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Maxwell's Equations in the Light of General Relativity

  1. Aug 24, 2010 #1
    In the realm of General Relativity one must use Maxwell's Equations in their covariant form[the ordinary derivatives in the traditional form should be replaced by the covariant derivatives].

    Now we select a point, A ,on the 4D spacetime surface and setup a "local inertial frame" on it by some suitable transformation.In the original frame we denote the axes by x1,x2,x3 and x0. In the transformed frame the corresponding axes are x1',x2',x3' and x0'.There is a small laboratory associated with the local inertial frame so that we may conduct experiments on electromagnetism. Interestingly in our local inertial frame we may use Maxwell's equations in their traditional form[ie we may use ordinary derivatives instead of covariant derivatives].

    Suppose Maxwell's equations yield a solution of the form stated below[for Bx1']

    Bx1'[x1,x2',x3',x0']= Ax1'^3 + B x2'^2+C x3'^2+D x0' +E --------------- (1)
    [1)Consideration of one such result will be sufficient for our purpose
    2)Bx1' denotes the x1'-component of the magnetic field ]

    Now let a dense object approach the laboratory[or let the laboratory approach it].This will upset the space-time metric changing,changing the value of the coefficients[g(mu,nu)] at A.We will have to set up a new "local inertial frame" at A by some other suitable transformation.We denote the axis of this new local inertial frame by x1'',x2'',x3'' and x0''. In the new frame Maxwell's Equations will again have the same traditional form .
    Consequently the form of the solution equations will not change though the individual entries may change in their values.
    The new solution,corresponding to equation(1), will obviously be of the form:

    B'x1''[x1'',x2'',x3'',x0'']= A' x1''^3 + B' x2''^2+C' x3''^2+D' x0'' +E' -------- (2)

    [B'x1'' denotes the x1'' component of the magnetic field]
    The constants in a differential equation are simply a reflection of the boundary conditions.
    A,B,C,D and E reflect the original boundary conditions. A',B',C',D' and E' reflect the transformed boundary conditions. In most typical problems the boundary conditions at infinity do not change and so the constants should remain the same.But in our case we are conducting our experiment in a limited region of space and within a small interval of time. Better, we keep the constants different.

    Equations (1) and (2) clearly bring out the fact that electric and magnetic fields should under effects of gravity.The manner in which the change should occur has also been indicated through the two equations.Only one equation[relating to the Bx1- component] from the solution set has been used to illustrate this point.

    If the electric and magnetic fields change under the influence of gravity ,we may use this effect to detect gravitational waves. It may be easier to detect changes in the electric or magnetic fields than to make a direct observation of changes in spacetime curvature due to gravitational effects.

    [We can always find a relationship between sets (x1',x2',x3'and x0') and (x1'',x2''x3'' and x0'') from the transformations (x1,x2,x3,x0)-->(x1',x2',x3'and x0') and (x1,x2,x3,x0)-->(x1',x2',x3'and x0')]
    Last edited: Aug 24, 2010
  2. jcsd
  3. Aug 24, 2010 #2
    Everybody knows the first part...an example is gravitational lensing.
    Yet your second part is not the technique being used to search for gravitational waves....at least not from what I have read...

    maybe because, for example, an airplane passing overhead would also have a gravitatioal influence....or even a lightning storm...

    As you likely know, here are the basic techniques:

    LISA and LIGO have been discussed on these forums...
  4. Aug 24, 2010 #3
    Well I can always suggest a new method.What's the harm?

    Some other facts:
    1)Changes in the electric and the magnetic fields in response to changes in space-time curvature[in a manner I have tried to show or by methods contrary to it] should have played an important role in the early cosmological processes. We must take care of them when in trying to trace back to those early events by theoretical or by experimental means.In fact they must have taken care of these considerations in the LHC experiments. The advisers/experts of the forum will be able to tell us a lot more on the points and we will definitely gain from such information.

    2) The manner in which the electric and magnetic fields should change if some spacecraft happens to pass near some dense object could also be an interesting issue in relation to this problem.
  5. Aug 24, 2010 #4


    Staff: Mentor

    I agree that there is no harm in suggesting alternate methods. What would be important would be a proposal of a specific experimental apparatus and an analysis of the sensitivity to gravitational waves compared to LIGO or LISA and an analysis of the sensitivity to confounding effects such as vibration, EMI, temperature, etc. I have no feeling one way or the other which method would wind up having more signal to noise in the end.
  6. Aug 24, 2010 #5
    that's how we all learn....and discovering the flaws in our own arguments....in fact most ideas scientists pursue are wrong, at least in their original form...so Ptolomy and those before him who thought stars were suspended on a heavenly globe, too, was wrong, Copernicus better, Galileo saw more witha telescope, then we discovered elliptical orbits,and only recently..was it just a 100 years ago, we realized OH my gosh, our galaxy is NOT the entire universe...there are OTHER galaxies.....and OMG (again) Hubble found they are accelerting in expansion and Vera Rubin studied them and realzied "their orbital spin does not conform with existing theory"..and voila dark matter was hypothesized...and on and on....

    The only group of people who NEVER make mistakes are politicians. Here in the forums, criticism of new ideas can be severe and unending....the trick is to tell which is legitimate and which is not.

    By the way, what is the LHC doing with regards to gravity??? Or does your comment refer to other work??
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