## Wednesday, November 24, 2010

### 1.7 Staircase Wit

1.7 Staircase Wit

In 1908 Minkowski delivered a famous lecture in which he argued that the relativistic phenomena described by Lorentz and clarified by Einstein might have been inferred from first principles long before, if only more careful thought had been given to the foundations of classical geometry and mechanics.

- This is 3 years after the publication of Einstein’s SR paper. The Sagnac X was performed in 1913, 5 years later than this lecture. If only the Sagnac result had preceeded the Minkowski abstraction of a non-existent space-time, perhaps the wrong turn of physics into mathematical speculation would have been avoided, saving a century of misguided effort.

Minkowski pointed out that special relativity arises naturally from the reconciliation of two physical symmetries that we individually take for granted. One is spatial isotropy.. The other is Galilean relativity... However, these transformations obviously do not leave the quantity x2 + y2 + z2 invariant.

- These 2 symmetries - that are taken for granted - are both discounted by the Sagnac-type experiments. Space can move (aether flow) , introducing a preferred direction and spatial anisotropy. The discovery of the absolute lab frame converts Galilean relativity to Galilean absolutism.

Finally, a reference frame co-moving with the aether does leave distance invariant.

..the lack of an invariant measure for the Galilean transformations prevents us from even assigning a definite meaning to 'orthogonality' between the time and space coordinates.

- Why should time be orthogonal to space? Time is a parameter used to quantify motion. Why the arbitrary conditions of orthogonality for the time parameter and spatial dimensions in the absence of experimental proof?

Since the velocity transformations leave the laws of physics unchanged, Minkowski reasoned...

- Not when aether - which is ubiquitous - is included.

..this [Lorentz transformation] appears to be the most natural (and almost the only) way of reconciling the observed symmetries of physical phenomena.

- What observed symmetries of physical phenomena?

As Minkowski said,

‘Such a premonition would have been an extraordinary triumph for pure mathematics. ..’

- Well said, but not as intended. There’s no connection with real tests of light speed and motion, making the Minkowski mathematical‘premonition’ devoid of physical importance.

The invariance of this quantity [s^2] under re-orientations is called spatial isotropy. It’s worth emphasizing that the invariance of s2 under these operations applies only if the x, y, and z coordinates are mutually orthogonal.

- No mention of the fact that the measurement process of this spatial abstraction is affected by the presence of aether.

The spatial isotropy of physical entities implies a non-trivial unification of orthogonal measures.

- Then the spatial anisotropy of physical entities as observed, involving aether, must not imply unification of orthogonal measures.

If an object is in motion (relative to the system of coordinates), then the coordinates of its endpoints are variable functions of time, so instead of the constant x1 we have a function x1(t), and likewise for the other coordinates.

- Interesting - so here we see that t is a parameter measuring the changing location of x, not an independent fourth dimension in an unknown direction.

..experience teaches us that equation (1) does apply to objects in motion.

- Not if space itself - aether - is moving.

..the combined symmetry covering states of uniform motion is valid only if the time component t is mutually orthogonal to each of the space coordinates.

- Time is incommensurate with space, like the attempted comparison of apples and oranges. If we use latitude and longitude and altitude for x, y, and z on the Earth’s surface, where do we put the time axis?

..we can only establish the physical orthogonality of coordinate axes based on physical phenomena.

- Then why is the physical phenomenon of aether ignored?

Evidently to establish orthogonality between space and time axes we need a physically meaningful measure of space-time distance, rather than merely spatial distance.

- This forces time to be a dimension, despite its parametric role in describing motion.

Using the logic above we could just as well look for converting space into the time ‘dimension’ by dividing x,y,z by c. Distances would then be measured in intervals of time that light travels!

Admittedly we could postulate a universal preferred reference frame for the purpose of assessing the complete separations between events, but such a postulate is entirely foreign to the logical structure of Galilean space and time, and has no operational significance.

- We do so postulate: the lab frame . Such an absolute postulate is entirely foreign to the logical structure of any flavor of relativity, and has no operational significance... Except being the preferred frame of reference for applying the laws of physics, according to Sagnac testing and the ALFA model.

The most natural supposition is that the squared spacelike intervals and the squared timelike intervals have opposite signs, so that they are mutually 'imaginary' (in the numerical sense).

- And in the cognitive sense.

Hence our proposed invariant quantity for a suitable class of repeatable physical processes extending uniformly from event 1 to event 2 is

s^2 = (x2-x1)^2 +(y2 - y1)^2 + (z2-z1)^2 - c^2(t2-t1)^2

- As noted, we could also choose to use an interval = s^2/c^2 with spatial units being x/c and t as is.

This quantity is invariant under any combination of spatial rotations and changes in the state of uniform motion, as well as simple translations of the origin in space and/or time.

- So is s^2/c^2

Minkowski remarked that,

" Thus the essence of this postulate may be clothed mathematically in a very pregnant manner in the mystic formula 300000 km = (-1)^.5 secs "

- More mythic than mystic, more puzzling than pregnant.

The significance of this 'mystic formula' continues to be debated,

- It’s clear enough - a real number equals an imaginary number. A contradiction, no matter how it’s sliced and diced.

..we cannot assume, a priori, that permittivity and permeability are invariant with respect to changes in reference frame.

- That seems to be another empirical consequence of the Sagnac and Fitzeau aether drag experiments.

Actually permeability is an assigned value, but permittivity must be measured, and the usual means of empirically determining permittivity involve observations of the force between charged plates.

Maxwell clearly believed these measurements must be made with the apparatus "at rest" with respect to the ether in order to yield the true and isotropic value of permittivity.

- This would be the co-moving aether frame, where aether is measured first from the lab frame. Too bad Maxwell wasn’t contemporaneous with Sagnac.

According to Maxwell's conception, if measurements of permittivity are performed with an apparatus traveling at some significant fraction of the speed of light, the results would not only differ from the result at rest, they would also vary depending on the orientation of the plates relative to the direction of the absolute velocity of the apparatus.

- If velocity is measured in the lab frame... exactly!

Of course, the efforts of Maxwell and others to devise empirical methods for measuring the absolute rest frame (either by measuring anisotropies in the speed of light or by detecting variations in the electromagnetic properties of the vacuum) were doomed to failure..

- Well, the doom of failure ended with the Sagnac results, didn’t it.

..even though it's true that the equations of electromagnetism are not invariant under Galilean transformations, it is also true that those equations are invariant with respect to every system of inertial coordinates.

- The Maxwell/Heaviside/Ampere EM laws and Newton’s laws are not invariant in any frame other than the ALFA model (absolute lab frame + flexible aether).

Maxwell's equations are suggestive of the invariance of c only because of the added circumstance that we are unable to physically identify any particular frame of reference for the application of those equations.

- We were unable to, until 1913 - the Sagnac X. The above statement was false since then.

...the empirical invariance of light speed with respect to every inertial system of coordinates

- Why ignore the empirical evidence that light speed varies with aether speed in the lab?

..the Minkowski structure of spacetime ... strongly supports Einstein's decision to base his kinematics on the light speed principle itself. (As in the case of Euclid's decision to specify a "fifth postulate" for his theory of geometry, we can only marvel in retrospect at the underlying insight and maturity that this decision reveals.)

- We marvel that Einstein refused to acknowledge or comment on the Sagnac results, which disproved SRT only 8 years after the SRT postulates were proposed. The policy of ignoring contradicting evidence is inherited by his modern mainstream fellow travelers.

One problem with this line of reasoning is that it's based on a principle (causality) that is not unambiguously self-evident.

- Effects without causes? Causality violated? Causality may not be unambiguously self-evident.. but it’s never been refuted/disproven. in reality - by testing of nature. In the speculative world of pure mathematics - disconnected from physicality - anything goes.

..causality and the directionality of time are far from being straightforward principles.

- Only the future can disprove these 2 assertions - the past hasn’t.

Every real number is finite, but it does not follow that there must be some finite upper bound on the real numbers.

- Nor does it follow that number - the abstraction of material quantity - has any relationship to the limit of real physical objects, space or time.

.we can't really say that Minkowskian spacetime is prima facie any more consistent with causality than is Galilean spacetime.

- Minkowskian spacetime will be covered in its own section.

If the spatial ordering of events is to have any absolute significance, in spite of the fact that distance can be transformed away by motion, it seems that there must be some definite limit on speeds.

- Why?

..the continuity and identity of objects from one instant to the next (ignoring the lessons of quantum mechanics) is most intelligible in the context of a unified spacetime manifold with a definite non-singular connection, which implies a finite upper bound on speeds.

- What is the argument for aether speed? How is a finite upper bound on aether speed ‘most intelligible’?

This is in the spirit of Minkowski's 1908 lecture in which he urged the greater "mathematical intelligibility" of the Lorentzian group as opposed to the Galilean group of transformations.

- So mathematical intelligibility has priority over the scientific method of testing against nature?

We have the fundamental principle that for any material object in any state of motion there exists a system of space and time coordinates with respect to which the object is instantaneously at rest and Newton's laws of inertial motion hold good (at least quasi-statically).

- Only if aether is ignored.

Only the Lorentzian transformation, given by setting k = 1, has completely satisfactory properties from an abstract point of view, which is presumably why Minkowski referred to it as "more intelligible".

- Abstract points of view are fine, if there’s a clear connection with reality.

.we can be persuaded to adopt such a postulate only by empirical facts.

- What desperation – accepting only postulates that are tested! Shocking.

In 1908 Minkowski delivered a famous lecture in which he argued that the relativistic phenomena described by Lorentz and clarified by Einstein might have been inferred from first principles long before, if only more careful thought had been given to the foundations of classical geometry and mechanics.

- This is 3 years after the publication of Einstein’s SR paper. The Sagnac X was performed in 1913, 5 years later than this lecture. If only the Sagnac result had preceeded the Minkowski abstraction of a non-existent space-time, perhaps the wrong turn of physics into mathematical speculation would have been avoided, saving a century of misguided effort.

Minkowski pointed out that special relativity arises naturally from the reconciliation of two physical symmetries that we individually take for granted. One is spatial isotropy.. The other is Galilean relativity... However, these transformations obviously do not leave the quantity x2 + y2 + z2 invariant.

- These 2 symmetries - that are taken for granted - are both discounted by the Sagnac-type experiments. Space can move (aether flow) , introducing a preferred direction and spatial anisotropy. The discovery of the absolute lab frame converts Galilean relativity to Galilean absolutism.

Finally, a reference frame co-moving with the aether does leave distance invariant.

..the lack of an invariant measure for the Galilean transformations prevents us from even assigning a definite meaning to 'orthogonality' between the time and space coordinates.

- Why should time be orthogonal to space? Time is a parameter used to quantify motion. Why the arbitrary conditions of orthogonality for the time parameter and spatial dimensions in the absence of experimental proof?

Since the velocity transformations leave the laws of physics unchanged, Minkowski reasoned...

- Not when aether - which is ubiquitous - is included.

..this [Lorentz transformation] appears to be the most natural (and almost the only) way of reconciling the observed symmetries of physical phenomena.

- What observed symmetries of physical phenomena?

As Minkowski said,

‘Such a premonition would have been an extraordinary triumph for pure mathematics. ..’

- Well said, but not as intended. There’s no connection with real tests of light speed and motion, making the Minkowski mathematical‘premonition’ devoid of physical importance.

The invariance of this quantity [s^2] under re-orientations is called spatial isotropy. It’s worth emphasizing that the invariance of s2 under these operations applies only if the x, y, and z coordinates are mutually orthogonal.

- No mention of the fact that the measurement process of this spatial abstraction is affected by the presence of aether.

The spatial isotropy of physical entities implies a non-trivial unification of orthogonal measures.

- Then the spatial anisotropy of physical entities as observed, involving aether, must not imply unification of orthogonal measures.

If an object is in motion (relative to the system of coordinates), then the coordinates of its endpoints are variable functions of time, so instead of the constant x1 we have a function x1(t), and likewise for the other coordinates.

- Interesting - so here we see that t is a parameter measuring the changing location of x, not an independent fourth dimension in an unknown direction.

..experience teaches us that equation (1) does apply to objects in motion.

- Not if space itself - aether - is moving.

..the combined symmetry covering states of uniform motion is valid only if the time component t is mutually orthogonal to each of the space coordinates.

- Time is incommensurate with space, like the attempted comparison of apples and oranges. If we use latitude and longitude and altitude for x, y, and z on the Earth’s surface, where do we put the time axis?

..we can only establish the physical orthogonality of coordinate axes based on physical phenomena.

- Then why is the physical phenomenon of aether ignored?

Evidently to establish orthogonality between space and time axes we need a physically meaningful measure of space-time distance, rather than merely spatial distance.

- This forces time to be a dimension, despite its parametric role in describing motion.

Using the logic above we could just as well look for converting space into the time ‘dimension’ by dividing x,y,z by c. Distances would then be measured in intervals of time that light travels!

Admittedly we could postulate a universal preferred reference frame for the purpose of assessing the complete separations between events, but such a postulate is entirely foreign to the logical structure of Galilean space and time, and has no operational significance.

- We do so postulate: the lab frame . Such an absolute postulate is entirely foreign to the logical structure of any flavor of relativity, and has no operational significance... Except being the preferred frame of reference for applying the laws of physics, according to Sagnac testing and the ALFA model.

The most natural supposition is that the squared spacelike intervals and the squared timelike intervals have opposite signs, so that they are mutually 'imaginary' (in the numerical sense).

- And in the cognitive sense.

Hence our proposed invariant quantity for a suitable class of repeatable physical processes extending uniformly from event 1 to event 2 is

s^2 = (x2-x1)^2 +(y2 - y1)^2 + (z2-z1)^2 - c^2(t2-t1)^2

- As noted, we could also choose to use an interval = s^2/c^2 with spatial units being x/c and t as is.

This quantity is invariant under any combination of spatial rotations and changes in the state of uniform motion, as well as simple translations of the origin in space and/or time.

- So is s^2/c^2

Minkowski remarked that,

" Thus the essence of this postulate may be clothed mathematically in a very pregnant manner in the mystic formula 300000 km = (-1)^.5 secs "

- More mythic than mystic, more puzzling than pregnant.

The significance of this 'mystic formula' continues to be debated,

- It’s clear enough - a real number equals an imaginary number. A contradiction, no matter how it’s sliced and diced.

..we cannot assume, a priori, that permittivity and permeability are invariant with respect to changes in reference frame.

- That seems to be another empirical consequence of the Sagnac and Fitzeau aether drag experiments.

Actually permeability is an assigned value, but permittivity must be measured, and the usual means of empirically determining permittivity involve observations of the force between charged plates.

Maxwell clearly believed these measurements must be made with the apparatus "at rest" with respect to the ether in order to yield the true and isotropic value of permittivity.

- This would be the co-moving aether frame, where aether is measured first from the lab frame. Too bad Maxwell wasn’t contemporaneous with Sagnac.

According to Maxwell's conception, if measurements of permittivity are performed with an apparatus traveling at some significant fraction of the speed of light, the results would not only differ from the result at rest, they would also vary depending on the orientation of the plates relative to the direction of the absolute velocity of the apparatus.

- If velocity is measured in the lab frame... exactly!

Of course, the efforts of Maxwell and others to devise empirical methods for measuring the absolute rest frame (either by measuring anisotropies in the speed of light or by detecting variations in the electromagnetic properties of the vacuum) were doomed to failure..

- Well, the doom of failure ended with the Sagnac results, didn’t it.

..even though it's true that the equations of electromagnetism are not invariant under Galilean transformations, it is also true that those equations are invariant with respect to every system of inertial coordinates.

- The Maxwell/Heaviside/Ampere EM laws and Newton’s laws are not invariant in any frame other than the ALFA model (absolute lab frame + flexible aether).

Maxwell's equations are suggestive of the invariance of c only because of the added circumstance that we are unable to physically identify any particular frame of reference for the application of those equations.

- We were unable to, until 1913 - the Sagnac X. The above statement was false since then.

...the empirical invariance of light speed with respect to every inertial system of coordinates

- Why ignore the empirical evidence that light speed varies with aether speed in the lab?

..the Minkowski structure of spacetime ... strongly supports Einstein's decision to base his kinematics on the light speed principle itself. (As in the case of Euclid's decision to specify a "fifth postulate" for his theory of geometry, we can only marvel in retrospect at the underlying insight and maturity that this decision reveals.)

- We marvel that Einstein refused to acknowledge or comment on the Sagnac results, which disproved SRT only 8 years after the SRT postulates were proposed. The policy of ignoring contradicting evidence is inherited by his modern mainstream fellow travelers.

One problem with this line of reasoning is that it's based on a principle (causality) that is not unambiguously self-evident.

- Effects without causes? Causality violated? Causality may not be unambiguously self-evident.. but it’s never been refuted/disproven. in reality - by testing of nature. In the speculative world of pure mathematics - disconnected from physicality - anything goes.

..causality and the directionality of time are far from being straightforward principles.

- Only the future can disprove these 2 assertions - the past hasn’t.

Every real number is finite, but it does not follow that there must be some finite upper bound on the real numbers.

- Nor does it follow that number - the abstraction of material quantity - has any relationship to the limit of real physical objects, space or time.

.we can't really say that Minkowskian spacetime is prima facie any more consistent with causality than is Galilean spacetime.

- Minkowskian spacetime will be covered in its own section.

If the spatial ordering of events is to have any absolute significance, in spite of the fact that distance can be transformed away by motion, it seems that there must be some definite limit on speeds.

- Why?

..the continuity and identity of objects from one instant to the next (ignoring the lessons of quantum mechanics) is most intelligible in the context of a unified spacetime manifold with a definite non-singular connection, which implies a finite upper bound on speeds.

- What is the argument for aether speed? How is a finite upper bound on aether speed ‘most intelligible’?

This is in the spirit of Minkowski's 1908 lecture in which he urged the greater "mathematical intelligibility" of the Lorentzian group as opposed to the Galilean group of transformations.

- So mathematical intelligibility has priority over the scientific method of testing against nature?

We have the fundamental principle that for any material object in any state of motion there exists a system of space and time coordinates with respect to which the object is instantaneously at rest and Newton's laws of inertial motion hold good (at least quasi-statically).

- Only if aether is ignored.

Only the Lorentzian transformation, given by setting k = 1, has completely satisfactory properties from an abstract point of view, which is presumably why Minkowski referred to it as "more intelligible".

- Abstract points of view are fine, if there’s a clear connection with reality.

.we can be persuaded to adopt such a postulate only by empirical facts.

- What desperation – accepting only postulates that are tested! Shocking.

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