Maxwell’s Equations Maxwell’s Equations from Special Relativityfrom Special Relativity

James G. O’BrienJames G. O’BrienPhysics Club PresentationPhysics Club PresentationUniversity Of ConnecticutUniversity Of Connecticut

Thursday October 30th , 2008Thursday October 30th , 2008

HistoryHistory

In the beginning, there was Maxwell…In the beginning, there was Maxwell…

Well, really Faraday, Gauss, Ampere, Biot, Savart, and Coulomb.Well, really Faraday, Gauss, Ampere, Biot, Savart, and Coulomb.

HistoryHistory

In 1905, Einstein Unified the separate In 1905, Einstein Unified the separate theories of Electricity and Magnetism into theories of Electricity and Magnetism into one law, the Electromagnetic Theory.one law, the Electromagnetic Theory.

In 1905 the famous paper, entitled “On the Electrodynamics of moving bodies” was published by Albert Einstein, forever revolutionizing the way physicists not only view E+M, but the entire basis of the underlying principles of physics.

Special RelativitySpecial Relativity

On the surface, SR is a simple theory with only two postulates, one On the surface, SR is a simple theory with only two postulates, one philosophical and one physical. The combination of the two challenges philosophical and one physical. The combination of the two challenges the standard theory of the measurements of all observations.the standard theory of the measurements of all observations.

1. (Philosophical) : The Laws Of Physics are invariant, or the same in all INERTIAL (non-accelerating) reference frames.

2. (Physical) : The speed of light c, is the same in ALL reference frames.

Moral of the story..if c is constant, and c=d/t, distance and time are no longer consistent as two separate entities…they must both change to accomodate the constancy of c.

Evolution of DimensionalityEvolution of Dimensionality

In order to accommodate the postulates of SR, we must treat time and space on the same footing, namely treat time as another coordinate, placing our existence in a 4-D space-time. This can be done mathematically by defining now, 4-vectors, where the zeroth component is in the time direction, viz.

As the new displacement 4-vector. Similarly, we now have a definition of absolute distance, namely:

Consequences of SRConsequences of SR

We can see that treating time and space on the same footing leads to two separate We can see that treating time and space on the same footing leads to two separate types of times, depending on the observer. This can be seen by some fun algebra:types of times, depending on the observer. This can be seen by some fun algebra:

Consequences of SRConsequences of SRPlacing this new restriction on space and time being treated together, it means that ordinary 3 vectors must become 4 vectors, and thus, ordinary total derivatives now must be taken over space and time as:

Consequences of SRConsequences of SRA note about this 4-space tensor notation. The metric tensor of special relativity, can be used to raise and lower indices (not very important for now, but you can treat it for the most part as matrix algebra where: (note, repeated indices means a summation)

since

Remove HistoryRemove HistoryNow lets suppose Maxwell and the others never wrote down the Theory of Electricity and Magnetism. Instead, all we had was some understanding that there existed a Vector Field A, which interacted at long range.

The Question we wish to answer, is are the postulates of special relativity (and therefore the consequences which follow) enough to derive the entire electromagnetic theory, namely, Maxwell’s Equations.

***IMPORTANT: Again, we are only assuming the postulates of SR, the existence of a four dimensional vector field A, and a souce, which again must be a four vector, which we shall conveniently call, J.

The Continuity Equation:The Continuity Equation:

We can make the first leap with four dimensional notation quite easily. We can observe the fact that the source J, is created by moving particles, which obey the continuity equation (derived from a simple application of the divergence theorem).

Staring at this long enough, we realize that if we make a 4-vector out of the scalar density and the 3-vector J, such that:

ContinuingContinuingWe can see that this process will be very important, namely the construction of a 4-vector from some scalar field and an ordinary three vector…basically how the original four vector of displacement was defined.

Now the question is as follows: What is the most general four dimensional field equation for a four vector field A up to second order in derivatives, with source four vector J?

The Field Equation:The Field Equation:The most general field equation thus must take the form of:

And recall that repeated indices imply a summation. Do not worry, we will write this out in components explicitly shortly in order to solve it. For example, the time only component, or the zeroeth component is:

The Field Equation cont..The Field Equation cont..

Let us assume A=A(r), spherically symetric, time independent for ease. Then lets assume we are far from the source, so the right hand side is zero. The field equation then results in:

The homogeneous solution is left to the viewer

Boundary ConditionsBoundary Conditions

Applying Boundary conditions, that the force be long range gives us the value for a. The field must be long range, so the negative exponent is ruled out since it decays too quickly. The positive exponent is ruled out since the field would get exponentially stronger as you moved away, so the only valid value for a, is zero!

Thus, the homogenous solution gave us the value for a. Now we can try to resolve b.

The Field Equation:The Field Equation:

Now becomes

Let us now resolve b by taking a derivative with respect to all coordinates of the above, as:

The Field Equation:The Field Equation:Now, rearranging derivatives since partials all commute, we see:

Which allows us to draw one of two conclusions:

The Field Equation:The Field Equation:

Becomes:

Which can be renamed as:

Where F is a tensor of rank 2. We will leave this equation for the moment, but we now have resolved our field equation to a mush simpler form.

Almost there…Almost there…

With these two equations, we can see that the following identity has to hold. Taking a derivative of the second, and taking linear combinations will force the following to be valid:

Proof:

So…So…Thus, we are now left with two equations, the general field equation for out vector field A, in terms of the current, and the previously proven identity due to the continuity.

and

Now let us actually compute some components of these and see what they are…

First, the Field Tensor can be computed via:

And we now make the relation between A as a 3 vector and a scalar (just like the current vector), as:

Definitions:Definitions:By making use of the relation between the physical fields which we can measure, and the scalar and vector potentials:

We can see that the definition given for F, yields the 16 different values for the two indices as:

The Source EquationThe Source Equation

Taking the zeroeth component (the time component) of the above, yields:

Which if we compare to the original Maxwell, gives us a value for e, namely that

But with this definition of e, we yielded Gauss’ law!

Using Components:Using Components:

Taking the first component (a spatial component) of the above, yields:

Which is the x component of Ampere’s Law!

Similarly, the second and third components of the above will yield the y and z components of Ampere’s Law respectivly.

The Source-less EquationThe Source-less Equation

Taking mu=1, nu=2, sigma=3, yields: (spatial components only)

Which is the Guass law for Magnetic Fields!

The Source-less EquationThe Source-less Equation

Taking mu=0, nu=2, sigma=3, yields: (one time and 2 spatial components)

Which is the x component of Faradays Law!

ConclusionsConclusionsWe have shown that the four Maxwell equations can be generated from the 2 space time equations,

and

Remember, these were derived from the most general field equation for a vector field A in the space time domain. Thus, this is the ONLY possible solution for the long range force described by a vector field.

This makes a very important statement about the structure of our universe at large. The vector field A has to give rise to the electromagnetic field, and it is the unique solution. Thus to explain other forces, such as gravity, we cannot use a vector field since it is already solved. This is why the other forces are explained by different field theories…such as gravity which is a tensor field, and the strong force which is a scalar field.

Maxwell’s Equations Maxwell’s Equations from Special Relativityfrom Special Relativity

Thank you!Thank you!