In the context of general relativity, it is indeed accurate to say that energy is a fundamental factor contributing to both time dilation and gravity. However, it's important to clarify the specific mechanisms involved in each case.
In general relativity, mass and energy are related through Einstein's famous equation, E = mc², which states that mass can be considered a form of energy. Therefore, when we talk about the effects of mass on time dilation and gravity, we are essentially referring to the effects of energy.
Regarding time dilation, it occurs due to the curvature of spacetime caused by mass and energy. The presence of mass or energy curves spacetime, and this curvature affects the flow of time. Clocks in regions of stronger gravitational fields or near massive objects will experience a slower passage of time compared to clocks in weaker gravitational fields or distant from massive objects.
On the other hand, gravity itself is also a consequence of the curvature of spacetime caused by mass and energy. In general relativity, massive objects create a curvature in spacetime, and other objects moving through that curved spacetime follow what we perceive as gravitational paths. The motion of objects, as well as the behavior of light, is influenced by the curvature of spacetime caused by the distribution of mass and energy.
So, while it is accurate to say that energy is a common factor contributing to both time dilation and gravity, it's essential to recognize the specific role of energy in each case. In time dilation, the curvature of spacetime affects the flow of time, while in gravity, the curvature of spacetime determines the paths of objects in the presence of mass and energy.