Gravimagnetic Submarine Warfare?
"The Question is: What is The Question?" John Archibald Wheeler
Metric Engineering Investigations 1.6
Special Relativity considerations: In a global inertial frame in Cartesian coordinates the frame-invariant small differential space-time interval ds obeys
ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2
Any Lorentz transformation to another inertial frame in uniform relative motion to the first preserves this Cartesian form. That is under the action of O(1,3) x^u -> x^u'
ds^2 = (cdt')^2 - dx'^2 - dy'^2 - dz'^2
A transformation to an accelerated noninertial frame in 1905 special relativity sense will formally look lie a transformation to curvilinear coordinates with the possibility of off-diagonal terms. Not so however for the trivial 3D change to spherical polar coordinates where
ds^2 = (cdt)^2 - dr^2 - r^2(dtheta^2 + sin^2thetadphi^2)
Example 1 uniformly accelerating noninertial frame in the Galilean limit gt/c << 1
t' ~ t
z' ~ z - (1/2)gt^2
dz = dz' + gtdt
dz^2 = dz'^2 + (gt/c)^2(cdt)^2 + 2(gt/c)dz(cdt)
So the important part of the metric in the z'-t plane is
(cdt)^2[1 - (gt/c)^2] - dz'^2 - 2(gt/c)dz(cdt)
The mixed space-time off-diagonal cross-term is a longitudinal "gravimagnetism" effect. In this case translational acceleration of the noninertial frame is a source of gravimagnetism.
Bz = 2gt/c points along the z-axis direction of translational acceleration.
Special relativity where gt/c -> 1 changes this to the hyperbolic motion problem using hyperbolic functions.
Example 2 Galilean relativity Wx'/c << 1 etc. limit of rotating noninertial frame about z-axis
x = x'cosWt - y'sinWt
y = x'sinWt + y'cosWt
dx = dx'cosWt - x'sinWtdt - dy'sinWt - y'cosWtdt
dy = dx'sinWt + x'cosWt + dy'cosWt - y'sinWtdt
ds^ = [1 - W^2(x'^2 + y'^2)/c^2](cdt)^2 + (2Wy'/c)dx'(cdt) -(2Wx'/c)dy'(cdt) - dx'^2 - dy'^2 - dz'^2
Note the inhomogeneous transverse gravimagnetism here from the physical rotation of the noninertial frame. That is
Bx'(y') = 2Wy'/c
By'(x') = 2Wx'/c
The gravimagnetic 3-vector B = goi points in the plane perpendicular to the axis of rotation. See Ray Chiao's "Gravity Radio" A(em).B(gravity) interaction Hamiltonian papers online for the application of gravimagnetism in rotating superconductors for the high efficiency transduction between gravity waves and electromagnetic waves with application to submarine warfare C^3 and a host of other applications to the cosmology of the Big Bang if it can be achieved. Note the "Stern-Gerlach" inhomogeneous gravimagnetic "potential well" for gyroscopes in this solution. What happens in a medium c/n when n >> 1. Will that enhance the strength of gravimagnetism?