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 the 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 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 imporantly of all.
The modern
inertial frame
is defined in
general relativity
discovered by Albert Einstein (1879--1955).
General relativity
is our best theory
of gravity
and the main theory ingredient
of cosmology.
To be specific,
the modern inertial frame
(as defined and described below) depends on the
strong equivalence
principle
of general relativity which
has been verified to a degree, but NOT as much as people would like.
Definition: An
inertial frame
is reference frame
defined inside a region of space with a
uniform external gravitational field
and that is unaccelerated with respect to the
free-fall trajectory in that
uniform external gravitational field.
Some explication is needed.
The common symbol is "g" put in vector notation: e.g.,
boldface vector notation g or, as yours truly often writes it in
HTML, the vector notation vec g.
The gravitational force
on a body in a small compared to
variations in vec g is just m(vec g), where m is the body's mass.
This is because the
uniform external gravitational field
pulls on every bit of a system equally.
The external gravitational field
of the definition just has to be uniform enough for your purposes or
the non-uniformities have to be correctable in your calculations.
So, in fact, exact
inertial frames are also
idealizations, but in many cases approached to high
accuracy/precision.
Some
reference frames are more
inertial than others:
there are varying degrees of approximation to exact
inertiality.
The center of mass
is a mass-weighted average position for a
system.
We explicate
center of mass
further below in
subsection Center of Mass.
The center of mass
follows the exact
free fall trajectory.
The reference frame
with this origin
cannot be taken to rotate with respect to the
free-fall trajectory
because that would be an
acceleration.
An acceleration is
change in speed
and/or direction.
We explicate
acceleration
further below in subsection
Acceleration, Force, and Inertial Frames.
Let's call the
inertial frame
with the origin
at the center of mass,
the center-of-mass
inertial frame.
It is an non-inertial frame
for that region.
One might suggest:
But the dead hand of tradition has stuck us with
inertial frames
and non-inertial frames.
To make concrete all that explication of
inertial frames
given above in subsection
Inertial Frames,
let's consider a few important examples of
inertial frames
The center of mass
does follow a
free-fall trajectory in
the external gravitational field
mainly due to the
Sun
and Moon.
Actually, this
external gravitational field
is NOT perfectly uniform across the
Earth.
The variation in the
external gravitational field
causes a slight time-varying stretching of the
Earth.
We call the effective stretching
force
the tidal force which,
among other things, causes
the tides.
It has vanishingly small effect on objects much smaller than the
Earth like us.
The ground
is actually a
rotating frame
relative to the
center-of-mass
inertial frame
of the Earth.
However, the acceleration
of the ground
is only about 5/1000 at most of the
acceleration due to gravity near the Earth's surface
(see Wikipedia: Gravity of Earth: Latitude).
So for most purposes, the
ground is suffiicently
inertial
to be considered an inertial frame.
Not all purposes.
It's NOT for long-range artillery
nor atmospheric circulation.
Non-inertial-frame
effects are important for those phenomena.
The effects can be treated by using
inertial forces
which are NOT
real forces,
but they act like
body forces pulling on every
bit of an object like
gravity.
The best known
inertial force
for rotating frames
is the centrifugal force
which tries to throw you off
playground merry-go-rounds
and carnival centrifuges.
Real forces have to be exerted to keep you
moving with the rotating frame.
The centrifugal force of
the Earth just acts as small correction
to gravity and is easily treated.
The Coriolis force
is another inertial force
for rotating frames.
The Coriolis force
for the Earth is the
inertial force
needed for
long-range artillery
and atmospheric circulation.
The Coriolis force
acts when you have a velocity
relative to a rotating frame.
Since the internal
gravitational field
is negligible, there is
weightlessness.
Actually, this
reference frame is
NOT
inertial
enough to be used as
an inertial frame
for accurate calculations.
To calculate the
orbit of the Moon adequately,
you have to take into account the
variations in the
gravitational field
of the Sun.
This reference frame
is very inertial
because the fixed stars
in their orbits
around the center of the
Milky Way
are pretty much all in the same
free-fall region
(i.e., region of approximately uniform external
gravitational field
due to the whole Milky Way).
For much of Solar-System
astronomy,
the inertial frame
of the fixed stars
is adequate for describing motion.
Actually, the
center-of-mass
inertial frame
of the Solar System
which is virtually the same as that of the
fixed stars, but it's
much easier to locate the
fixed stars than
that center of mass
of Solar System
directly.
By the by,
the center of mass
of an orbital system
has the special name
barycenter.
These are the most fundamental
free-fall frames
and can be called
the comoving frames of the expanding universe.
We explicate them further below in subsection
Comoving Frames.
To further explicate inertial frames,
we need to define what we mean by
acceleration
and
force.
An acceleration
is a change in speed AND/OR a change in direction.
These two kinds of change are illustrated in the two figures just below.
Now a force is a physical relationship
between bodies or between a body and
force field
(e.g., the gravitational field
and electromagnetic field)
that causes an
acceleration of a body relative
to all inertial frames.
To explicate:
Often the center-of-mass
inertial frame.
A force can depend on
mass (gravity),
electric charge
(the electromagnetic force),
relative position (gravity,
the electromagnetic force),
velocity
(the magnetic force),
and other things.
Note that Newton's laws of motion
are referenced to inertial frames.
It is just part of their statements---though this part is often omitted
in initial presentations of the laws to students.
What referenced to means is that the laws do NOT work if NOT applied relative to
inertial frames.
This does NOT mean the physical laws
are wrong somehow since they are explicity or implicitly formulated as
referenced to inertial frames.
General relativity
is a great exception to the rule that
physical laws are referenced
to inertial frames.
General relativity is the
other way around.
It tells us what
inertial frames are and where they
are.
See subsection
Inertial Frames above.
The expression
inertial frame
does NOT come up much in everyday speech, but
inertial frames
are actually really well known.
They are the
reference frames
we use for understanding most ordinary motion
both in the ordinary everyday empirical way used by everyone and all
biota
and in applying
Newtonian physics
as discussed above in subsection
Acceleration, Force, and Inertial Frames.
So there is nothing mysterious about
inertial frames
in everyday life, except the name.
They are just the
reference frames
in which your empirical sense of how things move is right.
Examples of inertial frames
(sufficiently inertial for most purposes)
are the
ground (as discussed above),
unaccelerating
planes,
trains,
and automobiles,
and the frames of unaccelerating
elevators.
See figure of a nice ground
inertial frame below.
Non-inertial frames
are reference frames
that are in acceleration
with respect to
inertial frames
as discussed above in subsection Inertial Frames.
If you are in acceleration
relative to one local inertial frame,
you are in acceleration relative
to them all since they are NOT in
acceleration relative to each other.
Non-inertial frames
are common too, but we tend to avoid being in them with our bodies
because they are tricky relative to our empirical sense of how things move.
But we have to be in them when we accelerate in a
vehicle of any kind
since the vehicle interior defines
an non-inertial frame.
Sometimes we like being in
non-inertial frames for fun like
playground merry-go-rounds---see the
figure below.
In non-inertial frames,
there can be
accelerations
without forces
just due to the motion of the
non-inertial frames.
There will be no accelerations
relative to any inertial frame
if there are no net forces acting on a body.
The funny effects in
non-inertial frames
can be accounted for by
inertial forces
which we discussed above in subsection
Important Examples of Inertial Frames.
In non-inertial frames,
you feel inertial forces
that try to throw you around relative to the
non-inertial frames.
If there were no real
forces acting on you,
Newton's 1st law of motion
would just keep you at a constant velocity
relative to
inertial frames.
You have to exert real forces
to counteract the inertial forces
in order to keep moving with the
non-inertial frames and
be accelerating with them relative to inertial frames.
Rotating reference frames
are particularly difficult to understand because they are actually a continuum
of different non-inertial frames.
See the animation below that dynamically illustrates
motion in a rotating reference frame.
Well:
To avoid convoluted discussion, we will
NOT keep mentioning the exceptional
rotating frames
that are inertial frames.
The
geocentric reference frame
is attached to the
Earth's surface
and has
the Earth's center as the origin
and extend to the whole
universe
which rotates around the
Earth once per day
in this reference frame.
See the Earth
and the geocentric reference frame
in the figure below
The universe
it is overwhelmingly NOT in an uniform external
gravitational field
with the
center of the Earth at its
center of mass.
No physical law
determines the rotational motion of
the universe around
the Earth in any direct sense.
Caption: Geocentric
Alien.
However, for astronomical measurement purposes
(i.e., astrometry), we often do use the
geocentric reference frame
for the whole universe
as we will discuss in IAL: The Sky.
Why? Well after all:
See the figure below for examples of
rotating astronomical objects.
Are there fundamental inertial frames?
Isaac Newton (1643--1727)
postulated that the average
frame of rest of the
fixed stars defined the
fundamental inertial frame
which he called absolute space.
See the fixed stars in the figure below.
But, of course, we now know that the fixed stars
are all very nearby in the Milky Way
and do NOT represent the observable universe.
So Newton's idea was wrong, but
the inertial frame of the
fixed stars is a very
good inertial frame for many purposes.
In fact, we do NOT now believe in
absolute space
in Newton's sense.
There is no single fundamental
inertial frame.
The expanding universe
gives us a different picture.
Note the
expanding universe
is the observable universe
and probably a lot more, but NOT necessarily the
whole universe whatever that is.
Maybe the whole universe is
the multiverse.
Yours truly has invented the expression
pan-universe for the
whole universe, but
yours truly doesn't think it will catch on.
Note that systems like you, me,
moons,
planets,
stars,
galaxies, and
probably most galaxy clusters
are NOT expanding.
But the space
between them is.
In general relativity,
space is a sort of stuff and
it can literally grow.
The two figures below illustrate the
expanding universe
which is a general scaling up of UNBOUND systems.
Nowadays from general relativity,
we do NOT believe in a single
fundamental inertial frame
(i.e., a fundamental fundamental
inertial frame).
We believe there are an infinite continuum of
fundamental inertial frames.
Each one attached to a point participating in the mean
expansion of the universe.
These can be called
the comoving frames of the expanding universe.
These reference frames
are actually in free fall,
and so are consistent with our discussion
of inertial frames given above
in subsection Inertial Frames.
The region of space
up to about the scale of
a galaxy cluster
or maybe a galaxy supercluster
can be approximated to some adequacy as a single
local basic
comoving frame.
That single big
comoving frame
can be used for structures bigger than in the smaller local
inertial frames
such as those described in subsection
Important Examples of Inertial Frames.
For the
observable universe
as whole, we are beyond the realm of validity of the
concept of
inertial frame
and use general relativity
in a more direct sense.
What the heck is center of mass
and why do WE (i.e., YOU) need to know about
it for a discussion of
inertial frames?---don't panic,
we'll
NEVER calculate a center of mass---we
just need to grok the
concept.
The figure below illustrates and explicates
center of mass.
Below is a figure illustrating a
gravitationally-bound system
of astronomical objects
orbiting the system
barycenter
(i.e., center of mass).
The centers of mass
for objects of sufficiently high symmetry are the obvious centers of symmetry
as the figure below illustrates.
One can find these centers of mass
by inspection.
For objects where
center of mass
CANNOT be found by
by inspection,
one can do a calculation
from the formula
for center of mass displayed in the figure shown
somewhere above.
However, there is a simple
empirical method for
finding the
center of mass for
rigid systems.
The method is illustrated in the figure below.
The center of mass can located
deceptively as shown in the figure below.
????
The Sun's mass
is 99.86 % of
the Solar System
mass
(see Wikipedia: Solar System:
Structure and composition).
This overwhelming dominance of the
Sun's mass
means that to good approximation the
Sun's center is
approximately the
barycenter
(i.e., orbital system
center of mass)
of Solar System
and can be used to define an
approximate inertial frame
which is good enough for many purposes.
However, the inertial frame of
fixed stars
is more inertial
and is easier to use in general.
Note that the gravitational force
the Sun is the
main determinant of the
structure of the Solar System.
It pulls the planets
into their
orbits---which means the
planets are in states
of acceleration.
If you recall
Newton's 3rd law,
you know that for every force there is an equal and opposite
force---but note these two
forces do NOT have to be on the same body, and so
just don't cancel out all the time.
Thus the
planets exert equal
gravitational forces on the
Sun to what the
Sun exerts on them.
So the
Sun should also be
accelerated in the frame of fixed stars.
But, as we will discuss
IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides,
acceleration
is proportional to the force
and INVERSELY proportional to
mass.
So the gravitational forces of the
planets
on Sun,
do NOT affect the motion of the Sun very much.
This has been a long story, but it explains why we say
"the planets orbit the Sun".
They orbit the
Sun
in our local-in-the-Milky Way
inertial frame:
i.e., the inertial frame
of the fixed stars.
The moons also
orbit their planets
in inertial frame
of the fixed stars.
However, it also true that they orbit
their planets
in the approximate
inertial frames
defined by the
planet
centers of mass---but
these
inertial frames
are NOT as inertial
as the frame of the fixed stars.
One must use the inertial frame
fixed stars
at least via
perturbation theory.
The adverb "physically" here having
the special meaning of "with-reference-to-a-local-approximate inertial-frame".
Now I know what you are thinking:
is there a fundamental reference
inertial frame?
For everyday life, the ground
serves as a basic reference
inertial frame.
All reference frames
NOT accelerated with respect to the
ground also serve
pretty well as
inertial frames.
But the ground is only an approximately
inertial frame.
It's actually, a
non-inertial frame
if looked at a closely.
In fact, exact
inertial frames
are elusive---as we will now discuss and as we illustrate in the figure below.
Caption: The Local Group of galaxies.
The Local Group comprises about
30 galaxies.
Not all objects are shown or labeled in the figure.
The Large Magellanic Cloud (LMC)
and Small Magellanic Cloud (SMC) are NOT shown
or NOT labeled---I guess they are too close to the
Milky Way.
There is no exact number of objects in the Local Group
since there may be still be
undiscovered dwarf galaxies hidden from
our view by the dust lane of the Milky Way and
also because the status of some objects is NOT clear:
dwarf galaxy or
globular clusters?
There are only three large galaxies, all
spirals:
The Local Group is gravitationally bound and
orbit the center of mass which is somewhere
between the Milky Way and
Andromeda.
The orbits are NOT closed and are probably somewhat chaotic.
Credit/Permission: ©
Richard Powell
2006
(uploaded to Wikipedia
by User:AndrewRT,
2006) /
Creative Commons
CC BY-SA 2.5.
Image linked to Wikipedia.
Nowadays, we believe there is NOT a single fundamental
inertial frame
like Newton's hypothesized
absolute space.
Rather we believe there is a continuum of
fundamental
inertial frames which
are frames that participate in the
mean expansion of the universe.
These frames can be called the
comoving frames of expanding universe
The comoving frames
are everywhere in a sense---one is definable at every point---but virtually no matter
is at rest with respect to them, except very rarely by accident.
Most matter is rotating with respect to them and also has a translation velocity with
respect to its local
comoving frames.
The expansion of the universe
is topic of IAL 31: Cosmology, but we
can give brief introduction here.
The expansion of the universe is the
growth of space---yes
space---between bound systems---but
NOT inside them---you and I are
NOT expanding, nor is the Milky Way
or other galaxies
or gravitationally bound
galaxy clusters.
But the space
between the bound systems is growing as the universe
scales up.
The figure and animation
below illustrate the expansion of the universe.
We can actually identify
the comoving frames
of the expanding universe
which are fundamental
inertial frames.
The barycenters
of most
galaxy clusters
are approximately
comoving frames---this
is a hypothesis which is verified
by the consistency of all modern cosmology.
We can also identify very exactly the local-to-us or nearby-to-us
comoving frames by observations.
Cosmologically remote astronomical objects
(i.e., galaxies
or quasars)
should be unrotating relative to the
local-to-us comoving frames in
modern theory cosmological theory,
and so define for us local unrotating frames.
Now the
cosmic microwave background (CMB)
is by modern theory cosmological theory
NOT Doppler shifted
in a comoving frame.
Fortunately, we do NOT usually have to work that hard.
Actually, inertial frame
is NOT a good name in yours truly's
opinion: it's NOT descriptive, it's hard for yours truly
to pronounce,
and yours truly when speaking is always confusing it with its opposite
the non-inertial frame.
The meme comes from
Animal Farm (1945).
See the figure below.
If a reference frame is
inertial enough for your purposes,
it is an inertial frame.
If NOT, it's a non-inertial frame.
There is NO hard line.
If you know the forces
acting on a body from known force laws, then
physical law
will predict the acceleration
relative to the inertial frame you are using.
The physical law
in the classical limit is
Newton's 2nd law of motion (AKA F=ma).
If you are NOT
in the classical limit,
you have to use
relativistic mechanics
and/or quantum mechanics.
Newtonian physics
is based on
Newton's 3 laws of motion,
Newton's law of universal gravitation,
and other force laws.
There is whole lot more
Newtonian physics
formalism developed on that basis too.
Actually, almost all physical laws
are referenced to
inertial frames.
The only exceptions are
rotating frames
NOT rotating with respect to
the local inertial frames,
but rather with respect to
reference
non-inertial
rotating frames.
We discussed the
odd effects in
rotating frames
above in subsection
Inertial Frames and Everyday Life
"Man is the measure of all things."
The Earth is our platform for
observing the whole universe,
and for locating things in the sky,
and so for these purposes,
the geocentric reference frame
is a convenient or natural
reference frame.
----Protagoras (c. 490--c. 420 BCE)
If the
Sun
suddenly disappeared, the planets
would fly away from each other in space
and never meet again because
the major source of gravity was gone:
gravity is proportional to mass.
The instructor can---if he remembers---do a demonstration with a swirling object.
The Earth-Moon system
neither the
inertial frame
of th Earth
center of mass
nor
the Earth-Moon system
center of mass
is sufficiently inertial
for an accurate/precise
calculated
Moon's orbit.
To NOT have to say inertial frame
all the time, we often just say
the planets physically orbit
the Sun
and the moons
physically orbit the planets.
The Doppler effect
is covered in IAL 7: Spectra
But we observe
Doppler shift
which we call the
CMB dipole anisotropy.
From that observation, we can
determine the translations with respect to the
local-to-us and nearby-to-us
comoving frames.
For very exact, modern studies of motions one actually does make use of
local comoving frames.