseudo-force. How do you know if it is caused by gravity exerted by a mass you don't see (and then it's true force) or by your system being non-inertial in an absolute sense (and then it's a pseudo-force)? – Luis Mendo Jan 25 '15 at 17:36
• You're asking all the right questions. The pseudo-forces arising from rotating coordinate systems all have an explicit dependence on position. Alas the ones arising from straight-line accelerating coordinate systems don't; they are just a ever present acceleration term (kinda like gravity in a near-the-surface-of-the-earth lab; but then gravity is a pseudo-force in GR). – dmckee --- ex-moderator kitten Jan 25 '15 at 17:39
• @dmckee We can identify the psuedo-forces arising from linear motion by their directional dependence. In an inertial frame the laws of physics are isotropic, i.e. independent of direction. Therefore we can identify an inertial frame as being one whose laws are isotropic and homogeneous. – 2012rcampion Jan 25 '15 at 20:36

Suppose you surround yourself with a sphere of test masses that are too small to have any significant gravitational field. You are in an inertial frame if the masses remain as a sphere and do not accelerate away from you. You do not need to refer to any other frames - this measurement is done entirely in your own frame and works even if you are in a sealed room (or spaceship!) with no way to see what is outside.

This works in both special and general relativity. For example the International Space Station is, from our point of view, accelerating towards the Earth because it's moving in a circle. However an astronaut on the ISS could do this experiment and would report that they are in an (approximately) inertial frame. The difference is that in GR the frame is only locally inertial. If the astronaut made the frame too big they would see the sphere deforming due to tidal forces.

You just need a different, more fundamental, definition of inertial frame. Landau and Lifshitz define it as "a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner", or within Newtonian dynamics, a frame of reference in which objects not acted by any force stay at rest or move with constant velocity. That inertial frames move with respect to each other with constant velocity is but a corollary of this definition.

• This sounds interesting. Can you elaborate how uniform movement is a corollary (or give a link where that is explained)? – Luis Mendo Jan 25 '15 at 17:00
• The Wikipedia page is a good place to start. Landau's first chapter has everything you want, including a rigorous deduction of the fact that inertial frames move with constant velocity based on the validity of Newton's first law in inertial reference frames. – andrepd Jan 25 '15 at 17:07
• Do you mean this book, chapter 1: The equations of motion? – Luis Mendo Jan 25 '15 at 17:10

I'd just like to add to the answer given by dmckee♦ regarding identifying inertial reference frames. The confusion you're having appears to be due to using a naive definition of an inertial reference frame. The first thing you need to understand is that gravity is another one of those pseudo-forces that dmckee♦ mentioned. Even if you ignore the Coriolis and centripetal forces that you experience on Earth (imagine it wasn't rotating), you are still in a non-inertial reference frame due to having to invoke gravity to explain why you're being attracted to the earth. This explains why someone standing on a non-rotating planet and someone in free fall are not moving at a constant velocity relative to each other, as it is only the person in free fall who is in an inertial reference frame. To frame this in terms of the simplified definition (where an accelerating reference frame is non-inertial), the person in free fall experiences zero proper acceleration whereas the person on the non-rotating planet does not.

You might be wondering how someone in free fall can be in an inertial reference frame if gravity needs to be invoked to explain their nonlinear motion around say, a planet. This is because someone in free fall isn't in a global inertial reference frame. They are only in an inertial reference frame locally, which is to say over a small area of space where the effect of gravity cannot be measured. You might also be wondering what causes planets to orbit stars if gravity is a fictitious force. The answer is that the planets are actually traveling in straight lines (meaning they're not being acted on by a force). The difference is that the straight lines occupy curved space and the paths traveled by the planets are known as geodesics. The curvature of this space is determined by the mass of the object occupying it.

Everything we can measure is relative, because measurement is comparision with a standard. Hence, there are, in a strict sense, no absolute frames in experimental science. The task is rather to make it clear in our mind against what we measure and specify our physical quantities, in order to be able to share our results and to build a common knowledge. Take the example of Newton's rotating bucket. If the bucket is closed with a cover, a state of rotation omega can be detected from inside the bucket by measuring the centrifugal force F = mr(omega)^2. This measurement is obviously not "relative to the fixed stars", but relative to the test mass m used to determine the state of rotation, the radius r being a mere factor of geometry. It is already interesting from a philosophical point of view that mass is used to determine a state of rotation; hence mass must be intrinsically linked to rotation.

• Just one puzzling thing in physics is absolute: the quantization of angular momentum, electric charge, spin, etc.! – Edgar Mueller Oct 1 '15 at 15:31

An inertial frame is one with respect to which Newtons second and first laws are valid.There is no ideal inertial frame in the universe although the heliocentric refrence frame fixed with the center of the sun can be regarded as an inertial frame with a high degree of accuracy. If we assume the heliocentric frame as an inertial one then all other frames moving with constant velocity with respect to it are also inertial.

Consider a number $N$ of bodies. It is generally impossible to fix their motion choosing your own motion with respect to all them. If $N=1$ you can, for instance, stay at rest with the body. But if $N>1$ you generally cannot.

However a remarkable physical fact happens.

If the $N$ bodies are far form each other and far from the other masses of the universe, there exists a reference frame where all them, simultaneously, move with constant velocity (generally different and depending on the body).

This is the physical content of inertia principle and defines the notion of inertial reference frame: a reference frame fulfilling the property above.

Next, it turns out that, given a such reference frame $I_0$, any another reference frame $I$ satisfies the same property (isolated bodies simultaneously move with constant velocity therein) if and only if $I$ moves with constant velocity with respect to $I_0$. This fact, differently form the physical one above, can be proved mathematically: it is nothing but a theorem.

Moreover, it can be mathematically proved that the set of transformations between orthonormal coordinates at rest with inertial reference frames is the well-known group, called the Galileian group.

Mathematically speaking, the principle of inertia selects an affine space structure in the spacetime, defined as the unique (up to isomorphisms) affine structure such that the Galileian transformation of coordinates (between inertial frames) are a subgroup of bijective affine transformations. The inertial motion of an isolated body is nothing but an affine geodesic parametrized by the absolute time, of that affine structure.

The affine structure defines an affine connection over the spacetime in the sense of differential geometry. Within this framework, the inertial forces turn out to be mathematically embodied in the Christoffel coefficients of the connection. From this viewpoint General Relativity and Newtonian dynamics are not so different.