Transforming Object Positions b/w Frames: A Procedure

[Killtech]

TL;DR Summary

How is the this normally handled?

Let's same I have an observer A and B that initially occupy the same point at $t=0$ but they have a relative velocity to each other.

Now let's assume there is an object C that moves in a circular motion around some point from A's frame. The initial condition/position is given (in A's frame).

What is the usual procedure to transform C into B's frame? Given that the SoS (surface of simultaneity) disagrees between A and B it seems to be quite a complicated task. My guess would be that the most general approach would be to calculate the entire trajectory of C, which in this case should form a helix in A's spacetime and then calculate the SoS of B and determine where they intersect. That would be where C is located at $t=0$ in B's frame, right?

I guess for objects moving with constant velocities and some other special cases there are formulas to simplify the calculation. But for a non-trivial movement this seems to be quite a lot of work. It's also quite annoying for the idea of an initial conditions which therefore are frame specific, i.e. not so "initial" from the perspective of other frame.

 

[Ibix]

What is the usual procedure to transform C into B's frame? Given that the SoS (surface of simultaneity) disagrees between A and B it seems to be quite a complicated task.

Yes. You write $x=x_0+R\sin(\omega t)$ and $y=y_0+R\cos(\omega t)$ and it's easy to write $x'(t)$ and $y'(t)$, but converting those expressions into $x'(t')$ and $y'(t')$ is transcendental (or however you spell that).

It's also quite annoying for the idea of an initial conditions which therefore are frame specific, i.e. not so "initial" from the perspective of other frame.

You are correct that in general a set of events that lie at $t=0$ do not lie at $t'=0$. But this doesn't preclude calling them "initial conditions". Generally you specify your experiment in some frame and specify that everything starts at time $t=0$. The fact that $t=0$ doesn't everywhere to transform to $t'=0$ doesn't mean that the $t'=\gamma vx/c^2$ plane isn't the start of the experiment. It's just that the experiment doesn't necessarily start everywhere at the same time in the primed frame.

Call them boundary conditions if it bothers you.

 

[Dale]

Summary:: How is the this normally handled?

What is the usual procedure to transform C into B's frame?

With a single worldline the easiest approach is to express things in terms of proper time before transforming. So first write the worldline in A's frame$$ r^\mu = (t, R \cos(\omega t), R \sin(\omega t), 0)$$ then calculate the proper time along the worldline$$ \frac{d\tau}{dt} = \sqrt{g_{\mu \nu} \frac{r^\mu}{dt} \frac{r^\nu}{dt}} $$ $$t = \frac{\tau}{\sqrt{1-R^2 \omega^2}}$$ Then express the worldline in terms of proper time $$ r^\mu = \left(\frac{\tau}{\sqrt{1-R^2 \omega^2}}, R \cos\left(\frac{\tau \omega}{\sqrt{1-R^2 \omega^2}}\right), R \sin\left( \frac{\tau \omega}{\sqrt{1-R^2 \omega^2}}\right),0\right)$$ Now that everything is in terms of proper time we can simply directly transform into the other frame $$r'^{\mu'}=\Lambda_{\mu}^{\mu'} r^\mu = $$ $$ \left( \frac{\frac{\tau}{\sqrt{1-R^2 \omega^2}}-R v \cos\left( \frac{\tau \omega}{\sqrt{1-R^2 \omega^2}} \right)}{\sqrt{1-v^2}} , \frac{-\frac{v \tau}{\sqrt{1-R^2 \omega^2}}+R \cos\left( \frac{\tau \omega}{\sqrt{1-R^2 \omega^2}} \right)}{\sqrt{1-v^2}} , R \sin \left( \frac{\tau \omega}{\sqrt{1-R^2 \omega^2}} \right) , 0 \right)$$

Of course, if you are dealing with fields or multiple worldlines then you may not be able to express everything in terms of proper time. In that case you would need to transform first, then solve for $t$ in terms of $t'$, and substitute back. There may be no analytical solution so it would have to be done numerically.

 

[Killtech]

Yes. You write $x=x_0+R\sin(\omega t)$ and $y=y_0+R\cos(\omega t)$ and it's easy to write $x'(t)$ and $y'(t)$, but converting those expressions into $x'(t')$ and $y'(t')$ is transcendental (or however you spell that).

yeah, well i'd still think one should lead explaining relativity with this concept first. On that level it's not even really about relativity at all but rather treating the time coordinate the same as any other allowing it to mix. That concept is just not what one would intuitively think of - and a trap i fell into. So a static 4 dim world instead of a dynamic 3 dim world it is.

If you see no one doing this while explaining relativity you won't think of it on your own until you run into a situation where you get things wrong and find out the issue the hard way.

Call them boundary conditions if it bothers you.

Yeah, right, that's a much better wording.

$$ \left( \frac{\frac{\tau}{\sqrt{1-R^2 \omega^2}}-R v \cos\left( \frac{\tau \omega}{\sqrt{1-R^2 \omega^2}} \right)}{\sqrt{1-v^2}} , \frac{-\frac{v \tau}{\sqrt{1-R^2 \omega^2}}+R \cos\left( \frac{\tau \omega}{\sqrt{1-R^2 \omega^2}} \right)}{\sqrt{1-v^2}} , R \sin \left( \frac{\tau \omega}{\sqrt{1-R^2 \omega^2}} \right) , 0 \right)$$

Hmm, wouldn't have though of going the extra mile by expressing it via proper time though i guess it's more general approach to quickly transform it into any frame. Will have to contemplate a little on this why this is more efficient.

 

[Ibix]

yeah, well i'd still think one should lead explaining relativity with this concept first.

What, that relativity allows some notion of spatial and timelike directions "rotating" towards each other? Quite a lot of treatments do start that way. Getting students to really believe it serms to be a hard problem, though.

Will have to contemplate a little on this why this is more efficient.

I wouldn't say it's more efficient. Dale's written $t(\tau)$ and $x(\tau)$ which can easily be transformed into $t'(\tau)$ and $x'(\tau)$. If that's what you need then that's fine, but if you want to ask the question "where is the object at time $t'$" (rather than "where is the object when its onboard clock reads $\tau$") then you have the same problem that you need to find $x'(t')$, which can be written $x'(\tau(t'))$ - and there is no closed form expression of $\tau(t')$.

As long as what you need is a parameterised worldline Dale's approach is an elegant solution since its parameter is frame invariant. But because the relationship between $t'$ and $\tau$ is more complicated than that between $t$ and $\tau$ it just moves the nasty maths in my approach around, it doesn't get rid of it.

 

[Dale]

it just moves the nasty maths in my approach around, it doesn't get rid of it.

I agree, and also it really only works conveniently when you are interested in just one worldline.

 

[anuttarasammyak]

Hmm, wouldn't have though of going the extra mile by expressing it via proper time though i guess it's more general approach to quickly transform it into any frame. Will have to contemplate a little on this why this is more efficient.

Say the relevant motion is discribed in system S as
$$x=f(t)$$
$$y=g(t)$$
$$z=h(t)$$
In S' system they are
$$x(x',y',z',t')=f(t(x',y',z',t'))$$
$$y(x',y',z',t')=g(t(x',y',z',t'))$$
$$z(x',y',z',t')=h(t(x',y',z',t'))$$
where x(x',y',z',t') is formula of x expressed by x',y',z',t' ,etc. You may get simple equations of x',y',z',t' or may not depending on what kind of transformation it is.

 

[Killtech]

What, that relativity allows some notion of spatial and timelike directions "rotating" towards each other? Quite a lot of treatments do start that way. Getting students to really believe it serms to be a hard problem, though.

I wouldn't have to ask for this example if it i could find it everywhere in the books. Yes, what's discussed is what all the means for equations but that is straight forward as they don't need to distinguish between time and space anyway. But that does help so very little when discussing actual experimental setups that need to be translated from one frame to another. We wouldn't need Bell's spaceship paradox if literature would start with this - and it's even more fun to see Bell run into just the very same pitfalls as me thinking of it as a purely physical effect (50 years after SRT was developed) - as that would hint towards an asymmetry between frames. And apparently Wikipedia dates the updated interpretation to 2009... so yeah, I'm not entirely so sure it's so obvious from how the theory is taught.

 

[Dale]

But that does help so very little when discussing actual experimental setups that need to be translated from one frame to another

Well, I agree with that except for the word “need”. The whole point of the first postulate is that you can use any frame. There is never an actual need to transform to another coordinate system.

In real scientific practice you use coordinates that make the expressions simpler. It would never make sense to do the coordinate transform you asked for because it made the expressions more complicated. Since this is never something that you would do in scientific practice, I don’t think that it would be a great idea to emphasize it in a curriculum.

To teach the process and motivation for doing coordinate transforms, it would be better to use real examples where transforming coordinates changes a problem from one with a numerical solution to one with an analytical solution.

 

[Killtech]

Well, I agree with that except for the word “need”. The whole point of the first postulate is that you can use any frame. There is never an actual need to transform to another coordinate system.

In real scientific practice you use coordinates that make the expressions simpler. It would never make sense to do the coordinate transform you asked for because it made the expressions more complicated. Since this is never something that you would do in scientific practice, I don’t think that it would be a great idea to emphasize it in a curriculum.

I agree, that for practical purposes it may seem to make little sense. Indeed, most of the time we just have one observer in the lab frame to worry about and the cases where we have multiple observers with their own setups between which we need to translate is specific to a few thought experiments.

Maybe my perspektive as a mathematician is the issue. For me it is most imperative to first get a good intuition of a theory which one only gets from studying all the edge cases first. Their practically is of no concern here, rather it's just to show off that naive intuition one has from other physics needs a big correction here. Gaining that solid understanding is actually of practical value. On the other hand focusing too much on practically relevant calculus feels a little to me like an education towards a human computer (hmm, come to think of it, it's the very thing pure mathematicians frown most upon).

 

[Dale]

Maybe my perspektive as a mathematician is the issue

You keep saying this, but I find it annoying. Physics is not mathematics and physicists are not mathematicians. We do not teach in a mathematical style and we do not value the same things that mathematicians value. If you come to a physicist judging things as a mathematician then you will be consistently disappointed.

Please leave off your “as a mathematician” comments. They sound judgmental and they are not helpful. We are not mathematicians and do not desired to be judged by standards that we are not even attempting to meet

 

[Killtech]

You keep saying this, but I find it annoying. Physics is not mathematics and physicists are not mathematicians. We do not teach in a mathematical style and we do not value the same things that mathematicians value. If you come to a physicist judging things as a mathematician then you will be consistently disappointed.

Please leave off your “as a mathematician” comments. They sound judgmental and they are not helpful. We are not mathematicians and do not desired to be judged by standards that we are not even attempting to meet

Sorry, you feel that way. But of course you are right, physicists and mathematicians are very different and focus on have different focus of their interests, which is perfectly fine. Truth be told, the issue is that there is simply not enough opportunities to get a mathematical view on all these topics which is why mathematicians interested in them have to ask physicists for the answers. Believe me, if there were math forums with subsections for physics, relativity and quantum mechanics i wouldn't be here.

As we think and do things very differently the path to enlightment is full of painful disappointments and annoyances (for both sides). Indeed it takes some perseverance to not simply give up. And whenever I find mathematicians interested in these topics i of course rather ask them then physicists. And i have to admit i observe a judgmental tone whenever they speak of the "dark side" (=physicist, and yes this is a actual quote of a math professor i asked for help once) and learning that David Hilberts quote that "physics becoming too difficult for physicsist" was not meant as a clever joke. Then again, this goes both ways as mathematicians asking mathematically motivated question don't always get the friendlies responses here on the Physics Forums. I guess this had rubbed off on me a little.

I guess there is a reason, why there aren't that many polymaths around in our modern days and i fear this is a big loss for both physics and math. I find that the problems that are present in physics are very interesting from a mathematical perspective, it just takes quite some translation to present them in a way they become understandable. People with different background and perspectives working on the same problems usually yields faster and better results then one only one group of people focusing on it.

But instead i observe a divide and subtle hostility between those worlds, with few people willing to go to the lengths of building bridges between them, willing to try to understand different perspectives other then their own.

Though, I must admit, that sometimes frustration gets the better of me and i might write something i wouldn't have otherwise. Sorry for that, my previous comment was uncalled for and didn't help to build bridges.

 

[Grasshopper]

I thought this interdisciplinary rivalry was just a joke. Would love to see Einstein and Hilbert in a cage fight.