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MP: Theoretische und Mathematische Grundlagen der Physik
MP 10: Poster
MP 10.9: Poster
Tuesday, March 8, 2005, 13:00–16:00, Poster TU B
Lattice Structure and Relativity — •helmut günther — FH Bielefeld FB Elektrotechnik, W.-Bertelsmann-Str.10, 33603 Bielefeld
We discuss a lattice model of W. Thirring’s transformation
x′ = (x − vt)/γ , t′= γ t, γ = √1 − v2/c2,
for relativistic spacetime with absolute simultaneity. Here the unprimed particular inertial frame Σo(x, t) is selected arbitrarily.
We consider an unbounded lattice representing Thirring’s particular frame. An infinite straight dislocation line represents the x-direction of a physical vacuum. The displacements q = q(x, t) perpendicular to that dislocation line satisfy the sine-Gordon equation
∂2q/∂ x2 −
1/c2∂2q/∂ t2 = sinq. The
soliton solutions of this equation represent the particles in that
lattice-vacuum.
We find length contraction lv/lo = γ and time dilatation
To/Tv = γ with respect to the lattice-vacuum by comparing
a static kink solution with a uniformly moving one as well as a
static breather solution with a uniformly moving one.
In a next step we identify the primed frame Σ′(x′, t′) with an observer which is at rest with respect to a moving kink or breather particle. By synchronising the clocks in Σ′ according to t′ = 0 for t = 0 in Σo we arrive at Thirring’s transformation. However,
synchronising the clocks according to t′ = x To/Tv − lo/lv/v in Σ′ for t = 0 in Σo leads to Lorentz transformation, and hence the lattice no longer represents a particular frame.