# Coordinate transformations and acceleration

• Trilairian
In summary, the conversation discusses the twin paradox and how acceleration plays a role in round trip scenarios. The idea of using transformation equations to remove the paradox is introduced, and the concept of choosing curvalinear coordinates for an accelerated frame is explained. The conversation also touches on the limits of this coordinate system and the potential confusion it may cause for students. Finally, the importance of including accelerated motion in special relativity courses is mentioned, along with the suggestion to cover it towards the end of the course.
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?

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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>.

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.

pervect said:
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.

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.

## 1. What is a coordinate transformation and why is it important in science?

A coordinate transformation is a mathematical process that converts the coordinates of a point from one coordinate system to another. It is important in science because it allows us to analyze and interpret data collected in different coordinate systems and compare them in a meaningful way. It also helps us to simplify complex systems and equations by translating them into a more convenient coordinate system.

## 2. How do you perform a coordinate transformation?

A coordinate transformation involves using a set of equations or transformation matrices to convert the coordinates of a point from one coordinate system to another. The specific equations or matrices used will depend on the type of coordinate system being transformed (e.g. Cartesian, polar, spherical) and the relationship between the two coordinate systems.

## 3. Can a coordinate transformation affect the acceleration of an object?

Yes, a coordinate transformation can affect the apparent acceleration of an object. This is because the acceleration of an object is dependent on the reference frame in which it is being measured. When transforming coordinates, the reference frame may also change, leading to a different apparent acceleration.

## 4. What is the difference between linear and angular acceleration?

Linear acceleration refers to the change in an object's speed over time in a straight line, while angular acceleration refers to the change in an object's rotational speed over time around a fixed point. Linear acceleration is typically measured in meters per second squared (m/s^2), while angular acceleration is measured in radians per second squared (rad/s^2).

## 5. How can coordinate transformations help us in understanding motion and forces?

Coordinate transformations play a crucial role in understanding motion and forces by allowing us to analyze and interpret data from different coordinate systems. They help us to determine the direction and magnitude of forces acting on an object, calculate the displacement, velocity, and acceleration of an object, and predict the future motion of an object based on its current state. They also allow us to translate complex motion and force equations into simpler forms, making it easier to analyze and understand.

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