# I Equations for observed distance/velocity in SR

1. Mar 31, 2017

### Arkalius

Hello everyone. I've just recently found this forum and it has been a lot of fun browsing around. I've recently taken a stronger interest in the topics of relativity in physics and have recently developed a much better understanding of SR (and somewhat of GR) than I'd had previously and its been fun exploring that understanding in various scenarios.

Of greater interest to me recently is how things appear to observers in relativistic situations when you consider the travel time of light. Many thought experiments and scenarios are described from the "measured" viewpoint, or as if each observer could witness all instantaneous events (from their frame) at the same moment. This has its uses, and it simplifies an already complicated subject, but I find it useful to explore what observers would actually see in reality.

To that end I'd come up with some simple equations that give a ratio of observed length and speed to the actual length and speed of an object moving toward or away from you. These are \begin{align} \frac 1 {1-\beta} \\ \frac 1 {1+\beta} \end{align} with (1) being for objects moving toward you, and (2) for objects moving away, and with $\beta = \frac v c$. These were great and all, but I wanted a more general equation that worked in 3d and 4d spacetime. In those, an object doesn't always move directly toward or away from you.

So, I came up with this more generic equation for this ratio: $$\gamma^2 \left( \beta \cos \alpha + \sqrt{1-\beta^2 \sin^2 \alpha} \right)$$ Here, $\alpha$ is the (actual) angle between the relative velocity vector and position vector. You will see that when that angle is 0 or $\pi$, the equation reduces to the two I have above. You simply multiply actual length or velocity by this value to get the observed length or velocity.

Another useful equation is for the observed angle of deflection and that is given by $$\alpha_{obs} = \alpha - \arcsin \left( \beta \sin \alpha \right)$$

I'd not seen equations like these listed anywhere that I'd looked in the past, and I found them useful for examining scenarios for how they would appear to the observers. It certainly brings a different perspective to the situation. Things moving toward us won't appear contracted, but rather stretched out. Also, things can appear to move toward us at many times the speed of light because of this. It also means nothing can appear to move directly away from us at more than half the speed of light either.

Anyway, I look forward to participating in more discussions on this forum and learning more interesting things about relativity and other topics.

2. Mar 31, 2017

### BvU

Hello Arkalius,

I advise you to first and foremost become familiar with the simpler cases of special relativity. Expressions are already complicated enough there. Learn about hte Lorentz transform and its accompanying phenomena. If you are fluent with those (conceptually and with the formalism), then it's still early enough to move on to the issues you are now messing with. I don't believe a single one of your expressions -- but you may ascribe that to my ignorance.

In the mean time play with the MIT game and wonder about this strange world. Have fun !

3. Mar 31, 2017

### FactChecker

Your equations look like they are just some sort of Dopler effect. That is not correct. You are missing the main point of SR. There are simple explanations of SR already in terms of the relativity of "simultaneity". You should pay more attention to them before trying to make your own equations.

4. Mar 31, 2017

### Arkalius

I'm already quite familiar with those. I feel like I've developed a more intuitive understanding of how Minkowski spacetime works, and understand Lorentz transformations almost instinctually now. There are scenarios I run into where I can't quite visualize it right and have to rely on a Minkowski diagram (my favorite tool for that currently is http://ibises.org.uk/Minkowski.html ) but the more scenarios I encounter the better at it I've gotten.

Well.. points for being direct I guess. I assure you, they should be accurate. I'm looking at all of the fun triangle drawings I have on my desk from the work I did on it right now, and they seem to fit with the examples of relativistic Doppler effects that I've seen.

5. Mar 31, 2017

### Arkalius

You're right that it is a kind of Doppler effect transformation. But, I think you might be missing the main point of my post, which is this: I understand the main point of SR, and wanted to move beyond, into understanding how to move from what the measured view of reality is in SR to what the observed view would be, factoring travel time of light. This view of things is in some ways less bizarre, and in others, moreso. For example, in a measured view, when accelerating away from distant objects, their clocks can appear to go backwards as our plane of simultaneity shifts. But, when looking at the observed view, this effect disappears, and clocks always tick in the forward direction. On the other hand, sufficient acceleration toward something will actually cause it to seem to stretch away further into the distance, and if you settle at a high enough velocity, make it appear as if you approach it faster than light

6. Mar 31, 2017

### A.T.

This might interest you:
http://www.spacetimetravel.org/

7. Mar 31, 2017

### Staff: Mentor

8. Mar 31, 2017

### pixel

9. Mar 31, 2017

### FactChecker

10. Mar 31, 2017

### PAllen

If you want feedback on your equations, you will need to define your observational model more. For example, it is not clear at all to me how it makes sense to talk about the apparent length of a ruler moving towards you parallel to its length. Is your formula possibly for a ruler turned perpendicular to this? Then, for defining what you 'see', you need to specify e.g. whether your idealization is a tiny spherical detector, versus a pinhole camera with a flat detector read in simultaneous captures of the movie camera frame. The latter introduces additional distortions compared to the former, and makes the orientation of the camera a necessary element of the specification.

Note there is a very cute trick that can be used for these problems. As long as you assume all objects are luminous at a standard frequency and intensity in their rest frame ( thus avoiding specifying lighting source position and motion), aberration plus Doppler can be used to get the exact appearance, including surface markings, without doing any form of ray tracing. This includes getting brightness and color correct.

Last edited: Mar 31, 2017
11. Mar 31, 2017

### Arkalius

Well, sure there are more complicated things to resolve if you're trying to model the precise appearance of a moving object, especially in 3d space. The distortions can be fairly strange as I understand, such as objects moving past you appearing somewhat rotated. I'm really only concerned with the more general aspects, the effects on apparent distance, speed, and general size in the direction of motion. That is what these equations are intended to help with. They are, in effect, a polar coordinate transformation.

But, I appreciate your input. Certainly the observed effects of relativistic motion can be quite bizarre. It's too bad we don't have access to macroscopic demonstrations of such things.

12. Mar 31, 2017

### Arkalius

Ah that does look pretty neat. It makes me think of Velocity Raptor, which is a web-based 2d puzzle game that uses special relativity and does kind of the same thing. The villain slows down the speed of light to 3mph so all of your motion is notably relativistic. The game then requires you to take advantages of the strange trappings of relativistic motion to overcome obstacles that would be impossible to surmount under normal circumstances. In fact, it is kind of what spurred my interest in the concepts of "observed" reality in SR. Later in the game, it moves from showing you a measured view of your environment to the observed view based on travel time of light, and the distortions there were quite wild, and I was fascinated by them and wanted to understand them better.

13. Mar 31, 2017

### PAllen

Right, but you have not specified enough for anyone but you to understand what your equations are supposed to mean. Having worked through both aberration and ray tracing computations of moving objects (thus experienced in the field), I have not a clue what your equations are supposed to describe.

14. Mar 31, 2017

### Arkalius

I see. The larger equation is a ratio between observed distance to measured distance of a moving object, as well as observed velocity to actual velocity. Take the measured distance between something moving relative to you with some velocity, and multiply by the value of the equation to get the distance the object appears to be from you at that instant. Similarly, multiply that by the object's velocity to see what velocity the object will appear to be moving. The 4th equation gives you the angle between the relative velocity vector and observed position vector of the moving object.

15. Mar 31, 2017

### PAllen

Define apparent distance and speed versus measured. You may think there are universal definitions of these, but that is not so. Give us yours, or a link to what you are using. Otherwise, no one can say if your equations are correct or not.

[edit: Let me be clear about how definition is so crucial. Suppose I define measured distance at a given time as the coordinates in a standard Minkowski inertial frame, with 'me' being the t axis throught the origin. Then, if an object is at some position, at some time, I define its apparent distance in terms of the angle subtended by the light from the object emitted at that position and time, when such light reaches me. If this angle is less than expected per rest dimensions for that distance, I am defining it as apparently further away. Using this definition (transverse angle subtended compared to expectation), then an object approaching me never has an apparent distance different from its measured distance. Rather than claim your equation is wrong, I simply want to know what definitions you are using.]

Last edited: Mar 31, 2017
16. Mar 31, 2017

### m4r35n357

17. Mar 31, 2017

### Arkalius

Measured distance is the $\Delta x$ in the spacetime interval equation $s^2 = \Delta x^2 - c^2\Delta t^2$, from our chosen frame. Apparent distance is how far the moving thing in question appears to be from that observer, based on the light information reaching him at that moment.

Actual velocity is just that. It is the velocity of the object of interest relative to the observer. Apparent velocity is how fast the object will appear to be traveling based on the light being received by the observer.

The angle $\alpha$ represents the angle between actual velocity vector and the actual position vector (from whence the aforementioned $\Delta x$ is derived). The calculated $\alpha_{obs}$ is the angle between actual velocity vector and the apparent position vector based on light being received at that moment.

Together these equations can tell you where a moving thing appears to be as told by light signals based on where they actually are in that instant for that observer, as well as telling you how fast the thing appears to be moving based on its actual velocity.

You have to consider some timing issues when using these equations. If a ship is stationary relative to you 9 light-days away, and then begins a journey to you at 0.9c, you won't see him traveling to you at 0.9c for 10 days. From your point of view he'll appear stationary for 9 days, and then seemingly approach at 9c for 1 day. If you just naïvely apply my equation at the moment he begins moving, it would suggest he appears to be 90 light days away at that point. So you have to be careful how you apply it.

18. Mar 31, 2017

### PAllen

You keep using terms (apparent distance) with no definition whatsoever. (also, see the edit to my post asking for this, for an example of the issue of no definition).

19. Mar 31, 2017

### Arkalius

I said "Apparent distance is how far the moving thing in question appears to be from that observer, based on the light information reaching him at that moment." Another way of putting it is how far the moving object actually was from you when it emitted the photons you are now observing. I suppose I can understand some amount of confusion, in that one cannot know the distance of a source of light merely by observing that light in an instant. You have to do other things like generate parallax etc. But it doesn't change the fact that the light you see "now" from a moving object doesn't reflect where that object is "now" (from your reference frame), but rather where it was when it emitted it. This all emerged from my interest in how one's actual view of everything around us distorts as a result of relativistic motion, spurred on from seeing a simulation of this distortion in a game. The observed distortion is significantly different than the actual spacetime distortion given by the Lorentz transformation, and this was fascinating to me.

20. Mar 31, 2017

### PAllen

This definition has precisely zero content for me. Sorry. By what model? Brightness? subtended angle? I have no idea.