Inertial frame of reference

Question

I read in my Physics textbook that an inertial frame of reference should not spin. Thus, Earth should not be used to calculate the position, velocity, or acceleration of satellites and rockets. Instead, it should be done using stars as the inertial frame of reference. However, don't stars spin about their $y$-axis too? Thus utilizing Earth as an inertial frame of reference should be ok.

"Strictly speaking, Newton’s laws of motion are valid only in a coordinate system at rest with respect to the 'fixed' stars. Such a system is known as a Newtonian, or inertial reference frame. The laws are also valid in any set of rigid axes moving with constant velocity and without rotation relative to the inertial frame; this concept is known as the principle of Newtonian or Galilean relativity."

I found this above sentence on Britannica but it just reinforces my point. From my understanding, the movement along the $x,y,z$ planes is not the problem, it's the rotation.

Thus my question is:
Why should the inertial frame of reference not spin?

Answer 1

Here is how an inertial frame of reference can be established for the case of a double star, using only the motion of that double star system.

The two stars making up that double star system are orbiting each other. Both stars are moving, but there is a point that is not accelerated: the common center of mass of the two stars. The two stars are orbiting the common center of mass.

The orbital motion of those stars doesn't have to be circular orbit. The orbital motion can be in the shape of an ellipse. The general case is called 'Kepler orbit'.

To prepare for the following I recapitulate some properties of the mathematical object 'ellipse': an ellipse has a major axis and a minor axis, and those to axes are perpendicular to each other

The shape of a Kepler orbit is an ellipse. A Kepler orbit has the following property: the orientation of the axes remains the same. Conversely, you can use that property to find the inertial frame of reference, using only information from within that star system. The inertial frame of reference is that coordinate syste where the axes of the orbits remain in the same orientation.

Our own solar system is a single star system, of course, but we can still apply the same type of reasoning.

If you use a non-rotating coordinate system for representing the motions of the planets, then all planets move according to a single law of gravity: Newton's law of gravity.

What would you get if you would use a coordinate system that, unknown to you, has a small rotation rate? Then the orbits of the planets would not line up. You would need to fudge the orbit of each planet, and for each planet the fudging would be slightly different from the other planets.

The reason I'm describing this:
In order to measure which coordinate system is the non-rotating coordinate system for the solar system it is sufficient to use orbit data from the solar system only; that's enough.

Historically all planetary motion has been measured with respect to the background of stars. But if the solar system would be inside some interstellar dust cloud that would hide all distant stars, even then astronomers would eventually find the universal law of gravity, allowing them to find from measurement the inertial frame of reference of the solar system as a whole.

Answer 2

From a Newtonian perspective, the earth and sun are not inertial frames since the earth orbits the sun (accelerates) and the sun orbits the galactic centre (accelerates) and the galaxy ...However they are "close enough" for most calculations

The standard statements of Newtons Laws are now more than 300 years old and they carry implicit assumptions that are not at all obvious (e.g. existence of a common time). An inertial frame is one where $F = ma$ where $F$ is the resultant force calculated from our knowledge of engineering. In an non-inertial frame "mysterious" forces appear (e.g. centrifugal force).

If $x$ is an inertial frame coordinate system, and $y$ is moving a constant velocity relative to $x$, then $y = x + v.t$. Differentiating $a_Y = a_X$ so $F = ma$ in both coordinate systems. I.e. y is an inertial frame coordinate system too. If $y = f(x)$ rotates with respect to $x$, the $y$ would not be a inertial frame coordinate system since $F = ma$ would no longer hold.

Answer 3

In practice, what constitutes a reference frame is an experimental question (depending on how accurately you can measure accelerations). . If you consider real forces (exerted by a physical entity or associated with gravity) it happens that in the so-called non-inertial frames, you have:

$$\boldsymbol{F}_{phys} \neq m\boldsymbol{a}$$

In order to restore the equality, you need fictious forces:

$$\boldsymbol{F}_{phys} + \boldsymbol{F}_{fict} = m\boldsymbol{a}$$

A frame with origin the center of mass of the sun is is very approximately inertial, since the spin of the sun around the center of mass of the galaxy takes hundreds of years to execute and leads to minuscule fictitious forces (Coriollis's force, centripetal force, ...).

On the other hand, there is no simple way to know whether a given reference frame is inertial or not. One theoretical possibility is to place an electric charge stationary with respect to the origin of the reference frame, since any acceleration will cause the electric charge to emit radiation (but this may have such a small intensity that it may not be easy to measure).

Finally, in general relativity the equation of motion is:

$$m\left(\frac{d^2 x^\mu}{d\tau^2} + \Gamma_{\sigma \nu}^{\mu} \frac{dx^\sigma}{d\tau}\cfrac{dx^\nu}{d\tau}\right) = F^\mu$$

being $\Gamma_{\sigma \nu}^{\mu} = 0$ for an inertial frame.