As I understand physics there is a limit for every equation in the way there must not be an infinite quantity present in it.So my question is related to the Lorentz transformations for a body that its mass increases due to its movement.Is there any experimental deduction that it should be a limit for the increase of the mass in the way that for higher and higer velocities the mass is not increasing so fast as it would do if there weren't some effect that might produce resistence for the mass to reach every thinkable amount?
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The "relativistic mass," $m_\text{rel} = \gamma m,$ is an outdated way to think about relativity. In modern discussions of relativity, we discuss the invariant mass $m$ instead. So your question is whether there are limits on the relativistic factor $\gamma = 1/\sqrt{1-v^2/c^2}$.
Within special relativity, there is no such limit. However, there is an environmental effect which limits the lifetime of extremely relativistic particles. In our universe we have the cosmic microwave background, a gas of photons remaining from the universe's hotter, denser past that have a thermal spectrum at a temperature of a few kelvin. That thermal equilibrium is present only in a privileged reference frame. (Cosmologists call it the "co-moving reference frame.") An object that is boosted with respect to the co-moving reference frame will see a warmer microwave background in the forward direction and a cooler CMB in the backward direction. If the object is electrically charged and boosted so that the temperature of the microwave background is above about 100 MeV, there are photons in the spectrum with enough energy to cause pion photoproduction, and the object will emit pions and their decay products and slow down with respect to the local co-moving CMB.
People who hunt for cosmic rays call this effect the GZK cutoff, which is a good search term. We do observe cosmic rays with energies above the GZK limit, but we conclude those particles must have been accelerated somewhere nearby, since the CMB is opaque to such particles at intergalactic length scales.
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