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Suppose a ball moving with 0.9c in the x-axis on a frame S' with axes parallel to frame S on which an observer sees the ball. Due to length contraction it appears as a vertical ellipsoid, when it moves in the vertical direction then as a horizontal ellipsoid, and when it moves at an angle 45$^\circ$ to both axes how does the ball now appears? I have two thoughts:

  • Spherical because we can get two equal components of speed in the two axes which give equal contraction
  • Diagonal Ellipsoid using the previous logic. enter image description here
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If by appears you mean what the ball looks like to someone watching it, then it always appears spherical. This rather surprising result happens because to work out what an observer actually sees you have to calculate how the light travels from the surface of the ball to the observer's eye, and the travel times from different points on the ball's surface are different. The end result of this (rather involved) calculation is that the ball always appears to the observer to be spherical.

Assuming you're not esking what the observer sees, but just what shape the ball is in our coordinate frame, then it is always contracted in the direction of travel. So a ball moving at 45º would look like:

Ball

We are free to arrange our axes as we want, so no matter what direction the ball is going I can rotate my axes so my $x$ axis lies along the direction of motion of the ball. If you look at the diagram above and imagine rotating your axes 45º anticlockwise then the ball will be moving along the new $x$ axis and therefore will be shortened along the new $x$ axis.

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