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An inertial frame of reference, in classical physics, is a frame of
reference in which bodies, whose net force acting upon them is zero,
are not accelerated; that is they are at rest or they move at a
constant velocity in a straight line.[1] In analytical terms, it is a
frame of reference that describes time and space homogeneously,
isotropically, and in a time-independent manner.[2] Conceptually, in
classical physics and special relativity, the physics of a system in
an inertial frame have no causes external to the system.[3] An
inertial frame of reference may also be called an inertial reference
frame, inertial frame, Galilean reference frame, or inertial
space.[citation needed]
All inertial frames are in a state of constant, rectilinear motion
with respect to one another; an accelerometer moving with any of them
would detect zero acceleration. Measurements in one inertial frame can
be converted to measurements in another by a simple transformation
(the
**Galilean transformation**

Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, in any region small enough for the curvature of spacetime and tidal forces[4] to be negligible, one can find a set of inertial frames that approximately describe that region.[5][6] In a non-inertial reference frame in classical physics and special relativity, the physics of a system vary depending on the acceleration of that frame with respect to an inertial frame, and the usual physical forces must be supplemented by fictitious forces.[7][8] In contrast, systems in non-inertial frames in general relativity don't have external causes, because of the principle of geodesic motion.[9] In classical physics, for example, a ball dropped towards the ground does not go exactly straight down because the**Earth**

Earth is rotating, which means the frame of reference of an observer on**Earth**

Earth is not inertial. The physics must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force.

Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, in any region small enough for the curvature of spacetime and tidal forces[4] to be negligible, one can find a set of inertial frames that approximately describe that region.[5][6] In a non-inertial reference frame in classical physics and special relativity, the physics of a system vary depending on the acceleration of that frame with respect to an inertial frame, and the usual physical forces must be supplemented by fictitious forces.[7][8] In contrast, systems in non-inertial frames in general relativity don't have external causes, because of the principle of geodesic motion.[9] In classical physics, for example, a ball dropped towards the ground does not go exactly straight down because the

Earth is rotating, which means the frame of reference of an observer on

Earth is not inertial. The physics must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force.

Contents

1 Introduction 2 Background

2.1 A set of frames where the laws of physics are simple 2.2 Absolute space

3 Newton's inertial frame of reference 4 Separating non-inertial from inertial reference frames

4.1 Theory 4.2 Applications

5 Newtonian mechanics
6
**Special**

Special relativity
7 General relativity
8 See also
9 References
10 Further reading
11 External links

Introduction[edit]
The motion of a body can only be described relative to something
else—other bodies, observers, or a set of space-time coordinates.
These are called frames of reference. If the coordinates are chosen
badly, the laws of motion may be more complex than necessary. For
example, suppose a free body that has no external forces acting on it
is at rest at some instant. In many coordinate systems, it would begin
to move at the next instant, even though there are no forces on it.
However, a frame of reference can always be chosen in which it remains
stationary. Similarly, if space is not described uniformly or time
independently, a coordinate system could describe the simple flight of
a free body in space as a complicated zig-zag in its coordinate
system. Indeed, an intuitive summary of inertial frames can be given:
in an inertial reference frame, the laws of mechanics take their
simplest form.[2]
In an inertial frame, Newton's first law, the law of inertia, is
satisfied: Any free motion has a constant magnitude and direction.[2]
**Newton's second law**

Newton's second law for a particle takes the form:

F

= m

a

,

displaystyle mathbf F =mmathbf a ,

with F the net force (a vector), m the mass of a particle and a the acceleration of the particle (also a vector) which would be measured by an observer at rest in the frame. The force F is the vector sum of all "real" forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. In contrast, Newton's second law in a rotating frame of reference, rotating at angular rate Ω about an axis, takes the form:

F

′

= m

a

,

displaystyle mathbf F '=mmathbf a ,

which looks the same as in an inertial frame, but now the force F′ is the resultant of not only F, but also additional terms (the paragraph following this equation presents the main points without detailed mathematics):

F

′

=

F

− 2 m

Ω

×

v

B

− m

Ω

× (

Ω

×

x

B

) − m

d

Ω

d t

×

x

B

,

displaystyle mathbf F '=mathbf F -2mmathbf Omega times mathbf v _ B -mmathbf Omega times (mathbf Omega times mathbf x _ B )-m frac dmathbf Omega dt times mathbf x _ B ,

where the angular rotation of the frame is expressed by the vector Ω
pointing in the direction of the axis of rotation, and with magnitude
equal to the angular rate of rotation Ω, symbol × denotes the vector
cross product, vector xB locates the body and vector vB is the
velocity of the body according to a rotating observer (different from
the velocity seen by the inertial observer).
The extra terms in the force F′ are the "fictitious" forces for this
frame, whose causes are external to the system in the frame. The first
extra term is the Coriolis force, the second the centrifugal force,
and the third the Euler force. These terms all have these properties:
they vanish when Ω = 0; that is, they are zero for an inertial frame
(which, of course, does not rotate); they take on a different
magnitude and direction in every rotating frame, depending upon its
particular value of Ω; they are ubiquitous in the rotating frame
(affect every particle, regardless of circumstance); and they have no
apparent source in identifiable physical sources, in particular,
matter. Also, fictitious forces do not drop off with distance (unlike,
for example, nuclear forces or electrical forces). For example, the
centrifugal force that appears to emanate from the axis of rotation in
a rotating frame increases with distance from the axis.
All observers agree on the real forces, F; only non-inertial observers
need fictitious forces. The laws of physics in the inertial frame are
simpler because unnecessary forces are not present.
In Newton's time the fixed stars were invoked as a reference frame,
supposedly at rest relative to absolute space. In reference frames
that were either at rest with respect to the fixed stars or in uniform
translation relative to these stars,
**Newton's laws of motion**

Newton's laws of motion were
supposed to hold. In contrast, in frames accelerating with respect to
the fixed stars, an important case being frames rotating relative to
the fixed stars, the laws of motion did not hold in their simplest
form, but had to be supplemented by the addition of fictitious forces,
for example, the
**Coriolis force**

Coriolis force and the centrifugal force. Two
interesting experiments were devised by Newton to demonstrate how
these forces could be discovered, thereby revealing to an observer
that they were not in an inertial frame: the example of the tension in
the cord linking two spheres rotating about their center of gravity,
and the example of the curvature of the surface of water in a rotating
bucket. In both cases, application of
**Newton's second law**

Newton's second law would not
work for the rotating observer without invoking centrifugal and
Coriolis forces to account for their observations (tension in the case
of the spheres; parabolic water surface in the case of the rotating
bucket).
As we now know, the fixed stars are not fixed. Those that reside in
the
**Milky Way**

Milky Way turn with the galaxy, exhibiting proper motions. Those
that are outside our galaxy (such as nebulae once mistaken to be
stars) participate in their own motion as well, partly due to
expansion of the universe, and partly due to peculiar velocities.[10]
The
**Andromeda galaxy**

Andromeda galaxy is on collision course with the
**Milky Way**

Milky Way at a
speed of 117 km/s.[11] The concept of inertial frames of
reference is no longer tied to either the fixed stars or to absolute
space. Rather, the identification of an inertial frame is based upon
the simplicity of the laws of physics in the frame. In particular, the
absence of fictitious forces is their identifying property.[12]
In practice, although not a requirement, using a frame of reference
based upon the fixed stars as though it were an inertial frame of
reference introduces very little discrepancy. For example, the
centrifugal acceleration of the
**Earth**

Earth because of its rotation about
the Sun is about thirty million times greater than that of the Sun
about the galactic center.[13]
To illustrate further, consider the question: "Does our Universe
rotate?" To answer, we might attempt to explain the shape of the Milky
Way galaxy using the laws of physics,[14] although other observations
might be more definitive, that is, provide larger discrepancies or
less measurement uncertainty, like the anisotropy of the microwave
background radiation or Big Bang nucleosynthesis.[15][16] The flatness
of the
**Milky Way**

Milky Way depends on its rate of rotation in an inertial frame
of reference. If we attribute its apparent rate of rotation entirely
to rotation in an inertial frame, a different "flatness" is predicted
than if we suppose part of this rotation actually is due to rotation
of the universe and should not be included in the rotation of the
galaxy itself. Based upon the laws of physics, a model is set up in
which one parameter is the rate of rotation of the Universe. If the
laws of physics agree more accurately with observations in a model
with rotation than without it, we are inclined to select the best-fit
value for rotation, subject to all other pertinent experimental
observations. If no value of the rotation parameter is successful and
theory is not within observational error, a modification of physical
law is considered, for example, dark matter is invoked to explain the
galactic rotation curve. So far, observations show any rotation of the
universe is very slow, no faster than once every 60·1012 years
(10−13 rad/yr),[17] and debate persists over whether there is any
rotation. However, if rotation were found, interpretation of
observations in a frame tied to the universe would have to be
corrected for the fictitious forces inherent in such rotation in
classical physics and special relativity, or interpreted as the
curvature of spacetime and the motion of matter along the geodesics in
general relativity.
When quantum effects are important, there are additional conceptual
complications that arise in quantum reference frames.
Background[edit]
A brief comparison of inertial frames in special relativity and in
Newtonian mechanics, and the role of absolute space is next.
A set of frames where the laws of physics are simple[edit]
According to the first postulate of special relativity, all physical
laws take their simplest form in an inertial frame, and there exist
multiple inertial frames interrelated by uniform translation: [18]

**Special**

Special principle of relativity: If a system of coordinates K is
chosen so that, in relation to it, physical laws hold good in their
simplest form, the same laws hold good in relation to any other system
of coordinates K' moving in uniform translation relatively to K.
— Albert Einstein: The foundation of the general theory of
relativity, Section A, §1

This simplicity manifests in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames have external causes.[3] The principle of simplicity can be used within Newtonian physics as well as in special relativity; see Nagel[19] and also Blagojević.[20]

The laws of Newtonian mechanics do not always hold in their simplest
form...If, for instance, an observer is placed on a disc rotating
relative to the earth, he/she will sense a 'force' pushing him/her
toward the periphery of the disc, which is not caused by any
interaction with other bodies. Here, the acceleration is not the
consequence of the usual force, but of the so-called inertial force.
**Newton's laws**

Newton's laws hold in their simplest form only in a family of
reference frames, called inertial frames. This fact represents the
essence of the Galilean principle of relativity:
The laws of mechanics have the same form in all inertial
frames.
— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 4

In practical terms, the equivalence of inertial reference frames means
that scientists within a box moving uniformly cannot determine their
absolute velocity by any experiment (otherwise the differences would
set up an absolute standard reference frame).[21][22] According to
this definition, supplemented with the constancy of the speed of
light, inertial frames of reference transform among themselves
according to the
**Poincaré group**

Poincaré group of symmetry transformations, of which
the Lorentz transformations are a subgroup.[23] In Newtonian
mechanics, which can be viewed as a limiting case of special
relativity in which the speed of light is infinite, inertial frames of
reference are related by the
**Galilean group**

Galilean group of symmetries.
Absolute space[edit]
Main article:
**Absolute space**

Absolute space and time
Newton posited an absolute space considered well approximated by a
frame of reference stationary relative to the fixed stars. An inertial
frame was then one in uniform translation relative to absolute space.
However, some scientists (called "relativists" by Mach[24]), even at
the time of Newton, felt that absolute space was a defect of the
formulation, and should be replaced.
Indeed, the expression inertial frame of reference (German:
Inertialsystem) was coined by Ludwig Lange in 1885, to replace
Newton's definitions of "absolute space and time" by a more
operational definition.[25][26] As translated by Iro, Lange proposed
the following definition:[27]

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

A discussion of Lange's proposal can be found in Mach.[24] The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojević:[28]

The existence of absolute space contradicts the internal logic of
classical mechanics since, according to Galilean principle of
relativity, none of the inertial frames can be singled out.
**Absolute space**

Absolute space does not explain inertial forces since they are related
to acceleration with respect to any one of the inertial frames.
**Absolute space**

Absolute space acts on physical objects by inducing their resistance
to acceleration but it cannot be acted upon.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5

The utility of operational definitions was carried much further in the special theory of relativity.[29] Some historical background including Lange's definition is provided by DiSalle, who says in summary:[30]

The original question, "relative to what frame of reference do the
laws of motion hold?" is revealed to be wrongly posed. For the laws of
motion essentially determine a class of reference frames, and (in
principle) a procedure for constructing them.
— Robert DiSalle
**Space**

Space and Time: Inertial Frames

Newton's inertial frame of reference[edit]

Figure 1: Two frames of reference moving with relative velocity

v →

displaystyle stackrel vec v

. Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.

Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid.[31] However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law. Newton viewed the first law as valid in any reference frame that is in uniform motion relative to the fixed stars;[32] that is, neither rotating nor accelerating relative to the stars.[33] Today the notion of "absolute space" is abandoned, and an inertial frame in the field of classical mechanics is defined as:[34][35]

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

Hence, with respect to an inertial frame, an object or body
accelerates only when a physical force is applied, and (following
**Newton's first law**

Newton's first law of motion), in the absence of a net force, a body
at rest will remain at rest and a body in motion will continue to move
uniformly—that is, in a straight line and at constant speed.
Newtonian inertial frames transform among each other according to the
**Galilean group**

Galilean group of symmetries.
If this rule is interpreted as saying that straight-line motion is an
indication of zero net force, the rule does not identify inertial
reference frames because straight-line motion can be observed in a
variety of frames. If the rule is interpreted as defining an inertial
frame, then we have to be able to determine when zero net force is
applied. The problem was summarized by Einstein:[36]

The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration. — Albert Einstein: The Meaning of Relativity, p. 58

There are several approaches to this issue. One approach is to argue
that all real forces drop off with distance from their sources in a
known manner, so we have only to be sure that a body is far enough
away from all sources to ensure that no force is present.[37] A
possible issue with this approach is the historically long-lived view
that the distant universe might affect matters (Mach's principle).
Another approach is to identify all real sources for real forces and
account for them. A possible issue with this approach is that we might
miss something, or account inappropriately for their influence,
perhaps, again, due to
**Mach's principle**

Mach's principle and an incomplete
understanding of the universe. A third approach is to look at the way
the forces transform when we shift reference frames. Fictitious
forces, those that arise due to the acceleration of a frame, disappear
in inertial frames, and have complicated rules of transformation in
general cases. On the basis of universality of physical law and the
request for frames where the laws are most simply expressed, inertial
frames are distinguished by the absence of such fictitious forces.
Newton enunciated a principle of relativity himself in one of his
corollaries to the laws of motion:[38][39]

The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line. — Isaac Newton: Principia, Corollary V, p. 88 in Andrew Motte translation

This principle differs from the special principle in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares with the special principle the invariance of the form of the description among mutually translating reference frames.[40] The role of fictitious forces in classifying reference frames is pursued further below. Separating non-inertial from inertial reference frames[edit] Theory[edit] Main article: Fictitious force See also: Non-inertial frame, Rotating spheres, and Bucket argument

Figure 2: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.

Figure 3: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.

Inertial and non-inertial reference frames can be distinguished by the absence or presence of fictitious forces, as explained shortly.[7][8]

The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations…. — Sidney Borowitz and Lawrence A Bornstein in A Contemporary View of Elementary Physics, p. 138

The presence of fictitious forces indicates the physical laws are not the simplest laws available so, in terms of the special principle of relativity, a frame where fictitious forces are present is not an inertial frame:[41]

The equations of motion in a non-inertial system differ from the
equations in an inertial system by additional terms called inertial
forces. This allows us to detect experimentally the non-inertial
nature of a system.
— V. I. Arnol'd: Mathematical Methods of Classical Mechanics
**Second**

Second Edition, p. 129

Bodies in non-inertial reference frames are subject to so-called
fictitious forces (pseudo-forces); that is, forces that result from
the acceleration of the reference frame itself and not from any
physical force acting on the body. Examples of fictitious forces are
the centrifugal force and the
**Coriolis force**

Coriolis force in rotating reference
frames.
How then, are "fictitious" forces to be separated from "real" forces?
It is hard to apply the Newtonian definition of an inertial frame
without this separation. For example, consider a stationary object in
an inertial frame. Being at rest, no net force is applied. But in a
frame rotating about a fixed axis, the object appears to move in a
circle, and is subject to centripetal force (which is made up of the
**Coriolis force**

Coriolis force and the centrifugal force). How can we decide that the
rotating frame is a non-inertial frame? There are two approaches to
this resolution: one approach is to look for the origin of the
fictitious forces (the
**Coriolis force**

Coriolis force and the centrifugal force). We
will find there are no sources for these forces, no associated force
carriers, no originating bodies.[42] A second approach is to look at a
variety of frames of reference. For any inertial frame, the Coriolis
force and the centrifugal force disappear, so application of the
principle of special relativity would identify these frames where the
forces disappear as sharing the same and the simplest physical laws,
and hence rule that the rotating frame is not an inertial frame.
Newton examined this problem himself using rotating spheres, as shown
in Figure 2 and Figure 3. He pointed out that if the spheres are not
rotating, the tension in the tying string is measured as zero in every
frame of reference.[43] If the spheres only appear to rotate (that is,
we are watching stationary spheres from a rotating frame), the zero
tension in the string is accounted for by observing that the
centripetal force is supplied by the centrifugal and Coriolis forces
in combination, so no tension is needed. If the spheres really are
rotating, the tension observed is exactly the centripetal force
required by the circular motion. Thus, measurement of the tension in
the string identifies the inertial frame: it is the one where the
tension in the string provides exactly the centripetal force demanded
by the motion as it is observed in that frame, and not a different
value. That is, the inertial frame is the one where the fictitious
forces vanish.
So much for fictitious forces due to rotation. However, for linear
acceleration, Newton expressed the idea of undetectability of
straight-line accelerations held in common:[39]

If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces. — Isaac Newton: Principia Corollary VI, p. 89, in Andrew Motte translation

This principle generalizes the notion of an inertial frame. For
example, an observer confined in a free-falling lift will assert that
he himself is a valid inertial frame, even if he is accelerating under
gravity, so long as he has no knowledge about anything outside the
lift. So, strictly speaking, inertial frame is a relative concept.
With this in mind, we can define inertial frames collectively as a set
of frames which are stationary or moving at constant velocity with
respect to each other, so that a single inertial frame is defined as
an element of this set.
For these ideas to apply, everything observed in the frame has to be
subject to a base-line, common acceleration shared by the frame
itself. That situation would apply, for example, to the elevator
example, where all objects are subject to the same gravitational
acceleration, and the elevator itself accelerates at the same rate.
Applications[edit]
Inertial navigation systems used a cluster of gyroscopes and
accelerometers to determine accelerations relative to inertial space.
After a gyroscope is spun up in a particular orientation in inertial
space, the law of conservation of angular momentum requires that it
retain that orientation as long as no external forces are applied to
it.[44]:59 Three orthogonal gyroscopes establish an inertial reference
frame, and the accelerators measure acceleration relative to that
frame. The accelerations, along with a clock, can then be used to
calculate the change in position. Thus, inertial navigation is a form
of dead reckoning that requires no external input, and therefore
cannot be jammed by any external or internal signal source.[45]
A gyrocompass, employed for navigation of seagoing vessels, finds the
geometric north. It does so, not by sensing the Earth's magnetic
field, but by using inertial space as its reference. The outer casing
of the gyrocompass device is held in such a way that it remains
aligned with the local plumb line. When the gyroscope wheel inside the
gyrocompass device is spun up, the way the gyroscope wheel is
suspended causes the gyroscope wheel to gradually align its spinning
axis with the Earth's axis. Alignment with the Earth's axis is the
only direction for which the gyroscope's spinning axis can be
stationary with respect to the
**Earth**

Earth and not be required to change
direction with respect to inertial space. After being spun up, a
gyrocompass can reach the direction of alignment with the Earth's axis
in as little as a quarter of an hour.[46]
Newtonian mechanics[edit]
Main article:
**Newton's laws**

Newton's laws of motion
Classical mechanics, which includes relativity, assumes the
equivalence of all inertial reference frames. Newtonian mechanics
makes the additional assumptions of absolute space and absolute time.
Given these two assumptions, the coordinates of the same event (a
point in space and time) described in two inertial reference frames
are related by a Galilean transformation.

r

′

=

r

−

r

0

−

v

t

displaystyle mathbf r ^ prime =mathbf r -mathbf r _ 0 -mathbf v t

t

′

= t −

t

0

displaystyle t^ prime =t-t_ 0

where r0 and t0 represent shifts in the origin of space and time, and
v is the relative velocity of the two inertial reference frames. Under
Galilean transformations, the time t2 − t1 between two events is the
same for all inertial reference frames and the distance between two
simultaneous events (or, equivalently, the length of any object, r2
− r1) is also the same.
**Special**

Special relativity[edit]
Main articles:
**Special relativity**

Special relativity and Introduction to special
relativity
Einstein's theory of special relativity, like Newtonian mechanics,
assumes the equivalence of all inertial reference frames, but makes an
additional assumption, foreign to Newtonian mechanics, namely, that in
free space light always is propagated with the speed of light c0, a
defined value independent of its direction of propagation and its
frequency, and also independent of the state of motion of the emitting
body. This second assumption has been verified experimentally[47] and
leads to counter-intuitive deductions including:

time dilation (moving clocks tick more slowly) length contraction (moving objects are shortened in the direction of motion) relativity of simultaneity (simultaneous events in one reference frame are not simultaneous in almost all frames moving relative to the first).

These deductions are logical consequences of the stated assumptions, and are general properties of space-time, typically without regard to a consideration of properties pertaining to the structure of individual objects like atoms or stars, nor to the mechanisms of clocks. These effects are expressed mathematically by the Lorentz transformation

x

′

= γ

(

x − v t

)

displaystyle x^ prime =gamma left(x-vtright)

y

′

= y

displaystyle y^ prime =y

z

′

= z

displaystyle z^ prime =z

t

′

= γ

(

t −

v x

c

0

2

)

displaystyle t^ prime =gamma left(t- frac vx c_ 0 ^ 2 right)

where shifts in origin have been ignored, the relative velocity is assumed to be in the

x

displaystyle x

-direction and the
**Lorentz factor**

Lorentz factor γ is defined by:

γ

=

d e f

1

1 − ( v

/

c

0

)

2

≥ 1.

displaystyle gamma stackrel mathrm def = frac 1 sqrt 1-(v/c_ 0 )^ 2 geq 1.

The
**Lorentz transformation**

Lorentz transformation is equivalent to the Galilean
transformation in the limit c0 → ∞ (a hypothetical case) or v →
0 (low speeds).
Under Lorentz transformations, the time and distance between events
may differ among inertial reference frames; however, the Lorentz
scalar distance s between two events is the same in all inertial
reference frames

s

2

=

(

x

2

−

x

1

)

2

+

(

y

2

−

y

1

)

2

+

(

z

2

−

z

1

)

2

−

c

0

2

(

t

2

−

t

1

)

2

displaystyle s^ 2 =left(x_ 2 -x_ 1 right)^ 2 +left(y_ 2 -y_ 1 right)^ 2 +left(z_ 2 -z_ 1 right)^ 2 -c_ 0 ^ 2 left(t_ 2 -t_ 1 right)^ 2

From this perspective, the speed of light is only accidentally a
property of light, and is rather a property of spacetime, a conversion
factor between conventional time units (such as seconds) and length
units (such as meters).
Incidentally, because of the limitations on speeds faster than the
speed of light, notice that in a rotating frame of reference (which is
a non-inertial frame, of course) stationarity is not possible at
arbitrary distances because at large radius the object would move
faster than the speed of light.[48]
General relativity[edit]
Main articles:
**General relativity**

General relativity and Introduction to general
relativity
See also:
**Equivalence principle**

Equivalence principle and Eötvös experiment
**General relativity**

General relativity is based upon the principle of equivalence:[49][50]

There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating. — Douglas C. Giancoli, Physics for Scientists and Engineers with Modern Physics, p. 155.

This idea was introduced in Einstein's 1907 article "Principle of
Relativity and Gravitation" and later developed in 1911.[51] Support
for this principle is found in the Eötvös experiment, which
determines whether the ratio of inertial to gravitational mass is the
same for all bodies, regardless of size or composition. To date no
difference has been found to a few parts in 1011.[52] For some
discussion of the subtleties of the Eötvös experiment, such as the
local mass distribution around the experimental site (including a quip
about the mass of Eötvös himself), see Franklin.[53]
Einstein’s general theory modifies the distinction between nominally
"inertial" and "noninertial" effects by replacing special relativity's
"flat" Minkowski
**Space**

Space with a metric that produces non-zero curvature.
In general relativity, the principle of inertia is replaced with the
principle of geodesic motion, whereby objects move in a way dictated
by the curvature of spacetime. As a consequence of this curvature, it
is not a given in general relativity that inertial objects moving at a
particular rate with respect to each other will continue to do so.
This phenomenon of geodesic deviation means that inertial frames of
reference do not exist globally as they do in Newtonian mechanics and
special relativity.
However, the general theory reduces to the special theory over
sufficiently small regions of spacetime, where curvature effects
become less important and the earlier inertial frame arguments can
come back into play.[54][55] Consequently, modern special relativity
is now sometimes described as only a "local theory".[56] "Local" can
encompass, for example, the entire
**Milky Way**

Milky Way galaxy: The astronomer
**Karl Schwarzschild**

Karl Schwarzschild observed the motion of pairs of stars orbiting each
other. He found that the two orbits of the stars of such a system lie
in a plane, and the perihelion of the orbits of the two stars remains
pointing in the same direction with respect to the solar system.
Schwarzschild pointed out that that was invariably seen: the direction
of the angular momentum of all observed double star systems remains
fixed with respect to the direction of the angular momentum of the
Solar System. These observations allowed him to conclude that inertial
frames inside the galaxy do not rotate with respect to one another,
and that the space of the
**Milky Way**

Milky Way is approximately Galilean or
Minkowskian.[57]
See also[edit]

Absolute rotation Diffeomorphism Galilean invariance General covariance

Local reference frame Lorentz invariance Newton's first law Quantum reference frame

References[edit]

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remains intact in another frame of reference, just because we choose
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**Karl Schwarzschild**

Karl Schwarzschild (2.2 MB PDF-file)

Further reading[edit]

**Edwin F. Taylor** and John Archibald Wheeler,
**Spacetime**

Spacetime Physics, 2nd ed.
(Freeman, NY, 1992)
Albert Einstein, Relativity, the special and the general theories,
15th ed. (1954)
Poincaré, Henri (1900). "La théorie de Lorentz et le Principe de
Réaction". Archives Neerlandaises. V: 253–78.
Albert Einstein, On the Electrodynamics of Moving Bodies, included in
The Principle of Relativity, page 38. Dover 1923

Rotation of the Universe

Julian B. Barbour; Herbert Pfister (1998). Mach's Principle: From
Newton's Bucket to Quantum Gravity. Birkhäuser. p. 445.
ISBN 0-8176-3823-7.
PJ Nahin (1999).
**Time**

Time Machines. Springer. p. 369; Footnote 12.
ISBN 0-387-98571-9.
B Ciobanu, I Radinchi Modeling the electric and magnetic fields in a
rotating universe Rom. Journ. Phys., Vol. 53, Nos. 1–2, P.
405–415, Bucharest, 2008
Yuri N. Obukhov, Thoralf Chrobok, Mike Scherfner Shear-free rotating
inflation Phys. Rev. D 66, 043518 (2002) [5 pages]
Yuri N. Obukhov On physical foundations and observational effects of
cosmic rotation (2000)
Li-Xin Li Effect of the Global Rotation of the Universe on the
Formation of Galaxies General Relativity and Gravitation, 30 (1998)
doi:10.1023/A:1018867011142
P Birch Is the Universe rotating? Nature 298, 451 - 454 (29 July 1982)
Kurt Gödel An example of a new type of cosmological solutions of
Einstein’s field equations of gravitation Rev. Mod. Phys., Vol. 21,
p. 447, 1949.

External links[edit]

Stanford Encyclopedia of Philosophy entry
Animation clip on
**YouTube**

YouTube showing scenes as viewed from both an
inertial frame and a rotating frame of reference, visualizing the
Coriolis and centrif