Is there a plausible basis for faster-than-light travel? (II)

FIG. 1 Basis of Variable tau.

Variable Speed of Light and Variable Time:

More fundamental than the warp drive concept is that of a variable speed of light, proposed by Joao Magueijo of Imperial College, London. Magueijo’s theory is that the speed of light is not always constant, so can sometimes (under special conditions in vacuo) exceed 186,000 miles per second. If so, then it might be used for practical applications. (Though one is left to ponder whether what might happen if, while in transit, the speed of light ratchets back to below 186,000 msp!)

Up to now, Magueijo’s theory has received little by way of attention, other than in a recent book he published, entitled ‘Faster Than The Speed of Light’. This is because he has found no referees willing to validate his assorted papers for publication. His book itself, alas, sheds little actual light on the physics issues, and is more in the way of a long personal account (including harangues) against his presumed oppressors and censors.

That being the case, it is still worthwhile to inquire how a variable speed of light theory might work. My own solution to this entails not variability in the speed proper, but rather a variability in time, which leads to variability in speed. (since v = d/t so as t -> 0, v -> oo for some finite distance d).

The concept of a variable time vector is not new. A. Achong first proposed a possible elastic temporal vector that could have uses in physics. He invoked an ansatz in which particles possess an internal (intrinsic) time vector arising from the internal structure of their individual constituents, including quarks. The individual time vectors for each particle are allowed to assume either (+) or (-) signs, depending on whether the constituent is associated with normal matter or anti-matter- and the direction of global time is determined by the dominant sign. [1]

A somewhat different take can be developed by using quantum set theory in relation to time, as originally developed by Finkelstein.[2] In line with this, we adopt time vectors such that there exist intrinisic variable magnitudes. I use the term tau (t) for these, based on the same unit from quantum set theory. According to this definition, 'one tau' is:[2]

10^-43 < τ < 10^-23

This is an extremely small time unit to be sure, but up to twenty orders of magnitude larger than the Planck time (10^-44s).

The tau, or variable tau, can be thought of as a ‘building block’ or element of proper time. In another guise, as the smallest conceivable increment of temporal difference. An aggregate or set of elemental taus yields the normal, standardized units such as ‘second’, ‘minute’, ‘hour’ and so on. The cautionary note here being that all these can vary in duration (from an extrinsic objective observer’s view) depending on what its tau component is at the time of measurement.
Tau changes, as well as the derivative unit times dependent on it, by virtue of tau expansion (Fig. 1)

This may most probably occur by distortions or discontinuities in fiber bundle sheafs at different levels. Tau expansion assumes that conservation of quantum probability currents[3] is valid throughout. This means that the area shown for the probability space must be equal throughout. Hence, if I stretch the tau, elongating it as shown in the upper view (Fig. 1), the Planck Length (L_P) will be contracted in order to compensate and keep the same phase space area.

Tau expansion is possible based on the well-known second postulate of quantum mechanics:

II. The independent variables x, p of classical mechanics are replaced by Hermitian operators x _op, p_ op. Operators corresponding to dependent variables (i.e. H ) are also assigned Hermitian operators. Thus we have:


x _op <-> x and p_ op(x) <-> -ih/2pi (d/ dx)


In the context of temporal uncertainty:

t _op <-> t and E _op(x) <-> -ih/2pi ( d/ dt)

and


delta E (delta) t > h/ 2pi

Which is the energy-time format for the Heisenberg Uncertainty Principle. That is, the simultaneous product of uncertainty in energy, by uncertainty in time, yields the Planck constant divided by 2 pi. Another way of putting this, is:

delta(t) > h-bar / delta (E)

Where the numerator, called ‘h-bar’ is equivalent to h/ 2pi. This inequality is telling us that we can expect, based on the Heisenberg Principle, uncertainties in the properties of time – as well as of space. (I.e. measuring position of an electron during a momentum measurement). This uncertainty is such that the temporal variation (dt ) increases as the energy variation associated with the particular region of space (dE) decreases. This leads directly to the inference of variable c, supposing there are two separate regions (1,2) for which:

delta (t2) > delta(t1)

so that, c (1) = D/ [delta(t1)] > c(2) = D/ [delta(t 2)]

In other words, the speed of light in region (1) exceeds that in region (2), all other things (e.g. distance D) being equal IF the times are not. (The longer time interval delta (t2) gives rise to slower light speed, for the same distance D covered). For concreteness, let:

delta (t2) = 0.11 sec, delta (t 1) = 0.10 sec

Then: V(1) = 1.1 c and V(2) = c, taking D = 186,000 miles

This is well and good, but we need to go deeper into the tau connection (tau expansion). First, as to physically how variable time is possible, we note that the creation of particles and energy can be spontaneous. Thus, the quantum of energy:

delta (E) > h -bar/ delta (t)

say appearing in pair production, can be wholly spontaneous. By extension, as James Gott and other physicists have noted, the same energy-time principle can be used to account for the spontaneous appearance of the cosmos. Based on some initial energy uncertainty, delta (E). It follows from this, that spontaneous time variation is also possible.

Finkelstein actually created an operator explicitly to vary time via ‘bracing’. The operator is called ‘the brace operator’, Br. To see how it works on an elementary level, select a quantum unit set (say of cardinality 1) over some sub-module S (1) of the Clifford algebra S, with basis B(1). Then it follows from application of Br, and its conjugate Br*:

Br* Br = 1

Br Br* = [unit]

Br* Br - Br Br* = [non-unit]

A more graphic way to see this in term of tau change is as follows:

After delta (t) = 1 τ, Br = { }

After delta (t) = 2 τ, Br = { { } }

After delta (t) = 3 τ, Br = { { { } } }

Notice that the brace creation increases arithmetically as the unit tau increases. The operation Br, equivalent to C(b) in Grassman space, generates an elemental tau (τ). It is easy to see from this that the inclusion of Br* is equivalent to generating a state projector [unit] τ_ 1 >. Conversely, brace annihilation reduces tau, viz.


Br* Br - Br Br* [ { { { } } } ] -> { { } } or 3 τ -> 2 τ. It can be shown that, just as say for quantum mechanics, Ψ > = Ψ 1 > + Ψ 2 > + Ψ 3 > + Ψ 4 >+ …….Ψ N >, so also in ‘tau mechanics’: τ > = τ1 > + τ2 > + τ3 > + τ4 >+ …….τN >, where the τ are resultants in an N-dimensional complex vector space (or Hilbert space). This leads naturally to a conception of an infinite ‘tau’ Hilbert space embedded in a Euclidean 4-space manifold (Think of each elemental τ lined up head to tail along an infinitely long 'box'). In effect, unit temporal vectors (taus) shown, extend all the way to unit temporal vectors (taus) shown, all the way to τ (oo).

What if a ship were to travel over a region in which variable tau applied, say in τ-Hilbert space from τ1 to τN? in particular, such that the tau gradient ( dτ / d x) decreased say by 10^-23 second each meter? Even traveling at constant velocity (by ship controls) we’d expect it to undergo acceleration on account of the changing tau.

Elementary calculation shows that it would take ~11 million light years displacement to attain a time differential of one second. 5.5 million light years for a half second. Whether such increments could translate into super-light velocity depends upon the initial velocity of the craft. Say the hypothetical craft was traveling at 200,000 km/s (or 2 c/3) initially. Then after 5.5. million light years it would be at: v = 200,000 km/s/ 0.5s = 400,000 km/s or 4c/ 3. This would increase to 800,000 km/s (8c/ 3) after another 2.75 million LY. Obviously, super-light speeds could be attained over much shorter distance spans if the tau gradient is higher. (For instance, the distance threshold for v > c would be 100 times less, if d τ / dx increased the same magnitude tau each centimeter.)

In the case depicted above, a temporal Br (brace) operator projects the unitary temporal state τ > into a multiple of the chosen state, Uτ_s > and U τ >.

Br = exp (iE) = cis E Br* = exp(-iE) = cis (-E)

It is this quantification basis, applied to a certain region of space, that allows for variable tau as I’ve described it. Of course, the discussion can easily be extended to two (or more) dimensions with appropriate modifications of the mathematics. This can be the subject for a future article.

Next: A Multiverse and Cosmic Shortcuts?


[1] Achong, A., 1984, Internal Time and Global Time, Proceedings of the 2nd Caribbean Physics Conference, Leo L. Mosely (Ed.)



[2] Finkelstein, D.: 1982, Quantum Sets and Clifford Algebras, in International Journal of Theoretical Physics, Vol. 21, Nos. 6/7, p. 489. (Reference to tau on p. 494).

[3] Think of the quantum wave function ι changing over time. Then the wave probability, P = Ψ Ψ* will also change. This can be thought of as a 'current' in time.