Frame of reference
The situation thus differs from Galilean relativity, in which all possible coordinate times are essentially equivalent.
For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference.
On the other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern.
In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related to space and time.
In a robot design, they could be angles of relative rotations, linear displacements, or deformations of joints.
Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized only by its state of motion.
An inertial frame of reference is defined as one in which all laws of physics take on their simplest form.
In Newtonian mechanics, a more restricted definition requires only that Newton's first law holds true; that is, a Newtonian inertial frame is one in which a free particle travels in a straight line at constant speed, or is at rest.
(All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.)
In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect metrology that is connected to the nature of the vacuum, and uses atomic clocks that operate according to the standard model and that must be corrected for gravitational time dilation.
In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules.
Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer.
The corresponding set of axes, sharing the rigid body motion of the frame
, coordinates are changed from R to R′ by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame.and this on the utility of separating the notions of
]:[6] As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made.
So frames correspond at best to classes of coordinate systems.and from J. D. Norton:[8] In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas.
The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods.
[…] Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.
This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…More recently, to negotiate the obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system.
