No, the increase in mass from time dilation in special relativity is not similar to the mass increase in a gravitational field. It's important to note that mass increase in a gravitational field is not a concept supported by modern physics.
In special relativity, according to Einstein's theory, as an object moves at speeds approaching the speed of light, its time slows down relative to an observer at rest. This time dilation effect is associated with an increase in the object's relativistic mass. However, it's crucial to understand that this increase in relativistic mass is purely a consequence of the object's energy increasing, not an actual increase in its rest mass. The object's rest mass remains the same.
On the other hand, in a gravitational field, such as near a massive object like a planet or a star, the concept of mass increase is not applicable in modern physics. In Einstein's theory of general relativity, the presence of mass or energy curves the fabric of spacetime, causing objects to follow curved paths. This curvature is what we perceive as the force of gravity. However, the mass of an object does not increase due to the gravitational field itself. The motion of objects in a gravitational field is determined by the curvature of spacetime, rather than an increase in mass.
The notion of mass increase in a gravitational field was a concept proposed in Newtonian physics, where it was suggested that an object's gravitational mass and inertial mass might be different. However, in modern physics, the equivalence principle of general relativity states that the gravitational mass and inertial mass are the same, and thus, the concept of mass increase due to gravity is not applicable.
To summarize, the increase in mass from time dilation in special relativity is a consequence of increased energy, while the concept of mass increase in a gravitational field is not supported by modern physics.