general relativity - Time Dilation Properties

I've read up on time dilation and how gravitational/kinematic time dilation work but I have not received a clear answer on how the two work together.

If you are observing something traveling at a speed that causes one second to dilate to .5 seconds, and it is in a position within a gravitational field that causes one second to tick by at .75 seconds for a non-moving object. Then what is the total time dilation effect being observed?

I've heard it is not additive but a product of the two, does that mean the effect would be:

$$(.5)*(.75)=.375$$

Time ticking by .375 seconds for every 1 second from the observers point of view?

Answer

Yes. Imagine you are observing a clock on a fast-moving ship in a gravitational well and think of what each of those statements is really saying:

1. 
   
   

   The ship's speed is such that 1 second of proper time takes 2 seconds of your time.

   
   


2. 
   

   The ship is in a gravitational field such that 1 second of proper time takes 3 seconds of your time.

   
   

How do we combine these? Let the clock on the ship tick 1 second (i.e. let 1 second of proper time elapse). Then this 1 second takes 2 seconds for an observer that is stationary with respect to you but who is also in the high gravitational field. Each of these 2 seconds (as measured by the stationary observer in the high gravitational field) takes 3 seconds of your time to elapse due to the gravitational time dilation.

Thus while you observe 1 second pass on the ship's clock, $2 \cdot 3=6$ seconds have passed for you.