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Linking String and Membrane theory to Quantum Mechanics and
Special Relativity equations, avoiding any Special Relativity
assumptions M.G. Hocking Materials Dept, Imperial College, London SW7 2BP
Website Page 9 Email: m.hocking@imperial.ac.uk (CONTINUED FROM PREVIOUS PAGE) Special Relativity Equations derived assuming absolute motion: Rest Mass; Length and Time Dilation; E = mc2 On the basis of the "Big Bang" theory with its
residual microwave radiation, it is concluded that there is an absolute
reference point of origin (the "Big Bang" site) in space. This
negates the First Principle of Special Relativity, which denies an absolute
reference point in space. A Big Bang
point of origin in 3-D space would also be accessible in higher dimensional
spaces. Although Special Relativity is an idealisation for gravity-free space, and so strictly does not apply to the Big Bang universe, it is nevertheless widely used in practice in physics and should not thus be “sheltered” from the existence of a point of origin of the Big Bang. To ingeniously avoid the problem (for relativity) of having a central reference point, the analogy is sometimes given of the universe being like a balloon being inflated, starting at a point (Big Bang origin), but later when large (when the universe had expanded) anyone anywhere on the surface of the balloon would think his location was the original centre. But if space pre-existed the Big Bang, this balloon model would be wrong. Who can say? A two-line derivation is given below, of the mass, time and
length dilation formulae of Special Relativity but without assuming any
relativity. 2-D space is not viable for the existence of life forms because
the complexity required for brain interconnections, digestive tracts, etc
requires 3-D. Simple calculations
show that electron orbitals in atoms would not be stable for dimensions
higher than 3, which makes only 3-D space The electrostatic force falls off as the inverse square of
distance in 3-D but it would fall off as the inverse cube in 4-D space (it
would then be too weak to bind electrons to their atoms). The inverse square
arises simply because a given flux through unit element of area on the
surface of a 3-D sphere is spread out in proportion to the square of the
radius, as the area of a 3-D sphere is 4πr²,
but the volume of the 4-D analogue of a sphere is proportional to r³. A 3-D elementary particle and derived particles like atoms and
molecules cannot make up a 4-D object, because they have no extension in the
direction of a 4th spatial dimension. So they
(and any larger body they constitute) are thus confined to 3-D space only and
so cannot enter 4-D space, with the one very localised exception described in
the section on Rest Mass below, as part of a very small amplitude
oscillation. For a larger scale
excursion into a 4th or higher dimensions, the
7 orders of coiling-up of the 7 higher dimensions in 3-D particles must be
reduced by 1 order, each time the next higher dimension is reached. Rest Mass In 3-D space, elementary particles which constitute molecules,
etc, are proposed in the Introduction above as being like gas bubbles in a
continuous medium (Dirac, 1962; Besant & Leadbeater, 1994). However, a continuous medium cannot be
described as a "fluid" because a fluid is able to flow and to thus
permits particles to move through it due to mobile atomic-size
"holes" in it (in the conventional An analogy is the common observation of a solid block of ice
which has tiny bubbles of air trapped in it - these bubbles are "locked
up solid" and cannot move at all. Thus it is proposed that 3-D elementary particles (bubbles) in
the continuous background medium can only have a zero velocity in it. Actual physical movement which is of
course commonly observed in 3-D space can then be postulated as occurring by
the following mechanism, which is necessarily similar to diffusion (being
movement through a medium). This
accords with the identical functional If 3-D space consists of a 3-dimensional continuous background
medium (Besant & Leadbeater, 1994) as explained above (Cf. air bubbles in
block of ice model) an elementary particle (bubble) would be unable to move
in any of the 3-dimensional directions.
But if it were able to jump out as part of an oscillation into a
higher spatial dimension where there is no such continuous medium, it could
then move and then land back in the 3-D space medium in a different place. An elementary particle might be rotating and vibrating
continuously (even if at rest in 3-dimensional space) in a path which takes
it continuously in and out of the fourth dimension (an effect similar to
zitterbewegung). "Zero-Point Energy" means that even at zero
degrees Kelvin "rest", a particle is still oscillating incessantly
(called "zitterbewegung", Ger. "trembling"). If the
energy (welling up from Jumps into the next higher dimension would only be possible for
elementary particles as the amplitude of an excursion into 4-D space would be
limited to the very small diameter of the coiled-up 4th dimension for 3-D particles, and not available to
large bodies, and it is called "quantum mechanical tunnelling" in
physics but not yet interpreted as involving jumps into 4-D space. If there
are higher dimensions, it would be very odd if they were not involved at all
in atomic-size processes. They cannot just apply to quark physics and nothing
else. Such a model leads immediately to Schroedinger's time and
distance equations (for a case with zero potential energy), as shown above.
It also provides a theory of rest mass, and leads to the same experimentally
verified equations of Special Relativity but for absolute motion. The
derivation is far simpler than that from Special Relativity. This absolute
motion derivation uses the assumption of The mass, length and time dilation equations are easily obtained
immediately by solving a Pythagorean triangle with sides moc, mv and the resultant mc (Fig. 2):
Fig. 2. (moc)2 + (mv)2 =
(mc)2 moc can be regarded as the momentum of creation of a particle at
rest in 3-dimensional space, due to an energy welling-up from the direction
of a fourth spatial dimension (which is at right angles to any 3-D
direction); mo is the rest mass of the resulting stationary particle in
3-dimensional space, which this force creates. If the particle is then made to move in 3-D space, by giving it
momentum in a So, from Fig. 2: (moc)2 + (mv)2
= (mc)2, which rearranges to mo = mÖ(1 – v2/c2). This is the well-known and experimentally verified
"relativistic" mass dilation formula but has been derived above for
absolute motion in only 2 lines and without assuming the two principles of
Special Relativity. Time Dilation Time dilation will also occur, because when a particle (e.g. a
meson) is jumping in the fourth dimension, its internal decay processes will
be frozen for the duration of that jump and so its lifetime will be
extended. The well-known time
dilation formula can then also be obtained immediately, as above, from a
Pythagorean triangle (Fig. 3) with sides to, tOUT and t, as explained below.
Fig. 3. t ² = to ² + tOUT ². To explain this, pursuing the analogy with diffusion, assume
that the motion of an elementary particle occurs by very short jumps
alternating with longer stationary periods, thus allowing any observed
overall velocity to be made up, modelled on the conventional mechanism of
diffusive jumps of atoms or ions through a lattice, from site to site. The identical functional forms of Fick’s
Second Law of Diffusion and Schroedinger’s Equation was discussed
earlier. In diffusion of an ion
through a lattice, a jump down a concentration gradient occurs when the chemical
energy gradient (Gibbs Free Energy) gives sufficient activation energy
for a jump to the next lattice position.
This model is used below for motion of a particle, where a mechanical
energy gradient drives it. There are only two velocities possible, zero for the periods at
rest in the 3-D world, and c for the periods when the energy constituting the
particle is moving in 4-D space. Any
actually observed overall velocity, v (0<v<c), is then made up of
rapidly alternating combinations of these two values. The moving particle
travels in a series of very small jumps each of which is at velocity c,
separated A moving atomic-size particle is thus a "particle"
when stationary and may appear to be an apparent "wave" (a
non-particle) when jumping. Light
photons alternately jump a distance λ in λ/c seconds followed by a
stationary instantaneous wait or appearance.
It is thought that photons (unlike gravitons) cannot move appreciably
away into 4-D and so are bound to continually intersect our 3-D world. Let the total stationary time (spent residing at successive
positions) be tIN and the total transition time (spent in
jumping between these positions) be tOUT. tIN is the inactive stationary time elapsing
between jumps, and can have any value (0 < tIN < oo). tOUT is the time taken for a transition or jump
between residences, and represents a non-material (non-particle, apparently
wavelike) condition in between the physical sites at which the moving
particle successively resides. It
means that there is no physical movement at all and that all actual movement
occurs during the time when the particle is in 4-D, by a series of
non-material (non-3-D) jumps. It is Consider now the motion of a mechanical clock which contains a
balance wheel. On the proposed theory, the balance wheel (=B) jumps have
their specific discrete B activations (see activation energy, above), but
when the clock (clock = C) as whole is also set in motion, specific discrete
C activations will occur additionally.
Any activation effect which becomes due to cause an imminent balance
wheel (B) jump during the course of a clock (C) jump, would be inoperative,
as the clock is "frozen" - already engaged in a jump and so its
balance wheel cannot also simultaneously move then. Consider the clock to be moving much faster than the balance
wheel rotations. Then the balance
wheel (B) jump frequency is comparatively very low and those B jumps which
arise during a regular C jump will be lost.
The consequent loss of some B jumps will (in effect) slow down the
balance wheel. Consider now the motion of a clock whose tick-tick period is
to at rest, which corresponds to tIN as defined above. Let
this clock travel with a constant overall velocity v and record the passage
of one tick-tick time period to during its travel through a certain distance
s. The total C jumping time (at
velocity c) which is non-material (being in 4-D), is not sensed or recorded
by the clock (by B jumps, as explained above) is tOUT , where tOUT = distance/velocity = s/c = vt/c
……………(iv). A stationary observer would have a total time t, in (iv) above,
elapsed on his watch, as being the time taken for the moving clock to travel
the distance s. Now, t > to or tIN due to the additional
time tOUT taken for the journey,
noticed only by the stationary observer, which must be added to to. This addition must be vectorial, because
as the moving clock does not sense or record tOUT there is no break (in its sensation of time) at which tOUT can be added in a scalar manner. tOUT and tIN have no component in common and must
thus be added as vectors at right angles (Fig. 3). This gives t ² = tIN ² + tOUT ² ………..(v), by Pythagoras' Theorem, or t ² =
to²
+ tOUT ². Substituting tOUT from (iv) above, to² =
t ² - v² t²/c², which is the well-known Time Dilation formula of Special Relativity, but all the assumptions of Special Relativity are avoided. This equation has been well-verified experimentally, e.g. by the increased lifetimes of decaying mesons which are moving very fast, compared with slow-moving mesons. The time dilation formula can also lead to an alternative
derivation of the mass dilation formula, already derived otherwise, above. Fitzgerald-Lorentz Length Contraction Similarly, the length of a moving body will contract (only in
the direction of travel) due to the interatomic cohesive forces pulling in
its length across planes of jumps when it is in 4-D space (where it is not
affected by 3-D electrostatic cohesive physical forces; only gravity can enter 4-D space and
gravity is not involved in cohesive forces). A similar Pythagorean triangle gives the well-known length
contraction equation. The length of a moving object is proportional to the
number of moving elements materially present ("IN") in it along any
given line in the direction of motion. The term "moving element" At rest, L = Lo and tIN = t. The number of planes (perpendicular to the direction of motion), of moving elements which are materially present (IN), at velocity v, is n = no(tIN/t), where no is the number of such planes present at rest (for which state tIN = t). no is proportional to Lo and n is proportional to L, where
n and L are number and length respectively, at a steady velocity v. Thus, from n = no(tIN/t) above, we have: L = Lo (tIN /t) = Lo Ö [t ² - tOUT ²] / t, using
equation (v) above, So L = LoÖ[1 - tOUT ²/ t ²], and then
using equation (iv) above we obtain: L = LoÖ(1 - v ²/ c²), which
is the Fitzgerald-Lorentz length contraction. An alternative approach also follows from the assumption that
when an object is travelling, some of the elementary particles constituting
it are engaged in a jump in 4-D and are thus "missing" from the 3-D
object, as suggested by the interpretation of Schroedinger's Equation given
earlier. Consider the number of elementary particles in a line in its direction
of travel to be no at rest and n at velocity v, where n < no as some of them are jumping.
n and no are their numbers in 3-D space. As mass in conserved, nomo = nm, [where m is the enhanced mass at velocity v given in mo = mÖ(1 – v2/c2) ] . Then, as no is proportional to Lo and n is proportional to L for a
line in the direction of motion of the object, Lomo = Lm
and so L = LoÖ(1 – v2/c2). This is the Fitzgerald-Lorentz length contraction. E = mc2
derivation The well known Relativity equation E = mc2 can also easily be obtained (for absolute
motion), by elementary algebra: From the Pythagoras triangle of the Rest Mass section above, (moc)2 = (mc)2 - (mv)2 . (See Fig. 1) Take differentials: 0
= 2c2 mdm - 2mv2dm
- 2vm2dv. Divide both sides by 2m: c2dm = v2dm + mvdv ………………..(vi) By definition, force is rate of change of momentum, so F = d(mv)
/ dt = m(dv/dt) + v(dm/dt). By definition, a force is also an energy field or gradient,
dE/ds, and velocity v = ds/dt, where s is distance. So dE = Fds = m(dv/dt)ds + v(dm/dt)ds = mvdv + v2dm. Compare this with equation (vi) above, to get: dE = c2dm, and so, integrating: E = mc2 (Einstein's Equation). The integration constant is zero, as E =0 when m = 0. Heisenberg's Uncertainty Principle Heisenberg's Uncertainty Principle takes on a new meaning: a
moving particle will actually spend most of its time at rest (punctuated by
very short times at c), but its experimentally observed velocity is measured
as v and so a measure of the uncertainty in its velocity at any instant will
be v - 0 = v. (This uncertainty depends on exactly when an observation is
made and so is in the mind or control of the observer and is not a property
of the particle.) From de Broglie's Equation,
mv is proportional to h/λ, and so the Uncertainty Principle becomes an
expression of de Broglie's Equation if λ is interpreted as the moving
particle's smallest jump length on the above diffusion model for motion. Spin An object in 3-D requires a rotation of 360° to return it to its original position,
but a bizarre 720° of rotation (not just
360°!) is required to bring fermions ("spin-½" particles,
such as protons) back to their original state. This is easily explained as follows, on the above model: For clarity, a 2-D / 3-D analogue will be used, instead of 3-D /
4-D. If a lower case letter "d" is lifted out of its 2-D
paper sheet and turned over in 3-D space and then put back as a "b",
then this would appear to a 2-D inhabitant to be a d to b vibration with only its antinodes (d
& b) being visible. If this d to
b vibration is analogous to Zero-Point Energy vibration, then if the
"d" is also spinning in 2-D (d to p to
d), then after 360° of spin in 2-D it
could have simultaneously rotated to a "b" by the 3-D
rotation, which means that the 360° spin in 2-D did not
return the "d" back to its initial state and that a further
360° of 2-D spin is needed by which time
the "b" would have rotated back to a "d" in
its simultaneous 3-D rotation. Thus a
bizarre (to a 2-D observer) 720° of spin is required
for a "d" spinning in 2-D space to return to its original
"d" state. With this preamble, for our case in 3-D space, an
observed (in 3-D) rotation of 720° is needed to return a proton to its original state, which can easily
be explained analogously to the example above. In 3-D to 4-D terms, this means that (to give an analogy) a
tennis ball spins in 3-D and 4-D simultaneously but after 360° of observed (in 3-D) rotation the ball
would be everted (i.e. having its fluffy side inside and smooth side outside,
without loss of the gas pressure which it contains) by the simultaneous 4-D
rotation and so clearly a further 360° of observed (in 3-D)
rotation would be needed for it to return (by further 4-D rotation) to its
original state with the fluffy side outside, making a total of 720°! This can only be understood in terms of the existence of 4-D
space and it happens routinely for elementary particles, which have access
to 4-D space. Note: A rotation in 3-D could only be perceived by a (hypothetical) 2-D observer
as a vibration (like Zero-Point Energy).
And a rotation in 4-D could only be
perceived by us (in our 3-D world) as a vibration (Zero Point Energy). Access of large objects to 4-D space is problematical. Eversion of tennis balls has been reported
anecdotally which is, of course, not scientifically acceptable, but there is
a report in Nature by Hasted et al (1975) of a refractory crystal of vanadium
carbide being removed from a sealed tube in laboratory conditions, without
any contact being made with the tube, which could only be feasible by
transfer out via 4-D space. References Besant A. and
Leadbeater C.W. (1994), http://www.4-D.org.uk Cottrell A.H., "Theoretical Structural Metallurgy",
Arnold (1960). Dirac P.A.M., "The Conditions for a Quantum Field Theory to
be Relativistic", Proc Royal Soc
(London) Series A 268, 57 (1962); see also Stedile E., "Quantum Aspects
of the fundamental Dirac Membrane Model", Int J Theor Phys 43,
385 (2004). Euler L. (18th Century). Feynman R.P., Leighton R.B. & Sands M.: "The Feynman Lectures on Physics", Volume 3, Addison-Wesley (1966). Hasted J.B., Bohm D.J., Bastin E.W., O'Regan P. and Taylor J.G.,
"Recent research at Birkbeck College, University of London", Nature
254, 470 (1975). Margenau H. & G.M. Murphy G.M.: "The Mathematics of
Physics & Chemistry", van Nostrand (1961). Moelwyn-Hughes E.A.,
"Physical Chemistry" (chapter "Mathematical Formulation of the
Quantum Theory"), Pergamon Press (1961). Moore W.J., "Physical Chemistry", Longmans (1962). Moussa S. & Hocking M.G., "Photo-inhibition of
localised corrosion of stainless steel in NaCl solution", Corrosion
Science 43, 2037 (2001). Schroedinger E.: "Quantisation as a problem of
Eigenvalues", Annalen der Physik 79, 372 (1926); and Schroedinger E.: "Quantisation as a
problem of Eigenvalues", Annalen der Physik 81, 135 (1926). [Quoted by M. Jammer in "Conceptual
Development of Quantum Mechanics", p. 372, 267, McGraw-Hill (1966).] CLICK HERE TO RETURN TO
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