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A new method involving the effective wave function is used to define the mass of a particle in a standard five-dimensional extension of general relativity. The mass is inversely proportional to the magnitude of the scalar field of the extra dimension. Since the scalar field is global and depends on the coordinates, this definition for the particle mass agrees with Machs Principle., Comment: Paul S. Wesson passed away on September 16, 2015. This paper, his 300th research paper, was submitted for publication before his death, and is posted here for posterity. A detailed obituary will be posted on the STM website in the near future

We give a new, wave-like solution of the field equations of five-dimensional relativity. In ordinary three-dimensional space, the waves resemble de Broglie or matter waves, whose puzzling behaviour can be better understood in terms of one or more extra dimensions. Causality is appropriately defined by a null higher-dimensional interval. It may be possible to test the properties of these waves in the laboratory., Comment: 15 pages, no figures

Recent criticism of higher-dimensional extensions of Einstein's theory is considered. This may have some justification as regards string theory, but is misguided as applied to five-dimensional theories with a large extra dimension. Such theories smoothly embed general relativity, ensuring recovery of the latter's observational support. When the embedding of spacetime is carried out in accordance with Campbell's theorem, the resulting 5D theory naturally explains the origin of classical matter and vacuum energy. Also, constraints on the equations of motion near a high-energy surface or membrane in the 5D manifold lead to quantization and quantum uncertainty. These are major returns on the modest investment of one extra dimension. Instead of fruitless bickering about whether it is possible to "see" the fifth dimension, it is suggested that it be treated on par with other concepts of physics, such as time. The main criterion for the acceptance of a fifth dimension (or not) should be its usefulness.

The cosmological constant (lambda) of general relativity is a natural consequence of embedding Einstein's theory in a five-dimensional theory of the type needed for unification. The exact 5D solution for lambda less than 0 shows waves in ordinary 3D space with properties similar to those of de Broglie or matter waves. Here the dispersion relation is derived for matter waves in a toy two-phase model, where regions with lambda less than 0 and lambda greater than 0 average on the large scale to lambda = 0, thus providing in principle a resolution of the cosmological-constant problem. A striking result of the analysis is that the dispersion relation is bimodal, with a well-defined window of high-frequency transmission which effectively defines the speed of light.

When the cosmological "constant" is derived from modern five-dimensional relativity, exact solutions imply that for small systems it scales in proportion to the square of the mass. However, a duality transformation implies that for large systems it scales as the inverse square of the mass.

In cosmologies with more than four dimensions, of the type required for unification, it is possible for signals to have velocities in excess of that of light. Using a five-dimensional model which otherwise agrees with observations, two subjects are reviewed: (a) An exact solution of the field equations which describes a 4D spacetime with a large cosmological constant and waves travelling in ordinary 3D space with velocities exceeding lightspeed. (b) An example where the 4D interval or proper time is modulated by the systematic variation of the scalar field associated with the fifth dimension, providing a simple signalling method. These and related consequences of higher-dimensional cosmology have significant implications for astrophysics, and especially the Search for Extraterrestrial Intelligence.

To better understand the scalar field typical of higher-dimensional extensions of general relativity, we analyse three classes of solutions. In all, the field equation for the extra dimension resembles the Klein-Gordon equation, and we evaluate the strength of the source. Our results show that the scalar field is coupled to matter, and may be regarded as generating it.

In unified field theories with more than four dimensions, the form of the equations of physics in spacetime depends in general on the choice of coordinates in higher dimensions. The reason is that the group of coordinate transformations in (say) five dimensions is broader than in spacetime. This kind of gauge dependence is illustrated by two examples: a cosmological model in general relativity and a matter wave in quantum theory. Surprisingly, both are equivalent by coordinate transformations to flat featureless five-dimensional space. This kind of transformity is of fundamental significance for the philosophy of physics.

Replacing 4D Minkowski space by 5D canonical space leads to a clearer derivation of the main features of wave mechanics, including the wave function and the velocity of de Broglie waves. Recent tests of wave-particle duality could be adapted to investigate whether de Broglie waves are basically 4D or 5D in nature.

Astronomy is a precise and relatively simple science because objects accelerate in a gravitational field at the same rate, irrespective of their composition. Galileo knew this, and Einstein took it as the basis for general relativity. Surprisingly, it is also a consequence of new theories that use a fifth dimension.

As an example of the unification of gravitation and particle physics, an exact solution of the five-dimensional field equations is studied which describes waves in the classical Einstein vacuum. While the solution is essentially 5D in nature, the waves exist in ordinary 3D space, and may provide a way to test for an extra dimension.

Five-dimensional relativity as an extension of general relativity has field equations that simplify considerably given the adoption of a new gauge. The result is a scalar field governed by the Klein-Gordon equation, in an empty spacetime with a cosmological constant governed by Einstein's equations. The main application is to the properties of massive particles in a vacuum-dominated universe., Comment: Changes from v.1: New last paragraph, Appendix

I give a compact, pedagogical review of our present understanding of the universe as based on general relativity. This includes the uniform models, with special reference to the cosmological 'constant'; and the equations for spherically-symmetric systems, in a particularly convenient form that aids their application to astrophysics. New ideas in research are also outlined, notably involving extra dimensions.

Since the 5D canonical metric embeds all 4D vacuum solutions of Einstein's equations, I review its application to the cosmological 'constant', quantized particles, deBroglie waves, scalar fields and wave-particle duality. There are several ways to ra-tionalize these things using an extra dimension. A possible explanation of wave-particle duality is that an observed particle manifests two isometries of flat 5D space in different 4D ways, one with waves and one without.

I give metrics and equations of motion in 5D general relativity, and use the conservation of momentum and conformal transformations to study the possible variability of particle masses and the cosmological 'constant'. It is feasible that all particles are photon-like and travel on null paths in 5D, that massive particles are perturbations of the extra dimension which intrude into 4D, and that the cosmological 'constant' is a 5D / 4D gauge term. In the simplest case, a particle has a local cosmological 'constant' whose magnitude is proportional to the square of the mass.

The basic quasi-Schwarzschild 5D objects known as solitons have a long history, which is reviewed. Then some material is added, leading to the inference that a soliton is a singularity in the geometry which represents a bivalent source of gravitational and scalar mass.

I outline a model where a massive particle in 4D spacetime follows a null (photon-like) path in 5D canonical (super-spherically-symmetric) space. This leads to wave-particle duality and quantization, along with other effects which show that it is possible to unify general relativity and wave mechanics in a simple fashion, given an extra dimension., Comment: Version 2

In 5D, I take the metric in canonical form and define causality by null-paths. Then spacetime is modulated by a factor equivalent to the wave function, and the 5D geodesic equation gives the 4D Klein-Gordon equation. These results effectively show how general relativity and quantum mechanics may be unified in 5D.

Campbell's theorem enables the embedding of 4D anti-deSitter space in 5D canonical space, so a particle becomes a wave in the extra dimension, running through spacetime. This model of wave-particle duality provides a new approach to particle mass.

When the proper time of general relativity is adjusted to reflect the possible existence of higher dimensions, small dynamical effects appear in spacetime of the sort usually associated with the cosmological constant, Hubble's Law and Heisenberg's relation.

Einstein theory can be embedded in Kaluza-Klein theory, and in particular all 4D vacuum solutions can be embedded in 5D (pure) canonical space where spacetime is independent of the extra coordinate. The uniqueness of 5D canonical space is quickly reproven, and its history and implications briefly reviewed. For 4D anti-deSitter space, the 5D space involves a wave running around spacetime, with an associated particle that has a local value of the cosmological 'constant' proportional to the square of the mass.

Astronomically, there are viable mechanisms for distributing organic material throughout the Milky Way. Biologically, the destructive effects of ultraviolet light and cosmic rays means that the majority of organisms arrive broken and dead on a new world. The likelihood of conventional forms of panspermia must therefore be considered low. However, the information content of dam-aged biological molecules might serve to seed new life (necropanspermia)., Comment: Accepted for publication in Space Science Reviews

When four-dimensional general relativity is embedded in an unconstrained man-ner in a fifth dimension, the physical quantities of spacetime can be interpreted as geometrical properties related to the extra dimension. It has become widespread to view the ten Einstein equations and the source terms of the energy-momentum tensor in this way. We now assign physical meanings to the other five equations involved. The scalar field acts like gravity, but concerns inertial as opposed to gravitational mass. The other four equations are conservation laws for 4D dynamics, but where the mass of a test particle is related to a local value of the cosmological 'constant'. Ways of testing these identifications are suggested.

The main results are reviewed of relating the scalar field of noncompactified Kaluza-Klein gravity to the Higgs field of particle physics. The embedding of 4D spacetime in a 5D manifold can result in a variable cosmological 'constant' and particle masses tiny compared to the Planck value.

The group of coordinate transformations for 5D noncompact Kaluza-Klein theory is broader than the 4D group for Einstein's general relativity. Therefore, a 4D quantity can take on different forms depending on the choice for the 5D coordinates. We illustrate this by deriving the physical consequences for several forms of the canonical metric, where the fifth coordinate is altered by a translation, an inversion and a change from spacelike to timelike. These cause, respectively, the 4D cosmological 'constant' to become dependent on the fifth coordinate, the rest mass of a test particle to become measured by its Compton wavelength, and the dynamics to become wave-mechanical with a small mass quantum. These consequendes of 5D covariance -- whether viewed as positive or negative -- help to determine the viability of current attempts to unify gravity with the interactions of particles.

We review the idea, due to Einstein, Eddington, Hoyle and Ballard, that time is a subjective label, whose primary purpose is to order events, perhaps in a higher-dimensional universe. In this approach, all moments in time exist simultaneously, but they are ordered to create the illusion of an unfolding experience by some physical mechanism. This, in the language of relativity, may be connected to a hypersurface in a world that extends beyond spacetime. Death in such a scenario may be merely a phase change.

All objects in 4D spacetime may in principle travel on null paths in a 5D mani-fold. We use this, together with a change in the extra coordinate and the signature of the metric, to construct a simple model of a classical universe and a quantized particle. There are notable implications for the Weak Equivalence Principle and the cosmological constant, and for astrophysics.

Massive particles on timelike paths in spacetime can be viewed as moving on null paths in a higher-dimensional manifold. This and other consequences follow from the use of Campbell's theorem to embed 4D general relativity in non-compactified 5D Kaluza-Klein theory. We now show that it is possible in principle to obtain the standard rule for quantization in 4D from the canonical metric with null paths in 5D. Particle mass can be wavelike, as suggested originally by Dirac, and other 4D/5D consequences are outlined.

Mach's Principle is usually taken to mean that the mass of a particle as measured locally is determined in some way by the other matter in the universe. This is difficult to formalize in 4D,but is feasible in 5D if the scalar potential of non-compactified Kaluza-Klein theory is interpreted as an inertial field. We therefore review 5D space-time-matter theory, but take the local particle mass to be defined by the integral of a global scalar field. This approach smoothly embeds general relativity, and leads to several new effects which can be tested.

We review the current status of dimensions, as the result of a long and controversial history that includes input from philosophy and physics. Our conclusion is that they are subjective but essential concepts which provide a kind of book-keeping device, their number increasing as required by advances in physics. The world almost certainly has more than the four dimensions of space and time, but the introduction of the fifth and higher dimensions requires a careful approach wherein known results are embedded and new ones are couched in the most productive manner.

By using a massless scalar field we examine the effect of an extra dimension on black hole radiation. Because the equations are coupled, we find that the structure of the fifth dimension (as for membrane and induced-matter theory) affects the nature of the radiation observed in four-dimensional spacetime. In the case of the Schwarzschild-de Sitter solution embedded in a Randall-Sundrum brane model, the extension of the black hole along the fifth dimension looks like a black string. Then it is shown that, on the brane, the potential barrier surrounding the black hole has a quantized as well as a continuous spectrum. In principle, Hawking radiation may thus provide a probe for higher dimensions., Comment: 9 pages, 3 figures, accepted for publication on MPLA

Higher-dimensional theories of the kind which may unify gravitation with particle physics can lead to significant modifications of general relativity. In five dimensions, the vacuum becomes non-standard, and the Weak Equivalence Principle becomes a geometrical symmetry which can be broken, perhaps at a level detectable by new tests in space.

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.

The Campbell-Magaard theorem is widely seen as a way of embedding Einstein's 4D theory of general relativity in a 5D theory of the Kaluza-Klein type. We give a brief history of theorem, present a short account of it, and show that it provides a geometrical frame for new physics related to the unification of the forces. Anderson's recent vituperative attack on Campbell's theorem (gr-qc/0409122)is errant.

We present an exact solution to the Einstein field equations which is Ricci and Riemann flat in five dimensions, but in four dimensions is a good model for the early vacuum-dominated universe., Comment: 6 pages; to appear in Journal of Mathematical Physics; v2: reference 3 corrected

Following fresh attempts to resolve the problem of the energy density of the vacuum, we reconsider the case where the cosmological constant is derived from a higher-dimensional version of general relativity, and interpret the gauge-dependence of $\Lambda $ as a dynamical effect. This leads to a relation between the change in $\Lambda $ and the line element (action) which is independent of gauge choices and fundamental constants: $d\Lambda ds^{2}=-3$. This implies that the (classical) vacuum is unstable, with implications for particle production.

When the cosmological constant of spacetime is derived from the 5D induced-matter theory of gravity, we show that a simple gauge transformation changes it to a variable measure of the vacuum which is infinite at the big bang and decays to an astrophysically-acceptable value at late epochs. We outline implications of this for cosmology and galaxy formation., Comment: 14 pages, no figures, expanded version to be published in Class. Quantum Grav

We review the implications of modern higher-dimensional theories of gravity for astrophysics and cosmology. In particular, we discuss the latest developments of space-time-matter theory in connection with dark matter, particle dynamics and the cosmological constant, as well as related aspects of quantum theory. There are also more immediate tests of extra dimensions, notably involving perturbations of the cosmic 3K microwave background and the precession of a supercooled gyroscope in Earth orbit. We also outline some general features of embeddings, and include pictures of the big bang as viewed from a higher dimension., Comment: 23 pages, 2 figures; to appear in Space Science Reviews; v3: typos corrected and minor changes to text, expanded derivation of fundamental mode added

We study a five-dimensional cosmological model, which suggests that the universe bagan as a discontinuity in a (Higgs-type) scalar field, or alternatively as a conventional four-dimensional phase transition., Comment: 10 pages, 2 figures; typo corrected in equation (18); 1 reference added; version to appear in International Journal of Modern Physics A

Heisenberg's uncertainty relation is commonly regarded as defining a level of unpredictability that is fundamentally incompatible with the deterministic laws embodied in classical field theories such as Einstein's general relativity. We here show that this is not necessarily the case. Using 5D as an example of dimensionally-extended relativity, we employ a novel metric to derive the standard quantum rule for the action and a form of Heisenberg's relation that applies to real and virtual particles. The philosophical implications of these technical results are somewhat profound., Comment: Accepted, J. Gen. Rel. and Grav., February 2004

The cosmological constant combined with Planck's constant and the speed of light implies a quantum of mass of approximately 2 x 10^{-65}g. This follows either from a generic dimensional analysis, or from a specific analysis where the cosmological constant appears in 4D spacetime as the result of a dimensional reduction from higher dimensional relativity (such as 5D induced-matter and membrane theory). In the latter type of theory, all the particles in the universe can be in higher-dimensional contact.

We clarify the status of two known solutions to the 5-dimensional vacuum Einstein field equations derived by Liu, Mashhoon & Wesson (LMW) and Fukui, Seahra & Wesson (FSW), respectively. Both 5-metrics explicitly embed 4-dimensional Friedman-Lemaitre-Robertson-Walker cosmologies with a wide range of characteristics. We show that both metrics are also equivalent to 5-dimensional topological black hole (TBH) solutions, which is demonstrated by finding explicit coordinate transformations from the TBH to LMW and FSW line elements. We argue that the equivalence is a direct consequence of Birkhoff's theorem generalized to 5 dimensions. Finally, for a special choice of parameters we plot constant coordinate surfaces of the LMW patch in a Penrose-Carter diagram. This shows that the LMW coordinates are regular across the black and/or white hole horizons., Comment: Reference added, 20 pages, 3 figures, REVTeX4

It is shown that the extra coordinate of 5D induced-matter and membrane theory is related in certain gauges to the inertial rest mass of a test particle. This implies that the Weak Equivalence Principle is a geometric symmetry, valid only in the limit in which the test mass is negligible compared to the source mass. Exact solutions illustrate this, and show the way to possible resolutions of the cosmological-constant and hierarchy problems.

Stated succinctly, the original version of the Campbell-Magaard theorem says that it is always possible to locally embed any solution of 4-dimensional general relativity in a 5-dimensional Ricci-flat manifold. We discuss the proof of this theorem (and its variants) in n dimensions, and its application to current theories that postulate that our universe is a 4-dimensional hypersurface Sigma_0 within a 5-dimensional manifold, such as Space-Time-Matter (STM) theory and the Randall & Sundrum (RS) braneworld scenario. In particular, we determine whether or not arbitrary spacetimes may be embedded in such theories, and demonstrate how these seemingly disparate models are interconnected. Special attention is given to the motion of test observers in 5 dimensions, and the circumstances under which they are confined to Sigma_0. For each 5-dimensional scenario considered, the requirement that observers be confined to the embedded spacetime places restrictions on the 4-geometry. For example, we find that observers in the thin braneworld scenario can be localized around the brane if its total stress-energy tensor obeys the 5-dimensional strong energy condition. As a concrete example of some of our technical results, we discuss a Z_2 symmetric embedding of the standard radiation-dominated cosmology in a 5-dimensional vacuum., Comment: Typos corrected, 13 pages, no figures, two column format

It is possible that null paths in 5D appear as the timelike paths of massive particles in 4D, where there is an oscillation in the fifth dimension around the hypersurface we call spacetime. A particle in 5D may be regarded as multiply imaged in 4D, and the 4D weak equivalence principle may be regarded as a symmetry of the 5D metric., Comment: 15 pages, in press in Phys. Lett. B

A null path in 5D can appear as a timelike path in 4D, and for a certain gauge in 5D the motion of a massive particle in 4D obeys the usual quantization rule with an uncertainty-type relation. Generalizations of this result are discussed in regard to induced-matter and membrane theory., Comment: 26 pages, in press in Class. Quant. Grav

We give relations for the embedding of spatially-flat Friedmann-Robertson-Walker cosmological models of Einstein's theory in flat manifolds of the type used in Kaluza-Klein theory. We present embedding diagrams that depict different 4D universes as hypersurfaces in a higher dimensional flat manifold. The morphology of the hypersurfaces is found to depend on the equation of state of the matter. The hypersurfaces possess a line-like curvature singularity infinitesimally close to the $t = 0^+$ 3-surface, where $t$ is the time expired since the big bang. The family of timelike comoving geodesics on any given hypersurface is found to have a caustic on the singular line, which we conclude is the 5D position of the point-like big bang., Comment: 11 pages, 5 figures, revtex4, accepted in Class. Quant. Grav

We present a rich class of exact solutions which contains radiation-dominated and matter-dominated models for the early and late universe. They include a variable cosmological ``constant'' which is derived from a higher dimension and manifests itself in spacetime as an energy density for the vacuum. This is in agreement with observational data and is compatible with extensions of general relativity to string and membrane theory. Our solutions are also typified by a non-singular ``big bounce'' (as opposed to a singular big bang), where matter is created as in inflationary cosmology., Comment: 17 pages, 2 figures, AASTEX. To appear in ApJ

An exact class of solutions of the 5D vacuum Einstein field equations (EFEs) is obtained. The metric coefficients are found to be non-separable functions of time and the extra coordinate $l$ and the induced metric on $l$ = constant hypersurfaces has the form of a Friedmann-Robertson-Walker cosmology. The 5D manifold and 3D and 4D submanifolds are in general curved, which distinguishes this solution from previous ones in the literature. The singularity structure of the manifold is explored: some models in the class do not exhibit a big bang, while other exhibit a big bang and a big crunch. For the models with an initial singularity, the equation of state of the induced matter evolves from radiation like at early epochs to Milne-like at late times and the big bang manifests itself as a singular hypersurface in 5D. The projection of comoving 5D null geodesics onto the 4D submanifold is shown to be compatible with standard 4D comoving trajectories, while the expansion of 5D null congruences is shown to be in line with conventional notions of the Hubble expansion., Comment: 8 pages, in press in J. Math. Phys

Technical results are presented on motion in N(>4)D manifolds to clarify the physics of Kaluza-Klein theory, brane theory and string theory. The so-called canonical or warp metric in 5D effectively converts the manifold from a coordinate space to a momentum space, resulting in a new force (per unit mass) parallel to the 4D velocity. The form of this extra force is actually independent of the form of the metric, but for an unbound particle is tiny because it is set by the energy density of the vacuum or cosmological constant. It can be related to a small change in the rest mass of a particle, and can be evaluated in two convenient gauges relevant to gravitational and quantum systems. In the quantum gauge, the extra force leads to Heisenberg's relation between increments in the position and momenta. If the 4D action is quantized then so is the higher-dimensional part, implying that particle mass is quantized, though only at a level of 10^{-65} gm or less which is unobservably small. It is noted that massive particles which move on timeline paths in 4D can move on null paths in 5D. This agrees with the view from inflationary quantum field theory, that particles acquire mass dynamically in 4D but are intrinsically massless. A general prescription for dynamics is outlined, wherein particles move on null paths in an N(>4)D manifold which may be flat, but have masses set by an embedded 4D manifold which is curved.