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## Main Question or Discussion Point

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

[tex]ct = \gamma ct' + \gamma \beta x'[/tex]

[tex]x = \gamma x' + \gamma \beta ct'[/tex]

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 [tex]\gamma [/tex] and [tex]\beta [/tex] be expressed in terms of t

[tex]dct = \gamma dct' + d(\gamma \beta x')[/tex]

[tex]dx = d(\gamma x') + \gamma \beta dct'[/tex]

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

[tex]ct = \int^{ct'}\gamma dct' + \gamma \beta x'[/tex]

[tex]x = \gamma x' + \int^{ct'}\gamma \beta dct'[/tex]

Now one can see manifestly that when one considers the accelerated observer at x

[tex]ct = \gamma ct' + \gamma \beta x'[/tex]

[tex]x = \gamma x' + \gamma \beta ct'[/tex]

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 [tex]\gamma [/tex] and [tex]\beta [/tex] 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:[tex]dct = \gamma dct' + d(\gamma \beta x')[/tex]

[tex]dx = d(\gamma x') + \gamma \beta dct'[/tex]

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

[tex]ct = \int^{ct'}\gamma dct' + \gamma \beta x'[/tex]

[tex]x = \gamma x' + \int^{ct'}\gamma \beta dct'[/tex]

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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