Calculating Angular Distance in SR

[yuiop]

Hi,
In a previous thread https://www.physicsforums.com/showthread.php?t=240886 we concluded that an observer moving away from a luminous object would see the luminosity as reduced by a factor of 1/(1+z)^n where n is 3 or 4 and z is defined as :

$$z= \frac{\sqrt{1+v/c}}{\sqrt{1-v/c}} -1$$

The question in this thread is how would the apparent angular size of an object be calculated?

For example if a observer was passing the Earth at a relative velocity of v/c moving away from the sun, would the angle subtended by the disk of the Sun appear to larger or smaller (and correspondingly nearer or further) than the angle measured by an observer that stationary with respect to the Earth?

First guess would be be that distances tangential to the relative motion are Lorentz transformed so that there would be no change in angular size. Further thought suggests that the moving observer would see the the Earth-Sun distance as length contracted and so she should see the Sun as subtending a larger angle so as to maintain the aspect ratio. In other words the moving observer considers himself to be nearer the Sun than the stationary observer and so she should see the Sun as a larger disk.

Relativistic aberration also suggests that an observer would see objects behind him as subtending a larger angle than when she is at rest with those same objects at the same distance.

Please ignore the effects of gravity to keep thing (relatively) simple :)

 

[Vanadium 50]

It's not trivial. To first order, the Lorentz contraction and the difference in light travel time cancel, but to higher orders they do not. c.f. "Terrell Rotation".

 

[yuiop]

It's not trivial. To first order, the Lorentz contraction and the difference in light travel time cancel, but to higher orders they do not. c.f. "Terrell Rotation".

I suspect when everything is taken into account then everything should exactly cancel out to any order or we would have a method to detect absolute motion.

 

[DrGreg]

I think Terrell Rotation is relevant only for 3D objects i.e. where different visible parts of the object are at different distances from the observer.

For this experiment I think we have to assume that the relevant visible parts (the edges) of the object are all the same distance (at any moment) from the observer. In that case, I believe the angle will be invariant i.e. the same to all observers.

To see this, turn the experiment on its head -- instead of two observers and a single object, consider instead one observer and two identical objects. For the thought experiment we can imagine the objects to be ghostly entities that can pass through each other without interaction. One object stationary relative to the observer, the other moving towards the observer.

Consider the moment when the two objects coincide. (Note this is possible only for 2D objects facing the observer, or the 2D perimeter of an object as postulated in paragraph 2 above, otherwise length contraction will not permit exact coincidence). At that moment, both objects are emitting photons from identical events in spacetime, so the photons the observer later sees from this position will be identical. In other words, the observer will later see both objects coinciding and therefore with the same angular size.

 

[yuiop]

I think Terrell Rotation is relevant only for 3D objects i.e. where different visible parts of the object are at different distances from the observer.

For this experiment I think we have to assume that the relevant visible parts (the edges) of the object are all the same distance (at any moment) from the observer. In that case, I believe the angle will be invariant i.e. the same to all observers.

To see this, turn the experiment on its head -- instead of two observers and a single object, consider instead one observer and two identical objects. For the thought experiment we can imagine the objects to be ghostly entities that can pass through each other without interaction. One object stationary relative to the observer, the other moving towards the observer.

Consider the moment when the two objects coincide. (Note this is possible only for 2D objects facing the observer, or the 2D perimeter of an object as postulated in paragraph 2 above, otherwise length contraction will not permit exact coincidence). At that moment, both objects are emitting photons from identical events in spacetime, so the photons the observer later sees from this position will be identical. In other words, the observer will later see both objects coinciding and therefore with the same angular size.

Hi Dr Greg,

I have now gone full circle and gone back to the original argument that there is no there is no difference in in angular size for the moving and stationary object at the same distance in Special Relativity in line with your argument and the observation that perpendicular lengths are not length contracted. My initial concerns about the impact of length contraction in the parallel direction and SR aberration are canceled out (I think) by the relativity of simultaneity. Thanks :)

CMB data suggests that the ratio of luminosity distance to angular distance is (1+z)^2. The earlier thread suggests that relative luminosity in SR is proportional to 1/(1+z)^4 when spectral index is taken into account and by the inverse square rule of radiation that equates to a luminosity distance of (1+z)^2 which in turn is inline with an SR angular distance of unity.

 

[Vanadium 50]

I suspect when everything is taken into account then everything should exactly cancel out to any order or we would have a method to detect absolute motion.

You suspect wrongly.

First, it doesn't cancel exactly. Terrell Rotation is a very real effect. Second, it depends only on the relative velocities of the observer and object being observed, so there is no mechanism to determine absolute motion.