Transcript Gravitational Faraday Effect Produced by a Ring Laser

Maxwell’s Equations
from Special Relativity
James G. O’Brien
Physics Club Presentation
University Of Connecticut
Thursday October 30th , 2008
History
In the beginning, there was Maxwell…
Well, really Faraday, Gauss, Ampere, Biot, Savart, and Coulomb.
History
In 1905, Einstein Unified the separate
theories of Electricity and Magnetism into
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 Relativity
On the surface, SR is a simple theory with only two postulates, one
philosophical and one physical. The combination of the two
challenges 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 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 SR
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:
Consequences of SR
Placing 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 SR
A 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 History
Now 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:
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:
Continuing
We 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 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..
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 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:
Now becomes
Let us now resolve b by taking a derivative with respect to all coordinates of
the above, as:
The Field Equation:
Now, rearranging derivatives since partials all commute, we see:
Which allows us to draw one of two conclusions:
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…
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…
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:
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 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:
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 Equation
Taking mu=1, nu=2, sigma=3, yields: (spatial components only)
Which is the Guass law for Magnetic Fields!
The 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!
Conclusions
We 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
from Special Relativity
Thank you!