General Relativity and Gravitation

, Volume 38, Issue 5, pp 937–944 | Cite as

Wave mechanics and general relativity: a rapprochement

Letter

Abstract

Using exact solutions, we show that it is in principle possible to regard waves and particles as representations of the same underlying geometry, thereby resolving the problem of wave-particle duality.

Keywords

Wave-particle duality 

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    There are several useful coordinate transformations which relate flat 5D manifolds and curved 4D ones. For example, the metrics \(dS^{2}=dT^{2}-d\sigma ^{2}-dL^{2}\) and \(dS^{2}=l^{2}dt^{2}-d\sigma ^{2}-t^{2}dl^{2}\) are related by the transformation \(T=t^{2}l^{2}/ 4+\ln ( t^{1/ 2}l^{-1/ 2})\), \(L=t^{2}l^{2}/4-\ln ( t^{1/ 2}l^{-1/ 2})\). The “standard” 5D cosmologies of Ponce de Leon [10] have metrics of the second-noted form, and may by coordinate transformations be shown to be 5D flat. The full transformations, including those for the spatial part, are given elsewhere (ref. 5, p. 49). The Billyard wave [14] may similarly be shown to be a coordinate-transformed version of de Sitter space, and flat in 5D. A generic discussion of cosmological models which are flat in 5D is due to McManus [12]. One of his solutions is a metric for a particle in a manifold whose 3D part is curved, which effectively generalizes the Billyard wave whose 3D part is flat. [See ref. 12, p. 4895, equation (30).] This can be seen by changing the 4D coordinates as discussed in the main text, which results in \(dS^{2}=( l/ L) ^{2}dt^{2}-( l/L) ^{2}( e^{it/ L}+ke^{-it/ L}) ^{2}[ \exp ( 2i\kappa _{x}x) dx^{2}+\text{etc}] +dl^{2}\). When the curvature constant k is zero, this gives back the Billyard wave. Another of the McManus solutions reproduces work by Davidson et al. [See ref. 11; and also ref. 12, p. 4893, equation (19).] This is effectively a 5D embedding of the 4D Milne model, and can be written as \(dS^{2}=dt^{2}-t^{2}d \sigma ^{2}-dl^{2}\), where \(d\sigma ^{2}\equiv ( 1+kr^{2}/4) ^{-2}\) with \(k=-1\). The transformation \(t\rightarrow l\,$sinh$( t/ L)\), \(l\rightarrow l\ $cosh$( t/ L)\) causes the metric to read \(dS^{2}= ( l/ L)^{2}dt^{2}-[l\text{ sinh} ( t/ L) ] ^{2}d\sigma ^{2}-dl^{2}\). This is quoted as (3) of the main text, and its local approximation is (4). The former has proper distances which vary as sinh$t$, whereas the latter has proper distances which vary as t. The former is typical of motion in flat 5D space, when the 4D proper time $s$ (as opposed to the 5D proper time S) is used as parameter [5, p. 169]. The latter is typical of motion in flat 4D space, when the ordinary time t is used as parameter [4, p. 205]. Both of the models used in the main text to illustrate the passage from particle to wave use metrics which are canonical in form, and there is a large literature on these. However, a more general class of metrics is given by \(dS^{2}=g_{\alpha \beta } ( x^{\gamma },l ) dx^{\alpha }dx^{\beta }+\epsilon \Phi ^{2} ( x^{\gamma },l ) dl^{2}\), where \(\epsilon =\pm 1\) and \(\Phi\) is a scalar field. Einstein's 4D equations are satisfied for this 5D metric if the effective or induced energy-momentum tensor is given by \( \displaylines{ 8\pi T_{\alpha \beta } =\frac{\Phi _{,\alpha ;\beta }}{\Phi }-\frac{\epsilon }{2\Phi ^{2}}\left\{ \frac{\Phi ,_{4}g_{\alpha \beta ,4}}{\Phi }-g_{\alpha \beta ,44}+g^{\lambda \mu }g_{\alpha \lambda ,4}g_{\beta \mu ,4}\right. \cr \left. -\frac{g^{\mu \nu }g_{\mu \nu ,4}g_{\alpha \beta ,4}}{2}+\frac{g_{\alpha \beta }}{4}\left[ g_{\;, 4}^{\mu \nu }g_{\mu \nu ,4}+ (g^{\mu \nu }g_{\mu \nu , 4})^{2}\right] \right\} }\). Here a comma denotes the partial derivative and a semicolon denotes the 4D covariant derivative. We have not discussed the matter which relates to the exact solutions (3), (10) of the main text because it is merely vacuum [5, 14]. But the matter properties of these and more complicated solutions may be evaluated for any choice of coordinates by using the noted expression.Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of WaterlooWaterlooCanada

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