**by Peter Hickman** **Email: peter.hickman1@ntlworld.com**

__ABSTRACT__

In this paper, the extension of Riemann geometry to include an asymmetric metric tensor is presented. A new co-variant derivative is derived, and used in the commutator of two co-variant derivatives of a vector. This leads to two equations which describe spin 1 bosons. The energy-stress tensor arises as a contraction of the curvature tensor, its divergence enables the number of dimensions of Space-Time to be determined. A weak field approximation gives potential equations for both massless and massive bosons of at least 206.6GeV/c^2.

**Affine Connection**

The invariant interval between two points on a Riemann manifold is | |

1.1 | |

Let be the co-variant derivative, the invariance of ds, requires | |

1.2 | |

For the affine connection to be determined by a metric tensor only, two cases arise: | |

Case I: The metric and affine connection are both symmetric | |

1.3 | |

With the conditions given by equation 1.2, and 1.3, the symmetric affine connection are the Christoffel Symbols, see reference [1] | |

1.4 | |

Case 2: The metric and affine connection are both asymmetric: | |

1.5 | |

With the conditions given by equation 1.2, and 1.5, the asymmetric affine connection is | |

1.6 | |

A
general affine connection can be formed from equations 1.4
and 1.6 | |

1.7 | |

where the imaginary part of the connection is asymmetric in m and n | |

It can be shown that using condition 1.2 with the connection 1.7, that the affine connection is | |

1.8 | |

where | |

1.9 | |

the asymmetric affine connection is completely asymmetric. |

**The Field Equations**

The commutator
of 2 co-variant derivatives of a complex vector using 1.8 (http://homepage.ntlworld.com/peter.hickman1/page1.htm) for the affine
connection gives | |

2.1 | |

where | |

By adding to both sides of equation 2.1 and equating the LHS to zero, and lowering a , two equations arise: | |

2.2 | |

2.3 | |

Contracting equation 2.3 by setting n =b gives | |

2.4 | |

In a geodesic frame, this reduces to | |

and since , it can be shown that is imaginary. Contracting equation 2.2 by letting b =s and n =r gives | |

2.5 | |

The vector can be eliminated from 2.4 and 2.5 by using the relations | |

| |

where are NxN matrices where to give | |

2.6 | |

2.7 | |

For , the euclidean metric, the matrices are found to satisfy the following, (see section 3 for the calculation of N) | |

where I is the 6x6 unit matrix. A solution for the i=1,2,3 matrices is | |

where the matrices are given by, see reference [2] | |

These are the matrices for Spin 1 particles, thus the vector is the wavefunction for spin 1 bosons. |

**Calculation of
N, dimensions of Space-Time**

The symmetric
connection in equations 2.6 (http://homepage.ntlworld.com/peter.hickman1/page2.htm)and 2.7 (http://homepage.ntlworld.com/peter.hickman1/page2.htm) can be eliminated
to yield | |

3.1 | |

Multiply the expression in parenthesis by hc gives the total energy E, | |

where from which it follows that | |

| 3.2 |

where N is the number of dimensions. Contracting equation 3.2 gives G=NS, so equation 3.2 can be written as | |

3.3 | |

which can be written as | |

3.4 | |

compare with Einsteins field equations of gravitation with a cosmological term, see reference [1] | |

gives | |

3.5 | |

With the aid of the following identities | |

| |

with f=3 to N, the following expression arises when calculating the divergence of | |

which vanishes if (1-3/N)=3/N, gives N=6, Space-Time is 6d | |

The divergence of equation 3.4 becomes | |

**The Weak Field
Approximation**

From equation
3.6 (http://homepage.ntlworld.com/peter.hickman1/page3.htm), the equation
for the current densities is | |

4.1 | |

where | |

The weak field approximation for the asymmetric metric tensor is | |

4.2 | |

where w is scalar field and | |

To an approximation, equation 4.1 is | |

4.3 | |

Using equation 4.2 for the asymmetric metric, applying the gauge condition , and dividing by w , equation 4.3 becomes | |

4.4 | |

With w =constant, equation 4.4 reduces to the classical wave-equation. In general equations 4.4 is for interactions mediated by massive bosons. For w constant, equation 4.3 can be written in a more familiar form: | |

where | 4.5 |

If | |

4.6 | |

Maxwells equations of Electro-magnetism are obtained. see reference [3] | |

For the special
case of w
=constant, equation 3.2 (http://homepage.ntlworld.com/peter.hickman1/page3.htm) is found to
be, to an approximation: | |

Compare with Einsteins field equations: see reference [1] | |

Gives The particular case for f=5, | |

Is the energy-stress tensor of the Electro-magnetic field. see reference [3] | |

Equations 4.3 consist of 4 sets of equations, for the index f=3 to 6. | |

It can be shown that there are 4 'Electric' fields, given by the following equations | |

4.7a | |

4.7b | |

4.7c | |

There are 4 'Magnetic' fields given by | |

Where i=1,2,3 | |

The following identification can be made : | |

are the colour potentials, the electro-magnetic scalar potential and the scalar potentials related to the weak interaction. |

**Rest Mass Of Bosons**

The rest-mass of bosons depends on the scalar field w , which is determined by the following equation | |

5.1 | |

A general solution in spherical co-ordinates is | |

5.2 | |

where are seperation constants, A is a constant of integration and | |

Applying the following conditions | |

| |

with l,m,n integers, gives the particular solution for w | |

5.3 | |

A general time-independent solution for the potentials is | |

| 5.4 |

where are constants of integration. | |

The integral of the 44-component of the energy-stress tensor for a Boson at rest is to an approximation with 2p r=l is | |

5.5 | |

Using Plancks relation for the energy , equation 5.5 gives | |

5.6 | |

Equate the gravitational field strength to a and | |

Substituting for w and let l=m=n, and finally solve for E gives | |

5.7 | |

Evaluating E for n=0,1,2 gives | |

E(0) = 4.87x10^27 eV | |

E(1) = 3.17x10^19 eV | |

E(2) = 206.6GeV | |

Thus, the existence of a massive boson, of 206.6GeV is predicted. | |

**The Cosmological
Term**

With N=6, equation
3.5 (http://homepage.ntlworld.com/peter.hickman1/page3.htm) for the cosmological
term is | |

6.1 | |

Using the Robertson-Walker metric see reference [1], the current value for the cosmological term is | |

6.2 | |

where | |

For k=0, and the present value of the cosmological term is | |

which agrees well with observations |

**Modified De Broglie
Momentum Equation**

It can be shown that | |

7.1 | |

7.2 | |

For ,
i=1,2,3, then and
in the weak field approximation equation 7.2 for the 4^{th}
component of momentum reduces to | |

7.3 | |

For , , and solving equation 7.3 gives | |

7.4 | |

For r/t=c and from equation 7.4 | |

7.5 | |

Since the LHS of equation 7.5 is positive, then ie greater than the | |

Planck Length which implies particles have a maximum size of 1.6x10^-35m |