Lisa Randall's book Warped Passages encouraged in me the idea for probabilistic dimensions. The subtitle of her work 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's Interval of space-time or a function of Euclid's 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. But theyve also multiplied the Interval by a damping factor.
A damping factor, in math, is the coefficient used to describe things like the diminishing troffs 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, travelling 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). But 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.
I'm just borrowing Randall's idea of multiplying the Interval by an inverse coefficient. Ive mentioned the damping coefficient, found in text-books on mechanics. But I first came across the use of the inverse exponent, multiplied by the exponential series, in elementary statistics. This describes the Poisson distribution of probabilities.
The Poisson distribution follows the values of the terms in the exponential series. This is the (highly convergent) series that sums to the exponent, an infinite number of about 2.718... (With pi, the exponent, "e" or "exp" is one of the two most famous infinite constants in mathematics. The exponent measures constancy in rate of change. In differential calculus, the derivative, of the exponent to the power of x, is itself.) It so happens that some rare events, say, 0, 1, 2, 3, 4, etc events per year, are well approximated by the first few terms of the Poisson distribution. A celebrated example is the number of soldiers killed per year in the Prussian army by horse kicks. The terms of the Poisson distribution gave a good approximation to the actual numbers.
The Poisson distribution is the model that decides whether rates of occurence of certain rare events are as theoreticly expected. There are also statistical tests of deviation from the expected norm, which can decide how probable that the deviations are significant, which might mean something unexpected and unexplained was going on.
Probability is generally calculated in terms of unity being the sum of all probabilities. Therefore the exponent has to be reduced from about 2.718 to one. This is done by multiplying by its inverse, or .3679 to four decimal places. As the first two terms in the exponential series are both one, then the adjusted probabilities of occurence for both zero events and one event, is about .37 to two decimal places, or just over one-third probability each ( for example that no soldiers or one Prussian soldier die from a horse kick) over a certain period. The third term, in the series, is one half, which means one half of .37, or rather more than one-sixth probability (for example that two soldiers die of horse kicks).
The point about the exponential series is that it rapidly falls off, so that the probability of three events is already reduced to one sixth of .37 or about .06, and for four events, the probability is one twenty-fourth of .37 or about .015.
On my previous web page about the Statistics of Space-time, I noticed the Interval form in the first two terms of a binomial distribution. This distribution is closely related to the Poisson distribution. In terms of politics, the binomial distribution (or its continuously-curved version, the normal distribution) represents a country where most people have middling views. Whereas, the exponential distribution is too extreme to be found in a large population, generally speaking. For instance, it would be most unlikely if a country existed where most people's views would start at one of the extremes, either right or left (but not both) and fall off steeply in numbers of people holding middle views, to virtually no-one holding views of the opposite extreme.
Just as the inverse exponent is used as a coefficient or multiple, in simple statistics, to adjust the exponential series sum to total probability one, so the binomial series sum is adjustable to one by multiplying the binomial series thru by the inverse of its sum. This makes each term in the binomial series a fractional probability that sums to total probability one.
The idea is very simple but just to show what I mean, take a binomial distribution (from Pascal's triangle) of 1, 3, 3, 1. This sums to 8. To turn 8 into a total probability of 1, we multiply thru each term in the series by the inverse of 8, or 1/8. Then, 1/8, 3/8, 3/8, 1/8 might represent the respective probabilities of 0, 1, 2, 3 occurences of a certain event.
In the previous web page, on Statistics of Space-time, a form like the contraction factor, found within the Interval formula, was considered as a kind of average (a geometric mean but with an arithmetic mean property) that expanded into a binomial distribution of terms ranging from an arithmetic mean term to a geometric mean term with terms in between that range from more or less one kind of average to the other.
The contraction factor, or its form, relates only to a mean with upper and lower end limits to velocities. In statistics, it is also possible to construct such a mean with intermediate values of velocity. Three velocities, instead of one, produce a binomial expansion, in its first two terms, reminiscent of the Interval, where the three velocities are on the three spatial co-ordinates.
The previous web page showed that the three velocities in the conventional Interval, say, u, v, w, (relatable by Pythagoras' theorem, in three dimensions, to a radius vector r) are not the same as the three velocities, say, f, g, h, scaling a geometric series, whose average term is its geometric mean.
In the conventional Interval, the velocity r has the Pythagorean relation, r² = u² + v² + w², and is given by equation (1):
I²/c² = t²{1 - ( u² + v² + w² )/c²} = t'²{1 - ( u'² + v'² + w'² )/c²}.
In the statistical Interval, velocity r is one range measure equivalent to
a fuller range measure in three velocities f, g, h.
That is equation (2):
(1-r²) = {(1-f²)(1-g²)(1-h²)}^1/3.
Taking the square roots of both sides would give them the forms of geometric means. Indeed, (1-r²)^1/2 is a form like special relativity's Fitzgerald-Lorentz contraction factor, which is indeed a special case of this form.
The conventional Interval can thus be re-formulated under this new definition of r, for equation (3):
I²/c² = t²{1 - ( f² + g² + h² )/c² + ( f²g² + f²h² + g²h² )/c^4 - ( f²g²h² )/c^6} = t'²{1 - ( f'² + g'² + h'² )/c² + ( f'²g'² + f'²h'² + g'²h'² )/c^4 - ( f'²g'²h'² )/c^6}.
Provided that velocity r has the same value in both equation (1) and (3),
the two equations should be equal. We have merely transformed the
conventional Interval into a new statistical form.
As explained on the page, Statistics of Space-time..., we could just as well
let f, g, h, equal u, v, w, respectively, and then re-define equation (2)
with a new value R, instead of r, that substitutes in u, v, w. This makes the
conventional Interval a statistical approximation to the first two terms.
This is the more interesting possibility, equation (9) from the previous
page; equation (4) here:
I²/c² = t²{1 - ( u² + v² + w² )/c² + ( u²v² + u²w² + v²w² )/c^4 - ( u²v²w² )/c^6}= t'²{1 - ( u'² + v'² + w'² )/c² + ( u'²v'² + u'²w'² + v'²w'² )/c^4 - ( u'²v'²w'² )/c^6}.
Equation (4) re-formulates dimensions of time and space into terms in a probability distribution, which suggests that somehow the measures of time and space may be probabilistic. This provides a possibly useful statistical development to classical mechanics, to which relativity theory aspired, in its deterministic framework of space and time or, if need be, space-time, from which mechanical motion could be definitively measured.
Another branch of physics, called thermodynamics, has long suggested that time is a probabilistic dimension. In theory, a smashed milk bottle could pick itself up again to its former wholeness. But the possibilities are so inconceivable, that if you saw it happen, you would know you were watching a film of the event in reverse. The probabilities are that order will degenerate into disorder, relatively speaking. You may build up islands of order in an ocean of chaos but you will cause more disorder than order, by the way.
This inability to fully conserve order from disorder, or energy from entropy, is a sign of time from past to future, or "time's arrow". Its practical meaning is that perpetual motion machines are impossible.
In classical mechanics and relativity, time doesnt have direction: the equations are in principle reversible. Even quantum mechanics theoreticly allows the perspective of particles moving back in time, within the strict limits of quantum uncertainty. Yet time's arrow does seem to point to a state of affairs not only in biological evolution but even cosmic evolution.
There is a joke by Arthur Eddington, that if someone said they had a theory to challenge Relativity, he would be most interested. But at a theory that contravened the second law of thermo-dynamics, (the so-called running down of the universe) he would sadly shake his head that there was no hope of its being true.
However, equation (4) could be treated as a probability distribution with the Interval itself standing for total or unitary probability. To make the Interval always one, we could multiply thru by the inverse of the Interval squared, 1/I². Thus equation (5):
1/c² = (1/I²)t²{1 - ( u² + v² + w² )/c² + ( u²v² + u²w² + v²w² )/c^4 - ( u²v²w² )/c^6} =
(1/I²)t'²{1 - ( u'² + v'² + w'² )/c² + ( u'²v'² + u'²w'² + v'²w'² )/c^4 - ( u'²v'²w'² )/c^6}.
We assume that c is set at unity, as it usually is in physics
calculations, so that 1/c² actually means just one, for the total
probability. The reason for not replacing c by one is to remind ourselves
that if c = 1, the other velocities are fractions of one.
When all the observed velocities are zero or so small compared to light
speed, they can be treated as of practicly no importance for this
calculation. When this happens, the first term in the curly brackets, which
is one, carries all the probability. It is the term multiplied from outside
the curly brackets by time squared. This signifies that in the case of zero
or very small velocities, the term denoting the time dimension has a
probability of one or very near one, which means certainty or very nearly
so.
Here we have the first term as the time dimension with a probabilistic measure. In the reduction of the Interval to the classical situation of insignificant speeds compared to that of light, we have a near certain probability for the time dimension, as there is a near certain probability for time's arrow in thermodynamics, but a probability nevertheless.
This tackles the problem of time appearing to have no direction, or arrow from past to future, in classical physics or relativistic physics. According to this statistical interpretation of the Interval, time is a "probability vector", as it is in thermodynamics. Only, in high energy physics, the probability of the arrow of time, flying in one direction only, may be reduced from near certainty.
The familiar certain probability for time decreases for relativistic
speeds, speeds of the observers' velocities approaching light speed.
To maintain the over-all probability of one, in equation (5) of the Interval
(squared) as a space-time probability distribution, then the inverse of the
Interval (squared) has to become a scaling factor. (Lisa Randall's
incomparably more sophisticated theory in cutting-edge physics involves the
use of an inverse coefficient as a scaling or "rescaling factor".)
To give a crude idea of this, suppose the velocities term, on the spatial dimensions, amounts to a third of light speed, to be subtracted from unity, on the "temporal" dimension. Because there is a minus sign between them, the sum is one minus a third, which equals two thirds for the Interval. But this is rescaled by multiplying by the inverse of the Interval at three-halfs, to adjust for total probability of one.
There could have been a snag here. To encode the space-time Interval in its first two terms, the required binomial series must take account of the negative sign which distinguishes time from the spatial dimensions. That results in a binomial series of alternating plus and minus signs. But a probability distribution must add up to one. If you have minuses in the series, then potentially a positive term may be more than one, for the sum of terms to equal one. You cannot have a probability greater than one, for that is more than the sum of all possibilities.
Fortunately, this doesnt seem to matter. As long as we treat the time term and the space term, the first two terms, as a joint space-time term, we dont have the impossibility of negative probabilities. At any rate, the velocities in the second term are always going to be some fraction of the light speed, and so less than one, as no massive object can ever attain light speed.
Tho, the second term velocities can be made light velocity with unity as
the term's value. Taken away from unity as the time value of the first term,
this leaves zero.
The clocks of observers aproaching light speed, appear to slow, to relative
observers. Till, at the speed of light, itself unreachable to material
bodies, "time must have a stop." Motion at light speed is timeless. At light
speed, time has no arrow: the probability is zero for time having a
direction.
To rescale to unitary probability, the rescaling factor would have to be
infinite.
At this point, we look at the third and fourth terms in the binomial
distribution of equation (5). They are still negligible values, even with
light speed squared for a numerator. At least, the positive term is greater
than the negative term, so there's no need to worry here about an overall
negative probability of the two terms, considered as another space-time
probability.
Suppose the numerators of the third and fourth terms are about light speed squared. That gives a joint value of about: 1/c² - 1/c^4, for the new Interval, since the first two terms recognisable as the conventional Interval cancel to zero. The inverse Interval coefficient automaticly adjusts the new Interval to one, being the only remaining space-time probability.
The previous page on Statistics of Space-time did consider the possibility
of the third and fourth terms as another joint term for an alternate average
or geometry of high curvature.
Also, the binomial distribution could be expanded with more than three
velocity variables. This doesnt necessarily mean higher dimensions than three
of space. It may just mean more refined gradations of curvature in the
greater number of terms in the binomial distribution. And with it, a greater
break-down of probabilities of more or less curved space-times.
(Lisa Randall discusses the fluidity of the number of dimensions in the different models of string theory. That is at the cutting edge of research, rather than in the most elementary terms, which are all I can offer.)
It might be asked what is the use of treating space-time geometries as probable averages of curvature. Well, the concept might act as a bridge or transition between relativity and quantum mechanics. Lisa Randall says:
General relativity works only when there are smooth gravitational fields encoded in a gradually curving spacetime. But quantum mechanics tells us that anything that can probe or influence the Planck scale length has huge momentum uncertainty. A probe with sufficient energy to probe the Planck scale length would induce disruptive dynamical processes, such as energetic eruptions of virtual particles, that would dash any hope of a general relativity description. According to quantum mechanics, at the Planck scale length, instead of a gradually undulating geometry, there should be wild fluctuations and loops and handles of spacetime branching off...
Nor does general relativity step aside to give quantum mechanics free rein, for at the Planck scale length gravity exerts a substantial force. Although gravity is feeble at the particle physics energies we are accustomed to, it is enormously powerful at the high energies required to explore the Planck scale length. (Keep in mind that the quantum mechanical relations tell us that while the Planck scale length is miniscule, the Planck scale energy is enormous.) ...
In fact, at the Planck scale energy, gravity constructs barriers that make conventional quantum mechanical calculations impossible. Anything sufficiently energetic to probe 10^-33 cm would be snapped up into a black hole that imprisons whatever enters. Only a quantum theory of gravity can tell us what is really going on inside.
Randall goes on to say the two theories cry out for a more fundamental theory. With respect to the above quotation, I can only make the simple point that it makes sense to consider the probabilities of different space-times occuring, as exemplified in a binomial or other statistical distribution of geometries. The break-down of continuous space-time into a quantum foam of abruptly changing curvatures, or indeed massive astronomic distortions of the fabric of space-time, might so be represented as probabilities of different space-time curvatures occuring.
Equation (1) as an approximation of equation (5) may be more fundamental. But the equality of equation (1) to equation (3) may also have its uses. That is to say the equality of one side of equation (1) to the other side of equation (3). Equation (1) is in standard terms of uniform motion but equation (3) implies acceleration, in the velocities defined in terms of a geometric series.
I dont know if or how such a reformulation of the Interval might work. Tho, the twin paradox is a case of one observer being accelerated, while the other stays in uniform motion.
Paul Davies' book, Space and Time in the Modern Universe, analyses the twin paradox to show that it isnt actually a paradox that one of the twins should age and the other not. It is actually the twin, who, in going on a space journey, has had to under-go an accelerative force, to get back to the other twin remaining and mouldering on Earth, that has survived the eons gone by on Earth.
There was a definitive experiment on a particle with a counter-part, in orbit, which gives a component of acceleration. This verified the slower aging of the orbiting particle. By the way, the accelerated observer feels himself to age in the normal way. There is no slowing down of time for him, from his point of view. As someone said about relativistic times, that is the catch.
In terms of their motions, the earth-bound twin and the accelerated twin respectively resemble the conventional Interval and the Interval given an accelerative treatment, via re-interpreting velocities as in a geometric series. But the point of the twin paradox is that the local times of the twins are found to be vastly different, when they re-unite in the same space.
Either equation (1) or equation (3) allow for this possibility, in the difference between times t and t'. Equating one side of equation (1) to the other side of equation (3) suggests somehow distinguishing, respectively, an observer in uniform velocity, from an accelerated observer.
And so, like an over-extended runner, or rather pedestrian, I come to a lame ending.
Richard Lung.
7 january 2007
minor addition, 18 february 2007.