Advances in Imaging and Electron Physics (eBook)

−ρ2dϑdθ2−1/211−ρ2dϑ/dθ2d2ϑdθ2+1pqdϑdθ+1q2sinϑ=0θ=tτ,ρ=Rcτ,q=Rτ2m0mmoB=τpτ,p=2m0mmoβξm=τmpτppq=τmpτ=2Rm0τξm,τp=R2m0mmoB,τmp=2m0Rξm

For ρ2(dϑ/dθ)2 → 0 one obtains from Eq.(34) the nonrelativistic limit of Eq.(29).

The initial conditions of Eq.(34) are ϑ(θ) = nϑ0 = ϑn and dϑ(0)/dθ = 0 just as for Eq.(29). The velocities υ(θ) and υy(θ) of Eqs.(30) and (31) become:

θc=−Rcτdϑθ=−ρdϑdθ

yθc=−ρdϑdθsinϑ

Plots of υy(θ)/c are shown in Fig.1.4-11 for p = 1/4, q = 1 or τ = τp, ρ = 4, and various values of ϑ(0) = ϑn = nπ/8. The relativistic limitation υ/c < 1 is not conspicuous but a comparison with Fig. 1.4-9 shows how the peaks of the plots for n = 5, 6, 7 have been flattened.

1.5 INFINITESIMAL AND FINITE DIFFERENCES FOR SPACE AND TIME

The discussion of finite or infinite divisibility of space and time has been going on for some 2500 years. Zeno of Elea (c.490 – c. 430 B.C.) advanced the paradoxes of the race between Achilles and the turtle or the arrow which cannot fly that were supposed to show that infinite divisibility of space and time was not possible. A quote from Aristotle (384 – 322 B.C.) shows that infinite divisibility and thus the concept of the continuum was a matter of discussion before he wrote his Physica:

Aristotle argued the concept of the continuum for space, time and motion so convincingly1 that it does not seem to have been challenged until Max Planck introduced quantum mechanics. Newton (1971) took this concept apparently so much for granted that he did not even mention it, even though he was very meticulous in listing and elaborating his assumptions. The differential calculus of Leibnitz and Newton made us carry the concept of infinite divisible one step further since we distinguish now between dividing a finite interval ΔX into denumerable or non-denumerable many intervals.

A widely held view of a space-continuum is summed up in a quote by Weyl that emphasizes that this concept came from mathematics:

The following two quotes give a good summary of our currently accepted thinking about space and time:

We avoid all questions of how spatial and time distances can be divided indefinitely since we never find a hint how such a division can be carried out experimentally or can be observed. Instead we replace the differentials dx, dt by arbitrarily small but finite differences Δx, Δt. They can always be equal or even smaller than the shortest observable distance. The difference between a finite distance of 10 −100 m and an infinitesimal distance dx is not directly observable. The question arises whether such small values of Δx would not have to yield the same results as differentials dx. This question was answered by Hölder (1887) who showed that difference equations and differential equations define different classes of functions. In particular the Gamma function satisfies the algebraic difference equation

X+1=XΓX,X=x/Δx

but no algebraic differential equation.

The example of the Gamma function shows that difference equations can define continuous and differentiable functions. The use of finite differences Δx, Δt does not imply that only discrete functions defined at integer multiples of Δx and Δt can be obtained as solution of a partial difference equation.

Although this result may be of limited interest to the practical physicist it contributes to the philosophy of science. Our inability to make observations at x and x + dx or t and t + dt prevents any proof that there is a physical space-time continuum, but it also prevents a proof that there is NO physical space-time continuum. The question whether there is or is not a space-time continuum is no more answerable within the confines of a science based on observation than the question how many angels can dance on the point of a needle. We can use mathematics as a tool in physics, but we cannot use it as a source of concepts that are beyond observation.

Let us take one more step in the direction of philosophy of science and quote from Einstein and Infeld (1938, p. 311):

When we use differential calculus and then permit this mathematical method to define physical space and space-time we make physics a special branch of mathematics. Since mathematics is a science of the thinkable while physics is a science of the observable, mathematics can never be more than a tool in physics or provide inspiration. The succesful solution of a physical problem by means of differential calculus only implies that the assumption of a mathematical continuum can yield results that correspond with physical observation, it does not imply the existence of a physical space-time continuum.

Let us see how this principle works for finite differences. We know from observation that we can only resolve finite space and time differences x and x + Δx or t and t + Δt. If we look what mathematical tools are available that satisfy this requirement we find the calculus of finite differences. This calculus does not define any physical concept of space or space-time. It works for continuous functions, like the Gamma function, but does not suggest any particular topology of space or space-time. We could go one step further and require that x and t are integer multiples of Δx and Δt: x = nΔx, t = mΔt. If we did this we would introduce a cellular space or space-time into physics; we would repeat the mistake we made with differential calculus and the space-time continuum. There is no physical reason to do so.

The use of the calculus of finite differences reduces the concepts of space and space-time to coordinate systems and moving coordinate systems, which are obviously human constructs in line with the quote of Einstein and Infeld above. This is discussed in some detail in a book by...