Tensors are built from vectors. They provide the mathematical structure used to describe states of many particles in quantum mechanics, as well as the structure to express general physical law in general relativity. The idea in their construction is remarkably straightforward. Tensors simplify physical laws and show that the magnetic force is just the electrostatic force after Lorentz transformation.

In information technology, tensors are represented as multi-dimensional arrays. Tensors have practical applications in any field in which different kinds of information is treated as tabulated data. They are best thought of as abstract arrangements of data, assembled by the choice of a human being, not as direct abstractions of physical reality. Spacetime tensors are built from spacetime vectors, and acquire properties from them, in such a way that we can express physical laws obeyed by measurable vector quantities by means of tensor laws.

For any vectors,

in

is a rank

Clearly, lowering an index then restoring it has no effect. So,

is the the Kronecker delta in any coordinates.

Tensors obey the same transformation law as vectors, applied to every index individually,

Other indexed quantities appear, known as

The

This holds for all

So,

Since this is true for all values of

Thus, the transformation law for

It is always possible to choose an orthogonal basis. In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In Minkowski Coordinates, the metric for Minkowski spacetime has

It follows immediately that, using unprimed Minkowski coordinates, switching

The

I will use square braces to

The

I will use round braces to denote the sum formed by cyclic permutation of the indices, e.g.,

Round, square and curly braces will also be used to denote sums of permutations of subsets of indices.

It can be shown by Lorentz transformation that the form of the Levi-Civita tensor is the same in any Minkowski coordinates with forward pointing time axis and right handed space axes.

for a small displacement

There is a modern fashion for coordinate free notation, in which tensorial equations are expressed without reference to coordinates. In this form tensor equations become invariant. I dislike this notation and generally avoid it. In particular, I dislike the fact that it often seems to be associated with metaphysical claims which are thought to give deeper insight into physics. I see such claims as based on a logical fallacy. One may express a law in a coordinate-free form because the law is true in all coordinate systems, but one should not deduce that coordinate-free quantities exist. There is a fundamental difference between saying that something is true in all coordinate systems, and saying that it is true in the absence of a coordinate system. In practice, empirical science requires that a coordinate system is established before a vector or a tensor can be quantified. Mathematically, there is no such thing as a finite dimensional vector space without a basis. One cannot talk about a vector or tensor as existing independently from its descriptions in different coordinate systems, but one can make statements about vectors and tensors which are true in all coordinate systems.

The simplest situation is a static electric field. In this case the 3-vector force is

For a field due to a moving charge, Faraday is found from Lorentz transformation, acting on both indices,

The Lorentz transformation for a boost in the 1-direction (i.e the

Multiplying out gives,

Contracting with a 4-current (

In the general case, Faraday is not represented simply through boosting a static field, but it is the result of fields generated by many particles, each one of which could be regarded as static in the rest frame of that particle. This would lead to a complicated expression, and we write Faraday using the resultant electric and magnetic fields,

Then the simple tensor law of force,

can be seen to be precisely the same as the usual form of the 3-vector Lorentz Force law,

Anyone who instinctively rebels against cross products, fields perpendicular to motion, and forces perpendicular to fields, may be satisfied that the magnetic force is really in the axis of the motion of the source of the field, and that the cross product is merely a part of a dot product in spacetime.

The argument here is not specific to the electromagnetic field. By applying it to gravity, Joseph Lense and Hans Thirring predicted frame dragging, a gravitomagnetic effect also known as the Lense-Thirring effect. Data from the first direct test of gravitomagnetism, the Gravity Probe B experiment, is currently under analysis.

IntroductionToTensors ↑ Gravity →

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