Relativity paradox: binary star image separation changes with observer velocity? - Forums

I was reading Project Hail Mary by Andy Weir and thought of a relativity paradox I haven't seen before.

Suppose you are imaging a pair of binary stars that are separated by 1 arcmin from Earth using a telescope of focal length of 1 meter. The stars on the sensor should be separated by 0.29 mm (1000mm * pi/180/60).

Now suppose an astronaut with the same imaging equipment is on a spaceship traveling at a speed of 0.87c towards the binary star system. Due to length contraction, everything in the direction of the spaceship is contracted by a factor of 2 (the gamma factor) observed by the astronaut. But the direction perpendicular to the spaceship's direction is unchanged. Now, if the astronaut observes the binary stars, the distance between the observer and the stars in astronaut's perspective is now half the distance in earthlings' perspective. So the astronaut would see the binary stars separated by 2 arcmin, and the star images on the sensor would be separated by 0.58 mm!

On the other hand, people on Earth would observer the telescope on the spaceship is half of its original length. So the image size should be half, too. But the size of the image on the sensor shouldn't depend on the observer since its orientation is perpendicular to the velocity of the spaceship. How do we resolve this paradox?

Edit: Well, I forgot about the headlight effect https://en.wikipedia.org/wiki/Relativistic_aberration

The stars in the traveling direction will actually be pinched together in the perspective of the astronaut. The image separation is actually smaller, not bigger.

This is what I work on, so I’ll jump in. Hope I do not embarrass myself, as sometimes this topic might be tricky. You’d be surprised how many physics professors fall into the traps of special relativity!

The resolution is relativistic aberration, not naive length contraction.

The key issue is that the observed angular separation of the two stars cannot be found by saying “the distance is Lorentz-contracted while the transverse separation stays the same, so the angle must get bigger.” That is not how observation works in special relativity. What the astronaut sees is determined by the directions of the incoming light rays, and those directions are transformed by the aberration formula.

For light coming from roughly straight ahead, the apparent angle shrinks by about sqrt((1-beta)/(1+beta)).

At v=0.87c, this factor is about 0.264. So a binary separated by 1 arcmin in Earth’s frame would appear only about 0.26 arcmin apart to the astronaut, not 2 arcmin. For a 1 m focal length telescope, the image spacing would therefore shrink from about 0.29 mm to about 0.077 mm.

The other part of the paradox also mixes frames. The telescope’s 1 m focal length is its proper focal length in its own rest frame. In that frame, the optics behave normally. From Earth’s frame, the telescope is moving, but you cannot analyze the image by just saying the telescope is length-contracted so the image scale must halve. You also have to transform the incoming light consistently. Once you do that, both frames agree on the actual detection events on the sensor.

This is also closely related in spirit to Terrell-Penrose rotation. When Einstein published special relativity in 1905, it was thought for a long time that the Lorentz contraction was a visible effect. Only 50 years later, Terrell showed that Lorentz contraction is invisible (the paper's title is “Invisibility of Lorentz contraction”). Surprisingly, it has such an easy proof. When I teach this class, I try to emphasize how easy things can be misunderstood for decades. In both cases, the mistake is to assume that what you see is just the Lorentz-contracted geometry. But a visual image is determined by light arriving at the observer, not by a simultaneous snapshot of positions in one frame.

In your binary-star example, the relevant effect is mainly aberration, which compresses angles toward the forward direction. In Terrell rotation, the key effect is that light from different parts of an extended object is emitted at different times before reaching the camera. So they are not the same phenomenon, but both are examples of why apparent visual appearance is not the same thing as Lorentz-contracted geometry.

So your edit is correct. The stars are pinched closer together in the astronaut’s forward view, not spread farther apart.