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- invitation to discuss general covariance
The equation of motion for a particle in a gravitational field is
ai = -Γijk vj vk
In inertial coordinates the Lorentz force is
mai = qFij vk
So it seems like F corresponds to Γ. Just like F is expressed in terms of the derivatives of A, the christoffel symbols are expressed in terms of derivatives of gij:
Γijk = 1/2 gli [ gil,k + gkl,i - gil,l ]
Fij = gli [ Aj,l - Al,j ]
So apparently gij are the gravitational potentials just like A is the electromagnetic potential. What is the precise form of this analogy? What is the analogue of the gauge invariance under A → A + dχ? Is it general covariance? And where does Rijkl fit into all of this?
ai = -Γijk vj vk
In inertial coordinates the Lorentz force is
mai = qFij vk
So it seems like F corresponds to Γ. Just like F is expressed in terms of the derivatives of A, the christoffel symbols are expressed in terms of derivatives of gij:
Γijk = 1/2 gli [ gil,k + gkl,i - gil,l ]
Fij = gli [ Aj,l - Al,j ]
So apparently gij are the gravitational potentials just like A is the electromagnetic potential. What is the precise form of this analogy? What is the analogue of the gauge invariance under A → A + dχ? Is it general covariance? And where does Rijkl fit into all of this?
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