# Red- / Blueshifting and relativistic length contraction at $0.8c$ - How would it look?

I am a CG artist working on a specific shoot, that I have no clear answer to how it would look. And I seek your help. :D

The idea for the shoot is a flyby of two Interstellar Vessels named ITV Europa and ITV Lea Sudux.

Europa is flying back to Earth while Sudux blast to where Europa is coming from. Thus I figured that the two would meet quite close to each other, as the two systems don't move with respect to each other. Thus the paths for both ships should be the same. But I am not sure about that.

Anyways, the shoot is supposed to be a GoPro-like view from the Sudux looking ahead. And this is where the problems start.

I know that the Universe at this kind of speed is probably reduced to the CMB, so the Skybox may look like this. But as far as I can tell, this is only true for what is in front of the Sudux. I am not sure how the rest would look. Or how much it is Blueshifted.

The next major issue is, what does the Europa look like? We know that we can not just add the two speeds. From this equation:

We can figure out, that from the Sudux´s POV, the apparent Velocity of the Europa should be equal to 13/14c or around 0.93c. And since Europa is flying in the other direction, all the Light emitted from her and her Shield´s should be redshifted. But I have exactly zero clues how the flyby would look. Now sure, it would be fast. Really fast. And since the ships themself are not that high, some might say you wouldn't see anything. But here is a catch. Each ship has 2 shields. One that is mounted to the main body and one that is way bigger and aerodynamic a few light seconds ahead of the craft. These shields are, in both cases, 3.5 km heigh. They are that high because of the ITV´s need to flip 180 degrees in interstellar space.

Thus my assumption is that the big shields would be the one thing you could even see at a distance. Although it would still go by really fast. In the best case, the shields might appear as a star Exponentially growing and then getting smaller once they are past each other.

Now of course, how much you see depends on the distance. I would assume there is a distance where you could actually follow the other ship with your eyes. For example, if the two are 380 000 km away from each other at there closest. The closer the two get, the bigger they appear but also the smaller the time window in which you can see anything. For now, I would like to keep the distance close so maybe 100 km?

So I guess there are really two connected questions.

The first one being: "How would the space around you look like if you were to go 0.8c in deep space?"

And the other one: "How would a ship passing by you at pretty much 1c look like if it were to go the other way?"

Maybe the answer to one is also the answer to the other, but again I don't know.

EDIT #1 Here are some WIP Pictures of the Situation. Not renders because, as it turns out, in deep space there really is no sun :D

This shows the 2 Stages of Modelling most things. One very low detail mock up to get the idea across and then the high detail one, which as you can see is still WIP. You can see the Multi Layered Shield of the Craft, the Radiators are still missing though.

This one shows what everything might look like, at least without fancy light. The Europa is around 150km away and once it is at its closest point, it will only be 100km away. The Main shield is already out of view at this point. Also, Human for scale.

• Hi! Maybe a picture of the situation will clarify the question even more (though you explained the problem very well!). Why is the abbreviation ITV and not ISV? Oct 9, 2020 at 11:27
• Hello there :D I hope the Clay Viewport Images helped a bit. @m4r35n357 gave me a really good video and i think it is possible to just create a programm that displays all the effects at this speed in an accurat way. Famous last words. ITV stands for Interstellar Transport Vehicle. I thought it made more sense as these ships main objective is to carry stuff and people from a to b. Oct 9, 2020 at 11:57
• Not renders because, as it turns out, in deep space there really is no sun Good point! That's what I always think when I see SF movies where you see spaceships floating in deep space, between the stars. "Where are the huge studio lights...?", I ask myself. Succes with your project!!! Oct 9, 2020 at 19:38

Surprisingly, 0.8c is not really fast enough to do anything visually freaky! You need to go much closer to the speed of light (according to some observer or "scene").

What you actually see is rather more complicated than the "compressed" scenery in most pop-sci "explanations" (looking at you, Mr Tompkins!). It is described by a combination of Doppler Effect (red/blue shift), and relativistic beaming (brighter in front, darker behind), aberration (things you have gone past can appear to the side in front of you!). All these effects are derived from the Lorentz Transformation, but the transformation is not the whole story.

Anyhow, enough bluster, there are a few artifacts on the internet that can give you some idea:

This really old video

A Slower Speed of Light (unfortunately now abandonware)

My videos

Of course there are plenty of others, but you should now have enough pointers to look around for yourself!

• Not gona lie, the old video is gold. It shows the effects which should be easy to copy inside the 3D Software. Although it will be a shi*show getting everything to work as it should the first time. As always with equations... I love the game. Really trippy. Is it ok if i get back to you with a basic Shader Setup ? Aka a look at how things would look according to my now updated understanding . Oct 9, 2020 at 12:11
• @ErikHall no point I'm afraid, I know nothing about shaders, I made my videos using POVray, a ray-tracer! povray.org Oct 9, 2020 at 17:27
• In that really old video, 0.8 c is apparently enough to see a whole bunch of relativistic effects, so I am confused at your statement that it is not fast enough to see anything interesting. Oct 9, 2020 at 19:36
• I think he meant this in terms of what you would see in deep space. In created a shader with all the effects as shown in the video and tbh, there aint a lot to see till .95c. The Red / Blueshifting is just about the only thing happening and even then, its not a lot at 0.8c. So really the lack of any light is the problem. But the Redshift looks really cool Oct 9, 2020 at 20:26
• @ErikHall Be warned the colors in "A Slower Speed of Light" are 100% wrong. Their Doppler shift simulation makes no sense at all. The colors in the first video (rendered by Backlight) seem more plausible. Oct 9, 2020 at 22:57

This problem is best treated from the rest frame of the Europa (the ship being photographed), not the Sudux (the ship taking the photographs).

In that frame, the Europa is at rest for a long period of time, and space around it is filled with a static pattern of light. Ordinary 3D renderers simulate a camera at rest somewhere in this space. You want a camera in motion somewhere in this space.

If you idealize the camera as a pinhole camera with an infinitesimal shutter time, then a moving camera and a stationary camera detect the same light if their pinhole is at the same location when the shutter opens. The actual photo looks different because of what happens to the light after it's passed the aperture. You can simulate this by transforming the light between the Europa and Sudux frames at the aperture, and doing a standard simulation of the camera interior.

To summarize: work out the position (in Europa's rest frame) where each frame of the video will be taken, render to a cube map with an ordinary 3D renderer, then "distort" it with aberration, Doppler shift, and brightening/dimming. This will get you an accurate simulated image without the need to import your models into some obscure relativistic renderer with limited features. You still need a simple relativistic ray tracer, but it only has to support a static skybox at infinity.

(Fisheye Quake is an old project that used this technique to simulate a different kind of lens distortion.)

To calculate the aberration and Doppler shift, for each pixel on the camera's sensor, calculate a unit vector representing a ray at the aperture using standard ray-tracing math, then add a time component equal to -1 (so you now have a lightlike four-vector), then Lorentz transform it from the Sudux frame to the Europa frame. The time component of the result is (the negative of) the Doppler shift factor, and the spatial components are the direction in which you should shoot the ray to the cube map. The brightening/dimming factor can probably be approximated accurately enough by the product of the angular difference between adjacent horizontal and vertical pixels. (This will cause vignetting even in a camera at rest, so you may want to compensate for that.)

If you want a starry background for the shot rather then just blackness, then render the ship to a transparent background, distort it, then composite it on top of a skybox that's distorted by the same algorithm, but using the speed of the Sudux relative to the galaxy instead of relative to the Europa.

Don't forget the gamma factor when calculating the position of the camera. In the Europa rest frame, the separation between the locations of adjacent frames is $$γvΔτ$$ where $$Δτ$$ is the proper time interval between frames (e.g. 1/60 second).

To correctly simulate the appearance of a Doppler shifted spectrum, you need to know the whole spectrum, or at least the part of it that's shifted into the visible range. You can't correctly simulate Doppler shift on RGB colors. You could try to correctly simulate the whole spectrum, or you could just fake it. The appearance of surfaces under arbitrarily colored lights can't be correctly simulated in RGB either, but people do it anyway.