Suppose we feed two copies of the same non zero vector into the metric. If the metric will only ever give us back positive numbers, we say that the metric is .
In this case, the metric is called a Riemannian Metric . More generally, when the metric may give a negative value, the metric is called Pseudo-Riemannian . In Special and General Relativity, Spacetime is assumed to have a pseudo-Riemmanian metric (more specifically, a Lorentzian Metric ).
The manifold may also be given an Affine Connection , which is roughly an idea of change from one point to another. If the metric doesn't "vary from point to point" under this connection, we say that the metric and connection are , and we have a Riemann-Cartan manifold. If this connection is also self-commuting when acting on a scalar function, we say that it is '''torsion-free''', and the manifold is a Riemannian manifold.
Once a local coordinate system is chosen, the metric tensor appears as a Matrix , conventionally denoted . The notation is conventionally used for the components of the metric tensor (i.e., the elements of the matrix). Note that in the following, we use the Einstein Summation Notation for implicit sums.
In a Riemannian manifold, the length of a segment of a curve parameterized by ''t'', from ''a'' to ''b'', is defined as:
:
The angle between two Tangent Vector s, and , is defined as:
: |
In tensor analysis, the metric tensor is often used to provide a objects i.e. "raise and lower indices."
This has a nice physical interpretation which is often glossed over. The metric tensor obviously has to do with measurement. We may ask, what is the scale for these measurements? A ''choice of basis'' defines the system of units on our manifold. The notions of contravariance and covariance correspond to quantities whose components transform "inversely" or "with" the coordinate system, hence the names. For example, consider
3 with the standard coordinate chart. If we transform the coordinate system by scaling the unit distance (say meters) down by a factor of 1000, the displacement vector (1,2,3) becomes (1000,2000,3000). On the other hand, if (1,2,3) represents a dual vector (for example, electric field strength), an object which takes a displacement vector and yields a scalar (in the example: potential difference in, say, volts), then the transformed coordinates become (0.001,0.002,0.003). What does the Euclidean metric on
3 do? (1,2,3) becoming (1000,2000,3000) makes sense because scaling down by 1000 takes meters to millimeters. For the field strength vector, (1,2,3) becoming (0.001, 0.002, 0.003) is a reflection of field strength going from volts ''per'' meter to volts ''per'' millimeter.
But what is the contravariant version of the field strength? How can we make a field strength vector's coordinates go from (1,2,3) to (1000,2000,3000)? The solution is to view the scale-down-by-1000 transformation as affecting the ''volts'' on the units V/m instead of the ''meters'' so that our new strength is measured in millivolts per meter. The metric tensor tells us precisely that we are still dealing with the same object, that is, it ''identifies the scaling of the basis vectors'' for the units "in the denominator" with a corresponding inverse change "in the numerator." Although somewhat trivial for
3, for general manifolds ''M'' it is very important since one can only define things locally. One can also imagine, for example, defining "funny units" on
3 which vary from point to point.
