Gravitational and movement related time dilation

[serp777]

Consider a massive object moving close to the speed of light. Imagine it travels close to a black hole, but does not enter the black hole. How do you calculate the exact time dilation experienced by the massive object's reference frame? How does time dilation due to movement stack with gravitational time dilation?

 

[grav-universe]

Imagine a distant static observer that observes a local static observer's clock at the same place that the massive object is moving. According to the distant observer, the local observer's clock is ticking at a rate that is sqrt(1 - 2 m / r) slower due to gravitational time dilation. Locally SR is valid so the static observer also measures the time of the massive object be kinematically sqrt(1 - (v/c)^2) slower. Since there is no relativity of simultaneity between the static observers, the distant observer will also agree that the rate of time of the moving massive object is sqrt(1 - (v/c)^2) slower than the local clock, while the local clock is sqrt(1 - 2 m / r) slower than the distant observer's own clock, so according to the distant observer that is static to the black hole, the total time dilation of the massive object is sqrt(1 - 2 m / r) sqrt(1 - (v/c)^2), where v is the locally measured speed of the object.