Coordinate transformations and acceleration

[Trilairian]

So often students question the validity of the twin paradox and how acceleration is involved in looking at round trip scenarios that I am asking why not just give them the tools to transform between the coordinates of an inertial frame and those of an accelerating frame. It is not hard to do and once the students understand the transformation equations they can see for themselves that there is no real paradox. This is a method of introduction that I suggest. After learing about the Lorentz transformations in the form
$$ct = \gamma ct' + \gamma \beta x'$$
$$x = \gamma x' + \gamma \beta ct'$$
Explain that the curvalinear coordinates of an accelerated frame should be chosen so that at least for infinitesimal displacements in proper frame time and proper frame distance from the accelerated observer who is placed at his systems origin, his coordinates should agree with an inertial frame observer who is instantaneously comoving with and nearby him.
His coordinates should then *at the origin* transform as above. Let $$\gamma$$ and $$\beta$$ be expressed in terms of t' as they are now variable and then one can introduce as a natural choice for the differential relation between the coordinates the following:
$$dct = \gamma dct' + d(\gamma \beta x')$$
$$dx = d(\gamma x') + \gamma \beta dct'$$
Noting that this satisfies all the above conditions.
Then simply find anti-derivatives for whatever proper time dependence you choose to give his velocity and you have
$$ct = \int^{ct'}\gamma dct' + \gamma \beta x'$$
$$x = \gamma x' + \int^{ct'}\gamma \beta dct'$$
Now one can see manifestly that when one considers the accelerated observer at x' = 0 even in round trips his watch accumulates the time dilation in accordance with special relativity even though when the acceleration is zero these equations become the ordinary Lorentz transformations and mutual time dilation must then be observed. It shouldn't be a problem introducing these in a special relativity chapter for a calculus based physics course. Why aren't we all doing this?

 

[pervect]

I think that one of the issues is, that having defined the coordinate system of an accelerated observer, one probably ought to explain the limits of the concept. This entails pointing out that it breaks down far away from the observer, that it's only a "local" coordinate system.

Trying to explain this at the same time one is first trying to explain the twin paradox would probably be too confusing. Imagine students asking "Do clocks beyond the Rindler horizon really run *backwards*? <shudder>.

 

[Trilairian]

I suppose I should note that other choices for the remote form of the curvalinear coordinates for the accelerated observer can be made, but this choice seems to yield the simpelest equation for the invariant line element according to accelerated frames. It results in
$$ds^{2} = (1 + \alpha x'/c^{2})^{2}dct'^{2} - dx'^{2} - dy'^{2} - dz'^{2}$$ where $$\alpha$$ is the accelerated frame observer's proper acceleration.

 

[Trilairian]

I think that one of the issues is, that having defined the coordinate system of an accelerated observer, one probably ought to explain the limits of the concept. This entails pointing out that it breaks down far away from the observer, that it's only a "local" coordinate system.

Trying to explain this at the same time one is first trying to explain the twin paradox would probably be too confusing. Imagine students asking "Do clocks beyond the Rindler horizon really run *backwards*? <shudder>.

So explicitely state that it is an arbitrary choice of curvalinear coordinates far from the accelerated observer. The horizon problem then is also easy to explain as this choice encorporates a remote time coordinate chosen to adjust for relative simultaneity as the accelerated frame observer switches from commoving with one inertial frame to another.

 

[pervect]

I do agree that it would be nice if special relativity courses covered accelerated motion. (At least, uniformly accelerated motion). This would probably best be covered near the end of the course, though.

As far as the twin paradox goes, I think it's better to stress the idea that simultaneity is relative than to get into the details of accelerated motion at that point.

Conceptually I think it's better to point out that simultaneity depends on the observer and his choice of coordinates, than to give a specific defintion of simultaneity based on one coordinate system (that of an accelerated observer) - especially when that particular coordinate system has some very counter-intuitive properties.