Tensors are Invariant

Tensors are physical quantities resulting from the sum of the ordered product of two or more vectors. Because they are constructed from invariant vectors, they are invariant quantities, independent of the observer or the coordinate system used to describe them. However, just as the components of a vector are coordinate dependent, so too are the components of a tensor coordinate dependent. In other words, the components of a tensor are relative quantities even though the tensor itself is invariant.

One of the simplest examples of a tensor in three-dimensional space is the area tensor. A second simple example is the spring constant tensor. Other 3-D examples can be found in books that discuss the moment-of-inertia tensor, the stress tensor, and others.

In four-dimensional spacetime, vectors bring together and show relationships between quantities previously thought to be unrelated. For example, the position 4-vector relates space with time and the mass 4-vector relates mass, energy, and momentum. Since tensors are constructed from vectors, four-dimensional tensors do the same, except on a grander scale. For example, the 4-D symmetrical matter tensor includes the 3-D stress tensor, the 3-D momentum density vector (momentum per unit volume), and the scalar mass-energy density (mass-energy per unit volume) among its components. And the 4-D anti-symmetric electromagnetic field tensor includes both the3-D electric field vector and the 3-D magnetic field vector.

Consequently, when one changes coordinate systems, these previously thought to be unrelated quantities are mixed together to generate the corresponding quantities in the new frame of reference. As a result, at relativistic speeds, stress, momentum, and energy are converted into one another. And even at non-relativistic speeds, changing electric fields induce magnetic fields and vice versa.

Furthermore, a single condition imposed upon a 4-D tensor results in several conditions imposed upon its components. Consequently, setting the divergence of the 4-D matter tensor equal to zero results in the 3-D laws of conservation of mass, energy, and momentum as well as Newton’s first and second laws. Similarly, setting the exterior derivative of the 4D electromagnetic field tensor equal to zero gives two of Maxwell’s electromagnetic field equations and setting the divergence of this 4-D tensor equal to the 4-D current density vector renders Maxwell’s other two field equations.

1. Area Tensor –.The three-dimensional non-symmetrical area tensor.

2. Spring Constant Tensor –.The three-dimensional symmetrical spring constant tensor.

3. Matter Tensor –.The four-dimensional symmetrical stress-momentum-energy-density tensor.

4. Electromagnetic Tensor –.The four-dimensional anti-symmetrical electromagnetic field tensor.

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