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Can a particular real force have different magnitude in different frames?

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Well, take the electromagnetic force...it has been shown that the induced magnetic field around moving charges is a relativistic reference frame effect. See this post How Special Relativity causes magnetism

Therefore a static electric force in one frame becomes a magnetic force in another.

However, from a Newtownian view, we either allow pseudoforces and say "yes" or only allow forces present in an intertial refernce frame and say NO, it only appears that way to a person (e.g., falling vs floating in space)

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Short answer: If you're talking about the spatial components of force, then yes, through a Lorentz transformation on the force four-vector.

By "real force" I assume you mean a non-inertial force, so that you are computing force from an inertial frame, or, more generally, in a freefall frame - i.e. a locally flat Minkowskian tangent space to the spacetime manifold spanned by a nonaccelerating, nonspinning "vierbein" frame so that the orthonormal unit vectors $\vec{e}_j$ fulfill $\nabla_{\vec{e}_0} \vec{e}_j$ where $\vec{e}_0$ is the timelike unit vector and $\vec{e}_1,\vec{e}_2,\vec{e}_3$ are the orthonormal space unit vectors) in the general relativistic context.

In this precise case, you therefore ask what are the Lorentz-invariant analogues of the quantities in Newton's second law? These are the four-velocity $\vec{v}$, the four-acceleration $\vec{v}$, the four-momentum $\vec{p}$ and four-force $\vec{f}$:

$$\vec{v}=\gamma(v)\,(c,\,v_x,\,v_y,\,v_z)$$ $$\vec{p}=m\,\vec{v}=(E/c,\,p_x,\,p_y,\,p_z)$$ $$\vec{a}={\rm d}_\tau \vec{v} = \gamma(v)\,{\rm d}_t \left[\gamma(v)\,(c,\,v_x,\,v_y,\,v_z)\right]$$ $$\vec{f}={\rm d}_\tau m\ \vec{v} = \gamma(v)\,{\rm d}_t\left[m\,\gamma(v)\,(c,\,v_x,\,v_y,\,v_z)\right]$$

where $m$ is the mass (i.e. rest mass - note that the "relativistic mass" $ \gamma(v)\,m$ is not a concept that is thought to be helpful for learning, these days) of the system in question), $v_x, v_y, v_z$ are the spatial nonrelativistic velocity components measured in the freefall frame, $p_x, p_y, p_z$ are the spatial nonrelativistic momentum components measured in the freefall frame, $v=\sqrt{v_x^2+v_y^2+v_z^2}$, $\gamma(v) = \left(\sqrt{1-(v^2/c^2)}\right)^{-1}$, $t$ the nonrelativistic time measured in the freefall frame and $\tau$ is the proper time $\tau = \gamma(v)\,t$.

The last of the relationships above $\vec{f}={\rm d}_\tau m\ \vec{v} = \gamma(v)\,{\rm d}_t\left[m\,\gamma(v)\,(c,\,v_x,\,v_y,\,v_z)\right]$ is Newton's second law generalized to the freefall frame.

A handy reference sheet is given by the Wikipedia "four vector" page.

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