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From the Lorentz Force to the Binet and Euler Equations: Gravity versus Vorticity

From the Lorentz force there appears a dichotomy between gravitation in terms of the attraction of central potentials and the Euler force dealing with pseudo forces in non-inertial frames. Such forces appear in the geodesic equations of general relativity, of which the Lense-Thirring effect is a classical example. Phenomena related to "frame-dragging" can described completely classical without any recourse to general relativity.

The Lorentz force gives the acceleration in an inertial frame as

d^2 r/dt^2= - grad V -dv/dt - dr/dt X w (1)

where V is a scalar potential, v the velocity in a rotating frame, and w =(e/m) B, the angular speed related to the magnetic field. The last term in the equation is commonly referred to the Lorentz force, and it is not the Coriolis force, v X w which can be made to disappear by an appropriate coordinate transformation like all the other pseudo-forces. The first two terms in the equation are usually referred to the electric field, where v is the vector potential.

Introducing the relation between the two velocities

dr/dt = v- w X r, (2)

into the equation gives

d^2 r/dt^2 = - grad V - dv/dt + v X w - w X w X r,

which can be rearranged to read,

d^2 r/dt^2 - w^2 r = - grad V - dv/dt + v X w. (3)

The left-hand side is the radial acceleration, and transforming to u=1/r, is can be compared to the Binet equation,

u" + u = F/h^2 u^2, (4)

where the force is F=grav V, and h=r^2 w, the conserved angular momentum. Thus, the Lorentz force separates into an equation for a central potential, where if F is the inverse-square law, the right-hand side of (4) is constant, and from (3) there results the Euler equation

dv/dt = - w X v = - (curl v) X v = -(v nabla) v + (1/2) grad v^2, (5)

where we introduce the fact that w = curl v, and div v =0. Taking the curl of both sides of (5)


Dw/Dt = dw/dt + (v nabla) w= (w nabla) v. (6)

The acceleration in a rotating frame is given by the sum of pseudo-forces plus the Lorentz force

dv/dt = w^2 r + 2(w X v) + dr/dt X w, (7)

where the first term on the right hand side is the centrifugal acceleration, with (w r)=0, and the second term is the Coriolis force. Equation (7) will be seen to be identical to the Euler equation (5) once the relation between the two velocities, (2), has been introduced.

Thus there is a clear dichotomy between scalar gravitation acting radially, and the pseudo-forces which can be described in terms of the Christoffel symbols of the second kind. In contrast to the forces that enter into the Binet equation, (4), the pseudo-forces in (7) can be made to vanish by a mere change of coordinates except for the last term, showing that the magnetic field has an effect even in an inertial frame.

When viewed in these terms, laws like the Biot-Savart law,

w = v x r/r^2,

is a mere expression of the angular momentum. Moreover, the so-called gravito-magnetic potential is simply the curl of the velocity which is the curl of the vector potential

A = I w s/r,

where I is the moment of inertia and s is the "height" of the vortex tube, which has an area of the base proportional to the moment of inertia. The velocity is the curl of A,

v = I w (s X r)/r^3,

and the gravito-magnetic potential is its curl. Letting W= I w s, it is given by

H = [W- 3(W e)e]/r^3

where e is the unit normal in the radial direction. H exemplifies the frame dragging phenomenon in general relativity, which essentially comprises the Lense-Thirring effect. We will have much more to say about this in an upcoming blog.

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