From patdolan@21:1/5 to All on Sun Sep 3 13:24:10 2023
Consider a distant observer traveling at .867 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. In this situation the Lorentz transforms inform us that gamma = 2. So in accordance with SR, and after taking
the relativistic doppler into account, from the observer's point of view it takes the earth twice as long--730.5 days--to complete one revolution around the sun.

The observer measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.

We therefore conclude that Kepler's 3rd law of planetary motion is ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all other frames. This is a blatant violation of the
first postulate of special relativity.

Note: Newtonian gravity is not assumed in this paradox. Invariant
spacetime curvature is assumed to be the cause of the earth's orbit
around the sun.

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• From patdolan@21:1/5 to patdolan on Sun Sep 3 13:31:39 2023
On Sunday, September 3, 2023 at 1:24:13 PM UTC-7, patdolan wrote:
Consider a distant observer traveling at .867 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. In this situation the Lorentz transforms inform us that gamma = 2. So in accordance with SR, and after taking
the relativistic doppler into account, from the observer's point of view it takes the earth twice as long--730.5 days--to complete one revolution around the sun.

The observer measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.

We therefore conclude that Kepler's 3rd law of planetary motion is ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all other frames. This is a blatant violation of the
first postulate of special relativity.

Note: Newtonian gravity is not assumed in this paradox. Invariant
spacetime curvature is assumed to be the cause of the earth's orbit
around the sun.
Let us see if we can fix Kepler 3 to make it relativity-proof. It takes a genius to try this--and to know when to cry "uncle"

See the last answer here: https://physics.stackexchange.com/questions/580388/does-keplers-3rd-law-of-planetary-motion-violate-the-first-postulate

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• From Bill@21:1/5 to patdolan on Sun Sep 3 13:54:20 2023
On Sunday, September 3, 2023 at 1:24:13 PM UTC-7, patdolan wrote:
Consider a distant observer traveling at .867 c ( 𝛾=2 ) relative to the solar system...
In his inertial frame of reference the earth's orbital velocity is only half the velocity
necessary to keep the earth in stable orbit...

Not true, the earth follows a helical geodesic trajectory through spacetime, and this helical geodesic is not intrinsically altered by being described in terms of a different system of coordinates. Also, the *extrinsic* curvature of the trajectory is
invariant, which may be surprising to you if you aren't taking the time component of the trajectory into account.

Invariant spacetime curvature...

Be careful... the *extrinsic* curvature of the trajectory is invariant under Lorentz transformation (which is essentially what you are applying to the Schwarzschild coordinates by switching to the background inertial coordinates in which the distant high
speed object is at rest, superimposed on the mildly curved spacetime surrounding the sun), but the components of the *intrinsic* curvature tensor of spacetime are not invariant under coordinate transformations, they change along with the components of
the metric tensor as expressed in terms of the different coordinate systems, in such a way that all the invariant intervals are, well, invariant.

Will the earth spiral into the sun?

No, describing the phenomena in terms of a different system of coordinates doesn't change the intrinsic phenomena, and doesn't alter any of the invariant intervals. For example, if you draw two chalk grids on a putting green, and describe the trajectory
of a putt going into the hole in terms of one coordinate system, it will also go into the hole in terms of the other coordinate system. Yes, the ball has different coordinates at the end, but the cup also has different coordinates, so the ball still goes
into the cup. The idea that changing the coordinate system used to describe the phenomena can somehow change the phenomena is wrong... and, no, this does not imply that local Lorentz invariance has no physical dynamical effects. The dynamical equations
of physics are locally Lorentz invariant, which is the physical content of special relativity.

Ridiculous! See Einstein's First vs. Kepler's Third, ibid.

I assume by "Einstein's First" you mean the principle of relativity, i.e., that the laws by which the states of physical systems undergo change take the same simple homogeneous and isotropic form in terms of any standard system of inertial coordinates,
and I assume that by "Kepler's Third" you mean Kepler's proposition that the squares of the angular orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. There's no conflict between either of these
and the explanation stated above, because the principle of relativity is fully satisfied by local Lorentz invariance (which is a cornerstone of general relativity), and Kepler's proposition is also satisfied in terms of both systems of coordinates (
because the periods are scaled in the same proportion).

What may be confusing you is that you may be thinking of Newton's elaboration and refinement of Kepler's propositions based on his concepts of instantaneous gravity and mass and force, combined with the assumption that inertial coordinate systems are
related by Galilean transformations, and the lack of accounting for the inertia of energy. There is indeed a conflict between those Newtonian concepts applied to relativistic velocities versus the relativistic account. But that simply implies that if
general relativity (which entails local Lorentz invariance) was wrong, then general relativity would be wrong. That's a syntactically correct statement but the premise is counterfactual and hence devoid of significance.

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• From JanPB@21:1/5 to patdolan on Sun Sep 3 14:17:38 2023
On Sunday, September 3, 2023 at 1:24:13 PM UTC-7, patdolan wrote:
Consider a distant observer traveling at .867 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. In this situation the Lorentz transforms inform us that gamma = 2. So in accordance with SR, and after taking
the relativistic doppler into account, from the observer's point of view it takes the earth twice as long--730.5 days--to complete one revolution around the sun.

The observer measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.

We therefore conclude that Kepler's 3rd law of planetary motion is ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all other frames. This is a blatant violation of the
first postulate of special relativity.

Note: Newtonian gravity is not assumed in this paradox. Invariant
spacetime curvature is assumed to be the cause of the earth's orbit
around the sun.

Exercise: find the error in the above reasoning.

--
Jan

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• From patdolan@21:1/5 to Bill on Sun Sep 3 14:26:11 2023
On Sunday, September 3, 2023 at 1:54:22 PM UTC-7, Bill wrote:
On Sunday, September 3, 2023 at 1:24:13 PM UTC-7, patdolan wrote:
Consider a distant observer traveling at .867 c ( 𝛾=2 ) relative to the solar system...
In his inertial frame of reference the earth's orbital velocity is only half the velocity
necessary to keep the earth in stable orbit...

Not true, the earth follows a helical geodesic trajectory through spacetime, and this helical geodesic is not intrinsically altered by being described in terms of a different system of coordinates. Also, the *extrinsic* curvature of the trajectory is
invariant, which may be surprising to you if you aren't taking the time component of the trajectory into account.

Invariant spacetime curvature...

Be careful... the *extrinsic* curvature of the trajectory is invariant under Lorentz transformation (which is essentially what you are applying to the Schwarzschild coordinates by switching to the background inertial coordinates in which the distant
high speed object is at rest, superimposed on the mildly curved spacetime surrounding the sun), but the components of the *intrinsic* curvature tensor of spacetime are not invariant under coordinate transformations, they change along with the components
of the metric tensor as expressed in terms of the different coordinate systems, in such a way that all the invariant intervals are, well, invariant.

Will the earth spiral into the sun?

No, describing the phenomena in terms of a different system of coordinates doesn't change the intrinsic phenomena, and doesn't alter any of the invariant intervals. For example, if you draw two chalk grids on a putting green, and describe the
trajectory of a putt going into the hole in terms of one coordinate system, it will also go into the hole in terms of the other coordinate system. Yes, the ball has different coordinates at the end, but the cup also has different coordinates, so the ball
still goes into the cup. The idea that changing the coordinate system used to describe the phenomena can somehow change the phenomena is wrong... and, no, this does not imply that local Lorentz invariance has no physical dynamical effects. The dynamical
equations of physics are locally Lorentz invariant, which is the physical content of special relativity.

Ridiculous! See Einstein's First vs. Kepler's Third, ibid.

I assume by "Einstein's First" you mean the principle of relativity, i.e., that the laws by which the states of physical systems undergo change take the same simple homogeneous and isotropic form in terms of any standard system of inertial coordinates,
and I assume that by "Kepler's Third" you mean Kepler's proposition that the squares of the angular orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. There's no conflict between either of these
and the explanation stated above, because the principle of relativity is fully satisfied by local Lorentz invariance (which is a cornerstone of general relativity), and Kepler's proposition is also satisfied in terms of both systems of coordinates (
because the periods are scaled in the same proportion).

What may be confusing you is that you may be thinking of Newton's elaboration and refinement of Kepler's propositions based on his concepts of instantaneous gravity and mass and force, combined with the assumption that inertial coordinate systems are
related by Galilean transformations, and the lack of accounting for the inertia of energy. There is indeed a conflict between those Newtonian concepts applied to relativistic velocities versus the relativistic account.

My psychological suspicions are raised by this sentence of your below, Legion. It is perhaps revealing of your though processes in terms of the utilizations of tautology. You apparently believe they are syntactically correct. Syntax and grammar are as
logical a pair of sciences as any other, and abhor tautology. No Legion, it is not even a grammatically correct statement. I importune you to try this sometime: make yourself stop typing in mid-paragraph and check whether your line of argumentation
amounts to "if special relativity is right, then special relativity must be right"

But that simply implies that if general relativity (which entails local Lorentz invariance) was wrong, then general relativity would be wrong. That's a syntactically correct statement but the premise is counterfactual and hence devoid of significance.

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• From patdolan@21:1/5 to JanPB on Sun Sep 3 14:55:04 2023
On Sunday, September 3, 2023 at 2:17:41 PM UTC-7, JanPB wrote:
On Sunday, September 3, 2023 at 1:24:13 PM UTC-7, patdolan wrote:
Consider a distant observer traveling at .867 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. In this situation the Lorentz transforms inform us that gamma = 2. So in accordance with SR, and after
taking the relativistic doppler into account, from the observer's point of view it takes the earth twice as long--730.5 days--to complete one revolution around the sun.

The observer measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.

We therefore conclude that Kepler's 3rd law of planetary motion is ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all other frames. This is a blatant violation of
the first postulate of special relativity.

Note: Newtonian gravity is not assumed in this paradox. Invariant spacetime curvature is assumed to be the cause of the earth's orbit
around the sun.
Exercise: find the error in the above reasoning.

--
Jan
Jan, do you find my supposed error concealed in Lunatic Legion's word-wall? Or do you have something else in mind?

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• From Bill@21:1/5 to JanPB on Sun Sep 3 16:32:48 2023
On Sunday, September 3, 2023 at 2:17:41 PM UTC-7, JanPB wrote:
Consider a distant observer traveling at .867 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. In this situation the Lorentz transforms inform us that gamma = 2. So in accordance with SR, and after
taking the relativistic doppler into account, from the observer's point of view it takes the earth twice as long--730.5 days--to complete one revolution around the sun.

The observer measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.

We therefore conclude that Kepler's 3rd law of planetary motion is ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all other frames. This is a blatant violation of
the first postulate of special relativity.

Note: Newtonian gravity is not assumed in this paradox. Invariant spacetime curvature is assumed to be the cause of the earth's orbit
around the sun.
Exercise: find the error in the above reasoning.

That's sort of a trick question, because there are actually about a dozen errors underlying the above. Just to take one at random...

The OP imagines that it's a violation of relativity for the orbit of an object to take a particularly simple form in terms of the rest frame of the central gravitating body. That, of course, is false. The principle of relativity signifies that the
overall solar system behaves the same way *intrinsically*, regardless of the rest frame of the center of mass of the solar system, but it does not say that the description of a physical system in a certain state of motion is the same in terms of every
system of coordinates. Duh.

The OP doesn't seem to have noticed that, for example, Kepler's FIRST proposition that planets move in closed ellipses in space is obviously only true (approximately) in terms of inertial coordinates in which the sun is more or less at rest. In terms of
other coordinate systems, the planets move in spatial helical or spiraling paths. And this is true in Newtonian physics as well, and yet the OP doesn't imagine that it contradicts Newtonian gravity. His brain just doesn't (can't?) grasp the distinction
between intrinsic and extrinsic attributes. And that's just one of about a dozen disabilities standing between him and even an elementary understanding of physics.

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• From patdolan@21:1/5 to Bill on Sun Sep 3 17:09:06 2023
On Sunday, September 3, 2023 at 4:32:50 PM UTC-7, Bill wrote:
On Sunday, September 3, 2023 at 2:17:41 PM UTC-7, JanPB wrote:
Consider a distant observer traveling at .867 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. In this situation the Lorentz transforms inform us that gamma = 2. So in accordance with SR, and after
taking the relativistic doppler into account, from the observer's point of view it takes the earth twice as long--730.5 days--to complete one revolution around the sun.

The observer measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.

We therefore conclude that Kepler's 3rd law of planetary motion is ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all other frames. This is a blatant violation of
the first postulate of special relativity.

Note: Newtonian gravity is not assumed in this paradox. Invariant spacetime curvature is assumed to be the cause of the earth's orbit around the sun.
Exercise: find the error in the above reasoning.
That's sort of a trick question, because there are actually about a dozen errors underlying the above. Just to take one at random...

The OP imagines that it's a violation of relativity for the orbit of an object to take a particularly simple form in terms of the rest frame of the central gravitating body. That, of course, is false. The principle of relativity signifies that the
overall solar system behaves the same way *intrinsically*, regardless of the rest frame of the center of mass of the solar system, but it does not say that the description of a physical system in a certain state of motion is the same in terms of every
system of coordinates. Duh.

The OP doesn't seem to have noticed that, for example, Kepler's FIRST proposition that planets move in closed ellipses in space is obviously only true (approximately) in terms of inertial coordinates in which the sun is more or less at rest. In terms
of other coordinate systems, the planets move in spatial helical or spiraling paths. And this is true in Newtonian physics as well, and yet the OP doesn't imagine that it contradicts Newtonian gravity. His brain just doesn't (can't?) grasp the
distinction between intrinsic and extrinsic attributes. And that's just one of about a dozen disabilities standing between him and even an elementary understanding of physics.
Legion! This is ridiculous. Kepler's coordinates were earth-centered. In Kepler's and Tyco's star tables the sun rotated about the earth and the planets from which he derived his three laws were rotating about the sun. Compound helical motions. His
planets were helixing all over the place! And at non-relativistic velocities to boot. This criticism of yours is empty.

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• From Bill@21:1/5 to patdolan on Sun Sep 3 18:58:11 2023
On Sunday, September 3, 2023 at 5:09:09 PM UTC-7, patdolan wrote:
The OP imagines that it's a violation of relativity for the orbit of an object to take a particularly simple form in terms of the rest frame of the central gravitating body. That, of course, is false. The principle of relativity signifies that the
overall solar system behaves the same way *intrinsically*, regardless of the rest frame of the center of mass of the solar system, but it does not say that the description of a physical system in a certain state of motion is the same in terms of every
system of coordinates. Duh.

The OP doesn't seem to have noticed that, for example, Kepler's FIRST proposition that planets move in closed ellipses in space is obviously only true (approximately) in terms of inertial coordinates in which the sun is more or less at rest. In terms
of other coordinate systems, the planets move in spatial helical or spiraling paths. And this is true in Newtonian physics as well, and yet the OP doesn't imagine that it contradicts Newtonian gravity. His brain just doesn't (can't?) grasp the
distinction between intrinsic and extrinsic attributes. And that's just one of about a dozen disabilities standing between him and even an elementary understanding of physics.

Kepler's coordinates were earth-centered.

What is wrong with you? How could the trajectory of the earth be an ellipse in terms of earth-centered coordinates? Did you get kicked in the head by a mule or something?

Consider a distant observer traveling at .867 c ( 𝛾=2 ) relative to the solar system...
In his inertial frame of reference the earth's orbital velocity is only half the velocity
necessary to keep the earth in stable orbit...

Not true, the earth follows a helical geodesic trajectory through spacetime, and this helical geodesic is not intrinsically altered by being described in terms of a different system of coordinates. Also, the *extrinsic* curvature of the trajectory is
invariant, which may be surprising to you if you aren't taking the time component of the trajectory into account.

Will the earth spiral into the sun?

No, describing the phenomena in terms of a different system of coordinates doesn't change the intrinsic phenomena, and doesn't alter any of the invariant intervals. For example, if you draw two chalk grids on a putting green, and describe the trajectory
of a putt going into the hole in terms of one coordinate system, it will also go into the hole in terms of the other coordinate system. Yes, the ball has different coordinates at the end, but the cup also has different coordinates, so the ball still goes
into the cup. The idea that changing the coordinate system used to describe the phenomena can somehow change the phenomena is wrong... and, no, this does not imply that local Lorentz invariance has no physical dynamical effects. The dynamical equations
of physics are locally Lorentz invariant, which is the physical content of special relativity.

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• From Paul Alsing@21:1/5 to Bill on Sun Sep 3 21:21:57 2023
On Sunday, September 3, 2023 at 6:58:14 PM UTC-7, Bill wrote:
On Sunday, September 3, 2023 at 5:09:09 PM UTC-7, patdolan wrote:

Kepler's coordinates were earth-centered.

What is wrong with you? How could the trajectory of the earth be an ellipse in terms of earth-centered coordinates? Did you get kicked in the head by a mule or something?

I think you are on to something here, Bill... kicked in the head by a mule would explain a LOT about Dolan...

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• From patdolan@21:1/5 to Paul Alsing on Sun Sep 3 23:08:41 2023
On Sunday, September 3, 2023 at 9:22:00 PM UTC-7, Paul Alsing wrote:
On Sunday, September 3, 2023 at 6:58:14 PM UTC-7, Bill wrote:
On Sunday, September 3, 2023 at 5:09:09 PM UTC-7, patdolan wrote:

Kepler's coordinates were earth-centered.

What is wrong with you? How could the trajectory of the earth be an ellipse in terms of earth-centered coordinates? Did you get kicked in the head by a mule or something?
I think you are on to something here, Bill... kicked in the head by a mule would explain a LOT about Dolan...
Muttonchops, have you come down with Legionnaire's disease too? Kep worked out his laws of planetary motion on the other planets. I merely applied those laws to the earth in my earth-sun scenario. Lunatic Legion has gone mad with the overload of
paradoxes I've sent his way today.

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• From Sylvia Else@21:1/5 to patdolan on Mon Sep 4 17:03:54 2023
On 04-Sept-23 6:24 am, patdolan wrote:

We therefore conclude that Kepler's 3rd law of planetary motion is
ONLY valid in the rest frame of the solar system. This famous law of
physics has a preferred FoR in which it applies and is false in all
other frames. This is a blatant violation of the first postulate of
special relativity.

If your analysis is correct, which I haven't checked, all this means is
that Kepler's law is not covariant, which in turn tells us that it
cannot be a correct law of nature. This is a problem for Kepler's law,
not for special relativity.

Sylvia.

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• From Maciej Wozniak@21:1/5 to Sylvia Else on Mon Sep 4 00:25:36 2023
On Monday, 4 September 2023 at 09:03:58 UTC+2, Sylvia Else wrote:
On 04-Sept-23 6:24 am, patdolan wrote:

We therefore conclude that Kepler's 3rd law of planetary motion is
ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all
other frames. This is a blatant violation of the first postulate of
special relativity.
If your analysis is correct, which I haven't checked, all this means is
that Kepler's law is not covariant, which in turn tells us that it
cannot be a correct law of nature.

nature".

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• From J. J. Lodder@21:1/5 to Sylvia Else on Mon Sep 4 10:25:36 2023
Sylvia Else <sylvia@email.invalid> wrote:

On 04-Sept-23 6:24 am, patdolan wrote:

We therefore conclude that Kepler's 3rd law of planetary motion is
ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all
other frames. This is a blatant violation of the first postulate of
special relativity.

If your analysis is correct, which I haven't checked, all this means is
that Kepler's law is not covariant, which in turn tells us that it
cannot be a correct law of nature. This is a problem for Kepler's law,
not for special relativity.

Sylvia.

(natural) Dimensional analysis tells you that immediately.
If we have T^2 \propto a^3 then the proportionality constant
must have the dimension of a length^-1, (or a time^-1)
since length and time have the same natural dimension.

Yes, it is that simple,

Jan

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• From Richard Hachel@21:1/5 to All on Mon Sep 4 12:24:42 2023
Le 04/09/2023 à 10:25, nospam@de-ster.demon.nl (J. J. Lodder) a écrit :

since length and time have the same natural dimension.

Yes, it is that simple,

Il fallait le dire tout de suite ; en fait, je croyais que c'était très compliqué moi.

Jan

R.H.

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• From Bill@21:1/5 to patdolan on Mon Sep 4 08:38:06 2023
On Sunday, September 3, 2023 at 11:08:43 PM UTC-7, patdolan wrote:
Consider a distant observer traveling at .867 c ( 𝛾=2 ) relative to the solar system...
In his inertial frame of reference the earth's orbital velocity is only half the velocity
necessary to keep the earth in stable orbit...

Not true, the earth follows a helical geodesic trajectory through spacetime, and this helical geodesic is not intrinsically altered by being described in terms of a different system of coordinates. Also, the *extrinsic* curvature of the trajectory is
invariant, which may be surprising to you if you aren't taking the time component of the trajectory into account. Misner, et al, illustrate this with a baseball and a bullet.

Invariant spacetime curvature...

No, the components of the spacetime (intrinsic) curvature tensor are not invariant, they are covariant along with the metric tensor, and of course all invariant intervals are preserved under any coordinate transformation. The *extrinsic* curvature of
the trajectory is invariant under Lorentz transformation (which is essentially what you are applying to the Schwarzschild coordinates around the sun by switching to the background inertial coordinates in which the distant high speed object is at rest,
superimposed on the mildly curved spacetime surrounding the sun), but that's different from the intrinsic curvature of spacetime.

Will the earth spiral into the sun?

No, describing phenomena in terms of a different system of coordinates doesn't change the intrinsic phenomena, and doesn't alter any of the invariant intervals. For example, if you draw two chalk grids on a putting green, and describe the trajectory of a
putt going into the hole in terms of one coordinate system, it will also go into the hole in terms of the other coordinate system. Yes, the ball has different coordinates at the end, but the cup also has different coordinates, so the ball still goes into
the cup.

The idea that changing the coordinate system used to describe the phenomena can somehow change the phenomena is wrong... and, no, this does not imply that local Lorentz invariance has no physical dynamical effects. The dynamical equations of physics are
locally Lorentz invariant, which is the physical content of special relativity.

Ridiculous! See Einstein's First vs. Kepler's Third, ibid.

I assume by "Einstein's First" you mean the principle of relativity, i.e., that the laws by which the states of physical systems undergo change take the same simple homogeneous and isotropic form in terms of any standard system of inertial coordinates,
and I assume that by "Kepler's Third" you mean Kepler's proposition that the squares of the angular orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. There's no conflict between either of these
and the explanation stated above, because the principle of relativity is fully satisfied by local Lorentz invariance (which is a cornerstone of general relativity), and Kepler's proposition is also satisfied in terms of both systems of coordinates (
because the periods are scaled in the same proportion).

You may be getting confused by thinking of Newton's elaboration and refinement of Kepler's propositions based on his concepts of instantaneous gravity, mass, and force, combined with the assumption that inertial coordinate systems are related by Galilean
transformations, and the lack of accounting for the inertia of energy. There is indeed a conflict between those Newtonian concepts applied to relativistic velocities versus the relativistic account, but you assured us that you were not talking about
Newtonian force of gravity, so (unless you were lying) we can dispense with this attempted evasion. Of course, if you wanted to treat gravity as a Newtonian force in the context of special relativity, it is easily explained on that basis as well, as has
been thoroughly explained to you before.

The fact that Galilean invariance differs from Lorentz invariance is not a new realization, so pointing it out is not particularly illuminating or meaningful. Yes, if special relativity were wrong, then special relativity would be wrong, but this
statement is devoid of significance, i.e., it does not constitute a reason for thinking special relativity is wrong, which was your objective.

Kepler's coordinates were earth-centered.

How could the trajectory of the earth be an ellipse (Kepler's first law) in terms of earth-centered coordinates? What is wrong with you? Did you get kicked in the head by a mule or something? Obviously Kepler's three regularities are expressed in terms
of coordinates in which the sun is (more or less) at rest. That was the whole point of the Copernican revolution, i.e., it would have been impossible to discern the simple laws of motion as long as people worked in terms of earth-centered coordinates,
in which the planets follow highly complex and irregular motions. The three regularities that Kepler noticed could only be discerned when the motions of the planets were expressed in terms of Sun-centered coordinates. Of course, the same regularities
apply in terms of those coordinates, regardless of the frame in which the sun is at rest (principle of relativity).

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• From patdolan@21:1/5 to Sylvia Else on Mon Sep 4 09:32:54 2023
On Monday, September 4, 2023 at 12:03:58 AM UTC-7, Sylvia Else wrote:
On 04-Sept-23 6:24 am, patdolan wrote:

We therefore conclude that Kepler's 3rd law of planetary motion is
ONLY valid in the rest frame of the solar system. This famous law of physics has a preferred FoR in which it applies and is false in all
other frames. This is a blatant violation of the first postulate of special relativity.
If your analysis is correct, which I haven't checked, all this means is
that Kepler's law is not covariant, which in turn tells us that it
cannot be a correct law of nature. This is a problem for Kepler's law,
not for special relativity.

Sylvia.
Sylvia, this answer of yours is exactly the same answer, given by the top relativity guru on physics.stackexchange. The guru's answer garnered the most points as the best answer to the Einstein Kepler Paradox. Below I print my response to the guru's

"Does anyone else share my opinion that the first answer is circular reasoning? The first answer tacitly accepts that Kepler's third law of planetary motion falsifies the principle of relativity because it is demonstrated to be valid only in the rest
frame of the solar system. In the face of that demonstration the poster reaches the conclusion that Kepler's Laws can't be laws anymore. Instead, they are relegated to the status of non-relativistic approximations of other, as yet undiscovered
relativistic laws--new laws which presumably will not falsify the principle of relativity. In lieu of these new laws of planetary motion are we not justified in remaining convinced that Kepler's third law has indeed falsified the principle of relativity?"

My entirely reasonable answer above was considered just to hot for physics.stackexchange; and so, was summarily taken down from that platform.

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• From patdolan@21:1/5 to Bill on Mon Sep 4 09:36:41 2023
On Monday, September 4, 2023 at 8:38:09 AM UTC-7, Bill wrote:
On Sunday, September 3, 2023 at 11:08:43 PM UTC-7, patdolan wrote:
Consider a distant observer traveling at .867 c ( 𝛾=2 ) relative to the solar system...
In his inertial frame of reference the earth's orbital velocity is only half the velocity
necessary to keep the earth in stable orbit...
Not true, the earth follows a helical geodesic trajectory through spacetime, and this helical geodesic is not intrinsically altered by being described in terms of a different system of coordinates. Also, the *extrinsic* curvature of the trajectory is
invariant, which may be surprising to you if you aren't taking the time component of the trajectory into account. Misner, et al, illustrate this with a baseball and a bullet.

Invariant spacetime curvature...

No, the components of the spacetime (intrinsic) curvature tensor are not invariant, they are covariant along with the metric tensor, and of course all invariant intervals are preserved under any coordinate transformation. The *extrinsic* curvature of
the trajectory is invariant under Lorentz transformation (which is essentially what you are applying to the Schwarzschild coordinates around the sun by switching to the background inertial coordinates in which the distant high speed object is at rest,
superimposed on the mildly curved spacetime surrounding the sun), but that's different from the intrinsic curvature of spacetime.
Will the earth spiral into the sun?
No, describing phenomena in terms of a different system of coordinates doesn't change the intrinsic phenomena, and doesn't alter any of the invariant intervals. For example, if you draw two chalk grids on a putting green, and describe the trajectory of
a putt going into the hole in terms of one coordinate system, it will also go into the hole in terms of the other coordinate system. Yes, the ball has different coordinates at the end, but the cup also has different coordinates, so the ball still goes
into the cup.
The idea that changing the coordinate system used to describe the phenomena can somehow change the phenomena is wrong... and, no, this does not imply that local Lorentz invariance has no physical dynamical effects. The dynamical equations of physics
are locally Lorentz invariant, which is the physical content of special relativity.
Ridiculous! See Einstein's First vs. Kepler's Third, ibid.

I assume by "Einstein's First" you mean the principle of relativity, i.e., that the laws by which the states of physical systems undergo change take the same simple homogeneous and isotropic form in terms of any standard system of inertial coordinates,
and I assume that by "Kepler's Third" you mean Kepler's proposition that the squares of the angular orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. There's no conflict between either of these
and the explanation stated above, because the principle of relativity is fully satisfied by local Lorentz invariance (which is a cornerstone of general relativity), and Kepler's proposition is also satisfied in terms of both systems of coordinates (
because the periods are scaled in the same proportion).
You may be getting confused by thinking of Newton's elaboration and refinement of Kepler's propositions based on his concepts of instantaneous gravity, mass, and force, combined with the assumption that inertial coordinate systems are related by
Galilean transformations, and the lack of accounting for the inertia of energy. There is indeed a conflict between those Newtonian concepts applied to relativistic velocities versus the relativistic account, but you assured us that you were not talking
about Newtonian force of gravity, so (unless you were lying) we can dispense with this attempted evasion. Of course, if you wanted to treat gravity as a Newtonian force in the context of special relativity, it is easily explained on that basis as well,
as has been thoroughly explained to you before.

The fact that Galilean invariance differs from Lorentz invariance is not a new realization, so pointing it out is not particularly illuminating or meaningful. Yes, if special relativity were wrong, then special relativity would be wrong, but this
statement is devoid of significance, i.e., it does not constitute a reason for thinking special relativity is wrong, which was your objective.
Kepler's coordinates were earth-centered.
How could the trajectory of the earth be an ellipse (Kepler's first law) in terms of earth-centered coordinates? What is wrong with you? Did you get kicked in the head by a mule or something? Obviously Kepler's three regularities are expressed in terms
of coordinates in which the sun is (more or less) at rest. That was the whole point of the Copernican revolution, i.e., it would have been impossible to discern the simple laws of motion as long as people worked in terms of earth-centered coordinates, in
which the planets follow highly complex and irregular motions. The three regularities that Kepler noticed could only be discerned when the motions of the planets were expressed in terms of Sun-centered coordinates. Of course, the same regularities apply
in terms of those coordinates, regardless of the frame in which the sun is at rest (principle of relativity).
me;LL (more equations; less Legion)

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• From Richard Hachel@21:1/5 to All on Mon Sep 4 17:00:56 2023
Le 03/09/2023 à 22:24, patdolan a écrit :
Consider a distant observer traveling at .867 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. In this situation the Lorentz transforms inform us that gamma = 2. So in accordance with
SR, and after taking the relativistic doppler into account, from the observer's
point of view it takes the earth twice as long--730.5 days--to complete one revolution around the sun.

The observer measures the major and minor axes of the earth's orbit around the
sun to be identical to its major and minor axes in the solar system's rest frame,
where the orbital period is only 365.25 days.

We therefore conclude that Kepler's 3rd law of planetary motion is ONLY valid in
the rest frame of the solar system. This famous law of physics has a preferred FoR
in which it applies and is false in all other frames. This is a blatant violation
of the first postulate of special relativity.

Note: Newtonian gravity is not assumed in this paradox. Invariant
spacetime curvature is assumed to be the cause of the earth's orbit
around the sun.

There are indeed problems if we calculate like that, and it is clear that
time passes half as fast as for the terrestrial observer.
But let's not forget one thing, depending on the position, there are
distance contractions.
And if we take a longitudinal position (the observer sees the solar system approaching like a disk seen from the front), there is certainly no
lateral contraction, but the path of revolution is no longer a
circumference, but a spiral very stretched towards the front, and
moderately stretched when seen from the rear.
It is therefore difficult to calculate the areas covered.
But that doesn't matter.
The areas are given by Kepler for a fixed inertial observer.
Afterwards, we can always do what we want...

R.H.

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• From Dono.@21:1/5 to patdolan on Mon Sep 4 10:03:37 2023
On Monday, September 4, 2023 at 9:32:56 AM UTC-7, patdolan wrote:

My entirely reasonable answer above was considered just to hot for physics.stackexchange; and so, was summarily taken down from that platform.

"to hot"? More like "too idiotic".

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• From Bill@21:1/5 to Sylvia on Mon Sep 4 10:16:31 2023
Sylvia wrote:
If your analysis is correct, which I haven't checked, all this means is that Kepler's law is not covariant...

He hasn't done any analysis, but it's been explained to him that Kepler's actual third law ("the squares of the angular orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits") is actually just as
valid on both systems of coordinates for the situation described, so the whole premise of his question is false.

It's also been explained to him that he is confusing Kepler's law with the Newtonian propositions such as m = r^3 w^2 that show how Kelper's laws are (approximately) true as a result of the Newtonian laws of mechanics combined with instantaneous
gravitational force, which are Galilean invariant, not Lorentz invariant. Now, the issue isn't really covariance, per se, because nearly any physical laws (including Newton's) can be expressed in covariant form, it is simply that Newton's laws (which
the OP is conflating with Kelper's third regularity) are Galilean invariant whereas general relativity is locally (and on large asymptotic scales) Lorentz invariant.

So, the OP's grand announcement is that, at relativistic speeds, Lorentz invariance is significantly inconsistent with Galilean invariance. At this point I would insert a "Duh", except that I don't think Duh adequately expresses the towering idiocy of
the belief that this is somehow a revelation, let alone that it represents an argument against general relativity.

On Monday, September 4, 2023 at 9:32:56 AM UTC-7, patdolan wrote:
[Sylvia's] answer tacitly accepts that Kepler's third law of planetary motion
falsifies the principle of relativity...

No, the point is that (1) Kepler's law actually IS (approximately) satisfied in both systems, and (2) you are conflating Kepler's law with Newton's laws, which are NOT even approximately valid when relativistic speeds are involved, and (3) this does not
falsify the principle of relativity (which is satisfied by both Newton's laws of general relativity), it falsifies Galilean invariance.

In the face of this, [Sylvia] reaches the conclusion that [Newton's laws] can't be laws anymore. Instead, they are relegated to the status of non-relativistic
approximations....

Yes! That is exactly right. Bravo.

of other, as yet undiscovered relativistic laws...

Nope, you went off the rails again. The laws that accurately cover the scenario you described are called general relativity, which of course is locally Lorentz invariant (entailing all the aspects of special relativity that you were hoping to discredit),
and these are not undiscovered at all. They are quite well known for over a century.

which presumably will not falsify the principle of relativity.

Well, general relativity implies that Lorentz invariance is valid locally, and also in the large scale asymptotically falt background, but yes, for purposes of thwarting your hopes and dreams, general relativity is pefectly consistent with everything you

LOL.

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• From patdolan@21:1/5 to Bill on Mon Sep 4 10:35:59 2023
On Monday, September 4, 2023 at 10:16:34 AM UTC-7, Bill wrote:
Sylvia wrote:
If your analysis is correct, which I haven't checked, all this means is that Kepler's law is not covariant...

He hasn't done any analysis, but it's been explained to him that Kepler's actual third law ("the squares of the angular orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits") is actually just as
valid on both systems of coordinates for the situation described, so the whole premise of his question is false.

It's also been explained to him that he is confusing Kepler's law with the Newtonian propositions such as m = r^3 w^2 that show how Kelper's laws are (approximately) true as a result of the Newtonian laws of mechanics combined with instantaneous
gravitational force, which are Galilean invariant, not Lorentz invariant. Now, the issue isn't really covariance, per se, because nearly any physical laws (including Newton's) can be expressed in covariant form, it is simply that Newton's laws (which the
OP is conflating with Kelper's third regularity) are Galilean invariant whereas general relativity is locally (and on large asymptotic scales) Lorentz invariant.

So, the OP's grand announcement is that, at relativistic speeds, Lorentz invariance is significantly inconsistent with Galilean invariance. At this point I would insert a "Duh", except that I don't think Duh adequately expresses the towering idiocy of
the belief that this is somehow a revelation, let alone that it represents an argument against general relativity.
On Monday, September 4, 2023 at 9:32:56 AM UTC-7, patdolan wrote:
[Sylvia's] answer tacitly accepts that Kepler's third law of planetary motion
falsifies the principle of relativity...

No, the point is that (1) Kepler's law actually IS (approximately) satisfied in both systems, and (2) you are conflating Kepler's law with Newton's laws, which are NOT even approximately valid when relativistic speeds are involved, and (3) this does
not falsify the principle of relativity (which is satisfied by both Newton's laws of general relativity), it falsifies Galilean invariance.

In the face of this, [Sylvia] reaches the conclusion that [Newton's laws] can't be laws anymore. Instead, they are relegated to the status of non-relativistic
approximations....

Yes! That is exactly right. Bravo.

of other, as yet undiscovered relativistic laws...

Nope, you went off the rails again. The laws that accurately cover the scenario you described are called general relativity, which of course is locally Lorentz invariant (entailing all the aspects of special relativity that you were hoping to discredit)
, and these are not undiscovered at all. They are quite well known for over a century.
which presumably will not falsify the principle of relativity.
Well, general relativity implies that Lorentz invariance is valid locally, and also in the large scale asymptotically falt background, but yes, for purposes of thwarting your hopes and dreams, general relativity is pefectly consistent with everything

LOL.
Really Legion, you MUST learn how to communicate with less verbiage. Shakespeare said that brevity is the soul of wit. It is also the soul of science. Equations too, should be short and employed very sparingly. This is because mathematics is like any
other language--it is capable of expressing a falsity just as convincingly as a truth. Don't believe me? Then have a slow read-through of Albert Einstein's theory of special relativity.

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• From Bill@21:1/5 to patdolan on Mon Sep 4 11:05:27 2023
On Monday, September 4, 2023 at 10:36:02 AM UTC-7, patdolan wrote:
[Disgracefully disregards the patient and thorough yet optimally succinct answers to his questions, and replies with:]
me;... (more equations...
Equations should be employed very sparingly....

We've established that any text less than 250 words you disregard as being not sufficiently explanatory for a novice such as yourself (using advanced words and concepts without including the elementary background), and any text greater than 250 words you
disregard as being too lengthy for a mentally-impaired individual such as yourself to read. With those two limitations, it's difficult to see how you can ever make any progress.

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• From patdolan@21:1/5 to Bill on Mon Sep 4 12:33:26 2023
On Monday, September 4, 2023 at 11:05:31 AM UTC-7, Bill wrote:
On Monday, September 4, 2023 at 10:36:02 AM UTC-7, patdolan wrote:
[Disgracefully disregards the patient and thorough yet optimally succinct answers to his questions, and replies with:]
me;... (more equations...
Equations should be employed very sparingly....