The velocity of body and the increase in relativistic mass

Question

When a spaceship moving in a space at a high speed its mass will be increase due to it's high velocity. If there's an astronaut inside this ship does his mass also increase or the spaceship protect the astronaut from this increase in mass.

Answer 1

The relativistic mass has the following expression: $$m=m_{0}\gamma=\frac{m_{0}}{\sqrt{1-\frac{v^2}{c^2}}}$$ If an object is moving with velocity $v$ relative to a reference frame $S$, its relativistic mass as seen from this reference frame is greater than its rest mass - as you can see from the formula above. But in your own frame of reference, your relativistic mass is equal to the rest mass because you're not moving with respect to yourself.

In the scenario that you have described, the relativistic mass of the spaceship and the relativistic mass of the astronaut increase with the same amount, but only relative to other frames of reference (that are not moving along with the spaceship). But he will not feel heavier. He will not experience any increase in mass because he's not moving relative to himself, so there is no relativistic mass in his frame of reference.

Working physicists do not use the concept of relativistic mass anymore. It is better to think in terms of energy-momentum. The total energy of a free particle is:

$$E^2=p^2c^2+m^2c^4$$

here $m$ is just mass. It is redundant to say rest mass. Also in high energy physics natural units are often used. In natural units the energy-momentum equation is:

$$E^2=p^2+m^2$$

so the mass of a particle is just the difference between its total energy and its momentum:

$$m^2=E^2-p^2$$

$m$ is a Lorentz invariant quantity meaning that is has the same value in every inertial reference frame. The energy and momentum in a particular reference frame may differ from the energy and momentum as seen from another reference frame, but their difference will always give the same quantity (the mass).

E.g. A photon is massless, so its energy is equal to its momentum (in every inertial reference frame).