The Electromagnetic Field Tensor
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We have become accustomed to conceiving the electromagnetic field as the manifestation of a pair of vector fields, one electric and the other magnetic. But, as our use of the word electromagnetic implies, we also conceive those vector fields as two different aspects of a single entity. Maxwell痴 Equations, by displaying an intimate relationship between the two fields, validate that concept. We now want to devise a mathematical description of that single entity.
We begin that task by writing a description of the fields in terms of their potentials and translating the potentials into their covariant tensor forms;
(in which m = 1,2,or 3) and
In addition we have the Lorenz condition,∂iAi=0. In those equations we have exhausted all sixteenth ways to combine the numerals 1,2,3,4 taken two at a time. That fact means that we could, if we wish, lay out the possibilities on a four-by-four matrix. As a first guess we have
If we want to represent the electromagnetic field with a matrix (a second-rank tensor), then we want the multiplication of that tensor by the four-current to yield the four-vector corresponding to the Lorentz force. The above matrix won稚 satisfy that criterion, but if we examine the components of the Lorentz force density, then we can see how to modify that matrix to obtain a proper electromagnetic field tensor.
If an electric charge of densityρ and an electric current of density j occupy the space also occupied by the fields in Equations 1 and 2, then the classical Lorentz force density consists of an electric component,
and a magnetic component,
In the above equations I have used the hatted lower-case ees to represent the unit vectors of our coordinate system, we tacitly understanding that system to be the standard Cartesian rectangular grid. I have also made use in Equation 4 of the fact that in the covariant four-potential A4=-ϕ/c.
At this point we need to attend to the distinction between covariant and contravariant tensors. That distinction, which has a geometric aspect, also has an algebraic aspect. The geometric aspect only significantly comes into play in General Relativity and we can ignore it in the case of the flat space and time of the Minkowski metric, which I use exclusively in this essay. The algebraic aspect helps us to carry out correct calculations by removing the bewilderment concerning the proper placement of the algebraic signs.
When we organize measurements, actual or anticipated, in a four-vector we usually create a covariant four-vector. When we multiply two four-vectors together one of them must have the covariant form and the other one must have the contravariant form. To convert a covariant four-vector into the equivalent contravariant four-vector we multiply it by the contravariant metric tensor, so we have
in which the subscript refers to the covariant form and the superscripts denote the contravariant forms. Also we use the Einstein convention of automatically summing over all values of any repeated indices. Nicely enough, because gikgkm=δim, both the covariant and contravariant forms of the Minkowski metric tensor have exactly the same matrix form; specifically,
Thus we have the covariant and contravariant forms of the electric four-current density,
Now, using the contravariant electric four-current density, we can combine Equations 4 and 5 into four equations that give us the components of the four-force density that the electromagnetic field exerts upon the four-current,
While the first three components clearly represent the force density vector, the fourth component represents the density of the rate at which the field does work upon the current. Now define the field tensor through the statement that
The elements of the field tensor consist of the coefficients of the four-current components in Equations 9, so we get
Also, noting the convention that the first index on a second-rank tensor refers to the row and the second index represents the column, we see that we can write that matrix, also known as the Faraday tensor, as
Thus we get an antisymmetric matrix, Fik=-Fki that properly represents the electromagnetic field.
If we multiply the covariant field tensor twice by the contravariant Minkowski-space metric tensor, row-wise and then column-wise, we get the contravariant field tensor,
If we multiply that tensor by the covariant field tensor, we get a Lorentz invariant
That formula instructs us to take the first row of the covariant tensor and multiply its elements, in the manner of a vector dot product, by the elements of the first row of the contravariant tensor; multiply the second row of the covariant tensor in the same way by the second row of the contravariant tensor; multiply the third covariant row by the third contravariant row; multiply the fourth covariant row by the fourth contravariant row; then add together those partial products to obtain
Because we have the energy density in an electromagnetic field as
we now have the basis for calculating the Lagrangian density of an electromagnetic field (see "The Electromagnetic Lagrangian Density").
Finally we want to work out the tensor version of Maxwell痴 Equations.
Those four equations relate the derivatives of the vectors representing the electric and magnetic fields to each other and to the densities of the electric charge and of the electric current existing in the space occupied by those fields. Dividing those equations into pairs, we have the field equations,
and the source equations,
Recalling that we have the contravariant four-current density as Jk=(ρv,ρc), we seek to translate Equations 18 into tensor form. For the first of Equations 18 we have
in which the index alpha takes only the values 1,2,3 (note that F44=0). For the second of Equations 18 we put both fields on the right side of the equality sign and write
Dividing Equation 19 by the speed of light and adding it to Equation 20 gives us Maxwell痴 two source equations as
Alternatively, consider what we get when we vary the electromagnetic Lagrangian density with respect to the potentials. In both vector and tensor form we have that Lagrangian density as
To extract equations of evolution from that function we need only apply to it the Euler-Lagrange operator configured for the appropriate field. Taking the variation of the action with respect to a variation of the electrostatic-potential field as our example necessitates that we write the Euler-Lagrange equations as
With appropriate algebraic rearrangement that equation becomes the expression of Gauss痴 law for the electric field, the first of Maxwell痴 Equations (and the first of Equations 18). In setting up that calculation I included only those terms of the Lagrangian density that involve the electrostatic potential because the other terms simply zero out under the differentiation. I also didn稚 bother with the case i=4 because the gradient of the electrostatic potential does not include a time-derivative term.
If we vary the action integral with respect to a variation in the electrotonic field, we get the Euler-Lagrange equations as
With appropriate algebraic rearrangement that equation becomes the expression of Ampere痴 law, the fourth of Maxwell痴 Equations (and the second of Equations 18). We can see easily how we obtain the second and third terms in that last expression. We have to work a little harder to see how we get the first term. To that end we apply the product rule of differentiation and then carry out a series of differentiations that yield the relevant vectors; thus we get
In this term the operator∂t drops out of the second line because the curl operator does not include a time derivative.
Now let痴 carry out the same derivation with tensors. In this case we have the variation of the action integral with respect to the four-potential, so we must write the Lagrangian density, insofar as possible, in terms of the four-potential. For the field part of the Lagrangian density we have
In going from the second line to the third I have exploited the fact that the indices are dummy numbers, so that the first and fourth terms on line two represent the same thing and likewise for the second and third terms on line two. We add that result to the interaction term (JiAi) and thus get the Euler-Lagrange equations as
Note that the factor of ｽ in the first term on the first line of that equation goes away because the differentiation produces two identical terms, one for the covariant differentiation and one for the contravariant differentiation (essentially the produce rule for derivatives). That equation corresponds to Equation 21, which we can see clearly when we recall to mind the fact that ε0μ0c2=1. If we then calculate the four-divergence of the result of that equation, we get the equation expressing the continuity (conservation) of electric charge.
Thus we obtain the tensor form of the inhomogeneous Maxwell Equations (Equations 18), also known as the equations of electricity, from the principle of least action. But then whence come the homogeneous Maxwell Equations (Equations 17), the equations of magnetism?
In Equation 27 we have the divergence of the field tensor. We can also write a kind of anti-divergence,
Permuting the three indices on that equation twice and adding the results to that equation gives us
We know that the function on the right side of that equation zeroes out perfectly because the differentiation operators commute with each other, so the fourth term cancels the first, the fifth term cancels the second, and the sixth term cancels the third. So how does that equation relate to Maxwell痴 Equations?
For all four values of the indices Equation 29 gives us sixty-four terms. If we consider only the indices representing the spatial coordinates (i,k,m=1,2, or 3), we see from the anti-symmetry of the field tensor that in any case in which two of the indices have the same value one term will equal zero and the other two will cancel each other in a perfectly trivial expression. For example, we might have
But if the indices all have different values, we get
Thus, we obtain the second of Maxwell痴 Equations, Gauss痴 law of the magnetic field (the first of Equations 17).
If we now include the fourth index value, that of the temporal coordinate, we get three combinations that begin with the index sets (1,2,4), (1,3,4), and (2,3,4). Thus we get three versions of Equation 29:
Multiplying each of those equations by the speed of light and the appropriate unit vector, multiplying the second equation by minus one, and adding the results yields
Faraday痴 law, the third of Maxwell痴 Equations (the second of Equations 17). Any other combinations of the indices will produce either a trivial zero, as in Equation 30, or a repetition of Equations 32.
That seems like a lot of effort to put into gaining something that we can do more easily with vector calculus. Indeed, the use of vectors makes the fields easier to envision, easier to manipulate mentally. So why bother with tensors? Simply put, we need a tensor version of electromagnetism because we will need to incorporate it into the theory of General Relativity, which Einstein created as an exercise in tensor calculus. Unless something like a vector version of General Relativity comes to light, we will need the ability to work out problems in electromagnetism in tensor form.
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