The definition of inertial coordinate system
Cleonis
From: Peter Enders
Date: 25 may 2007 19:15
> For me, the definitions of inertial frame and non-inertial forces
> appear like a circle...
I will present evidence that the definition of inertial frame, when
properly formulated, is not circular. I submit as evidence the fact
that it is possible to formulate laws of motion that serve as powerful
physics tools.
(The following reasoning is in terms of newtonian dynamics, but it
applies equally in the context of the special theory of relativity and
the general theory of relativity.)
I start with recapitulating Galileo's reasoning as to why it is
possible to compose laws of motion. Imagine you are in a cabin of a
boat that is in motion over perfectly smooth water. You are juggling,
or you are throwing darts, or something like that. Galileo argued: no
matter the velocity of the boat, the timing for juggling the balls or
the trowing of the darts is the same. This illustrates that there are
laws of motion, and to become a skilful juggler is the acquirement of
implicit knowledge (motoric knowledge) of how to work with the laws of
motion.
The grid of the dartboard serves as a reference, the darter's aim is
with respect to that reference system. One layer of 'the same' is that
it does not matter where in space the dartboard is positioned, the
laws of motion are the same. A second layer is that the velocity of
one boat relative to another does not matter either, the laws of
motion are the same.
Galileo's reasoning introduces an extensive equivalence class. Take
the set of all reference systems that have a uniform velocity relative
to each other: that set forms a class of coordinate systems with the
property that required skills (juggling skills) are identical for any
member of that equivalence class. This equivalence class is referred
to as 'the equivalence class of inertial coordinate systems'. The
large extend of this equivalence class is what makes formulating laws
of motion worthwile.
The next thing to show is that the equivalence class of inertial
coordinate system is in fact the only class for which it is possible
to formulate laws of motion at all.
When motion is mapped in a rotating coordinate system, two additions
to the equation of motion are necessary: the centrifugal term and the
coriolis term. Let the angular velocity of the rotating coordinate
system with respect to the inertial coordinate system be called
'omega'.
The centrifugal term: omega^2*r
The coriolis term: 2*omega*v
Both the centrifugal term and the coriolis term feature 'omega', the
angular velocity of the rotating coordinate system with respect to the
inertial coordinate system. That is, the centrifugal term and the
coriolis term refer back to the inertial coordinate system! This
illustrates that such a thing as a law of motion that doesn't refer to
the equivalence class of inertial coordinate systems simply doesn't
exist. No matter what you do, in the end you find yourself referring
to the inertial coordinate system.
Conclusions:
We have as evidence that it is possible to formulate laws of motion
that serve as powerful physics tools.
Laws of motion can be formulated if and only if motion is mapped with
respect to a member of the equivalence class of inertial coordinate
systems. This suffices as unequivocal definition of the equivalence
class of inertial coordinate systems.
(In GTR, the equations are composed in a formalism that is designed to
be independent of how the motion will later be mapped. Roughly said:
first you obtain a solution to the Einstein field equations, and then
you map that solution to a chosen coordinate system. You are free to
map either to an inertial system or to an non-inertial system, but if
you map to a non-inertial system you must specify the mapping's
acceleration with respect to the inertial coordinate system. This
shows that GTR does not dispense with the equivalence class of
inertial frames.)
Cleonis
Oh No
>I copy and paste from a previous thread
>From: Peter Enders
>Date: 25 may 2007 19:15
>
>> For me, the definitions of inertial frame and non-inertial forces
>> appear like a circle...
>
>I will present evidence that the definition of inertial frame, when
>properly formulated, is not circular. I submit as evidence the fact
>that it is possible to formulate laws of motion that serve as powerful
>physics tools.
>
Very neat argument. Can you take it as far as showing that Mach's
principle is silly?
What I meant by agreeing with the circularity in the definition was that
I see the first law of motion (N1) as being circular, in the sense that
a straight line (now a geodesic) is that which is defined locally by
inertial motion.
Regards
--
Charles Francis
moderator sci.physics.foundations.
substitute charles for NotI to email
Cleonis
> What I meant by agreeing with the circularity in the
> definition was that I see the first law of motion (N1) as
> being circular, in the sense that a straight line (now a
> geodesic) is that which is defined locally by inertial
> motion.
I like the following comparison: a dictionary describes the words of a
language - using words. Words to describe words. Still, a dictionary
is not merely circular. While being strongly self-referential, a
dictionary isn't exclusively self-referential. A dictionary
incorporates knowledge and understanding of the world and refers to
it.
The basic definitions and axioms of any theory of motion are strongly
selfreferential, and I think the sometimes arising suspicion that the
sets of definitions are merely circular is understandable.
Especially definitions that are only locally valid tend to be so
circular that they are empty; generally I think it's best to work with
definitions that can be scaled up to universal validity.
Cleonis
Oh No
being empty. But a definition can be circular and still capture
something of physical reality. I think this is true of the case in hand,
and is relevant to discussions of whether Mach's principle is contained
in gtr - since Einstein is supposed to have thought a lot about this
principle while developing gtr. If a geodesic is that which is defined
by local inertial motion, then whether a bucket is rotating is
determined entirely locally, not by reference to the universe at large,
so Mach's principle does not hold.
Ken S. Tucker
Suppose I draw a diagram,
(M1) and (M2) are Massive objects, the "=="
is a supporting scafold, and EL1 and EL2 are
enclose ELevators, and all components are at
relative rest, (not rotating).
(M1)=EL1=======+=======EL2=(M2) FIG1.
<= =>
(accelometers)
Observers in EL1 and EL2 will observe non-zero
readings on their accelometers, they are in non-inertial
Frames of Reference.
Next, elliminate the Masses and set EL1 and EL2
to rotate around "+",
rotating around +
EL1=======+=======EL2 FIG2
<= => (accelometers)
FIG2 has the same accelometer readings as FIG1.
The above is an easy example of how gravitational
accelerations and inertial accelerations are equivalent,
known as the Principle of Equivalence.
> (In GTR, the equations are composed in a formalism that is designed to
> be independent of how the motion will later be mapped. Roughly said:
> first you obtain a solution to the Einstein field equations, and then
> you map that solution to a chosen coordinate system. You are free to
> map either to an inertial system or to an non-inertial system, but if
> you map to a non-inertial system you must specify the mapping's
> acceleration with respect to the inertial coordinate system. This
> shows that GTR does not dispense with the equivalence class of
> inertial frames.)
Agreed, but Inertial FoR's are not special anymore.
In the above FIG's, we can let an observer in an Inertial
FoR be situated at the "+" ~ Center of Gravity.
The laws of Physics apply to all 3 observers equally,
and therefore their measurements are equally valid
and comparable by transformation of the components
of their respective measurements, that's General
Covariance.
> Cleonis
Yup Good Stuff
Ken S. Tucker
Oh No
>On Apr 30, 4:58 am, Cleonis <cleonis...@gmail.com> wrote:
>Suppose I draw a diagram,
>(M1) and (M2) are Massive objects, the "=="
>is a supporting scafold, and EL1 and EL2 are
>enclose ELevators, and all components are at
>relative rest, (not rotating).
>
> (M1)=EL1=======+=======EL2=(M2) FIG1.
> <= =>
>(accelometers)
>
>Observers in EL1 and EL2 will observe non-zero
>readings on their accelometers, they are in non-inertial
>Frames of Reference.
>
>Next, elliminate the Masses and set EL1 and EL2
>to rotate around "+",
>
> rotating around +
>EL1=======+=======EL2 FIG2
> <= => (accelometers)
>
>FIG2 has the same accelometer readings as FIG1.
>
>The above is an easy example of how gravitational
>accelerations and inertial accelerations are equivalent,
>known as the Principle of Equivalence.
Actually not. The principle of equivalence applies to linear
acceleration. You are ignoring coriolis forces.
>
>> (In GTR, the equations are composed in a formalism that is designed to
>> be independent of how the motion will later be mapped. Roughly said:
>> first you obtain a solution to the Einstein field equations, and then
>> you map that solution to a chosen coordinate system. You are free to
>> map either to an inertial system or to an non-inertial system, but if
>> you map to a non-inertial system you must specify the mapping's
>> acceleration with respect to the inertial coordinate system. This
>> shows that GTR does not dispense with the equivalence class of
>> inertial frames.)
>
>Agreed, but Inertial FoR's are not special anymore.
>In the above FIG's, we can let an observer in an Inertial
>FoR be situated at the "+" ~ Center of Gravity.
>The laws of Physics apply to all 3 observers equally,
>and therefore their measurements are equally valid
>and comparable by transformation of the components
>of their respective measurements, that's General
>Covariance.
>
Indeed, but we still have to interpret the laws of physics, and in
particular general covariance. There is a difference between laws which
apply in all coordinate systems, and laws which apply in a particular
inertial coordinate system, but which can be expressed in all coordinate
systems. General relativity, and general covariance, gives us the
latter, not the former. Inertial frames remain special. These are frames
in which the laws have a particularly simple form, essentially that
momentum is always conserved.
Ken S. Tucker
> Thus spake Ken S. Tucker <dynam...@vianet.on.ca>
>
>
>
> >On Apr 30, 4:58 am, Cleonis <cleonis...@gmail.com> wrote:
> >Suppose I draw a diagram,
> >(M1) and (M2) are Massive objects, the "=="
> >is a supporting scafold, and EL1 and EL2 are
> >enclose ELevators, and all components are at
> >relative rest, (not rotating).
>
> > (M1)=EL1=======+=======EL2=(M2) FIG1.
> > <= =>
> >(accelometers)
>
> >Observers in EL1 and EL2 will observe non-zero
> >readings on their accelometers, they are in non-inertial
> >Frames of Reference.
>
> >Next, elliminate the Masses and set EL1 and EL2
> >to rotate around "+",
>
> > rotating around +
> >EL1=======+=======EL2 FIG2
> > <= => (accelometers)
>
> >FIG2 has the same accelometer readings as FIG1.
>
> >The above is an easy example of how gravitational
> >accelerations and inertial accelerations are equivalent,
> >known as the Principle of Equivalence.
>
> Actually not. The principle of equivalence applies to linear
> acceleration. You are ignoring coriolis forces.
No, the Coriolis accelerations, are a tidal measure
(see Weinberg's G&C, Eq.(6.10.1).
I specified acccelometers in the ELevators. The idea
being is that over the small finite region of said EL's
tidal forces are nil. Secondly, by using an acclerometer
at a point, your objection disappears.
> >> (In GTR, the equations are composed in a formalism that is designed to
> >> be independent of how the motion will later be mapped. Roughly said:
> >> first you obtain a solution to the Einstein field equations, and then
> >> you map that solution to a chosen coordinate system. You are free to
> >> map either to an inertial system or to an non-inertial system, but if
> >> you map to a non-inertial system you must specify the mapping's
> >> acceleration with respect to the inertial coordinate system. This
> >> shows that GTR does not dispense with the equivalence class of
> >> inertial frames.)
>
> >Agreed, but Inertial FoR's are not special anymore.
> >In the above FIG's, we can let an observer in an Inertial
> >FoR be situated at the "+" ~ Center of Gravity.
> >The laws of Physics apply to all 3 observers equally,
> >and therefore their measurements are equally valid
> >and comparable by transformation of the components
> >of their respective measurements, that's General
> >Covariance.
>
> Indeed, but we still have to interpret the laws of physics, and in
> particular general covariance. There is a difference between laws which
> apply in all coordinate systems, and laws which apply in a particular
> inertial coordinate system, but which can be expressed in all coordinate
> systems.
No, the purpose of General Covariance (and thus GR),
was to utilize General Covariance to make of FoR's
equal.
> General relativity, and general covariance, gives us the
> latter, not the former. Inertial frames remain special. These are frames
> in which the laws have a particularly simple form, essentially that
> momentum is always conserved.
A great achievement of AE's was to generalize
Conservation of Energy -Momentum in 4D, 1st
in SR, then later in GR by generalizing the Lorentz
Transform to become General Covariance, that
includes inertial FoR's and all non-inertial FoR's,
and the math of tensors is nearly prefect for that.
Best Regards
Ken S. Tucker
Oh No
>>
>> Actually not. The principle of equivalence applies to linear
>> acceleration. You are ignoring coriolis forces.
>
>No, the Coriolis accelerations, are a tidal measure
Get on a children's roundabout. When you move, you will feel a force
sideways to the motion.
>I specified acccelometers in the ELevators. The idea
>being is that over the small finite region of said EL's
>tidal forces are nil.
What I call tidal forces are due to geometric differences between one
point and another. There are no tidal forces by my definition within
your diagram, but there are Coriolis forces on anything moving.
>Secondly, by using an acclerometer
>at a point, your objection disappears.
Coriolis forces act at a point on anything moving in rotating
coordinates. If you keep everything static, they will not appear, but
the laws of physics concern movement, so it is wrong to ignore them.
>
>>
>> Indeed, but we still have to interpret the laws of physics, and in
>> particular general covariance. There is a difference between laws which
>> apply in all coordinate systems, and laws which apply in a particular
>> inertial coordinate system, but which can be expressed in all coordinate
>> systems.
>
>No, the purpose of General Covariance (and thus GR),
>was to utilize General Covariance to make of FoR's
>equal.
>
That is what I have been discussing with Harry. Read Cleonis post again,
carefully. This is a subtle enough point in general relativity and
widely misunderstood, especially by physicists who mistake maths for
physics. Inertial frames remain special. In inertial frames, forces have
a direct mechanistic cause. They are "contact" forces. As a result, the
laws of physics take a simple form. General covariance allows us to
express those laws in a unified way in any frame. But that is a
mathematical statement, and it comes at the cost of including forces
which do not have a direct mechanistic cause, "inertial forces". The PoE
says that gravity is an inertial force. Where I disagreed with Harry is
that I don't think Einstein misunderstood this point, though loads of
physicists do, including Weinberg, whom you cite. If they understood it,
they would not be looking for a graviton or trying to find unification
via string theory.
>> General relativity, and general covariance, gives us the
>> latter, not the former. Inertial frames remain special. These are frames
>> in which the laws have a particularly simple form, essentially that
>> momentum is always conserved.
>
>A great achievement of AE's was to generalize
>Conservation of Energy -Momentum in 4D, 1st
>in SR, then later in GR by generalizing the Lorentz
>Transform to become General Covariance, that
>includes inertial FoR's and all non-inertial FoR's,
>and the math of tensors is nearly prefect for that.
>Best Regards
Momentum is not conserved in non-inertial frames. That is why fictitious
forces have to be invented in order to restore covariance.
harry
"Ken S. Tucker" <dyna...@vianet.on.ca> wrote in message
news:1177957067.7...@u30g2000hsc.googlegroups.com...[...]
Attention: different people and theories use the word "inertial" with
different meanings.
> Next, elliminate the Masses and set EL1 and EL2
> to rotate around "+",
>
> rotating around +
> EL1=======+=======EL2 FIG2
> <= => (accelometers)
>
> FIG2 has the same accelometer readings as FIG1.
>
> The above is an easy example of how gravitational
> accelerations and inertial accelerations are equivalent,
> known as the Principle of Equivalence.
[..]
> Inertial FoR's are not special anymore.
> In the above FIG's, we can let an observer in an Inertial
> FoR be situated at the "+" ~ Center of Gravity.
> The laws of Physics apply to all 3 observers equally,
> and therefore their measurements are equally valid
> and comparable by transformation of the components
> of their respective measurements, that's General
> Covariance.
Do you define for example
a) coordinate systems K and K' that extend from EL1 to EL2, but that are 1D
and don't go beyond EL1 and EL2?
Or, instead, do you define
b) 3 coordinate systems with near-zero length?
In case a), one just has to throw a ball to discover the difference between
the frames.
In case b), inertial frames are special because the laws of motion are valid
over significant distances.
Regards,
Harald
harry
"Cleonis" <cleon...@gmail.com> wrote in message
news:1177929217....@l77g2000hsb.googlegroups.com...
>I copy and paste from a previous thread
> From: Peter Enders
> Date: 25 may 2007 19:15
>
>> For me, the definitions of inertial frame and non-inertial forces
>> appear like a circle...
Cleonis, below you make an interesting point, explaining why inertial frames
are preferred for descriptions of motion.
However, that does not tell me what your definition of inertial frames is;
and thus I can't even judge if I find that circular or not!
Please specify.
Thanks,
Harald
Ken S. Tucker
> Thus spake Ken S. Tucker <dynam...@vianet.on.ca>
>
>
>
> >> Actually not. The principle of equivalence applies to linear
> >> acceleration. You are ignoring coriolis forces.
>
> >No, the Coriolis accelerations, are a tidal measure
>
> Get on a children's roundabout. When you move, you will feel a force
> sideways to the motion.
>
> >I specified acccelometers in the ELevators. The idea
> >being is that over the small finite region of said EL's
> >tidal forces are nil.
>
> What I call tidal forces are due to geometric differences between one
> point and another. There are no tidal forces by my definition within
> your diagram, but there are Coriolis forces on anything moving.
In this ref,
http://en.wikipedia.org/wiki/Coriolis_effect#Formula
The vector product w x v = coriolis effect.
In my FIG 2, I specify v=0, the accelometers are merely
weigh scales.
> >Secondly, by using an acclerometer
> >at a point, your objection disappears.
>
> Coriolis forces act at a point on anything moving in rotating
> coordinates. If you keep everything static, they will not appear, but
> the laws of physics concern movement, so it is wrong to ignore them.
True, however in our idealism, we may extend
the distance between the ELevators indefinitely,
and tend to create - within the confines of the EL,
the equivalence of a gravitational field.
> >> Indeed, but we still have to interpret the laws of physics, and in
> >> particular general covariance. There is a difference between laws which
> >> apply in all coordinate systems, and laws which apply in a particular
> >> inertial coordinate system, but which can be expressed in all coordinate
> >> systems.
>
> >No, the purpose of General Covariance (and thus GR),
> >was to utilize General Covariance to make of FoR's
> >equal.
>
> That is what I have been discussing with Harry. Read Cleonis post again,
> carefully. This is a subtle enough point in general relativity and
> widely misunderstood, especially by physicists who mistake maths for
> physics. Inertial frames remain special. In inertial frames, forces have
> a direct mechanistic cause. They are "contact" forces. As a result, the
> laws of physics take a simple form. General covariance allows us to
> express those laws in a unified way in any frame. But that is a
> mathematical statement, and it comes at the cost of including forces
> which do not have a direct mechanistic cause, "inertial forces". The PoE
> says that gravity is an inertial force. Where I disagreed with Harry is
> that I don't think Einstein misunderstood this point, though loads of
> physicists do, including Weinberg, whom you cite.
Weinberg might disagree with Einstein on somethings,
though not a lot I have learned, so I tend to use either
of their references, though carefully, being mindful of
their subtle differences, they're a generation apart.
> If they understood it,
> they would not be looking for a graviton or trying to find unification
> via string theory.
My opinion leans your way, but recently Mr. Yablon
breathed new life into the "string theory", so I'm keeping
an open mind on that solution technique.
> >> General relativity, and general covariance, gives us the
> >> latter, not the former. Inertial frames remain special. These are frames
> >> in which the laws have a particularly simple form, essentially that
> >> momentum is always conserved.
>
> >A great achievement of AE's was to generalize
> >Conservation of Energy -Momentum in 4D, 1st
> >in SR, then later in GR by generalizing the Lorentz
> >Transform to become General Covariance, that
> >includes inertial FoR's and all non-inertial FoR's,
> >and the math of tensors is nearly prefect for that.
> >Best Regards
>
> Momentum is not conserved in non-inertial frames. That is why fictitious
> forces have to be invented in order to restore covariance.
I'll do further research online, and try to find
references in support of the ideas.
> Regards
> Charles Francis
Thanks and Regards
Ken S. Tucker
Oh No
>> That is what I have been discussing with Harry. Read Cleonis post again,
>> carefully. This is a subtle enough point in general relativity and
>> widely misunderstood, especially by physicists who mistake maths for
>> physics. Inertial frames remain special. In inertial frames, forces have
>> a direct mechanistic cause. They are "contact" forces. As a result, the
>> laws of physics take a simple form. General covariance allows us to
>> express those laws in a unified way in any frame. But that is a
>> mathematical statement, and it comes at the cost of including forces
>> which do not have a direct mechanistic cause, "inertial forces". The PoE
>> says that gravity is an inertial force. Where I disagreed with Harry is
>> that I don't think Einstein misunderstood this point, though loads of
>> physicists do, including Weinberg, whom you cite.
>
>Weinberg might disagree with Einstein on somethings, though not a lot I
>have learned, so I tend to use either of their references, though
>carefully, being mindful of their subtle differences, they're a
>generation apart.
Unlike Harry, I don't find what Einstein said very clear in the
references he cites, but I do agree with Einstein that the use of the
word "real" is problematic and confusing in this context. Weinberg,
otoh, puts forward a view of general relativity in his introduction
which I completely disagree with. However, I am told that, having tried
to dispense with a geometrical view of general relativity his
mathematical account is geometrical.
>
>> If they understood it,
>> they would not be looking for a graviton or trying to find unification
>> via string theory.
>
>My opinion leans your way, but recently Mr. Yablon breathed new life
>into the "string theory", so I'm keeping an open mind on that solution
>technique.
This was a somewhat different context, that of strong interactions not
gravity. I do not know enough of the mathematical detail of string
theory to comment, but Jay's application did seem more reasonable than
the usual context.
Ken S. Tucker
> Thus spake Ken S. Tucker <dynam...@vianet.on.ca>
>
>
>
> >> That is what I have been discussing with Harry. Read Cleonis post again,
> >> carefully. This is a subtle enough point in general relativity and
> >> widely misunderstood, especially by physicists who mistake maths for
> >> physics. Inertial frames remain special. In inertial frames, forces have
> >> a direct mechanistic cause. They are "contact" forces. As a result, the
> >> laws of physics take a simple form. General covariance allows us to
> >> express those laws in a unified way in any frame. But that is a
> >> mathematical statement, and it comes at the cost of including forces
> >> which do not have a direct mechanistic cause, "inertial forces". The PoE
> >> says that gravity is an inertial force. Where I disagreed with Harry is
> >> that I don't think Einstein misunderstood this point, though loads of
> >> physicists do, including Weinberg, whom you cite.
>
> >Weinberg might disagree with Einstein on somethings, though not a lot I
> >have learned, so I tend to use either of their references, though
> >carefully, being mindful of their subtle differences, they're a
> >generation apart.
>
> Unlike Harry, I don't find what Einstein said very clear in the
> references he cites, but I do agree with Einstein that the use of the
> word "real" is problematic and confusing in this context.
I think Einstein had a lot of domestic problems
and tried to convert his GR tensors into "word
salad" at that time, I think he's near the best,
and tried to communicate his ideas, but's rather
hard to explain 81/9 = 9 to people who can't count.
> Weinberg,
> otoh, puts forward a view of general relativity in his introduction
> which I completely disagree with. However, I am told that, having tried
> to dispense with a geometrical view of general relativity his
> mathematical account is geometrical.
He does a rather dry candid account on pg.147 in
G&C, "9 The Geometric Analogy", to explain his
"disappointment" with the geometric interpretation.
IMHO, he was focused on the continuum when he
wrote that, and if he would have taken the relational
approach you - and I think too - to relativity he may
have written otherwise.
It looks to me, that by embracing tensors, GR may
have incorporated too much Newton and not enough
relativity, because relativity is, at it's heart, a theory
of relations, while Newton's theory is a field theory.
> >> If they understood it,
> >> they would not be looking for a graviton or trying to find unification
> >> via string theory.
>
> >My opinion leans your way, but recently Mr. Yablon breathed new life
> >into the "string theory", so I'm keeping an open mind on that solution
> >technique.
>
> This was a somewhat different context, that of strong interactions not
> gravity. I do not know enough of the mathematical detail of string
> theory to comment, but Jay's application did seem more reasonable than
> the usual context.
"strong interactions not gravity"
At some point in time, we'll need to account for
the spectrum of the masses of the fermions and
baryons. An objective Jay has stated (mine too)
is why is the proton mass ~ 1860 x electron mass.
Rewrite that as,
proton gravitation ~ 1860 x electron gravitation,
then Jay either finds G_uv = T_uv falls out, or
replaces it with something better.
> Charles Francis
Cheers
Ken S. Tucker
Cleonis
> > The next thing to show is that the equivalence class of inertial
> > coordinate system is in fact the only class for which it is
> > possible to formulate laws of motion at all. When motion is mapped
> > in a rotating coordinate system, two additions to the equation of
> > motion are necessary: the centrifugal term and the coriolis term.
> > Let the angular velocity of the rotating coordinate system with
> > respect to the inertial coordinate system be called 'omega'.
> >
> > The centrifugal term: omega^2*r
> > The coriolis term: 2*omega*v
> >
> > Both the centrifugal term and the coriolis term feature 'omega',
> > the angular velocity of the rotating coordinate system with
> > respect to the inertial coordinate system. That is, the
> > centrifugal term and the coriolis term refer back to the inertial
> > coordinate system! This illustrates that such a thing as a law of
> > motion that doesn't refer to the equivalence class of inertial
> > coordinate systems simply doesn't exist. No matter what you do, in
> > the end you find yourself referring to the inertial coordinate
> > system.
On May 1, 11:34 am, "harry" <harald.vanlintelButNotT...@epfl.ch>
wrote:
> you make an interesting point, explaining why
> inertial frames are preferred for descriptions of motion.
> However, that does not tell me what your definition of inertial
> frames is; and thus I can't even judge if I find that circular or
> not!
Personally, I never use the expression 'preferred for describing
motion'. To me that sounds just as odd as saying 'Conservation of
energy is the preferred hypothesis'.
In order to formulate laws of motion one must recognize the
equivalence class of inertial coordinate systems. Formulating laws of
motion and recognizing the equivalence class of inertial coordinate
systems are two sides of the same coin. Each makes the existence of
the other possible.
The definition of the equivalence class of inertial coordinate
systems:
The equivalence class of inertial coordinate system is the class of
coordinate systems that is singled out by the laws of motion. (This is
illustrated with the example above: the centrifugal term and the
coriolis term refer to the inertial coordinate system. All laws of
motion refer to the equivalence class of inertial coordinate systems.)
Proof that the above definition is not circular: the power of the laws
of motion as physics tools; if the definition would be purely
circular, the laws would be empty tautologies.
The above definition is the most fundamental one. More superficially,
some implications/properties can be listed. For example: each member
of the equivalence class of inertial coordinate systems has the
following property: inertia is uniform in all directions. That is,
when something is accelerated with respect to an inertial coordinate
system, the same force results in the same acceleration in all
directions.
The following example illustrates how important inertial frames are
for GTR.
In november 1915, Einstein had returned to equations he had discarded
in 1913, when in collaboration with Grossmann he had developed the
theory that is nowadays referred to as the 'Entwurf theory'.
Having rushed the november 1915 field equations to publication,
Einstein set out to calculate what Mercury perihelion precession was
predicted by the november 1915 field equations. Using techniques
developed earlier, Einstein obtained an answer in a matter of weeks.
Einstein was extatic with joy to find that the november 1915 field
equations predict that Mercyry's perihelion of Mercury will precess 43
seconds of arc per century with respect to the inertial frame of
reference.
This example also helps to illustrate the distinction between what can
be classified as artefact and what can be classified as physics taking
place. The perihelion precession of Mercury is (according to GTR)
physics taking place; it is accounted for by the field equations.
Mercury's perihelion precession cannot be "transformed away". Mapping
Mercury's orbit to a coordinate system that co-rotates with Mercury's
perihelion precession is not an option, for if you do so then all the
other planets will have anomalous orbits.
The inertial coordinate system is the coordinate system in which all
of the planets motions are free of anomalies (as defined by the laws
of motion).
Cleonis
Peter
> ...
> In order to formulate laws of motion one must recognize the
> equivalence class of inertial coordinate systems. Formulating laws of
> motion and recognizing the equivalence class of inertial coordinate
> systems are two sides of the same coin. Each makes the existence of
> the other possible.
For the sake of methodical rigor let me comment that this is not generally
true. Aristotle stated laws of motion without inertial systems. For our
universe, however, momentum conservation holds true and this is related to
the equivalence of all reference systems that move with constant velocity
each against another. (Huygens derived the momentum conservation law from the
assumption that the laws of impact are the same on the shelf and on a ship
moving with constant velocity.)
> The definition of the equivalence class of inertial coordinate
> systems:
> The equivalence class of inertial coordinate system is the class of
> coordinate systems that is singled out by the laws of motion.
Yes, this is the crucial question:
Are the laws of motion determined by the space(-time) or is the space(-time)
determined by the dynamics of the bodies/particles and their interactions?
Peter
Oh No
>Cleonis <cleon...@gmail.com> writes:
>
>> ...
>> In order to formulate laws of motion one must recognize the
>> equivalence class of inertial coordinate systems. Formulating laws of
>> motion and recognizing the equivalence class of inertial coordinate
>> systems are two sides of the same coin. Each makes the existence of
>> the other possible.
>
>For the sake of methodical rigor let me comment that this is not generally
>true.
I agree with Cleonis.
>Aristotle stated laws of motion without inertial systems.
Aristotle's ideas were both badly formulated and incorrect.
>For our
>universe, however, momentum conservation holds true and this is related to
>the equivalence of all reference systems that move with constant velocity
>each against another.
This is not a good formulation. Momentum is only conserved in inertial
reference frames.
>(Huygens derived the momentum conservation law from the
>assumption that the laws of impact are the same on the shelf and on a ship
>moving with constant velocity.)
There is more to conservation of momentum than impact. N1 also expresses
conservation of momentum.
>
>> The definition of the equivalence class of inertial coordinate
>> systems:
>> The equivalence class of inertial coordinate system is the class of
>> coordinate systems that is singled out by the laws of motion.
>
>Yes, this is the crucial question:
>Are the laws of motion determined by the space(-time) or is the space(-time)
>determined by the dynamics of the bodies/particles and their interactions?
>
Indeed, that is the crucial question. I am not sure we can use the word
dynamics in this context - to me dynamics already assumes space-time. I
base my ideas on the thought that space-time is determined by particles
and their interactions. I think this is expressed in Einstein's field
equation, and also in geodesic motion.
Peter
> Thus spake Peter <end...@dekasges.de>
> >Cleonis <cleon...@gmail.com> writes:
> >
> >> ...
> >> In order to formulate laws of motion one must recognize the
> >> equivalence class of inertial coordinate systems. Formulating laws of
> >> motion and recognizing the equivalence class of inertial coordinate
> >> systems are two sides of the same coin. Each makes the existence of
> >> the other possible.
> >For the sake of methodical rigor let me comment that this is not generally
> >true.
> I agree with Cleonis.
I guess that you both have in mind 'to formulate _correct_ laws of motion
_for our universe (as far as we know it)_'.
Galilei invariance holds true, when Newton's equation of motion holds true.
But one doesn't need first to assume Galilei invariance before one can derive
Newton's equation of motion. Euler needed only the homogeneity and isotropy
of space for his ansatz that the change of velocity is proportional to the
external force.
> >Aristotle stated laws of motion without inertial systems.
> Aristotle's ideas were both badly formulated and incorrect.
You know it better than me. I had in mind his law v=const, if F=const and
that light/heavy bodies move upward/downward. This is fine for the motion of
a spherical body in a fluid with Stokes friction. Thus, _there are_ laws of
motion without inertial system.
I just wish to stress how careful one has to formulate such issues - and I'm
indebted to you for your clarifications :-)
> >For our
> >universe, however, momentum conservation holds true and this is related to
> >the equivalence of all reference systems that move with constant velocity
> >each against another.
>
> This is not a good formulation. Momentum is only conserved in inertial
> reference frames.
Agreed.
I had in mind the following:
- Assume an inertial system with Cartesian axes x,y,z;
- assume a non-inertial system that rotates with constant angular velocity
around the z-axis of that;
- assume a 2nd non-inertial system that rotates with the same constant
angular velocity around the z-axis of the inertial system and, additionally,
moves with constant speed along the x-axis of it.
Are these 2 non-inertial systems equivalent?
> >(Huygens derived the momentum conservation law from the
> >assumption that the laws of impact are the same on the shelf and on a ship
> >moving with constant velocity.)
>
> There is more to conservation of momentum than impact. N1 also expresses
> conservation of momentum.
Yes. I was like to evoke a less known example which, what's more, stems from
the time before the 'Principia'.
> >> The definition of the equivalence class of inertial coordinate
> >> systems:
> >> The equivalence class of inertial coordinate system is the class of
> >> coordinate systems that is singled out by the laws of motion.
> >Yes, this is the crucial question:
> >Are the laws of motion determined by the space(-time) or is the
> space(-time)
> >determined by the dynamics of the bodies/particles and their interactions?
> Indeed, that is the crucial question. I am not sure we can use the word
> dynamics in this context - to me dynamics already assumes space-time. I
> base my ideas on the thought that space-time is determined by particles
> and their interactions. I think this is expressed in Einstein's field
> equation, and also in geodesic motion.
I agree. 'dynamics' (science of forces; example: Newton's Laws) was meant in
contrast to kinematics (science of motion in space without looking at its
causes; example: Einstein's 1905 foundation of special relativity).
Best wishes,
Peter
Ken S. Tucker
> On May 1, 12:41 am, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
> > Momentum is not conserved in non-inertial frames. That is why fictitious
> > forces have to be invented in order to restore covariance.
>
> I'll do further research online, and try to find
> references in support of the ideas.
Ok, I'm a bit weak, but have a glance at the geodesic...
http://en.wikipedia.org/wiki/Geodesic_%28general_relativity%29#Mathematical_expression
It says, the covariant derivative of the 4 velocity
"U^u" is zero, I'm write that in terms of the
absolute derivative (which is simpler) as,
DU^u / ds = 0,
from that, the geodesic is derived in GR, as
the ref demo's.
I'll cheat a bit by using a unit mass "m" =1, so
that m*U^u = p^u = 4-momentum, thus, (the ds
parameter is understood)...
Dp^u = 0 .
I would accept that as a Generally Covariant
conservation of Momentum.
Do you think I'm on *thin ice* ?
Cleonis
> > For the sake of methodical rigor let me comment that this is not
> > systems.
Charles Francis wrote:
> Aristotle's ideas were both badly formulated and incorrect.
I go along with Peter's rigorous approach.
Aristotle's rules of thumb do apply in their own proper context.
The distinction 'correct' versus 'incorrect' is hardly helpful here.
Newtonian dynamics has been superseded by relativistic dynamics,
nonetheless we do not regard newtonian dynamics as 'incorrect'. It's
just that today we are aware of certain limitations to the range of
applicability of newtonian dynamics. Applied within its proper
context, newtonian physics is a powerful system to understand physics
taking place.
>From a rigorously non-biased point of view: Aristotle's rules of thumb
are correct; the range of applicability is everyday life only -
strongly limited as compared to the type of motion laws that were
introduced by Galilei.
Cleonis
Oh No
>Peter Enders wrote:
>> > For the sake of methodical rigor let me comment that this is not
>> > generally true. Aristotle stated laws of motion without inertial
>> > systems.
>
>Charles Francis wrote:
>> Aristotle's ideas were both badly formulated and incorrect.
>
>I go along with Peter's rigorous approach.
>Aristotle's rules of thumb do apply in their own proper context.
Peter's version of Aristotle's laws is charitable. Specifically his
laws stated that heavier objects fall faster, the speed being
proportional to weight, that the speed of an object is in direct
proportion to the applied force, and he deduced that a vacuum cannot
have exist, because if it did, since it has zero density, all bodies
will fall through it at infinite speed.
>
>The distinction 'correct' versus 'incorrect' is hardly helpful here.
>Newtonian dynamics has been superseded by relativistic dynamics,
>nonetheless we do not regard newtonian dynamics as 'incorrect'. It's
>just that today we are aware of certain limitations to the range of
>applicability of newtonian dynamics. Applied within its proper
>context, newtonian physics is a powerful system to understand physics
>taking place.
This is quite different. It is perfectly possible to derive Newton's
laws in approximation from gtr and from quantum theory. They are
therefore correct with an appropriate context. That is not so for
Aristotle's laws. That is the distinction between science (Newton) and
not-science (Aristotle).
Cleonis
> > The distinction 'correct' versus 'incorrect' is hardly helpful
> > here. Newtonian dynamics has been superseded by relativistic
> > dynamics, nonetheless we do not regard newtonian dynamics as
> > 'incorrect'. It's just that today we are aware of certain
> > limitations to the range of applicability of newtonian dynamics.
> > Applied within its proper context, newtonian physics is a powerful
> > system to understand physics taking place.
> >
> > thumb are correct; it's just that the range of applicability is
> > everyday life only - strongly limited as compared to the type of
> > motion laws that were introduced by Galilei.
Charles Francis wrote:
> This is quite different. It is perfectly possible to derive
> Newton's laws in approximation from gtr and from quantum theory.
> They are therefore correct with an appropriate context. That is
> not so for Aristotle's laws. That is the distinction between
> science (Newton) and not-science (Aristotle).
It seems to me that the criterium for distinguishing between science
and not-science that you offer here can be rephrased as follows:
A theory is scientific if it is a limiting case of the theory that
superseded it.
That's a bit awkward. In order to learn whether the currently reigning
theory is scientific, you'd have to patiently wait until it is
eclipsed by a successor-theory.
I endorse the following point of view:
A discipline (such as astronomy) reaches a stage of being a scientific
discipline when a community of scientists reaches consensus on how to
proceed in order to increase the body of knowledge. Having a common
perspective, a common framework to organize the perceptions into
overseeable chunks, they are in a position to stimulate each others
development, which translates to development of the discipline as a
science.
By contrast: if the individuals working in a certain area do not
gravitate towards a common framework to organize the perceptions, then
all the development remains individual development, which means that
effectively there is no community effort, no shared body of
knowledge.
Cleonis
Oh No
>> Cleonis wrote:
>> > The distinction 'correct' versus 'incorrect' is hardly helpful
>> > here. Newtonian dynamics has been superseded by relativistic
>> > dynamics, nonetheless we do not regard newtonian dynamics as
>> > 'incorrect'. It's just that today we are aware of certain
>> > limitations to the range of applicability of newtonian dynamics.
>> > Applied within its proper context, newtonian physics is a powerful
>> > system to understand physics taking place.
>> >
>> > From a rigorously non-biased point of view: Aristotle's rules of
>> > thumb are correct; it's just that the range of applicability is
>> > everyday life only - strongly limited as compared to the type of
>> > motion laws that were introduced by Galilei.
>
>Charles Francis wrote:
>> This is quite different. It is perfectly possible to derive
>> Newton's laws in approximation from gtr and from quantum theory.
>> They are therefore correct with an appropriate context. That is
>> not so for Aristotle's laws. That is the distinction between
>> science (Newton) and not-science (Aristotle).
>
>
>It seems to me that the criterium for distinguishing between science
>and not-science that you offer here can be rephrased as follows:
>A theory is scientific if it is a limiting case of the theory that
>superseded it.
Lol. I wouldn't actually have described that as a criterion, but I do
think it is a feature.
>
>That's a bit awkward. In order to learn whether the currently reigning
>theory is scientific, you'd have to patiently wait until it is
>eclipsed by a successor-theory.
Quite. That is why it is not a criterion.
>I endorse the following point of view:
>A discipline (such as astronomy) reaches a stage of being a scientific
>discipline when a community of scientists reaches consensus on how to
>proceed in order to increase the body of knowledge. Having a common
>perspective, a common framework to organize the perceptions into
>overseeable chunks, they are in a position to stimulate each others
>development, which translates to development of the discipline as a
>science.
I, on the other hand, think that consensus has been historically of very
little use in ascertaining whether something constitutes "knowledge". We
do not for example, normally regard alchemy as science, and nor was
there knowledge that the earth was flat, just because there was a
consensus.
>By contrast: if the individuals working in a certain area do not
>gravitate towards a common framework to organize the perceptions, then
>all the development remains individual development, which means that
>effectively there is no community effort, no shared body of
>knowledge.
Your definition omits such things as logical consistency and empirical
evidence. These are the true arbiters of whether something is science,
even if only one person percieves them correctly. Einstein's 1905 papers
were science even before they were published.