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Consider two frames: the rest frame that has a stationary point charge and the moving frame which has a moving point charge.

In the rest frame the point charge has electric field lines pointing radically outward. The charge can be imagined to be surrounded by a spherical shell. In the moving frame, by length contraction, this sphere is deformed into a spheroid. The field lines still points directly away from the charge, but the perpendicular components of the field are enhanced by the gamma factor.

1. If I understand correctly, the electric field and so electric force will tend to be more vertical and also stronger in the moving frame than in the rest frame, except for the field at the x-axis which is the same in both frames?

2. To calculate this electric field, can I simply use pythagorean theorem to calculate this electric field? For example, to calculate the components of the electric field in the rest frame at an angle of 30°, we use sine and cosine to determine the components of the field. Now that I know the vertical component of the field, I also know the vertical component in the moving frame(enhcnaed by gamma factor). The electric field parallel will be the same in both frames. So now that I know vertical component and horizontal component in the moving frame, can I use pythagorean theorem to calculate the resulting field in the moving frame?

3. I suppose this does not apply to gravitational field of moving masses?