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This is how modern relativistic dynamics is formulated.

The equation of motion for general relativity is four-vector force equals mass times four-vector acceleration

[tex]F^{\mu } = mA^{\mu }[/tex]

I have already shown in another thread that in special relativity four-vector acceleration [tex]A[/tex] is related to coordinate acceleration [tex]a[/tex] and coordinate velocity [tex]v[/tex] by

[tex]A^{\mu } = \gamma ^{2}(a^{\mu } + \gamma ^{2}v^{\mu }(\mathbf{v}\cdot \mathbf{a}/c^{2}))[/tex]

(Im choosing [tex]\mathbf{v}\cdot \mathbf{a}[/tex] to represent the ordinary three component dot product of coordinate velocity and coordinate acceleration and using v and a to represent those even when I choose to give them a fourth element as indicated by the greek index.)

In special relativity four-vector force is the derivative of four-vector momentum with respect to proper time:

[tex]F^{\mu } = dp^{\mu }/d\tau [/tex]

From time dilation this can be written

[tex]F^{\mu } = \gamma dp^{\mu }/dt [/tex]

But the coordinate time derivative of momentum is the ordinary force so

[tex]F^{\mu } = \gamma f^{\mu } [/tex]

Now using the results in the first equation one arrives at

[tex]f^{\mu } = \gamma m a^{\mu } + \gamma ^{3}mv^{\mu }(\mathbf{v}\cdot \mathbf{a}/c^{2}))[/tex]

This last equation is the correct dynamics equation for special relativity expressed in terms of mass which does not change with speed m and coordinate velocity and coordinate acceleration yielding the ordinary force.

The equation of motion for general relativity is four-vector force equals mass times four-vector acceleration

[tex]F^{\mu } = mA^{\mu }[/tex]

I have already shown in another thread that in special relativity four-vector acceleration [tex]A[/tex] is related to coordinate acceleration [tex]a[/tex] and coordinate velocity [tex]v[/tex] by

[tex]A^{\mu } = \gamma ^{2}(a^{\mu } + \gamma ^{2}v^{\mu }(\mathbf{v}\cdot \mathbf{a}/c^{2}))[/tex]

(Im choosing [tex]\mathbf{v}\cdot \mathbf{a}[/tex] to represent the ordinary three component dot product of coordinate velocity and coordinate acceleration and using v and a to represent those even when I choose to give them a fourth element as indicated by the greek index.)

In special relativity four-vector force is the derivative of four-vector momentum with respect to proper time:

[tex]F^{\mu } = dp^{\mu }/d\tau [/tex]

From time dilation this can be written

[tex]F^{\mu } = \gamma dp^{\mu }/dt [/tex]

But the coordinate time derivative of momentum is the ordinary force so

[tex]F^{\mu } = \gamma f^{\mu } [/tex]

Now using the results in the first equation one arrives at

[tex]f^{\mu } = \gamma m a^{\mu } + \gamma ^{3}mv^{\mu }(\mathbf{v}\cdot \mathbf{a}/c^{2}))[/tex]

This last equation is the correct dynamics equation for special relativity expressed in terms of mass which does not change with speed m and coordinate velocity and coordinate acceleration yielding the ordinary force.

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