The Evans Equations of Unified Field Theory - Alpha Institute for ...

The Metric
The metric of special relativity is the Minkowski metric. A metric is a map
in a spacetime that defines the distance and also some mathematical functions of
the spacetime, like simple addition and multiplication. If one looks around the
region in which they find themselves, a metric exists. It is the reality within which
they find themselves and is a simple mapping. It establishes a distance from
every point to every other. In relativity the metric is typically designated as ηµν =
( -1, 1, 1, 1) and is multiplied against distances. At times it appears as (+1, -1, -1,
-1). For example, (dt 2 , dx 2 , dy 2 , and dz 2 ) give the distances between two four
dimensional points. Then ηαβ (dt 2 , dx 2 , dy 2 , and dz 2 ) = -dt 2 + dx 2 + dy 2 + dz 2
The invariant distance is ds where
ds 2 = -dt 2 + dx 2 + dy 2 + dz 2 (2)
Alternatively, letting dx stand for dx, dy, and dz combined, ds 2 = dx 2 -dt 2 .
This distance is sometimes called the “line element.” In relativity – both
special and general – the spacetime metric is defined as above. When
calculating various quantities, distance or momentum or energy, etc., the metric
must be considered. Movement in time does change quantities just as
movement in space does.
We will see the equation g(a, b) or simply g used. It is the metric
tensor and is a function of two vectors. All it means is the dot, more precisely the
inner) product of the two vectors combined with the spacetime metric. The result
is a 4-dimensional distance as shown in Figure 1-9 using curved Riemann
geometry
instead of the Euclidean in the figure. (See the Glossary under Metric
Tensor.)
The nature of the metric becomes very important as one moves on to
general relativity and unified field theory. The metric of quantum theory is the
same as that of special relativity. It is a good mathematical model for
approximations, but it is not the metric of our real universe. Gravitation cannot
be described. In the same manner, Riemann geometry describes gravitational
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