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Anamitra
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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')]
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')]
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