Relativistic Force: Transforming Forces Between Reference Frames

[Lostinthought]

hi,
i was tryin to learn some relativistic particle dynamics and came across reltivistic force
$$m\frac{d^{2}x}{d\tau^{2}}$$,how does this help in transforming forces between reference frames?i dnt understand how this wrks since the vale of velocity changes with time

 

[Bill_K]

Lostinthought, this is the most useful way to write F = ma in special relativity because it's manifestly covariant. That is, the position vector x = (x, ct) is a 4-vector. The mass m is a scalar, and the proper time τ is a scalar. Therefore dx/dτ and d²x/dτ² are also 4-vectors, and when you write F = m d²x/dτ² you know that F will be a 4-vector too.

If you tried to express the same equation in terms of the coordinate time derivative d/dt you would not only have to deal with explicit factors of γ, but even a term involving dγ/dt.