Rideout and Wallden (2009, 5) prefer to call this “naive spatial distance” because even also this construction is ultimately not satisfactory: x, y are always in the interval \([w_i, z_i]:= \{ u \in C | w_i \prec_C u \prec_C z_i\}\). Consequently, there is usually ...
The introduction of spacetime and its invariant interval has practical and predictive value, as seen in examples such as GPS and satellite communications. The invariance of spacetime intervals has been supported by empirical evidence, similar to other experiments in physics. While not all aspects...
(or electric) part of the curvature tensor implies that there exist > 0 such that no conjugate point along γ , in the parameter interval [t0, t2 + ε), can occur to σt0 . This, with reference to the claims in the proof of Proposition 3.1 of [40], implies that to any point q...
a, Coloured points denote the radial dependence of the azimuthal velocityvθfor six vortex configurations distinguished by the drive (propeller) frequency. Each point is obtained by averaging over a 2.5-mm radial interval. Radial velocity component is approximately zero across all instances. Best fits...
More precisely, M is equipped with a (pseudo-)distance function d (x, y) interpreted as the 4-interval between the two points x and y: a negative d2(x, y) gives the time measured by a clock in inertial motion between x and y; a positive d2(x, y) gives the proper length of ...
in other words, the fractal curve is homeomorphic to the unit interval. As such it's a manifold! Aug 29, 2005 #20 mccrone 100 0 "what if spacetime looked like ordinary familiar 4D spacetime at our scale, and the scale of things like atoms and quarks and stuff, but what if it ...
Now, it might be asked, if the morals we draw all flow from examples of non-unitarily implementable dynamics, why resort to QFT on curved spacetime (or, as in one of the examples we present below, curved time slices of flat spacetime)? Does not Haag's theorem show that such examples ...
culations of the self-force in actual examples have been restricted to very few simple cases, such as static charges in the Schwarzschild and Reissner-Nordstr¨om geometries [7,5]. A general formal framework for obtaining equa- tions of motion for a test particle in a curved spacetime ...
Now, if the interval of time that S embraces is large enough so as to explore the minimum frequency in the spectrum (now we can really call ωn[t˜] a frequency), then it is clear that we can talk about the modes (17) as forming an orthonormal basis of modes with well-defined ...
We assume γ is given in a proper time parameterization as a smooth function γ : I → R, where I is a possibly unbounded open interval of R and denote the four-velocity of the curve by u = γ˙ . The curve forms a small sampling domain, with the proper time parameterization as a...