- #1

jcap

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For example if we have a particle with inertial mass ##m_i## and charge ##q## moving in flat spacetime in an electromagnetic field ##\vec{E},\vec{B}## with relativistic 3-velocity ##\vec{v}## then its equation of motion with respect to its proper time ##\tau## is

$$q(\vec{E}+\vec{v}\times\vec{B})=m_i\frac{d\vec{v}}{d\tau}.\tag{1}$$

In curved spacetime the equation of motion ##(1)## becomes

$$q\ {F^\mu}_\nu\ v^\nu=m_i\Big(\frac{dv^\mu}{d\tau}+\Gamma^\mu_{\rho\sigma}\ v^\rho\ v^\sigma\Big)\tag{2}$$

where ##{F^\mu}_\nu## is the electromagnetic tensor, ##v^\mu## is the 4-velocity of the particle and ##\Gamma^\mu_{\rho\sigma}## is the metric connection.

Neither Eqn ##(1)## nor Eqn ##(2)## actually explain

*why*it takes a force ##\vec{F}=m_i\ \vec{a}## in order to impart an acceleration ##\vec{a}## to an object with an inertial mass ##m_i##.

Is this correct?