According to the theory of special relativity, the mass of an object does indeed increase with increasing velocity. This concept is often referred to as "relativistic mass." However, it's important to note that the concept of relativistic mass is not commonly used in modern physics, and instead, the focus is on the object's rest mass, which remains constant regardless of its velocity.
In special relativity, the relationship between an object's relativistic mass, its velocity, and energy is described by the famous equation E = mc², where E is the object's energy, m is its relativistic mass, and c is the speed of light in a vacuum. As an object approaches the speed of light, its relativistic mass increases, and so does its energy.
Now, regarding the warping of surrounding spacetime, the increase in relativistic mass alone does not directly cause a significant change in the curvature of spacetime. The curvature of spacetime is primarily determined by the distribution of mass and energy within it, rather than the mere increase in an object's mass due to its velocity.
However, it is important to consider that as an object's velocity increases, its energy and momentum also increase, which can have effects on spacetime. According to Einstein's field equations of general relativity, the presence of energy and momentum, including relativistic mass, contributes to the curvature of spacetime. Thus, a highly energetic or massive object moving at high speeds could potentially affect the curvature of spacetime in its vicinity.
In practical terms, however, the effects of relativistic mass on the curvature of spacetime are usually negligible unless the object in question is extremely massive and moving at velocities approaching the speed of light. In most everyday scenarios, the warping of spacetime due to relativistic mass is not significant enough to be observed or have a noticeable impact.