The following paper was taken from [http://msu.edu/~micheals/roadmap.xml Gravitation and Elementary Particles], a publication of the [http://msu.edu/~micheals Faraday Group]. The article is posted on by the author - for Gravity, Recent alternative theories.
3. Temporal Curvature
A New View of Gravity – A Distributed Compression of Time
Salvatore G. Micheal, Faraday Group, firstname.lastname@example.org, 11/17/2007
Y0, the elasticity of space, is defined and calculated. Linear strain is calculated for electrons and protons. In the process, after a few assumptions, a new relation between temporal curvature and spatial curvature is established. Needed work is reviewed.
From the previous paper on frame-dragging, we invented a new relation between mass and the linear strain of space:
λ0 = Y0μ0ε0(Δl/l) (1)
mass per unit length (implicit) is linearly related to extension through the three parameters of space: elasticity, permeability, and permittivity
We had some trouble defining an appropriate Y0, the elasticity of space. Recall that the basic constraint on Y0 is that it must be consistent between elementary particles (and of course its units must agree with the equation above). Let's make a few standard assumptions which should not cause too much of a ruckus. Of course, those must be verified (or at least – not disproved) – as the consequences of those assumptions must also be verified. Until now, we have not made the 'per unit length' explicit. Let's do that and assign the Planck-length:
λ0/lP = Y0μ0ε0(Δl/l) (2)
This is a place to start and we'll follow a similar convention when the need arises. Let's replace lambda with the standard notation and move lP to the other side:
m0 = (Y0lP)μ0ε0(Δl/l) (3)
Multiply by unity (where tP is the Planck-time):
m0 = (Y0lPtP)μ0ε0(Δl/ltP) (4)
Now, the first factor on the RHS is 'where we want it' (units are in joule-seconds). And, the fact we had to 'contort' the extension by dividing it by the Planck-time should not prove insurmountable to deal with later. Finally, let's assume the first factor is equal to the magnitude of spin of electrons and protons, Ä§/2:
m0 = (Ä§/2)μ0ε0(Δl/ltP) (5)
By our last assumption, Y0 = Ä§/2lPtP ≈ 6.0526*1043 N. To simplify and isolate the extension:
m0 = (Ä§/2c2)(Δl/l)(1/tP) (6)
> (Δl/l) (2c2tP/Ä§)m0 = 2(tP/Ä§)E0 (7)
So, the linear strain of space due to internal stress is directly related to rest-energy through a Planck-measure. Later, if space allows (pun intended), we will show that (7) reduces to an even simpler form involving only two factors. If our assumptions hold, the numerical values for (7), for electrons and protons respectively, are approximately:
8.3700*10-23 and 1.5368*10-19.
The values are dimensionless – per the definition of linear strain. The meaning is: 'locally', space is expanded (linearly) by the fractions above (assumed in each dimension). What exactly locally means – will have to be addressed later. The numerical value of Y0 is extremely high as expected. All this says is: space is extremely inelastic. The numerical values for âˆ†l/l will have to be investigated – perhaps as suggested in the previous paper.
Let's deal with our assumptions first. The notions of Planck-time and Planck-length are associated with 'minimum measures' conventionally. Anything less is considered physically meaningless. If there is a fundamental limit on our precision in measuring things, we consider those to be lower bounds. If we could make a 'meter stick' with a length of the Planck-length or a clock that 'ticked' per Planck-time, that would be the limit of our technology – physically imposed by the nature of our Universe. So, to use them above is not a huge stretch of our 'belief system'. Our first assumption, to employ 'mass per Planck-length', is not implying we assume electron masses are actually divided into small parts of m0/lP. It simply means that's the limit of our measuring ability – and that we associate a linear change in space (for now) with that minimum measure.
Conventionally, we think of m0, E0, Ä§, c, and tP as fixed. If any of them varied, that would throw physics into chaos, right? But that is exactly what quantum mechanics has tried to cope with since inception: the seemingly statistical variation of m0/E0 about some modal value. Fortunately for science, Ä§ and c do not seem to vary statistically.
The fact we had to introduce tP above in order to simplify the expression for extension, is only the completion of another expression of uncertainty. That's the conventional view. Another perspective is to view that change in space per unit time. There are two further ways to view that: as the propagation of the gravity wave of a newly minted particle – or – as the locally changing extension over time. If we tentatively adopt the latter view, this provides a natural/integrated explanation of uncertainty. The only 'problem' is that the linear increase in extension cannot go on forever. It must necessarily oscillate. The simplest form of modeling that is with a saw-tooth wave (and slope ±âˆ†l/l). We could get a little 'fancier' and model with a sinusoid. The critical factors are: amplitude and wavelength. Amplitude is associated with the variation in rest-mass/energy. Wavelength is associated with the choice of period: Planck-time, de Broglie 'period', Compton-period, or relativistic-period? The first appears too small (and arbitrary), the second is not properly defined for particles at rest, the third does not account for relativistic effects, so we are left with the fourth. The fourth is based on the third but takes into account time-dilation.
For consistency with relativistic-mass, relativistic-energy is defined as:
E Ä§ω E0/γ (8)
where omega is the relativistic-angular-frequency and gamma = sqrt(1-(v/c)2). For consistency with time-dilation, relativistic-period must be lengthened:
T = T0/γ (9)
where T0 is the Compton-period of a particle at rest. Let's repeat equation seven here for convenience:
(Δl/l) (2c2tP/Ä§)m0 2(tP/Ä§)E0 (7)
If we notice that heavier particles have larger extensions (comparing protons and electrons), we can replace every variable above with its relativistic counterpart (let's also give the extension a new name, X):
X (2c2tP/Ä§)m 2(tP/Ä§)E (10)
But because of (8), (10) can be rewritten:
X 2tPω 4πtP/Tγ2 (11)
relativistic-extension is two times the Planck-time times relativistic-angular-frequency which is also equal to the ratio of Planck-time to relativistic-period through a solid angle! (gamma-squared is a scaling factor from the relation ν≡1/Tγ2.)
For particles at rest, (11) reduces to:
X0 = 4πtP/T0 (12)
extension is the ratio of Planck-time to period through a solid angle
You can't get much more intuitive and simpler than that!
One way to think of gravity is as curved space. Another way to think of gravity is as curved time (only). An object in a circular orbit (around Earth) is following a 'straight line' path (of least action) through curved space – or – is following a path of same temporal curvature. An object in free-fall is following a straight-line path to the maximum of spatial curvature – or – is following a path to the maximum of temporal curvature. Gravity can be analyzed exclusively as a distributed compression of time. (All trajectories can be treated as a linear combination of those two orthogonal trajectories. They are fundamentally different in terms of temporal curvature. All extended objects experience a gradient on different parts of their extension – it’s not just the ‘steepness of the hill’ which pulls them down. In the same way, time is infinitesimally slower on the ‘low side’ of an object in orbit. Objects move to maximize time-dilation.)
The analysis above has shown that, with a few assumptions, there’s an equivalence between spatial and temporal curvatures. So, another way of looking at particles is as:
charged twists of space
and localized compressions of time.
What 'local' means still needs to be defined (not in a tautological way) precisely. A preference needs to be established – in viewing curvature – such that characteristics of space-time (such as Maxwell's relations) are more easily exhibited. Those characteristics need to be derived from (1). The other theoretical tasks need to be performed (set in the previous paper). The two experiments from the previous paper need to be performed. If there is indeed a deterministic oscillation in mass/energy/extension, that needs to be experimentally verified. A small joke was forgotten to be placed in the previous paper: “Don't cross the beams .. Never cross the beams!” ;)