Understanding Gravitational Time Dilation in Classical Mechanics

[sirius0]

Homework Statement

I am self studying Chow's Classical Mechanics. I have realized that I am at my best if I leave no stone unturned

Homework Equations

$$\hat e_t . \hat e_t = 1$$
Understood

$$d(\hat e_t. \hat e_t)/dt =0$$
Also understood

But $$2\hat e_t .d\hat e_t/dt=0$$
Not understood (tex was first time too)

 

[tiny-tim]

Hi sirius0!

(hmm … tex not bad … but you could use use \frac{}{} and \cdot )

$$\frac{d}{dt}(\hat e_t. \hat e_t)\ =\ 0$$

Use the chain rule, then a.b = b.a:

$$\hat e_t \cdot\frac{d\hat e_t}{dt}\ +\ \frac{d\hat e_t}{dt} \cdot\hat e_t\ =\ 0$$

 

[sirius0]

Hi sirius0!

(hmm … tex not bad … but you could use use \frac{}{} and \cdot )

$$\frac{d}{dt}(\hat e_t. \hat e_t)\ =\ 0$$

Use the chain rule, then a.b = b.a:

$$\hat e_t \cdot\frac{d\hat e_t}{dt}\ +\ \frac{d\hat e_t}{dt} \cdot\hat e_t\ =\ 0$$

Oh thanks a good clue, explains where the two comes from! Pathetically I have to have a quick re-visit to the chain rule but at least I remember what it looked like and I think I can mop it up from here. Good tip on the tex too.

 

[sirius0]

I am now up to pp 20 of Chow's book. I am puzzled as the text states for cylindrical coordinates that $$\hat r =p\hat e_p +z\hat e_z$$. But I thought that any 3D coordinate system needed three dimensions. Should the above have been $$\hat r =p\hat e_p +z\hat e_z+\phi e_$$.

 

[tiny-tim]

I am now up to pp 20 of Chow's book. I am puzzled as the text states for cylindrical coordinates that $$\hat r =p\hat e_p +z\hat e_z$$. But I thought that any 3D coordinate system needed three dimensions. Should the above have been $$\hat r =p\hat e_p +z\hat e_z+\phi e_$$.

Hi sirius0!

I don't have the same book, but I assume that $\hat{\bold{e}}_p$ is two-dimensional and variable (just as $\hat{\bold{r}}$ is three-dimensional ), and that only $\hat{\bold{e}}_z$ is fixed.

 

[sirius0]

Hi sirius0!

I don't have the same book, but I assume that $\hat{\bold{e}}_p$ is two-dimensional and variable (just as $\hat{\bold{r}}$ is three-dimensional ), and that only $\hat{\bold{e}}_z$ is fixed.

Understood progressing well for now...
Thank you.

 

[sirius0]

The whole reason for my self study is in order to relearn SR and learn GR in an initial manner. The notation conventions and clear thinking were an issue so I turned back to classical mechanics. I have gotten distracted from this agenda on another forum and have been looking again at SR. I have a long path ahead WRT GR, tensors Riemann groups etc.
But as a result of the distraction I made an assumption regarding time dilation and gravity.
This is what I came up with using SR. Is it familiar or even right I wonder? Am I bordering on to GR via SR?

$$\Delta t \sqrt{\frac{GM}{rC^2}+1} = \Delta t'$$

 

[sirius0]

Just had a look http://en.wikipedia.org/wiki/Gravitational_time_dilation" . There is something of a resemblance but I don't appear to be right. However there must be something to be learned from this.