Determing the physical geometry
of our Universe

J. Boguta

boguta@digihara.com

Abstract

The purpose of this work is to determine the physical geometry of our Universe from observational data. General theory of relativity assumes that the physical geometry of our Universe is such that all geometric interactions are metric in nature. This work studies a Universe where this is not the case. The main aim is to construct a principled theory of geometric interactions that goes beyond general relativity, but satisfies the same principles. This theory is developed on two principles: Galileo-Einstein's observation that in the presence of geometric interactions a coordinate system exists in which all bodies experience the same free fall, and also on Penrose's quanta identity principle. It assures the local validity of pauli and bose statistics. The equations of motion for this theory are inferred from the second Bianchi identity. In this theory of geometric interactions dark forms of energy–dark mater and dark energy–emerge as excitations of a new geometric degree of freedom. This affine degree of freedom measures the difference between the straightest and metrically optimal paths between two points.  In spherically symmetric geometry–as could be assumed for some galaxies– it is shown that this theory has geometric singularities that are not metric. Properties of model galaxies with such singularities are studied and compared with observational data.  The gradient flow of this field theory is briefly studied.

1. Introduction

Einstein himself remarked that the physical nature of our Universe's geometry, whether it be Riemannian, non-Riemannian, or of some other type, is an experimental question. Surprisingly, he did not put that question to a test himself, but assumed, ab inition, this geometry to be pseudo Riemannian. The reason for such an assumption lies in  the observable fact that one, constant, measure for distance, as well as for time, exists througout our immediate solar environment. A universal scale of measure also assures that in general relativity the energy of an electron does not depend on its past world history. General relativity satisfies Penrose's quanta identity principle explicitly.

The assumption that scales of measure are constant in our Universe could  be false. This work develops a theory  in which the scale of measure is no longer universal. This theory is based on two geometric principles having direct physical meaning.  The first is that in the presence of geometric interactions all bodies accelerate at the same rate. This is Galileo's observation. Einstein's observation, the very key one, is that there always exists a coordinate system in which all bodies are in free fall. This gives a geometric interpretation for the gravitational force in general theory of relativity. The second principle is Penrose's quanta identity principle–locally all electrons are identical. This principle is not true for general spaces. The validity of this principle requires, at the very minimumm,  that the physical manifold have an intrinsic scalar field  assciated  with it.

The above two principles are realized in a number of model theories about the connection coefficients  of our Univers–. Galileo's observation,  that all bodies accelerate the same,  implies that the connection is symmetric (=) . Penrose quanta identity principle imples that

Where is the determinant of the metric.

From the definition the Ricci tensor   its antisymmetric is,

If one is seeking a theory of geometric interactions in which the source term is the symmetric energy momentum tensor then one must insist that  =0 and noting that the curl of a gradient always vanishes, Eq (1). is obtained.  A theory of geometric interactions with a symmetric energy-momentum tensor  is possible if it satisfies Penrose's quanta identity principle.

A key tool to investigate the possible equations of motion is the second Bianchi identity. It was repeatedly emphasised by Wheeler that this identity, which follows from a purely geometric constraint, reveals what the structure of general relativity equations of motion must be. In the same way, this identity will be used to study a number of model theories. For this identity to be useful, in pseudo remannian space, or  in a more general space, a number of key symmetries of the corresponding Riemann tensor must be true. All these conditions turn out to be true when Penrose quanta identity principle holds. By studying the second Bianchi identity of theories satisfying the above two principles it is shown that this scalar field has an exponential potential Λ .  This  field and its exponential potential are the dark forms of energy–dark matter and dark energy. Its origin is geometric.

For a constant scalar field  φ the above theory reduces to that of general relativity with a cosmological constant. It can significantly differ from it when the scalar field has singularities. That such singularites can exist is clear from the observation that the scalar field  φ is a masquearading dilaton (but geometric in nature). In diliton theory it is known that an exponential dilaton potential can lead  to peculiar solutions of the field equations.  In particular, there are solutions which have a finite Ricci tensor value , but do not reach flat asymptotic values as in Schwartzschield case. In particular, φ→log(r).  Such theories, as emphasized by Witten, must break down at large distances due to quantum effects.

One way to study these quatum effects is by means of renormalization flow. This is too hard. As a first approximation one can study the gradient flow of the field equations.  The singularities of this flow reflect the topology of our Universe.  If one were allowed to flow past the singularity, then there is the unusual situation that regions that are well separated at large distance scales may become neighbors viewed microscopically {pereleman reference}.

2. Observational data about galaxies

Observations confirm  the following energy partition in our Universe: 73% dark energy,  23% dark matter and 4% regular matter. This indicates that the overwhelming part of energy is in dark form. Matter, which we would consider as energy of the universe such as protons, electrons, photons and so on, makes up just a small part of the total energy. Observational data also shows that motions distant from the luminous region of a galaxy can not be accounted for by the sole gravitational action of visible matter. To explain  observational data, the presence of a large amount of dark matter in a huge halo around the galaxy is inferred. A similar deficit is also seen in galaxy clusters.

Recent advances in the ability to make observations on a few pc scale has leads to new insight into the properties of dark matter.  It has been know for some time that the black hole mass  scales with absolute blue luminosity of the host bulge for spiral galaxies or the whole elliptical galaxy. A further result was reported by {Gebhard}, and by {Ferrarese} on the properties of the host galaxy. They found a very tight relation between the mass of the black hole and the stellar velocity dispersion for the bulge of a spiral galaxy and the whole elliptical one.

A further significant  result was recently reported by {Ferrareze}. For galaxies studied, the mass of the black hole is correlated with the mass of that galaxy's dark matter halo.

This is a rather suprising result. The black hole is confined to a very small volume at the center of the galaxy, while affecting the properties of dark matter halo reaching distances very far away from it. There is no direct causal relation between the two.

Observational results show persuasively that at the center of galaxies there appears to be a singularity. This singularity is assumed to be a black hole, since general theory of relativity has only metric singularities. There is no direct observational evidence that the
singularity is a metric one. If it assumed, as this work does, that dark form of –dark matter and dark energy–are the manifestations of the geometric field φ, then a natural relation between the singularity and the halo is obtained by assuming that the singularity is not metric, but affine–a singularity of  φ  itself. The same affine field is the cause of both the halo and the singularity. The amount of visible matter will set the distance scale.

3. The physical geometry of our Universe

The central  object of space is its connection Γ. It completely specifies the geometry of the manifold.  The aim of this work is to infer  the connection of the Universe as revealed by observational data and assumed principles.

Though itself not a tensor, the connection Γ determines the meaning of change by specifying what is  the effect of a covariant derivative ▽ when acting on tensors and spinors. The action  of ▽ on a  contravariant vector  ξ is

The connection also determines the meaning of a straight path between two points–it is that path which parallel transports its tangent vector.

where α is a function on the curve. A path satisfying this equation can be reparametrized and cast into the standard form, where τ is the affine parameter

The aim is to determine the connection coefficients from assumed principles and observational data.

A. Galileo-Einstein principle

Galileo observed that in the presence of a geometric force all bodies accelerate at the same rate, independent of their composition. Einstein observed (his happiest day in life) that such bodies, besides having the same rate of acceleration, allow the construction of a coordinate system where all such bodies are in free fall. This can be taken to mean that the gravitational force is geometric in nature. The simplest example of this are the motions of bodies in a free falling elevator. For an observer in the elevator, all bodies appear to satisfy Newon's first law of motion.  The requirement that there exists such a coordinate system places a strong constraint on the connection coefficients . .

From the equation for the geodesic it is seem, that free fall occurs at a point P  when  = 0. This condition can be satisfied  if and only if the connection coefficients symmetric

The necessary condition is shown by observing the transformation properties of the connection coefficients .

Setting = 0 shows that that must be symmetric. The sufficient condition can be shown by expanding around the point

From the Eq. (7) it follows that = 0.  The coordinate system in which the fall is free is called the geodesic coordiantes. Such coordinates are assured to exist only if the connection is symmetric.  

A symmetric connection has other important geometric properties.  Suppose a parallelogram is formed by parallel transporting a segment along a geodesic.  

The symmetry of the connection coefficient allows allows for the definition of the Riemann tensor , expressed in local coordinates by:

where is a symmetric connection.

The second Bianchi identity is the linchpin of the present work as well.  For it to be useful, in general relativity, or in a broader theory of geometric interactions, it must satisfy a number of permutational identies, to be discussed below. The validity of these identities is assured by Penrose's quantum identity principle. It severely restricts the possible physical geometries of our Universe.

B. Qunta identity principle

             C. The second Bianchi identity
               and the field equations

Created by Mathematica  (January 10, 2006)