Are curvilinear coordinates inertial? At 1:46:34 of this lecture by Frederic Schuller, Inertial coordinates are defined as ones which satisfy the following equation:



I am confused by the above equation because it would imply any curvilinear coordinate eg: polar is not inertial. I thought 'inertial' meant the frame is at rest w.r.t some absolute space/ absolute time.
Could someone explain how to make this consistent with previous knowledge?
 A: Indeed, it is valid to consider that polar coordinates are non-inertial. You should be aware that the term “reference frame” does not have one unique meaning. Different authors may use it to mean different things. Three common meanings are as follows:

*

*a reference frame is a coordinate system which is an invertible and continuous mapping from spacetime to R4.


*a reference frame is a tetrad which is an ordered set of four vector fields on spacetime, one timelike and three spacelike, all orthogonal to each other, and each of unit length.


*a reference frame is a physical collection of clocks and rulers used to measure the position and time of physical events in some lab or other region.
There are even authors that have more unusual or niche definitions of the term. As a result of the different definitions you can get conflicting descriptions, so it is important to know how different authors use the term.
In your question, it looks like the author may be using the term to refer to the coordinate system. With that definition it would be perfectly reasonable to say that polar coordinates are non-inertial. After all, if you write the laws of physics in polar coordinates they will have a different form than if you used Cartesian coordinates.
Just as there are different definitions for “reference frame” there are also different definitions for “inertial frame”. Three common meanings might be:

*

*an inertial frame is one where the laws of physics take their “standard” form


*an inertial frame is one where proper acceleration is equal to coordinate acceleration


*an inertial frame is one where any non/interacting objects moves in a straight line at a constant speed
Again, you will need to see how each specific author uses the term. These various definitions are usually equivalent in certain circumstances, but they can differ in some cases. Your book could be using either meaning 1) or 2).
As far as I know, no scientist today would define an inertial reference frame as one that is at rest with respect to some absolute space. I think that you probably will need to let go of that concept
A: Reference frames can be inertial or non inertial. Coodinate systems are not reference frames unless the frame is somehow being tied to the coordinate system. Does the book explain how the frame is attached? If not,  the book is making a non-standard definition.
