# How will the length of a moving rod change for an observer 'seeing' it?

I was studying length contraction and it considered the following scenario.

A rod is moving at a velocity $$v$$ with respect to a frame $$S$$. A frame $$S'$$ observes the rod stationary and thus measures proper length $$l_0$$ of the rod. Now, if the observer in $$S$$ frame measures the length $$l$$ of rod at same time t, then it can be shown that $$l_0=\gamma l$$

It is further mentioned that if the $$S$$ observer looks at the moving rod, it won't see it as shorter. It further says that

If the time that is required for the light from each point on the rod to reach the observer’s eye is taken into account, the overall effect is that of making the rod appear as if it is rotated in space.

I don't know what it means and how to prove it. Also, will a similar situation occur with time dilation ?

I'd be grateful if someone could help me with this.

• en.wikipedia.org/wiki/Terrell_rotation Commented Nov 26, 2019 at 12:48
• @probably_someone Thanks for the article. But still, frankly speaking, it is beyond my understanding and I would be grateful if someone could explain me. Commented Nov 26, 2019 at 15:14

There are two separate effects which should not be confused, one being length contraction and the other being Terrell rotation.

Length contraction is the reduction in the measured length of an object when it is measured in a frame moving relative to it.

Terrell rotation is a change to the apparent length and orientation of an object when viewed by an observer moving relative to it.

The distinction between the two definitions I have given relies upon an understanding of the difference between viewing and measuring an object. You can get a feel for this by imagining the difference in relation to a stationary object. Imaging you have a train stationary alongside a platform. You might measure its length at 200m, say. Its apparent length will depend upon the position from which you view it. If you stand close to one end of the train and look along it, it will appear foreshortened, an effect of perspective.

The Terrell effect is analogous to the foreshortening effect of perspective on a stationary object.

Calculating the effect of terrell rotation is not straightforward.

• "Terrell rotation is a change to the apparent length and orientation of an object when viewed by an observer moving relative to it." . If a rod is moving, the rod should be Lorentz - contracted. If the camera (or observer) is moving, the photo-film should be Lorentz contracted. At first glance pictures should be different, on the other hand it is relativity. For example, if moving camera takes picture at that instant when it is just opposite the center of the rod, rays from the ends of the rod towards aperture and from aperture to film form similar triangles.
– user139020
Commented Jan 16, 2020 at 12:30

First I suggest you absolutely ignore any reference to how long it takes the light to get to an observer who sees something. Observers in Special Relativity use special imaginary cameras to make their observations. These cameras build their images from rays of light that all left at the same time, rather than our physical cameras that record rays that all arrive at the same time.

Now to get the observations right. The observer moving with the rod sees its proper length. That doesn't change for him even if his speed changes as long as he stays with it.

The observer that sees it passing, lying in the direction of travel, sees a shorter version of it. Other observers moving in the same direction but at different speeds will see other versions of it with different lengths, all shorter than its proper length.

If the rod happened to be a train, and the passengers all had carefully synchronised their watches on the train, the original observer would see the time on the watches of passengers in the front of the train showing earlier times than those at the rear. He sees a shorter rod because the time (for the rod) at the front is earlier than at the rear. He is seeing a time-smeared version of it. Other observers will see different time-smearing and different lengths.

As well as time dilation and Lorentz contraction, you need the relativity of simultaneity to fully understand what is going on. It all starts with Einstein's thought experiments.

The internet is your friend here. Look for material that matches your own way of understanding.

If you then want to understand what an observer would see without the magic camera, you have to add another time calculation for each point of the rod (or other object) to allow for the time it takes the light to reach your eye. This means the more distant parts will be seen at an earlier time in their motion than the nearer parts, and this is what causes the distortions referred to as Terrel rotation.

A straight rod pointing in the direction of motion will seem shorter or longer as a result of this distortion, but in this special case there will be no bending or rotation.

• No -- the answer is indeed Terrell rotation as probably_someone pointed out. Unfortunately I'm not qualified to provide the explanation either... Commented Jan 16, 2020 at 1:27