Gravitomagnetism

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Edward Harris published a paper in the American Journal of Physics concerning the analogy between gravity (within the context of general relativity) and electrodynamics. [1] The abstract reads, in part 

Starting from the equations of general relativity, equations similar to those of the electromagnetic theory are derived. It is assumed that the particles are slowly moving (v << c), and the gravitational field is sufficiently weak that non-linear terms in Einstein's equations can be neglected. For static fields, the analogy to electrostatics and magnetostatics is very close.

The equation of motion in general relativity for a particle in free-fall (i.e. the 4-force on the particle is zero) is given by

The quantities

are the called the Christoffel symbols of the first kind and

are called the Christoffel symbols of the second kind. The 4-velocity of the falling particle can be expressed as

where g º dt/dt and v^k º (c, v). The 4-momentum is define as

where m º gm₀ We will be concerned only with the nth component (n = 1, 2, 3) of this equation. Eq. (1) may be written in terms of these quantities as

Expand the second term on the left hand side of Eq. (5) to give

Write the metric tensor, gab, as

where h_ab = diag(1,-1,-1,-1) is the Minkowski metric. Assume the particles are moving slow enough such that if the speed is v then terms in v² can be ignored and v^av^b » 0. Assume also that the field is weak, i.e. |h_ab | << 1 . Then Eq. (5) may be approximated as

To find Christoffel symbols we note that the Minkowski metric is used to raise indices in the weak field approximation. Substituting Eq. (8) into Eq. (2) gives, using the Minkowski metric to raise the index l

First find 

where f_bl are defined as

Write f_bl as

The first Christoffel symbol to evaluate is

Contracting with velocity gives

The first term on the right hand side of Eq. (15) has the value

Eq. (15) thus becomes

We can now write Eq. (9) as

Next find

We can now evaluate the first Christoffel symbol on the right side of Eq. (19).

Substituting Eq. (21) into Eq. (19) gives

The first term on the right side of Eq. (22) is referred to as the gravitoelectric field while the second term on the right is referred to as the gravitomagnetic field.

Note: Since I have not yet interpreted the last two terms (i.e. I don’t have an intuitive understanding of them) then this web page should be considered unfinished. It is online so as to demonstrate where the gravitoelectric field and gravitomagnetic field come from. 

References: 

[1] Analogy between general relativity and electrodynamics for slowly moving particles in weak gravitational field, Edward G. Harris, Am. J. Phys. 59(5), May 1991

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