The continuous tensor calculus

Question

It is usual to have tensors in general relativity with discret index in the integer numbers. I propose to have index in the real numbers so that we can have for example:

$$\tilde A^t =\int_{-\infty}^{+\infty} A^{t'} (\frac{\partial \tilde x^t}{\partial x^{t'}}) dt'$$

The differential of a function is:

$$df=\int_{-\infty}^{+\infty} \frac{\partial f}{\partial x^t} dx^t dt$$

We have also:

$$\int_{-\infty}^{+\infty}(\frac{\partial \tilde x^t}{\partial x^{t'}})(\frac{\partial x^{t'}}{\partial \tilde x^{t''}})dt'=\delta (t-t'')$$

The sums are replaced by integrals. The points of the manifold are replaced by smooth functions. The coordinates are:

$$x^t(f)=f(t)$$

Can we make Einstein general relativity with continuous tensor calculus?

Comments

You are proposing an "index set" $I\subset{\Bbb R}$ with coordinates as mappings $x:I\rightarrow{\Bbb R}, x\mapsto x(t)\equiv x^t$. In particular you chose $I=(-\infty,+\infty)$. A question that occurs to me w.r.t. your first equation (coordinate transformation) is: What is the meaning of $\partial {\tilde x}^t / \partial x^{t'}$? Is this a well-defined object? By definition it should be the derivative of the value of the function $\tilde x$ at point $t$ w.r.t. the value of another function, $x$, at point $t'$. Once this is clarified, the next question is: Does the integral exist?

Similar questions can be asked w.r.t. ${\rm d}x^t$ and $\partial f / \partial x^t$.

Regarding your final equation:

If you have $x^t(f)=f(t)$, what then is ${\tilde x}^t(f)$?

Assuming such issues can be clarified, one could also consider different measures in the integral or fractal index sets. A further possibility is arbitrary index sets (countable and non-countable) with a definition of summation as known for families in a normed vector space.

This approach might also stimulate further thought on different concepts of dimension.

Given that spacetime has (or appears to have) four dimensions (Why? I don't know. Quantum decoherence?), the question is of the relevance of this approach to general relativity. For some procedures treating the number of dimensions as a continuous quantity might be useful or even a prerequisite.