In the framework of special relativity, the equation for the change in gravitational potential energy due to a change in height is given by:
ΔU = m * g * Δh * (1 - (v^2 / c^2))^(-1/2)
Where: ΔU is the change in gravitational potential energy m is the mass of the object g is the acceleration due to gravity Δh is the change in height v is the velocity of the object c is the speed of light in a vacuum
This equation takes into account the effects of time dilation and length contraction that occur at relativistic speeds. It shows that the change in gravitational potential energy depends not only on the change in height but also on the velocity of the object.
In the framework of general relativity, which provides a more comprehensive description of gravity, the equation for the change in gravitational potential energy becomes more complex and is described by the Einstein field equations. These equations involve the curvature of spacetime caused by the distribution of mass and energy. The change in gravitational potential energy due to a change in height is intimately connected to the overall geometry of spacetime and is not easily expressed in a simple equation like in special relativity.