4. Dependent Variable, Independent Variable

 

The Lorentz factor in special relativity is used in both cases as follows. One deals with momentum and the other deals with energy.

 

$r'=\gamma{r}, dr'=\gamma{dr}$

$\partial{r'}=\gamma\partial{r}$

 

γ is a function of speed, which in turn is a function of distance (r) and time (t). Then, are γ and r independent of each other? In other words, there is a question as to whether it should be treated as a partial differential rather than a total differential. When treated as total derivative, it takes the form r = ar’, v/c = b (a, b: constant). Although this is quantized, it can be thought of as being differentiable. In mathematics, it appears that the term ‘differential manifold’ is used for this. Since this also holds true from a partial differential perspective, v and r can be treated as independent variables. In that sense, the differential manifold can be seen as arising from the difference between the total derivative and the partial derivative as long as I know.

 

This phenomenon also appears in the derivative of light velocity.

$$\frac{\partial(r\hat{e_r})}{\partial{t}}=v_\theta\hat{e_\theta}, \frac{\partial(r\hat{e_\theta})}{\partial{t}}=-v_r\hat{e_r}$$

This can be expressed like figure1, 2. Here, $\theta$ is thought to be the curved angle in tensor field. In figure 1, the surface is not smooth, but must be differentiable according to the above equation.

curvilinear coordinate

 

Reference: General Relativistic Quantum Mechanics