: Can I apply equations of Special Relativity like time dilation, length contraction.... to an object which is moving on a circle like the Moon or a satellite?

If you're careful special relativity can be used to deal with such accelerated states to an extent, but in restricting yourself to special relativity you must first be able to distinguish the inertial frames from among the accelerated ones. Then you can consider any local region of the accelerated frame to be a frame comprised of little boosts between many different instantaneously inertial frames. For instance take an observer with a clock whose time is represented by t' to be accelerating and another observer with an identical clock whose time is represented by t who is in an inertial frame. As the accelerated clock ticks for a small time interval dt' the inertial frame observer observes that the time for that clock's ticks to go by according to the inertial frame's clock is different by dt = gdt' where g is a function of variable velocity. Otherwise, there are two main problems you face when using special relativity in accelerated circumstances. The first is that you prefer inertial frames and so you must be able to determine what frames are inertial in the first place. The second is that when doing calculations for remote processes from the accelerated frame you must be able to account for the lack of simultaneity between the different inertial frames comprising your accelerated frame. For instance consider the traditional presentation for the twin paradox. One twin stays on earth while the other makes a round trip to another star. The ship is taken to be in an inertial frame on the way there and in another inertial frame on the way back. When they meet back the twin who stayed behind aged the most. Using the above equation its simple enough to show but that is using the earth as the inertial frame for the calculation to find how their final ages compare. In order to do the calculation from the accelerated frame while restricting our math to special relativity we must account for a lack of simultaneity component that occurs during the traveling twin's transition from the inertial frame going away from the earth to the inertial frame going back toward the earth. Just before the acceleration the traveling twin observes the earth twin is say 23 years old but just after the acceleration lasting just a moment the accelerated twin observes the earth based twin is say 28 years old. That time must be added into the calculation in order to do the math from the accelerated frame when only using special relativity. In general relativity you have the advantage that you needn't prefer inertial frames the way we did here. Instead of considering a lack of simultaneity as above you can due the physics from the accelerated frame as a single good frame of its own with the consideration of the gravitational field involved. By gravitational field I mean consideration of the form of the geometry of space-time contained by the metric tensor according to this frame. From the ship frame, instead of the ship accelerating, a gravitational field arises accelerating the earth and the ships rockets engines are being used to keep it from accelerating in the presence of this gravitational field. Then calculation for the aging of the earth based twin during earth's acceleration can be done from the ship frame using the geometry of the space-time and the geodesic equation. The disadvantage in this case for using general relativity is that the math for this calculation is more complicated.

For Cartesian coordinates in special relativity the space-time geometry or gravitational field can be read off the invariant interval ds2 = dct2 - dx2 - dy2 - dz2

For the case of an accelerating spaceship accelerating in the x direction we can consider the ship to be held still by the engines in a gravitational field accelerating the earth by introducing the gravitational field(space-time geometry) contained in the following invariant interval

ds2 = (1 + ax/c2)2dct2 - dx2 - dy2 - dz2

Above a is the gravitational acceleration as a function of time t. (These coordinate are now the ship frame coordinates not the earth based frame.)

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