How does orbital velocity time dilation combine with gravity near a black hole?

[RonaldG]

I've been trying to wrap my head around the concept of gravitational time dilation near a supermassive black hole, specifically how time would practically pass for an observer in a stable orbit versus one falling in. My confusion is whether the orbital velocity's time dilation effect would be additive or multiplicative with the immense gravitational effect from the black hole itself.

[Andrew_R]

I picture it this way: gravity drags down the rate of clocks the deeper you go. If the object is also moving fast, relativity adds another slowdown. In GR you don’t add the two factors, you compute the rate dτ/dt along the actual path. For a stable circular orbit the velocity is fixed by gravity, so the net slowdown isn’t two separate numbers but a single factor that depends on radius and the orbit.

[EllaL]

On a napkin, I tried to write dτ/dt as a product of a gravitational redshift factor and a Doppler-like factor, and it felt more natural than adding. Still, keeping track of which frame you compare to at infinity trips me up.

[TimothyGL]

Two clocks at the same radius with different worldlines won’t tick the same. A free-falling clock versus a hovering clock at the same radius will have different rates when you compare to a distant observer. The orbiting one has extra kinetic dilation; the falling one doesn’t have the same steady velocity. So again not a simple sum.

[PatrickL]

Is the real snag about what you mean by time passing from afar, or locally? I keep bumping into that idea when I try to pin it down.