Inertial frames are the tricky topic. They are NOT so hard to understand, but there seems NO short way to explicate them. One just has to grasp all their aspects at once in a big picture---and then they make sense---or so yours truly hopes.
Yours truly admits to having rewritten the discussion of them 10 times or more.
Here we give the somewhat abstract summary explication of inertial frames---which is longish---then in following file sections, we expand on it with diagrams and animations.
This means all inertial frames are unrotating with respect to each other.
Actually, a qualification is needed in that there may be reference frames that are inertial frames (in a sense) rotating with respect to the observable universe in very strong gravitational fields such as near black holes. But yours truly CANNOT find any reference that elucidates this qualification. It is hinted at by Wikipedia: Inertial frame of reference: General relativity. Yours truly will usually NOT refer to the qualification again.
One can quibble about whether there are other physical laws NOT referenced to inertial frames, but yours truly thinks the quibbling is a matter of perspective or may amount to saying you are NOT using inertial frames in some definitional sense when effectively you are using them.
The centers of mass (CMs) are the actual points in exact free fall in an external gravitational field: i.e., the net gravitational field due all sources external to the system.
Yours truly calls such systems center-of-mass free-fall inertial frames (COMFFI frames). It just seems natural to use the term COMMFI frame to mean the reference frame itself AND the system of astronomical objects used to define it.
However, such a part (call it part A) has its own center of mass (CM) and constitutes its own COMMFI frame with the other parts constituting the sources of part A's external gravitational field.
There is, in fact, a whole hierarchy of COMMFI frames in the observable universe.
A perfectly uniform external gravitational field is ideal since it CANNOT effect the motions of the astronomical objects relative the center of mass of the COMMFI frame.
In particular, a perfectly uniform external gravitational field CANNOT change the total angular momentum of a COMMFI frame about the center of mass (CM): i.e., the COMMFI frame has conservation of angular momentum.
If you need to analyze motions of astronomical objects outside of the COMMFI frame you are using, then you should probably use a larger COMMFI frame in the hierarchy of COMMFI frames that includes those astronomical objects.
No actual center of mass (CM) exactly participates in the mean expansion of the universe, but the centers of mass (CMs) of galaxy clusters and field galaxies (i.e., galaxies not in gravitationally-bound systems) do approximately.
Note the term "local" is used variously in
physics jargon.
Here we mean at the same place or in the same
COMMFI frame.
Note absolutely positively different
COMMFI frame
comoving frames
are NOT local with respect to each other.
Such frames
are in "acceleration" with respect to each other
in the expansion-of-the-universe motion,
but are still
inertial frames.
Any location on the surface of
the Earth
or any planet
is an approximate
inertial frame
for most purposes.
Thus, you can reference
Newton's laws of motion
to the ground for most purposes.
But NOT for all purposes: see sections
The Centrifugal Force of the Earth's Rotation
The Coriolis Force of the Earth's Rotation
below.
What are reference frames?
Just a set of coordinates one lays down on space or, including time, on spacetime as we say in relativity speak.
See the figure below (local link / general link: frame_reference_spacetime.html) for an example reference frame which includes a time coordinate.
Inertial frames are reference frames to which all physical laws in the actual observable universe are referenced with respect to, except general relativity (GR), as aforesaid in section Summary of Inertial Frames.
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
including those that are effective inertial frames.
This means acually all local observers measure the same
vacuum light speed: it is invariant as well
as fastest.
Points to explicate and/or expand on:
UNDER RECONSTRUCTION BELOW: everything is right I think, but there are repetitions
What are inertial frames?
They are free-fall frames
under gravity
unrotating with respect to the
observable universe:
i.e., to the bulk mass-energy
of observable universe)
(see Wikipedia:
Inertial frame of reference: General relativity).
How do you do physics
with non-inertial frames?
Well, you can just NOT use them and use
inertial frames instead.
There are always local inertial frames
wherever you are.
Inertial forces
are sometimes
called fictitious forces,
but yours truly deprecates that
term because
inertial forces
act just like
gravity
on sufficiently small scales.
A key point is that
inertial forces
are linearly dependent on mass.
just like gravity.
In fact, 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 many important cases where it is convenient.
Inertial frames
are illustrated in the figure below
(local link /
general link: frame_inertial_weightless.html).
Our current cosmological paradigm
of the expanding universe
(which is general to almost all currently though-of
cosmological theories
including the favored
Λ-CDM model)
tells us that the bulk
mass-energy
of observable universe is
NOT in rotation at least
in any sense we normally understand by the term
rotation.
A shorthand,
we say that the observable universe
is NOT in rotation.
Thus, we can say there is such a thing as absolute rotation:
i.e., rotation relative to the
observable universe.
How do we measure absolute rotation?
The accurate/precise
way at present is relative to cosmologically remote
quasars
whose peculiar velocities
relative to the mean
expansion of the universe
are believed to be negiligible from our perspective
on Earth.
Such measurements establish the
International Celestial Reference Frame
(ICRF).
However, at a lower, but often very adequate, level of
accuracy/precision
one can use the average array of the
traditional fixed stars
which are just the
stars that you see
at night.
The array of fixed stars
do actually have some absolute rotation, but for most, but NOT all, purposes
it is negligible.
Before the advent of modern
cosmology
(circa 1900--1930)
rotation relative to the array of
fixed stars was taken
as an exact measure absolute rotation
by most astronomers.
Of course,
the fixed stars individually
were known or assumed to have
peculiar velocities
since the time of
Isaac Newton (1643--1727), but
on average they were thought to be
at rest in
absolute space.
We will explicate
absolute space
below in the section
Absolute Space and Comoving Frames.
Note that many people and yours truly
occasionally say "relative to the
fixed stars"
as a synonym for relative to the
observable universe.
This is just a traditional usage and
yours truly is trying to get out of the habit of using it.
Absolute space
was hypothesized by
Isaac Newton (1643--1727)
to be the fundamental
inertial frame
(and the one in which the
fixed stars
[which were all the
stars in his age]
were at rest on average)
and only
reference frames
NOT accelerated with respect to it were
true inertial frames.
Now yours truly likes the perspective that
Newtonian physics
is a true emergent theory.
It is exactly true in the
classical limit.
But NOT
absolute space.
That was always a wrong hypothesis.
However, practitioners of
celestial mechanics
assuming absolute space
from Newton
on until the advent of
general relativity
in 1915 and even a bit later (see below)
still got the right answers for calculations of celestial motions
of the Solar System
and observable
multiple star systems.
Why?
They used the fixed stars
for defining absolute rotation (as we discussed in
section
What Do We Mean by Rotation With Respect to the Observable Universe?)
and that was adequate for their level of
accuracy/precision.
They then treated
the free-fall frames
defined by the
centers of mass
of their systems
unrotating relative to the fixed stars
as non-inertial frames
converted to inertial frames
by the use of inertial forces.
This procedure as far as the celestial motions they were dealing with
gives exactly the right answers since
converted non-inertial frames
are also
inertial frames.
Now Newton and those
other old practitioners of
celestial mechanics
could equally well have anticipated
the general relativity
perspective of
free-fall frames
unrotating with respect to the
observable universe
(which for them was the
fixed stars) all
being true
inertial frames
(unneeding of any conversion using
inertial forces),
but they
did NOT do so.
General relativity, of course,
tells us that its perspective on
inertial frames
is the correct one for the
observable universe.
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 thoroughly and most sincerely discarded.
So there is NO
absolute space
in the sense used by Newton.
What is there instead?
Free-fall frames
unrotating with respect to the
observable universe
and participating the mean
expansion of the universe
are now considered the most
basic inertial frames
or the most fundamental
inertial frames.
They are called comoving frames.
The centers of mass
of most
galaxy groups and clusters
and most field galaxies
define inertial frames
that are good approximations to
comoving frames: i.e., they coincide
approximately with exact comoving frames.
For the
expansion of the universe,
see the figure below
(local link /
general link: expanding_universe.html).
Furthermore on
comoving frames,
there are two fine points:
Because the two dark energy forms
act the same in the
Friedmann-equation (FE) models,
they are, as a shorthand,
often just collectively referred to as
Lambda
since the capital
Greek letter
Λ (pronounced Lambda) is the
symbol for
the cosmological constant.
It may be the neither of the
two dark energy forms
are the true cause of
acceleration of the universe.
Hopefully, we will find the true cause someday.
The cosmological constant Λ
(or whatever is causing the
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 importance to an analysis.
In fact, we can measure our local motions relative to our
local comoving frame
to good accuracy/precision
by measurement of the
cosmic microwave background (CMB).
The
CMB
is just electromagnetic radiation (EMR)
in the
microwave band (fiducial range 0.1--100 cm).
It strongest in the
energy/frequency/wavelength bands
∼ 1--22 cm
and has one of the most perfect
blackbody spectra
found in nature
(see Wikipedia:
Cosmic microwave background: Features).
It is a relic of the Big Bang era
speaking loosely.
It permeates the observable universe
and, according to
Big Bang cosmology
(which is highly trusted),
it 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 (ℓ=1);
Caltech:
Description of CMB Anisotropies).
The
CMB dipole anisotropy
is further explicated in
file
cmb_dipole_anisotropy.html.
We will NOT elaborate on the
CMB here.
But we can give some local velocities
determined using it:
If you attach reference frames
to the
centers of mass
of astronomical objects
and said reference frames
are
unrotating with respect to the
observable universe:
i.e., to the bulk mass-energy
of observable universe)
(see Wikipedia:
Inertial frame of reference: General relativity),
then those reference frames
are
inertial frame and
yours truly calls them
center-of-mass free-fall inertial (COMFFI) frames.
The observable universe
contains a whole hierarchy of
center-of-mass free-fall inertial (COMFFI) frames
which hierarchy is illustrated in the cartoon in the figure below
(local link /
general link: frame_reference_hierarchy_astro.html).
By rotating frames,
we mean those rotating
observable universe.
Rotating frames
are non-inertial frames,
but NOT simple ones.
Every small region in them over a short enough time scale is a simple
non-inertial frame
(i.e., a reference frame
accelerated relative to a local
inertial frame)
but overall they are a continuum of such simple
non-inertial frames.
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 an ordinary
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 below the narrative section
Ground Free-Fall Inertial Frames (GFFI) Frames),
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
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 Coriolis Force of the Earth's Rotation.
For important example of the
Coriolis force
at work, see below the figure section
The Coriolis Force of the Earth's Rotation.
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.
It and anything at rest
in the vertical direction are NOT obviously falling.
But actually, they are
free falling
with the center of mass
of the Earth in
the COMFFI frame
defined by that center of mass.
But because the
Earth is
in rotation
relative to the
observable universe,
the surface at every point is
NOT an
inertial frame.
But for most ordinary purposes, it is approximately
an inertial frame,
and so any point on the Earth
can be used to define
an inertial frame
for most purposes: e.g., for using
Newtonian physics
for most purposes.
The reason is that the acceleration
of the ground relative to the
Earth's
COMFFI frame
is actually small compared to the
Earth surface acceleration due to gravity (fiducial value 9.8 m/s**2)
and other relevant accelerations.
In fact, the effects of the ground NOT being exactly
an inertial frame
are treated using the
inertial forces
the centrifugal force
(see below the figure section
The Centrifugal Force of the Earth's Rotation)
and the Coriolis force
(see below the figure section
The Coriolis Force of the Earth's Rotation).
But for most ordinary purposes, you do NOT need to make use of those
inertial forces.
So for most ordinary purposes, you do treat the
ground as an inertial frame.
Yours truly, as a
nonce name, calls
approximate ground
inertial frames
ground free-fall inertial (GFFI) Frames---but
GFFI frames will probably
NOT catch on.
For an example of a picturesque
GFFI frame,
see the figure below
(local link /
general link: alpine_tundra.html).
An explication of the basics of
rotating frames
is given in the figure below
(local link /
general link: frame_rotating.html).
An explication of how the
figure of the Earth
is affected by the
centrifugal force
due to the Earth's rotation
is given in the figure below
(local link /
general link: earth_oblate_spheroid.html).
An explication of how
weather
is affected by the
Coriolis force
due to the Earth's rotation
is given in the figure below
(local link /
general link: mechanics/coriolis_force.html).
The Foucault pendulum
and how it demonstrates the
Earth's rotation
relative to the
observable universe
is explicated the figure below
(local link /
general link: pendulum_foucault.html).
UNDER CONSTRUCTION BELOW
As a qualification, one should say under
gravity and
the cosmological constant force.
However, the
cosmological constant force
(whatever it is exactly) is only important on the cosmological scale and we will NOT
usually mention this qualification again.
Note that unrotating with respect to the
observable universe means that all
inertial frames
are unrotating with respect to each other.
Actually, a qualification is needed in that there may be
reference frames
that are inertial frames
(in a sense)
rotating with respect to the
observable universe
in very strong gravitational fields
such as near black holes.
But yours truly CANNOT find any reference that elucidates this
qualification.
It is hinted at by
Wikipedia:
Inertial frame of reference: General relativity).
Yours truly will usually NOT refer to the qualification again.
If a reference frame
is in acceleration
relative to a free-fall frame
or in rotation with respect
observable universe
(which is actually an accelerated motion),
it is a
non-inertial frame.
"Local" in physics means in the same place
or nearly the same place.
The term is used elastically.
On the other hand, you can convert
non-inertial frames
into
inertial frames
using special frame-dependent forces
called inertial forces.