Determinations of the motion of the sun with respect to the extra-galactic nebulae have involved a K term of several hundred kilometers which appears to be variable. Explanations of this paradox have been sought in a correlation between apparent radial velocities and distances, but so far the results have not been convincing. The present paper is a re-examination of the question, based on only those nebular distances which are believed to be fairly reliable.

Distances of extra-galactic nebulae depend ultimately upon the application of absolute-luminosity criteria to involved stars whose types can be recognized. These include, among others, Cepheid variables, novae, and blue stars involved in emission nebulosity. Numerical values depend upon the zero point of the period-luminosity relation among Cepheids, the other criteria merely check the order of the distances. This method is restricted to the few nebulae which are well resolved by existing instruments. A study of these nebulae, together with those in which any stars at all can be recognized, indicates the probability of an approximately uniform upper limit to the absolute luminosity of stars, in the late-type spirals and irregular nebulae at least, of the order of M (photographic) = −6.3.1 The apparent luminosities of the brightest stars in such nebulae are thus criteria which, although rough and to be applied with caution, furnish reasonable estimates of the distances of all extra-galactic systems in which even a few stars can be detected.

Finally, the nebulae themselves appear to be of a definite order of absolute luminosity, exhibiting a range of four or five magnitudes about an average value M (visual) = −15.2.1 The application of this statistical average to individual cases can rarely be used to advantage, but where considerable numbers are involved, and especially in the various clusters of …

A Relation Between Distance and Radial Velocity Among Extra-Galactic Nebulae

A new scalar‐tensor theory of gravitation is formulated in a modified Riemannian manifold in which both the scalar and tensor fields have intrinsic geometrical significance. This is in contrast to the well‐known Brans‐Dicke theory where the tensor field alone is geometrized and the scalar field is alien to the geometry. The static spherically symmetric solution of the exterior field equations is worked out in detail.

A Scalar-Tensor Theory of Gravitation in a Modified Riemannian Manifold

A closed‐form exact solution to the field equations of a scalar‐tensor theory, formally similar to the Brans‐Dicke theory, is obtained. It is shown that the present theory predicts the same effects, within observational limits, as the Einstein theory.

Scalar‐Tensor Theory of Gravitation in a Lyra Manifold

We study a chiral operator algebra of conformal field theory and quantum deformation of the finite-dimensional Lie group to obtain the definition of (T
*
G)
 t
 and its representation. The closeness of the Kac-Moody algebras, constituting the chiral operator algebra of a typical (and generic) conformal field theory model, namely the WZNW model, and quantum deformation of corresponding finite-dimensional Lie groupG has become more and more evident in recent years [1–5]. This in particular prompts further investigation of the differential geometry of such deformations. The notion of tangent and cotangent bundles is basic in classical differential geometry. It is only natural that the quantum deformations ofTG andT
*
G are to be introduced alongside those forG itself. Physical ideas could be useful for this goal. Indeed, theT
*
G can be interpreted as a phase space for a kind of a top, generalizing the usual top associated withG=SO(3). The classical mechanics is a natural language to describe differential geometry, whereas the usual quantization is nothing but the representation theory. In this paper we put corresponding formulas in such a fashion that their deformation becomes almost evident, given the experience in this domain. As a result we get the definition of (T
*
G)
 t
 and its representation (t is the deformation parameter). To make the exposition most simple and formulas transparent we shall work on an example ofG=sl(2) and present results in such a way that the generalizations become evident. We shall stick to generic complex versions, real and especially compact forms requiring some additional consideration, not all of which are self-evident.

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( T * G ) t : A toy model for conformal field theory

It is pointed out that the cosmologies based on Lyra's manifold, with constant gauge-vector which have been studied in the literature, will either include a creation field and be equal to Hoyle's creation field cosmology, or contain a special vacuum fluid which together with the gauge-vector term may be considered as a cosmological term. In the latter case the solutions are equal to the general relativistic cosmologies with a cosmological term.

Cosmologies based on Lyra's geometry

A static cosmological model of the universe based onLyra's modified Riemannian Geometry is proposed. The red-shift of spectral lines from extra-galactic nebulae is shown to be a consequence of an inherent geometrical property of the model independent of expansion. The model is similar to the staticEinstein model but shows a red-shift and has a finite density even without the introduction of a cosmological constant.

A static cosmological model

It is shown that Weyl's geometry and an apparently similar geometry suggested by Lyra are special cases of manifolds with more general connections. The difference between the two geometries and their relationship with Riemannian geometry are clarified by giving a global formulation of Lyra's geometry. Finally the outline of a field theory based on the latter geometry is given.

On Weyl and Lyra Manifolds

Fields and/or particles

In Riemannian geometry the autoparallels associated with the affine connexion coincide with the geodesies which arise from the metric. This is not the case in a modification of Riemannian geometry suggested by Lyra. A sufficient condition that the two classes of curves coincide is obtained.

D. K. Sen

Papers

A Scalar-Tensor Theory of Gravitation in a Modified Riemannian Manifold

A static cosmological model

On Weyl and Lyra Manifolds

Fields and/or particles

On geodesics of a modified Riemannian manifold