Exactly Why C is impossible for Massive Objects?

[Agerhell]

I know, but a change in velocity doesn't have to be a change in speed.

That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?




So would γ increase to a change in direction?

For the case of electromagnetism and a small particle of charge q and mass "invariant mass" m accelerated in an electromagnetic field classically we have:

\frac{d}{dt}(m\bar{v})=q(\bar{E}+\bar{v}\times\bar{B})

Assuming "m" not to vary with time we get:

\frac{d\bar{v}}{dt}=\frac{q}{m}(\bar{E}+\bar{v}\times\bar{B})

Instead assuming that the classical momentum m\bar{v} in the first expression above should be replaced by its relativistic counterpart m\gamma\bar{v} where

\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

and solving the differential equation we get:



\frac{{\rm d}\bar{v}}{{\rm d}t}=\frac{q(\bar{E}\cdot\hat{v})}{m\gamma}\left(1-\frac{v^2}{c^2}\right)\hat{v}-\frac{q}{m\gamma}((\bar{E}+\bar{v}\times\bar{B})\times\hat{v}) \times \hat{v}.

As you can se there is a factor of 1/\gamma also in the second term on the right side. This means that it is harder to accelerate a particle in an electromagnetic field also in the direction transverse to its velocity, such as for a circular orbit.

 

[harrylin]

No, that's why Sommerfeld added the famous clarifying footnote to that statement in Einstein's paper, saying "to the first approximation", by which he meant quasi-statically, which correctly establishes the physical basis, i.e., the coordinate systems Einstein is referring to are those in which mechanical inertia is homogeneous and isotropic.[..]

Sorry for this little off-topic question, but are you sure that it was Sommerfeld?
I assumed that it was Einstein who authorised that translation footnote, and so I have been wondering if Einstein had forgotten what he meant, as I was pretty sure that Einstein meant reference coordinate systems as used for Newton's mechanics, just as you also explain (and there is nothing "approximate" about that); according to me it should have been "are supposed to hold good". So, if it was not Einstein but someone else who put that footnote there, then that mystery is solved for me.

 

[harrylin]

[..]
That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?
[..]
So would γ increase to a change in direction?

γ is a function of speed, so that a particle that moves at constant speed in a circular orbit has constant γ and thus constant inertia.
Moreover, a particle can be deflected sideways in such a way that its speed remains constant. In such cases its kinetic energy remains constant, and thus also its inertia.