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## Main Question or Discussion Point

Are the accelaration and forces in different inertial referance frame equal ?

- Thread starter arpon
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Are the accelaration and forces in different inertial referance frame equal ?

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Ibix

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No.Are the accelaration and forces in different inertial referance frame equal ?

The post by Ibex suggests why, though when you factor in all the different relativistic effects, it's probably too hard to figure out all the details on one's own.

I'm not sure where the best reference is, Wiki has some discussion of the issues at http://en.wikipedia.org/wiki/Mass_in_special_relativity#Transverse_and_longitudinal_mass

The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest massmoving in thexdirection with velocityvand associated Lorentz factoris

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I think your question has two different answers, depending on how you define "acceleration" and "force", and depending on what you mean by two vector quantities being equal.Are the accelaration and forces in different inertial referance frame equal ?

Here are two different ways to define the acceleration of an object:

- Coordinate acceleration.
- "Proper" acceleration (which basically is acceleration relative to objects in freefall)

- Component-wise equality.
- Covariant equality.

The changes that happen to vectors (actually, 4-vectors, because in Special Relativity, there are 4 components to vectors; besides the usual 3 spatial components, there is also a time component) when you change reference frames can be thought of in the same terms. You can think of it as the vector as remaining constant, and only your description of the vector changes.

Coordinate acceleration definitely changes when you change reference frames, no matter how you slice it. Proper acceleration as a 4-vector can be thought of as remaining the same in all inertial reference frames, and the only thing that changes is the description of the 4-vector in terms of components.

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Is there any straight forward way to prove these equations by using special relativity ?I'm not sure where the best reference is, Wiki has some discussion of the issues at http://en.wikipedia.org/wiki/Mass_in_special_relativity#Transverse_and_longitudinal_mass

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It's straight-forward, but tedious.Is there any straight forward way to prove these equations by using special relativity ?

I should point out, though, that the notion of "relativistic mass" is not used in modern physics, at all. So I really consider the effort to derive "transverse" and "longitudinal" mass to be a complete waste of effort. What's much more worth-while is to learn the modern way of doing Special Relativity, which is to use 4-vectors, whose components transform in a straight-forward way under a change of coordinates.

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