# 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?

• "traveling at a speed that causes one second to dilate to .5 seconds" -- Relative to what? Time dilation can only be defined relative to some choice of inertial reference frame in SR, where it depends only on the velocity relative to the chosen frame, but all large-scale GR coordinate systems are non-inertial, so there's no reason to expect any simple relationship between velocity relative to that coordinate system and time dilation relative to that coordinate system. – Hypnosifl Jan 10 '15 at 1:07

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.

No you cannot just multiply together a gravitational time dilation and relativistic time dilation to get a total time dilation. The time dilation is calculated by comparing the elapsed proper time for the two observers you are comparing.

I discuss this in my answer to How does time dilate in a gravitational field having a relative velocity of v with the field?, where I calculate the total time dilation for an object moving radially in a gravitational field. In fact I compare the time dilation with the answer you get by just multiplying the GR and SR dilations and show that they differ.

You might also be interested in What is the correct formula for gravitational time dilation for a satellite in a circular orbit?, which discusses the time dilation for an orbiting satellite. Also A clock in freefall, which discusses the time dilation for an object falling freely into a black hole.

• John, can you check my math on physics.stackexchange.com/questions/158212/… – Joe Jan 11 '15 at 0:02
• I thought the link in my comment above drew an interesting connection between both gravitational and kinematic time dilation. I just don't know how to speak without using math why the connection exists. – Joe Jan 11 '15 at 0:06
• @Joe: that's an accidental coincindence. In Reissner-Nordstrom or Kerr/Kerr-Newman black holes the cooincidence you noticed doesn't exist. – John Rennie Jan 11 '15 at 6:14
• So the math is wrong? – Joe Jan 11 '15 at 13:56
• @Joe: there's nothing wrong with your working, but you're thinking there must be some physicaly significance to the similarity in the time dilation equations, but there isn't. It's just coincidence. – John Rennie Jan 11 '15 at 16:28