Note:

This post uses a lot of "Tex" formatting. I never get "Tex" right the first time. My apologies if you see it mangled. I expect to be editing it in trial and error mode for a while.

Note:

To understand this post you need to have a knowledge of Newtonian physics, linear algebra and at least introductory level Special Relativity.

Subject:

I'm trying to understand the difference between two SR Transformations that apply when transforming between Inertial and Accelerating reference frames. So far this is what I have.

The first transformation appears in two web sites. One site appears to be from the University of California Riverside.

https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

The two dependent equations (4) and (6) say;

$$ d = \frac{c^2}{a} \left(cosh\frac{a \tau}{c}-1 \right) $$

$$ t = \frac{c}{a}sinh\left(\frac{a\tau}{c} \right) $$

Where:

c = speed of light

a = a uniform acceleration as experienced by the object and indicated by an accelerometer attached to the object (Proper Acceleration).

(t, d) = space time coordinates of an event using the inertial reference frame.

$$ \tau $$ = time coordinate of an event using the accelerating coordinate system.

The other site is from Wikipedia.

https://en.wikipedia.org/wiki/Acceleration_(special_relativity)

Two two dependent equations at 6a say;

$$ x(\tau) = \frac{c^2}{\alpha} \left(cosh\frac{\alpha \tau}{c}-1 \right) $$

$$ t(\tau) = \frac{c}{\alpha}sinh\left(\frac{\alpha\tau}{c} \right) $$

Where:

c = speed of light

$$ \alpha $$ = a uniform acceleration as experienced by the object and indicated by an accelerometer attached to the object (Proper Acceleration).

(t, x) = space time coordinates of an event using the inertial reference frame.

$$ \tau $$ = time coordinate of an event using the accelerating coordinate system.

Note 1:

As with the Lorentz Transformation, the coordinates x and t are really differentials. It is the distance traveled, $$ x_{final} - x_{initial} $$ , and elapsed time, $$ t_{final} - t_{initial} $$ , that is being transformed. The values of x and t alone apply only when the origins of the reference frames initially coincide, $$ x_{initial} = 0 $$ , $$ t_{initial} = 0 $$ .

Note 2:

When one is transforming position and time coordinates with respect to an accelerating reference frame to coordinates with respect to an inertial reference frame (this case), the time component of the first set of coordinates, $$ \tau $$ , is called Proper Time and the time component of the second set, t, is called Coordinate Time. Proper Time is the time recorded on the accelerating object's clock. It is frame invariant. Coordinate Time is the time component of the space-time coordinates of the event with respect to the observer (the inertial object in this case). It is not frame invariant. If one flips the transformation then the assignment of which is the Proper Time and which is the Coordinate Time also flips.

Note 3:

When describing the motion of a point object, there are two kinds of acceleration in relativity. Proper Acceleration was just defined above. There is also Coordinate Acceleration, which is defined in the traditional way as dv/dt, the change in the observed velocity of an object divided by the change in that observer's time. Proper Acceleration is frame invariant. Coordinate Acceleration is not. These transformations, and this discussion, applies to problems where Proper Acceleration is given and is constant. This discussion does not apply to problems where Coordinate Acceleration is given and is constant. (An analogy would be that there are two kinds of length in relativity, Proper Length and Coordinate Length.) When an accelerating object's velocity is momentarily 0 relative to some inertial frame, the Coordinate Acceleration relative to that frame is equal to the Proper Acceleration. For other circumstances, if one value is known, the other can be calculated.

The two transformations are the same. They just use different notation. Actually I'm not sure they can be called transformations. They don't map the coordinates of time and position between two reference frames. I don't see time and position with respect to an accelerating reference frame mapping to time and position with respect to an inertial reference frame. What I see is time with respect to an accelerating reference frame mapping to time and position with respect to an inertial reference frame. If these equations are a transformation, they are not generic. They are for the special case where the position of the object with respect to the accelerating reference frame is 0. If that is true, it has important implications when applying these transformations to the traditional Bell's Spaceships problem, the one with two rockets. One rocket can be declared to start at position 0 with respect to the accelerating reference frame. OK. But then the other must start with a nonzero value for position. It must have an offset and neither version of the above transformation applies.

Compare the above to a more generic transformation from the following textbook.

Basic Relativity (Chapter 8)

By Richard A. Mould

Copyright 1994 Springer-Verlag, New York, NY

This textbook says (using my personal notation);

$$ x(t_{accel}, x_{accel}) = \frac{-c^{2}}{a_{proper}} + \left(x_{accel} + \frac{c^2}{a_{proper}}\right)cosh\left(\frac{a_{proper}ct_{accel}}{c^2}\right) $$

$$ ct(t_{accel}, x_{accel}) = \left(x_{accel} + \frac{c^2}{a_{proper}}\right)sinh\left(\frac{a_{proper}ct_{accel}}{c^2}\right) $$

Where:

c = speed of light

$$ a_{proper}$$ = a uniform acceleration as experienced by the object and indicated by an accelerometer attached to the object (Proper Acceleration).

(ct, x) = space time coordinates of an event using the inertial reference frame.

$$ (ct_{accel}, x_{accel}) $$ = space time coordinates of an event using the accelerating coordinate system.

Note 4:

$$ x_{accel} $$ in the above transformation is, I think, the distance to the accelerating object using the accelerating coordinate system, $$ S_{accel} $$ . The object is at rest using that coordinate system so $$ x_{accel} $$ does not change with time. $$ x_{accel} $$ may be thought of as the offset in $$ S_{accel} $$ . In the case where the origin of the accelerating coordinate system is initially at rest and coincides with the origin of the inertial reference frame, S, then $$ x_{accel} $$ , is also the initial distance using the coordinates of S (meaning initially, $$ x_{accel} = x $$ ).

A comparison to the Lorentz Transformation

Recall that the Lorentz Transformation applies between two inertial reference frames.

$$ x'(t,x) = \gamma(x -\beta ct) $$

$$ ct'(ct,x) = \gamma(ct-\beta x) $$

Where:

$$ \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} $$

$$ \beta = v/c $$

(ct, x) = space time coordinates of an event using one of the inertial reference frames

(ct', x') = space time coordinates of an event using the other inertial reference frame

Note that the above transformation is generic. You can transform any space time coordinates (x, ct) from the unprimed reference frame to the primed reference frame (x', ct').

Now consider the following transformation.

$$ x'(t) = \gamma(-\beta ct) $$

$$ ct'(ct) = \gamma(ct) $$

I expect you would immediately recognize that the above transformation is a special case of the Lorentz Transformation, one where x = 0. Similarly, I believe the relativistic transformation from Wikipedia and the ucr web sites are copied from a derivation where the initial conditions are that the two reference frames coincide and the accelerating object is at that coincident point. I know the relativistic transformation derived by Richard Mould allows the accelerating object to have an initial offset because the derivation is provided in the textbook. (All require the initial velocity to be zero.)

I'm going to end my initial post here. I've covered a lot of material and I'm not 100% sure I have it all correct. If anyone has knowledge of these transformations and sees a mistake please comment.

P.S.

I'd like to keep this discussion free of space-time diagrams. Space-time diagrams are visual aids. The way they are drawn and the way they are interpreted is determined by the math they are intended to describe. My experience with space-time diagrams is that people get bogged down arguing about how to draw them and how to interpret them and never get to the source math.

This post uses a lot of "Tex" formatting. I never get "Tex" right the first time. My apologies if you see it mangled. I expect to be editing it in trial and error mode for a while.

Note:

To understand this post you need to have a knowledge of Newtonian physics, linear algebra and at least introductory level Special Relativity.

Subject:

I'm trying to understand the difference between two SR Transformations that apply when transforming between Inertial and Accelerating reference frames. So far this is what I have.

The first transformation appears in two web sites. One site appears to be from the University of California Riverside.

https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

The two dependent equations (4) and (6) say;

$$ d = \frac{c^2}{a} \left(cosh\frac{a \tau}{c}-1 \right) $$

$$ t = \frac{c}{a}sinh\left(\frac{a\tau}{c} \right) $$

Where:

c = speed of light

a = a uniform acceleration as experienced by the object and indicated by an accelerometer attached to the object (Proper Acceleration).

(t, d) = space time coordinates of an event using the inertial reference frame.

$$ \tau $$ = time coordinate of an event using the accelerating coordinate system.

The other site is from Wikipedia.

https://en.wikipedia.org/wiki/Acceleration_(special_relativity)

Two two dependent equations at 6a say;

$$ x(\tau) = \frac{c^2}{\alpha} \left(cosh\frac{\alpha \tau}{c}-1 \right) $$

$$ t(\tau) = \frac{c}{\alpha}sinh\left(\frac{\alpha\tau}{c} \right) $$

Where:

c = speed of light

$$ \alpha $$ = a uniform acceleration as experienced by the object and indicated by an accelerometer attached to the object (Proper Acceleration).

(t, x) = space time coordinates of an event using the inertial reference frame.

$$ \tau $$ = time coordinate of an event using the accelerating coordinate system.

Note 1:

As with the Lorentz Transformation, the coordinates x and t are really differentials. It is the distance traveled, $$ x_{final} - x_{initial} $$ , and elapsed time, $$ t_{final} - t_{initial} $$ , that is being transformed. The values of x and t alone apply only when the origins of the reference frames initially coincide, $$ x_{initial} = 0 $$ , $$ t_{initial} = 0 $$ .

Note 2:

When one is transforming position and time coordinates with respect to an accelerating reference frame to coordinates with respect to an inertial reference frame (this case), the time component of the first set of coordinates, $$ \tau $$ , is called Proper Time and the time component of the second set, t, is called Coordinate Time. Proper Time is the time recorded on the accelerating object's clock. It is frame invariant. Coordinate Time is the time component of the space-time coordinates of the event with respect to the observer (the inertial object in this case). It is not frame invariant. If one flips the transformation then the assignment of which is the Proper Time and which is the Coordinate Time also flips.

Note 3:

When describing the motion of a point object, there are two kinds of acceleration in relativity. Proper Acceleration was just defined above. There is also Coordinate Acceleration, which is defined in the traditional way as dv/dt, the change in the observed velocity of an object divided by the change in that observer's time. Proper Acceleration is frame invariant. Coordinate Acceleration is not. These transformations, and this discussion, applies to problems where Proper Acceleration is given and is constant. This discussion does not apply to problems where Coordinate Acceleration is given and is constant. (An analogy would be that there are two kinds of length in relativity, Proper Length and Coordinate Length.) When an accelerating object's velocity is momentarily 0 relative to some inertial frame, the Coordinate Acceleration relative to that frame is equal to the Proper Acceleration. For other circumstances, if one value is known, the other can be calculated.

The two transformations are the same. They just use different notation. Actually I'm not sure they can be called transformations. They don't map the coordinates of time and position between two reference frames. I don't see time and position with respect to an accelerating reference frame mapping to time and position with respect to an inertial reference frame. What I see is time with respect to an accelerating reference frame mapping to time and position with respect to an inertial reference frame. If these equations are a transformation, they are not generic. They are for the special case where the position of the object with respect to the accelerating reference frame is 0. If that is true, it has important implications when applying these transformations to the traditional Bell's Spaceships problem, the one with two rockets. One rocket can be declared to start at position 0 with respect to the accelerating reference frame. OK. But then the other must start with a nonzero value for position. It must have an offset and neither version of the above transformation applies.

Compare the above to a more generic transformation from the following textbook.

Basic Relativity (Chapter 8)

By Richard A. Mould

Copyright 1994 Springer-Verlag, New York, NY

This textbook says (using my personal notation);

$$ x(t_{accel}, x_{accel}) = \frac{-c^{2}}{a_{proper}} + \left(x_{accel} + \frac{c^2}{a_{proper}}\right)cosh\left(\frac{a_{proper}ct_{accel}}{c^2}\right) $$

$$ ct(t_{accel}, x_{accel}) = \left(x_{accel} + \frac{c^2}{a_{proper}}\right)sinh\left(\frac{a_{proper}ct_{accel}}{c^2}\right) $$

Where:

c = speed of light

$$ a_{proper}$$ = a uniform acceleration as experienced by the object and indicated by an accelerometer attached to the object (Proper Acceleration).

(ct, x) = space time coordinates of an event using the inertial reference frame.

$$ (ct_{accel}, x_{accel}) $$ = space time coordinates of an event using the accelerating coordinate system.

Note 4:

$$ x_{accel} $$ in the above transformation is, I think, the distance to the accelerating object using the accelerating coordinate system, $$ S_{accel} $$ . The object is at rest using that coordinate system so $$ x_{accel} $$ does not change with time. $$ x_{accel} $$ may be thought of as the offset in $$ S_{accel} $$ . In the case where the origin of the accelerating coordinate system is initially at rest and coincides with the origin of the inertial reference frame, S, then $$ x_{accel} $$ , is also the initial distance using the coordinates of S (meaning initially, $$ x_{accel} = x $$ ).

A comparison to the Lorentz Transformation

Recall that the Lorentz Transformation applies between two inertial reference frames.

$$ x'(t,x) = \gamma(x -\beta ct) $$

$$ ct'(ct,x) = \gamma(ct-\beta x) $$

Where:

$$ \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} $$

$$ \beta = v/c $$

(ct, x) = space time coordinates of an event using one of the inertial reference frames

(ct', x') = space time coordinates of an event using the other inertial reference frame

Note that the above transformation is generic. You can transform any space time coordinates (x, ct) from the unprimed reference frame to the primed reference frame (x', ct').

Now consider the following transformation.

$$ x'(t) = \gamma(-\beta ct) $$

$$ ct'(ct) = \gamma(ct) $$

I expect you would immediately recognize that the above transformation is a special case of the Lorentz Transformation, one where x = 0. Similarly, I believe the relativistic transformation from Wikipedia and the ucr web sites are copied from a derivation where the initial conditions are that the two reference frames coincide and the accelerating object is at that coincident point. I know the relativistic transformation derived by Richard Mould allows the accelerating object to have an initial offset because the derivation is provided in the textbook. (All require the initial velocity to be zero.)

I'm going to end my initial post here. I've covered a lot of material and I'm not 100% sure I have it all correct. If anyone has knowledge of these transformations and sees a mistake please comment.

P.S.

I'd like to keep this discussion free of space-time diagrams. Space-time diagrams are visual aids. The way they are drawn and the way they are interpreted is determined by the math they are intended to describe. My experience with space-time diagrams is that people get bogged down arguing about how to draw them and how to interpret them and never get to the source math.

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