Is dτ Invariant Under Transformations Beyond Lorentz in Special Relativity?

[strangerep]

I don't think you can change x₀ to y₀ without specifying a function y(x). So I don't think you can go from the integral on the left to the middle integral.

Yes I can. Both steps are valid, provided both $x_0$ and $y_0$ are somewhere between $-\infty$ and $+\infty$.

Anyway...

Although I'm happy to help you understand the content of published papers, I'm not going to follow you down a rabbit hole into crackpot land.

 

[friend]

Yes I can. Both steps are valid, provided both $x_0$ and $y_0$ are somewhere between $-\infty$ and $+\infty$.

Anyway...

Although I'm happy to help you understand the content of published papers, I'm not going to follow you down a rabbit hole into crackpot land.

I certainly hope it's not a "rabit hole". I was trying to find the simplest construction that relates a metric to a field. Then maybe that could be used in the derivation of both the fields of QFT and curvature of GR. And it occurs to me that fields consist of individual values at each point in a space. And the minimum section of space is a infinitesimally small flat portion at the point of interest. A metric is inherently needed to do integration. So what integration process uses the least portion of space and picks out individual field values? That would be the Dirac delta function. Intuitively that seems to me like a good place to start when trying to relate space and fields in terms of their smallest constituents. One would think that the integration of the dirac delta being one is a true statement independent of any physical reason to use it. So if it does prove useful in deriving physics, then we will have succeeded in deriving physics from inherent truth.

As I recall, the integration of the dirac delta is suppose to be one no matter how small the integration interval, as long as the interval contains the zero of the dirac delta's argument. And if the interval of integration is not infinite but is small instead, then my objection to your comments holds.

But even with an infinite interval of integration, I wonder if there must be a diffeomorphism between the x and y coordinates. For it seems that the field in both coordinates needs to be sufficiently well behaved in order to do the integration. Does that mean we should be able to construct a smooth and invertible function between x and y, a diffeomorphism?