If you want to know the value of the electromagnetic field at some point in space P at time t_{1}, I assume that since EM is a relativistic theory, it should be possible to derive it using only the value of the field (along with charges, but let's say we are dealing with fields in free space) at an earlier time t_{0}, in a sphere of radius c*(t_{1} - t_{0}), with the field outside the sphere unknown. Does anyone know of any books or papers that show what the equations for this would look like, and how they would be derived, preferably at a not-too-advanced level? In my EM class we did study the "retarded potential" method which is based on the past light cone, but it requires looking at the contribution from every charge whose path crosses the past light cone at any point in the arbitrarily distant past, whereas I'm thinking of a method that only looks at a cross-section of the light cone at a particular earlier time t_{0}, and which deals with the field at that time rather than looking exclusively at the sources of the field (charges). Another question related to this: if you know the initial conditions I in the sphere at time t_{0} (both the E and B fields, and I think you'd need their instantaneous time-derivatives to determine their later evolution too), I wonder if you could use the following approximation to calculate the time-evolution, an approximation which would hopefully approach the exact solution in the limit considered in step 4: 1. Divide the whole volume of the sphere up into a lattice of cubes (or any other space-filling volume), and find the average value of the fields in I within each cube. 2. Create a new set of initial conditions I' where the fields have uniform values within each individual cube (with discontinuities at the boundaries between cubes), equal to the averages found in step 1. 3. Compute the time-evolution of I'[/b] using the superposition principle--the total value of the field at a later time is just the sum of the contributions from a large number of separate initial conditions, each one consists of a single cube filled with uniform fields surrounded by an initial field of 0 everywhere outside that cube. 4. Then, consider the limit as the size of the cubes goes to zero. I would guess that in this limit, the contribution from each cube might approach some fairly simple form, perhaps identical to the contribution from a proportionally small spherical region of uniform field with an initial field of 0 everywhere outside the small spherical region. Probably the time-evolution of such an initial condition would just be a sphere of nonzero field expanding at the speed of light, though I don't know whether the field would be uniform in the expanding sphere or concentrated at the surface. Could this work as a way of deriving the time-evolution, or is there some problem I haven't considered?
Perhaps you are thinking of something like Huygens' Principle. Given the field distribution across a closed surface, you can then calculate the fields of the propagated field. You're asking about projecting forward knowing the fields on a given surface in, what, the Minkowski space I assume? This seems to be equivalent to saying that you know the field distribution of a wavefront at a specific point in time. So Huygens' Principle is perfectly valid in this situation.
But Huygen's principle doesn't give exact solutions to Maxwell's equations, does it? I was under the impression it involved some approximations used in optics. Also, I realized there was a mistake in the second part of my question, about approximating the initial conditions: it shouldn't actually be possible to have an initial condition in free space consisting of a cube filled with uniform E and B fields, and 0 field outside, because in this case the E field lines wouldn't be able to terminate on a charge.
Kirchhoff's theorem seems to state similar idea: http://en.wikipedia.org/wiki/Kirchhoff_integral_theorem In this theorem, only the fields on the surface of the sphere in the past time t0 are needed. Similar equation should be valid in which fields at future time 2t1 - t0 are used (due to symmetry of the wave equation). Kirchhoff's theorem is discussed shortly in Mandel&Wolf, Principles of Optics, sec. 8.3.1. Regarding the second part, I think some book on numerical methods for EM field may help you. A friend of mine found this book very useful: Alain Bossavit: Computational electromagnetism http://butler.cc.tut.fi/~bossavit/ (scroll down to the section "Books", the line "This one, out of print, ... "
Thanks. But about Kirchhoff's theorem, I was under the impression that pretty much all of optics was based on a "scalar theory" which only approximates the full Maxwell equations, since it ignores polarization--looking around a little on google books, p. 145 of this book says "since the approach is essentially scalar, Kirchhoff's approach cannot account for the polarization of the EM field." It does go on to say, though, that "A vector equivalent of the scalar Kirchhoff's formulation was introduced by Stratton and Chu [10] and Schelkunoff [11]", so maybe I should look into those--I wonder if the altered approach is still approximate or if it gives (in principle) exact solutions to the Maxwell equations given some set of initial conditions.
The theorem itself is exact - it is derived from the wave equation. Usually it is stated for scalar function, but since each component of the field separately obeys the same wave equation, the theorem is valid for the whole vector field. The Kirchhoff theory of diffraction is further development of this idea, which however is only approximate. The difficulty is that we never know the field on the boundary surface exactly, so Kirchhoff made some further assumptions and these make his resulting theory approximate.
JesseM, what you are looking for are Jefimenko's equations: http://en.wikipedia.org/wiki/Jefimenko's_equations These are derived directly from Maxwell's equations, and show explicitly the dependence on values within the past lightcone only. In fact, since EM waves travel at the speed of light, one sees that the EM field now depends only on disturbances strictly on the past lightcone.
Jefimenko's equations were part of what I meant by the "retarded potential" method (using the potential at the "retarded time" which is distance/c earlier than the current time), see my earlier comment for why I was looking for something a little different:
But the different components aren't independent, they are related by the cross products in the Maxwell equations. So it isn't clear that applying the Kirchhoff theorem to each component individually would produce an exact solution that would respect the complete Maxwell equations--have you seen a reference saying that it would? I was just wondering about the question from a theoretical point of view, using the same sort of theoretical assumptions about the properties of bound charges/currents in volumes and surfaces, that are typically made in finding solutions to the Maxwell equations, leaving aside the question of whether these assumptions precisely describe real materials.
I am not sure what you mean. The wave equation for the electric field, for example, consists of three independent wave equations for three components of the field. Each component is a scalar function of ##x,y,z## and thus independently obeys Kirchhoff''s theorem. For example, to determine ##E_x## at some point inside some surface, only ##E_x##, its time derivative ##\partial_t E_x## and its gradient ##\partial E_x /\partial n## on the enclosing surface is needed. Now I realize that this is probably less spectacular than we imagined first - the knowledge of the fields on the surface is not enough, the knowledge of gradients seems necessary too. But still it is true that knowledge of the quantities on the surface is sufficient.
OK, I see--I hadn't remembered that in a vacuum, Maxwell's equations are completely equivalent to the wave equations for E and B. What about a situation other than a perfect vacuum, though--are there any cases (like the case where at least one media is a conductor, or at least one is dispersive) where Maxwell's equations wouldn't be equivalent to a set of independent equations for each component of E and B, and so Kirchhoff's theorem wouldn't be sufficient to predict the exact dynamics given initial conditions in a cross section of the past light cone? Or can Kirchhoff's theorem be used (in principle) to get exact predictions for the evolution of the field, given initial conditions, in any region where there may be any combination of media containing bound charges, as long as there are no free charges to consider?
If you consider the presence of matter, this will of course affect the behavior of the field. In general the EM field is not sufficient to describe the state of matter, so the evolution of the field will be much more complicated and the Kirchhoff theorem does not apply. One still has wave equations for the fields though, but with nonzero sources (charge and current density). The effect of these is usually considered to depend only on their values at past light cone, so again something like the Kirchhoff theorem should work, provided you include these sources among the quantities known at the boundary of the sphere.
I think this has already been dealt with, but Huygen's Principle is exact, although the form that I have used may not be the original formulation. I have worked with it under the case of the Equivalence Principle in electromagnetics which states that we can represent any wave as a set of equivalent sources impressed on a closed surface (although I've only worked with the theory in terms of time-harmonic waves).