Special Relativity says that as an object’s speed gets closer to the speed of light, its length (ie its linear measurement from front to back in the direction of travel) reduces; this is called length contraction. The formula for this contraction is quite simple: L = L₀ x sqrt(1-(v/c)²), where L₀ is the length at rest , c is the speed of light in a vacuum, and v is the object’s velocity.

Consider two observers, one at rest in a stationary reference frame, the other moving with some constant velocity v₁ relative to the first observer, and a cube moving past them from left to right at relativistic speed, say 0.99c.

What do the observers see? Is the cube more flattened for the stationary observer or the moving observer? Or do they see the same thing?

They see differing things. That’s why it’s called relativity.

To the object in relative motion, they see no difference in themselves, but to the “fixed” observer, they see the relativistic effects.

What observers would actually see is a bit complicated because of the differences in travel time of the rays of light coming from different parts of the object.

Terrell rotation

Terrell rotation or Terrell effect is the name of a mathematical and physical effect. Specifically, Terrell rotation is the distortion that a passing object would appear to undergo, according to the special theory of relativity if it were travelling a significant fraction of the speed of light. This behaviour was described independently by both James Terrell and Roger Penrose in pieces published in 1959, though the general phenomenon was noted already in 1924 by Austrian physicist Anton Lampa.

Due to an early dispute about priority and correct attribution, the effect is also sometimes referred to as the Penrose–Terrell effect, the Terrell–Penrose effect or the Lampa-Terrell-Penrose effect.

Terrell’s and Penrose’s papers pointed out that although special relativity appeared to describe an “observed contraction” in moving objects, these interpreted “observations” were not to be confused with the theory’s literal predictions for the visible appearance of a moving object. Thanks to the differential timelag effects in signals reaching the observer from the object’s different parts, a receding object would appear contracted, an approaching object would appear elongated (even under special relativity) and the geometry of a passing object would appear skewed, as if rotated.

For images of passing objects, the apparent contraction of distances between points on the object’s transverse surface could then be interpreted as being due to an apparent change in viewing angle, and the image of the object could be interpreted as appearing instead to be rotated. A previously-popular description of special relativity’s predictions, in which an observer sees a passing object to be contracted (for instance, from a sphere to a flattened ellipsoid), was wrong.

Terrell’s and Penrose’s papers prompted a number of follow-up papers, mostly in the American Journal of Physics, exploring the consequences of this correction. These papers pointed out that some existing discussions of special relativity were flawed and “explained” effects that the theory did not actually predict – while these papers did not change the actual mathematical structure of special relativity in any way, they did correct a misconception regarding the theory’s predictions.

Also see Can You See the Lorentz–Fitzgerald Contraction? Or: Penrose-Terrell Rotation from the Usenet Physics FAQ.

Here’s a crude video that illustrates Penrose-Terrell rotation and other SR optical effects.

Optical Effects of Special Relativity

btm said:

Special Relativity says that as an object’s speed gets closer to the speed of light, its length (ie its linear measurement from front to back in the direction of travel) reduces; this is called length contraction. The formula for this contraction is quite simple: L = L₀ x sqrt(1-(v/c)²), where L₀ is the length at rest , c is the speed of light in a vacuum, and v is the object’s velocity.

Consider two observers, one at rest in a stationary reference frame, the other moving with some constant velocity v₁ relative to the first observer, and a cube moving past them from left to right at relativistic speed, say 0.99c.

What do the observers see? Is the cube more flattened for the stationary observer or the moving observer? Or do they see the same thing?

In the case of the “moving observer” moving in the roughly the same direction as the cube relative to the “stationary observer”, the cube is more flattened for the stationary observer.

When the “moving observer” is moving at a direction not even remotely in the same direction as the cube then, as PM 2Ring has implied, things start to get complicated.