← Introduction to 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.
Definition: A tensor space is a vector space built from other vector spaces using three simple rules:
Rule 1. If xi and y j are vectors, then the object consisting of the products of the coefficients, xiy j, is a tensor.
Rule 2. If xij and yij are tensors and a and b are scalars, then axij + byij is a tensor.
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
For any basis vector,
, in an n
-dimensional vector space, V1
, and any basis vector,
, in an m
-dimensional vector space, V2
, make a new object
, and V2
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,
, there is an object in T
So, the coefficients of
are simply the products of the coefficients of
. The converse is not the case. There are tensors formed from sums of products which cannot be factorised into vectors from the original spaces.
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
Definition: The tensor product of n vector spaces, V2, … Vn
is a rank n tensor space.
Tensors are built from vectors, and vector properties apply to each index of a tensor individually. We can apply the index lowering operator, or metric
to change a contravariant index to a covariant one, and the raising operator gij
to restore it to contravariance, e.g. for a Rank 3 tensor, Aijk
, we have,
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 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 xi
is a tensor, then Aijk
is a tensor.
: since xiAijk
is a tensor
This holds for all xi
obeys the tensor transformation law. The quotient theorem holds if Aijk
is replaced with a quantity with any number of upstairs or downstairs indices.
The Metric Tensor
To preserve the scalar product under general coordinate transformation we require that, for any vectors x
Since this is true for all values of xi'
and y j'
, we have
Thus, the transformation law for gij
is as for covariant vectors for each suffix. Similarly, the transformation law for gij
is as for contravariant vectors for each superfix. Thus, the metric
, 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 g00 = 1
, g11 = g22 = g33 = −1
, and all other entries equal to zero. In spherical coordinates
Minkowski spacetime has metric with g00 = 1
, g11 = −1
, g22 = −r2
, g33 = −r2sin2θ
and all other entries equal to zero.
It follows immediately that, using unprimed Minkowski coordinates, switching m
makes no difference to
Hence, in any coordinates, gi'j' = gj'i'
. In other words, g
is symmetrical under interchange of indices.
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, Ai
, from the rank 3 tensor Aijk
by contracting the second and third indices thus, Ai = Aijj
.We also talk of contracting an index of one tensor quantity with an index of another. The contraction of the second index of Bij
with the first of Cij
is a rank 2 tensor, BijC 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.,
The 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.,
The 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.,
Round, square and curly braces will also be used to denote sums of permutations of subsets of indices.
The Levi-Civita Tensor
The Levi-Civita tensor is defined in Minkowski Coordinates
to be a totally anti-symmetric tensor, with ε0123 = 1
. Because of antisymmetry, εijkl = 0
whenever any two indices are the same, and εijkl = ±1
whenever all the indices are different. We have
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.
A tensor valued function on a manifold
is a tensor field
. A vector field is a rank 1 tensor field. The value of a tensor field at a particular point is a member of a vector or tensor space defined at that point. If we add components of a tensor defined at one point with those defined at another, we are adding members of different tensor spaces. The result is not a tensor. In particular, the partial derivative
of a tensor field, x → f(x)
for a small displacement dxi
in the i
-direction, involves the subtraction of a tensor, f(x)
, defined at x
, from one, f(x + dxi)
, defined at x + dxi
. The partial derivative of a tensor field is not, in general, a tensor.
The Principle of General Covariance
Definition: An equation is covariant if it has the same form in any coordinates, up to the naming of coordinate axes.
The general principle of relativity
states Local laws of physics are the same irrespective of the coordinate system used to quantify them.
Vectors are not invariant, as their coordinate representation changes with the coordinate system. Relationships between vectors, defined from the dot product, are unchanged by coordinate transformation. Such relationships are said to be covariant
. Similarly, relationships between tensors are covariant. The form of the general principle of relativity most directly applicable to classical physics is the principle of general covariance, The equations of physics have tensorial form
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.
To apply the principle of general covariance
in the formulation of physical law we have to find vector and tensor quantities suitable for the description of physics. The 4-vector quantity describing a charged particle is current, 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 = (γq, γqv)
, where γ
is the Lorentz factor
Definition: The electromagnetic field is described by the Faraday tensor, F.
To describe the electromagnetic force acting on a charged particle we need to contract J
with a tensor representing the electromagnetic field. This tensor is called Faraday
. The result of contracting J
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:
The simplest situation is a static electric field. In this case the 3-vector force is qE = (qEx, qEy, qEz)
. We can then write the Faraday tensor, using antisymmetry to determine the other components.
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 x
Contracting with a 4-current ( Jt,  Jx,  Jy,  Jz)
gives a relativistic adjustment to the electrical force acting on the charge Jt
, and also introduces a force in the x
-direction (the direction opposite to the boost) with terms proportional to Jy
, i.e. perpendicular to the components of current in those directions. We identify this with magnetic force.
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 = (Ex, Ey, Ez)
and B = (Bx, By, Bz)
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
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 →