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Electromagnetism and Special Relativity Johar M. Ashfaque We view electromagnetism through the lens of relativity. We will find that observers in different frames will disagree on what they call electric fields and what they call magnetic fields. They will observe different charge densities and different currents. But all will agree that these quantities are related by the same Maxwell equations. I.
A LIGHTENING REVIEW OF SPECIAL RELATIVITY
We begin with a lightening review of the relevant concepts of special relativity. The basic postulate of relativity is that the laws of physics are the same in all inertial reference frames. The theory of special relativity tells us how things look to observers who are moving relative to each other. Consider the first observer who sits in an inertial frame S with space-time coordinates (ct, x, y, z) while the second observer sits in an inertial frame S0 with space-time coordinates (ct0 , x0 , y 0 , z 0 ). If S0 is taken to be moving with speed v in the x-direction relative to S then the coordinate systems are related by the Lorentz boost v v 0 0 x = γ x − ct andct = γ ct − x c c while y 0 = y and z 0 = z with c being the speed of light and γ being the ubiquitous factor s 1 γ= . 1 − v 2 /c2
A.
Four-Vectors
To make things easier and useful, the space-time coordinates can be packaged in 4-vectors with indices running from µ = 0 to µ = 3 X µ = (ct, x, y, z),
µ = 0, 1, 2, 3.
Note that the index is a superscript rather than a subscript. This will bear great importance in the discussion which follows. A general Lorentz transformation is a linear map from X to X 0 of the form (X 0 )µ = Λµν X ν . Here Λ is a 4 × 4 which obeys the matrix equation ΛT ηΛ = η ⇐⇒ Λρµ ηρσ Λσν = ηµν where ηµν = diag(+1, −1, −1, −1). The solutions to ΛT ηΛ = η ⇐⇒ Λρµ ηρσ Λσν = ηµν fall into two classes. The first class is simply rotations. Given a 3×3 rotation matrix R obeying RT R = 1, we can construct a Lorentz transformation Λ by embedding R in the spatial part 1 0 0 0 0 0 R 0 These transformations describe how to relate the coordinates of two observers who are rotated with respect to each other. The second class is that of Lorentz boosts. These are transformations appropriate for observers moving relative to each other. The Lorentz transformation v v 0 0 x = γ x − ct andct = γ ct − x c c
2 is equivalent to γ −γv/c 0 0 γ 0 0 −γv/c 0 0 1 0 0 0 0 1
There are similar solutions associated to boosts along y and z axes. The beauty of 4-vectors is that they make it very easy to write down invariant equations. These are things which all observers, regardless of their reference frame, can agree on. To construct these we take the inner product of two 4-vectors. For example, the square of a the distance from the origin to some point in space-time labelled by say X is X · X = X µ ηµν X ν = c2 t2 − x2 − y 2 − z 2 which is the invariant interval. Similarly, if we were to compute the inner product of two 4-vectors X and Y then X · Y = X µ ηµν Y ν which is also Lorentz invariant.
B.
Proper Time
Here we review how other kinematical variables like velocity, momentum and acceleration can be incorporated into 4-vectors. Suppose that in some frame, the particle traces out a world-line. The idea is to parameterize this path in such a way that all observers agree upon. The natural choice is that of proper time τ , the duration of time experienced by the particle itself. So if one was sitting in some frame, watching some particle move with an old-fashioned Newtonian 3-velocity u(t), then it is simple to show that the relationship between time t and the proper time τ of the particle is given by dt = γ(u). dτ The proper time allows us to define the 4-velocity and the 4-momentum. Suppose that the particle traces out a path X(τ ) in some frame. Then the 4-velocity is dX c U= =γ . dτ u Similarly, the 4-momentum is P = mU where m is the rest mass of the particle. We write E/c P = p where E = mγc2 is the energy of the particle and p = γmu is the 3-momentum in special relativity. The importance of U and P lies in the fact that they are 4-vectors. Since all observers agree on τ , the transformation law of U and P are inherited from X and therefore the inner products of U and P are guaranteed to be Lorentz invariant.
C.
Indices Up, Indices Down
For all the 4-vectors introduced so far, we always place the space-time index µ = 0, 1, 2, 3 as a superscript rather than as a subscript ct µ X = . x
3 This is because the same object with an index down, Xµ would mean something different. We define Xµ =
ct . −x
With this convention, the Minkowski inner product can be written using the usual convention of summing over repeated indices as X µ Xµ = c2 t2 − x · x. In contrast, X µ X µ = c2 t2 + x2 is something one would not write down in the context of special relativity as it looks very different to observers in different inertial frames. There is a natural way to think of Xµ in terms of X µ using the Minkowski metric. The following is trivially true Xµ = ηµν X ν . This means that we can think of the Minkowski metric as allowing us to lower indices. To raise the indices up again, we need to find the inverse of the Minkowski metric. It so happens that the Minkowski metric is its own inverse which simply means η µρ ηρν = δ µν and we can write X ν = η νµ Xµ .
II.
CONSERVED CURRENTS
Recall taht the charge density ρ(x, t) and the current density J(x, t) are related by the continuity equation ∂ρ +∇·J =0 ∂t which simply tells us that the charge is locally conserved. The continuity equation is already fully consistent with relativity. To see this, we first notice that the charge and current density sit nicely together in a 4-vector ρc J = . J µ
The continuity equation then takes the particularly simple form m ∂µ J µ = 0. This equation is Lorentz invariant.
III.
GAUGE POTENTIALS AND ELECTROMAGNETIC TENSOR
Under Lorentz transformations, electric and magnetic fields will transform into each other.
4 A.
Gauge Invariance and Relativity
Recall the following equations of electrostatics and magnetostatics ∇ × E = 0 ⇒ E = −∇φ ∇·B=0⇒B =∇×φ where φ is the scalar potential and A is the vector potential. However, these expressions can not hold in general. We know that when B and E change with time, the two source-free Maxwell equations are ∇×E+
∂B =0 ∂t
and ∇ · B = 0. Nonetheless, it is still possible to use the scalar and vector potentials to solve both these equations. The solutions are ∂A E = −∇φ − ∂t and B=∇×A where φ = φ(x, t) and A = A(x, t). We can always shift A → A + ∇χ and B remains unchanged. However, this requires a compensating shift of φ ∂χ ∂t with χ = χ(x, t). These are gauge transformations. They reproduce our earlier gauge transformation for A while also encompassing constant shifts in φ. Now we can define a 4-vector φ/c µ A = . A φ→φ−
In terms of this new definition, the gauge transformations take the particularly nice form Aµ → Aµ − ∂ µ χ where χ is any function of space and time. B.
The Electromagnetic Tensor
We now have all the ingredients necessary to determine how the electric and magnetic fields transform. From the 4-derivative ∂µ = (∂/∂(ct), ∇) and the 4-vector Aµ = (φ/c, −A), we can form the antisymmetric tensor Fµν = ∂µ Aν − ∂ν Aµ . This is constructed to be invariant under gauge transformations. We have Fµν → Fµν + ∂µ ∂ν χ − ∂ν ∂µ χ = Fµν . This already suggests that the components involve E and B fields. We find that 0 Ex /c Ey /c Ez /c −Bz By −Ex /c 0 Fµν = −Ey /c Bz 0 −Bx −Ez /c −By Bx 0 which is called the electromagnetic tensor. This is a tensor because it is constructed from objects Aµ , ∂µ and ηµν which themselves transform nicely under the Lorentz group. This means that Fµν must transform as F 0µν = Λµρ Λνσ F ρσ .
5 IV.
THE MAXWELL EQUATIONS
We now have the machinery to write down the Maxwell equations in a way which manifestly compatible with special relativity. They take the particularly simple form ∂µ F µν = µ0 J ν ,
∂mu F̃ µν = 0
where F̃ µν is the dual electromagnetic tensor. The Maxwell equations are not invariant under the Lorentz transformations. Having said that the equations are covariant under the Lorentz transformations. This simply means that an observer in a different frame of reference will mix everything up. So although the observers will disagree on what these things are they will all agree on how they fit together. This is what it means for an equation to be covariant: the ingredients change but the relationship between them stays the same.
V.
THE LORENTZ FORCE LAW
In the Newtonian world, the equation of motion for a particle moving with velocity u and momentum p = mu is dp = q(E + u × B). dt We want to write this equation in the 4-vector notation in a way that makes it clear how all the objects change under Lorentz transformations. The relativistic version involves the 4-momentum P µ , the proper time τ and the electromagnetic tensor F µν . The electromagnetic force acting on a point particle of charge q can then be written as dP µ = qF µν Uν dτ where the 4-velocity is Uµ = and the 4-momentum is P = mU .
c dX µ =γ dτ u