# ← Introduction to Tensors ↑ →

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.

### Tensor Space

**Definition:**A

*tensor space*is a vector space built from other vector spaces using three simple rules:

**Rule 1.**If

*x*and

^{i}*y*are vectors, then the object consisting of the products of the coefficients,

^{ j}*x*, is a tensor.

^{i}y^{ j}**Rule 2.**If

*x*and

^{ij}*y*are tensors and

^{ij}*a*and

*b*are scalars, then

*ax*+

^{ij}*by*is a tensor.

^{ij}**Rule 3.**Addition and multiplication by scalars obey the rules of vector space.

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.

### The Tensor Product

*n*-dimensional vector space,

*V*

_{1}, and any basis vector, , in an

*m*-dimensional vector space,

*V*

_{2}, make a new object (

*V*

_{1}, and

*V*

_{2}may be the same vector space). The

*mn*objects of the form can be represented as

*mn*× 1 column vectors, each with a 1 in one position and a zero in all others (this is how they are represented in computer memory). Introduce the usual rules of matrix addition and multiplication by scalars. The

*mn*-dimensional vector space built like this is a tensor space. Call this tensor space

*T*. The objects in

*T*are tensors. The objects are a basis for

*T*. We write .

**Definition:**is the

*tensor*(or

*outer*)

*product*.

For any vectors,

*V*

_{1}and

*V*

_{2}, there is an object in

*T*:

### Tensor Rank

Rank 2 tensors are formed from the product of two vector spaces, and have coefficents with two vector indices. One may find the tensor product of any number of vector spaces in exactly the same way. Rank*n*tensors have coefficients with

*n*indices.

**Definition:**The

*tensor product*of

*n*vector spaces,

*V*

_{2}, …

*V*,

_{n}*n*tensor space.

### Index Gymnastics

**Definition:**An equation is

*covariant*if it has the same form in any coordinates, up to the naming of coordinate axes.

*g*to change a contravariant index to a covariant one, and the raising operator

_{ij}*g*to restore it to contravariance, e.g. for a Rank 3 tensor,

^{ij}*A*, we have,

^{ijk}Tensors obey the same transformation law as vectors, applied to every index individually,

*non-tensors*. One can raise and lower the indices of non-tensors, using

*g*, but non-tensors are distinguished from tensors because their indices do not satisfy vector transformation laws (any indexed quantity satisfying vector transformation laws can be written as a sum of basis elements and is therefore a tensor). Some treatments even “define” tensors by their transformation properties. In my view, that obscures the mathematical structure of a tensor space.

The

*quotient theorem*states that if, for any vector

*x*,

^{i}*x*is a tensor, then

^{i}A_{ijk}*A*is a tensor.

_{ijk}*Proof*: since

*x*is a tensor

^{i}A_{ijk}*x*. So,

^{i}*A*obeys the tensor transformation law. The quotient theorem holds if

_{ijk}*A*is replaced with a quantity with any number of upstairs or downstairs indices.

_{ijk}### The Metric Tensor

To preserve the scalar product under general coordinate transformation we require that, for any vectors*x*and

*y*,

*x*and

^{i'}*y*, we have

^{ j'}*g*is as for covariant vectors for each suffix. Similarly, the transformation law for

_{ij}*g*is as for contravariant vectors for each superfix. Thus, the metric,

^{ij}*g*, is a tensor, known as the

*metric tensor*.

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

*g*

_{00}= 1,

*g*

_{11}=

*g*

_{22}=

*g*

_{33}= −1, and all other entries equal to zero. In spherical coordinates Minkowski spacetime has metric with

*g*

_{00}= 1,

*g*

_{11}= −1,

*g*

_{22}= −

*r*

^{2},

*g*

_{33}= −

*r*

^{2}sin

^{2}θ and all other entries equal to zero.

It follows immediately that, using unprimed Minkowski coordinates, switching

*m*and

*n*makes no difference to Hence, in any coordinates,

*g*. In other words,

_{i'j'}= g_{j'i'}*g*is symmetrical under interchange of indices.

### Contraction

A lower rank tensor is sometimes created from a higher one by*contraction*. This means setting upper and lower indices equal and using the summation convention. For example, we might wish to define a vector,

*A*, from the rank 3 tensor

^{i}*A*by contracting the second and third indices thus,

^{ij}_{k}*A*.We also talk of contracting an index of one tensor quantity with an index of another. The contraction of the second index of

^{i}= A^{ij}_{j}*B*with the first of

_{ij}*C*is a rank 2 tensor,

^{ij}*B*.

_{ij}C^{ jk}### Symmetry and Antisymmetry

A rank 2 tensor is*symmetric*if it is unchanged under the interchange of the indices, A rank 2 tensor is

*antisymmetric*if it reverses sign under the interchange the indices I will use curly braces to

*symmetrise*a tensor to any order by finding the sum formed by permutating the indices, e.g.,

*symmetric part*is found by dividing by the number of terms, i.e.

*n*! if there are

*n*indices. A tensor is

*totally symmetric*if it is equal to its symmetric part.

I will use square braces to

*antisymmetrise*a tensor by finding the alternating sum, formed by permutating the indices and subtracting odd permutations, e.g.,

*antisymmetric part*is found by dividing by the number of terms, i.e.

*n*! if there are

*n*indices. A tensor is

*totally antisymmetric*if it is equal to its antisymmetric part. A tensor is the sum of its symmetric and antisymmetric parts. It is easily shown that symmetry and antisymmetry are preserved by coordinate transformations — if a tensor is (anti)symmetric in one frame, it is (anti)symmetric in any frame.

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

### The Levi-Civita Tensor

The Levi-Civita tensor is defined in Minkowski Coordinates to be a totally anti-symmetric tensor, with ε_{0123}= 1. Because of antisymmetry, ε

_{ijkl}= 0 whenever any two indices are the same, and ε

_{ijkl}= ±1 whenever all the indices are different. We have

### Tensor Fields

**Definition:**A

*field*is a function on a coordinate system or a manifold.

*x → f*(

*x*),

*dx*in the

^{i}*i*-direction, involves the subtraction of a tensor,

*f*(

*x*), defined at

*x*, from one,

*f*(

*x*+

*dx*), defined at

^{i}*x*+

*dx*. The partial derivative of a tensor field is not, in general, a tensor.

^{i}### Electromagnetic Force

In formulating of physical laws we use vector and tensor quantities. The 4-vector quantity describing a charged particle is current,*J*.

*J*contains information about both-the charge and the motion of the particle. For a stationary charge the 3-current is zero, so the 4-current is (

*q*, 0). The general form of

*J*is found by Lorentz transformation and has the form

*J*= (γ

*q*, γ

*qv*), where γ is the Lorentz factor.

**Definition:**The electromagnetic field is described by the

*Faraday*tensor,

*F*.

*J*with a tensor representing the electromagnetic field. This tensor is called Faraday,

*F*. The result of contracting

*J*with

*F*is a vector, force. So, Faraday is a rank 2 tensor. Faraday should express both the force acting on a charged particle, and the equal and opposite reactive force exerted by the particle on its environment. If Faraday is an antisymmetric tensor, contracting with one index will give the force on the particle and contracting with the other will give the reactive force. We write down the 4-vector law of force:

*qE*= (

*qE*,

_{x}*qE*,

_{y}*qE*). We can then write the Faraday tensor, using antisymmetry to determine the other components.

_{z}*x*-direction),

*J*,

_{t}*J*,

_{x}*J*,

_{y}*J*) gives a relativistic adjustment to the electrical force acting on the charge

_{z}*J*, and also introduces a force in the

_{t}*x*-direction (the direction opposite to the boost) with terms proportional to

*J*and

_{y}*J*, i.e. perpendicular to the components of current in those directions. We identify this with magnetic force.

_{z}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,

*E*= (

*E*,

_{x}*E*,

_{y}*E*) and

_{z}*B*= (

*B*,

_{x}*B*,

_{y}*B*),

_{z}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 →