# How does one interpret the relative motion of an object in orbit as it compares to to the object it is orbiting?

How does one interpret the relative motion of an object in orbit as it compares to to the object it is orbiting?

In flat spacetime, it's pretty easy to determine relative motion. If Alice sees Bob as traveling at 100mph, and herself as stationary - Bob sees Alice as traveling at 100mps and himself as stationary. It's perfectly reciprocal.

How does this principle work with gravity introduced?

Assume Alice is at the center of gravity of a large, non-rotating planet, and Bob is in a perfectly circular orbit:

1.) Alice can tell the distance to Bob.

2.) Alice can verify she is not rotating, and at what speed she would need to be rotating to keep facing bob.

Using the above two observations, Alice determines bob is moving at 100mph.

Now, Bob is under the impression he is in an inertial frame. Does Bob view Alice as moving at 100mph relative to him? Or does Bob view Alice as stationary? Intuitively, it seems like Bob would view Alice as stationary.

EDIT FOR CLARITY:

More precisely, the combined formula for time dilation based on gravity and relative speeds can be written as:

(1−2GM/rc2−v2/c2)^(-1/2)

which reduces to: (1−v2/c2)^(-1/2) when gravity is not present

and reduces to: (1−2GM/rc2)^(-1/2) when relative motion is not present

so when Alice and Bob are calculating the rate at which they observe each other's time as passing, what v (velocity) should Alice record for Bob, and vice versa, in order for the formula to accurately match observations.

Please directly answer this portion of the question before explaining "why", to avoid vague explanations without a clear answer of which velocities should be recorded in order for the formula to match observations.

• Just make a sketch , with the situation some small time apart, than you see th two relative motions. Commented Sep 18, 2023 at 15:47
• Trula, Please answer the question. If I "sketch" this out, Alice sees herself as not moving, and bob as moving 100mph. Bob does not see the reciprocal of this. He sees himself as not moving (because freefall is an inertial frame), and he also sees Alice as stationary (always distance "x" to his left, perhaps rotating in his frame of reference). Is this right, in terms of relativity (for relative time calculations) Commented Sep 18, 2023 at 16:47