What happens if the moving frame in special relativity is non-inertial?

We have the lorentz transformation equations for relating inertial frame of references.

What will the transformation equations be if the frames are non inertial?

Is there any "pseudo-force like thing" in special relativity?

• Three distinct questions here with three distinct answers. Jul 1 '18 at 20:10
• Sorry,now I have edited the question. Jul 1 '18 at 20:18
• Not to be snarky with you, but you have read through this: en.wikipedia.org/wiki/Non-inertial_reference_frame ?
– user198207
Jul 1 '18 at 20:26
• For a linear acceleration: en.m.wikipedia.org/wiki/Rindler_coordinates ; For non-zero y and z speeds, still Special Relativity (search for General Lorentz Transformation or Poincare Symmetry); for rotation you may need to use General Relativity. Fictitios forces are gravity in non inertial frames according to the equivalence principle. Jul 1 '18 at 21:01

SR (special relativity) theory describes inertial reference frames at constant relative velocity. Nevertheless a non inertial frame can be tracked in an inertial reference frame via a continuous set of inertial reference frames instantaneously at rest with the accelerated frame.

You choose as stationary frame an inertial frame (in which SR assumptions hold) and then you measure the non inertial frame with the Lorentz transformation, however with the relative velocity $v$ and the Lorentz factor $\gamma$ referring to the inertial reference frame at each instant in time at rest with the accelerated frame.

Note: By the way, this is also how to explain the twin paradox.

• Thank you so much!Can you tell me name of a book which has this theory explained in more details?perhaps mathematically too? Jul 2 '18 at 14:29
• It is not a theory on top of SR. Simply you track the accelerated frame in that way. As for the mathematics, it is the Lorentz transformation. Of course, the velocity $v$ and the Lorentz factor $\gamma$ will vary at each instant in time. Jul 2 '18 at 17:03

If non-inertial is taken to mean that the frame changes the speed of its movement with respect to an observer even when the direction of its motion remains the same, then it does not belong in special relativity. When a frame changes its speed with respect to an observer, consider in special relativity this observer and a location in the frame a large distance from this observer in the direction of the motion of the non-inertial frame. Due to the change in speed with respect to the observer the relativistic effects give the location a large amount of change in its distance from the observer with respect to the observer. This change in distance is larger for distances that are larger. The problem occurs when the distance is large enough to have the change in distance due to relativistic effects on distance faster with respect to the observer than the speed of light. Similar problems occur with special relativity when there is acceleration between inertial frames with different speeds. The change due to the relativistic effect in distances between locations depends on the change in speed. Also clocks separated in distance in the direction of the motion have a change due to relativistic effects on time related to how much they are out of synchronization. This also depends on the change in speed. These changes due to relativistic effects increase as the distance between the locations and between the clocks are larger with respect to the observer.