In this insert, we explicate the basics of inertial frames: correctly, clearly, concisely---the last only in a sense.
Our discussion mostly assumes the classical limit where Newtonian physics applies and relativistic effects are vanishingly small. Occasionally, relativistic effects are mentioned, but a full discussion of them is beyond our scope.
Some terms from Newtonian physics and astronomy have to be introduced without explanation or much thereof. If one tries to explicate everything all at once, the explications goes on forever. To some degree, one just has to swim in Newtonian physics and astronomy.
In order to have an undistracted concise explication, we have put this explication in the Narrative Sections which have NO figures. Following the Narrative Sections, we have the Figure Sections which complement/supplement the Narrative Sections.
A reference frame
is a coordinate system
used for describing locations and motions of
(physical) objects
in space.
Usually (including in the astrophysical realm), the
coordinate system
is attached to some kind of
object
(in the astrophysical realm, an
astronomical object).
For an example of a
reference frame
for spacetime, see
below the figure section
An Illustration of a Spacetime Reference Frame.
REST OF THIS part need RECONSTRUCTION
A reference frame
is just a coordinate system (or some simpler kind of reference system if you want to think that way)
that one lays on all space or some subset of all space.
It allows you to locate objects in space and describe their
motion (in physics jargon,
their kinematics).
Usually, there is a specified origin for
a reference frame.
There are infinitely many
reference frames
that one can use:
they can have different
origins,
different kinds of coordinates
(e.g., Cartesian coordinates,
spherical coordinates),
and be in any kind of motion.
See the figure below
(local link /
general link: frame_reference_spacetime.html)
for an example reference frame
which includes a time coordinate.
Reference frames
can be just geometrical descriptions or they can be
attached to some physical object, and/or have a definite physical
nature which is the case for
inertial frames
most importantly of all.
There is a special class of
reference frames
that have definite physical nature:
inertial frames.
They are NOT just arbitrarily defined
coordinate systems
in space.
Almost all
physical laws
are referenced to
inertial frames.
By referenced, we mean
locations and motions
of particles,
objects,
system,
etc.
in formulae
that embody those physical laws
must be relative to
inertial frames
in order for those
physical laws
to be obeyed.
Note that such physical laws
are NOT wrong in
non-inertial frames
because they are NOT defined relative to them.
The prime exception to the rule
that physical laws
are referenced to
inertial frames
is general relativity
(presented 1915
by Albert Einstein (1879--1955))
which tells us what
inertial frames are
in our modern understanding.
There was an older understanding
in Newtonian physics
in the orginal sense due to
Isaac Newton (1643--1727)
which discuss in a narrative section below???.
Are there other exceptions?
Yours truly is NOT really sure.
It may be a matter of perspective.
For example,
2nd law of thermodynamics
is thought to hold in almost any imagined
world no matter what its
physics.
But in the actual world, it seems likely that
one would only apply it relative to
inertial frames.
Here is an incomplete list the
the physical laws
and/or physics theories
which must be referenced to
inertial frames:
Newtonian physics
(in our modern understanding),
classical electromagnetism,
special relativity,
quantum mechanics,
quantum electrodynamics,
quantum field theory,
etc.
A key example physical law
that must be referenced to
inertial frames
is that the
vacuum light speed c = 2.99792458*10**5 km/s
≅ 2.998*10**5 km/s ≅ 3*10**5 km/s ≅ 1 ft/ns
is the fastest physical speed
relative to all local
inertial frames.
Points to explicate:
Luckily, all
reference frames
at least on sufficiently small scales
in space
and time
(i.e., spacetime
in relativity speak)
can be inertial frames
if you choose to regard them as such
by invoking
inertial forces
(which we explicate in narrative sections below???) when NEEDED.
Inertial forces
account for
accelerations
relative free-fall
inertial frames
(which we explicate in narrative sections below???).
Inertial forces
are sometimes
called fictitious forces,
but yours truly deprecates that
term because
inertial forces
act just like
gravity
on sufficiently small scales.
It is an
axiom
of general relativity
that inertial forces
and gravity have
a fundamental likeness.
So using inertial forces
is NOT a trick, it is a perspective that
may be taken if it is convenient to do so.
There are important cases where it is convenient.
We consider some in narrative sections below???.
When we say inertial frame
without qualification, we usually mean
free-fall
inertial frame.
The other case is a
non-inertial frame
converted to an
inertial frame by the
introduction of
inertial forces.
We discuss
converted inertial frames
and inertial forces
in narrative sections below????
To explicate further explicate
free-fall
inertial frames,
let's first limit the discussion to
reference frames
defined by
objects.
Trying to discuss
reference frames
that are just arbitrary
coordinate systems
in space
is tedious and, in yours truly's opinion, pretty useless
at the level of this insert.
First, general relativity
dictates that a point in
free fall
in a gravitational field
defines a free-fall
inertial frame:
i.e., an inertial frame
that moves with the point.
Second, consider a
physical system
on which the only external force
acting is gravity.
The
center of mass
(i.e., the mass-weighted
mean position)
of the physical system
is in exact free fall
as dictated by
Newtonian physics.
So the
center of mass
(or a point at rest with
respect to it) defines
the origin
for a free-fall
inertial frame
for the physical system.
Below we will
consider 3
important examples of
free-fall
inertial frames.
For the sake of conciseness, we them
weightless free-fall inertial (WFFI) frames
(section Weightless Free-Fall Inertial (WFFI) Frames,
center-of-mass free-fall inertial (COMFFI) frames,
(section
Center-of-Mass Free-Fall Inertial (COMFFI) Frames,
and
ground free-fall inertial frames (GFFI) frames
(section
Ground Free-Fall Inertial Frames (GFFI) Frames).
An important point about
free-fall
inertial frames
is that they do NOT rotate with respect to the
observable universe
(i.e., to the bulk mass-energy
of observable universe)
and therefore do NOT rotate with respect to each other
(see Wikipedia:
Inertial frame of reference: General relativity).
The last statement shows that there is an
absolute rotation
at least in
observable universe.
Yours truly likes to
specify
absolute rotation
by saying
"rotation relative to the
observable universe".
However, a more conventional expression meaning the same thing for
most purposes is
"rotation relative
In modern times, one can use
the fixed stars
to measure
absolute rotation
approximately.
For more exact measurements, one uses
distant
extragalactic sources,
mainly quasars
(see
Wikipedia:
International Celestial Reference Frame).
Isaac Newton (1643--1727)
posited there was an
absolute space
in which the
fixed stars
(which were all the
stars) were
at rest
at least on average.
Absolute space
was the single fundamental
inertial frame.
All reference frames
NOT
accelerated relative to
absolute space
were also inertial frames.
The theory
of absolute space
continued to be held by some up to the
1920s.
The observational discovery of the
expanding universe
in 1929
by Edwin Hubble (1889--1953)
and its theoretical understanding in terms of the
Friedmann-equation (FE) models
derived from
general relativity
caused absolute space
to be discarded.
Are there fundamental
inertial frames?
Yes. They are
reference frames
that participate in the
mean expansion of the universe
and they are called
comoving frames.
The centers of mass
of field galaxies and
galaxy groups and clusters
approximately coincide with
comoving frames.
Furthermore, the
cosmic microwave background (CMB)
is isotropic when
viewed in a
comoving frame.
In
non-comoving frames,
the CMB
is distorted by
a direction-varying Doppler shift
due to the motion of that
non-comoving frame
relative to the local
comoving frame.
For observers on Earth,
this direction-varying Doppler shift
is called
CMB dipole anisotropy
(see
Wikipedia: CMB dipole anisotropy;
Caltech:
Description of CMB Anisotropies).
The CMB dipole anisotropy
is further explicated below in
the figure section
The CMB Dipole Anisotropy
Now the
comoving frames
are
free-fall
inertial frames, but
there are two qualifications:
The effect causing
acceleration of the universe
is usually unimportant on scales much less than that of
the observable universe
and is NOT usually mentioned unless it is of importantance to an analysis.
The simplest
free-fall
inertial frames
to consider are
reference frames
in free fall
under in a uniform external
gravitational field
with negligible internal
gravitational field.
Because of the negligible internal
gravitational field,
the objects
in these
free-fall
inertial frames are
weightless:
hence the description
weightless free-fall inertial (WFFI) frames.
Examples of
WFFI frames
include
spacecraft
coasting or orbiting
in outer space
or a falling elevator
in vacuum.
In both cases, to be exact, it is the
center of mass
(i.e., the mass-weighted
mean position)
that is in exact free fall
since internal forces cancel out for its motion
due to
Newton's 3rd law of motion.
So the
center of mass
(or a point at rest with
respect to it) is the most accurate choice for the
origin
of a WFFI frame.
However, if a physical structure
defining a
WFFI frame is massive
compared to objects
it contains
and rigid, then any point in the structure could be used for the
origin
of the inertial frame.
Note that in WFFI frames,
everything is weightless
because everything is in
free fall
(when not acted by forces other
than gravity), NOT
because gravity is
zero or negligible
in general.
For example,
in the low-Earth-orbit
the gravitational field
of the Earth
is only slightly smaller than
it is on the Earth's surface.
However, a
spacecraft and its
contents
are perpetually free falling toward the
Earth, but
they keep missing the Earth
because they have a sideways
velocity: in
physics jargon,
they have angular momentum.
Literally, being
in a non-escape orbit
is to be perpetually free falling
because you keep missing the
orbited astro-body.
For an illustration of
WFFI frame,
see below figure section
An Illustration of Weightless Free-Fall Inertial (WFFI) Frames.
The concept of
center-of-mass free-fall inertial (COMFFI) frames
is actually very general.
The WFFI frames
of narrative section
Weightless Free-Fall Inertial (WFFI) Frames
are a special case of
COMFFI frames.
However, the concept of
COMFFI frames is most obviously
applied to
astrophysical systems consisting multiple
astro-bodies
that can individually be considered as
point masses.
Such systems
are the subject of celestial mechanics
and N-body simulations.
For COMFFI frames,
the net external
gravity
determines the motion of
center of mass
(AKA barycenter in this
context).
If the
COMFFI frames
under consideration
is sufficiently well isolated from
other systems,
only the internal
gravitational fields
are important for the motions relative to the
barycenter.
If it is NOT, then the
tidal force of
those other systems
must be considered.
A chosen COMFFI frame
is most useful when the
tidal forces
are small or negligible.
More details on COMFFI frames
are given the below figure section
Hierarchy of Center-of-Mass Free-Fall Inertial Frames (COMFFI Frames) Illustrated.
By rotating frames,
we those rotating
observable universe.
Rotating frames
are non-inertial frames,
but simple ones.
Every small region in them over a short enough time scale is a simple
non-inertial frame,
but over all they are a continuum of such simple
non-inertial frame.
Nevertheless, they can be converted to
inertial frames
easily in the
classical limit
by invoking two special
inertial forces:
the centrifugal force
and the Coriolis force.
The centrifugal force
is that "force" that tries to throw you off
carnival centrifuges.
In the rotating frame
it is an outward pointing
body force trying to throw every bit
of you outward and a real
force has to be exerted on
you to hold you in position.
From the perspective of the
approximate inertial frame
of the ground
(i.e., a GFFI frame:
see narrative section
Ground Free-Fall Inertial Frames Frames below),
you are just trying to move at a uniform
velocity
in a straight line per
Newton's 1st law of motion.
The Coriolis force is a bit trickier
and arise when you have
velocity relative to
rotating frames.
Both
the centrifugal force
and the Coriolis force
are important in understanding the internal motions
of planets
and stars
which are always
rotating frames.
More details on on
rotating frames
are given below in the figure section
Rotating Frames Explicated.
For an important example of the
centrifugal force
at work, see below the figure section
The Centrifugal Force on the Earth.
For important example of the
Coriolis force
at work, see below the figure section
The Coriolis Force on the Earth.
One of the things that is obvious is that
the ground anywhere on
Earth
is NOT in
free fall
in the way you ordinarily think of
free fall.
UNDER CONSTRUCTION
For an example of a picturesque
GFFI) Frame,
see the below figure section
An Example of a Picturesque Ground Free-Fall Inertial (GFFI) Frame.
But every is NOT so hard since, in fact, any
reference frame
on a sufficiently small scale in
space
and time
(i.e., in spacetime)
is an inertial frame if
you choose to analyze it as an
inertial frame.
If choose NOT, then that is because you can
analyze it as
an accelerated
reference frame
relative to some other
reference frame
analyzed as an
inertial frame.
Accelerated
reference frames
in the sense just used are
non-inertial frames.
Usually ovewhelming convenience in analysis dictates whether to analyze
a reference frame
as an inertial frame
or a non-inertial frames.
You need some experience with special cases to
know how to make the convenient choice for analysis---but that is what
introductory physics courses are for.