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.