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In this insert, we introduce the basics of reference frames relevant to physics.
In particular, we cover inertial frames, celestial frames (CEFs), comoving frames, inertial forces (especially, the centrifugal force and the Coriolis force), and the tidal force.
What are reference frames?
Just a set of coordinates
you lay down on
geometrical space
(e.g., the
(3-dimensional physical) space
which includes 3-dimensional space
of outer 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 time
coordinates.
Here we offer a classification of
reference frames
relevant to physics:
Note the qualification is required by
general relativity
and is NOT present in
pure Newtonian physics as it
is NOW understood.
You 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.
Our modern understanding of
inertial frames
has only been available since
1915 when
general relativity
was introduced by
Albert Einstein (1879--1955)
(see Wikipedia: History of general relativity).
So the original conception of
inertial frames
in Newtonian physics
due to Isaac Newton (1643--1727)
is obsolete.
There can be internal
gravitational fields
due to system
of objects
you include
inertial frame
you are considering.
There is choice for that
inertial frame
which is usually made for simplicity in analysing the internal motions
of the included objects.
However, one treats non-uniformities in the external
gravitational field by means of
tidal forces
in the inertial frame.
A tidal force is just the difference
in gravity between two points.
If the tidal forces are
comparably large or larger than
the internal (to the
inertial frame)
gravitational forces,
then you should probably find another
inertial frame for
analyzing the system.
One way of NOT being in
free fall is to be
in acceleration relative
to local inertial frame.
"Local" meaning coincident or nearly coincident in
space.
Physics
relative to
non-inertial frames
can be done by treating them as
effective inertial frames
(see below).
In fact, yours truly thinks this is the only way to treat
non-inertial frames
though there may be some treatments that disguise this fact.
Now it just seems natural sometimes to use the term
celestial frame
as a synonym
for the system
of astro-bodies
it is used analyze.
As usual context
determines the precise meaning of
celestial frame.
The intenral forces alluded to in the last paragraph
are usually almost entirely
gravity when one is talking of
discrete astro-bodies.
Other forces are usually
just astronomical perturbations.
Of course, inside discrete
astro-bodies
(e.g., moons,
planets,
stars)
other forces notably
the pressure force will be important.
The ideal case actually virtually holds, for example,
for planetary systems
and multiple star systems.
They are usually so small that the external
gravitational field
is close to be uniform over them, and thus
tidal forces are negligible.
In the limit that the
external gravitational field
has only
linear variation over
a celestial frame,
the center of mass of the
celestial frame
does free fall
with the local
gravitational field.
However, as the variation of the
external gravitational field departs from
linear variation, this is NO
longer true.
The center of mass
does NOT free fall
with the local
gravitational field.
This means that the
celestial frame
is actually an
effective
inertial frame
(see below subsection
Effective
Inertial Frames (IEFs)).
However, as we discuss in subsection
Effective
Inertial Frames (IEFs),
there is NO fundamental difference between
effective
inertial frames
and inertial frames
without qualification.
The center of mass of a
celestial frame
is remains a good origin
for the celestial frame
for Newtonian physics in any case.
Explication:
Solutions to the
Friedmann equation
dictate that the
observable universe
should in general expansion: a literal growth of
space according to
general relativity.
If the mass of the
observable universe
were spread out uniformly through the
observable universe,
there would just be a general scaling up of the distances between all points
participating in the
expansion of the universe
and every point would be
an local
inertial frame:
local meaning right where the point is.
These are
inertial frames are
indeed
free-fall frames (FFFs)
in a general relativity.
They are what we call
comoving frames.
If you put a
test particle
(i.e., a particle of negligible
mass)
at rest in a
comoving frame
in a uniform
density
observable universe
and only allowed
gravity
the dark energy to act on,
it would stay at rest
in the
comoving frame
and participate in the mean
expansion of the universe.
There is still a mean
expansion of the universe
that happens
between bound systems
(most obviously
gravitationally bound systems).
Bound systems do NOT expand.
Also there are still local
inertial frames
(i.e., free-fall frames (FFFs))
at every point in
space and
test particle
acted only by the local
gravitational force
(and dark energy)
would be free fall.
However, in general those local
inertial frames will NOT be
comoving frames:
the test particle
motion with them is NOT participating in the mean
expansion of the universe,
except on average.
Nevertheless, there are still
comoving frames
at every point in space
and they still define useful
inertial frames, just
not local ones.
They define average
inertial frames
over large regions of
space: to be useful for analysis
they should be larger than
galaxy clusters.
In fact,
comoving frames
are very large
celestial frames
of the
effective
inertial frame variety
(see above subection
Celestial Frames (CEFs)
and below subsection
Effective
Inertial Frames (IEFs)).
Note for clarity in discussion, we use the term
comoving frame
to mean the point in
space
participating in the mean
expansion of the universe.
Yours truly does NOT much use
the term
comoving cosmic rest frame
since it seems tricky to explain.
Such velocities are called
peculiar velocities---but note
the term
peculiar velocity is used in
for any velocity relative to any
overall motion, NOT just for
velocities relative to local
comoving frame.
The measurement
of peculiar velocities
(velocities relative to local
comoving frames)
have to be done non-locally: i.e., they must be done using
astronomical objects remote from
the comoving frames
since the
comoving frames
do NOT define
local inertial frames,
and so there is NO local information about
the comoving frames.
Peculiar velocities
are important in understanding the
large-scale structure of the universe.
Important ones are those for the
centers of mass of the largest
astro-bodies:
e.g., those for galaxies,
galaxy groups,
and galaxy clusters.
Another important
peculiar velocity
is that of the Sun:
it is 369.82(11) km/s in the direction of
constellation
Crater
(Wikipedia:
Cosmic microwave background: Features).
The Sun's
peculiar velocity
is measured from the
cosmic microwave background
(CMB) dipole---which we will will NOT expand on here.
But note the Sun's
peculiar velocity
(relative to the local
comoving frame)
is important for cosmology,
but it does NOT tell us anything about local motions since
comoving frames
do NOT define local
inertial frames.
For the Sun,
the actual relevant
inertial frame for local motions
is
the celestial frame
of the Milky Way.
The Sun
has average orbital velocity
around the center of mass
of the Milky Way
of ∼ 230 km/s.
It is this orbital velocity
that is determined by
Newtonian physics.
These are the
celestial frames
of galaxy clusters,
galaxy groups,
and field galaxies
(i.e., galaxies NOT in
gravitationally bound systems:
i.e.,
NOT in galaxy clusters and
galaxy groups).
The motions of these large-scale
celestial frames,
in fact,
reveal the
expansion of the universe
and allow us to study it through
cosmic time.
We "average" over their
peculiar velocities
for this research.
The use of
inertial forces is NOT
just a trick.
General relativity
tells us there is a fundamental likeness
between inertial forces
and gravity.
So effective
inertial frames
are often just called
inertial frames, unless you want
to emphasize that
inertial forces
are being used.
So the expression
approximate inertial frame
would only be used if you need to make point of an
approximate inertial frame
being an
approximate inertial frame.
Strictly, speaking exact
inertial frames are an
ideal limit that can NEVER reached absolutely exactly.
However, many inertial frames
are so close to that ideal limit that you almost never would call them
approximate inertial frames.
An important example of
approximate inertial frames
are any points on the surface of a
pressure-supported
rotating
spherical
astro-body in
free fall: e.g.,
any points on the surface of the
Earth.
For a shorthand, let's call such
astro-bodies
spherical astro-bodies.
Spherical astro-bodies,
in fact, constitute a
celestial frames
with theirs centers of mass
at their centers.
However, if the rotation
of the spherical astro-body
is sufficiently slow, any point on its surface
will define a local inertial frame
(i.e., a local
approximate inertial frame)
for most purposes.
For example, for most purposes any point on the surface
of the Earth defines
a local inertial frame for most purposes.
The case of the Earth is
further explicated in the figure below
(local link /
general link: frame_inertial_free_fall_2b.html).
When can't you?
For long-range gunnery,
for some weather phenomena
(e.g., anticyclones
and cyclones which
are affected by the Coriolis force),
precise measurements of the
Earth's gravity
(which are affected by the
centrifugal force),
very delicate small-scale operations
(e.g., a Foucault pendulum
which depends on the Coriolis force),
and probably other cases.
Yours truly at this instant in
cosmic time
(2024
Aug27)
believes what
yours truly
calls
celestial frames (CEFs)
are a good way to understand
reference frames
used in astrophysics
in the classical limit.
The definition of
celestial frames
is given above in subsection
Celestial Frames (CEFs).
To go beyond the
classical limit
requires
special relativity
and
general relativity.
In the classical limit,
we use Newtonian physics
and certain restrictions apply:
The classical limit
sounds very restrictive, but, in fact, pretty much everything from
cosmic dust to
the
large scale structure
of the universe
can be analyzed as in the
classical limit in good to
excellent approximation
depending on the case.
Actually, the observable universe
as whole can be treated using
Newtonian physics
in that the
Friedmann equation
can be derived from
Newtonian physics
plus special hypotheses.
We will NOT go into that here.
The behavior of black holes close to
black holes CANNOT
be dealt with by Newtonian physics,
but black holes from far enough
from them can be treated as
point mass
sources of gravity
by Newtonian physics.
To conclude this section,
celestial frames are very
general inertial frames
for celestial mechanics, but
they are NOT completely general.
Complete generality is beyond yours truly's scope of knowledge
and is probably unnecessary for understanding
most reference frames
used in astrophysics.
More Features of the hierarchy of celestial frames:
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 celestial frame.
In particular,
a perfectly uniform
external gravitational field
CANNOT change the total
angular momentum of
celestial frame
about the center of mass (CM): i.e., the
celestial frame
has conservation of angular momentum.
If you need to analyze motions of astronomical objects
outside of the celestial frame
you are using, then you should probably use a larger
celestial frame
in the hierarchy of
celestial 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.
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 section
Rotating Frames Explicated.
For an important example of the
centrifugal force
at work, see below the section
The Coriolis Force of the Earth's Rotation.
For important example of the
Coriolis force
at work, see below the 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
celestial 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
celestial 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 section
The Centrifugal Force of the Earth's Rotation)
and the Coriolis force
(see below the 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
UNDER CONSTRUCTION BELOW
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.
Except dark energy
acts externally importantly
on celestial frames.
But its importance is almost entirely to cause
the acceleration of the
expansion of the universe:
the general scaling up of all distances between
bound systems.
This effect is the background to all our discussions, and so we do NOT
have to mention it often.
Note saying an astro-body is in
free fall means its
center of mass is
in free fall---but, in general,
in averaged gravitational field
NOT in the gravitational field at the
center of mass though often
the gravitational field at the
center of mass approximates the
averaged gravitational field to
high accuracy.
But any point on the surface of a spherical
astro-body
is rotating relative
to the observable universe,
and thus in acceleration
since it is NOT
in straight line
motion relative to the
celestial frame
of the spherical astro-body.
Usually, you can just use
Newtonian physics in
such lcoal inertial frames
without worrying about them NOT being exactly
inertial frames.