Quadratic damping of the Interval.
Amplitude Symmetry of the GM Interval canceling a "damping" term.
After-note: an inverse exponential Interval for higher dimensional unification of forces.
A few non-technical references.
Starting with the Interval, I, the commonly observed space-time measurement,
for two observers, O and O', of a given event, from their differently measured
local view-points, at velocities significantly approaching light speed, and
where velocities, u and u', can be in one dimension or a resultant vector of up
to three dimensions.
Equation 1)
I² = t²(c² - u²) = t'²(c² - u'²).
Where t and t' are the two observers different local times, and c is the commonly observed speed of light. For convenience, consider only one observer, eqn. 2)
I² = t²(c² - u²) = {(tc)² - x²}.
To put the equation in a factorial form, add the terms, 2x² - 2xct, to both sides. Eqn. 3)
I² + 2x² - 2xct = (tc)² - x²) + 2x² - 2xct.
Factorising for eqn. 4)
I² - 2x(ct - x) = {(tc) - x}²
Therefore, eqn. 5)
I²/{(tc) - x}² - 2x(ct - x)/{(tc) - x}² - 1 = 0.
And:
I²/{(tc) - x}² - 2x/{(tc) - x} - 1 = 0.
This equation can be solved for 1/(tc - x) by the well-known quadratic equation formula.
Given that:
ay² + by + c = 0,
y = {-b ± (b² - 4ac)^1/2}/2a.
Substituting in the quadratic formula for eqn. 6)
1/(tc - x) = {+2x ±(4x² + 4I²)^1/2}/2I²
= {x ±(x² + I²)^1/2}/I²
There are two alternative solutions, which form a range, from which an average value solution can be found, in the form of the geometric mean [GM] of the range limit solutions of 1/(tc-x).
Eqn. 7)
[GM]1/(tc-x) =
<[{x +(x² + I²)^1/2}/I²][{x -(x² + I²)^1/2}/I²]>1/2
= {x² - (x² + I²)}^1/2/I²
= {(-I²)^1/2}/I² = ±iI/I² = ±i/I.
Therefore, the geometric mean of the factor, 1/(ct-x) is itself a range or dimension or positive and negative scale of the Interval on an imaginary axis, which means at ninety degrees, to the (real axis) Interval.
Thus it is possible to take a geometric mean of this new Interval range. That means taking a geometric mean of the geometric mean value of factor, 1/(ct-x), which is to say a second order geometric mean (signified as [GM]²).
Eqn. 8)
[GM]²1/(ct-x) = {(-i/I)(i/I)}^1/2
= (+1/I²)^1/2 = ±1/I.
Thus, the second order GM of the factor, 1/(ct-x), represents a real axis of the (inverse) Interval, whereas the first order GM represents the imaginary axis of the (inverse) Interval.
The Interval equation, when transformed into an equation between the second order geometric means of observers O and O' looks like this:
[GM]²1/(ct-x) = [GM]²1/(ct'-x') = ±1/I.
This draws on eqn. 8, only stated for one observer, so as to include another observer, both of whose locally distinct observations share the Interval value, ±1/I, which is an oscillation of the inverse Interval from positive "crest" to negative "trof" making a "wave."
Suppose this quadratic equation solution exercise with the factorised Interval is repeated for the conventional Interval.
Re-write eqn, 2:
I² = (tc)² - x²), as:
I²/{(tc)² - x²} - 1 = 0.
This equation has a second order term, 1/{(tc)² - x²}, and a constant, 1, but no first order term, 1/(ct-x). and therefore zero coefficient, b, in the quadratic equation formula. So, eqn. 9:
1/(ct-x) = {0 ±(0 + 4I²)^1/2}/2I²
= ±2I/2I² = ±1/I.
Eqn. 10):
[GM] 1/(ct-x) = {(1/I)(-1/I)}^1/2
= (1/I²)^1/2 = ±i/I.
Thus, eqn. 10 agrees with eqn. 7.
The second order Geometric Mean, of factor 1/(ct-x), is eqn 11:
[GM]²1/(ct-x) = {(+i/I)(-i/I)}^1/2 = ±1/I.
Thus, eqn. 11 agrees with eqn. 9.
Of course, they should agree, because the equations 2 and 4, they derive from, are equivalent.
But they are not equal. Equation 4 is no longer an equation of the (conventional) Interval. It is the Interval minus an extra first order term, in (ct-x), as a damping factor. It could equally well have been added as an amplifying factor.
To see how the first order term in a quadratic equation can be a damping (or amplifying) term, the algebra for the Fibonacci series is derived from this form. In the formula for solving quadratic equations, a Fibonacci series can be obtained by one, of all the possible ajustments to the coefficient of the first order term, to come out with the square root of five. With this in the formula, the Fibonacci numbers, approximately 0.62 and 1.62, (with the unique property of being each others inverse) may be derived.
These numbers represent the ratios of the successive terms of the Fibonacci series, which increasingly accurately approximate the Fibonacci numbers. When these are graphed, they produce a zig-zag "curve" in which the zig-zags get successively smaller and more flattened out, like the subsidance of angular "ripples." In other words, the Fibonacci series graph is like a discrete form of wave damping.
Compare the pair of second order quadratic equations, with or without a first order term, with a likewise pair of second order differential equations. This compares, respectively, algebra with calculus. Apparently, the difference between the two branches of maths is the difference between the discrete and the continuous treatment of variables.
The ordinary wave equation, whose algebraic equivalent is eqn. 2, is a second order derivative and a constant. It expresses a wave, in simple harmonic motion, that is a uniform wave with a constant amplitude of crests and trofs.
The damped wave equation, whose algebraic equivalent is equation 5, includes a first order derivative, as the damping factor. It is usually solved in terms of the equation for a sine wave, multiplied by an exponent to a negative power, which makes the otherwise uniform wave undergo a diminution like successive waves in decreasing ripples, on a pond, known as exponential decay.
In fact, sine waves themselves can be expressed in terms of exponential functions, known as Euler equations, which simplify trigonometry. Circular motion is more conveniently expressed in polar co-ordinates than rectilinear co-ordinates.
The complex variable in rectilinear co-ordinates, z = x + iy, maps any point (by convention) on the upper right quadrant of a plane. This converts to polar co-ordinates as: z = r(cosQ + i.sinQ), with circle radius, r, and angle Q (measured between the horizontal x-axis and any angle of elevation, from horizontal, of the radius. (The radius is any straight line from circle origin to circumference.)
Then, the Euler relation is: z = r.e^iQ, where the exponent, e, is an
infinite number constant of about 2.718...
So, the damped wave equation could be expressed as a multiple of two exponents,
one with a negative power and the other with an imaginary power. As multiplied
exponents have their powers added, this would result in an exponent with a
power that is a complex variable.
Likewise, the Interval can be pictured as a constant radius, located by any arbitrarily placed rectilinear co-ordinates. These stand for all the possible different locations by which observers might see a given event, occuring at the position, where the Interval radius meets the circumference. The Interval is the common magnitude and direction, in space and time (space-time) measured by all observers in uniform relative motion, significantly approaching light speed.
That is a pictorial representation of the Interval as expressed in eqn. 2. But equation 5 includes a damping factor which means that the equation is of a damped or decreasing Interval.
One could imagine the Interval following a succession of events moving round the circumference of the circle. Observers could follow these events from their differently positioned co-ordinate systems. (We would say that the x-axis of each co-ordinate system is at a different angle, with respect to, say, an x-axis, assigned by convention as the base axis, common to all observer axes. Those different angles between the base axis and observers x-axes are known as their respective phases.)
We could draw a separate but corresponding rectilinear or Cartesian graph to a regular orbit of events on the Interval circle (in polar co-ordinates). Instead of a rotation, its track, on the rectilinear graph, would now look like a simple harmonic wave. The equal amplitude, of its crests and trofs, which is the magnitude of the Interval (constant), are measured by the positive and negative vertical rectilinear axis. At the same level as the origin or center of the circle, the zero point is at the equilibrium level of the wave, marked out by the rectilinear horizontal axis.
Suppose that the events, on the Interval circle, do not follow a constant revolution but begin to spiral inwards. Then the corresponding rectilinear graph would trace out a damped wave. This would be the graph of a decreasing Interval.
Successively taking the geometric mean of the (inverse) factor, 1/(ct-x), is equivalent to successively multiplying the (inverse) Interval by the negative operator, -i = i^3, which signifies the operation of three turns thru a ninety degrees (or a 270 degrees) change of the co-ordinates.
Adding the second and first order geometric mean Intervals derives a complex Interval form:
[GM]²1/(ct-x) ± [GM]1/(ct-x) = ±1/I ± ±i/I = ±(1±i)/I.
An after-note refers to physicist Lisa Randall. She used an exponential damping factor on the Interval, in connection with work on a full force of gravity, being wrapped-up in hidden dimensions, to explain why it appears much weaker than the other three known forces of nature.
In the conventional Interval, the observers are inter-changable, without
changing the value of the Interval. The conventional Interval has the symmetry
of uniform velocity in a straight line. Thus, the Interval measure stays
unchanged, as the common space-time measure, for all observers with different
local space and time measures.
Measurement inter-changes, which do not affect the result, are known as
symmetry transformations.
The geometric mean of two polar limits is either of the other two polar limits, at right angles to them. The average values are inter-changable with the limit values, or the representative values with the represented values. (This was explained in my study of a caldera model.) The real and imaginary dimensions have statistical symmetry.
The Interval is a complex variable circular function, in perpendicular and bi-polar real and imaginary variables, like the four points of the compass. Whereas electro-magnetism, consists of positive and negative electricity, which is bi-polar and magnetism, which is monopolar.
An analogy with democratic representation might illustrate the difference. A representative democracy consists of a parliament and people, that is representatives and represented. The representatives are the averages of popular opinion that cover a political spread from Left to Right. The representatives may be positive or negative, meaning government or opposition.
While the People cannot represent the Parliament, tho the Parliament can
represent the People, the latter are in a one-way or monopolar relation of
being represented but not representative.
Since electricity is bi-polar or of positive and negative charge, and magnetism
is mono-polar, Representative Parliament is analgous to the electric and the
Represented People analgous to the magnetic.
Suppose the People are also allowed to be representative and Parliament in
turn becomes represented. This can happen when politicians cannot agree (say,
are split or polarised between Left and Right) on some issue, and leave a
decision to the public in a referendum. (This actually happened in 1975, when
the UK Labour government couldn't agree with itself over staying in the Common
Market, and so tossed the decision to the British people.)
In that case, with a combination of direct democracy, and representative
democracy, there is a symmetry, of representation and represented, between
People and Parliament.
In physics jargon, the Interval possesses "invariance." The Interval is a measure, in the dimensions of directional distance, or displacement, common to all observers. (It is often said to be a four-dimensional measure of space-time.)
The Interval shows that local observers give different space and time
measures from their different co-ordinate systems, which are merely out of
phase with each other. Their only difference is an angle of rotation between
the different co-ordinate systems, about a common origin. Thus, the jargon
describes the Interval as having "space-time rotational symmetry."
The Interval invariance is despite any rotations between local co-ordinate systems of measurement. The change of angle between these local co-ordinate systems, leaves the conventional Interval the same. A change or transformation, that leaves something the same, is called a symmetry transformation.
The Interval is rotation-invariant or phase-invariant. The Interval is the
same measure, it has symmetry, despite turning round local observers different
space and time co-ordinate systems - like a circle or sphere is symmetrical,
because it looks the same from turning about its center.
When the geometric mean is taken of the positive and negative solutions of a quadratic equation, any first order term is automaticly canceled in the working. That is why it does not matter whether the Interval is kept in its conventional form, analgous to a simple harmonic wave form, or whether it is qualified, with an extra term, analgous to a damping or amplification factor, in its quadratic equation, changing the amplitude of the wave form.
A first-order term means a variable which is not multiplied by itself, unlike a second order term, which is a variable squared. A first-order term is analgous to a damping term in a wave equation that reduces the amplitude of the wave. The damping term would not change the geometric mean of the oscillation, it would just change the amplitude range limits of the oscillation.
The geometric mean of a damped Interval amplitude remains unchanged. The
amplitude is the height of a wave or corresponding depth of a trof. Such a wave
can be plotted from the rotation of a circle, whose radius (the Interval) is
the same length as the amplitude.
So, I suggest that just as the Interval is invariant, under rotation, or by
phase difference between local observers co-ordinate systems, it might also be
considered as invariant by amplitude change , or equivalently by radius
magnitude change.
This seems logical. If one changing property of a wave, that is phase, leaves a given law unchanged, why can not its other property, amplitude? Equivalently, if one property of a circle, rotation, may leave some law invariant, why should not the other property, radius?
The Interval, having rotation symmetry is only part of the picture. This is just a direction symmetry. A vector, like the Interval, has both direction and magnitude.
This essay has shown that the Interval also possesses symmetry, under change of magnitude, as a circle radius or wave amplitude. The Interval has complete vector symmetry of radius magnitude, as well as rotation direction. Or equivalently the Interval symmetry is of both amplitude magnitude and wave direction.
A symmetry operation is one that leaves something, it works on, unchanged. It displays an invariance in the world, which implies conservation of something. The Noether theorem derives conservation laws from symmetry.
The Interval, which has rotation symmetry, implies the conservation of angular momentum.
However, a damping factor or amplifying factor is changing the energy put in
the system. In the Michelson-Morley experiment, there is a light beam reflected
in a linear back and forth motion, with respect to earth motion. This change in
velocity of the light beam relative to earth motion is a relative
acceleration.
Since light has mass of motion, in principle, its moving with and against earth
motion will increase and decrease its mass. (In practise, of course, such
effects could only be measured on an astronomical scale.)
The earth-aligned light beam experiences a 180 degrees turn of its
earth-aligned motion, which maintains its alignment of zero degrees rotation
from linear alignment. But there has been a change in linear momentum.
The cross-ways beam, in the M-M experiment, experiences a 180 degrees reflection of its earth-perpendicular motion, which maintains (or conserves) its 90 degrees angular momentum but there has been a directional change in angular momentum. Tho, no change in mass magnitude, because not moving in line with or against the earth.
The earth-crossways reflected light beam changes its direction relative to earth, analgously to a boat, making a perpendicular crossing from one bank to another. (The energy effect of the stream, skewing the perpendicular crossing, like a boat from one bank to the other, is canceled by the two-way journey.)
Thus, in principle, the Michelson-Morley experiment (MMX) is a conservation of momentum, thru change in linear magnitude equated to change in angular direction. This MMX, which theoreticly exemplifies this conservation law of vector momentum, may be calculated in terms of the Minkowski Interval.
The Interval rotation symmetry implies conservation of angular momentum. Therefore, the Interval vector symmetry of rotation direction and radius magnitude implies a conservation law of vector momentum.
It is perhaps worth bearing in mind, that altho physicists speak of linear and rotational laws of momentum conservation, the Michelson-Morley experiment is implicitly about change in momentum. The earth-aligned light beam (light energy with mass in motion) changes its velocity (and, in principle, its mass) relative to the earth, on reflection.
Lisa Randall popularised her work, in Warped Passages. The subtitle is "Unravelling the Universe's Hidden Dimensions."
In the Mathematical Notes, at the end of the book, which she makes look a lot easier than they are, note 36 gives a modified formula for Minkowski Interval of space-time or a function of Euclid geometry of three spatial dimensions minus a fourth dimension of time. Randall and her colleague, Raman Sundrum modified the Interval into five dimensions, by adding a fourth spatial dimension to the other four dimensions. They also multiplied the Interval by a damping factor.
A damping factor, in math, is the coefficient used to describe things like the diminishing trofs and crests of a series of water waves, such as from a stone thrown in a pond, or the decrease in vibrations of a spring after it has been stretched and let go.
The damping factor is the inverse of an exponent to some power. In this case, the power is in terms of a fifth dimension (as a fourth spatial dimension) multiplied by a constant. This factor makes the strength of gravitational interaction fall off exponentially in the fifth dimension, traveling between areas ("branes") bounding that higher dimensional space. Randall refers to the factor as a "warp factor" which measures the warping of space by the presence of highly concentrated gravitational mass.
Gravity is extraordinarily weak compared to the other three forces of nature (that have become known, not too helpfully, as the strong, weak, and electromagnetic forces). This disparity might be explained, if gravity is as strong as the other forces on one brane, but being confined there, only interacts weakly with the other three forces on another brane, the area of our own experience of nature. Randall explains:
"...extra dimensions can be hidden either because they are curled up and small, or because spacetime is warped and gravity so concentrated in a small region that even an infinite dimension is invisible. Either way, whether dimensions are compact or localised, spacetime would appear to be four-dimensional everywhere, no matter where you are."
There are many variations on this possible scenario, including the
possibility of unification of the four forces of nature at comparable strengths
in higher dimensional space.
Also Randall has developed the idea of different numbers of dimensions too far
away to see, on the scale of the universe, in contrast to the older idea of
extra dimensions being rolled up too small to see.
This chapter does not pretend to be an authoritative account of symmetry and conservation law. Victor J Stenger, in The Comprehensible Cosmos (2006) supplies that very well. I thought I understood some of the earlier part of this popular exposition fairly well at the time but had long since forgotten reading it.
Heinz Pagels: two popularisations, regardable as modern classics: Perfect Symmetry (1982); The Cosmic Code (1985).
Richard Feynman: The Character of Physical Laws. (1965).
Richard Lung.
19, 20 & 24 december 2009 & note, 15 november 2013.
Completely re-written june to 3 july 2015.
This page extensively corrected and clarified: 01/07/2018.